I. Stochastic models for turbulence A. Test model B. Statistics of turbulent solutions in physical space C. Turbulent Rossby waves. II. Filtering linear stochastic PDE models with instability and model error A. Two-state Markov jump process B. Idealized spatially extended turbulent systems with instability C. The mean stochastic model for filtering D. Sparse regularly spaced observations E. Numerical performance of the filters with and without model error 1/28 Introduction to stochastic models for turbulence and filtering with instability and model error June 29, 2014 Short Course in High Dimensional Filtering Warwick Mathematics Institute Nan Chen Center for Atmosphere Ocean Science (CAOS) Courant Institute of Mathematical Sciences New York University
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I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
1/28
Introduction to stochastic models forturbulence and filtering withinstability and model error
June 29, 2014Short Course in High Dimensional FilteringWarwick Mathematics Institute
Nan ChenCenter for Atmosphere Ocean Science (CAOS)
Courant Institute of Mathematical SciencesNew York University
Di Qi
I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
2/28
Main reference
• Andrew J. Majda and John Harlim, Filtering Complex Turbulent Systems,Cambridge University Press (2012). [Chapter 5 and 8]
Supplementary materials
• Lecture notes of Professor Majda’s 2013 graduate course [Lecture 1 – 5]http://www.cims.nyu.edu/~chennan/CourseNotes2013.html
• Andrew J. Majda, Introduction to PDEs and Waves for the Atmosphere andOcean, Courant Lecture Notes Vol. 9, American Mathematical Society & CourantInstitute of Mathematical Sciences (2002).
• The statistical equilibrium distribution exists provided F(x, t) = 0and is a Gaussian with zero mean.
• The equilibrium variance defines the climatological energyspectrum and is given by
Ek =σ2
k
2γ(ik).
• The real part of the temporal correlation function in the statisticalsteady state is given by
Real [Rk(τ)] ≡ Real [〈(uk(t)− uk)(uk(t)− uk)∗〉]
= Eke−γ(ik)τ cos(ωkτ).
• γ(ik) defines the correlation time, γ(ik)−1. iω = p(ik) defines ωk,the oscillation frequency at wavenumber k.
I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
8/28
Calibrating the noise level for a turbulent signal.From observation or lab experiments, the energy Ek and correlationtime γ(ik)−1 at wavenumber k are known. Since Ek = σ2
k/(2γ(ik)),then the noise level is given by
σk = (2γ(ik)Ek)1/2.
Typical turbulent spectra involve power laws, such as
Ek = E(k) = Eo|k|−β , |k| > 1.
• β = 5/3 is Kolmogorov spectrum which generates a fractalrandom field.
• β = 0 corresponds to a white noise spectrum.
For the stochastic advection-diffusion equation, γ(ik) = d + µk2. Thenthe noise level is determined by
σk = 21/2E1/2o |k|−β/2(d + µ|k|2)1/2.
If the viscosity µ 6= 0, then at small spatial scales |k| 1
• noise increases if β < 2, and
• noise decreases if β > 2.
I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
9/28
B. Statistics of turbulent solutions in physical space.
∂u(x, t)∂t
= P(∂
∂x
)u(x, t)− γ
(∂
∂x
)u(x, t) + F(x, t) + σ(x)W(t).
Define ek(x) = e2πikx and p(ik) = p(ik)− γ(ik). Assume
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
11/28
C. Turbulent Rossby waves.Barotropic Rossby waves with phase varying only in the north-southdirection have the dispersion relation
ωk =β
kand it is natural to assume uniform damping
γ(ik) = d > 0
representing Ekmann friction. From observations on scales of order ofthousands of kilometers these waves have a k−3 energy spectrum
Ek = k−3.
• At midlatitude θ = 45o,
β = 2Ω cos(45o)/sec = 8.91/day
• The natural frequency is given by
ωk =8.91
kso the lowest wavenumber k = 1 has an oscillation period ofroughly 17 hours.
• The parameter d = 1.5 is chosen such that the decorrelationtime is three days which corresponds to realistic weatherpredictability.
I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
12/28
Figure 1: Statistics of turbulent Rossby waves.
I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
13/28
II. Filtering linear stochastic PDE modelswith instability and model error
A. Two-state Markov jump process
0 50 100 150 200 250−0.5
0
0.5
1
1.5
2
2.5
sst
sun
Switching rate:
ν :sst → sun,
µ :sun → sst.
Switching times:
P(Tst ≤ t) = 1− eνt,
P(Tun ≤ t) = 1− eµt.
Expectation:
E[X] =µsst + νsun
ν + µ.
Equilibrium distribution
(peqst , p
equn) =
(µ
ν + µ,
ν
ν + µ
).
I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
14/28
Complex models with regime switching
du = [(−γ(t) + iωu)u + f (t)]dt + σudWu,
γ satisfies two-state Markov jump process.
• The unresolved process: a two-state Markov jump process.• Intermittency.• Example: Rossby wave with baroclinic instability.
0 50 100 150 200 250−1
−0.5
0
0.5
1
0 50 100 150 200 250−1
−0.5
0
0.5
1
0 50 100 150 200 250
0
1
2
t
10−5
100
105
−1
−0.5
0
0.5
1PDF in log scale
TrueGaussian fit
10−5
100
105
−1
−0.5
0
0.5
1
Sample trajectory
Real part of u
Imag part of u
γ
Figure 2: True signal, in which γ is generated from a two state markov process, where the stable phaseis d+ = 2.27 and the unstable phase is d− = −0.04.
