Defects, Coarsening and Turbulence Navdeep Rana - TIFR ...

Incompressible Polar Active Matter: Defects,Coarsening and Turbulence

A thesis

Submitted to theTata Institute of Fundamental Research, Mumbai

for the degree ofDoctorate of Philosophy in Physics

by

Navdeep Rana

Tata Institute of Fundamental Research

Tifr Center for Interdisciplinary SciencesHyderabad, India

September, 2021

To the open-source community

Publications relevant to the thesis

1. Navdeep Rana, Pushpita Ghosh, and Prasad Perlekar, “Spreading of nonmotile bacteria

on a hard agar plate: Comparison between agent-based and stochastic simulations”.

In Physical Review E 96, 052403 (2017).

2. Navdeep Rana and Prasad Perlekar, “Coarsening in the 2D incompressible Toner-Tu

equation: Signatures of turbulence”. In Physical Review E 102, 032617 (2020).

3. Navdeep Rana and Prasad Perlekar, “Phase-ordering, topological defects, and turbu-

lence in the 3D incompressible Toner-Tu equation”. In arxiv:2106.03383

4. Navdeep Rana, Rayan Chatterjee, Sriram Ramaswamy, and Prasad Perlekar, “Dense sus-

pensions of polar active particles: Stability and Turbulence”. Manuscript under

preparation.

5. Navdeep Rana, Gaurav Garg, Prathu B. Tiwari, Sayak Bhowmick and Prasad Perlekar,

“Turbulence on a DGX Station : a GPGPU pseudo-spectral solver”. Manuscript

under preparation.

Other publications

1. Rayan Chatterjee, Navdeep Rana, R. Aditi Simha, Prasad Perlekar and Sriram Ramaswamy,

“Inertia drives a flocking phase transition in viscous active fluids”. In arxiv:1907.03492.

To appear in Physical Review X.

| iii

doi.org/10.1103/PhysRevE.96.052403

doi.org/10.1103/PhysRevE.102.032617

http://arxiv.org/abs/2106.03383


Acknowledgments

I am grateful for all the support my thesis advisor Prasad Perlekar has provided me through-

out my doctoral research. I thank him for being patient with me and help me understand

things better. I would like to extend my gratitude towards my collaborators Akshi Gupta,

Debjani Paul, Gaurav Garg, Pushpita Ghosh, Purnima Jain, Rayan Chatterjee, Sriram

Ramaswamy and Vikash Pandey. I enjoyed working with them a lot.

I thank my thesis committee members Smarajit Karmakar and Surajit Sengupta for

their helpful advice throughout the years. I have learnt a lot (dare I say) from the courses

taught by Mustansir Barma, N.D. Hari Dass, Prasad Perlekar, Rama Govindarajan, Sagar

Chakraborty, Shubha Tiwari, Smarajit Karmakar, Sriram Ramaswamy, Subodh Shenoy and

Tamal Das. I am indebted to Gaurav Garg for teaching me Cuda programming.

I thank my teachers Ashwani Kumar, Balraj Bandral, Kanchan Bala, Kanta Sharma,

Madhu Sharma, Neena Awasthi, Sudhir Awasthi, S. K. Soni, V. K. Vats and Yuvraj Sharma

for their constant support in my school and college years.

Most of the work in this thesis would not have been possible without the efforts of open-

source developers, who have spent countless hours creating excellent quality software that

I used daily. To mention a few, I thank the developers of Vim, Neovim, Linux, Cinnamon,

Paraview, Gnuplot, Numpy, Scipy and Matplotlib.

I will fondly remember the time I have spent with the theory group in TIFR Hyderabad

discussing physics and myriad other things. A special thanks to Kabir for all the coffee

table discussions.

I want to express my gratitude to TIFR Hyderabad and the Department of Atomic

Energy for funding support. I thank TIFR Hyderabad HPC facility for computational

resources and Kalyan, Suman and Srinidhi for technical support.

PhD life would not have been so good without my friends in Hyderabad and home.

Anusheela, Arpita, Debankur, Delta, Jose, Kallol, Mukul, Nikhita, Pankaj, Pappu, Raju,

Sharada, Sumit, Tapas, Vikash and Vishnu made my times memorable in TIFR Hyderabad.

| v

Friends at home, Abhimanyu, Abhishek, Aruna, Ashish, Bharti, Jaswinder, Manav, Nitin,

Priyanka, Rakesh and Vivek, have always supported me. A special thanks to the members

of TIFR-H sports groups and The GrimCamRiPper for all the games and plays. I will

surely miss putting up lively performances with Plain Blue Jeans.

Finally, this would not have been possible without the constant support from my family.

I thank them for always looking out for me and for their constant care and love.

vi |

Contents

1 Introduction 1

1.1 Active Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Hydrodynamic formalism: Polar order parameter . . . . . . . . . . . . . . . 4

1.3 Hydrodynamic formalism: Momentum conservation . . . . . . . . . . . . . . 5

1.3.1 Dry active matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.2 Wet active matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Hydrodynamic formalism: Number conservation . . . . . . . . . . . . . . . . 10

1.4.1 Malthusian active matter . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4.2 Incompressible active matter . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Topological defects in polar active systems . . . . . . . . . . . . . . . . . . . 12

1.5.1 Incompressible topological defects . . . . . . . . . . . . . . . . . . . . 13

1.6 Activity in bacteria colonies growing on hard substrates . . . . . . . . . . . . 15

1.7 A guide to this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Coarsening in the two-dimensional incompressible Toner-Tu equation 19

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Dimensionless ITT equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Vortex solution for the ITT equation . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Coarsening dynamics of the ITT equation . . . . . . . . . . . . . . . . . . . 24

2.6 Vortex merger dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.7 Energy dissipation rate and energy spectrum . . . . . . . . . . . . . . . . . . 27

2.7.1 Energy dissipation rate . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.7.2 Energy dissipation rate and the coarsening length scale . . . . . . . . 28

2.7.3 Energy spectrum and enstrophy budget . . . . . . . . . . . . . . . . 31

2.8 Structure functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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2.9 Effect of noise on the coarsening dynamics . . . . . . . . . . . . . . . . . . . 35

2.10 Coarsening in ITT versus bacterial turbulence . . . . . . . . . . . . . . . . . 36

2.11 Pseudo-spectral algorithm for the 2D ITT equation . . . . . . . . . . . . . . 36

2.12 Comparison with Nek5000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.13 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Phase ordering, defects, and turbulence in the 3D incompressible Toner-Tu

equation 41

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Direct numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.1 Average velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.2 Excess free energy surfaces and topological defects . . . . . . . . . . 45

3.3.3 Defect clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3.4 Velocity gradient invariants . . . . . . . . . . . . . . . . . . . . . . . 47

3.4 Energy spectrum and the phase ordering length scale . . . . . . . . . . . . . 49

3.4.1 Low Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4.2 High Reynolds number: Turbulence in the 3D ITT equation . . . . . 52

3.4.3 Structure function analysis . . . . . . . . . . . . . . . . . . . . . . . 53

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4 Dense suspensions of polar active particles 55

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.1 Equations for dense suspensions . . . . . . . . . . . . . . . . . . . . . 59

4.3 Linear stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3.1 Linear stability phase diagram . . . . . . . . . . . . . . . . . . . . . 64

4.4 Non-dimensional equations of motion . . . . . . . . . . . . . . . . . . . . . . 65

4.5 Numerical studies in two dimensions . . . . . . . . . . . . . . . . . . . . . . 66

4.5.1 Numerical verification of linear stability analysis . . . . . . . . . . . . 66

4.6 Turbulence in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.6.1 Correlation functions and correlation length . . . . . . . . . . . . . . 70

4.6.2 Energy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.7 Turbulence in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 72

viii |

4.8 Comparison with Malthusian suspensions . . . . . . . . . . . . . . . . . . . . 74

4.8.1 Linear stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.8.2 Non equilibrium steady states . . . . . . . . . . . . . . . . . . . . . . 74

4.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5 Population fluctuations in growing bacteria colonies 77

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2 Construction of the stochastic continuum model . . . . . . . . . . . . . . . . 81

5.2.1 Nutrient-Bacteria (NB) models . . . . . . . . . . . . . . . . . . . . . 82

5.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3.1 Numerical integration scheme . . . . . . . . . . . . . . . . . . . . . . 84

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.4.1 Snapshots of growing colonies . . . . . . . . . . . . . . . . . . . . . . 87

5.4.2 Front speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4.3 Morphological behavior . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.5 Comparison with the agent-based model . . . . . . . . . . . . . . . . . . . . 91

5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6 Turbulence on DGX architecture: a GPGPU pseudospectral solver 95

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2 Pseudospectral algorithm for the Navier-Stokes equation . . . . . . . . . . . 99

6.2.1 Integration scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.2.2 Computing the nonlinear term . . . . . . . . . . . . . . . . . . . . . 100

6.2.3 Aliasing errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.2.4 Dealiasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.3 Overview of the DGX architecture . . . . . . . . . . . . . . . . . . . . . . . 103

6.4 Earlier attempts on porting FFT algorithms to multi-GPU architecture . . . 105

6.5 Pseudospectral algorithm on the DGX architecture . . . . . . . . . . . . . . 106

6.6 The FFT Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.6.1 Data Layout for the in place transforms . . . . . . . . . . . . . . . . 106

6.6.2 FFT benchmarks on DGX architecture . . . . . . . . . . . . . . . . . 107

6.6.3 Strong scaling for FFT on DGX architecture . . . . . . . . . . . . . . 108

6.6.4 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.7 The Pseudospectral Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

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6.8 Validating the solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.9 Memory requirements of the Navier-Stokes solver . . . . . . . . . . . . . . . 116

6.10 Performance of the Navier-Stokes solver . . . . . . . . . . . . . . . . . . . . 116

6.10.1 Computational cost of the FFTs . . . . . . . . . . . . . . . . . . . . 117

6.11 Limitations of the GPGPU solver . . . . . . . . . . . . . . . . . . . . . . . . 119

6.12 Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7 Conclusions and future directions 121

References 125

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Chapter 1

Introduction

1.1 Active Matter

Collectively moving animals are a sight to behold. Starling flocks show intricately coordi-

nated motion over large length scales [1]. Other animal species, from wildebeest herds to

fish schools and bacterial colonies, also exhibit similar collective motion [2–13]. All these

living systems share a common characteristic; they consist of individuals that continuously

consume energy and self-propel. Such systems are broadly classified as active matter. Ac-

tivity is not limited to living organisms. Artificial active systems are readily realized in

controlled laboratory settings, for example, rods on a vibrating surface [14, 15], Janus par-

ticles [16], and two-dimensional electron-gas driven with microwaves [17]. Fig. 1.1 shows

various realizations of active matter.

The energy required for self-propulsion can either be internally stored or taken up from

the environment. For example, bacteria and birds move at the expense of nutrients [8–10].

Polar rods on a vertically shaken plate move on the horizontal plane using the energy taken

up from the external driving. For a single self-propelled particle, the direction of motion is

determined by the particle’s orientation. However, in a collectively moving phase (flocking),

an individual’s dynamics is determined by its interactions with the neighbours. On its own,

a single bacterium exhibits run-and-tumble motion but moves coherently with others in a

colony to form vortical flows [10].

Collective motion emerges spontaneously in active systems [18]. A key defining char-

acteristic of emergent collective motion is the presence of orientational order over length

scales larger than an individual. Orientational order can either be truly long-ranged or

quasi long-ranged. Truly long-ranged order spans the entire system, as is observed in a

| 1

Figure 1.1: Top row: Collectively moving animals. (Left) A flock of rosy starlings [19] (Right) Schooling

anchovies [20]. Bottom row: Experimental realization of active matter. (Left) Fluorescence microscopy

image of the Microtubule-Kinesin network [21]. (Middle) Quasi-two-dimensional suspensions of Bacillis

Subtilis, scale bar is 50𝜇𝑚, and the inset shows a zoomed-in area from the same image [10], Copyright

(2012) National Academy of Sciences. (Right) Polar rods suspended in a spherical bead sea on a horizontal

plate shaken vertically [22]. Images are used with permission from Wensink et al. [10], Tan et al. [21], Kumar

et al. [22] and Wikimedia commons.

uniformly moving bird flock whereas quasi long-ranged order is restricted to length scales

smaller than the system size. Bacterial suspensions show quasi long-ranged order, where

coherent structures (vortices) ∼ 10 − 20 times larger than a single bacterium are observed,

but no system-wide order is present [10].

For systems in thermal equilibrium, the equations of motion at the microscopic level

are time-reversible [23, 24]. The principle of detailed balance tells us that the transitions

between the microscopic states are pairwise balanced, which rules out the possibility of any

steady-state phase space currents [24–27]. Along with the symmetries and conservation

laws, two universal rules derived from these fundamental postulates govern the dynamics of

equilibrium systems: (a) Principle of universal probability distribution: At thermal equilib-

rium, the steady-state probabilities are given by the Boltzmann distribution 𝑒−𝛽𝐹 , where 𝐹is the Helmholtz free energy, 𝛽 = 1/𝑘𝐵𝑇 , 𝑇 is the Temperature, and 𝑘𝐵 is the Boltzmann’s

2 | Introduction

constant. (b) Fluctuation-dissipation theorem: The relation between the fluctuations in a

system and the system’s response to said fluctuations.

Continuous energy intake at an individual’s level breaks the time-reversal symmetry at

the microscopic level and drives active matter out of equilibrium. Consequently, steady

states in active matter do not obey the principle of detailed balance and exhibit constant

mean energy and momentum fluxes [28]. The absence of detailed balance implies that the

fluctuation-dissipation theorem is not valid for such systems. Owing to the continuous

driving, we cannot treat activity as a small perturbation to an equilibrium system. The

equations of motion for an active system are then governed solely by the conservation laws

and the symmetries. The precise nature of non-equilibrium steady states will vary from

system to system [28], but common features like collective motion, steady-state currents, and

topological defects shared by a variety of active systems suggests that a general statistical

framework, independent of the microscopic details, is possible for active systems.

In this thesis, we study the statistical and dynamical properties of dense collections

of polar self-propelled particles using the coarse-grained hydrodynamic description. Active

hydrodynamics, pioneered by Toner and Tu [4], Marchetti et al. [18], Simha and Ramaswamy

[29] and others, has proven to be quite successful in understanding various properties of

active matter. It focuses on the large-time, long-wavelength (average) behaviour of slow

variables of a dynamical system that do not relax to their steady-state values in a finite

time [30, 31]. Examples of slow variables are densities of conserved quantities and broken

symmetry variables. A well known hydrodynamic equation is the equation of continuity

which says that if the total mass in a system is conserved, the local mass density 𝜌(𝐱, 𝑡)can only be altered via density currents. For a simple fluid, mass conservation reads [32]

𝜕𝑡𝜌 + ∇ ⋅ (𝜌𝐮) = 0, (1.1)

where 𝐮 is the fluid velocity.

In the first part of the thesis, we focus on the coarsening dynamics of dense polar ac-

tive matter in the absence of momentum conservation. In the second part, we explore the

stability of the aligned state to perturbations in bulk suspensions of active polar particles,

where the system’s total momentum is conserved. In the final part, we study the spreading

of a bacterial colony growing on a hard agar plate, where the energy-intake does not lead

to self-propulsion, but drives birth-death processes. The rest of this chapter surveys the hy-

drodynamic formalism and previously known results for active systems under consideration

in this thesis. We conclude the chapter with a brief guide to the thesis.

Hydrodynamic formalism: Polar order parameter | 3

1.2 Hydrodynamic formalism: Polar order parameter

Collective motion in active matter is characterized by orientational order, which arises from

alignment interactions between the self-propelled particles. These interactions are either

mechanical, for example, in polar rods [2, 22], or behavioural as observed in a bird flock

[2, 3, 18].

Orientational alignment can either be polar or apolar. Polar individuals have a preferred

sense of direction along the alignment axis that apolar individuals lack. Bird flocks, Bacteria

and Fish schools are polar systems, whereas Microtubules are apolar. Apolarity can also

manifest on macroscopic scales when polar individuals rapidly switch the direction of self-

propulsion as observed in the colonies of Myxobacteria [33].

Figure 1.2: (a) Polar and (b) apolar (nematic) orientational order. Reflection symmetry for the local

orientation is only available for the apolar orientational order. Rotating an arrow by 180∘ does not lead to

the same state, rotating a symmetric rod does.

We are interested in polar active systems, where a vector order parameter 𝐩 measures

the extent of orientational order [2, 29, 34]. 𝐩 = 0 everywhere represents a disordered

phase, whereas 𝐩 = 1 throughout implies a perfect orientational alignment. Orientational

order in active systems emerges spontaneously and a priori, there is no preferred direction

of alignment. A flock can end up orienting in any direction with equal probability [2, 4,

35]. By virtue of this rotational symmetry, the ordered state is invariant under uniform

rotations. The transverse component of the order parameter 𝐩⟂ is then a slow variable with

an infinitely long relaxation time [36]. Note that the longitudinal component of the order

parameter field is not a slow variable as there is no symmetry preventing it from relaxing

back quickly [36].

The reader should note that not all active systems show alignment interactions. For

example, spherical self-propelled colloids do not align with their neighbours [37]. Such

systems are described by a scalar order parameter, namely the density difference between the

liquid and gaseous phase, and show motility induced phase separation [37, 38]. Alignment

interactions are also absent in chiral active fluids, where the activity manifests as a self-

spinning at a constant rate in two dimensions [39]. For a discussion on Nematic, Scalar,

4 | Introduction

and Chiral active systems, we refer the reader to the articles of Ramaswamy [2], Marchetti

et al. [18], Cates and Tailleur [37], Cates and Tjhung [38], and Fürthauer et al. [39].

1.3 Hydrodynamic formalism: Momentum conservation

Based on how the background fluid is treated we can classify active matter into two cate-

gories: (i) dry active matter and (ii) wet active matter.

1.3.1 Dry active matter

Active systems where the background fluid is ignored in the hydrodynamic formalism are

called dry. Typical examples of dry active matter are bird flocks and granular rods on a

vibrating surface. In quiescent conditions, the surrounding air exerts negligible force on a

bird, and we need not consider the motion of the air to understand the dynamics of the bird

flock. For polar rods on a vibrating surface, there is no background fluid present. Another

classic example of a dry active system is the microscopic Vicsek model, where a polar

individual moves at a constant speed and aligns in the average direction of its neighbours,

albeit with some rotational error (noise) [35]. Since the background fluid is ignored in

the hydrodynamic theory, the total momentum of the active particles and the fluid is not

conserved.

The hydrodynamic equations of motion (also known as the Toner-Tu theory in literature)

for dry active matter with conserved number of particles1 are: [4–6, 40]

𝜕𝑡𝑐 + ∇ ⋅ (𝐩𝑐) = 0,

𝜕𝑡𝐩 + 𝜆(𝐩 ⋅ ∇)𝐩 + 𝜆2(∇ ⋅ 𝐩)𝐩 + 𝜆3∇(|𝐩|2) = (𝛼 − 𝛽|𝐩|2)𝐩 − ∇Π − 𝐩(𝐩 ⋅ ∇Π2)

+ 𝐷∇2𝐩 + 𝐷1∇(∇ ⋅ 𝐩) + 𝐷2(𝐩 ⋅ ∇)2𝐩 + 𝜼.(1.2)

Here 𝑐(𝐱, 𝑡) is the local concentration, 𝜆 terms arise from the self-propulsion, Π and Π2

are the pressure terms, 𝐷𝑖 are the diffusion terms, and 𝜼 is the rotational noise. The active

driving term (𝛼−𝛽|𝐩|2) tries to maintain the magnitude of the velocity field at 𝑝0 = √𝛼/𝛽provided 𝛼, 𝛽 > 0. The diffusion terms with coefficients 𝜈, and 𝐷𝑖 represent the tendency

of active particles to follow their neighbors.

In the absence of momentum conservation, (1.2) lack Galilean invariance and contains

terms that are otherwise not allowed for equilibrium systems. For example, for a fluid1We discuss number conservation in active matters in details in Section 1.4

Hydrodynamic formalism: Momentum conservation | 5

described by the Navier-Stokes equation, Galilean invariance forces 𝜆 = 1, 𝜆2 = 𝜆3 = 0,

and the anisotropic pressure term Π2 is forbidden [32].

Figure 1.3: Phase diagram for polar dry active systems [34]. Three distinct phases are observed based on

the strength of rotational noise and the mean particle concentration (density), (i) Homogeneous disordered

phase, (ii) Coexistence phase, where dense bands of particles are observed amidst a disordered gaseous

phase, and (iii) An orientationally ordered phase where the entire flock moves collectively. Reprinted with

permission from Chaté [34].

Order-disorder transition

The Toner-Tu hydrodynamic theory and its microscopic variant, the Vicsek model, show

that dry active systems exhibit three distinct phases based on the strength of rotational noise

and the mean particle concentration. At low concentration or high noise, a homogeneous

disordered phase is observed. As the concentration is increased while keeping the noise fixed

(or the noise is decreased keeping the concentration fixed), a coexistence phase emerges,

where dense collectively moving bands are observed amidst a disordered gaseous phase. At

low noise or high concentration an orientationally ordered phase emerges, where the entire

flock moves collectively.

While the order-disorder transition is well understood within the framework of Toner-Tu

theory and the Vicsek model, the coarsening dynamics from the disordered gas-like phase to

the orientationally ordered liquid-like phase is yet to be explored fully. A key challenge in

understanding coarsening in dry active systems arises from the fact that the concentration

and the velocity field are strongly coupled [34, 41–43]. Indeed Mishra et al. [41] used both

the density and the velocity correlations to study coarsening in the TT equations. The

6 | Introduction

authors observed that the coarsening length scale grew faster than equilibrium systems

with the vector order parameter, and argued that the advective nonlinearity accelerates the

coarsening dynamics. However, how nonlinearity alters energy transfer between different

scales remains unanswered. In Chapters 2 and 3 we study the coarsening dynamics of

incompressible polar active matter in two and three dimensions, respectively. In the dense

limit, the fact that the order parameter is the only hydrodynamic variable allows us to

characterize the role of advective nonlinearity in the coarsening dynamics. We show that the

coarsening proceeds via repeated defect merger, and turbulence accelerates the coarsening

dynamics.

1.3.2 Wet active matter

In wet systems, the dynamics of the background fluid flow is explicitly taken into account

[29]. Suspensions of self-propelling swimmers like Escherichia coli [8, 10] and Chlamy-

domonas are typical examples of wet active matter. Here, the swimmers generate stresses

that churn the surrounding fluid. In turn, the fluid flow alters the swimmer orientation and

velocity in a momentum conserving fashion [18, 29, 38].

In the limit of constant suspension density 𝜌, the total mass conservation equation (1.1)

reduces to the incompressibility constraint ∇ ⋅ 𝐮 = 0 for the suspension velocity 𝐮. Further,

momentum conservation gives us the following equations of motion [29, 38]:

𝜕𝑡(𝜌𝐮) = −∇ ⋅ 𝚺,

𝜕𝑡𝐩 + (𝐮 + 𝑣0𝐩) ⋅ ∇𝐩 = 𝜆𝐒 ⋅ 𝐩 + 𝛀 ⋅ 𝐩 + Γ𝐡 + ℓ∇2𝐮,

𝜕𝑡𝑐 + ∇ ⋅ [(𝐮 + 𝑣1𝐩) 𝑐] = 0,

(1.3)

where 𝚺 = 𝑃 𝐈 + 𝜌𝐮𝐮 − 𝜇(∇𝐮 + ∇𝐮𝑇 ) + 𝚺𝑎 + 𝚺𝑟 is the total stress tensor. 𝑃𝐈, 𝜇(∇𝐮 +∇𝐮𝑇 ), and 𝜌𝐮𝐮 are the familiar pressure, viscous and inertial stresses of a Newtonian fluid,

respectively. 𝚺𝑎 = 𝜎𝑎(𝑐)𝐩𝐩 + 𝛾𝑎(𝑐) (∇𝐩 + ∇𝐩𝑇 ) and 𝚺𝑟 = 𝜆+𝐡𝐩 + 𝜆−𝐩𝐡 + ℓ(∇𝐡 + ∇𝐡𝑇 )are the active and restoring stresses arising from swimming activity [18, 29, 38], where

𝜎𝑎(𝑐) > 0(< 0) for extensile (contractile) swimmers [see Fig. 1.4], and 𝛾𝑎 determines the

polar contribution to the active stress.

In the 𝐩 equation, 𝑣0𝐩 is the local velocity of the suspended particles, 𝜆 is the flow

alignment parameter, 𝐒 and 𝛀 are the symmetric and anti-symmetric parts of the velocity

gradient tensor ∇𝐮. 𝐡 = −𝛿𝐹/𝛿𝐩 is the molecular field conjugate to 𝐩 derived from the


free-energy functional

𝐹 = ∫ 𝑑3𝑟 [𝐾2 (∇𝐩)2 + 1

4(𝐩.𝐩 − 1)2 − 𝐸𝐩 ⋅ ∇𝑐] . (1.4)

𝐹 favors a uniform ordered state with a unit magnitude. For simplicity, we choose a single

Frank constant 𝐾, which penalizes gradients in 𝐩 [44]. 𝐸 favors the alignment of 𝐩 to up

or down gradients of 𝑐 according to its sign. Γ is the rotational mobility for the relaxation

of the order parameter field, and ℓ governs the lowest-order polar flow-coupling term [45].

𝑣1 is the speed at which the order parameter advects the concentration field.

Simha-Ramaswamy instability in wet suspensions

In the Stokesian limit, viscosity dominates over inertia, and the Reynolds number which

measures their relative strength is very small. In this regime, Simha and Ramaswamy [29]

have shown that ordered states in wet polar suspensions are unstable to small perturba-

tions. For a perfectly aligned state, the net fluid flow generated by the active stress cancels

completely. However, small perturbations to the aligned state lead to a net local fluid flow,

which in turn amplifies the perturbations and destabilizes the orientational order. Extensile

(contractile) suspensions are unstable to bend (splay) perturbations. Fig. 1.4 illustrates the

instability.

Figure 1.4: An illustration of Simha-Ramaswamy instability [2, 29]. (a) Fluid flow around an extensile

(pusher) and a contractile (puller) swimmer. Escherichia coli are extensile, whereas Chlamydomonas are

contractile. (b) Bend instability to the aligned state 𝐩 = �� for a collection of extensile swimmers. (c) Splay

instability to the aligned state 𝐩 = 𝑦 for a collection of contractile swimmers. Reprinted with permission

from Ramaswamy [2].

Meso-scale turbulence

Active systems like bacteria suspensions, where the Reynolds number is very small, are well

described by Stokesian hydrodynamics. The typical size of an Escherichia coli bacterium is

8 | Introduction

around 5𝜇𝑚, and it swims at an average speed of 10𝜇𝑚/𝑠, which sets the Reynolds number

on its scale at 10−5 − 10−4 [10, 46]. At such small Reynolds numbers, Simha-Ramaswamy

instability results in complex, chaotic flows. These chaotic flows are characterized by the

absence of global collective motion; instead, coherent structures (vortices) with sizes much

larger than a single individual are observed [8–10, 47–51]. The phenomenon is known as

active turbulence or meso-scale turbulence [see Fig. 1.1].

Microscopic driving at an individual’s level sets the statistical properties of meso-scale

turbulence different from classical hydrodynamical turbulence characterized by universal

features at high Reynolds numbers. A constant energy flux over a wide range of length

scales and a power-law spectrum with a universal exponent are the hallmark features of

three-dimensional hydrodynamic turbulence [52]. On the other hand, the properties of

meso-scale turbulence vary with the system’s parameters. For example, Wensink et al.

[10] measured the energy spectrum of the chaotic flows in quasi-two and three-dimensional

bacterial suspensions. The energy spectrum peaks at the correlation length (typical vortex

size) and shows power-law scaling at both larger and smaller length scales, albeit with a

tiny scaling range. Further experiments [16, 53] and numerical studies [10, 47] have shown

that the scaling exponents are not universal and depend on different parameters.

Simha-Ramaswamy instability tells us that the aligned state of active swimmers cannot

persist at low Reynolds numbers. However, collectively moving swimmers are frequently

observed in bulk fluid regimes far away from the Stokesian limit. For such swimmers,

particularly when the Reynolds number at an individual’s level is of the order of unity,

both the inertial and viscous forces play an essential role in determining the dynamics. In

Chapter 4 we study the stability of the ordered state in dense suspensions of polar active

swimmers, taking inertia explicitly into account. We show how large enough inertia can

stabilize the orientational order. We characterize the properties of the emergent Spatio-

temporal chaos in the regimes where inertia fails to stabilize the orientational order.


1.4 Hydrodynamic formalism: Number conservation

So far, we have considered active matter where the total number of particles is conserved,

and the particle concentration is a slow variable. Bird flocks and fish schools are good exam-

ples of number conserving active systems. Self-propulsion implies that the order parameter

𝐩 couples with the concentration fluctuations and serves as a concentration current. A

peculiar consequence arising from this coupling is Giant Number Fluctuations in a number

conserving active systems [4, 5, 15, 40, 54]. Unlike equilibrium systems where the density

fluctuations scale as O(√

𝑁), where 𝑁 is the number of particles, the concentration fluctu-

ations in active systems can be as large as the mean, i.e., √⟨𝛿𝑁2⟩ ∝ 𝑁 [54]. Giant number

fluctuations lead to the formation of concentration bands in active systems as observed in

experiments and numerical simulations [35].

We can ignore concentration fluctuations in an active systems for (a) Malthusian flocks

where the birth-death processes restore the concentration quickly to its equilibrium value,

and (b) incompressible flocks where the fluctuations are small compared to the mean value

of the concentration.

