introduction to turbulent dynamical systems in complex …qidi/turbulence16/intro.pdfintroduction to...

Introduction to Turbulent Dynamical Systems inComplex Systems

Di Qi, and Andrew J. Majda

Courant Institute of Mathematical Sciences

Fall 2016 Advanced Topics in Applied Math

Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 10, 2016 1 / 38

Lecture 10: Introduction to Turbulent Dynamical Systemsin Complex Systems

1 General turbulent dynamical systems for complex systemsBasic issues for prediction, uncertainty quantification, and state estimationStochastic toolkit for UQDetailed structure in complex systemsEnergy conservation principles

2 Prototype examples of complex turbulent dynamical systemsTurbulent dynamical systems for complex geophysical flows: one-layer modelThe Lorenz ’96 model as a turbulent dynamical systemStatistical triad models, the building blocks of turbulent dynamical systemsMore rich examples of complex turbulent dynamical systems


Outline




General complex turbulent dynamical systems

Consider a general dynamical system, perhaps with noise, written in the Ito senseas given by

du

dt= F (u, t) + σ (u, t) W (t) . (1)

General properties

The turbulent dynamical systems live in a large dimensional phase spaceu ∈ RN , with N 1;

σ is an N × K noise matrix and W ∈ RK is K -dimensional white noise;

The noise often represents degrees of freedom that are not explicitly modeledsuch as the small-scale surface wind on the ocean;

Typically one thinks about the evolution of smooth probability density p (u, t)associated with (1).


Challenges in predicting turbulent systems

du

dt= F (u, t) + σ (u, t) W (t) .

The turbulent dynamical systems are characterized by a large dimensional phasespace u ∈ RN , with N 1

The phase space contain a large dimensional of unstable directions measuredby

I the number of positive Lyapunov exponents; orI non-normal transient growth subspace.

All these linear instabilities are mitigated by energy-conserving nonlinearinteractions that transfer energy from linearly unstable modes to the stableones

I Increase of energy in the linearly unstable modes is balanced by the nonlinearinteractions;

I The linear stable modes dissipate the energy transferred from the unstablemodes;

I A statistical steady state is reached for the complex turbulent system.


Challenges for uncertainty quantification in turbulentdynamical systems

Uncertainty quantification (UQ) deals with the probabilistic characterization of all thepossible evolutions of a dynamical system given an initial set of possible states as well asthe statistical characteristics of the random forcing or parameters.

Sources of uncertaintyInternal instabilities

Initial and boundary conditions

Approximations on the model

Limited observations and data

Computational challengesNon-Gaussian statistics

Large dimensional phase-space

Non-stationary dynamics

Wide range of scales

Major Goal:Obtain accurate statistical estimates such as change in the mean and variance forthe key statistical quantities in the nonlinear response to changes in externalforcing or uncertain initial data.


Imperfect models with model error

The efforts in UQ are hampered by the inevitable model errors and the curse ofensemble size for complex turbulent dynamical systems

Perfect Model :du

dt= F (u, t) + σ (u, t) W (t) , u ∈ RN ,

Imperfect Model :duM

dt= FM (uM , t) + σM (uM , t) W (t) , uM ∈ RM .

Practical complex turbulent models often have a huge phase space withN = 106 to N = 1010;

Model errors typically arise from lack of resolution M N compared withthe original perfect model.

I The perfect model is too expensive to simulate directly;I The lack of physical understanding of certain physical effects;I The noise σM is often non-zero and judicious chosen to mitigate model errors.


Monte-Carlo ensemble prediction

For chaotic turbulent dynamical systems, single predictions often have littlestatistical information and Monte-Carlo ensemble predictions are utilized

pL (u (x , t)) =L∑

j=1

pjδ (u− uj (t)) , where duj = F (uj , t) dt,

with initial data p0 (u) ∼=L∑

j=1

p0,jδ (u− u0,j (t)) , uj |t=0= u0,j .

