Special Relativistic Inviscid Fluid Turbulence in 2+1 · 2018. 7. 11. · John Ryan Westernacher-Schneider, Luis Lehner Guelph-Waterloo Physics Institute October 25, 2013 JRWSSpecial

Special Relativistic Inviscid Fluid Turbulence in2+1

John Ryan Westernacher-Schneider, Luis Lehner

Guelph-Waterloo Physics Institute

October 25, 2013

JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1

Motivation

Broad: probe gravitational physics through fluid/gravityduality conjecture.

Specific: test analytic results by Fouxon, Oz (2009)

JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1

Background: Five-Thirds Law

Etot =

∫E (k)dk

[E ] =energy

mass= (

length

time)2

[dk] =1

length

⇒ [E (k)] =length3

time2

JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1

Background: Five-Thirds Law

Ansatz: energy flow ε scale-free (ε 6= ε(k)), so E (k) ∝ εpkq, where

[ε] =energy

mass

1

time=

length2

time3

[k] =1

length

JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1

Background: Five-Thirds Law

Now match units on both sides of E (k) ∝ εpkq:

length3

time2=

length2p−q

time3p

⇒ p =2

3

⇒ q = −5

3

Thus, E (k) ∝ k−5/3... this is true in the inertial range.Scale-free enstrophy flow leads instead to E (k) ∝ k−3.

JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1

Five-Thirds Law

Champagne (1978)JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1

Relativistic Concerns

length contraction ... whose k are we talking about?

geometrized units ... are dimensional arguments still valid?

JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1

Our Setup

spacetime dimension = 2+1 ... computationally inexpensive,useful for 4D gravity

perfect fluid ... T ab = (P + ρ)uaub + Pηab

equations of motion given by conservation of T ab, ∂aTab = 0

edges of computational grid are periodically identified ... i.e. a2-torus

JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1

Lack of scale implies an equation of state...

To be useful for fluid/gravity duality, we impose that T ab has nointrinsic scale. This implies tracelessness, T a

a = 0. Thus,

0 = T aa = (P + ρ)uau

a + Pδaa

= −(P + ρ) + 3P

⇒ P =1

2ρ.

JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1

Turbulence onset through destabilization...

JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1

Five-Thirds law is recovered up to v ≈ 0.6c ...

Carrasco, Lehner et al. (2012)

JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1

Fouxon, Oz results...

Impose random forcing with stationary statistics: ∂aTab = f b.

Pick two points: label ~0 and ~r .

Suppose |~r | is in the inertial range.⟨T0j(~0, t)Tij(~r , t)

⟩∝ ri , no sum on j.

〈.〉 means ensemble average.

JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1

Preparation for testing Fouxon, Oz results...

Argument relies on approximations. It’s worth checkingnumerically.

Must satisfy mathematical assumptions as best as possible.

Simplify ensemble average by imposing spatiallyhomogeneous, isotropic statistics for f.

Must restrict forcing to narrow range of scales to allow |~r | tobe far from forcing scale.

JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1

How we construct f

Build it in Fourier space, then Fourier transform to real space...Real condition: f ∗(~k) = f (−~k)

JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1

Non-triviality of the task...

Previous non-relativistic simulations of forced turbulence arevery different.

Usually incompressible... ⇒ ∇ · ~v = 0... This simplifiesequations analytically, and vorticity is forced directly.

JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1

Unforeseen difficulty

JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1

Summary

Kolmogorov scalings seem to hold in relativistic regime.

Satisfying assumptions of Fouxon and Oz results requiresforcing the fluid.

Appropriate forcing method is not given straightforwardly byprevious work.

Work in progress...

JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1

End

Thanks for listening.

JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1