1.4.1 Malthusian active matter

If the number of active particles can be altered locally by birth and death processes, con-

centration fluctuation is no longer a slow variable and drops out of the hydrodynamic

description [6, 55]. Consider for example, a bacterial colony growing in a nutrient-rich

environment. Ignoring any spatial inhomogeneities, the bacteria concentration follows the

logistic equation [13, 56, 57]𝑑𝑐𝑑𝑡 = 𝛾𝑐 (1 − 𝑐) , (1.5)

where 𝛾 is the growth rate. Linear stability analysis tells us that the steady-state 𝑐 = 1is stable and small perturbations to this state relax exponentially. If the growth rate is

large enough such that the perturbations relax at time scales smaller than the time scales

of collective motion, the concentration can be assumed constant. Such systems are called

Malthusian.

1.4.2 Incompressible active matter

In the dense limit, where the average concentration of active particles is large, fluctuations

are small and can be ignored [58]. This situation arises in various real world systems like

10 | Introduction

dense bacterial suspensions with short-ranged repulsive interactions [10, 47], in microflu-

idic experiments of self-propelled colloids [59], and in systems with scale-free, long-ranged

repulsive interactions like bird flocks [3, 58, 60].

Number conservation in the constant concentration limit implies the incompressibility

constraint on the order parameter, ∇ ⋅ 𝐩 = 0. In the dry limit, order parameter 𝐩 is the

only hydrodynamic variable with the following equation of motion [58]:

𝜕𝑡𝐩 + 𝜆(𝐩 ⋅ ∇)𝐩 = −∇Π + 𝜈∇2𝐩 + (𝛼 − 𝛽|𝐩|2)𝐩 + 𝜼, (1.6)

where the pressure term Π enforces the incompressibility constraint. Chen et al. [58, 61, 62]

have shown that for such systems the coexistence phase [see Section 1.3] is ruled out and

the order-disorder transition becomes continuous and belongs to a new universality class.

For dense wet systems, the suspension velocity 𝐮 and the order parameter 𝐩 are the

only hydrodynamic variables. Both satisfy the incompressibility criteria and are governed

by the following equations of motion

𝜌 (𝜕𝑡𝐮 + 𝐮 ⋅ ∇𝐮) = −∇𝑃 + 𝜇∇2𝐮 − ∇ ⋅ 𝚺𝑎 − ∇ ⋅ 𝚺𝑟

𝜕𝑡𝐩 + (𝐮 + 𝑣0𝐩) ⋅ ∇𝐩 = −∇Π + 𝜆𝐒 ⋅ 𝐩 + 𝛀 ⋅ 𝐩 + Γ𝐡 + ℓ∇2𝐮.(1.7)

In Chapter 4 we show how incompressibility couples with the splay-bend modes of the

perturbations to the ordered state and alters its stability. We find that aligned states in

dense suspensions of contractile swimmers are stable, whereas the extensile suspensions can

still destabilize. Incompressibility also limits the allowed topological defect solutions for the

order parameter field which we discuss in the next section.

Hydrodynamic formalism: Number conservation | 11

1.5 Topological defects in polar active systems

Topological defects are zeroes of the order parameter field which cannot be removed by

a continuous deformation of the order parameter [63]. They play an important role in

determining the behaviour of many systems; for example, unbinding defect pairs in two

dimensions lead to the Berezinskii-Kosterlitz-Thouless phase transition observed in systems

varying from super-fluids to two-dimensional crystals [64]. Topological defects are also

crucial in determining the dynamical properties of active systems. Defect unbinding in

active nematics causes non-equilibrium phase transitions and gives rise to chaotic flows

[64–66]. In Myxococcus xanthus colonies, topological defects lead to layer formation [33].

In Chapters 2 and 3 we will show how repeated merger of topological vortices drives the

coarsening dynamics in dense dry active matter. Further, in Chapter 4 we show how vortices

suppress a flocking phase transition from a defect-ordered state to a phase turbulent state

in inertial, dense suspensions of polar active matter.

At the core of a topological defect the order parameter vanishes (𝐩 = 0), and at distances

larger than the core size it varies slowly in space [63]. For the order parameter with 𝑛components in 𝑑 dimensions, 𝐩 = 0 implies that the defect core’s dimensionality is 𝑑 − 𝑛[63, 67]. For polar active systems the order parameter has as many components as the

dimensionality of the system and hence only point defects are allowed. Further, defects in

polar systems are characterized by an integer topological charge (or the winding number)

𝑚 [63, 67].

In two dimensions, the topological charge 𝑚 is defined as the total change in the orien-

tation of the order parameter along a loop encircling the defect core

𝑚 = 12𝜋 ∮ d𝜙

d𝑠 d𝑠, (1.8)

where ∮ represents integration over the closed loop and 𝜙 is the orientation of the order

parameter field [63, 68]. The simplest functional form of the order parameter field for a

topological defect with charge 𝑚 is

𝐩 = 𝑔(𝑟) [cos (𝑚𝜃 + 𝜃0) 𝑥 + sin (𝑚𝜃 + 𝜃0) 𝑦] . (1.9)

Here 𝑔(𝑟) is the magnitude of the order parameter which only depends on 𝑟 = √𝑥2 + 𝑦2,

𝑔(0) = 0, 𝜃 = tan−1(𝑦/𝑥) and 𝜃0 is a constant phase factor [63]. From (1.9) it is evident

that at all topological defects with different 𝜃0 have the same topological charge. In Fig. 1.5

we plot the configurations of topological defects for 𝑚 = ±1 and various 𝜃0.

12 | Introduction

Figure 1.5: Orientation of the order parameter 𝐩 around topological defects with different 𝑚 and 𝜃0 in

two dimensions. (a) An outward aster, (b) an inward aster, (c) a vortex, and (d) a saddle. Different values

of phase factor 𝜃0 leads to uniform rotation by 𝜃0 of the saddle.

In three dimensions the topological charge enclosed by a closed surface Ω is defined as

[69, 70]

𝑚 = 14𝜋 ∮ 𝐽 (𝜃(𝐬), 𝜙(𝐬)) 𝑑Ω(𝐬), (1.10)

where 𝐽 (𝜃(𝐬), 𝜙(𝐬)) is the Jacobian of angles 𝜃 and 𝜙 that specify the orientation of the order

parameter, 𝐬 is the generalized coordinate on the surface Ω and ∮ represents the integration

over Ω. Fig. 1.6 shows the 𝐩 field for various topological defects in three dimensions.

Figure 1.6: Topological defects for the polar order parameter in three dimensions. (a) A hedgehog with a

+1 charge, (b) A hedgehog with a −1 charge. (c) An outwards spiralling hedgehog with +1 charge and (d)

An inwards spiralling inwards hedgehog with −1 charge.

1.5.1 Incompressible topological defects

Incompressibility restricts the topological defects allowed for the order parameter field. In

two dimensions, asters and spirals are ruled out as they have a non-vanishing divergence.

Imposing ∇ ⋅ 𝐩 = 0, it can be easily shown that a topological vortex (𝑚 = 1, 𝜃0 = 𝜋/2)

[see Fig. 1.5(c)] is the only allowed solution of the functional form (1.9). Note that incom-

Topological defects in polar active systems | 13

pressibility does not rule out all other possible defect solutions. Defect solutions with a 𝜃dependent |𝐩| are still allowed. In Chapter 2 we will give an example of one such topological

defect: the incompressible saddle with a −1 charge.

Figure 1.7: Topological defects for the incompressible order parameter in three dimensions. On the left

we show streamlines of the order parameter field for a −1 charged hedgehog, whereas on the right is a +1charged hedgehog. Pseudocolor map shows the magnitude of the order parameter in normalized units. The

defects were obtained by numerically integrating (1.6) with 𝜆 = 0 [see Chapter 3 for more details] .

In three dimensions, purely diverging hedgehogs are once again ruled out by the incom-

pressibility constraint. The only possibility left are the topological defects where the order

parameter field pointing inwards in two directions and outwards in the third one (or vice

versa), such that ∇⋅𝐩 = 0 is satisfied. In Fig. 1.7 we show the order parameter field around

incompressible topological defects in three dimensions.

14 | Introduction

1.6 Activity in bacteria colonies growing on hard substrates

So far, we have described active systems that constitute of self-propelled particles. However,

activity is not limited to self-propulsion and emergent collective behaviour and can manifest

in various other forms in driven systems. For example, an isolated fully coated catalytic

colloid does not self-propel and only exhibits anomalous diffusion [71, 72]. In a collection

of these colloids, activity arises in the nature of the effective interactions between different

particle species [71].

Another system where motility is of little importance is a bacterial colony growing on

a hard agar surface. The agar surface provides a highly damped environment, leaving

bacteria motility ineffective [11, 12]. Instead, the colony expands at the expense of constant

energy intake in the form of nutrients. Interactions between individuals in such systems

can lead to various collective phenomena, such as spatial segregation of well-mixed alleles in

an expanding population, invasion dynamics, and morphological transitions under varying

nutrient conditions [13]. In Chapter 5, we study the effect of population fluctuations and

nutrient availability on the morphology of a growing bacterial colony.

Activity in bacteria colonies growing on hard substrates | 15

1.7 A guide to this thesis

This section provides a summary of the rest of the chapters in this thesis.

In Chapter 2, we investigate the coarsening dynamics in two-dimensional dry incom-

pressible polar active matter, where the order parameter is the only dynamical variable

[58, 62]. We show that coarsening proceeds via vortex merger events, and the dynamics

crucially depend on the Reynolds number Re. For low Re, the coarsening process has sim-

ilarities to Ginzburg-Landau dynamics. On the other hand, for high Re, coarsening shows

signatures of turbulence. In particular, we show the presence of an enstrophy cascade from

the inter-vortex separation scale to the dissipation scale. Although the coarsening dynamics

is Re dependent, we show that defects are uniformly distributed throughout the domain,

and dynamical scaling holds at all Re.

In Chapter 3, we study coarsening dynamics in three-dimensional dry incompressible

polar active matter. As was observed in two-dimensions, the transient states en route to

the global order are turbulent. We observe a forward energy cascade and a Kolmogorov

energy spectra and once again, turbulence accelerates the coarsening dynamics. However,

the defect distribution changes as we vary Re. At low Re defects are uniformly distributed

but show clustering at high Re. Further, dynamical scaling holds only at low Re and we

observe that multiple, interacting length scales govern the coarsening dynamics at high Re.

In Chapter 4, we study dense wet suspensions of active polar particles in two and three-

dimensions. We investigate the instabilities of the aligned state to small perturbations

and show how inertia can stabilize the orientational order in incompressible suspensions of

extensile swimmers. We find that a non-dimensional parameter 𝑅 characterizes the stability

of the aligned state. At small 𝑅, the instabilities in the ordered state exhibit a growth rate

proportional to O(𝑞). After a threshold value of 𝑅 = 𝑅1, the instabilities grow at a rate

proportional to O(𝑞2). Past a second threshold value 𝑅 = 𝑅2, the flock is stable. We further

characterize the properties of the spatio-temporal chaos resulting from the instabilities. We

show that, for all 𝑅 < 𝑅2, the flow is riddled with topological vortices with no global order

in sight. Further, the inter-defect spacing grows with 𝑅 and in two dimensions, appears to

diverge at 𝑅 = 𝑅2.

In Chapter 5 we focus on bacterial colonies growing on a hard agar surface. We investi-

gate how population fluctuations and nutrient availability can affect the morphology of grow-

ing bacterial colony. We find that the population fluctuations and nutrient-dependent bac-

16 | Introduction

teria diffusion are sufficient to cause the morphological transition from finger-like branched

fronts to smooth fronts upon increasing nutrient concentration.

In Chapter 6 we present the numerical methods used in this thesis. The chapter focuses

on a general-purpose GPU based (GPGPU) pseudospectral solver for the Navier-Stokes

equation in three dimensions. First, we describe the pseudospectral algorithm and its

GPGPU implementation. We then discuss the performance of the pseudospectral algorithm

on a high bandwidth GPGPU architecture. We will show how the high bandwidth GPGPU

architecture is an ideal platform to perform discrete numerical simulations of the Navier-

Stokes equation at moderate resolutions of size 5123 − 20483 in three dimensions.

In the Chapter 7 we conclude the thesis and outline possible future research directions.

A guide to this thesis | 17

Chapter 2

Coarsening in the two-dimensional incompress-ible Toner-Tu equation

In this chapter, we investigate the coarsening dynamics in the two-dimensional, incompress-

ible Toner-Tu equation. We show that coarsening proceeds via vortex merger events, and the

dynamics crucially depend on the Reynolds number Re. For low Re, the coarsening process

has similarities to Ginzburg-Landau dynamics. On the other hand, for high Re, coarsening

shows signatures of turbulence. In particular, we show the presence of an enstrophy cascade

from the inter-vortex separation scale to the dissipation scale.

2.1 Introduction

Active matter theories have made remarkable progress in understanding the dynamics of

active suspension of polar particles (SPP) such as fish schools, locust swarms, and bird

flocks [2, 18, 73]. The particle based Vicsek model [35] and the hydrodynamic Toner-Tu

(TT) equation [5] provide the simplest setting to investigate the dynamics of SPP. Variants

of the TT equation have been used to model bacterial turbulence [10] and pattern forma-

tion in active fluids [74–77]. An important prediction of these theories is the presence of

a liquid-gas-like transition from a disordered gas phase to an orientationally ordered liquid

phase [2, 34, 37]. This picture is dramatically altered if the density fluctuations are sup-

pressed by imposing an incompressibility constraint. Chen et al. [58, 61], using dynamical

renormalization group studies, showed that for the incompressible Toner-Tu (ITT) equation

the order-disorder transition becomes continuous. The near ordered state of the wet SPP

on a substrate or under confinement [45, 58, 59] belongs to the same universality class as

| 19

the two-dimensional (2D) ITT equation.

Investigating coarsening dynamics from a disordered state to an ordered state in sys-

tems showing phase transitions has been the subject of intense investigation [78–83]. In

active-matter coarsening has been studied either in systems showing motility-induced phase

separation [37, 84] or for dry aligning dilute active matter (DADAM) [34, 41–43]. A key

challenge in understanding coarsening in DADAM comes from the fact that the density

and the velocity field are strongly coupled to each other. Indeed, in [41] the authors used

both the density and the velocity correlations to study coarsening in the TT equation. The

authors observed that the coarsening length scale grew faster than equilibrium systems with

the vector order parameter and argued that the accelerated dynamics are because of the

advective nonlinearity in the TT equation. However, how nonlinearity alters energy transfer

between different scales remains unanswered.

The incompressible limit, where the velocity field is the only dynamical variable, provides

an ideal platform to investigate the role of advection. Therefore, in this paper, we investigate

coarsening dynamics using the ITT equation [58]:

𝜕𝑡𝐮 + 𝜆𝐮 ⋅ ∇𝐮 = −∇𝑃 + 𝜈∇2𝐮 + (𝛼 − 𝛽|𝐮|2) 𝐮. (2.1)

Here 𝐮(𝐱, 𝑡) is the velocity (or the order parameter) field at position 𝐱 and time 𝑡, 𝜆 is

the advection coefficient, 𝜈 is the viscosity, and (𝛼 − 𝛽|𝐮|2) 𝐮 is the active driving term with

coefficients 𝛼, 𝛽 > 0. The pressure 𝑃(𝐱, 𝑡) enforces the incompressibility criterion ∇⋅𝐮 = 0.

We do not consider the random driving term in (2.3) because we are interested in

coarsening under a sudden quench to zero noise. For 𝜆 = 0 and in the absence of pressure

term, (2.3) reduces to the Ginzburg-Landau (GL) equation. On the other hand, (2.3)

reduces to the Navier-Stokes (NS) equation on fixing 𝛼 = 0, 𝛽 = 0, and 𝜆 = 1. Since most

of the dry active matter studies are done on a substrate, we investigate coarsening in two

spatial dimensions.

2.2 Model

We begin by writing down the hydrodynamic equations for dry active matter on a substrate,

which acts as a momentum sink and provides a preferred frame of reference1. The governing

equations of motion for the coarse grained velocity field 𝐮(𝐱, 𝑡) and the density field 𝜌(𝐱, 𝑡)are determined by the conservation laws and the symmetries of the system [4, 36, 40]2. For

1Particles move relative to the surface.2For an excellent pedagogical review see [36].

20 | Coarsening in the two-dimensional incompressible Toner-Tu equation

dry active matter, in the absence of any birth and death processes, there is only one conser-

vation law; the density remains conserved. The system also possesses complete rotational

symmetry as the active particles are equally likely to move in any direction on the substrate.

The equations of motion then are

𝜕𝑡𝜌 + ∇ ⋅ (𝐮𝜌) = 0,

𝜕𝑡𝐮 + 𝜆(𝐮 ⋅ ∇)𝐮 + 𝜆2(∇ ⋅ 𝐮)𝐮 + 𝜆3∇(|𝐮|2) = (𝛼 − 𝛽|𝐮|2)𝐮 − ∇𝑃1 − 𝐮(𝐮 ⋅ ∇𝑃2)

+ 𝜈∇2𝐮 + 𝐷1∇(∇ ⋅ 𝐮) + 𝐷2(𝐮 ⋅ ∇)2𝐮 + 𝜼.(2.2)

As the system is not Galilean invariant, (2.2) contains terms that are otherwise not allowed

for equilibrium systems. For example, for a fluid described by the Navier-Stokes equation,

Galilean invariance forces 𝜆 = 1, 𝜆2,3 = 0, and the anisotropic pressure term (𝑃2) is

forbidden. The active driving term (𝛼 − 𝛽|𝐮|2) tries to maintain the magnitude of the

velocity field at 𝑈 = √𝛼/𝛽 provided 𝛼, 𝛽 > 0. The diffusion terms with coefficients 𝜈, and

𝐷𝑖 represent the tendency of active particles to follow their neighbors. 𝜼 is the noise term

which represents fluctuations in the system. All the parameters, in general, are functions

of the density field 𝜌 and the magnitude of the velocity field |𝐮|.For dense systems on a substrate, we can assume that the particle density is uniform and

constant. In this case, the conservation equation reduces to the incompressibility constraint

∇ ⋅ 𝐮 = 0 that is enforced by the pressure term −∇𝑃 , where 𝑃 = 𝑃1 + 𝜆3|𝐮|2. 𝜆2 and

𝐷1 terms also drop out due to the incompressibility constraint and the parameters 𝜆, 𝛼, 𝛽,

and 𝜈 will only depend on the magnitude of the local velocity |𝐮|, for simplicity we assume

them to be constants. We further drop the anisotropic pressure and diffusion terms to keep

our analysis simple and arrive at the incompressible Toner-Tu equation (ITT)

𝜕𝑡𝐮 + 𝜆(𝐮 ⋅ ∇)𝐮 = −∇𝑃 + 𝜈∇2𝐮 + (𝛼 − 𝛽|𝐮|2)𝐮 + 𝜼. (2.3)

Since we are interested in coarsening dynamics of disordered configurations quenched to

zero temperature, we will ignore the noise term in the following discussion. We discuss the

effect of the noise on the coarsening dynamics in Section 2.9.

2.3 Dimensionless ITT equation

In this section, we write down the dimensionless form of the ITT equation and enumerate

the dimensionless numbers that govern its characteristics. We rescale the space 𝐱′ → 𝐱/𝐿,

Dimensionless ITT equation | 21

the time 𝑡′ → 𝛼𝑡, the velocity field 𝐮′ → 𝐮/𝑈 , and the pressure 𝑃 ′ → 𝑃/𝛼𝐿𝑈 , to get

𝛼𝑈𝜕𝑡′𝐮′ + 𝜆𝑈2

𝐿 𝐮′ ⋅ ∇′𝐮′ = −𝛼𝑈∇′𝑃 ′ + 𝜈𝑈𝐿2 ∇′2𝐮′ + (𝛼 − 𝛽𝑈2|𝐮′|2) 𝑈𝐮′.

Here 𝐿 is the box length and 𝑈2 = 𝛼/𝛽. Ignoring the primed index for convenience, we

arrive at the dimensionless form of the ITT equation

𝜕𝑡𝐮 + ReCn2𝐮 ⋅ ∇𝐮 = −∇𝑃 + Cn2∇2𝐮 + (1 − |𝐮|2) 𝐮.

Here Re ≡ 𝜆𝐿𝑈/𝜈 is the Reynolds number, Cn ≡ ℓ𝑐2/𝐿2 is the Cahn number, and ℓ𝑐 =

√𝜈/𝛼 is the length scale above which fluctuations in the homogeneous disordered state

𝐮 = 0 are linearly unstable3. The ITT equation is then entirely characterized by Re and

Cn.

2.4 Vortex solution for the ITT equation

In this section, we will discussion the topological defect solutions for the ITT equation. The

2D Ginzburg-Landau equation

𝜕𝑡𝐮 = Cn2∇2𝐮 + (1 − |𝐮|2)𝐮, (2.4)

allows for topological defect solutions of the form

𝐮(𝑟, 𝜃) = 𝑓(𝑟) [cos(𝑚𝜃 + 𝜙) 𝑥 + sin(𝑚𝜃 + 𝜙) 𝑦] . (2.5)

Here (𝑟, 𝜃) are the polar coordinates, ( 𝑥, 𝑦) are the unit vectors in Cartesian coordinates,

𝜙 is a constant phase, and 𝑚 is the winding number of the defect. Although, any integer

winding number satisfies (2.4), defects with higher winding number (|𝑚| > 1) are unstable

and decay down to defects with 𝑚 = ±1 [63, 85]. In Fig. 1.5 we plot the velocity field for

different configurations for 𝑚 = ±1. To get the governing equation for 𝑓(𝑟), we set 𝑚 = 1,

and 𝜙 = 0 in (2.5) . The equation for the radial component of the velocity field readily

gives

Cn2 (𝑓 ′′ + 𝑓 ′

𝑟 − 𝑓𝑟2 ) + (1 − 𝑓2)𝑓 = 0. (2.6)

Here the superscript ′ indicates derivatives with respect to 𝑟, and the boundary conditions

are 𝑓(0) = 0, and 𝑓 ′(1) = 0.

For the ITT equation, the nonlinear advection term and the incompressibility constraint

impose additional restrictions on the allowed defect solutions. In particular, ∇ ⋅ 𝐮 = 0 rules3Alternatively, ℓ𝑐 is also the core radius of the vortex defect.


out all other solutions for 𝑚 = 1 except when 𝜙 = ±𝜋/2. It means that vortices are the only

positively charged solutions allowed for the ITT equation. For 𝑚 = −1, no incompressible

solutions of the form (2.5) exist, although other topological charges with 𝑚 = −1 are not

ruled out.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7r

0.0

0.2

0.4

0.6

0.8

1.0f

(r)

Cn = 1.0× 10−1

Cn = 3.2× 10−2

Cn = 1.0× 10−2

Figure 2.1: Plot of 𝑓(𝑟) vs 𝑟 for different values of Cn.

Consider now the radially symmetric velocity field of an isolated unbounded vortex

𝐮(𝐱, 𝑡) ≡ 𝑓(𝑟) 𝜃, where 𝜃 is the unit vector along the angular direction. Substituting in the

ITT equation, we get the following equations

(𝑓 ′′ + 𝑓 ′

𝑟 − 𝑓𝑟2 ) = 1

Cn2 (𝑓2 − 1)𝑓,

𝑃 (𝑟) = ReCn2 ∫𝑟

0

𝑓2(𝑠)𝑠 𝑑𝑠.

(2.7)

The ITT equation thus admits vortex solutions with pressure being a function of radius 𝑟only. Note that equation for 𝑓(𝑟) does not depend on Re and is identical to the equation of

an isolated defect in the Ginzburg-Landau equation (2.6). In Fig. 2.1 we plot the numerical

solution of 𝑓(𝑟) for different values of Cn. For Cn << 1, a regular perturbation analysis

reveals that 𝑓(𝑟) → 𝐴𝑟(1 − 𝑟2/8Cn2).

Vortex solution for the ITT equation | 23

2.5 Coarsening dynamics of the ITT equation

We will now present the results from our study of the coarsening dynamics of (2.3). We use

a pseudospectral method in the stream function-vorticity formulation [86, 87] to perform

direct numerical simulation (DNS) of (2.3) in a periodic square box of side length 𝐿, and

discretize the simulation domain with 𝑁2 collocation points. Unless stated otherwise, we

set 𝐿 = 2𝜋 and 𝑁 = 2048. For details of the numerical methods, see Section 2.11.

(a)

(b)

Figure 2.2: Pseudocolor plots of the vorticity field 𝜔 = 𝑧 ⋅ ∇ × 𝐮 superimposed on the velocity streamlines

at different times for (a) Re = 2𝜋 × 102, and (b) Re = 2𝜋 × 104 in the coarsening regime.

To investigate the coarsening dynamics of the ITT equation, we initialize our simu-

lations in a disordered configuration of randomly oriented velocity vectors drawn from a

Gaussian distribution with zero mean and standard deviation 𝜎 = 𝑈/3. The pseudocolor

plot of the vorticity field in Fig. 2.2(a) and (b) shows different stages of coarsening at

low (Re = 2𝜋 × 102) and high (Re = 2𝜋 × 104) Reynolds number respectively. During

the coarsening, vortices merge, and the inter-vortex spacing continues increasing. For low

Re [see Fig. 2.2(a)], the dynamics in the coarsening regime resembles defect dynamics in

the Ginzburg-Landau equation [79, 82, 88]. On the other hand, for high Re, the vorticity

snapshots resemble 2D turbulence. In particular, similar to vortex merger events in 2D

[89, 90], it is easy to identify a pair of co-rotating vortices undergoing a merger and the

surrounding filamentary structure. Earlier studies on the vortex merger in two-dimensional

Navier-Stokes equations have showed that the filamentary structures formed during the

merger process lead to an enstrophy cascade. Because the ITT equation structure is similar


to NS equations we expect that the vortex merger at high Re will also lead to an enstrophy

cascade.

2.6 Vortex merger dynamics

To investigate the merger of two co-rotating vortices, we perform a DNS of an isolated

vortex-saddle-vortex configuration at various Reynolds numbers. We use a square domain

of area 𝐿2 = 4𝜋2 and discretize it with 𝑁2 = 40962 collocation points. Furthermore, to

minimize the effect of periodic boundaries, we set 𝛼 = −10 for 𝑟 > 0.9𝐿/2 and keep 𝛼 = 1otherwise, where 𝑟 ≡ √(𝑥 − 𝐿/2)2 + (𝑦 − 𝐿/2)2. This ensures that the velocity decays to

zero for 𝑟 ≥ 0.9𝐿/2. The initial condition constitutes a saddle at the center of the square

domain, and two vortices placed at coordinates [(𝐿 − 1)/2, 𝐿/2] and [(𝐿 + 1)/2, 𝐿/2]. It is

important to note that

• Similar to the GL equation [63, 88, 91], vortices in ITT have a topological charge,

• Similar to the NS equation [92], the ITT equation has an advective nonlinearity and

the presence of pressure leads to non-local interactions.

In Fig. 2.3(a)-(e), we plot vorticity contours during different stages of the vortex merger for

different Re. Since the saddle is at equal distance away from the two vortices, its position

does not change during evolution. For low Re = 0, the vortex dynamics has similarities

to the over-damped motion of defects with opposite topological charge in the Ginzburg-

Landau equation. Vortices get attracted to the saddle and move along a straight-line path.

On increasing Re ≥ 2𝜋 × 102, similar to Navier-Stokes, advective nonlinearity in the ITT

becomes crucial. Not only are the vortices attracted to the saddle, but they also go around

each other. The flexure of the vortex trajectory also depends on the Reynolds number.

Thus a vortex merger event in the two-dimensional ITT equation has ingredients both from

the NS and the GL equations.

In Fig. 2.4(a) we plot the inter-vortex separation 𝑑(𝑡) versus time for different Re. Be-

cause of long-range hydrodynamic interactions due to incompressibility, the merger dynam-

ics is accelerated even for Re = 0. The inter-vortex separation decreases as 𝑑(𝑡) ∼ 1/√

𝑡[see Fig. 2.4(b)] in contrast to the much slower 𝑑(𝑡) ∼ √𝑡0 − 𝑡 observed in the GL dy-

namics [63, 93]. On increasing the Re number, inertia becomes dominant, vortices rotate

around each other, and 𝑑(𝑡) decreases in an oscillatory manner. The time for the merger 𝑡0

decreases with increasing Re [see Fig. 2.4(c)].

Vortex merger dynamics | 25

Figure 2.3: (a)-(e) Contour plots of the vorticity field 𝜔 at various times during the merger process for

different values of the Reynolds number Re = 0, 2𝜋 × 102, 2𝜋 × 103, 𝜋 × 104, and 2� × 104.

0.00 0.25 0.50 0.75 1.00t/t0

0.0

0.2

0.4

0.6

0.8

1.0

d(t/t

0)

(a)

Re = 0Re = 2π × 102

Re = 2π × 103

Re = π × 104

Re = 2π × 104

10−1 100

t/t0

10−1

100

d(t/t

0)

(b)

Re = 0t−1/2

0 2 4 6Re ×104

10

30

50

70

t 0

(c)

Figure 2.4: (a) Plot of inter-vortex distance 𝑑(𝑡) vs time 𝑡 at various Reynolds numbers. The time axis

is scaled by the merger time 𝑡0. (b) Log-log plot of 𝑑(𝑡) vs 𝑡 for Re = 0, the black dashed line shows the

1/√

𝑡 scaling. (c) Plot of merger time 𝑡0 versus Re. As Re increases, merger time decreases.


2.7 Energy dissipation rate and energy spectrum

To further quantify coarsening dynamics, we conduct a series of high-resolution DNS (𝑁 =2048) of the ITT equation by varying Re while keeping Cn = 1/(100𝐿) fixed. For en-

semble averaging, we evolve 48 independent realizations at every Re. We monitor the

evolution of the energy spectrum 𝐸𝑘(𝑡) ≡ 12 ∑𝑘−1/2≤𝑞<𝑘+1/2⟨|��𝐪(𝑡)|2⟩ and the energy

dissipation rate (or equivalently the excess free energy) 𝜖(𝑡) ≡ ⟨2𝜈 ∑𝑘 𝑘2𝐸𝑘(𝑡)⟩. Here

��𝐤(𝑡) ≡ ∑𝐱 𝐮(𝐱, 𝑡) exp(−𝑖𝐤 ⋅ 𝐱), 𝑖 =√

−1, and the angular brackets indicate ensemble

average 4.