The curse of ensemble size arise for practical predictions since N is hugewhile L = O (100) is small by computational limitation;

With model errors using lower resolution M, L can be increased but modelerrors can swamp this gain in statistical accuracy;

It is a grand challenge to devise methods that make judicious model errorsthat lead to accurate predictions and UQ.


State estimation (data assimilation, or filtering)2. Basic idea of filtering.

In analysis step, Bayesian formula is utilized:

p(um+1|vm+1) p(um+1)p(vm+1|um+1)

In linear and Gaussian case, the optimal filter is the so-called Kalman filter.

(See Majda-Harlim book for more details.)

46 / 48Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 10, 2016 9 / 38

Examples for complex dynamical systems with uncertaintyExamples of continuous systems

Geophysical fluid flows Plasma flow High Reynolds numbers

Examples of discrete systems

2

Large or ∞ dimensional systems with uncertainties

Examples of continuous systems

Examples of discrete systems

Thermal transport in heterogeneous media

Water waves spectrum evolution Geophysical fluid flows

Structural systems US power network Systems biology


Distribution and density function Distribution and density functions

7Monday, September 19, 2011


Moments of random variables4. Moments.

Consider 1-D case for simplicity. The n-th moment and central moment of X aredefined respectively by

µn = E [X n] =

Z 1

1xnp(x)dx , n 1,

µn = E [(X µ)n] =

Z 1

1(x µ)np(x)dx , n 2.

Statistics Description Moment Formula

Mean the central tendency 1-st moment µ

Variance spreading out 2-nd central moment 2 = µ2

Skewness the asymmetry 3-rd normalized central moment µ3/3

Kurtosis the “peakedness” 4-th normalized central moment µ4/4

−5 0 5 100

0.1

0.2

0.3

0.4

0.5

0.6PDF with different skewness (linear scale)

Skewness = 3.26Skewness = 1.75Skewness = 0.78Skewness = 0

−5 0 5 10

10−4

10−2

100

Logarithm scale

−10 −5 0 5 100

0.1

0.2

0.3

0.4

0.5PDF with different kurtosis (linear scale)

Kurtosis = 9Kurtosis = 7Kurtosis = 5Kurtosis = 3Kurtosis = 2

−10 −5 0 5 10

10−4

10−2

100

Logarithm scale

fat−tails

sub−Gaussian 5 / 48


Stochastic processes Stochastic processes

Definitions:

Definition: [Wiener process (Brownian motion) ]Real-valued stochastic process W(t) such that

Sample paths of the Wiener process are, with probability one, nowhere differentiable

(with probability one)

Wi W (ti) W (ti1)

E(W 2i ) = ti


Ito Stochastic differential equations

Stochastic process X satisfies the Ito SDE

if for all

conditions (Lipschitz and growth cnds.) on the functions A and B for existence and uniqueness of path-wise solutions (see, e.g., Oksendal 2000)

Stieltjes integral Ito integral

X(t) = X(t0) +

Z t

t0

A(X(s), s)ds +

Z t

t0

B(X(s), s)dW (s)

Definition: [Ito stochastic integral]

where the limit is taken in the mean-square sense.

X nonanticipating

t, t0

m.s. limn!1

Xn = X ) limn!1

h(Xn X)2i = 0

Z t

t0

B(X(s), s)dW (s) := m.s. limn!1

nX

j=1

B(X(tj1), tj1)(W (tj) W (tj1))

+

t > t0 > 0, X(t0) = X0



Basic properties of Ito calculus Basic properties of Ito calculus

Ito formula dX = a(X, t)dt + b(X, t)dW (t)Assume that



The Fokker-Planck equation The Fokker Planck equationDescribes evolution of probability density associated with a stochastic process which satisfies an SDE

Scalar case

Consider a function f. Then the mean evolution

Integrate by parts, note that f arbitrary and by definition

Multivariate case

Fokker-Planck equation

C2

12Monday, September 19, 2011Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 10, 2016 15 / 38

Stationary solutions of scalar Fokker-Planck equation Stationary solutions of scalar Fokker Planck equation