I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
15/28
B. Idealized spatially extended turbulent systems with instability.
∂u(x, t)∂t
= P(∂
∂x
)u(x, t)− γ
(∂
∂x
)u(x, t) + F(x, t) + σ(x)W(t).
• Assume F(x, t) = 0 and consider a 2π-periodic domain.• Discretize the model at 2N + 1 equally spaced grid points.• Each Fourier mode, uk, solves the linear Langevin equation
duk(t) = [p(ik)− γ(ik)]uk(t)dt + σkdWk(t).
Set p(ik) = iωk = i8.91k−1 to mimic barotropic Rossby wave.The damping coefficients γ(ik) are assumed to be:
such that the two-state Markov process spends on average of 10 daysin the stable regime and 5 days in the unstable regime. The averageddamping for all the modes are 1.5.
I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
16/28
Figure 3: Panels 1-4: evolution of both stable u1 and unstable u5 modes together with the correspondingdamping parameters. Panel 5: evolution of the total energy.
I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
17/28
Figure 4: Upper panel: total energy E(t); lower panel: damping regime.
I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
18/28
Figure 5: Spatial pattern for a turbulent system of the externally forced barotropic Rossby wave equationwith instability through intermittent negative damping. Note the coherent wave train that emerges duringthe unstable regime.
I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
19/28
Time-averaged energy spectrum
• Modes 1 − 3: 89% of the total energy.
• Modes 4 − 5: 10% of the total energy.
• Modes 6 − N: 1% of the total energy.
Energy averaged over the unstable regime(∼ 1/3 of the total time)
• Modes 1 − 3: 75% of the total energy.
• Modes 4 − 5: 24% of the total energy.
• Modes 6 − N: 1% of the total energy.
Figure 6: Energy spectrum for the model with switching damping.
I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
20/28
C. The mean stochastic model for filtering.
A simple approach for filtering signals with intermittent instability is touse the mean stochastic model (MSM), which is based on the twoequilibrium statistical quantities,
• the energy spectrum, and
• the damping time
The MSM reads
duk(t)dt
= (−d + iωk)uk(t) + σkWk(t),
where d = 1.5 is the averaged damping constant for all modes.
I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
21/28
D. Sparse regularly spaced observations.• The system is discretized at 2N + 1 equally spaced grid points.• The observations are taken at 2M + 1 regularly spaced grid
points, where M ≤ N.• Define the ratio of the total # of mesh points to the # of obs,
P =2N + 12M + 1
.
Features:• Modes belonging to different aliasing sets are completely independent.
• Modes within one aliasing set are coupled through the observation.
vl,m =∑
k∈A(l)
uk,m + σol,m, |l| ≤ M,
A(l) = k|k = l + (2M + 1)q, |k| ≤ N, q ∈ Z.
• The filtering problem is reduced into 2M + 1 independent filteringproblems; each involves P components in each aliasing set.
Example. Consider N = 52 and P = 5.
• 2N + 1 = 105 model grid points; the obs are only at (2N + 1)/P = 21spatial locations.
• Only M = 10 primary (most energetic) modes, where 2M + 1 = 21.
• The aliasing set A(1) = 1, 22, 43,−20,−41 and the primary mode inA(1) is mode 1.
I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
22/28
Models:• Perfect model:
duk(t) = (−γk(t) + iωk)uk(t)dt + σkdWk(t),
where γ(ik) switches between ds and dw for modes 1− 3, betweend+ and d− for modes 4− 5, and γk(t) = d for modes 6− N.
• KF: Kalman filter of perfect model.• KFME: Kalman filter with model error.• RFDKF: Reduced Fourier domain Kalman filter of perfect model.• RFDKFME: Reduced Fourier domain Kalman filter with model error.
Remarks:• RFDKF always trusts the dynamics for all the aliased modes
2 < i < P and only the primary mode is filtered.• When P ≥ 15, the switching
modes 4-5 are not theprimary modes and RFDKFsets these modes toalways be the prior meanstate with constant meandamping d = 1.5.
P # of obs # of aliasing sets M1 105 523 35 175 21 107 15 715 7 321 5 2
I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
23/28
E. Numerical performance of the filters with andwithout model error.
Figure 7: Filter performance for P = 1 at T = 5000.
I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
24/28
Figure 8: Filter performance for P = 5 at T = 5000.
I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
25/28
Figure 9: Filter performance for P = 15 at T = 5000.
I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
26/28
Figure 10: RMS error as a function of observation time ∆t for ro = 0.1 and various values of P. Thesolid lines correspond to the KF with exact damping, dashed lines correspond to the KF with model error.The solid lines with pluses correspond to RFDKF and the dashes with pluses correspond to RFDKF withmodel error.
I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error
27/28
Figure 11: RMS error as a function of observation variance ro for ∆t = 0.1 and various values of P. Thesolid lines correspond to the KF with exact damping, dashed lines correspond to the KF with model error.The solid lines with pluses correspond to RFDKF and the dashes with pluses correspond to RFDKF withmodel error.
I. Stochasticmodels forturbulenceA. Test model
B. Statistics of turbulentsolutions in physical space
C. Turbulent Rossby waves.
II. Filtering linearstochastic PDEmodels withinstability andmodel errorA. Two-state Markov jumpprocess
B. Idealized spatiallyextended turbulent systemswith instability
C. The mean stochasticmodel for filtering
D. Sparse regularly spacedobservations
E. Numerical performanceof the filters with andwithout model error