2.7.1 Energy dissipation rate

Figure 2.5: (a) Plot of the energy dissipation rate 𝜖(𝑡) vs time at various Reynolds numbers. The early

time evolution of 𝜖(𝑡) is well approximated by (2.8) (solid black line). At late times, 𝜖(𝑡) decays as

𝜖(𝑡) ∼ 𝑡−𝛿 ln(𝑡) (black solid lines) with 𝛿 obtained using a least-squares fit. (b) Plot of Re vs 𝛿 and the fit

𝛿 ∼ 1 + 0.46 ln(Re/Re∗) at higher Re. Re∗ = 3.16 × 103 is marked by a vertical black dashed line. For

Re → 0, consistent with Ginzburg-Landau scaling, we obtain 𝛿 → 1.

The time evolution of the energy dissipation rate 𝜖(𝑡) is shown in Fig. 2.5(a). For the

initial disordered configuration, because the statistics of velocity separation is Gaussian, we

approximate the fourth-order correlations in terms of product of second-order correlations

to get the following equation for the early time evolution of the energy spectrum [47]

𝜕𝑡𝐸𝑘(𝑡) ≈ [2𝛼 − 8𝛽𝐸(𝑡)]𝐸𝑘(𝑡) − 2𝜈𝑘2𝐸𝑘(𝑡), (2.8)

where 𝐸(𝑡) = ∑𝑘 𝐸𝑘(𝑡). In Fig. 2.5 we show that the early-time evolution of the energy

dissipation rate 𝜖(𝑡) obtained from (2.8) is in good agreement with the DNS.4The energy spectrum 𝐸𝑘 and the structure factor 𝑆𝑘 are related to each other as 𝐸𝑘 = 𝑘𝑑−1𝑆𝑘.

Energy dissipation rate and energy spectrum | 27

For late times, coarsening proceeds via vortex (defect) mergers. For GL equations in

two dimensions, Refs. [91, 94] show that 𝜖(𝑡) ∝ 𝑡−1 ln(𝑡). In our simulations, we find that

𝜖(𝑡) ∝ 𝑡−𝛿 ln(𝑡), where 𝛿 is now Re dependent. For low Re, where the effect of the advective

nonlinearity can be ignored, we recover GL scaling (𝛿 → 1 as Re → 0). For high Re,

coarsening dynamics is accelerated with 𝛿 = −2.71 + 0.46 ln(Re) [see Fig. 2.5(b)].

2.7.2 Energy dissipation rate and the coarsening length scale

We now discuss the relationship between the energy dissipation rate, the defect number

density, and the coarsening length scale. The coarsening length scale [82, 83, 88, 95, 96]

L(𝑡) ≡ 2𝜋 ∑𝑘 𝐸𝑘(𝑡)∑𝑘 𝑘𝐸𝑘(𝑡) (2.9)

has been used to monitor inter-defect separation during the dynamics.

We identify defects from the local minima of the |𝐮| field in our DNS of the ITT equation

and define the defect number density as 𝑛(𝑡) ≡ N𝑑(𝑡)/𝐿2, where N𝑑 denotes the number

of defects at time 𝑡 5. In Fig. 2.6, we show that in the coarsening regime 𝑛(𝑡) ∝ L−2(𝑡) ∝𝜖(𝑡)/ ln(𝑡) for low Re = 2 × 102 as well as high Re = 2 × 104. As discussed above, the

energy dissipation rate decays as 𝜖(𝑡) ∼ 𝑡−𝛿 ln(𝑡) in the coarsening regime. Similar to GL

dynamics, we find that 𝑛(𝑡) ∝ L−2(𝑡) even for the ITT equation. However, both 𝑛(𝑡) and

L−2(𝑡) show a power-law decay (𝑛 ∝ L−2 ∼ 𝑡−𝛿) without any logarithmic correction.

A purely geometrical argument can be constructed to explain the observed relation

between 𝑛(𝑡) and L(𝑡). As we start our simulations from a disordered configuration, defects

are expected to be uniformly distributed over the entire simulation domain. In Fig. 2.7(a),

we plot the radial distribution function [98]

𝑔(𝑟) ≡ 12𝜋𝑟d𝑟𝑛(𝑡) ∑

𝑖≠𝑗𝛿(𝑟 − 𝑟𝑖𝑗). (2.10)

Here 𝑟𝑖𝑗 = |𝐫𝑖 − 𝐫𝑗|, 𝐫𝑖 are the defect coordinates and d𝑟 is the bin width used to calculate

𝑔(𝑟). Consistent with our assumption above, we find 𝑔(𝑟) = 1, indicating defects are

uniformly distributed in the coarsening regime. Then following Refs. [99, 100] we get 𝑅(𝑡) =1/2√𝑛(𝑡), where 𝑅(𝑡) is the average nearest-neighbor distance at time 𝑡. Consistent with

the dynamic scaling hypothesis [79], in Fig. 2.7(b) and 4(c) we show that L(𝑡) ∝ 𝑅(𝑡) in

the coarsening regime. Using this we get L(𝑡) ∝ 1/√𝑛(𝑡) independent of Re.

5We use scikit-image library [97] to identify local minima


Figure 2.6: Plots comparing the time evolution of 𝑛(𝑡), L(𝑡), and 𝜖(𝑡) for (a) Re = 2𝜋 × 102, and (b)

Re = 2𝜋 × 104. The curves are vertically shifted to highlight identical scaling behavior [𝑛(𝑡) ∝ L−2(𝑡) ∝𝜖(𝑡) ln(𝑡) ∝ 𝑡−𝛿] in the coarsening regime.

For systems with topological defects, the energy dissipation rate (or the excess free

energy) is proportional to the defect number density 𝑛(𝑡) [63, 79, 91, 94]. Thus, consistent

with Fig. 2.6, we get L(𝑡) ∝ 1/√𝜖(𝑡) (apart from the logarithmic factor).


Figure 2.7: (a) Plot of the radial distribution function 𝑔(𝑟) for Re = 2𝜋 × 102 at time 𝑡 = 40 and

Re = 2𝜋 × 104 at time 𝑡 = 10 in the coarsening regime. The dashed black line indicates theoretical

prediction 𝑔(𝑟) = 1 for uniformly distributed points. Plots showing L(𝑡)/𝑅(𝑡) for (b) Re = 2𝜋 × 102 and

(c) Re = 2𝜋 × 104. L(𝑡)/𝑅(𝑡) is fairly constant in the coarsening regime (shaded region).


2.7.3 Energy spectrum and enstrophy budget

The plots in Fig. 2.8 show the energy spectrum 𝐸𝑘 versus 𝑘 at different times for low

Re = 2𝜋 × 102 and high Re = 2𝜋 × 104. In both cases, the energy spectrum in the

coarsening regime show a power law scaling 𝐸𝑘(𝑡) ∝ 𝑘−3. We find that consistent with

the dynamic scaling hypothesis [79], the scaled spectrum collapses between wave numbers

𝑘L ≡ 1/L and 𝑘ℓ𝑐≡ ℓ−1

𝑐 for low Re. At high Re the collapse is between 𝑘L and the

dissipation wave number 𝑘𝑑 [see Fig. 2.8(b,inset)].

The observed 𝑘−3 scaling for the energy spectrum can appear because of (i) the mod-

ulation of the velocity field around the topological defects (Porod’s tail) [88], and (ii) the

enstrophy cascade, similar to two-dimensional turbulence, due to the advective nonlinearity

in (2.3).

To investigate the dominant balances between different scales, we use the scale-by-scale

enstrophy budget equation

𝜕𝑡Ω𝑘(𝑡) + 𝑇𝑘(𝑡) = −2𝜈𝑘2Ω𝑘(𝑡) + F𝑘(𝑡), (2.11)

where Ω𝑘 ≡ 𝑘2𝐸𝑘 is the enstrophy, F𝑘(𝑡) ≡ 𝑘2(��−𝑘 ⋅ 𝐟𝑘 + ��𝑘 ⋅ 𝐟−𝑘) is the net enstrophy

injected because of active driving, 𝑇𝑘 ≡ 𝑑𝑍𝑘(𝑡)/𝑑𝑘 is the enstrophy transfer function, and

𝑍𝑘 ≡ ∑𝑁/2|𝐪|≤|𝐤| ��𝐪 ⋅ (𝐮 ⋅ ∇𝜔)−𝐪 is the enstrophy flux.

The classical theory of 2D turbulence [101–106] assumes the presence of an inertial range

with constant enstrophy flux at scales smaller than the forcing scale and larger than the

dissipation scale. Indeed, for high Re = 2𝜋 × 104, in Fig. 2.8(a) we confirm the presence

of a positive enstrophy flux 𝑍𝑘 between wave number 𝑘L ≡ 1/L corresponding to the inter-

vortex separation and the dissipation wave number 𝑘𝑑 ≡ (8𝜈3/𝑍𝑚)−1/6 for 2 ≤ 𝑡 < 30 in the

coarsening regime. As the coarsening proceeds, the region of positive flux becomes broader

and 𝑘L shifts to smaller wave numbers but the maximum value of the flux 𝑍𝑚(𝑡) decreases

[see Fig. 2.8(a),inset]. In Fig. 2.8(b) we plot different terms in the enstrophy budget equation

(2.11). We find that the active driving primarily injects enstrophy (F𝑘 > 0) around wave

number 𝑘L but, unlike classical turbulence, it is not zero in the region of constant enstrophy

flux (𝑘L < 𝑘 < 𝑘𝑑). Viscous dissipation is active only at small scales 𝑘 ≥ 𝑘𝑑. At late times

𝑡 > 30, the enstrophy flux is negligible [see Fig. 2.8(a),inset].

For low Re, the enstrophy transfer 𝑇𝑘 is negligible and the enstrophy dissipation D𝑘(𝑡)balances the injection because of the active driving F𝑘(𝑡) [see Fig. 2.8(b,inset)]. Therefore,

the 𝑘−3 scaling in the energy spectrum [see Fig. 2.8(a)] is due to Porod’s tail.


Figure 2.8: Time evolution of the energy spectra for (a) Re = 2𝜋 × 102 and (b) Re = 2𝜋 × 104. Inset:

The scaled energy spectrum 𝑘L𝐸𝑘(𝑡) versus 𝑘/𝑘L shows an excellent collapse between wave numbers 𝑘L

and 𝑘ℓ𝑐(𝑘𝑑) for Re = 2𝜋 × 102 (Re = 2𝜋 × 104), confirming the dynamical scaling hypothesis. The wave

numbers 𝑘ℓ𝑐and 𝑘𝑑 at different times are marked by vertical dashed lines (same color as the spectra).


Figure 2.9: (a) Plot of the enstrophy flux 𝑍𝑘(𝑡)/𝑍𝑚(𝑡) versus 𝑘 at Re = 2𝜋×104 for different times in the

coarsening regime. Wave numbers 𝑘L and 𝑘𝑑 are marked with vertical dashed lines (same color as the main

plot). Inset: Time evolution of 𝑍𝑚(𝑡). (b) Enstrophy budget: Plot of the transfer function T𝑘 ≡ 𝑑𝑍𝑘/𝑑𝑘,

enstrophy injection due to the active driving F𝑘, and the enstrophy dissipation D𝑘 = −2𝜈𝑘2Ω𝑘 for

Re = 2𝜋 × 104 and at time 𝑡 = 7 in the coarsening regime. Inset: Plot of different terms in the enstrophy

budget for low Re = 2𝜋 × 102 and at time 𝑡 = 25 in the coarsening regime.


2.8 Structure functions

The real-space measures of enstrophy flux in 2D turbulence is the following exact relation

for the inertial range scaling of the third-order velocity structure function:

𝑆3(𝑟, 𝑡) = 18𝑍𝑘∼1/𝑟𝑟3. (2.12)

Here 𝑆3(𝑟, 𝑡) ≡ ⟨[𝛿𝑟𝑢]3⟩, 𝛿𝑟𝑢 ≡ [𝐮(𝐱 + 𝐫, 𝑡) − 𝐮(𝐱, 𝑡)] . 𝐫, and the angular brackets indicate

spatial and ensemble averaging [107, 108]. In the statistically steady turbulence, the enstro-

phy flux 𝑍𝑘 is constant in the inertial range and is equal to the enstrophy dissipation rate.

During coarsening in ITT, we observe a nearly uniform flux 𝑍𝑘 for 𝑘L ≤ 𝑘 ≤ 𝑘𝑑, albeit with

a decreasing magnitude [see Fig. 2.8(a)]. Therefore, for ITT we choose 𝑍𝑘∼1/𝑟 = 𝑍𝑚(𝑡) in

(2.12). In Fig. 2.10, we show the compensated plot of 𝑆3(𝑟, 𝑡) in the coarsening regime and

find the inertial range scaling to be consistent with the exact result (2.12).

10−1 100

r

10−7

10−5

10−3

S3(r,t

)/Zm

(t)

t = 05t = 07t = 09

Figure 2.10: Plot of the third-order velocity structure function 𝑆3(𝑟, 𝑡) scaled by the maxima of enstrophy

flux 𝑍𝑚(𝑟) for 𝑡 = 5 − 9 in the coarsening regime. The dashed black line shows the theoretical prediction

𝑆3(𝑟)/𝑍𝑚(𝑡) = 18 𝑟3 for comparison.


2.9 Effect of noise on the coarsening dynamics

To investigate the effect of noise on the coarsening dynamics, we add a Gaussian noise

𝜼(𝐱, 𝑡) to the ITT equation [58],

𝜕𝑡𝐮 + 𝜆𝐮 ⋅ ∇𝐮 = −∇𝑃 + 𝜈∇2𝐮 + 𝐟 + 𝜼, (2.13)

where ⟨𝜼(𝐱, 𝑡)⟩ = 0 and ⟨𝜂𝑖(𝐱, 𝑡)𝜂𝑗(𝐱′, 𝑡′)⟩ = 𝐴𝛿𝑖𝑗𝜹(𝐱 − 𝐱′)𝛿(𝑡 − 𝑡′), where 𝐴 controls the

noise strength. In Fig. 2.11, we show that the evolution of the energy dissipation rate 𝜖(𝑡)for Re = 2𝜋 × 104, averaged over 16 independent noise realizations, remains unchanged for

different values of 𝐴 = 0, 0.1, and 0.01. Clearly, the presence of noise in the ITT equation

does not alter the coarsening dynamics.

100 101

t

10−3

10−2

ε(t)

A = 0A = 10−2

A = 10−1

Figure 2.11: Plot comparing the evolution of the energy dissipation rate at different noise strengths for

Re = 2𝜋×104. For ensemble averaging, we evolve 16 independent realizations at 𝐴 = 10−1 and 𝐴 = 10−2.

Effect of noise on the coarsening dynamics | 35

2.10 Coarsening in ITT versus bacterial turbulence

Bacterial turbulence (BT) refers to the chaotic spatio-temporal flows generated by dense

suspensions of motile bacteria [8, 10]. The dynamics of a turbulent bacterial suspension is

modeled by the ITT equation, albeit with the viscous dissipation in ITT replaced with a

Swift-Hohenberg-type fourth-order term to mimic energy injection due to bacterial swim-

ming [10, 47, 109–111],

𝜕𝑡𝐮 + 𝜆𝐮 ⋅ ∇𝐮 = −∇𝑃 − 𝜈∇2𝐮 − Γ∇4𝐮 + 𝐟, (2.14)

where 𝜈 > 0 and the parameter Γ > 0.

In contrast to BT (2.14) , the ITT is a model of flocking dynamics. Indeed the ho-

mogeneous, ordered state is a stable solution of the ITT (2.3) but not of BT (2.14). Fur-

thermore, (2.14) and its variants show an inverse energy transfer from small scales to large

scales, whereas during coarsening in ITT we observe a forward enstrophy cascade from the

coarsening length scale L to small scales.

2.11 Pseudo-spectral algorithm for the 2D ITT equation

In this section, we describe the numerical methods used for the direct numerical simulations

(DNS) of the ITT equation(2.3). We use a pseudo-spectral method in the stream function-

vorticity formulation [86, 87] to perform DNS of (2.3) in a periodic square box of side

length 𝐿, and discretize the simulation domain with 𝑁2 collocation points. Unless stated

otherwise, we set 𝐿 = 2𝜋 and 𝑁 = 2048. The stream function-vorticity formulation of the

ITT equation reads as

𝜕𝑡𝝎 + 𝜆𝐮 ⋅ ∇𝝎 = 𝜈∇2𝝎 + 𝛼𝝎 − 𝛽∇ × (|𝐮|2𝐮). (2.15)

Here 𝝎 ≡ 𝑧 ⋅ ∇ × 𝐮 is the vorticity field, 𝐮 = (𝜕𝑦𝜓, −𝜕𝑥𝜓), and 𝜓 satisfies the Laplace

equation 𝜔 = ∇2𝜓. In Fourier space, (2.15) is

𝜕𝑡𝝎 + 𝜆 (𝐮 ⋅ ∇𝝎) = (𝛼 − 𝜈𝑘2) 𝝎 − 𝛽𝐤 × (|𝐮|2𝐮), (2.16)

where 𝝎(𝐤, 𝑡) ≡ ∑𝐱 𝝎(𝐱, 𝑡) exp(−𝑖𝐤 ⋅ 𝐱) is the Fourier transform of the vorticity field. We

discretize time with steps of size Δ𝑡 and use a second-order exponential time differencing

scheme (ETD2)[112] for time integration.


ETD schemes integrate the −𝜈𝑘2𝝎 term explicitly, and the other terms are approximated

by their weighted sum at the time steps 𝑛 and 𝑛 − 1. The ETD2 discretized form of (2.16)

is

𝝎(𝐤, 𝑡 + Δ𝑡) = 𝝎(𝐤, 𝑡)𝑒𝑐Δ𝑡 + (1 + 𝑐Δ𝑡)𝑒𝑐Δ𝑡 − 1 − 2𝑐Δ𝑡Δ𝑡𝑐2 𝐹(𝐤, 𝑡) + 1 + 𝑐Δ𝑡 − 𝑒𝑐Δ𝑡

Δ𝑡𝑐2 𝐹(𝐤, 𝑡 − Δ𝑡).(2.17)

Here 𝑐 = −𝜈𝑘2 and 𝐹(𝐤, 𝑡) = 𝜆𝐮 ⋅ ∇𝝎 − 𝛽(𝐤 × |𝐮|2𝐮). Since 𝐹(𝐤, 𝑡 − Δ𝑡) is not known at

𝑡 = 0, we integrate (2.16) at the first time step using an ETD1 scheme

𝝎(𝐤, 𝑡 + Δ𝑡) = 𝝎(𝐤, 𝑡)𝑒𝑐Δ𝑡 + 𝑒𝑐Δ𝑡 − 1𝑐 𝐹(𝐤, 𝑡). (2.18)

In the limit 𝑐 = 0, for example for 𝑘 = 0, ETD1 scheme reduces to the Euler scheme and

ETD2 scheme reduces to the Adam-Bashford scheme.

The mean vorticity in the simulation domain is zero; ⟨𝜔⟩ = ⟨𝜕𝑥𝑢𝑦 − 𝜕𝑦𝑢𝑥⟩ = 06, as a

result 𝐮 computed from the stream function only contains the fluctuating part of the local

velocity, and the mean velocity 𝐮 in the stream function-vorticity formulation is also zero.

To account for the global long-range order that may arise during the evolution, we evolve

the mean velocity separately. We integrate (2.3) over the entire simulation domain, to get

the equation of motion for the mean velocity

𝜕𝑡𝐮 = 𝛼𝐮 − 𝛽|𝐮|2𝐮. (2.19)

𝐮 is then updated using (2.19) and added to the fluctuating 𝐮 computed from the stream

function in real space before computing 𝐮 ⋅ ∇𝝎 at each time step. In Algorithm. 1, we

outline our numerical algorithm.

6⟨𝜕𝑓/𝜕𝑥𝑖⟩ = 0 for any periodic function 𝑓.

Pseudo-spectral algorithm for the 2D ITT equation | 37

Algorithm 1: Pseudo-spectral algorithm for ITT

𝐮 ← 𝐮0 ; ! Set the initial condition.

𝐮 ← FFT[𝐮] ; ! Take the velocity to Fourier space.

𝝎 ← 𝐤 × 𝐮 ; ! Compute vorticity in Fourier space.

begin time loop

𝐮 ← (𝑖𝑘𝑦𝑘2 𝝎, −𝑖𝑘𝑥

𝑘2 𝝎) ; ! Compute velocity from vorticity.

𝐮 ← IFFT[𝐮]𝝎 ← IFFT[𝝎] ; ! Bring velocity and vorticity to physical space.

𝐮 ← 𝐮 + 𝐮 ; ! Add the mean velocity to the fluctuating part.

𝐝𝐮 ← 𝐮 × 𝝎 + 𝛼𝐮 − 𝛽|𝐮|2𝐮 ; ! Compute the R.H.S and store it in 𝐝𝐮array.

𝐮 ← 𝐮 + Δ𝑡 (𝛼𝐮 − 𝛽|𝐮|2𝐮) ; ! Evolve the mean using a simple Euler

method.

𝐝𝐮 ← FFT[𝐝𝐮] ; ! Take the nonlinear term back to the Fourier space.

𝑑𝝎 ← 𝐤 × 𝐝𝐮 ; ! Take curl

𝝎 ← ETD[𝝎, 𝑑𝝎] ; ! Integrate

end


2.12 Comparison with Nek5000

We numerically integrate the ITT equation for a test case using Nek5000 to verify the

correctness of our pseudo-spectral algorithm. Nek50007 is a finite element solver designed

to solve a wide class of partial differential equations on meshes of arbitrary shapes. In

Fig. 2.12 we plot the snapshots from Nek5000 test runs for Re = 2𝜋×104 at different stages

of coarsening. Once again, we observe vortices emerging at the onset of the coarsening. With

time, similar to our pseudo-spectral runs, we find that the inter-defect spacing increases and

number of vortices decreases. Our results are then independent of the choice of discretization

and numerical integration method.

Figure 2.12: Pseudocolor plots of the vorticity field 𝜔 = 𝑧 ⋅∇×𝐮 superimposed on the velocity streamlines

at different times for Re = 2𝜋×104 in the coarsening regime. These snapshots were obtained by numerically

integrating (2.3) using Nek5000.

2.13 Conclusions

To conclude this chapter, we have investigated the coarsening dynamics of the ITT equation

in two dimensions. We find that the dynamics at all high Reynolds number is governed

by repeated vortex mergers and at high Re, turbulence accelerates the coarsening process.

Vortices being topological defects, and the presence of non-local interactions due to the

pressure term gives the vortex dynamics in the ITT equation a complex character. The

merger processes take elements both from the GL equation as well as the NS equation.

At low Reynolds number, where we can ignore the effect of advective nonlinearity, the

coarsening dynamics is governed solely by the topological nature of the vortices. At high

Re, coarsening shows signatures of 2D turbulence. We find that at all Re, the free energy

decays as a power-law in time with a logarithmic correction, 𝜖(𝑡) ∝ 𝑡−𝛿 ln 𝑡. We find that

the exponent 𝛿 itself Using least-square fitting we find 𝛿 ≈ 1+0.46 ln (Re/Re∗) We find that7https://nek5000.mcs.anl.gov/

Comparison with Nek5000 | 39

https://nek5000.mcs.anl.gov/

the exponent 𝛿 itself is an increasing function of Re. As Re → 0, 𝛿 → 1, and we recover the

GL scaling. We find that at all Re, the energy spectrum shows a power-law scaling of 𝑘−3.

At low Re, the scaling appears due to the Porod’s law. At high Re, we show the presence

of a forward enstrophy cascade, which extends the power law scaling to much higher wave

numbers as compared to low Re. Finally, we verify the exact relation for the third-order

velocity structure-function at high Re.


Chapter 3

Phase ordering, defects, and turbulence in the3D incompressible Toner-Tu equation

We investigate coarsening dynamics of the incompressible Toner-Tu equation in three di-

mensions. We show that the dynamics is characterized by Reynolds number Re. At all

Re, coarsening proceeds via defect merger events. At low Re, the dynamics is similar to

the Ginzburg-Landau equation. We find a unique growing length scale viz. the inter-defect

spacing and dynamical scaling holds. At high Re, turbulence alters the coarsening dynam-

ics. In particular, we observe a forward energy cascade and multi-scaling similar to classical

three-dimensional turbulence.

3.1 Introduction

Phase ordering (or coarsening) refers to the dynamics of a system from a disordered state

to an orientationally ordered phase with broken symmetry on a sudden change of the con-

trol parameter [63, 113]. Biological systems such as a fish school or a flock of birds show

collective behavior - an otherwise randomly moving group of organisms start to perform co-

herent motion to generate spectacularly ordered patterns whose size is much larger than an

individual organism [2, 18, 73]. Although the exact biological or environmental factors that

trigger such transition depend on the particular species, physicists have successfully used

the theory of dry-active matter to study disorder-order phase transition in these systems

[4, 34, 114]. Theoretical studies have revealed that for an incompressible or a Malthu-

sian flock, where we can ignore density fluctuations, the order-disorder phase transition is

continuous [58, 60–62].

| 41

In classical spin systems with continuous symmetry, domain walls or topological defects

are crucial to the growth of order and the equilibrium phase transition [63, 113]. Several

studies have highlighted the role of defects in the phase-ordering dynamics in spin sys-

tems [70, 115–117]. Interestingly, suppression of defects in the two-dimensional (2D) XY

model [117] and the three-dimensional (3D) Heisenberg model [70] destroys the underlying

phase-transition, and the system remains ordered at all temperatures.

In dry-active matter, only recent studies have started to explore the role of defects in

phase-ordering. We investigated phase-ordering in the two-dimensional (2D) incompress-

ible Toner-Tu (ITT) equation in Chapter 2 [see also[118]]. Our study revealed that the

phase-ordering proceeds defect merger events, in a manner similar to the planar XY model.

At high Reynolds number, the merger dynamics had similarities with vortex mergers in

2D hydrodynamic turbulence. More recently, experiments [119] investigated coarsening dy-

namics in 2D dry-active matter and found that, consistent with our results described in

two-dimensions, phase-ordering proceeds via the merger of topological defects.

In this chapter, we investigate the phase-ordering dynamics of the 3D ITT equation [58],

𝜕𝑡𝐮 + 𝜆𝐮 ⋅ ∇𝐮 = −∇𝑃 + 𝜈∇2𝐮 + 𝐟 + 𝜼, (3.1)

where 𝐮(𝐱, 𝑡) ≡ (𝑢𝑥, 𝑢𝑦, 𝑢𝑧), and 𝑃(𝐱, 𝑡) are the velocity and the pressure fields, 𝐟 ≡(𝛼 − 𝛽|𝐮|2) 𝐮 is the active driving term with coefficients 𝛼, 𝛽 > 0, 𝜆 is the advection

coefficient, and 𝜈 is the viscosity. The incompressibility constraint ∇ ⋅ 𝐮 = 0 relates the

velocity to the pressure. Note that 𝑢 = 0 and 𝑢 = 𝑈 with 𝑈 ≡ √𝛼/𝛽 are the unstable

and stable homogeneous solutions of the ITT equation. We study the order-parameter

dynamics in a cube of length 𝐿 and use periodic boundary conditions. As we are interested

in phase-ordering under a sudden quench from a disordered configuration to zero noise, we

once again switch off the random driving term (𝜼 = 0). As mentioned in Chapter 2, for

𝛽 = 0 and 𝜆 = 1, Eq. (3.1) reduces to the linearly forced Navier-Stokes equation [120, 121],

whereas it reduces to the Ginzburg-Landau (GL) equation [88] for 𝜆 = 0 and in the absence

of pressure term. Therefore, similar to the GL equation we expect topological defects to

play crucial role in phase-ordering dynamics of the ITT equation.

Using high-resolution numerical simulations, we show that phase-ordering in the 3D

ITT equation proceeds via defect merger. The Reynolds number Re ≡ 𝜆𝑈𝐿/𝜈 controls the

merger dynamics. For small Re, the ordering dynamics have similarities with the three-

dimensional Heisenberg model. On the other hand, for large Re, we show turbulence drives

the evolution and speeds up phase-ordering. In particular, we observe a Kolmogorov scaling

42 | Phase ordering, defects, and turbulence in the 3D incompressible Toner-Tu equation

in the energy spectrum over a range of length scales, and a detailed analysis of the velocity

structure-functions reveals multi-fractal scaling.

The rest of the chapter is organised as follows. In Section 3.2 we discuss the details

of our direct numerical simulations. In Section 3.3 we present the results for the mean

order and the energy dissipation rate at low and high Re. In Sections 3.4.1 and 3.4.2 we

characterize the coarsening at low and high Re respectively. We conclude the chapter in

Section 3.5.

3.2 Direct numerical simulations

The dimensionless form of the ITT equation is [see Chapter 2 and [118]]:

𝜕𝑡𝐮 + ReCn2𝐮 ⋅ ∇𝐮 = −∇𝑃 + Cn2∇2𝐮 + (1 − |𝐮|2) 𝐮, (3.2)

where Re ≡ 𝜆𝑈𝐿/𝜈 is the Reynolds number with 𝑈 ≡ √𝛼/𝛽, and Cn = √𝜈/𝛼𝐿2 is the

Cahn number.