LFP peq(x) = 0

Consider linear SDE

with

Fokker-Planck equation

Thus, implies 0

Gaussian density



Linear (unforced) Langevin equation Linear Langevin equation (scalar unforced case)

with

Second-order statistics

Equilibrium statistics

Equilibrium density

huieq = 0 V areq(u) =2

2

Path-wise solutions



Detailed structure in complex systems

The system setup will be a finite-dimensional system of, u ∈ RN , with lineardynamics and an energy preserving quadratic part

du

dt= L [u] = (L− D) u + B (u,u) + F (t) + σk (t) Wk (t;ω) . (2)

L being a skew-symmetric linear operator L∗ = −L, representing the β-effect ofEarth’s curvature, topography etc.

D being a negative definite symmetric operator D∗ = D, representing dissipativeprocesses such as surface drag, radiative damping, viscosity etc.

B (u, u) being the quadratic operator which conserves the energy by itself so that itsatisfies B (u, u) · u = 0.

F (t) + σk (t) Wk (t;ω) being the effects of external forcing, i.e. solar forcing,seasonal cycle, which can be split into a mean component F (t) and a stochasticcomponent.


Energy conserving principle

Consider the energy E = 12 |u|

2;

The skew-symmetric operator makes no contribution

u∗ · Lu = 0;

The symmetric operator D represents strict dissipation so that

u∗ · Du ≤ −d |u|2 , d > 0.

dE

dt=

d

dt

(1

2|u|2)

= (Du · u) + F · u

≤ −d

2|u|2 +

1

2d|F|2

= −dE +1

2d|F|2 .

The elementary inequality a · b ≤ |a|2

2 + |b|22 has been used, the Gronwall

inequality guarantees the global existence of bound smooth solutions.Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 10, 2016 19 / 38

Local existence and uniqueness theorem

Definition

Let O ∈ RN be an open set. A function F : O → RN is said to be Lipschitz on Oif there exists a constant K such that

|F (Y )− F (X )| ≤ K |Y − X | ,

for all X ,Y ∈ O.

TheoremConsider the initial value problem

X = F (X ) , X (0) = X0

where X0 ∈ RN . Suppose that F : O → RN is Lipschitz. Then there exists a > 0and a unique solution

X : (−a, a)→ RN

of this differential equation satisfying the initial condition.


Extending solutions

Definition

Let O ∈ RN be open, and let F : O → RN be Lipschitz. Let Y (t) be a solutionof X = F (X ) defined on a maximal open interval J = (α, β) ⊂ R with β <∞.Then given any compact set C ⊂ O, there is some t0 ∈ (α, β) with Y (t0) /∈ C.

This theorem says that if a solution Y (t) cannot be extended to a larger timeinterval, then this solution leaves any compact set in O. This implies that, ast → β, Y (t) accumulates on the boundary of O.Using the energy equation, Gronwall’s inequality implies

dE

dt≤ −dE +

1

2d|F|2

⇒ E (t) ≤ E (0) e−dt +1

2d

ˆ t

0

e−d(t−s) |F (s)|2 ds < C .


Outline




One-layer model

The one-layer geophysical flow on a periodic domain T2 = [−π, π]2 is given by

dq

dt+∇⊥ψ · ∇q = D (∆)ψ + f (x) + Wt ,

q = ∆ψ − F 2ψ + h (x) + βy .

q is potential vorticity, ψ is stream function, and the flow byu = ∇⊥ψ = (−∂yψ, ∂xψ);

The operator D (∆) =∑l

j=1 (−1)jγj ∆

j stands for a general dissipationoperator;

f (x) is the external deterministic forcing, the random forcing Wt is aGaussian random field;

βy is the β-plane approximation of the Coriolis effect, and h (x) is theperiodic topography;

The constant F = L−1R , where LR =

√gH0/f0 is the Rossby radius which

measures the relative strength of rotation to stratification.


