Introduction to stochastic models for turbulence …qidi/filtering18/Lecture8.pdfDeveloping stochastic models for turbulence Features of the dynamics: Turbulent and energetic at the

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

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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









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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).

http://www.cims.nyu.edu/~chennan/CourseNotes2013.html









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IntroductionDeveloping stochastic models for turbulence

Features of the dynamics:

• Turbulent and energetic at the smallest mesh scales.• Known climatological spectrum.

Example:

• Earth’s global scale: 20, 000 ∼ 40, 000 km.• Mesh spacing 10 ∼ 50 km.• Random and chaotic energy on 10 km scales due to chaotic motion of clouds,

topography and boundary layer turbulence which are unresolved.

Simplest models for representing turbulent fluctuations:

• Replacing nonlinear interaction by additional linear damping andstochastic white noise forcing.

Procedure:

• Linearize the complex PDE at a constant-coefficient background

∂u(x, t)∂t

= P(∂

∂x

)u(x, t) + F(x, t) .

• Add additional damping and white noise forcing

∂u(x, t)∂t

= P(∂

∂x

)u(x, t)−γ

(∂

∂x

)u(x, t) + F(x, t)+σ(x)W(t).









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Filtering linear stochastic models with instability and model error

A major difficulty in accurate filtering of noisy turbulent signals with manydegrees of freedom is model error. Sources of the model error:

• Discrete numerical solvers of a continuous system.• Parametrization of the important physical processes.• Reduced filtering strategies.

• [In this talk] Intermittently unstable turbulent processes.• True/perfect model

duk(t) = (−γk(t) + iωk)uk(t)dt + σkdWk(t).

• Imperfect model

duk(t) = (−γk + iωk)uk(t)dt + σkdWk(t).

Check the filter performance for plentifully and sparsely observedsignals with the Fourier domain Kalman filter (FDKF) and reducedFDKF.

0 50 100 150 200 250−1

0

1

10−5

100

105

−1

0

1PDF in log scale

TrueGaussian fit

0 50 100 150 200 250

0

1

2

t

γ

Real part of u

2.27

−0.04









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I. Stochastic models for turbulenceA. Test model.

∂u(x, t)∂t

= P(∂

∂x

)u(x, t)− γ

(∂

∂x

)u(x, t) + F(x, t) + σ(x)W(t),

u(x, 0) = u0(x).

The problem is non-dimensionalized to a 2π-periodic domain.

P(∂

∂x

)eikx = p(ik)eikx, p(ik) = iωk,

γ

(∂

∂x

)eikx = γ(ik)eikx, γ(ik) > 0.

The 2π-periodic solution is expanded in Fourier series

u(x, t) =∞∑

k=−∞

uk(t)eikx, u−k = u∗k ,

where uk(t) for k > 0 solves the scalar complex-coefficient stochasticODEs

duk(t) = [p(ik)− γ(ik)]uk(t)dt + Fk(t)dt + σkdWk(t), uk(0) = uk,0.









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Example. The stochastically forced dissipative advection equation.

∂u(x, t)∂t

= −c∂u(x, t)∂x

− du(x, t) + µ∂2u(x, t)∂x2 + F(x, t) + σ(x)W(t).

In this example,

p(ik) = iωk = −ick,

γ(ik) = d + µk2.

where d ≥ 0, µ ≥ 0 and at least one of them is nonzero.

Remark. In many geophysical applications p(ik) is not necessarily apolynomial. In quasi-geostrophic equations where p(ik) is given byp(ik) = ik

k2+F .









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Recall the ODE for the Fourier component k,

duk(t) = [p(ik)− γ(ik)]uk(t)dt + Fk(t)dt + σkdWk(t), with γ(ik) > 0.

• 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.









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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.









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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

u(x, t) =∑|k|≤N

uk(t)ek(x).

Then each uk satisfies

duk(t) = [p(ik)− γ(ik)]uk(t)dt + Fk(t)dt + σkdWk(t),

= p(ik)uk(t)dt + Fk(t)dt + σkdWk(t) with γ(ik) > 0.

• The ensemble mean of u(x, t) is

〈u(x, t)〉 =

⟨∑|k|≤N

uk(t)ek(x)

⟩=∑|k|≤N

〈uk(t)〉ek(x)

=∑|k|≤N

(uk(0)ep(ik)t +

∫ t

0Fk(s)ep(ik)(t−s)ds

)ek(x).









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• General spatio-temporal correlation function

R(x, x′, t, t′) ≡ 〈[u(x, t)− 〈u(x, t)〉][u(x′, t′)− 〈u(x′, t′)〉]∗〉.

• Covariance

Var[u(x, t)] = R(x, x, t, t) =∑|k|≤N

σ2k

2γ(ik)

(1− e−2γ(ik)t

).

• Temporal correlation function

R(x, x, t, t′) =∑|k|≤N

σ2k

2γ(ik)ep(ik)(t−t′)e−γ(ik)(t′−t)(1− e−2γ(ik)t)

• Spatial correlation at fixed time

R(x, x′, t, t) =∑|k|≤N

σ2k

2γ(ik)

(1− e−2γ(ik)t

)ek(x− x′).









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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.









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Figure 1: Statistics of turbulent Rossby waves.









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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

).









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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.









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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:

• Modes 1− 3: dw = 1.3 (weak) and ds = 1.6 (strong),• Modes 4− 5: d+ = 2.27 (stable) and d− = −0.04 (unstable),• Modes 6− N: d = 1.5,

with the switching rates

Sst → Sun and Ss → Sw : ν = 0.1,

Sun → Sst and Sw → Ss : µ = 0.2,

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.









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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.









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Figure 4: Upper panel: total energy E(t); lower panel: damping regime.









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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.









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Time-averaged energy spectrum

• Modes 1 − 3: 89% 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 6 − N: 1% of the total energy.

Figure 6: Energy spectrum for the model with switching damping.









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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.









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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.









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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.

• Imperfect model with model error (MSM)

duk(t) = (−d + iωk)uk(t)dt + σkdWk(t),Different filters:

• 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









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E. Numerical performance of the filters with andwithout model error.

Figure 7: Filter performance for P = 1 at T = 5000.









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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.









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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.









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Thank you

Documents

Introduction to stochastic models for turbulence …qidi/filtering18/Lecture8.pdfDeveloping stochastic models for turbulence Features of the dynamics: Turbulent and energetic at the