We use a pseudospectral method to perform direct numerical simulation (DNS) of

Eq. (3.2) in a tri-periodic cubic box of length 𝐿 = 2𝜋. We discretize the box with 𝑁3 collo-

cation points with 𝑁 = 1024 and use a second-order exponential time differencing scheme

for time integration [112]. We decompose the velocity field into its mean 𝐕(𝑡) = ⟨𝐮⟩ and

fluctuating part 𝐮′(𝑥, 𝑡) ≡ 𝐮(𝑥, 𝑡) − 𝐕(𝑡) to investigate the ordering dynamics, where the

angular brackets denote spatial averaging. Along with the velocity field, we monitor the

evolution of the energy spectrum

𝐸𝑘(𝑡) ≡ 12 ∑

𝑘− 12 ≤𝑝<𝑘+ 1

2

|��𝐩(𝑡)|2, (3.3)

and the energy E(𝑡) = ∑𝑁/2𝑘=1 𝐸𝑘(𝑡). Here, ��𝐤(𝑡) ≡ ∑𝐱 𝐮(𝐱, 𝑡) exp(−𝑖𝐤 ⋅ 𝐱) is the Fourier

transform of the velocity field with 𝑖 =√

−1 [118]. Note that the total energy is 𝐸(𝑡) ≡∑𝑁/2

𝑘=0 𝐸𝑘(𝑡)12𝑉 2(𝑡) + E(𝑡). The flow is initialized with a disordered configuration with

𝐸𝑘(𝑡 = 0) = 𝐴𝑘2 and 𝐴 = 10−8. In our DNS, we choose Cn = 1/(2𝜋 × 102), 𝑈 = 1, and

vary Re.

Direct numerical simulations | 43

20 40 60 80 100t

10−4

10−3

10−2

10−1

100

V(t

)

(a)

10−1 100

t

10−4

10−2

V(t

)

et

Re=0Re=2π × 102

Re=5π × 103

Re= π × 104

Re=2π × 104

20 40 60 80t

0.1

0.2

0.3

0.4

0.5

E(t)

(b)

10−1 100

t

10−3

10−1

E(t)

t−32 e2t

Re=0Re=2π × 102

Re=5π × 103

Re= π × 104

Re=2π × 104

Figure 3.1: Time evolution of (a) the mean velocity 𝑉 (𝑡) and (b) the fluctuating energy E(𝑡) for different

Re. (Insets) Zoomed in plots compare early time-evolution with the corresponding theoretical prediction

(dashed black lines).

3.3 Results

3.3.1 Average velocity

In Fig. 3.1, we plot the magnitude of the mean velocity 𝑉 (𝑡) and the fluctuation energy

E(𝑡) = ∑𝑁/2𝑘=1 𝐸𝑘(𝑡). At early times, the nonlinearities in Eq. (3.1) can be ignored and we


arrive at the following time evolution equation of the energy spectrum [47, 118]

𝜕𝑡𝐸𝑘(𝑡) ≈ 2(1 − Cn2𝑘2)𝐸𝑘(𝑡). (3.4)

Using Eq. (3.4) and the initial condition for the energy spectrum we obtain 𝑉 (𝑡) ∼ exp(𝑡)and E ∼ exp(2𝑡)/𝑡3/2 [see Fig. 3.1]. The departure from the early exponential growth of 𝑉 (𝑡)and E(𝑡) marks the onset of the phase-ordering regime. We observe that 𝑉 (𝑡) approaches

the ordered state faster by increasing the Reynolds number. On the other hand, E(𝑡) first

increases, attains a plateau and then decreases. The plateau region and the peak value of

E(𝑡) decrease with increasing Reynolds. We show later in Section 3.4.2 that the shorter

plateau region is an indicator of strength of the forward energy cascade.

3.3.2 Excess free energy surfaces and topological defects

For the ITT equations, the excess free-energy per unit volume is ℎ ≡ Cn2|∇𝐮|2/2 and

the defects are identified as velocity nulls, i.e. spatial locations where 𝐮 = 𝟎. For a 𝑛-

component order parameter in 𝐷-dimensions, the dimensionality of the defects is 𝑛 − 𝐷[63]. Since 𝑛 = 𝐷 = 3 for us, the ITT equation permits unit magnitude topological

charge. We locate defects using the algorithm prescribed by Berg and Lüscher [69] that

has been successfully used to study: (a) the role of defects in the 3D Heisenberg transition

[70], and (b) coarsening dynamics in the 3D Ginzburg-Landau equations [116]. Similar

algorithms have also been used to identify vector nulls in magnetohydrodynamics [122, 123]

and homogeneous, isotropic turbulence [124].

In Fig. 3.2(a,b) we show the time evolution of the iso-ℎ surfaces overlaid with defect

positions during phase-ordering for low and high-Re. The streamline plots of pair of oppo-

sitely charged defects undergoing merger are shown in Fig. 3.2(c) [Re = 0] and Fig. 3.2(d)

[Re = 2𝜋 × 104]. At low Re, in Fig. 3.2(a), we show that the iso-surfaces are primar-

ily localized around the lines joining oppositely charged defects. The evolution resembles

phase-ordering in the 3D Ginzburg-Landau equations [115, 116]; we shall explore this fur-

ther in Section 3.4. In contrast, at high-Re we observe tubular structures similar to fluid

turbulence [125] and the defects reside in the proximity of these tubes [see Fig. 3.2(b)].

Results | 45

Figure 3.2: Iso-ℎ surfaces overlaid on the defect positions marked by colored spheres (blue : -1, yellow

: +1) for (a) Re = 0 and (b) Re = 𝜋 × 104 at different stages in the coarsening regime. We show only

a subdomain of size (𝜋/2)3 from the simulation box for better visual representation. Streamlines of two

neighbour defects undergoing merger at (c) Re = 0 and (d) Re = 𝜋 × 103. (e) More snapshots of merger

event at Re = 0.


3.3.3 Defect clustering

In Fig. 3.3 we plot the radial distribution function

𝑔(𝑟) ≡ 14𝜋𝑟2𝑑𝑟𝑛(𝑡) ∑

𝑖≠𝑗𝛿(𝑟 − 𝑟𝑖𝑗), (3.5)

where 𝑟𝑖𝑗 = |𝐫𝑖 −𝐫𝑗|, 𝐫𝑖 are the defect coordinates, and 𝑑𝑟 is the bin width used to calculate

𝑔(𝑟). At Re = 0, 𝑔(𝑟) is constant for all values of 𝑟, implying that the defects are uniformly

distributed throughout the domain. A small increase in 𝑔(𝑟) at very small 𝑟 is likely due

to defect pairs undergoing merger events. On the other hand, at Re = 𝜋 × 104, 𝑔(𝑟)shows a peak at smaller 𝑟, implying that the defects are clustered. Indeed, our snapshots

in Fig. 3.4(b) show the same. Furthermore, the visible clustering of defects at high-Re is

consistent with the observed clustering of vector nulls in fluid turbulence [124].

0.5 1.0 1.5 2.0r

0

5

10

g(r

)

(a) Re = 0Re = π × 104

Figure 3.3: Radial distribution function 𝑔(𝑟) vs. 𝑟 at Re = 0 and Re = 𝜋 × 104. Defects are clustered at

Re = 𝜋 × 104.

3.3.4 Velocity gradient invariants

The spatial structures of fluid flows are often characterized by the invariants 𝑄 ≡ −Tr(𝐀2)/3and 𝑅 ≡ −Tr(𝐀3)/3 of the velocity gradient tensor 𝐀 ≡ ∇𝐮. For high-Re fluid turbulence,

the joint probability distribution function 𝑃(𝑅, 𝑄) resembles an inverted tear-drop [126–

129]. In the R-Q plane, regions above the curve (27/4)𝑅2 + 𝑄3 = 0 are vortical, whereas

those below are extensional [129]. From the flow structures around topological defects [see

Fig. 3.2(c,d)], it is easy to identify that a positive (negative) topological charge would have

𝑅 < 0(> 0).In Fig. 3.4, we plot the joint probability distribution function 𝑃(𝑅, 𝑄) for Re = 0 and

Re = 2𝜋 × 104 at a representative time in the phase-ordering regime. By overlaying the 𝑄

Results | 47

−150 −75 0 75 150−20

0

20

Q/〈S

ijSij〉

(a)+1−1

−30 −15 0 15 30

R/ 〈SijSij〉3/2−20

0

20

Q/〈S

ijSij〉

(b)+1−1

10−2

10−1

100

101

102

103

104

10−1

100

101

102

Figure 3.4: Contour plot of the joint probability distribution 𝑃(𝑅, 𝑄) for (a) Re = 0, and (b) Re = 𝜋×104

in the coarsening regime. 𝑄 and 𝑅 values are normalized by ⟨𝑆𝑖𝑗𝑆𝑖𝑗⟩, where 𝐒 is the symmetric part of

the velocity gradient tensor 𝐀. Blue + (Green −) signs mark the position of +1(-1) defect on the 𝑅 − 𝑄plane. In (b), the curve 27𝑅2 + 4𝑄3 = 0 is shown by the dashed black line.

and 𝑅 values at the location of topological defects on the 𝑃(𝑅, 𝑄), as expected, we find

that the negative (positive) defects occupy the region with 𝑅 > 0(< 0). For Re = 0, we

find symmetric 𝑃 (𝑅, 𝑄) located primarily in the region 𝑄 > 0, indicating that the flow

structures are vortical. In contrast, for Re = 2𝜋 × 104 we observe that 𝑃(𝑅, 𝑄) has a tear-

drop shape. The tail region (𝑄 > 0 and 𝑅 < 0) indicates strongly dissipative extensional

flow regions (which also carry a negative charge) reminiscent of fluid turbulence.


3.4 Energy spectrum and the phase ordering length scale

A unique length scale typically describes the dynamics of systems undergoing phase-ordering,

this is often referred to as the dynamic scaling hypothesis. In Chapter 2 we have shown

that at all Re, there exists a unique growing length scale, namely the inter-defect separa-

tion, that determines the coarsening dynamics. In the following sections we investigate the

validity of this hypothesis for phase-ordering in the 3D ITT equation for low and high-Re.

3.4.1 Low Reynolds number

In Fig. 3.5, we plot the energy spectrum. With time, the peak of the spectrum shifts towards

small-wave numbers, indicating a growing length scale often defined as [88, 88, 91, 94, 113,

118, 130]

L(𝑡) ≡ 2𝜋 ∑𝑘 𝐸𝑘(𝑡)∑𝑘 𝑘𝐸𝑘(𝑡) . (3.6)

For low Re → 0, consistent with the Ginzburg-Landau scaling, we observe L(𝑡) ∼√

𝑡 [see

Fig. 3.5(b)] [79]. The rescaled energy spectrum 𝑘L𝐸𝑘 versus 𝑘/𝑘L collapses onto a single

curve for different times [see Fig. 3.5(a)], and we observe Porod’s scaling 𝐸𝑘L ∝ (𝑘L)−4

for 𝑘L(𝑡) > 1 due to the presence of defects [79]. Note that the energy spectrum and the

structure factor are related to each other as 𝐸(𝑘, 𝑡) = 4𝜋𝑘2𝑆(𝑘, 𝑡).

Dynamical scaling hypothesis

The plot in Fig. 3.5(b) shows that the average minimum inter-defect separation 𝑅(𝑡) ∝ L(𝑡),validating the dynamical scaling hypothesis. For uniformly distributed defects, we expect

L(𝑡) ∝ 𝑛(𝑡)−1/3, where 𝑛(𝑡) is the defect number density [99, 100, 118]. We verify the same

in Fig. 3.5(b,inset).

Energy spectrum and the phase ordering length scale | 49

100 101 102

k/kL

10−8

10−5

10−2

kLE

k(t

)

k−4(a)

100 101 102k

10−9

10−6

10−3

Ek(t

)

t = 10t = 50t = 100t = 200

10 50 90 130 170t

0.5

1.0

1.5

2.0

R(t)

×10−1

(b)

101 102

10−1

100

n(t)−1/3

ε(t)−1/2

Re = 0Re = 2π × 102

Figure 3.5: (a) Scaled energy spectrum 𝑘L𝐸𝑘(𝑡) versus 𝑘/𝑘L for Re = 0 at different times. For 𝑘 ≫ 𝑘L,

we observe Porod’s scaling 𝐸𝑘(𝑘) ∼ 𝑘−4. Inset: Time evolution of the energy spectrum. (b) Evolution

of average minimum inter-defect separation 𝑅(𝑡) for Re = 0 and Re = 2𝜋 × 102. Dashed lines show the

evolution of L(𝑡) (scaled for comparison with 𝑅(𝑡)). Inset: Plot showing L(𝑡)𝑛(𝑡)1/3 versus 𝑡 at Re = 0.


100 101 102

k

10−6

10−4

10−2

Ek(t

)

k−5/3

(a)

t = 3t = 6t = 15t = 30t = 42

100 101 102

k

0.0

0.2

0.4

0.6

0.8

1.0

Πk(t

)/Πmax(t

)

(b)

Re = 5π × 103, t = 80Re = π × 104, t = 40Re = 2π × 104, t = 20

0 20 40 60 80 100t

10−2

10−1

Πmax(t

)

(c)

Re = 5π × 103

Re = π × 104

Re = 2π × 104

Figure 3.6: (a) Evolution of the energy spectrum at Re = 𝜋×104. Dashed black line shows the Kolmogorov

scaling 𝑘−5/3. (b) Energy flux at different Re in the coarsening regime. At higher Re, the wavenumber range

over which we observe the energy flux increases. (c) Evolution of the maximum of energy flux Π𝑚𝑎𝑥(𝑡) at

various Re.

Energy spectrum and the phase ordering length scale | 51

3.4.2 High Reynolds number: Turbulence in the 3D ITT equation

In Fig. 3.6(a) we show the time evolution of the energy spectra for high Re = 5𝜋 × 103.

At early times, similar to small Re, the peak of the spectrum shifts towards small 𝑘. At

intermediate times which correspond to the plateau region in Fig. 3.2 (b), we observe a

Kolmogorov scaling 𝐸𝑘 ∼ 𝑘−5/3 [Fig. 3.6(b)]. According to the theory of homogeneous

turbulence, a crucial feature of Kolmogorov scaling is the existence of a region of constant

energy flux Π𝑘 ≡ 𝜆 ∑𝑁/2|𝐩|≤|𝐤| ��𝐩 ⋅ (𝐮 ⋅ ∇𝐮)−𝐩. We evaluate Π𝑘 at a representative time 𝑡 = 20

at high Re in the phase-ordering regime and find that it remains nearly constant between

wave-numbers corresponding to the coarsening scale 𝑘L ∼ 2𝜋/L and the dissipation scale

𝑘𝜂 ∼ (𝜈3/Π𝑘)1/4 [Fig. 3.6(b)]. In Fig. 3.6(c), we show the time evolution of Π𝑚𝑎𝑥(𝑡) ≡𝑚𝑎𝑥[Π𝑘(𝑡)]. The time range, over which Π𝑚𝑎𝑥(𝑡) is nearly constant, coincides with the

plateau region in E(𝑡). On reducing the Re, the region of constant energy flux and the

value of Π𝑚𝑎𝑥 decreases indicating a reduction in the cascade efficiency.

Thus the following picture of phase-ordering emerges: active driving 𝛼 − 𝛽|𝐮|2 injects

energy primarily at large length scales, which is then redistributed to small scales by an

energy cascade due to the advective nonlinearity. For scales larger than the typical size

of an eddy, qualitatively, turbulent stirring leads to increase in viscosity which leads to a

faster phase-ordering. Finally, at late stages, we observe regions of turbulence interspersed

with growing patches of order (Fig. 3.7).

Figure 3.7: Pseudocolor plot of the velocity magnitude (z=𝜋 plane) along with the velocity streamlines

during the late stages of phase-ordering for Re = 2𝜋 × 104.


3.4.3 Structure function analysis

Turbulent flows are characterised by velocity structure functions

𝑆𝑝(𝑟) = ⟨{[𝐮(𝐱 + 𝐫) − 𝐮(𝐱)] ⋅ 𝐫}𝑝⟩ , (3.7)

where 𝐫 is the separation vector, and 𝐫 ≡ 𝐫/|𝐫|. In the inertial range, we expect 𝑆𝑝(𝑟) ∼ 𝑟𝜁𝑝 .

The plot of the third-order structure function in Fig. 3.8(a) is found to be in excellent agree-

ment with the exact result 𝑆3(𝑟, 𝑡) = −45Π𝑚𝑎𝑥(𝑡)𝑟 [52, 128]. Furthermore, in Fig. 3.8(b) we

show that the exponents 𝜁𝑝 for 𝑝 = 2, 4 and 6 are consistent with the She-Leveque formula

[131] indicating that the phase-ordering at high Re is accompanied by multiple interacting

scales. Thus, unlike for low Re, dynamic scaling is not valid for phase-ordering in the ITT

at high-Re.

10−1 100r

10−2

10−1

−S3(r,t

)/Πm

(t)

(a)10−1 100

r10−2

10−1

100

−S3(r,t

)/Πm

(t)

t = 50t = 70t = 90

t = 25t = 35t = 45

10−2 10−1 100

r

10−1

100

Sp(r

)rζ p

(b) p = 2 p = 4 p = 6

Figure 3.8: (a) Plot of the third-order velocity structure function −𝑆3(𝑟, 𝑡) (with a negative sign) scaled by

maxima of energy flux Π𝑚(𝑟) at different times for Re = 𝜋×104. For comparison, we show the theoretical

prediction −𝑆3(𝑟)/Π𝑚(𝑡) = 45 𝑟 by dashed black line. Inset: Similar plot for Re = 5𝜋×103. Scaling range

is small at lower Re. (b) Scaled velocity structure functions (−1)𝑝𝑆𝑝(𝑟)𝑟𝜁𝑝 vs 𝑟 for 𝑝 = 2, 4, and 6, where

𝜁𝑝 is the She-Leveque exponent [131].

Conclusions | 53

3.5 Conclusions

In this chapter, we have studied the phase-ordering dynamics of the 3D ITT equation. We

found that similar to 2D, coarsening proceeds via repeated defect merger. At low Re, the

defects are uniformly distributed throughout the domain and the coarsening dynamics is

characterized by a unique growing length scale. On the other hand, at high Re, defects

are clustered and the advective nonlinearities alter the coarsening dynamics. We find that

transient states enroute to global order show characteristics of three-dimensional turbulence.

In particular, we observe a near constant energy flux in the coarsening regime, with the

turbulent showing multi-fractal intermittent nature.


Chapter 4

Dense suspensions of polar active particles

In this chapter, we study dense, wet suspensions of active polar particles in two and three

dimensions. We first investigate the instabilities of the aligned state to small perturbations

and show how inertia can stabilize the orientational order. Further, we perform high resolu-

tion direct numerical simulations and characterize the nature of the turbulent states arising

in such systems.

4.1 Introduction

Self-propelled particles in a bulk fluid display a variety of dynamical phenomenon, from

chaotic flows to ordered collective motion [2, 8, 10, 18, 49, 51, 132, 133]. Of particular

interest to us are suspensions of polar active particles.

At low Reynolds number, viscous forces dominate over the inertial forces, such suspen-

sions are well described by Stokesian hydrodynamics, where the inertia is ignored com-

pletely. For example, the typical size of an Escherichia coli bacterium is around 5𝜇𝑚 and

it swims at an average speed of 10𝜇𝑚/𝑠, which sets the Reynolds number on its scale at

Re ∼ 10−5 − 10−4 [10, 46]. Bacterial suspensions are then overwhelmingly Stokesian. In

this limit, Simha and Ramaswamy [29] have shown that orientationally ordered states of

swimmers are unstable to small perturbations. The instability arises as follows: A perfectly

aligned state of active swimmers in quiescent fluid results in uniform uni axial stress. Per-

turbations on this state generate spatially varying stresses, which give rise to a net local

fluid flow. The resulting flow alters the alignment of the swimmers and can amplify the

perturbations, destabilizing the orientational order. Suspensions of extensile (contractile)

swimmers are unstable to bend (splay) perturbations [see Fig. 1.4 in Chapter 1] .

| 55

Simha-Ramaswamy instability at low Re results in lively chaotic flows widely observed

in bacterial suspensions [8, 10]. The phenomenon is also known as active turbulence or meso-

scale turbulence [see [10] and Fig. 1.1 in Chapter 1]. These chaotic flows are characterized

by the absence of global orientational order; instead, coherent structures (vortices) with

size much larger than a single individual but smaller than the experimental domain are

observed [8–10, 47–51].

Activity is not limited to low Re swimmers and aligned swimmers moving collectively

are frequently observed in bulk fluid in regimes that are far away from the Stokesian limit.

Clearly these swimmers are successful in escaping the Stokesian instability which governs

the fate of low Re swimmers. We can group these flocks into two categories: (a) Flocks of

swimmers where Re at an individual’s scale is very large (Re ≫ 1), such as sardine schools.

In this case, inertial forces are orders of magnitude larger than the viscous forces and hence

inviscid hydrodynamic interactions alone determine the dynamics of such flocks [18, 134].

(b) Flocks of swimmers with Re ∼ O(1), like brine shrimp, larval squids, fish schools where

individuals of size O(𝑚𝑚) are found moving at a speed O(𝑚𝑚𝑠−1) [134] in water. For such

swimmers inertial and viscous forces are comparable to each other and play an essential

role in determining the dynamics.

In this chapter, we consider the latter type of swimmers, and ask the following ques-

tion: Can inertia stabilize the ordered state in extensile suspensions against the Stokesian

instability? We focus on dense suspensions, where fluctuations in active particle concentra-

tion are small compared to its average value and are ignored [58]. This situation arises in

various experimental systems like dense bacterial suspensions with short-ranged repulsive

interactions [10, 47], in microfluidic experiments with self-propelled colloids [59], and in

systems with long-ranged repulsive interactions like bird flocks [58, 60] where the density is

maintained at an optimal level.

We show that indeed, inertia can stabilize the ordered state with respect to small

perturbations. Our linear stability analysis reveals that a single dimensionless parame-

ter 𝑅 ≡ 𝜌𝑣20/2𝜎0, where 𝜌 is the suspension mass density, 𝑣0 is the self-propulsion speed of

active particles, and 𝜎0 is the force-dipole density, governs the stability of the suspension to

small perturbations. For 𝑅 < 𝑅1, where 𝑅1 = 1 + 𝜆 and 𝜆 is the flow-alignment parameter,

we find an inviscid instability where the most unstable pure bend modes grow at a rate

I(𝜔) ∝ 𝑞. For 𝑅1 < 𝑅 < 𝑅2, we find that the pure bend perturbations grow at a rate

I(𝜔) ∝ 𝑞2. Finally for 𝑅 > 𝑅2, orientational order is stable. Our numerical studies in two

56 | Dense suspensions of polar active particles

dimensions show that the chaotic flows at all 𝑅 < 𝑅2 are riddled with topological vortices

and we do not observe any global order. Inter-defect spacing grows as we increase R and

appears to diverge at 𝑅 ∼ 𝑅2.

The rest of chapter is organized as follows. In Section 4.2 we write down the equations

of motion for dense suspensions in the constant concentration limit. In Section 4.3 we

outline our linear stability analysis results. In Section 4.5 we present the results of our

numerical simulations in two dimensions and in Section 4.7 we present the results from our

three-dimensional simulations. Finally we contrast our results for dense suspensions with

our earlier study of Malthusian suspensions [55, 135].

Introduction | 57

4.2 Equations of motion

The equations of motion for a suspension of polar self-propelled particles (SPP) described

by the hydrodynamic velocity field 𝐮(𝐱, 𝑡), the polar order parameter field 𝐩(𝐱, 𝑡), and the

concentration 𝑐(𝐱, 𝑡) are [18, 29, 45, 55, 136]

𝜌 (𝜕𝑡𝐮 + 𝐮 ⋅ ∇𝐮) = −∇𝑃 + 𝜇∇2𝐮 − ∇. (𝚺𝑎 + 𝚺𝑟) ,

𝜕𝑡𝐩 + (𝐮 + 𝑣0𝐩) ⋅ ∇𝐩 = 𝜆𝐒.𝐩 + 𝛀.𝐩 + Γ𝐡 + ℓ∇2𝐮, and

𝜕𝑡𝑐 + ∇ ⋅ [(𝐮 + 𝑣1𝐩)𝑐] = 0.

(4.1)

Here 𝜌 is the suspension density, 𝜇 is the fluid viscosity, and the hydrodynamic pressure

term 𝑃(𝐱, 𝑡) enforces incompressibility ∇ ⋅ 𝐮 = 0. 𝐒 and 𝛀 are the symmetric and anti-

symmetric parts of the velocity gradient tensor ∇𝐮, 𝜆 is the flow alignment parameter, 𝑣0𝐩is the local self-propulsion velocity of the active particles, and 𝑣1𝐩 is the velocity at which

the polar order parameter advects the concentration field.

𝚺𝑎 = 𝜎𝑎(𝑐)𝐩𝐩 + 𝛾𝑎(𝑐) (∇𝐩 + ∇𝐩𝑇 ) (4.2)

is the intrinsic stress associated with the swimming activity, where 𝜎𝑎 > 0(< 0) is the force-

dipole density for extensile (contractile) swimmers and 𝛾𝑎 determines the polar contribution

to the active stress [29, 55, 136].

𝚺𝑟 = 𝜆+𝐡𝐩 + 𝜆−𝐩𝐡 + ℓ(∇𝐡 + ∇𝐡𝑇 ) (4.3)

is the reversible thermodynamic stress [137], 𝜆± = (𝜆 ± 1)/2, and 𝐡 = −𝛿𝐹/𝛿𝐩 is the

molecular field conjugate to 𝐩, derived from a free-energy functional

𝐹 = ∫ 𝑑3𝑟 [𝐾2 (∇𝐩)2 + 1

4(𝐩.𝐩 − 1)2 − 𝐸𝐩 ⋅ ∇𝑐] (4.4)

that favors a aligned order parameter state with unit magnitude. For simplicity, we choose

a single Frank constant 𝐾, which penalizes gradients in 𝐩 [44]. 𝐸 favors the alignment of

𝐩 to up or down gradients of 𝑐, according to its sign. Γ is the rotational mobility for the

relaxation of the order parameter field, and ℓ governs the lowest-order polar flow-coupling

term [45]. All the coefficients in our hydrodynamic description are phenomenological in

nature. In particular, 𝜎𝑎(𝑐) and 𝛾𝑎(𝑐) are functions of the concentration 𝑐 and in the limit

𝑐 → 0 are proportional to 𝑐.


4.2.1 Equations for dense suspensions

In the constant concentration limit (𝑐 = 𝑐0), number conservation implies the incompress-

ibility constraint on the order parameter ∇ ⋅ 𝐩 = 0 [58]. Further, all the phenomenological

parameters take their values at 𝑐0. In our earlier study on Malthusian flocks [55] we have

shown that 𝛾𝑎 and ℓ do not alter the nature of the inviscid instability and only change

the coefficients of O(𝑞2) terms in the dispersion relation. In coming sections, we will show

that only splay-bend modes couple to the number-conservation. Twist-bend modes are de-

termined solely by the 𝐮 and 𝐩 dynamics. Thus, the nature of instability of these modes

is identical for number conserving, Malthusian, and dense suspensions. 𝛾𝑎 and ℓ will then

have similar effects in the constant concentration limit as well and we set them to zero to

simplify our analysis. Our effective equations of motion then are

𝜌 (𝜕𝑡𝐮 + 𝐮.∇𝐮) = −∇𝑃 + 𝜇∇2𝐮 − 𝜎0𝐩 ⋅ ∇𝐩 − ∇ ⋅ (𝜆−𝐩𝐡 + 𝜆+𝐡𝐩) ,

𝜕𝑡𝐩 + (𝐮 + 𝑣0𝐩).∇𝐩 = −∇Π + 𝜆𝐒.𝐩 + 𝛀.𝐩 + Γ𝐡,(4.5)

where 𝐡 = 𝐾∇2𝐩 + (1 − |𝐩|2) 𝐩, 𝜎0 = 𝜎𝑎(𝑐0) and the additional pressure-like term Πenforces the incompressibility constraint ∇ ⋅ 𝐩 = 0.

Equations of motion | 59

4.3 Linear stability analysis

We now discuss the linear stability analysis of dense suspensions, which was carried in

collaboration with Rayan Chatterjee.

We analyse the stability of the ordered state 𝐮 = 0, 𝐩 = 𝑥, which is a steady state solu-

tion of (4.5) to small perturbations. We define the ordering direction to be parallel to 𝑥 and

denote the unit vector in the plane perpendicular to 𝑥 by ⟂, as shown in Fig. 4.1. General

deformations around an ordered state can be decomposed into three basic deformations of

the order parameter: Splay, Bend and Twist [38, 63, 68] which are shown in Fig. 4.1.

Figure 4.1: (Top row) A schematic diagram of the ordered state and imposed perturbations. (Bottom row)

Primary deformations of the polar order parameter.

The linearized equations for small perturbations 𝛿𝐮 ≡ (𝛿𝑢𝑥, 𝜹𝐮⟂) and 𝛿𝐩 ≡ (𝛿𝑝𝑥, 𝜹𝐩⟂)are

𝜌𝜕𝑡𝛿𝑢𝑥 = −𝜕𝑥𝑃 + 𝜇∇2𝛿𝑢𝑥 − 𝜕𝑥 (𝜎0𝛿𝑝𝑥 + 𝜆+𝐾∇2𝛿𝑝𝑥 − 2𝜆𝛿𝑝𝑥) ,

𝜌𝜕𝑡𝜹𝐮⟂ = −∇⟂𝑃 + 𝜇∇2𝜹𝐮⟂ − 𝜕𝑥 (𝜎0𝜹𝐩⟂ + 𝜆+𝐾∇2𝜹𝐩⟂) ,

𝜕𝑡𝛿𝑝𝑥 = −𝜕𝑥Π − 𝑣0𝜕𝑥𝛿𝑝𝑥 + Γ𝐾∇2𝛿𝑝𝑥 − 2Γ𝛿𝑝𝑥 + 𝜆𝜕𝑥𝛿𝑢𝑥,

𝜕𝑡𝜹𝐩⟂ = −∇⟂Π − 𝑣0𝜕𝑥𝜹𝐩⟂ + Γ𝐾∇2𝜹𝐩⟂ + 𝜆+𝜕𝑥𝜹𝐮⟂ + 𝜆−∇⟂𝛿𝑢𝑥.