One-layer modelThe idealized geophysical flow can produce a remarkable number of realisticphenomena:

I the formation of coherent jets and vortices;I direct and inverse turbulent cascades;I statistical bifurcations between jets and vortices.

Numerical experiments and statistical approximations are only possible orvalid for large finite times;Recent paper1 proves with full mathematical rigor a unique smooth invariantmeasure which attracts all statistical initial data at exponential rate.

1Majda & Tong, Ergodicity of truncated stochastic Navier-Stokes with deterministic forcingand dissipation, JNLS, 2015


Lorenz ’96 system

L-96

dui

dt= ui−1 (ui+1 − ui−2)− ui + Fi , i = 0, 1, ..., J − 1.

usually take J = 40 to simulate mid-latitude turbulence;

energy conservation term B (u,u) · u = 0 (nonlinear advection);

negative definite term D = −I (dissipation);

system is invariant under translations, steady state statistics become spatiallyhomogeneous, thus Fourier basis a natural choice;

external noise is 0, so uncertainty builds from unstable modes of linearizeddynamics Lv .


FLinear Analysis Mean Statistics Mixing

kbegin kend Reω u E (u) Ep λ1 N+ Tcorr

5 5 11 0.08 1.6344 53.4276 110.0132 1.02 12 8.238 4 12 0.1004 2.3418 109.6825 265.0064 1.74 13 6.704

16 4 12 0.0876 3.0869 190.5900 797.2113 3.945 16 5.594

0 2 4 6 8 10 12 14 16 18 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

wavenumber

Pe

rce

nta

ge

of

En

erg

y %

F = 5

0 2 4 6 8 10 12 14 16 18 200.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

wavenumber

Pe

rce

nta

ge

of

En

erg

y %

F = 8

0 2 4 6 8 10 12 14 16 18 200.02

0.03

0.04

0.05

0.06

0.07

0.08

wavenumber

Pe

rce

nta

ge

of

En

erg

y %

F = 16

space

tim

e

F = 5

5 10 15 20 25 30 35 400

5

10

15

1.61

1.62

1.63

1.64

1.65

1.66

space

tim

e

F = 8

5 10 15 20 25 30 35 400

5

10

15

2.3

2.31

2.32

2.33

2.34

2.35

2.36

2.37

2.38

2.39

space

tim

e

F = 16

5 10 15 20 25 30 35 400

5

10

15

3

3.02

3.04

3.06

3.08

3.1

3.12

3.14

3.16

3.18

3.2


Triad system: building block of turbulenceThe three-dimensional triad system with a quadratic part that is both divergencefree and energy preserving, u = (u1, u2, u3)T

du

dt= (L + D) u + B (u, u) + F (t) + σk (t) Wk (t;ω) .

L =

0 λ12 λ13

−λ12 0 λ23

−λ13 −λ23 0

, B (u, v) =

B1u3v2

B2u1v3

B3u2v1

, D =

−γ1

−γ2

−γ3

, Σ =

σ1

σ2

σ3

energy conservation : B1 + B2 + B3 = 0.

Condition for Gaussian invariant measure

σ21

2γ1=

σ22

2γ2=

σ23

2γ3= E , peq = C−1 exp

(−2E−1 ‖u‖2

).

Strong nonlinear regime with energy cascade

I λ12 = 0, λ13 = 0, λ23 = 0,

I B1 = 2,B2 = −1,B3 = −1,

I γ1 = 0.001, γ2 = 1, γ3 = 1,

I σ21 = 1.41, σ2

2 = 0.04, σ23 = 0.04.

The first component is weakly dampedand strongly forced by noise while theother two components are stronglydamped and weakly forced.


Figure 4: Strongly nonlinear regime with energy cascade: Full-system statistics predicted with

direct Monte-Carlo in the original system. In the right plots the stready state conditional probability

density functions of pu1u2u3 are shown as well as 2D scatter plots.