(4.6)

Note that the perturbations also satisfy the incompressibility criteria, ∇ ⋅ 𝛿𝐮 = 0, and

∇ ⋅ 𝛿𝐩 = 0. To proceed further, we consider monochromatic perturbations of the form

⎛⎜⎝

𝜹𝐮𝜹𝐩

⎞⎟⎠

= ⎛⎜⎝

��

⎞⎟⎠

𝑒𝑖(𝐪⋅𝐱−𝜔𝑡), (4.7)


where 𝐪 ≡ (𝑞𝑥 𝑥 + 𝐪⟂⋅ ⟂) is the perturbation wavevector and 𝜔 = R(𝜔) + 𝑖I(𝜔), and

R(𝜔) and I(𝜔) are the real and imaginary parts of 𝜔. Ordered state will be unstable to

perturbations if I(𝜔) > 0. Substituting (4.7) in (4.6) gives

(−𝑖𝜌𝜔 + 𝜈𝑞2) ��𝑥 = −𝑖𝑞𝑥𝑃 − 𝑖(𝜎0 − 𝜆+𝐾𝑞2 − 2𝜆)𝑞𝑥 𝑝𝑥

(−𝑖𝜌𝜔 + 𝜈𝑞2) ��⟂ = −𝑖𝐪⟂𝑃 − 𝑖(𝜎0 − 𝜆+𝐾𝑞2)𝑞𝑥��⟂

(−𝑖𝜔 + Γ𝐾𝑞2 + 𝑖𝑣0𝑞𝑥) 𝑝𝑥 = −𝑖𝑞𝑥Π − 2Γ 𝑝𝑥 + 𝑖𝜆𝑞𝑥��𝑥

(−𝑖𝜔 + Γ𝐾𝑞2 + 𝑖𝑣0𝑞𝑥) ��⟂ = −𝑖𝐪⟂Π + 𝑖(𝜆+𝑞𝑥��⟂ + 𝜆−𝐪⟂��𝑥),

(4.8)

In the 𝛿𝑝𝑥 equation, the leading order term independent of the wavevector 𝐪 is −2𝛿𝑝𝑥.

In the absence of pressure term Π, perturbations parallel to the ordering direction decay

exponentially and are rendered fast [55]. Π couples the longitudinal and transverse pertur-

bations and 𝛿𝑝𝑥 is no longer a fast variable. However, incompressibility constraints on 𝐮and 𝐩 allows us to proceed in terms of transverse perturbations only, for which we have

(−𝑖𝜌𝜔 + 𝜇𝑞2)��⟂ = 2𝑖𝜆𝑞𝑥𝐪⟂ ⋅ ��⟂

𝑞2 𝐪⟂ − 𝑖𝑞𝑥 (𝜎0 − 𝜆+𝐾𝑞2) ��⟂

(−𝑖𝜔 + Γ𝐾𝑞2 + 𝑖𝑣0𝑞𝑥)��⟂ = −2Γ𝐪⟂ ⋅ ��⟂𝑞2 𝐪⟂ + 𝑖𝜆+𝑞𝑥��⟂.

(4.9)

Out of the four eigenvectors of (4.9), two are parallel to 𝐪⟂ and represent the coupled

splay-bend modes. The other two are perpendicular to 𝐪⟂ and represent the twist-bend

modes. Twist deformations are three-dimensional and hence the twist-bend modes do not

exist for two-dimensional systems.

Twist-bend modes

Taking the cross product of (4.9) with 𝐪⟂ gives us the following dispersion relation for

twist-bend modes

⎡⎢⎣

−𝑖𝜌𝜔 + 𝜇𝑞2 𝑖𝑞 cos 𝜙(𝜎0 − 𝜆+𝐾𝑞2)−𝑖𝜆+𝑞 cos 𝜙 −𝑖𝜔 + Γ𝐾𝑞2 + 𝑖𝑣0𝑞 cos 𝜙

⎤⎥⎦

⎡⎢⎣

𝐪⟂ × ��⟂

𝐪⟂ × ��⟂

⎤⎥⎦

= 0, (4.10)

where 𝑞𝑥 = 𝑞 cos 𝜙 and 𝐪⟂ = 𝑞 sin 𝜙. From the above equation we get

2𝜔𝑡± = 𝑣0𝑞 cos 𝜙 − 𝑖𝜇+

𝜌 𝑞2 ± 1𝜌√(𝜌𝑣0𝑞 cos 𝜙 + 𝑖𝜇−𝑞2)2 − 4𝜌𝜆+𝑞2 cos2 𝜙(𝜎0 − 𝜆+𝐾𝑞2),

(4.11)

where 𝜇± = 𝜇 ± Γ𝐾𝜌. A large wavelength (small 𝑞) expansion gives

2𝜔𝑡± = 𝑣0 cos 𝜙 (1 ± √1 − 𝑅1

𝑅 ) 𝑞 + 𝑖𝜇𝜌

⎛⎜⎜⎝

± (1 − 𝛽)√1 − 𝑅1

𝑅

− (1 + 𝛽)⎞⎟⎟⎠

𝑞2, (4.12)

Linear stability analysis | 61

0 1 2q

0

2

4

6

I(ω

)×10−3

R = 0.3R = 0.4R = 0.5

0.0 0.5 1.0q

−2.5

0.0

2.5

5.0

I(ω

)

×10−4

R = 2.0R = 3.0R = 4.0

Figure 4.2: Growth rate I(𝜔) vs. 𝑞 for pure bend modes at various 𝑅 in (a) regime A, and (b) regime B.

Note that the growth rate is an order of magnitude smaller in regime B as compared to regime A.

where we have defined the dimensionless numbers 𝑅 ≡ 𝜌𝑣20/2𝜎0, 𝑅1 ≡ 1+𝜆 and 𝛽 = Γ𝐾𝜌/𝜇.

From (4.12), we find that the 𝑅 − 𝛽 space has three distinct regimes:

• Regime A : For 𝑅 < 𝑅1 and any value of 𝛽, we have

1 − 𝑅1𝑅 < 0.

In this regime, twist-bend modes are unstable with a growth rate I(𝜔) ∝ 𝑞.

• Regime B : For 𝑅1 < 𝑅 < 𝑅2, where 𝑅2 ≡ 𝑅1 (1 + 𝛽)2 /4𝛽, we have

(1 − 𝛽)√1 − 𝑅1

𝑅

− (1 + 𝛽) > 0.

In this regime, twist-bend modes have a growth rate I(𝜔) ∝ 𝑞2.

• Regime C Finally for 𝑅 > 𝑅2, twist bend modes are stable.

In Fig. 4.2, we plot the growth rate I(𝜔) vs. 𝑞 computed from (4.11) for various values of

𝑅 in both regime A and regime B. It is clear from the plots that a range of unstable wave

numbers exist in both regime A and B. Further, the growth rate decreases with increasing

𝑅, and is order of magnitude smaller in regime B as compared to regime A.

Note that the small-𝑞 expansion (4.12) is not valid for pure twist modes (𝜙 = 𝜋/2). The

dispersion relation in this case reduces to 𝜔𝑡± = − 𝑖

2𝜌 (𝜇+ ∓ 𝜇−) 𝑞2 which implies pure twist

modes are stable to linear perturbations. The result is not surprising, as we know that the

extensile swimmers go unstable via bend deformations.


Splay-bend modes

Taking the dot product of (4.9) with 𝐪⟂ gives us the following dispersion relation for splay-

bend modes

⎡⎢⎣

−𝑖𝜌𝜔 + 𝜇𝑞2 𝑖𝑞 cos 𝜙(𝜎0 − 𝜆+𝐾𝑞2 − 2𝜆 sin2 𝜙)−𝑖𝜆+𝑞 cos 𝜙 −𝑖𝜔 + Γ𝐾𝑞2 + 𝑖𝑣0𝑞 cos 𝜙 + 2Γ sin2 𝜙

⎤⎥⎦

⎡⎢⎣

𝐪⟂ ⋅ ��⟂

𝐪⟂ ⋅ ��⟂

⎤⎥⎦

= 0 (4.13)

We then have

2𝜔𝑠± = 𝑣0𝑞 cos 𝜙 − 𝑖𝜇+

𝜌 𝑞2 − 2𝑖Γ sin2 𝜙

± 1𝜌

√(𝜌𝑣0𝑞 cos 𝜙 + 𝑖𝜇−𝑞2 − 2𝑖𝜌Γ sin2 𝜙)2 − 4𝜌𝜆+𝑞2 cos2 𝜙(𝜎0 − 𝜆+𝐾𝑞2 − 2𝜆 sin2 𝜙).(4.14)

Substituting 𝜙 = 0 for the two-dimensional pure bend mode in (4.14) gives

2𝜔± = 𝑣0𝑞 − 𝑖𝜇+𝜌 𝑞2 ± 1

𝜌√(𝜌𝑣0𝑞 + 𝑖𝜇−𝑞2)2 − 4𝜌𝜆+𝑞2(𝜎0 − 𝜆+𝐾𝑞2). (4.15)

The above expression is the same as (4.11) with 𝜙 = 0. The stability of two-dimensional

pure bend modes then follows from the discussion for (4.11).

For 𝜙 > 0, the situation is slightly different. First, the incompressibility constraint

eliminates the splay part of the deformations by an equal and opposite contribution since

𝜕𝑥𝛿𝑝𝑥 + ∇⟂ ⋅ 𝜹𝐩⟂ = 0. Second, for all non-zero 𝜙 it has a 𝑞 independent stabilizing effect

and the relaxation rate does not vanish in the large wavelength limit. Specifically at 𝑞 → 0we have one non vanishing eigenvalue

2𝜔𝑠− = −4𝑖Γ sin2 𝜙. (4.16)

For small but nonzero 𝜙, splay component of splay-bend modes is small and these modes

should go unstable in a similar fashion to the pure bend mode, albeit with a smaller growth

rate and diminished unstable 𝑞 range. In Fig. 4.3 we plot I(𝜔) vs. 𝑞 for various values of 𝜙at small 𝑅 = 0.5 and verify that indeed it is the case.

Linear stability analysis | 63

0 1 2 3q

−5

0

5

I(ω

)

×10−3

R=0.5

φ = 0◦φ = 4◦

φ = 8◦φ = 12◦

Figure 4.3: Growth rate I(𝜔) vs. 𝑞 for splay-bend modes at various 𝜙 for fixed 𝑅 = 0.5. With increasing

𝜙 the 𝜔 decreases, and the splay-bend modes show instabilities only small 𝜙.

4.3.1 Linear stability phase diagram

In Fig. 4.4, we show the 𝑅 − 𝛽 stability diagram for extensile suspensions highlighting

various stable and unstable regimes.

0.0 0.5 1.0 1.5 2.0β

0.8

1.0

1.2

1.4

1.6

1.8

2.0

R

R1

R2

Regime A

Regime B

Regime C

Figure 4.4: 𝑅 − 𝛽 phase diagram obtained from linear stability analysis showing the three distinct regimes

for pure bend modes for extensile suspensions. Note that the phase diagram is identical for suspensions in

two and three dimensions.


4.4 Non-dimensional equations of motion

Linear stability analysis have already given us two non-dimensional numbers, 𝑅 and 𝛽which determine the stability of the ordered state [see Fig. 4.4]. By rescaling the space

𝐱′ → 𝐿𝐱, the time 𝑡′ → 𝑇 𝑡, the pressure terms 𝑃 ′ → 𝜌𝑈2𝑃 , Π′ → Π/𝑈 and the velocity

field 𝑢′𝑖 → 𝑈𝑢𝑖, where 𝑈 = 𝐿/𝑇 we get

𝜕𝑡𝑢𝑖 + 𝑢𝑗𝜕𝑗𝑢𝑖 = −𝜕𝑖𝑃 + 𝜇𝜌𝑈𝐿𝜕2

𝑗 𝑢𝑖 − 1𝜌𝑈2 𝜎0𝑝𝑗𝜕𝑗𝑝𝑖,

𝜕𝑡𝑝𝑖 + (𝑢𝑗 + 𝑣0𝑈 𝑝𝑗) 𝜕𝑗𝑝𝑖 = −𝜕𝑖Π + 𝜆−𝑝𝑗𝜕𝑖𝑢𝑗 + 𝜆+𝑝𝑗𝜕𝑗𝑢𝑖 + Γ𝐾

𝑈𝐿𝜕2𝑗 𝑝𝑖 + Γ𝐿𝑎

𝑈 (1 − 𝑝2𝑗 ) 𝑝𝑖.(4.17)

Here we have ignored the prime index for brevity. Further, choosing 𝑈 = 𝑣0 and

ℓ𝜎 = 𝜇/√𝜌𝜎0 as our velocity and length scales respectively we get

𝜕𝑡𝑢𝑖 + 𝑢𝑗𝜕𝑗𝑢𝑖 = −𝜕𝑖𝑃 + 1√𝑅𝜕2

𝑗 𝑢𝑖 − 12𝑅𝑝𝑗𝜕𝑗𝑝𝑖,

𝜕𝑡𝑝𝑖 + (𝑢𝑗 + 𝑝𝑗) 𝜕𝑗𝑝𝑖 = −𝜕𝑖Π + 𝜆−𝑝𝑗𝜕𝑖𝑢𝑗 + 𝜆+𝑝𝑗𝜕𝑗𝑢𝑖 + 𝛽√2𝑅 [𝜕2

𝑗 𝑝𝑖 + 𝜃𝛼 (1 − 𝑝2

𝑗 ) 𝑝𝑖] ,(4.18)

where we have defined the other dimensionless numbers 𝛼 = 𝐾𝜌/𝜇2, and 𝜃 = 𝑎/𝜎0.

Non-dimensional equations of motion | 65

4.5 Numerical studies in two dimensions

We perform discrete numerical simulations (DNS) of (4.5) on a square domain of area 𝐿2

discretized over 𝑁2 collocation points. We use a pseudo-spectral method for the velocity

equation and a fourth-order central finite-difference scheme for the order parameter equation.

For time marching, we use a second-order Adam-Bashford scheme [112].

Table 4.1 enumerates all our simulation parameters. We first verify our linear stability

analysis results (prefix LSA) and then undertake high resolution numerical studies at vari-

ous values of 𝑅 and fixed 𝛽 = 10−2 to characterize the turbulent states arising from the

instabilities of the ordered state (prefix SPP).

𝐿 𝑁 𝑣0(×10−2) 𝑅 ≡ 𝜌𝑣20/2𝜎0

LSA1 2𝜋 64 2.52 0.05LSA2 2𝜋 64 15.81 2.00SPP1 10𝜋 1024 3.16 0.05SPP2 40𝜋 2048 3.16 0.20 − 0.40∗, 0.22SPP3 80𝜋 4096 3.16 0.5SPP4 160𝜋 4096 3.16 0.6 − 0.9†

SPP5 160𝜋 4096 3.16 1.4, 4.0

Table 4.1: Value of parameters used in DNS in two dimensions. 𝜌 = 1, 𝜆 = 0.1, 𝜇 = 0.1, 𝐾 = 10−3,

and Γ = 1 are kept fixed for all runs. Prefix LSA is for linear stability analysis runs, whereas SPP is for

turbulence runs. ∗With increments of 0.05. †With increments of 0.1.

4.5.1 Numerical verification of linear stability analysis

To verify our linear stability analysis results, we perform DNS of (4.5) in two dimensions

around perturbed ordered states. We start with pure bend perturbations 𝐮 = 𝐴 cos 𝐪 ⋅ 𝐫 𝑦and 𝐩 = 𝑥+𝐵 cos 𝐪 ⋅𝐫 𝑦 and monitor the growth of the perturbations ( Table 4.1, runs LSA1

and LSA2). We set the perturbation amplitudes 𝐴 = 𝐵 = 10−3 and choose the perturbation

wavevector 𝐪 parallel to 𝑥. This particular initial state ensures incompressibility.

In Fig. 4.5 we plot the growth rate I(𝜔) at different 𝑞 in regime A (𝑅 = 0.05), and

regime B (𝑅 = 2.00) and compare it with the growth rate obtained from simulations. For

𝑅 = 0.05, the exponential growth occurs at a much larger rate than the oscillations as is

evident from amplitude vs. time plot. For 𝑅 = 2.00 since the growth rate is much smaller


than the oscillation frequency, we observe both the oscillatory and the exponential growth

behaviour of the perturbations much more clearly.

0 2 4 6q

−2

−1

0

1

2

3

I(ω

)

×10−2

A

B

0 50 150t

10−4

10−3

10−2

10−1

A

0 100 300t

B

R = 0.05R = 2.00

Figure 4.5: Comparison of growth rates I(𝜔) for pure bend modes obtained from the linear stability analysis

(black lines) with simulations ( black circles) for (A) 𝑅 = 0.05, and (B) 𝑅 = 2.00. Insets: Growth of the

perturbation amplitude 𝛿𝐩 with time, for (A) (𝑞 = 1, 𝑅 = 0.05), and (B) (𝑞 = 1, 𝑅 = 2.00). Dashed

black lines show the growth rate obtained from the linear stability analysis.

Numerical studies in two dimensions | 67

4.6 Turbulence in two dimensions

We now discuss the statistical properties of the chaotic flow arising from the instabilities

arising from the perturbations on the ordered state. To achieve a statistically steady state,

we choose a series sum of monochromatic perturbations with wavenumbers 𝐪𝑖=1…𝑀 ,

𝐮 = 𝐴𝑀

∑𝑖=1

cos(𝐪𝑖 ⋅ 𝐫) 𝑦

𝐩 = 𝑥 + 𝐵𝑀

∑𝑖=1

cos(𝐪𝑖 ⋅ 𝐫) 𝑦.(4.19)

We set 𝐴 = 𝐵 = 10−3 and pick 𝐪𝑖 suitably from the unstable regime of the dispersion

relation [see Fig. 4.5 and Fig. 4.2]. We monitor the time-evolution of perturbed states and

investigate the statistical properties of the velocity and the order parameter field in the

turbulent steady states.

In Fig. 4.6 we show the streamlines of the order parameter field 𝐩 in the statistically

steady state at different values of R. In both regimes A and B, the order parameter field is

riddled with topological vortices and saddles, with no global order in sight. As pointed out

in earlier chapters, asters and spirals are ruled out by the incompressibility constraint. As

we increase 𝑅, the inter-defect separation increases and at the same time the defect number

density decreases. For uniformly distributed defects in two dimensions, one expects that

the defect density is inversely proportional to the square of the inter-defect spacing [see

Chapter 2 and [99, 100]].

To further verify that the turbulent states lack global order, we compute the magnitude

of the polar order parameter |⟨𝐩⟩| in the statistical steady state at different values of 𝑅,

where ⟨…⟩ denotes spatio-temporal averaging. Note that, |⟨𝐩⟩| = 0 for disordered states,

whereas |⟨𝐩⟩| = 1 for a perfectly aligned state. In Fig. 4.7 we plot |⟨𝐩⟩| with increasing 𝑅.

As expected, |⟨𝐩⟩| is consistent with zero in both the unstable regimes.


0 20π 40π0

10π

20πR = 0.05

0 20π 40π0

10π

20πR = 0.20

0 20π 40π0

10π

20πR = 0.35

0 20π 40π0

20π

40πR = 0.90

0 20π 40π0

20π

40πR = 1.40

0 20π 40π0

20π

40πR = 4.00

−3

−2

−1

0

1

2

3

∇× p

Figure 4.6: Order parameter streamlines superimposed over pseudocolor plot of ∇ × 𝐩 highlighting topo-

logical vortices at different values of R. As we increase 𝑅, inter-defect separating grows.

10−2 10−1 100 101

R

0.0

0.2

0.4

0.6

0.8

1.0

|〈p〉|

R1 R2

10−2 10−1 100 1010

2

4×10−2

R1

Figure 4.7: Average order | ⟨𝐩⟩ | at different 𝑅. We do not observe any order in regime A or regime B.

Ordered states are stable to perturbations in green shaded region. Inset: Zoomed in region of the same

plot.

Turbulence in two dimensions | 69

4.6.1 Correlation functions and correlation length

We compute the correlation function for the order parameter field

𝐶(𝑟) = ⟨𝐩 (𝐱 + 𝐫) ⋅ 𝐩 (𝐱)⟩⟨𝐩 (0)2⟩

, (4.20)

where ⟨…⟩ denote spatio-temporal averaging in the steady state. In Fig. 4.8(a) we plot 𝐶(𝑟)vs. 𝑟 at different values of 𝑅. Consistent with our order parameter snapshots, we find that

correlations grow as we increase 𝑅. 𝐶(𝑟) collapses on a single curve when plotted vs. 𝑟/𝜉[see Fig. 4.8(a,inset)] . We fit the functional form

𝐶(𝑟) = 𝑒−( 𝑟𝜉 )𝛿

, (4.21)

to extract 𝛿 and the correlation length 𝜉 from 𝐶(𝑟). For large 𝑅, we find that 𝛿 is close to one.

In Fig. 4.8(b) we plot the inverse correlation length 𝜉 at various values of 𝑅. 𝜉 grows with 𝑅and from the intercept of the linear fit on the 1/𝑅 axis it appears to diverge at 𝑅 = 𝑅2. Note

that a configuration of order parameter field with 𝜉 → ∞ is practically indistinguishable

from a perfectly aligned state on simulation boxes of finite size. To properly establish the

growing nature of 𝜉 at large 𝑅, further finite-size scaling studies are required.

0 20 40 60r

0.00

0.25

0.50

0.75

1.00

C(r

)

0 2 4

r/ξ

0.0

0.5

1.0

C(r/ξ

)

R = 0.35R = 0.60R = 0.90R = 1.40

0 1 2 3 4 51/R

0.0

0.2

0.4

0.6

0.8

1/ξ

1/R11/R2

ξLinear Fit

Figure 4.8: (a) Steady state correlation function 𝐶(𝑟) for different values of R. Inset: Collapse of correlation

functions, when distance is scaled by the correlation length 𝜉. (b) Plot of inverse correlation length 1/𝜉versus 1/𝑅. Correlation length stays finite as 𝑅 approaches 𝑅1. From the intercept of the linear fit on the

1/𝑅 axis, we conclude that 𝜉 diverges around 𝑅 ≈ 𝑅2.


4.6.2 Energy Spectrum

We define the shell-averaged energy spectra for the velocity and the order parameter field

as𝐸𝐮(𝑞) = ∑

𝑞− 12 ≤|𝐦|<𝑞+ 1

2

|��𝐦|2, and

𝐸𝐩(𝑞) = ∑𝑞− 1

2 ≤|𝐦|<𝑞+ 12

|��𝐦|2,(4.22)

where ��𝐦 and ��𝐦 are the Fourier coefficients of the velocity 𝐮 and the order parameter 𝐩fields. In Fig. 4.9, we plot 𝐸𝐮(𝑞𝜉) and 𝐸𝐩(𝑞𝜉) for different values of 𝑅. Consistent with

out correlation function plots, we find that the spectra collapses onto single curves. Order

parameter spectra shows two distinct power law scaling regimes:

𝐸𝐩(𝑞𝜉) =⎧{⎨{⎩

𝑞2 for 2𝜋𝐿 < 𝑞 < 2𝜋

𝜉

𝑞−3 for 2𝜋𝜉 < 𝑞 < 2𝜋

ℓ𝜎,

(4.23)

where ℓ𝜎 = 𝜇/√𝜌𝜎0, and 𝐿 determines the smallest wavenumber available in the simulation

domain. At large 𝑅 we observe a small departure from the 𝑞−3 scaling. Note that the

𝑞−3 scaling is consistent with Porod’s law [see Chapter 2 and [79]]. 𝐸𝐮(𝑞𝜉) shows a peak

around 𝑞𝜉 as well, and has a steeper slope than 𝐸𝐩(𝑞𝜉) for 𝑞𝜉 > 1. At large values of 𝑞,

𝐸𝐮(𝑞) ∼ 𝑞−2.5, whereas at smaller values of 𝑅, 𝐸𝐮(𝑞𝜉) decays rapidly and the power law

scaling is not very clear.

10−1 100 101

qξ

10−7

10−5

10−3

10−1

Eu(qξ)/E

m u

q−2.5

ξqσ

R = 0.35R = 0.60R = 0.90R = 1.40

10−1 100 101

qξ

10−7

10−5

10−3

10−1

Ep(qξ)/E

m p

q−3q2

ξqσ

R = 0.35R = 0.60R = 0.90R = 1.40

Figure 4.9: (a) Kinetic energy spectrum 𝐸𝐮(𝑞𝜉) and (b) order parameter energy spectrum 𝐸𝐩(𝑞𝜉) for

different values of R in both regime A and regime B. We scale the spectra by their respective peak values.

The order parameter spectra shows Porod’s scaling [79] for smaller 𝑅 for 1 < 𝑞𝜉 < 𝑞𝜎𝜉.

Turbulence in two dimensions | 71

4.7 Turbulence in three dimensions

We now discuss the results of our DNS in three dimensions. The most unstable modes for 2D

and 3D suspensions are the same i.e. the pure bend modes, hence the 𝑅 − 𝛽 phase diagram

is the same for both the cases. We initialize our simulations with perturbed ordered states

and characterize the properties of the resulting turbulent flows. In Table 4.2 we enumerate

our simulation parameters.

As was the case in two dimensions, we find that the order parameter develops chaotic

defect ridden configurations in both the regimes A and B. These defects have similar nature

to the low Re defects discussed in Chapter 3 [see Fig. 3.4] . In Fig. 4.10(a,b) we show defect

positions over the simulation domain and streamlines around a subset of defects for 𝑅 = 0.1.

Fig. 4.10(c,d) shows the same for 𝑅 = 1.2.

Figure 4.10: (a) Topological defects at 𝑅 = 0.1 (blue: +1, yellow: -1). (b) Zoomed in smaller red box

shows streamlines around a subset of topological defects. (c,d) Similar plot for 𝑅 = 1.2. As we increase 𝑅,

inter-defect spacing grows.


𝐿 𝑁 𝑅 ≡ 𝜌𝑣20/2𝜎0

SPP3D1 4𝜋 128 0.1SPP3D2 8𝜋 256 0.2, 0.3SPP3D3 16𝜋 512 0.6, 0.9, 1.2

Table 4.2: Parameters used in numerical simulations in 3D. 𝜌 = 1, 𝜆 = 0.1, 𝜇 = 0.1, 𝐾 = 10−3, Γ = 1and 0 = 3.16 × 10−2 are kept fixed for all runs.

In Fig. 4.11(a), we show the steady-state energy spectrum for the order parameter field.

The spectrum peaks at 𝑞𝜉 ≈ 1 and does not show any significant power law scaling. Similar

to 2D, with increasing 𝑅 we observe that the inter-defect spacing grows [see Fig. 4.11(b)].

However our current numerical simulations at large 𝑅 are restricted to smaller system sizes.

Studies with larger system sizes are required to properly resolve the growing inter-defect

spacing and characterize the statistical properties of the order parameter field. This remains

to be explored in future studies.

100 101

qξ

10−8

10−6

10−4

10−2

Ep(qξ)/E

m p

(a)

R = 0.10R = 0.30R = 0.60R = 0.90R = 1.20

0.2 0.4 0.6 0.8 1.0 1.2R

1.0

1.2

1.4

1.6

ξ

(b)

Figure 4.11: (a) Energy spectrum for the order parameter field at various values of 𝑅 in 3D. The spectrum

peaks at 𝑞𝜉 ≈ 1 and does not show any significant power law scaling. (b) Inter-defect spacing [or average

minimum separation, see Chapters 2 and 3] computed from the defect positions grows with 𝑅.

Turbulence in three dimensions | 73

4.8 Comparison with Malthusian suspensions

In an earlier study [55] [see also Section 1.4 and [135]], we have studied the stability of

ordered state in Malthusian suspensions where the birth-death processes occur at time

scales smaller than the time scales of collective motion, and can restore the concentration

to its equilibrium value quickly. Concentration is a fast variable in such systems and is

ignored in the hydrodynamic description. The equations of motion are

𝜌 (𝜕𝑡𝐮 + 𝐮.∇𝐮) = −∇𝑃 + 𝜇∇2𝐮 − ∇.𝚺,

𝜕𝑡𝐩 + (𝐮 + 𝑣0𝐩).∇𝐩 = 𝜆𝐒.𝐩 + 𝛀.𝐩 + Γ𝐡,

∇ ⋅ 𝐮 = 0.

(4.24)

Here all the phenomenological parameters take their values at the equilibrium concentration

maintained by the birth-death processes. Note that the order parameter is not incompress-

ible for Malthusian suspensions. We now highlight the similarities and differences between

dense and Malthusian suspensions.

4.8.1 Linear stability analysis

For a suspension described by (4.1), concentration fluctuations couple only to the two-

dimensional splay-bend modes [55, 135]. Any constraints imposed on the number conser-

vation equation will then also affect only these modes. Thus, the linear stability of the

twist-bend modes and crucially the most unstable pure bend modes is independent of the

dynamics of the concentration fluctuations. The differences between the Malthusian and

dense suspensions arise in the form of allowed topological defects and the nature of steady

states in regimes A and B, which we describe below.

4.8.2 Non equilibrium steady states

For dense suspensions, vortices and saddles are the only allowed topological defects. On the

other hand, asters are the dominant structures for Malthusian suspensions. In Fig. 4.12 we

show various structures observed in the steady-state at different values of 𝑅 for Malthusian

and dense suspensions. Malthusian suspensions show a non-equilibrium flocking transition

from aster-riddled defect turbulent states to aligned but phase turbulent states as R crosses

from regime A to B. We do not observe any such transition for incompressible suspensions,

where we observe vortices even in regime B. The absence of phase-turbulence is corroborated


Figure 4.12: Comparison of structures observed in the order parameter field 𝐩 for Malthusian and dense

suspensions at various 𝑅 in two dimensions. Malthusian suspensions show asters in regime A and phase

turbulence in regime B, where dense suspensions show vortices in both regime A and regime B. Both are

stable in regime C.

by the fact that the correlation length diverges at 𝑅 = 𝑅1 for the Malthusian case, and at

𝑅 = 𝑅2 for dense suspensions.

4.9 Conclusions

This chapter studied dense wet suspensions of polar active particles in two and three di-

mensions. We investigated the instabilities of the aligned state to small perturbations and

showed how suspension inertia could stabilize the orientational order. We found that a

non-dimensional parameter 𝑅 characterizes the stability of the aligned state. For 𝑅 < 𝑅1,

perturbation amplitude grows at a rate proportional to the magnitude of the perturbation

wavevector. For 𝑅1 < 𝑅 < 𝑅2, we observed a growth rate proportional to the square of

the magnitude of the perturbation wavevector. We then performed high resolution direct

numerical simulations and characterized the properties of the chaotic flows arising from the

instabilities. We showed that in two and three dimensions, for all 𝑅 < 𝑅2, the flow is rid-

dled with topological vortices with no global order insight. In contrast to their Malthusian

counterparts, which show a flocking transition from defect-turbulence to phase-turbulence

states at 𝑅 = 𝑅1, dense suspensions show a non-equilibrium flocking transition from defect-

turbulence to perfectly aligned states at 𝑅 = 𝑅2.

Conclusions | 75

Chapter 5

Population fluctuations in growing bacteria colonies

In earlier chapters, we have investigated systems where the activity manifests as self-

propelling particles. In this chapter, we focus on a colony of nonmotile bacteria growing

on a hard agar surface. Here, the activity arises not from the particle motility, but the

birth-death processes. At the colony front, bacteria repeatedly grow and divide at the expense

of nutrients and push each other in the process. These steric repulsions between the bacteria

result in an expanding colony. Population fluctuations intrinsic to any birth-death process

become essential for such systems, especially at the growing front where the number of

organisms is small compared to the colony’s interior. We investigate how these fluctuations

and nutrient availability can affect the growing colony’s morphology. We find that the

population fluctuations and nutrient-dependent bacteria diffusion are sufficient to cause the

morphological transition from finger-like branched fronts to smooth fronts upon increasing

nutrient concentration.

5.1 Introduction

Pattern formation is one of the fascinating feature observed in a broad range of natural

phenomena [18, 138, 139]. For example, colonies of bacteria growing on a Petri dish exhibit

a large variety of intricate spatial patterns ranging from compact circular growth, concentric

rings to long branched patterns [11, 13, 140–145]. The colony morphology depends on

various factors such as nutrient concentration, cell motility, birth-death dynamics, and

other chemical and physical variables [146–154].

In Fig. 5.1, we show various morphological patterns observed in Escherichia Coli colonies

growing on a hard agar surface in different environment settings. The colonies show compact

| 77

circular growth, rough circular growth, and branched finger-like patterns.

(a) (b) (c)

1 cm

Figure 5.1: Various colony morphologies of an Escherichia Coli colony growing on a hard agar surface. (a)

A compact colony with a smooth front, (b) a compact colony with rough front, and (c) a colony showing

branched fingers. Raw colony images provided by Akshi Gupta and Debjani Paul , IIT Mumbai.

The classic experiment of Wakita et al. [140] obtained the phase-diagram of Bacillus

subtilis colony morphology as a function of nutrient and agar concentration [see Fig. 5.2].

They identified the following five primary morphologies in the growing colonies.

• Diffusion Limited Aggregation (DLA, type-A) — The colonies growing on a hard

agar surface, in nutrient-poor conditions show thick branched fingers extending in all

directions. Here the colony growth is governed by the nutrients diffusing towards the

growing front [140, 155].

• Eden-like, (type-B) – In a nutrient-rich environment on a hard agar plate, the colony

growth is compact, with no branching, but the colony front shows small undulations.

• Concentric Ring-like, (type-C) – The colonies growing on an agar surface with inter-

mediate hardness, at high nutrient concentration, show concentric ring-like shapes,

consisting of numerous thin branches (similar to type-E) and an inhomogeneously

growing front which shows DLA like undulations [156].

• Homogeneous Spreading, (type-D) — In a nutrient-rich environment on a soft agar

plate, the colonies grow homogeneously in all directions, producing compact circular

shape with a uniform front.

• Dense Branching Morphology (DBM, type-E) — In nutrient-poor conditions on a

semi-solid agar, the colonies show Dense Branching Morphology made up of a large

number of thin branches, but the growing front stays uniform in all directions.

78 | Population fluctuations in growing bacteria colonies

Figure 5.2: Phase diagram of colony morphology for a Bacillus subtilis colony as a function of nutrient

concentration 𝐶𝑛 and agar concentration 𝐶𝑎. The red arrow shows the morphological transition from DLA-

like to Eden-like (type-A to type-B) upon varying nutrient concentration 𝐶𝑛 at high agar concentration

(small 1/𝐶𝑎). Reprinted with permission from Wakita et al. [140]. ©(1994) The Physical Society of Japan.

Similar morphological patterns are also observed in growing yeast colonies [157]. Several

studies [156, 158–163], since then, have proposed mathematical models to understand the

origin of observed morphological patterns. These models can be broadly classified into two

categories:

• Agent-based models — In these models, each bacteria is treated as an individual.

The local interactions among bacteria and their interaction with the substrate de-

termine the growing colony’s collective spatio-temporal behavior. These interactions

arise from the mechanical forces exerted by bacteria on each other and the substrate

as they grow and divide on the expanse of nutrients. Since the modeling is at an

individual’s level, agent-based models are used to study a broad range of problems in

population dynamics. For example, agent-based models have been utilized to explore

the role of demographic noise and genetic drifts in the bacteria population [13, 164].

Farrell et al. [165] have used an agent-based model to explore mechanically-driven

growth of nonmotile rod-like bacteria in an expanding colony that undergoes transi-

tions from circular to branched morphologies with varying nutrient consumption rate

or nutrient concentration. Recently, Ghosh et al. [166] have studied mechanical-driven

Introduction | 79

spontaneous phase-segregation of nonmotile, rod-shaped bacteria in the presence of

self-secreted extracellular polymeric substances in a growing biofilm.

• Reaction-Diffusion models — In such models, we treat bacteria density and nutri-

ent concentration as fields described by a set of continuum equations. Diffusive terms

model the bacteria and nutrient motility, and the birth-death processes are modeled

with a reaction term. Perhaps the most widely used reaction-diffusion equation is the

Fisher-Kolmogorov-Piscunoff-Petrovsky equation (FKPP) [56, 57]

𝜕𝑡𝜌 = 𝐷∇2𝜌 + 𝛾𝜌 (1 − 𝜌) , (5.1)

where 𝜌 is the bacteria density fraction, 𝛾 is the growth rate, and 𝐷 is the bacteria

diffusivity. (5.1) has been successfully used to model homogeneous spreading of bac-

teria on a soft-agar plate and in a nutrient-rich environment. Several studies have

incorporated the effect of the nutrient concentration and bacteria motility by coupling

(5.1) with an additional equation for each of these variables to obtain different mor-

phological patterns discussed above [11, 13, 140–145]. Studies designed to investigate

the role of demographic noise use the stochastic variants of (5.1) [13, 57, 167–170].

In this chapter, we explore how the morphology of a colony of nonmotile bacteria growing

on a hard agar substrate changes with varying initial nutrient concentration. Our study is

designed to mimic the experiments of Wakita et al. [140], which show a transition from DLA-

like morphology (type-A) to Eden-like patterns (type-B) when initial nutrient concentration

is increased at a fixed (high) agar concentration (shown by the red arrow in Fig. 5.2).

Earlier studies [11, 143, 163, 171, 172] attribute observed morphological patterns to

substrate properties such as irregularities of the agar substrate [171], substrate hardness

that depends on the agar concentration [11], and local lubrication created by the bacteria

[163]. These models ignore the population fluctuations which are intrinsic to birth-death

processes, and describe the colony growth in a mean-field setting. Recent studies [13, 173,

174] have however shown that fluctuations play an important role in determining the growth,

competition, and cooperation in growing bacterial colonies. Population fluctuations become

essential at the growing front, where the number of bacteria is very small compared to the

colony interior, and cannot be ignored. In particular, Kessler and Levine [173] show that

the stochastic noise leads to diffusive instabilities in the otherwise homogeneously growing

front.


We present here a stochastic continuum model, to study the role of initial nutrient

concentration on the spreading of nonmotile bacteria colony on a hard agar surface. Our

model takes into account the population fluctuation and we assume that the substrate is uni-

form, lacks any inhomogeneities, and ignore substrate-bacteria interaction. Our numerical

experiments show that population fluctuations and nutrient-dependent bacteria diffusivity

destabilize the front and lead to the formation of finger-like patterns in nutrient-deprived

conditions. We find that increasing initial nutrient concentration leads to faster growing

colonies, and the front speed agrees with the mean-field predictions. The front structure

undergoes a transition from a branching pattern to an Eden pattern on increasing the initial

nutrient condition.

In the rest of the chapter, we discuss our continuum model, numerical methods and

results in details.

5.2 Construction of the stochastic continuum model

In this section we outline the construction of the continuum model for a reaction-diffusion

bacteria-nutrient system [56, 57, 160, 175]. We model the growth and division of the bacteria

on the expense of the nutrients using the following reaction

𝐵 + 𝐹 → 𝐵 + 𝐵 at a rate 𝑘, (5.2)

where 𝐵 denotes bacteria and 𝐹 denotes nutrient. A bacterium eats a unit nutrient at a

rate 𝑘 per unit time and divides into two. Ignoring the spatial variations and defining 𝜌𝐵(𝑡)and 𝑐(𝑡) as the bacteria and nutrient number density at time 𝑡, the bacteria density fraction

𝜌(𝑡) = 𝜌𝐵/(𝜌𝐵 + 𝑐) obeys the equation

𝑑𝜌𝑑𝑡 = 𝛾𝜌 (1 − 𝜌) + 𝜇√𝜌 (1 − 𝜌)𝜂 (𝑡) . (5.3)

Here 𝛾 = 𝑘 (𝜌𝐵 + 𝑐) is the growth rate, 𝜇 controls the noise strength and 𝜂(𝑡) is a Gaussian

white noise with ⟨𝜂(𝑡)⟩ = 0, ⟨𝜂 (𝑡) 𝜂 (𝑡′)⟩ = 𝛿 (𝑡 − 𝑡′), and the angular brackets indicate

averaging over noise realizations.

In a spatially extended system where the bacteria are allowed to diffuse and the nutri-

ent concentration is uniform everywhere, (5.3) leads to the stochastic Fisher-Kolmogorov-

Piscunoff-Petrovsky (sFKPP) equation

𝜕𝑡𝜌 = 𝐷∇2𝜌 + 𝛾𝜌 (1 − 𝜌) + 𝜇√𝜌 (1 − 𝜌)𝜂(𝐱, 𝑡) (5.4)

Construction of the stochastic continuum model | 81

where 𝐷 is the diffusivity and 𝜂(𝐱, 𝑡) is Gaussian white noise with ⟨𝜂(𝐱, 𝑡)⟩ = 0, ⟨𝜂(𝐱, 𝑡)𝜂 (𝐱′, 𝑡′)⟩ =𝜹 (𝐱 − 𝐱′) 𝛿 (𝑡 − 𝑡′).

In the absence of population fluctuations (𝜇 = 0), (5.4) reduces to the mean-field FKPP

equation (5.1), which admits wave solutions of the form 𝜌(𝐱, 𝑡) = 𝑤 (𝐱 − 𝑉 𝑡). The front

propagation velocity 𝑉 satisfies the criterion 𝑉 ≥ 2√𝐷𝛾, and for sufficiently sharp initial

fronts the minimum speed 𝑉 = 2√𝐷𝛾 is selected [56, 57, 167]. For the sFKPP equation

(5.4), the stochastic term leads to corrections in the selected speed [167, 176, 177]. In the

weak noise limit, where the dimensionless noise strength 𝜇/ (𝐷𝛾)1/4 < 1, the selected front

velocity for the sFKPP equation is 𝑉 ≈ √𝐷𝛾 [2 − 𝜋2/(4 log2 𝜇)], whereas in the strong

noise limit where 𝜇/ (𝐷𝛾)1/4 > 1 the front velocity is 𝑉 = 2𝐷𝛾/𝜇2 [167].

5.2.1 Nutrient-Bacteria (NB) models

We now write our nutrient-bacteria (NB) models in which, similar to (5.2), bacteria con-

sumes nutrient and divides at a rate 𝛾 per unit biomass while diffusing through space. In

the mean-field setting, the bacteria-nutrient dynamics can be described by the diffusive

Fisher-Kolmogorov equations [56, 57, 160]:

𝜕𝑡𝑐 = 𝐷∇2𝑐 − 𝛾𝜌𝐵𝑐,

𝜕𝑡𝜌𝐵 = 𝐷𝐵∇2𝜌𝐵 + 𝛾𝜌𝐵𝑐.(5.5)

Here, 𝜌𝐵(𝐱, 𝑡) is the bacterial number density and 𝑐 (𝐱, 𝑡) is the nutrient concentration at

position 𝐱 and time 𝑡, 𝐷𝐵 and 𝐷 are diffusion coefficient of bacteria and nutrient, and total

number density over the entire domain 𝐿−2 ∫ [𝜌𝐵(𝐱, 𝑡) + 𝑐(𝐱, 𝑡)] 𝑑𝑥𝑑𝑦 remains conserved.

Variants of the NB model (5.5) but with more complicated reaction and diffusion terms

have been used earlier to investigate the transition from type-A to type-B [56, 160, 163].

However, these mean-field models ignore the role of population fluctuations in the system.

We incorporate population fluctuations in the NB model by adding to it a multiplicative

noise term similar to the sFKPP equation (5.4) [167, 170, 178]. The stochastic NB (sNB)

model that we use, written in terms of the total density 𝜌𝑇 (𝐱, 𝑡) ≡ 𝜌𝐵(𝐱, 𝑡) + 𝑐(𝐱, 𝑡) and

the bacterial number density 𝜌𝐵(𝐱, 𝑡) is

𝜕𝑡𝜌𝑇 = 𝐷𝐵∇2𝜌𝐵 + 𝐷∇2𝑐,

𝜕𝑡𝜌𝐵 = 𝐷𝐵∇2𝜌𝐵 + 𝛾𝜌𝐵 (𝜌𝑇 − 𝜌𝐵) + 𝜇√𝜌𝐵(𝜌𝑇 − 𝜌𝐵)𝜂(𝐱, 𝑡).(5.6)

The first equation simply tells us that the total number density can only vary due to the

diffusion of nutrient and bacteria (5.2) conserves the total number of particles. The second

equation describes the growth of the colony on the expanse of limited available nutrients.


In the sNB model, even in the absence of nutrients, the bacteria will diffuse through

space. However, in a more realistic case the motility of the colony might depend upon

nutrient concentration as well, wherein scarce nutrient conditions will lead to very less

or no movement at all. The diffusion of passive nutrients remains unchanged. Following

Golding et al. [160], we incorporate this effect by using a non-linear nutrient dependent

bacteria diffusivity to get the sNBNL model

𝜕𝑡𝜌𝑇 = 𝐷𝐵∇ ⋅ (𝑐∇𝜌𝐵) + 𝐷∇2𝑐, and

𝜕𝑡𝜌𝐵 = 𝐷𝐵∇ ⋅ (𝑐∇𝜌𝐵) + 𝛾𝜌𝐵 (𝜌𝑇 − 𝜌𝐵) + 𝜇√𝜌𝐵 (𝜌𝑇 − 𝜌𝐵)𝜂(𝐱, 𝑡).(5.7)

In what follows, we present a systematic study of how bacterial front speed and mor-

phology is modified because of nutrient concentration, population fluctuations, bacterial

diffusivity, and nutrient diffusivity using direct numerical simulations (DNS) of sNB and

sNBNL models.

5.3 Numerical methods

We perform simulations of sNB and sNBNL models on a square domain of length 𝐿 dis-

cretized with 𝑁2 collocation points. We use a second-order centered finite-difference scheme

to numerically evaluate the spatial derivatives and impose Neumann boundary conditions

on all sides of the simulation domain. For time marching, we use a variant of the operator

splitting scheme proposed in Refs [175, 179] as described below. We initialize the bacteria

and nutrient number density as

𝜌𝐵 (𝐱, 0) = 12 [1 − tanh {𝑎 (𝑦 − 𝑏)}] ,

𝑐 (𝐱, 0) = 𝐶0 {1 − 𝜌𝐵(𝐱, 0)} .(5.8)

The constants 𝑎 and 𝑏 fix the width and the position of the colony front, and 𝐶0 fixes

the initial nutrient concentration.

Constants in the simulations: In the macroscopic experiments of colony growth, the

number of bacteria is significant. To mimic the same, we fix the strength of population

noise to a small value 𝜇 = 5 × 10−2 in all our simulations. Our simulations are then in the

weak noise limit, where 𝜇/ (𝐷𝐵𝛾)1/4 < 1. We also fix 𝐷 = 10−1 to mimic the slow diffusion

of nutrients on a hard agar plate and choose our timescale such that 𝛾 = 1.

Numerical methods | 83

5.3.1 Numerical integration scheme

Stochastic partial differential equations are tricky to integrate numerically. Consider for

example the first order Euler discretization scheme for the zero-dimensional version (without

the diffusion term) of the sFKPP equation (5.4)

𝜌(𝑡 + Δ𝑡) = 𝜌(𝑡) + Δ𝑡𝜌(𝑡) (1 − 𝜌(𝑡)) + 𝜇√Δ𝑡𝜌(𝑡) (1 − 𝜌(𝑡))N (0, 1). (5.9)

Here Δ𝑡 is the time step and N (0, 1) is a normal random deviate. 𝜌(𝑡 + Δ𝑡) can take

unphysical, negative values if N (0, 1) is large and negative, especially near the unstable

fixed point 𝜌(𝑡) = 0 where the noise term dominates the dynamics [175]. Clearly, the above

discretization is not suitable to numerically integrate (5.4).

We now outline the algorithm to numerically integrate the sFKPP equation, that is

based on the operator-splitting algorithm proposed by Dornic et al. [175]. The idea is to

split the sFKPP equation into deterministic and stochastic parts

𝑑𝜌𝑑𝑡 = 𝜇√𝜌 (1 − 𝜌)𝜂(𝐱, 𝑡),

𝜕𝑡𝜌 = 𝐷∇2𝜌 + 𝛾𝜌 (1 − 𝜌) .(5.10)

To integrate the stochastic part we sample the random deviates not from the Gaussian

distribution, but directly from the Fokker-Planck equation associated with the stochastic

part of the sFKPP equation.

In other words, given 𝜌(𝑡) = 𝜌 at time 𝑡, we generate a random number 𝜌⋆ distributed

according to the conditional transition probability density function (PDF)

Prob {𝜌(𝑡 + Δ𝑡) = 𝜌⋆|𝜌(𝑡) = 𝜌} ,

where Δ𝑡 is the time step. We then use 𝜌⋆ to integrate the deterministic part using the

following discretized form

𝜌𝑖,𝑗(𝑡 + Δ𝑡) = 𝜌𝑖,𝑗 + Δ𝑡 (𝐷𝜌𝑖+1,𝑗 + 𝜌𝑖−1,𝑗 + 𝜌𝑖,𝑗+1 + 𝜌𝑖,𝑗−1 − 4𝜌𝑖,𝑗

Δ𝑥2 + 𝜌𝑖,𝑗 (1 − 𝜌𝑖,𝑗)) ,(5.11)

where 𝜌𝑖,𝑗 = 𝜌⋆(𝐱 = [𝑖Δ𝑥, 𝑗Δ𝑥]) and Δ𝑥 is the grid spacing (equal in both 𝑥 and 𝑦-

direction).

We now show how to obtain the probability distribution function for the stochastic part

of the sFKPP equation. Since fluctuations are crucial near the unstable 𝜌 = 0 state, we

approximate the multiplicative noise term √𝜌(1 − 𝜌) as Θ(1/2 − 𝜌) × √𝜌 + (𝜌 ↔ 1 − 𝜌) to


obtain the simplified stochastic equation [175],

𝑑𝜌𝑑𝑡 = 𝜇√𝜌𝜂(𝐱, 𝑡). (5.12)

The Fokker-Planck equation associated with (5.12) for 𝑃(𝜌, 𝑡) = Prob {𝜌(𝑡) = 𝜌|𝜌(0) = 𝜌∘}and its solution are [175, 179]

𝜕𝑡𝑃 (𝜌, 𝑡) = 𝜇2

2 𝜕2𝜌 [𝜌𝑃 (𝜌, 𝑡)] ,

𝑃 (𝜌, 𝑡) = 𝛿(𝜌)𝑒−𝜆𝜌∘ + 𝜆𝑒−𝜆(𝜌∘+𝜌)√𝜌∘𝜌 𝐼1 (2𝜆√𝜌∘𝜌) .

(5.13)

Here 𝜆 = 2𝜇2𝑡 , 𝛿(𝜌) is the Dirac delta function, and 𝐼1 is the modified Bessel function of

the first kind of order 1. Using the Taylor expansion of the modified Bessel function [180]

𝐼1(𝑥) = 12𝑥

∞∑𝑘=0

1𝑘!

(𝑥2/4)𝑘

(𝑘 + 1)! , (5.14)

we can write 𝑃(𝜌, 𝑡) as [175]

𝑃(𝜌, 𝑡) = 𝛿(𝜌)𝑒−𝜆𝜌∘ + 𝜆𝑒−𝜆(𝜌∘+𝜌)𝜆𝜌∘∞

∑𝑘=0

1𝑘!

(𝜆2𝜌∘𝜌)𝑘

(𝑘 + 1)!

= 𝛿(𝜌)𝑒−𝜆𝜌∘ +∞

∑𝑘=0

(𝜆𝜌∘)𝑘+1𝑒−𝜆𝜌∘

(𝑘 + 1)!𝜆𝑒−𝜆𝜌 (𝜆𝜌)𝑘

𝑘!

= 𝛿(𝜌)𝑒−𝜆𝜌∘ +∞

∑𝑛=1

(𝜆𝜌∘)𝑛𝑒−𝜆𝜌∘

𝑛!𝜆𝑒−𝜆𝜌 (𝜆𝜌)𝑛−1

(𝑛 − 1)! ,

=∞

∑𝑛=0

Prob (Gamma[𝑛] = 𝜆𝜌)Prob (Poisson[𝜆𝜌∘] = 𝑛)

(5.15)

where 𝑛 = 𝑘 + 1, and we have used the following definitions of Poisson and Gamma distri-

butions [175, 181]Prob (Poisson[𝑥] = 𝑛) ≡ 𝑥𝑛𝑒−𝑥

𝑛! ,

Prob (Poisson[𝑥] = 0) ≡ 𝑒−𝑥,

Prob (Gamma[𝑛] = 𝜈) ≡ 𝑒−𝜈𝜈𝑛−1

(𝑛 − 1)! ,

Prob (Gamma[0] = 𝜈) ≡ 𝛿(𝜈).

(5.16)

Using (5.15) the solution of (5.12) is the random number 𝜌⋆ generated as [175, 181]

𝜌⋆ = 𝑟Gamma[𝑟Poisson[𝜆𝜌∘]]/𝜆, (5.17)

where 𝑟Poisson[𝜆𝜌∘] is a random number generated from the Poisson distribution with mean

𝜆𝜌∘, and 𝑟Gamma[𝑟Poisson[𝜆𝜌∘]] is a random number generated from the Gamma distribution

Numerical methods | 85

with shape 𝑟Poisson[𝜆𝜌∘]. In Fig. 5.3 we show that the PDF of the random numbers generated

according to (5.17) matches well with the theory (5.13). In our simulations of the sNB and

sNBNL model, we use GNU Scientific Library [182] to sample the random numbers from

the Gamma and Poisson distributions.

0.0 0.5 1.0 1.5 2.0 2.5 3.0ρ

0.00

0.25

0.50

0.75

1.00

1.25

P(ρ,t

)

AnalyticalRandom Numbers

Figure 5.3: Comparison between the PDF of the random numbers generated according to (5.17), and the

functional form 𝑃(𝜌, 𝑡) given in (5.13). 𝜌∘ = 1, 𝜆 = 20.

The above method outlined for the sFKPP equation can be easily extended to numeri-

cally integrate both sNB and sNBNL models.


5.4 Results

We now present the results from our numerical simulations of sNB and sNBNL models.

5.4.1 Snapshots of growing colonies

In Fig. 5.4, we show colony morphologies for various values of bacteria diffusivity for sNB

model with initial nutrient concentration set to unity. As we decrease 𝐷𝐵, the front width

which is proportional to √𝐷𝐵/𝛾, decreases and undulations start to form at the colony

front. At low 𝐷𝐵 the colony grows with a rough front.

In Fig. 5.5, we show colony profiles at various nutrient concentration 𝐶0 for both sNB

and sNBNL models at low bacteria diffusivity 𝐷𝐵 = 5 × 10−4. In both the models, we

observe a rough growing front. While for the sNB model, the front stays compact even at

small nutrient concentration, the sNBNL model shows a transition from rough branched

fingers to a compact front as we increase 𝐶0.

In the next section, we systematically quantify the colony morphology for both the

models.

DB = 10−1 DB = 10−2 DB = 10−3 DB = 5× 10−4

0.0

0.2

0.4

0.6

0.8

1.0

Figure 5.4: Pseudo-color heatmap of the bacteria density fraction 𝜌𝐵/(𝜌𝐵 + 𝑐) for the sNB model at

different times but comparable colony size. We vary bacteria diffusivity keeping 𝐶0 = 1, 𝐿 = 10, 𝑁 = 1000fixed. As we reduce 𝐷𝐵, the front becomes sharp and shows a transition from a smooth to rough but

compact profile.

Results | 87

Figure 5.5: Density profiles at different times but comparable colony size for (a) sNB model and (b) sNBNL

model at various nutrient concentration. In the sNB model, we observe a rough but compact front at all

values of 𝐶0. Whereas in the sNBNL model, we observe a transition from branched finger-like front to a

compact rough front as we increase 𝐶0.

5.4.2 Front speed

An initial linear inoculation of bacteria 𝜌𝐵(𝐱, 0) spreads outward in 𝑦-direction by consum-

ing nutrients. The speed of this growing colony can be calculated as

𝑉 ≡ 𝑑𝑑𝑡⟨ 1

𝐿 ∫Ω

𝜌𝐵(𝐱, 𝑡)𝜌(𝐱, 𝑡) 𝑑Ω⟩. (5.18)

Here the integral is over the entire simulation domain. In Fig. 5.6, we plot front speed 𝑉versus initial nutrient concentration 𝐶0 for various values of bacteria diffusivity 𝐷𝐵 for the

sNB model. Since we are in the weak noise limit 𝜇/(𝐷𝐵𝛾)1/4 < 1, we expect 𝑉 ∼ 2√𝐷𝐵𝐶0

with a small logarithmic correction [167, 173]. Although the colony morphology changes on

changing 𝐶0 and 𝐷𝐵, we find that the front speed obtained from our numerical simulations

matches the mean field prediction 𝑉 ∼ 𝐶0 well.

The plot in Fig. 5.7 shows that for the sNBNL model, the front speed scales linearly

with the initial nutrient concentration for 𝐶0 ≥ 3. At large values of 𝐶0, up to the leading

order we can approximate the nonlinear diffusion term 𝐷𝐵∇⋅(𝑐∇𝜌𝐵) as 𝐷𝐵𝐶0∇2𝜌𝐵. Thus

by making an analogy with the sNB model we expect 𝑉 ∼ 𝐶0.


Figure 5.6: Plot of the front velocity 𝑉 (scaled with 2√𝐷𝐵) versus initial nutrient concentration 𝐶0

obtained from DNS of sNB model on a log-log scale. The black line show the expected mean-field 𝑉 ∼ √𝐶0

scaling. At small 𝐶0, where the population fluctuations are important, the front velocity is significantly

lower than the mean-field prediction. 𝐿 = 32, 𝑁 = 3200, 𝐷 = 10−1, 𝛾 = 1, 𝜇 = 5 × 10−2.

Figure 5.7: Plot of the front velocity 𝑉 (scaled with 2√𝐷𝐵) versus initial nutrient concentration 𝐶0

obtained from DNS of sNBNL model. Black line shows linear scaling 𝑉 ∼ 𝐶0. 𝐿 = 64, 𝑁 = 4096, 𝐷 =10−1, 𝛾 = 1, 𝜇 = 5 × 10−2.

Results | 89

5.4.3 Morphological behavior

Our numerical simulations show that population fluctuations give rise to diffusive insta-

bilities in the propagating front [173] resulting in various morphological patterns that are

absent in the mean-field equations. As we have shown in Figs. 5.4 and 5.5, the front tran-

sitions from a branched fingered one to a smooth one upon increasing the initial nutrient

concentration at low bacteria diffusivity. We quantify the front undulations by measuring

the roughness of the growing front [165, 183, 184]

𝜎ℎ(𝑡) = ⟨[ℎ(𝑥, 𝑡) − ℎ]2⟩1/2

. (5.19)

Here ℎ(𝑥, 𝑡) is the height of the front, the bar means spatial average in 𝑥 direction and

angular brackets denote ensemble average. In Figs. 5.8 and 5.9 we plot roughness versus

time at different nutrient concentration for sNB and sNBNL models respectively. We find

that roughness increases upon decreasing 𝐶0.

For the sNB model, similar to Ref. [174], we find that 𝜎ℎ(𝑡) ∼ 𝑡1/3. In addition, 𝜎ℎ(𝑡)shows a similar scaling with initial nutrient concentration 𝜎ℎ(𝑡) ∼ 𝐶1/3

0 [See Fig. 5.8(b)].

On the other hand in the sNBNL model, the dynamics of the front structure dramatically

alters on varying the nutrient concentration. Small values of 𝐶0 gives rise to more prominent

finger like patterns and we find that 𝜎ℎ(𝑡) ∼ 𝑡 for the sNBNL model. On increasing 𝐶0,

finger like growth transitions into a smooth and compact front [see Fig. 5.9].

0.1 0.3 0.5 0.7 0.9

ts = VL t

0.25

0.50

0.75

1.00

1.25

1.50

σh(ts)

×10−1

C0 = 1C0 = 2

C0 = 4C0 = 8

C0 = 10C0 = 18

0.05 0.10 0.20 0.40 0.80

ts = VL t

0.05

0.10

0.20

σh(ts)C

1/3

0

t1/3

C0 = 1C0 = 2C0 = 4

C0 = 8C0 = 10C0 = 18

Figure 5.8: (a) Roughness 𝜎ℎ(𝑡) versus time (scaled with 𝑉 /𝐿) at different initial nutrient concentration

𝐶0 for the sNB model. (b) Plot of 𝜎ℎ(𝑡)𝐶1/30 versus time (scaled), showing data collapse over 𝑡1/3 line,

which is in agreement with the exponent observed for the sFKPP equation in nutrient rich conditions [174].