In Figure 4 we present the statistics of the system as computed by the direct Monte-Carlo

method. Thus, in the statistical steady state regime we have one mode carrying most of the

system energy and two low-energy modes absorbing energy from the rst one. This energy transfer

property is manifested by the third-order moment hu1u2u3i (shown in Figure 4) whose negativevalue indicates the energy transfer from the rst mode to the other two. This ow of energy is also

illustrated by the nearly two-dimensional character of the joint probability density function (see [15]

for a rigorous connection of the energy transfer properties and the dimensionality of the probability

measure). This strongly nonlinear regime with energy cascade has a Fokker-Planck equation with

an elliptic generator with a smooth three dimensional steady state probability density which is

nearly two dimensional

Note that since this is a system where only one mode has important variance while the other

two are much weaker in terms of energy one may expect that it is an ideal candidate for single-mode

order-reduction. However this is not the case. In Figure 3 we present the results of the single-mode

reduction and we compare with direct Monte-Carlo simulation. We observe that we have very large

discrepancies both in the rst and second-order statistics. These discrepancies are much larger in

magnitude than the order of the ignored variance.

The answer comes from the essentially irreducible character of the steady state probability

measure which does not allow for any approximation by a single mode system (or even two mode

reduction - see the results in Section 5.3). This is because the two low-energy modes play an

important role in the global dynamics by extracting energy from the high-energy mode and strongly

dissipating it. In particular, this non-Gaussian cross-correlation structure is responsible for the

persistent energy transfer from mode 1 to modes 2 and 3 which contrary to the magnitude of the

modes 2 and 3 is very important. Ignoring one or both of these modes completely destabilizes

the energy uxes and causes large errors in the eigenvalues of the covariance and subsequently the

estimated mean.

15

Figure: Strongly nonlinear regime with energy cascade: full-system statistics predicted with

direct Monte-Carlo in the original system. In the right plots the steady state conditional pdfs of

pu1u2u3 .

This instructive example illustrates very clearly that the small magnitude of theuncertainty in specific directions may not always be an efficient criterion to neglectdynamics. This is especially the case for turbulent flows where energy transfers have tobe modeled even if the modes are associated with weak energy.


Intuition about energy transfer

2.3 Statistical Triad Models, the Building Blocks of Complex Turbulent Dynamical Systems 9

where ‘’ is the cross-product, L 2 R3, and the nonlinear term

B(u,u) =

0@

B1u2u3B2u3u1B3u1u2

1A ,

with B1 + B2 + B3 = 0, so that u · B(u,u) = 0. They are the building blocks ofcomplex turbulent dynamical systems since a three-dimensional Galerkin trunca-tion of many complex turbulent dynamics in (1.5)–(1.7) have the form in (2.4), inparticular the models from Sections 2.1 and 2.2. A nice paper illustrating the factfor many examples in the geosciences is [53]; the famous three-equation chaoticmodel of Lorenz is a special case of this procedure. The random forcing togetherwith some damping represent the effect of the interaction with other modes in aturbulent dynamical system that are not resolved in the three dimensional subspace[121, 122, 123]. Stochastic triad models are qualitative models for a wide varietyof turbulent phenomena regarding energy exchange and cascades and supply impor-tant intuition for such effects. They also provide elementary test models with subtlefeatures for prediction, UQ, and state estimation [49, 105, 51, 156, 157].

Elementary intuition about energy transfer in such models can be gained by look-ing at the special situation with L = D = F = s 0 so that there are only thenonlinear interactions in (2.4). We examine the linear stability of the fixed point,u = (u1,0,0)T . Elementary calculations show that the perturbation du1 satisfiesddu1

dt = 0 while the perturbations du2,du3 satisfy the second-order equation

d2

dt2 du2 = B2B3u21du2,

d2

dt2 du3 = B2B3u21du3,

so that

there is instability with B2B3 > 0 andthe energy of du2,du3 grows provided B1 has (2.5)

the opposite sign of B2 and B3 with B1 +B2 +B3 = 0.