𝐷𝐵 = 5 × 10−4, 𝐿 = 32, 𝑁 = 3200, 𝐷 = 10−1, 𝛾 = 1, 𝜇 = 5 × 10−2.


0.0 0.2 0.4 0.6 0.8 1.0

ts = VL t

0

1

2

3

σh(ts)

C0 = 3C0 = 4C0 = 5

C0 = 6C0 = 7C0 = 8

Figure 5.9: Roughness 𝜎ℎ(𝑡) versus time (scaled with 𝑉 /𝐿) at different initial nutrient concentration 𝐶0

for the sNBNL model. For small values of 𝐶0, 𝜎ℎ(𝑡) grows linearly with time (dashed black lines) and

saturates for higher values of 𝐶0. Sudden drops in 𝜎ℎ(𝑡) are due to merging of different branches. Data is

from a single realization. 𝐷𝐵 = 5 × 10−4, 𝐿 = 64, 𝑁 = 4096, 𝐷 = 10−1, 𝛾 = 1, 𝜇 = 5 × 10−2.

5.5 Comparison with the agent-based model

We now compare our results with that of an agent-based model used to study the role of

initial nutrient concentration and population fluctuations in colonies growing on hard agar

surfaces. Agent-based simulations and their analysis was carried by Dr. Pushpita Ghosh

[For details of these simulations and results, see Rana et al. [185]].

The agent-based model consists of two-dimensional rod-shaped bacteria of unit width

and variable length 𝑙 on a uniform frictionless surface. Each bacteria grows on the expense

of nutrients which diffuse through the space. After a growing bacteria reaches a certain

critical length 𝑙𝑚𝑎𝑥, it stops growing further and divide at a rate 𝑘𝑑𝑖𝑣 into two independent

daughter cells of equal lengths. The orientation of the daughter cells can be different than

that of the mother cell owing to various environmental factors, for example slight bending

of the cells, inhomogeneities on the surface, etc. To mimic the same, each daughter cell is

given a small random orientational kick right after the division. Note that this also prevents

the cells from growing in long filament-like structures.

In the model, individual bacteria interact with each other via repulsive elastic contact

forces. As we are interested in expanding colonies of nonmotile growing colonies, we consider

Comparison with the agent-based model | 91

Figure 5.10: Representative snapshots from the agent-based simulations of colonies growing at different

initial nutrient concentration 𝐶0 = 50, 100, 200, and 300 on a simulation box of size 𝐿𝑥 = 𝐿𝑦 = 200. As

𝐶0 is increased, the colonies show a transition from branched finger-like growth to a smooth and compact

growth.

the over-damped dynamics with the following equations of motion [165]

𝐫 = 1𝜁𝑙𝐅,

𝜔 = 12𝜁𝑙3 𝜏.

(5.20)

Here 𝐫, 𝜃, and 𝜔 ≡ 𝜃 are the center of mass position, the orientation, and the angular

velocity of a bacterium. 𝐅 is the net repulsive force on an individual and 𝜏 is the torque.

The force between two rods is approximated as the repulsive force between two disks with

their centers placed on the major axis of the rods at positions such that the distance between

them is the smallest [165]. If the shortest distance between the two disks is 𝑑, then the

overlap is ℎ = 1−𝑑 (provided 𝑑 < 1). The magnitude of the force is then 𝐹 = 𝐸ℎ3/2, where

𝐸 is the strength of the repulsive interaction between the rods. The direction of the force

is taken parallel to the shortest distance vector. We do not consider explicit translational

or rotational diffusivity for the bacteria and the position and orientation changes only

because of the steric repulsions between growing bacteria according to (5.20). Unlike in

Ref. [165], the nutrient resources are limited and initially kept fixed and uniform, to mimic

a Petri-dish like set up. We find that growth and morphological dynamics of growing

colony depends upon the interplay of local nutrient availability, nutrient diffusivity and

mechanical interactions. Colonies growing on a nutrient rich substrate show a rapid growth

and a smooth front morphology whereas, those growing on a nutrient deficient substrate

show slower growth and branched or finger-like structures.

In Fig. 5.11 we plot colony roughness obtained from our Agent-based simulations at

various nutrient concentration in a box of size 𝐿𝑥 × 𝐿𝑦. We find that similar to the sNBNL

model, the front roughness grows linearly with time [185].


3 6 9 12 15tkdiv

0.4

0.6

0.8

1.0

1.2

1.4

σf(tkdiv

)/Ly

×10−2

C0 = 050C0 = 100C0 = 200

C0 = 300C0 = 350C0 = 400

Figure 5.11: Colony roughness 𝜎𝑓 (scaled by 𝐿𝑦) with time (scaled by the division rate 𝑘𝑑𝑖𝑣) obtained

from agent-based model at different nutrient concentration. We find that similar to the sNBNL model, the

front roughness grows linearly with time, as shown by dashed black lines.

5.6 Conclusions

We have studied the role of nutrient concentration and population fluctuations on a growing

bacterial colony on a hard agar plate using continuum simulations and compared our results

with an agent-based model. We find a qualitative agreement between the two methodologies.

The main conclusions of our study are:

• Initial nutrient concentration has profound effect on colony growth leading to mor-

phological changes. A systematic change of initial nutrient concentrations from lower

to higher values causes transition of the colony periphery leading to the formation of

finger-like to branched-like structure to smoother front.

• Roughness of the colony front decreases with increase in initial nutrient concentration.

In particular, for small values of 𝐶0, both sNBNL model and agent-based models show

that at low nutrient concentration roughness grows linearly with time ( 𝜎ℎ ∝ 𝑡).

• Front speed of the colony increases as a function of initial nutrient concentration and

follows the mean field prediction for the sNB model 𝑉 ∼ √𝐶0 [173] and sNBNL

model 𝑉 ∼ 𝐶0. These predictions are in qualitative agreement with our agent-based

model.

• Our continuum simulations indicate that population fluctuations, intrinsic to birth-

Conclusions | 93

death processes can explain the transition from branched finger-like growth to smoothly

growing colonies on increasing initial nutrient concentration.

Although our present model only considers bacterial growth in a mono layer on surface,

there is a definite scope to extend our model in three-dimensions to study the growth

dynamics of bacteria forming biofilm-like structures. Bacteria growth and development

in three-dimensions might lead to complex morphologies as an outcome of interactions of

bacteria with surface and extracellular matrix [186]. Moreover, it would also be interesting

to investigate the spatio temporal dynamics of coexisting species using the ideas we present

here.


Chapter 6

Turbulence on DGX architecture: a GPGPU pseu-dospectral solver

In this chapter, we present a general purpose GPU based (GPGPU) pseudospectral solver

for the Navier-Stokes equation. First, we describe the pseudospectral algorithm and its

GPGPU implementation. We then discuss the performance of the pseudospectral algorithm

on the DGX architecture. We will show how the DGX architecture with its powerful GPU

cards and high bandwidth NVLINK communication switches is an ideal platform to perform

discrete numerical simulations of the Navier-Stokes equation at moderate resolutions of size

5123 − 20483.

6.1 Introduction

Finally, there is a physical problem that is common to many fields, that is very

old, and that has not been solved. It is not the problem of finding new fundamental

particles, but something left over from a long time ago –over a hundred years.

Nobody in physics has really been able to analyze it mathematically satisfactorily

in spite of its importance to the sister sciences. It is the analysis of circulating or

turbulent fluids.

– Richard Feynman

Turbulence is often touted as the last unsolved problem of classical physics. Several re-

searchers have tried to understand it both via experimental and analytical means. Perhaps,

the most significant breakthrough was achieved by Kolmogorov in his seminal work in the

early 1940s [187, 188]. Kolmogorov’s theory of turbulence assumes the fluid flow to be sta-

| 95

Energy Cascade

Dissipation Scale

Injection Scale

Figure 6.1: Pictorial representation of the energy cascade in three-dimensional turbulence. Energy is

injected at large length scales L. The nonlinear advection then transfers the energy from larger length scales

to the smaller ones while maintaining a constant energy flux. At the dissipation length scale 𝜂, viscosity

dissipates the energy. The separation between the injection scale and the dissipation scale increases with

the Reynolds number as L = Re3/4𝜂.

tistically homogeneous and isotropic at large Reynold’s numbers where inertia dominates

over viscous forces. The main features of Kolmogorov’s theory are:

• Energy is injected at large length scales L, and is dissipated by viscosity at scales

smaller than the Kolmogorov scale 𝜂 [52].

• At intermediate length scales, energy is transferred via nonlinear advection while

maintaining a constant energy flux.

• The separation between the injection scale and the dissipation scale increases with

the Reynolds number as L = Re3/4𝜂.

In Fig. 6.1 we show the pictorial representation of Kolmogorov’s theory of three-dimensional

turbulence. Several researchers have made further refinement within this framework, most

notable being the multi-fractal formalism [52], but a complete theoretical understanding of

the problem has eluded everyone to this date. It then becomes essential to aid the analyti-

cal and experimental efforts with direct numerical simulations (DNS) of the Navier-Stokes

equation. John von Neumann was the first person to emphasize the importance of perform-

ing numerical simulations for turbulence. In a report to the office of Naval Research [189],

he writes:

96 | Turbulence on DGX architecture: a GPGPU pseudospectral solver

...our intuitive relationship to the subject is still too loose—not having succeeded

at anything like deep mathematical penetration in any part of the subject, we are

still quite disoriented as to the relevant factors, and as to the proper analytical

machinery to be used.

Under these conditions there might be some hope to “break the deadlock” by extensive, but

well-planned, computational efforts. It must be admitted that the problems in question are

too vast to be solved by a direct computational attack, that is, by an outright calculation of a

representative family of special cases. There are, however, strong indications that one could

name certain strategic points in this complex, where relevant information must be obtainable

by direct calculations. If this is properly done, and the operation is then repeated on the

basis of broader information then becoming available, etc., there is a reasonable chance of

effecting real penetrations into this complex of problems and gradually developing a useful,

intuitive relationship to it. This should, in the end, make an attack with analytical methods,

that is truly more mathematical, possible.

– John von Neumann

He further remarks on the spatial resolution required and the capabilities of machines

of that era,

... even a linear resolution of 1:8 would require following 83 = 512 spatial

points and a linear resolution of 1:20 would already call for 203 = 8000 points.

The machines which one can hope to use in the immediate future will hardly be

able to exceed the first limit and none that is in sight for several years to come is

likely to get much beyond the second one. ...

– John von Neumann

As Neumann pointed out, it was hopeless to even perform a simulation with a resolution

of 83 grid points at that time, and it remained so until the computation capacity of the

machines improved dramatically. Simulations with large enough resolution required to

understand the dynamics of the turbulent flows became possible only after Patterson and

Orszag introduced the pseudospectral methods in the late 1960s and early 1970s [190–194].

Leveraging the newly introduced Cooley-Turkey algorithm for the Fourier transforms [195],

they performed the first pseudospectral simulations with a spatial resolution of 323 on a

cubic domain of side 2𝜋 and achieved a Taylor-microscale Reynolds number Re𝜆 ∼ 20 [190].

Since then, the computational capacity of the machines has increased by order of mag-

nitudes, and we have come far ahead of the 83 simulations Neumann proposed in the 1950s.

Introduction | 97

The advent of parallel computing using the MPI-framework has allowed the researchers to

conduct high-resolution DNSs of the Navier-Stokes equation, which have played a crucial

role in improving our understanding of turbulent flows. In this regard, the 40963 DNS of

fluid turbulence by Kaneda et al. [125] is a landmark. Further attempts have been made to

increase the grid-resolution and to the best of our knowledge, Iyer et al. [196] have achieved

the highest resolution of 163843 using the pseudospectral algorithm.

While the distributed memory clusters with thousands of CPU cores, communicating

via MPI-framework, dominated the high-performance computing (HPC) in the 1990s and

2000s, recent advances in HPC have come in terms of general-purpose GPU (GPGPU)

computing. Over the past decade, several algorithms like Molecular dynamics, Lattice-

Boltzmann methods have been successfully ported over to GPU based architecture with

impressive performance gains. Researchers have also tried to port FFT algorithms onto

distributed-GPU machines but with poor results. Unlike Lattice-Boltzmann, Molecular

Dynamics or Finite-difference based methods for solving partial differential equations, where

only the neighbor CPUs or GPUs communicate with each other, computing FFTs requires

all-to-all communication, which is the primary reason behind poor performance observed

for distributed-GPU FFT algorithms.

The issue is further aggravated by the fact that on traditional distributed-GPU ma-

chines, GPU-GPU communication takes place via CPUs. To communicate with a remote

GPU, first, the GPU transfers data to its host CPU via PCIe channels, which then commu-

nicates with the host CPU of the remote GPU via MPI interconnect, and finally, the remote

CPU transfers the data to the remote GPU. As a result of these significant communication

bottlenecks, roughly 80% of the total execution time for the FFTs on distributed GPU ma-

chines is spent in the communication phase alone, and the maximum achieved performance

is < 1% of the peak performance [197–199].

Recently introduced DGX architecture attempts to alleviate these issues. On a DGX

machine, the GPUs communicate directly via high bandwidth NVLINK switches, thus

eliminating the need to communicate with the CPUs. Further, each GPU card has a

significantly larger computational capacity and device memory than the earlier available

cards. In this chapter, we show how with enhancements on both the computational and

communication sides, DGX architecture presents as an ideal platform to perform moderate

resolution DNS of the Navier-Stokes equation.

The rest of the chapter is organized as follows. We begin by outlining the theory behind


the pseudospectral algorithm. Then we present a brief overview of the DGX architecture and

discuss the performance of FFT algorithms. We then describe the GPGPU implementation

of the pseudospectral algorithm and show how it performs on the DGX architecture. We

conclude the chapter by highlighting significant advantages and limitations of the DGX

architecture and a roadmap of our future work.

6.2 Pseudospectral algorithm for the Navier-Stokes equation

The dynamics of Newtonian flows is described by the Navier-Stokes equation [52, 200, 201]

𝜕𝑡𝐮 = −∇𝑃 + 𝐮 × 𝝎 + 𝜈∇2𝐮 + 𝐟, (6.1)

where 𝐮(𝐱, 𝑡) is the velocity field, 𝝎(𝐱, 𝑡) = ∇ × 𝐮 is the vorticity field, 𝜈 is the viscosity,

the pressure term 𝑃(𝐱, 𝑡) enforces incompressibility criterion ∇ ⋅ 𝐮 = 0, and 𝐟 is a suitable

driving term which injects energy in the system.

Our goal is to numerically integrate (6.1) on a periodic cubic domain of side 𝐿 discretized

over 𝑁3 collocation points. We use a pseudospectral algorithm [190, 192, 202] where a vector

field 𝐮(𝐱, 𝑡) is represented as a discrete Fourier series expansion 1

𝐮(𝐱, 𝑡) = 1𝑁 ∑

𝐤��𝐤𝑒𝑖𝐤⋅𝐱, (6.2)

where 𝑥𝛼 ∈ 𝐿𝑁 {0, 1, 2 … , 𝑁 − 1}, 𝑘𝛼 ∈ 2𝜋

𝐿 {−𝑁2 + 1, −𝑁

2 + 2, … , 𝑁2 }, and the Fourier am-

plitudes ��𝐤 are defined as ��𝐤 = F𝐤 {𝐮} ≡ ∑𝐱 𝐮(𝐱, 𝑡)𝑒−𝑖𝐤⋅𝐱. In Fourier space (6.1) is

written as𝜕𝑡��𝐤 = −𝑖𝐤𝑃𝐤 + 𝑊𝐤 − 𝜈𝑘2��𝐤 + 𝐟𝐤,

𝐤 ⋅ ��𝐤 = 0,(6.3)

where 𝑊𝐤 = F𝐤 {𝐮 × 𝝎}, 𝑃𝐤 = F𝐤 {𝑃}, 𝐟𝐤 = F𝐤 {𝐟}, and 𝑘2 = 𝐤 ⋅ 𝐤. Taking dot product

of (6.3) with 𝐤 and using the incompressibility constraint 𝐤 ⋅ ��𝐤 = 0 we get

𝑃𝐤 = 𝑖𝐤 ⋅ (𝑊𝐤 + 𝐟𝐤) /𝑘2. (6.4)

Substituting 𝑃𝐤 back in (6.3) we obtain for 𝐤 ≠ 0

𝜕𝑡��𝐤 = −𝜈𝑘2��𝐤 + P ⋅ (𝑊𝐤 + 𝐟𝐤) , (6.5)

where P = I − �� is the projection operator, and �� = 𝐤/𝑘. For 𝐤 = 0 (6.3) simplifies to

𝜕𝑡��0 = 𝑊0 + 𝐟0. (6.6)1Or a scalar field 𝜙(𝐱, 𝑡) as 𝜙(𝐱, 𝑡) = 1

𝑁 ∑𝐤 𝜙𝐤𝑒𝑖𝐤⋅𝐱.

Pseudospectral algorithm for the Navier-Stokes equation | 99

6.2.1 Integration scheme

We use a second order exponential scheme (ETD2) [112] which relies on exact integration

of the linear terms present in (6.5). Below we outline the derivation of ETD2 scheme. We

begin by rewriting (6.5) as

𝜕𝑡��𝐤 = 𝑐��𝐤 + 𝐹𝐤(𝑡), (6.7)

where 𝑐 = −𝜈𝑘2 and 𝐹𝐤(𝑡) = P ⋅ (𝑊𝐤 + 𝐟𝐤). Multiplying (6.7) with the integrating factor

𝑒−𝑐𝑡 and integrating from 𝑡 to 𝑡 + Δ𝑡 yields

��𝐤(𝑡 + Δ𝑡) = ��𝐤(𝑡)𝑒𝑐Δ𝑡 + 𝑒𝑐Δ𝑡 ∫Δ𝑡

0𝑒−𝑐𝑡′𝐹𝐤(𝑡 + 𝑡′)𝑑𝑡′. (6.8)

For the second order scheme (ETD2), we use the following approximation for 𝐹𝐤(𝑡 + 𝑡′),

𝐹𝐤 (𝑡 + 𝑡′) = 𝐹𝐤(𝑡) + 𝑡′

Δ𝑡 (𝐹𝐤(𝑡) − 𝐹𝐤(𝑡 − Δ𝑡)) + O (Δ𝑡2) ∀ 𝑡′ ∈ [0, Δ𝑡]. (6.9)

Solving the integral in (6.8) with the above approximation gives us the ETD2 scheme for

the Navier-Stokes equation,

��𝐤(𝑡 + Δ𝑡) = ��𝐤(𝑡)𝑒𝑐Δ𝑡 + (1 + 𝑐Δ𝑡)𝑒𝑐Δ𝑡 − 1 − 2𝑐Δ𝑡Δ𝑡𝑐2 𝐹𝐤(𝑡) + 1 + 𝑐Δ𝑡 − 𝑒𝑐Δ𝑡

Δ𝑡𝑐2 𝐹𝐤(𝑡 − Δ𝑡),(6.10)

with a local truncation error 5Δ𝑡3 𝐹𝐤(𝑡)/12. Note that (6.10) requires the knowledge of

both 𝐹𝐤(𝑡) and 𝐹𝐤(𝑡 − Δ𝑡) at all times. Since 𝐹𝐤(𝑡 − Δ𝑡) is not known at 𝑡 = 0, we use a

first order scheme (ETD1)

��𝐤(𝑡 + Δ𝑡) = ��𝐤(𝑡)𝑒𝑐Δ𝑡 + 𝑒𝑐Δ𝑡 − 1𝑐 𝐹𝐤(𝑡), (6.11)

to perform the first integration step that is derived from the approximation that 𝐹𝐤(𝑡) stays

constant in the interval 𝑡 and 𝑡 + Δ𝑡,

𝐹𝐤 (𝑡 + 𝑡′) = 𝐹𝐤(𝑡) + O (Δ𝑡) ∀ 𝑡′ ∈ [0, Δ𝑡]. (6.12)

6.2.2 Computing the nonlinear term

Using the Fourier amplitudes of the velocity 𝐮(𝐱) = ∑𝐩 ��𝐩𝑒𝑖𝐩⋅𝐱 and the vorticity field

𝝎(𝐱) = ∑𝐪 𝝎𝐪𝑒𝑖𝐪⋅𝐱, we can write the nonlinear term 𝐮 × 𝝎 as

(𝐮 × 𝝎)𝛼 = 𝜖𝛼𝛽𝛾𝑢𝛽𝜔𝛾

= 𝜖𝛼𝛽𝛾 ∑𝐩,𝐪

��𝐩,𝛽��𝐪,𝛾𝑒𝑖(𝐩+𝐪)⋅𝐱. (6.13)


Fourier transform of the above equation gives

𝑊𝐤,𝛼 = F {(𝐮 × 𝝎)𝛼} = 𝜖𝛼𝛽𝛾 ∑𝐩+𝐪=𝐤

��𝐩,𝛽��𝐪,𝛾, (6.14)

where 𝛿 (𝐩 + 𝐪 − 𝐤) is the Kronecker-Delta function.

From (6.14) it is evident that the nonlinear advection term is a convolution in the

Fourier space and computing it will require O (𝑁6) operations. In contrast, computing the

nonlinear term in the real space will only need O (𝑁3) operations. Thus the nonlinear term

can be computed in the real space, and then transformed to the Fourier space to perform

numerical integration provided efficient methods for the Fourier transforms exist. Orszag

and Patterson [193] realized that expensive convolutions could be avoided by leveraging

Fast Fourier Transform (FFT) algorithms which require O (𝑁3 log 𝑁3) operations. In the so

called Pseudospectral method, at each time step the velocity and vorticity fields are brought

back to the real space and the nonlinear term is then computed via simple multiplication.

It is then transformed to the Fourier space to perform the time integration. While FFTs

reduce the computational cost of the pseudospectral algorithm, computing the nonlinear

term in the Fourier space leads to aliasing errors in the pseudospectral algorithm, which we

describe in the next section.

6.2.3 Aliasing errors

In pseudospectral methods, aliasing errors arise due to the nonlinear advection term. To

see that consider the following expression for the nonlinear term [see (6.14)]

𝑊𝐤,𝛼 = 𝜖𝛼𝛽𝛾 ∑𝐩,𝐪

��𝐩,𝛽��𝐪,𝛾 [ 1𝑁 ∑

𝐱𝑒𝑖(𝐩+𝐪−𝐤)⋅𝐱] . (6.15)

Since the exponential is periodic on [0, 2𝜋], we can add or subtract 𝑁𝐛, where the compo-

nents of 𝐛 take integer values, from 𝐩 + 𝐪 − 𝐤 and the expression will remain unchanged.

𝑊𝐤,𝛼 = 𝜖𝛼𝛽𝛾 ∑𝐩,𝐪

��𝐩,𝛽��𝐪,𝛾 [ 1𝑁 ∑

𝐱𝑒𝑖(𝐩+𝐪−𝐤±𝐛𝑁)⋅𝐱]

= 𝜖𝛼𝛽𝛾 ∑𝐩,𝐪

��𝐩,𝛽��𝐪,𝛾𝛿 (𝐩 + 𝐪 − 𝐤 ± 𝐛𝑁)

= 𝜖𝛼𝛽𝛾 ∑𝐩+𝐪=𝐤±𝐛𝑁

��𝐩,𝛽��𝐪,𝛾.

(6.16)

Thus, the mode 𝐩 + 𝐪 can alias as the mode 𝐤 ± 𝐛𝑁 . The only values components of

𝐛 can take is either zero or one, as for all other possibilities the aliased wavenumber will

lie outside of the available wavenumber range. Then we can write 𝐛 = ∑𝛼 𝐈𝛼 𝐞𝛼, where

Pseudospectral algorithm for the Navier-Stokes equation | 101

𝛼 ∈ {1, 2, 3}, 𝐼𝛼 are indicator functions which take values zero or one, and 𝐞𝛼 are the unit

vectors in three dimensions. Out of all the terms, 𝐛 = 0 gives us the actual expression of

the nonlinear term and the rest of the seven possible values of 𝐛 leads to aliasing terms as

listed below.

𝑊𝐤,𝛼 = 𝜖𝛼𝛽𝛾 ∑𝐩+𝐪=𝐤

��𝐩,𝛽��𝐪,𝛾

+ 𝜖𝛼𝛽𝛾 ⎡⎢⎣

∑𝐩+𝐪=𝐤±𝑁��1

+ ∑𝐩+𝐪=𝐤±𝑁��2

+ ∑𝐩+𝐪=𝐤±𝑁��3

⎤⎥⎦


+ 𝜖𝛼𝛽𝛾 ⎡⎢⎣

∑𝐩+𝐪=𝐤±𝑁��1±𝑁��2

+ ∑𝐩+𝐪=𝐤±𝑁��1±𝑁��3

+ ∑𝐩+𝐪=𝐤±𝑁��2±𝑁��3

⎤⎥⎦


+ 𝜖𝛼𝛽𝛾 ∑𝐩+𝐪=𝐤±𝑁��1±𝑁��2±𝑁��3

��𝐩,𝛽��𝐪,𝛾.

(6.17)

6.2.4 Dealiasing

We use the “Two-third” truncation rule [192] to eliminate the aliasing errors, where all the

wavenumbers with magnitude larger than 𝑘2dealias = (2

3 ⋅ 𝑁2 )2 = 𝑁2

9 are removed. Other

Dealiasing algorithms are also available which retain slightly higher number of modes than

the Two-third rule, for a discussion see [202, 203].


6.3 Overview of the DGX architecture

In this section we present a brief overview of the DGX architecture and highlight the

advantage it has over traditional distributed-GPU architecture.

A DGX machine consists of multiple GPUs hosted on a single node which communicate

with each other through high bandwidth NVLINK channels. The total bandwidth of the

NVLINK channels depends on the connection type and is order of magnitude higher than

the PCIe based communication channels. We test our implementation of the pseudospectral

solver on three different DGX machines, whose specifications are listed in Table 6.1.

• DGX2 : DGX2 consists of 16 Tesla-V100 GPUs. All GPUs are fully connected with

six NVLINK connections per GPU, which sets the total bidirectional bandwidth of

the NVLINK connection to 300 GBps. Each V100 GPU has 32 GiB of device memory

and a peak double-precision performance of 7.8 TFLOPs.

• DGX-A100 : DGX-A100 consists of eight Tesla-V100 GPUs with 12 NVLINK con-

nections per GPU which gives a total bidirectional bandwidth of 600 GBps. Each

A100 GPU has 40 GiB of device memory and a peak double-precision performance of

9.75 TFLOPs.

• DGX-Station : DGX-Station is the smallest machine with DGX-architecture with

four Tesla-V100 GPUs. There are 4 NVLINKs per GPU which sets the total band-

width at 200 GBps. The connection topology for the DGX-Station is non-uniform.

GPU pairs GPU0-GPU3 and GPU1-GPU2 have 2 NVLINK connections each, whereas other

GPU pairs GPU0-GPU1, GPU0-GPU2, GPU1-GPU3, GPU2-GPU3 have only single NVLINK

connection between them as shown in Fig. 6.2. Better connected GPUs show higher

performance on both FFT and DNS benchmarks as compared to the other pairs.

Name GPU RAM/GPU TFLOPs/GPU # GPUs NVLINKs/GPU Bandwidth

DGX-Station V100 32 GiB 7.8 8 4 200∗ GBps

DGX2 V100 32 GiB 7.8 16 6 300 GBps

DGX-A100 A100 40 GiB 9.75 8 12 600 GBps

Table 6.1: Specifications of different machines with DGX architecture. ∗Non-uniform topology.

As compared to traditional distributed-GPU machines where the GPU-GPU commu-

nication takes place via CPUs, on a DGX machine the GPUs communicate directly via

Overview of the DGX architecture | 103

GPU0 GPU1

GPU3 GPU2

Figure 6.2: NVLINK topology for DGX-Station. Four Tesla-V100 are connected through high bandwidth

NVLINK connections. The connection topology is non-uniform. The GPU pairs GPU0-GPU3 and GPU1-GPU2

have two NVLINK connections, which results into twice the bandwidth as compared to GPU pairs GPU0-

GPU1, GPU0-GPU2, GPU1-GPU3 and GPU2-GPU3.

NVLINK connections. Further, NVLINK supports Peer-to-Peer access which allows access-

ing remote GPU device memory from within the compute kernels. These remote accesses

are hidden with compute work, which can also help in achieving performance gains.


6.4 Earlier attempts on porting FFT algorithms to multi-GPU archi-

tecture

Earlier works have implemented distributed FFT on CUDA enabled GPUs [197–199] but

with poor performance. Nukada et al. [197] show scaling results for up to 768 M2050 Fermi

GPUs for a 20483 size transform using a self-written CUDA FFT implementation. They

achieved a maximum performance which was around 1.2% of the peak performance.

Czechowski et al. [198] designed diGPUFFT which is based on P3DFFT. Although the

library was designed for the experimental validation of theoretical complexity analysis of

three-dimensional FFTs and what implications it has on the design of high performance

exa-scale architectures, once again the authors report a maximum performance which is

less than 1% of the peak performance.

Gholami et al. [199] used both cuFFT and FFTW to compute Fourier transforms on

machines with multiple GPUs distributed across various nodes. The authors were able

to reduce the PCIe communication overhead with novel transpose techniques, but even

with reduced communications, their scaling results show that to offset the communication

overhead incurred while moving from a single GPU to many, one needs a large number of

GPUs to achieve performance similar to a single GPU. Specifically they show that for a

transform of size 256 × 515 × 1024, minimum eight K40 GPUs are required to achieve a

performance comparable to a single K40 GPU.

The common thread in all these earlier studies is the severe communication bottlenecks.

Since the GPUs on different nodes communicate to each other through CPUs, communica-

tion time makes up for more than 70 − 80% of the total execution time, which results in

poor performance.

In the following sections we will show how DGX architecture overcomes these challenges

and is able to deliver a maximum performance for FFT and DNS simulations which is

15 − 20% of the peak performance.

Earlier attempts on porting FFT algorithms to multi-GPU architecture | 105

6.5 Pseudospectral algorithm on the DGX architecture

We now describe the implementation of the pseudospectral solver for the Navier-Stokes

equation for DGX architecture. The solver is written in cudafortran dialect of the Fortran

language. It has two core components:

• The FFT Core, where we define easy to use wrapper functions to perform multi-GPU

FFTs of three dimensional data using the cuFFT library.

• The Pseudospectral Core, which leverages the FFT core to numerically integrate the

Navier-Stokes equation.

Apart from these two core components, our solver contains multi-gpu finite difference

routines which we have used to perform three-dimensional simulations discussed in Chap-

ter 4 . It also contains various analysis routines that can compute common physical quan-

tities like total energy, energy spectrum, energy budget, and handles input and output of

data.

6.6 The FFT Core

We use the cuFFT library to compute Fourier transforms. Specifically, we use the multi-

GPU, in place “Real-to-Complex” (forward) and “Complex-to-Real” (backward) transform

methods, which overwrite the input array with the transformed data. Using in place trans-

form methods helps us to reduce the memory requirements, which can become large very

quickly in three dimensions. See Section 6.9 for the memory requirements for pseudospectral

methods.

6.6.1 Data Layout for the in place transforms

Data layout for the in place cuFFT transforms is determined as follows. We discretize the

velocity field on a cubic box of side length 𝐿 with 𝑁3 collocation points. Wavevectors

available on this domain have components 𝑘𝛼 = 2𝜋𝐿 {−𝑁

2 + 1, −𝑁2 + 2, … , 𝑁

2 − 1, 𝑁2 }. As

the velocity is a real physical variable, its Fourier amplitudes satisfy the Hermitian property

��−𝐤 = ��∗𝐤,

where ��−𝐤 is the complex conjugate of ��𝐤. Thus we only need to store amplitudes for half

the total possible modes, which amounts to storing (𝑁/2+1)𝑁2 complex numbers. Since a


tuple of two real numbers is used to represent a complex number, we need 2(𝑁/2+1)𝑁2 =(𝑁 + 2)𝑁2 real numbers to store the Fourier transform of 3D data of size 𝑁3.

We then allocate 3 arrays of size (𝑁 + 2)𝑁2 each of which can store a component of

the velocity field and its Fourier amplitude. In real space, the first 𝑁3 elements of each

array store a component of the velocity field and the rest of the 2𝑁2 elements are unused.

In Fourier space, the whole array stores the complex Fourier amplitudes for (𝑁/2 + 1)𝑁2

modes.

The data layout is better understood for a one-dimensional in place “Real-to-Complex”

transform of size 𝑁 , where an array of size 𝑁 + 2 is allocated which stores the real and

complex data in the following way.

1 2 3 4

u(1) u(2) u(3) u(4) u(N-1) u(N) 0 0

N-1 N N+1 N+2Array indices ⇒

Real field ⇒1 2 3 4 N-1 N N+1 N+2

u(k=0) u(k=1) u(k=N/2-1) u(k=N/2)

Array indices ⇒

Complex field ⇒

Accessing elements in Fourier space efficiently

As discussed in the last section, we use same arrays to store both the velocity field and

its Fourier amplitudes. The real valued velocity field at the grid point i,j,k is accessed

as u(i,j,k), whereas the real and the imaginary parts of the complex Fourier amplitudes

for the wavevector 𝑘𝑥, 𝑘𝑦, 𝑘𝑧 can be accessed as u_real = u(2*i-1,j,k), and u_imag =

u(2*i,j,k).

Instead of accessing the complex Fourier amplitudes in the manner described above, we

typecast the array into Fortran native complex datatype which allows for a cleaner and

efficient access. Typecasting also allows us to express the Fourier space components of our

algorithm in a closer form to the mathematical notation and reduces the possibility of errors

in the source code.

6.6.2 FFT benchmarks on DGX architecture

We now show benchmark results for in place forward and backward FFTs of double-precision

data. We measure the time taken for a pair of forward and backward transform for a problem

size 𝑁3 at different number of GPUs used. In Fig. 6.3 we plot execution time for a pair

of transforms (forward and backward) for two different problem sizes 𝑁 = 1024 and 2048,

The FFT Core | 107

and compare the performance of cuFFT on DGX architecture with MPI-enabled P3DFFT

library on a BlueGeneP machine.

At a problem size 𝑁 = 1024, a single GPU on DGX architecture outperforms 212 cores

of the BlueGeneP machine. Further, we observe a good scaling when we use two or more

GPUs. At a problem size 𝑁 = 2048, it takes 216 cores of the BlueGeneP to break even

with 16 GPUs on DGX2. We do not observe any significant performance gain when we use

two GPUs instead of one for the following reason. On a multi-GPU architecture the total

execution time 𝑇 is a sum of the computation time 𝑇𝑐𝑜𝑚𝑝 and the communication time

𝑇𝑐𝑜𝑚𝑚. For benchmarks on a single GPU, no communication takes place and the execution

time is solely determined by 𝑇𝑐𝑜𝑚𝑝. On two GPUs and onwards, while the computation

time per GPU decreases, communication between the GPUs also contribute to the total

execution time. In our benchmarks, this additional communication time compensates for

any gains achieved on the computational side on two GPUs and thus the total performance

gains are small.

1 2 4 8 16p

10−2

10−1

100

101

Tim

e(s)

N=1024 (a)

DGX2DGX-A100DGX-StationBlueGeneP

4 8 16p

10−1

100

101

Tim

e(s)

N=2048 (b)

DGX2DGX-A100BlueGeneP

27 29 211 213 215

#cores

210 212 214 216

#cores

Figure 6.3: Execution time per iteration in seconds for a pair of FFT (inverse and forward) for (a) 𝑁 = 1024and (b) 𝑁 = 2048. On the lower 𝑥-axis, we vary the number of GPUs on different DGX architecture, and

on the upper 𝑥-axis we show the number of cores used on the BlueGeneP machine. At 𝑁 = 10243, 16

GPUs on DGX2 outperform 215 cores on the BlueGeneP machine, and at 𝑁 = 20483 BlueGeneP machine

breaks even with 16 GPUs of DGX2 at 216 cores.

6.6.3 Strong scaling for FFT on DGX architecture

We now show the strong scaling results for our FFT benchmarks. We define the speedup

𝑆(𝑝) as execution time on 𝑝 GPUs divided by the execution time on a single GPU. If 𝑇 (𝑝)


1 2 4 8 16p

100

Sp

eed

up

p0.62

N=512 (a)

DGX2DGX-A100DGX-Station

1 2 4 8 16p

100

Sp

eed

up

p0.81

N=1024 (b)


Figure 6.4: Strong scaling 𝑆(𝑝) ∝ 𝑝𝛿 observed with cuFFT for (a) 𝑁 = 512 and (b) 𝑁 = 1024. Larger

problem size shows better scaling.

denotes time taken on 𝑝 GPUs, then

𝑆(𝑝) = 𝑇 (𝑝)𝑇 (1) . (6.18)

In Fig. 6.4 we show the strong scaling results for two different problem sizes 𝑁 = 512and 𝑁 = 1024. We observe that 𝑆(𝑝) ∝ 𝑝𝛾, at both the problem sizes for 𝑝 ≥ 2. At

𝑁 = 1024 we observe a better scaling with an exponent 𝛾 = 0.81 as compared to 𝑁 = 512where 𝛾 = 0.62. This is due to the fact that a 𝑁 = 1024 transform is approximately eight

times more computationally expensive than a 𝑁 = 512 transforms. At 𝑁 = 512, 𝑇𝑐𝑜𝑚𝑚

dominates the execution time, whereas at 𝑁 = 1024, more compute work is available

to efficiently hide the remote communication time, which results in better performance.

Similarly, for the DGX-A100 the time taken at 8 GPUs is larger than the time taken at 4

GPUs, which tells us that there’s not much compute work going on and 𝑇𝑐𝑜𝑚𝑚 dominates

the total execution time.

6.6.4 Performance

In Fig. 6.5 we show the maximum performance achieved per GPU (𝑃𝑚𝑎𝑥) for our FFT

benchmarks on different machines. We calculate 𝑃𝑚𝑎𝑥 as

𝑃𝑚𝑎𝑥 = 𝑁𝑜𝑝𝑠𝑇 (𝑝)𝑝

1𝑃𝑝𝑒𝑎𝑘

× 100,

where 𝑁𝑜𝑝𝑠 is the number of operations performed for a problem size 𝑁 , 𝑇 (𝑝) is the total

execution time, 𝑝 is the number of GPUs used, and 𝑃𝑝𝑒𝑎𝑘 is the peak performance of a

single GPU in TFLOPS [see Table 6.1 for more values]. On a single GPU, we can achieve

10 − 15% of the peak performance, whereas, for two or more GPUs, we observed 5 − 10%

The FFT Core | 109

of the peak performance. DGX-A100 shows roughly two times better performance than

DGX2 and DGX-Station, mainly because of its higher bidirectional bandwidth.

Note that to calculate 𝑃𝑚𝑎𝑥, we have used the theoretical value 𝑁𝑜𝑝𝑠 = 5𝑁3 log2 𝑁3.

However, as shown in Fig. 6.5(b), cuFFT performs a slightly larger number of computations

(roughly 10 − 40%) than the theoretical value, with the actual number depending on the

transform size. Thus, the true values of the maximum performance achieved are slightly

higher than what is reported in Fig. 6.5. For example, for a transform of size 5123 on a

single GPU on DGX-A100, the calculated 𝑃𝑚𝑎𝑥 is ∼ 15%, whereas the actual value is closer

to ∼ 20%.

20 21 22 23 24

p

5

10

15

Per

form

ance

(%)

DGX-StationDGX2DGX-A100

26 27 28 29

N

107

108

109

1010

1011

Op

erat

ion

s

26 27 28 29

N

0

15

30

45

Diff

eren

ce(%

)

2.5N3 logNCounted

Figure 6.5: (a) Maximum performance achieved 𝑃𝑚𝑎𝑥 per GPU (as percentage of the peak performance)

for FFT on different machines. Various symbols denote different problem sizes. (b) Comparison of actual

double-precision operations performed with the theoretical value 2.5𝑁3 log2 𝑁3 for a single forward trans-

form at different problem size 𝑁. cuFFT performs a slightly larger number of operations at all problem

sizes. Inset shows the percentage difference between the theoretical value and the actual number of opera-

tions. On average, cuFFT performs 20% more operations.


6.7 The Pseudospectral Core

We now describe the different steps of our GPGPU implementation of the pseudospectral

algorithm.

1. Initialization : The simulation is setup here. Required array are allocated, initial

conditions are set and ETD integration factors are computed.

2. Velocity in the Fourier space : Beginning of the time loop. If the velocity field was

initialized in real space, it is transformed to Fourier space before proceeding further.

3. Spectral analysis (Optional) : As the velocity is in Fourier space, spectral analysis,

for example, calculation of the energy spectrum is carried out at this step.

4. To real space : The velocity and vorticity arrays are transformed back to the real

space.

5. Real analysis, Snapshots (Optional): The velocity and vorticity fields are in real space

at this time step, and any real space analysis is performed here. Additionally, snap-

shots of the velocity field are also stored.

6. Compute the nonlinear term: Nonlinear term is computed and stored in the array

containing the vorticity field.

7. Add real space forces (Optional): Any additional forcing terms, which are better rep-

resented in the real space are added to the array containing the nonlinear term at this

step.

8. To Fourier space: The array containing the nonlinear term and additional real space

forces is transformed to the Fourier space.

9. Add Fourier space forces (Optional): Any additional forcing terms, which are better

represented in the Fourier space are added to the array containing the nonlinear term

at this step.

10. Integration: Time integration is performed at this step.

First, we apply the projection operator on the array containing the nonlinear term

and additional forces to get 𝐹𝐤(𝑡), and then the velocity is updated appropriately.

Additionally, 𝐹𝐤(𝑡) is stored away for use in the next integration time step. At this

The Pseudospectral Core | 111

stage, the velocity field is still in real space and needs to be transformed to the Fourier

space before we can perform the integration. A careful analysis of the ETD2 scheme

tells us that we can avoid these additional transforms by simply storing ��𝐤 along with

the 𝐹𝐤(𝑡) at the earlier time step with relevant factors as shown below.

The ETD2 scheme (6.12) at time step 𝑡 is written as,

��𝐤(𝑡 + Δ𝑡) = 𝑋1��𝐤(𝑡) + 𝑋2𝐹𝐤(𝑡) + 𝑋3𝐹𝐤(𝑡 − Δ𝑡), (6.19)

where 𝑋1, 𝑋2, 𝑋3 are the ETD2 factors. After updating the velocity field at time

step 𝑡, instead of storing 𝐹𝐤(𝑡) we store 𝑋1��𝐤(𝑡 + Δ𝑡) + 𝑋3𝐹𝐤(𝑡). Then, at time step

𝑡 + Δ𝑡, the ETD2 scheme reads as

��𝐤(𝑡 + 2Δ𝑡) = 𝑋1��𝐤(𝑡 + Δ𝑡) + 𝑋3𝐹𝐤(𝑡) + 𝑋2𝐹𝐤(𝑡 + Δ𝑡), (6.20)

since we have already have 𝑋1��𝐤(𝑡 + Δ𝑡) + 𝑋3𝐹𝐤(𝑡) stored away with us from the

last time step, there’s no need to transform the velocity field to Fourier space.

Fig. 6.6 shows a flow chart of the pseudospectral algorithm. Light blue rectangular

blocks make up the core time integration loop of the algorithm and the green elliptical

blocks are optional.


Initialize

simulation

Velocity in

Fourier space

Compute vorticity

𝝎𝐤 = 𝐤 × ��𝐤To real space

Compute the nonlinear

term 𝐮 × 𝝎To Fourier space

𝑊𝐤 = F𝐤 {𝐮 × 𝝎}Integration

𝑡 = 𝑡 + Δ𝑡

Spectral

Analysis

Real analysis,

Snapshots

Add real

space forces

Add Fourier

space forces

Figure 6.6: Flow chart for the pseudospectral algorithm. For a description of different blocks see the main

text.

The Pseudospectral Core | 113

6.8 Validating the solver

To validate the algorithm, we numerically integrate Shapiro flow [204], an exact solution of

the Navier-Stokes equation. The Shapiro flow at any time 𝑡 is given as

𝐮(𝐱, 𝑡) = 𝐴𝑘2 + 𝑙2

⎡⎢⎢⎢⎣

−𝜆𝑙 cos (𝑘𝑥) sin (𝑙𝑦) sin (𝑚𝑧) − 𝑚𝑘 sin (𝑘𝑥) cos (𝑙𝑦) cos (𝑚𝑧)+𝜆𝑘 sin (𝑘𝑥) cos (𝑙𝑦) sin (𝑚𝑧) − 𝑚𝑙 cos (𝑘𝑥) sin (𝑙𝑦) cos (𝑚𝑧)

(𝑘2 + 𝑙2) cos (𝑘𝑥) cos (𝑙𝑦) sin (𝑚𝑧)

⎤⎥⎥⎥⎦

exp (−𝜈𝜆2𝑡),

(6.21)

where 𝐴 is the amplitude, 𝑘, 𝑙, and 𝑚 are integers, and 𝜆2 = 𝑘2 + 𝑙2 + 𝑚2. Shapiro flow

satisfies the incompressibility criterion ∇ ⋅ 𝐮 = 0 at all times. Further, the vorticity is

parallel to the velocity field, i.e. 𝝎 = 𝜆𝐮, which sets the nonlinear advection term 𝐮 × 𝝎 to

zero. Since there is no coupling between various modes, the solution is stationary in space

as shown in Fig. 6.8. We plot the contours of the magnitude of the vorticity field 𝝎 for a

test case (𝐴 = 2, 𝑘 = 𝑙 = 𝑚 = 1, 𝜈 = 2 × 10−2, 𝐿 = 2𝜋, 𝑁 = 128) at 𝑡 = 0 and 𝑡 = 10.

While the magnitude of the vorticity decay with time, the spatial structures do not evolve.

From (6.21) it is clearly seen that the total kinetic energy for the Shapiro flow 𝐸(𝑡) =12 ⟨𝐮(𝐱, 𝑡)2⟩ decays exponentially with time because of the viscous dissipation as 𝐸(𝑡) =𝐸(0) exp−2𝜈𝜆2𝑡. The solver can then be easily validated by checking the decay of the total

kinetic energy for the Shapiro flow as shown in Fig. 6.7.

0 10 20 30 40 50t

10−3

10−2

10−1

En

ergy

E(0)e−2νλ2t

E(t)

Figure 6.7: Decay of the total energy 𝐸(𝑡) with time as computed from the simulations. Dashed black line

shows the theoretical estimate. 𝐴 = 2, 𝑘 = 𝑙 = 𝑚 = 1, 𝜈 = 2 × 10−2, 𝐿 = 2𝜋, 𝑁 = 128.


Figure 6.8: Contours of the magnitude of the vorticity |𝝎| showing spatial structures of the Shapiro flow

at time 𝑡 = 0 and 𝑡 = 10. The magnitude of the vorticity decays with time [See corresponding colormaps],

but the structures are stationary in space.

Validating the solver | 115

6.9 Memory requirements of the Navier-Stokes solver

To solve the Navier-Stokes equation using the pseudospectral method with a second order

ETD2 scheme, we need the following nine arrays of system size (𝑁 + 2)𝑁2.

• Three arrays to store the components of velocity field 𝐮 and their Fourier amplitudes.

• Three arrays to store the components of the nonlinear term 𝐮 × 𝝎 and their Fourier

amplitudes.

• Three arrays to store the components of 𝐹𝐤(𝑡 − Δ𝑡).

Additional memory of O (𝑁2) is required for housekeeping tasks like storing ETD scheme

factors, etc. Finally, cuFFT requires work arrays to compute Fourier transforms. The size

of the work array depends upon the transform size and the number of GPUs used and

is usually one to three times larger than the transform data itself. By default, cuFFT

allocates separate work arrays for different plans but as our forward and backward Fourier

transforms operate on the same arrays, we allocate a shared work array for both the plans

to reduce the memory requirements. For double-precision simulations, each array element

is eight bytes in size, which sets the total memory required for the pseudospectral solver to

80 − 100𝑁3 + O (𝑁2) bytes.

6.10 Performance of the Navier-Stokes solver

We now show the performance of the Navier-Stokes solver. In Fig. 6.9 we show the how the

execution time (in seconds) varies with number of GPUs used for two problem sizes 𝑁 = 512and 𝑁 = 1024. In Fig. 6.10 we plot the corresponding speedups as defined in (6.18). Our

benchmarks for the NSEXACT test case exhibit features similar to our cuFFT benchmarks.

Notably, very small performance gains are observed on two GPUs, the larger problem size

𝑁 = 1024 shows better scaling with number of GPUs, and DGX-A100 outperforms other

DGX machines.


1 2 4 8 16p

10−1

Tim

e(s)

N=512 (a)


4 8 16p

10−1

100

Tim

e(s)

N=1024 (b)


Figure 6.9: Execution time per iteration in seconds for NSEXACT at (a) 𝑁 = 512 and (b) 𝑁 = 1024.

We observe good speedup as number of GPUs (𝑝) are varied. Note that due to memory requirements, we

cannot perform simulations for 𝑁 = 1024 on one or two GPUs.

1 2 4 8 16p

100

Sp

eed

up

p0.57

N=512 (a)


4 8 16p

100

Sp

eed

up

p0.86

N=1024 (b)


Figure 6.10: Strong scaling 𝑡 ∝ 𝑝𝛿 observed for NSEXACT for (a) 𝑁 = 512 and (b) 𝑁 = 1024 as number

of GPUs is varied.

6.10.1 Computational cost of the FFTs

At each time step, we need to perform nine Fourier transforms to integrate the Navier-

Stokes equation 2. Six backward FFTs on the components of the velocity field to compute

the nonlinear term in real space and three forward transforms to take the nonlinear term

to Fourier space for time integration.

As the rest of the algorithm is embarrassingly parallel, as it does not require any GPU-

GPU communication, we expect that computing FFTs should take up most of the execution

time for the pseudospectral solver. To verify the same, we calculate the percentage of time

2Except the first time step, which requires three additional Fourier transforms before the integration

step.

Performance of the Navier-Stokes solver | 117

spent computing the FFTs in the pseudospectral algorithm as

FFT Share(p) = 92

𝑇𝑓𝑓𝑡(𝑝)𝑇𝑛𝑠𝑒𝑥𝑎𝑐𝑡(𝑝) × 100,

where 𝑇𝑓𝑓𝑡(𝑝) is the time taken to compute a pair of FFTs at 𝑝 number of GPUs for a

problem size 𝑁 , and 𝑇𝑛𝑠𝑒𝑥𝑎𝑐𝑡(𝑝) is the time taken to perform one integration step of the

NSEXACT test case at 𝑝 GPUs and for the same problem size. In Fig. 6.11 we show the

FFT Share (in percentage) for 𝑁 = 512 and 𝑁 = 1024. As expected, we find that a

significant amount of time (≈ 75 − 85%) is spent in computing the FFTs only.

70

80

90N=512

1 2 4 8 16p

70

80

90N=1024

FF

TS

har

e(%

)

DGX-StationDGX2DGX-A100

Figure 6.11: Time spent in computing the FFTs for the NSEXACT test case for two problem sizes 𝑁 = 512and 𝑁 = 1024 at various number of GPUs used (𝑝). On average 70-85% of the total execution time is

spent in computing the FFTs.


6.11 Limitations of the GPGPU solver

As we have shown in this chapter, pseudospectral algorithms perform well on DGX architec-

ture, which, owing to its powerful GPU cards and high bandwidth NVLINK communication

channels, is an ideal platform for performing moderate resolution DNS (5123 − 20483) of

the Navier-Stokes equation. However, there are some limitations of using pseudospectral

algorithms on DGX architecture, which we address now.

• Architecture specific: The current implementation of our solver is specific to DGX-

architecture only, where multiple GPUs reside on a single node and communicate via

NVLINK communication. With few tweaks, the implementation should also work

on multiple GPUs without NVLINK support, but the performance gains rely heav-

ily on the high-speed communication channels between the GPUs. Further, GPUs

distributed over multiple nodes are not supported yet.

• Memory requirements: As discussed earlier, the total memory required to perform a

simulation at a 𝑁3 resolution is around 80−100𝑁3 +O (𝑁2) bytes. For example, for

the NSEXACT test case at 𝑁3 = 10243 resolution, around 90 GiB of device memory

is required. A single Tesla-V100 has 32 GiB device memory, and we need four such

GPUs to perform the simulations at this resolution. Often, the fluid flow is coupled

with other physical fields, in which case memory requirements can increase two or

three-folds, which severely reduces the maximum achievable resolution. For example,

in wet incompressible polar active runs performed in Chapter 4, we need 18 additional

system size arrays, which brings the total memory requirements to 220−240𝑁3 bytes.

For these simulations, the maximum resolution we can achieve on DGX-Station then

reduces to 𝑁3 = 5123.

• Ease of development: While multiple libraries are available for CUDA programming

in C/C++, the support for cudafortran is not as extensive. The situation is improv-

ing rapidly, and one can still use the Fortran-C bindings to write wrappers for any

C/C++ libraries which are not available in cudafortran. Although it is doable, it

does introduce some friction and is not suitable for beginners. In addition, some al-

gorithms are easier to write on CPUs and require extensive efforts to port them over

to GPUs. OpenACC standard attempts to alleviate these issues and is designed to

simplify parallel programming of heterogeneous CPU/GPU systems.

Limitations of the GPGPU solver | 119

6.12 Conclusions and future work

Regardless of the limitations mentioned above, GPGPU pseudospectral methods show ex-

cellent performance and strong scaling as we increase the number of GPUs used on DGX

architecture. Thus, DGX architecture is an ideal platform to perform the DNS of the

Navier-Stokes equation on moderate resolutions of 5123 − 20483. The algorithm can be

easily ported to two dimensions, where the memory scales with the resolution as O (𝑁2),

and one can achieve very high-resolution simulations viz. 327682 − 655362.

Overall, memory requirements are the major bottleneck to perform high-resolution sim-

ulations in three dimensions. In the future, we will focus on adding support for GPUs

distributed over multiple nodes, which will allow us to achieve higher resolutions but at

the expense of performance. Further, we will couple the pseudospectral solver with particle

tracking methods and mesh evolution algorithms.


Chapter 7

Conclusions and future directions

In this thesis, we have studied the statistical and dynamical properties of incompressible

polar active matter.

In Chapters 2 and 3 we studied the coarsening dynamics of the ITT equation at various

Re in 2D and 3D, respectively. In particular, we have shown that coarsening proceeds

via repeated defect merger events. In 2D, defects are uniformly distributed at all Re,

and a unique growing length scale characterizes the coarsening dynamics. At low Re, the

coarsening dynamics proceeds similarly to the XY model, whereas at high Re, we observed a

forward enstrophy cascade and showed that turbulence accelerates the coarsening dynamics.

In 3D, the spatial distribution of defects depends on Re. At low Re, similar to 2D, defects

are uniformly distributed, at high Re we observed defect-clustering in the intense vorticity

regions. We observe Kolmogorov scaling in the energy spectrum over a range of length

scales and a forward energy cascade. While low Re coarsening is governed by a unique

growing length scale, at high Re our structure-function analysis indicates the presence of

multiple interacting length scales.

In our coarsening studies, we have focused on the constant concentration limit. A

straight-forward direction would be to study the coarsening dynamics in the presence of

concentration fluctuations. Indeed, earlier studies [34, 41–43] on the coarsening dynamics

of the Toner-Tu equations with concentration fluctuations have reported that both the

velocity and coarsening fields are coupled and coarsen simultaneously but with different

scaling exponents. Further, the coarsening proceeds faster compared to equilibrium systems.

However, a detailed study of the energy transfer and defect dynamics in systems with

concentration fluctuations is still lacking.

In Chapter 4 we studied dense suspensions of extensile swimmers in two and three

| 121

dimensions. We investigated the instabilities of the aligned state to small perturbations

and established how inertia can stabilize the orientational order. We found that a non

dimensional parameter 𝑅 characterizes the stability of the aligned state. At small 𝑅, the

instabilities in the ordered state exhibit a growth rate proportional to O(𝑞). For 𝑅 > 𝑅1,

the instabilities grow at a rate proportional to O(𝑞2). Past a second threshold 𝑅2, the

flock is stable. Further, we performed high-resolution direct numerical simulations and

characterize the properties of the spatio-temporal chaos arising from the instabilities. We

showed that in two dimensions, for all 𝑅 < 𝑅2, the flow is riddled with topological vortices

with no global order in sight. The correlation length (or the inter-defect spacing) grows with

𝑅 and appears to diverge at 𝑅2. Our preliminary DNS in 3D showed that bulk suspensions

also exhibit defects for 𝑅 < 𝑅2. To characterize the properties of growing inter-defect

separation and the order-disorder transition requires numerical studies on larger system

size, which is another future direction to look forward to.

In Chapter 4 and Chatterjee et al. [55] we have shown that the most unstable modes for

the number-conserving, Malthusian, and dense suspensions are identical, but the statistical

properties of the turbulent states differ widely. For the Malthusian case, we observed an

order-disorder transition from defect-turbulent states to phase-turbulent states at 𝑅 = 𝑅1.

In dense suspensions, defect-turbulent states persist all the way up to 𝑅 = 𝑅2. How

incompressibility suppresses the transition from defect-ridden states to phase-turbulence

is still remains to be addressed. Further, how the inclusion of concentration fluctuations

alters the nature of steady states and the transition remains yet to be studied. We also

look forward to the experimental validation of our linear stability and numerical studies.

In general, in Chapters 2 to 4 we have shown that topological defects play a crucial role

in determining the dynamics of incompressible polar active systems. While much is known

about the nature of topological defects for equilibrium systems, interest in active defects

is relatively new, and much remains unexplored. A recent review article by Shankar et al.

[64] has highlighted the advances in this direction. The complex nonlinear terms in the

equations of dry and wet polar active systems alter the nature of defects. For example, in

the Malthusian suspensions [55], we have shown that due to the presence of self-propulsion

term asters are preferred over spirals and the saddles have string-like structures. Defect

dynamics in active systems is another area of interest.

In Chapter 5 we focused on a colony of nonmotile bacteria growing on a hard agar sur-

face. Here, the activity arises not from the particle motility, but the birth-death processes.

122 | Conclusions and future directions

At the colony front, bacteria repeatedly grow and divide at the expense of nutrients and

push each other in the process. The colony thus expands because of the steric repulsions

between the bacteria. While the interior of the colony is densely populated, population

fluctuations intrinsic to any birth-death process become essential for such systems at the

growing front where the number of organisms is small. We investigated how these fluctua-

tions and nutrient availability can affect the growing colony’s morphology. We found that

the population fluctuations and nutrient-dependent bacteria diffusion are sufficient to cause

a morphological transition from finger-like branched fronts to smooth fronts with increasing

nutrient concentration.

In Chapter 6 we presented a general purpose GPU based (GPGPU) pseudospectral

solver for the Navier-Stokes equation. We showed how the DGX architecture is an ideal

platform to perform discrete numerical simulations of the Navier-Stokes and related equa-

tions at moderate resolutions of size 5123 −20483. Our GPGPU pseudospectral solver in its

current state is designed for multiple GPUs hosted on a single node. To tackle much more

computationally intensive problems, for example the characterization of the order-disorder

transition in wet suspensions in 3D, we plan to support multi-node multi-GPU architecture

in near future.

Finally, we have used coarse-grained hydrodynamic equations in our numerical studies of

incompressible polar active systems. An alternative approach is to use agent-based models

of active systems, where each particle is modelled at the microscopic level. Concentration

fluctuations are intrinsic to these models. For example, the coarsening dynamics described

in Chapter 2 can also be studied with motile rods moving on frictional substrates [165].

| 123

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