The elementary analysis in (2.5) suggests that we can expect a flow or cascade ofenergy from u1 to u2 and u3 where it is dissipated provided the interaction coefficientB1 has the opposite sign from B2 and B3.

We illustrate this intuition in a simple numerical experiment in a nonlinear regimewith a statistical cascade. For the nonlinear coupling we set B1 = 2, B2 = B3 = 1so that (2.5) is satisfied and L 0,F 0 for simplicity. We randomly force u1 witha large variance s2

1 = 10 and only weakly force u2,u3 with variances s22 = s2

3 =0.01 while we use diagonal dissipation D with d1 = 1 but the stronger dampingd2 = d3 = 2 for the other two modes. A large Monte Carlo simulation with N =1105 is used to generate the variance of the statistical solution and the probabilitydistribution function (pdf) along the coordinates in Figure 2.2. These results show astatistical steady state with much more variance in u1 than u2 and u3 reflecting the


Quantitative models2.4 More Rich Examples of Complex Turbulent Dynamical Systems 11

2.4.1 Quantitative Models

A) The truncated turbulent Navier-Stokes equations in two or three space dimen-sions with shear and periodic or channel geometry [143].

B) Two-layer or even multi-layer stratified flows with topography and shears inperiodic, channel geometry or on the sphere [130, 171, 94]. These models in-clude more physics like baroclinic instability for transfer of heat and generalizethe one-layer model discussed in Section 2.1. There has been promising nov-el multiscale methods in two-layer models for the ocean which overcome thecurse of ensemble size for statistical dynamics and state estimation called s-tochastic superparameterization. See [110] for a survey and for the applications[65, 66, 67, 68, 64, 63] for state estimation and filtering. The numerical dynamicsof these stochastic algorithms is a fruitful and important research topic. The endof Chapter 1 of [130] contains the formal relationship of these more complexmodels to the one-layer model in Section 2.1.

C) The rotating and stratified Boussinesq equations with both gravity waves andvortices [124, 171, 94].

There are even more models with clouds and moisture which could be listed. Next isthe list of qualitative models with insight on the central issues for complex turbulentdynamical systems.

2.4.2 Qualitative Models

A) The truncated Burgers-Hopf (TBH) model: Galerkin truncation of the inviscidBurgers equation with remarkable turbulent dynamics with features predicted bysimple statistical theory [119, 125, 120]. The models mimic stochastic backscat-ter in a deterministic chaotic system [2].

B) The MMT models of dispersive wave turbulence: One-dimensional models ofwave turbulence with coherent structure, wave radiation, and direct and inverseturbulent cascades [116, 23]. Recent applications to multi-scale stochastic su-perparameterization [66], a novel multi-scale algorithm for state estimation [62],and extreme event prediction [35] are developed.

C) Conceptual dynamical models for turbulence: There are low-dimensional modelscapturing key features of complex turbulent systems such as non-Gaussian in-termittency through energy conserving dyad interactions between the mean andfluctuations in a short self-contained paper [115]. Applications as a test model fornon-Gaussian multi-scale filtering algorithms for state estimation and prediction[91] will be discussed in Section 5.4.

It is very interesting and accessible to develop a rigorous analysis of these modelsand also the above algorithms.


Two-layer baroclinic model

We consider the Phillips model in a barotropic-baroclinic mode formulation withperiodic boundary condition given by

dq

dt= L (q) + B (q,q) ,

with q = (qψ, qτ )T =(∇2ψ,∇2τ − λ2τ

)T, and the quadratic operator

B (q1,q2) = −(

J (ψ1, q2,ψ) + J (τ1, q2,τ )J (ψ1, q2,τ ) + J (τ1, q2,ψ) + ξJ (τ1, q2,τ )

)

as well as the linear operator

L (q) =

(− (1− δ) r∇2

(ψ − a−1τ

)− U ∂

∂x∇2τ − β ∂ψ∂x√δ (1− δ)r∇2

(ψ − a−1τ

)− β ∂τ∂x − U ∂

∂x

(∇2ψ + λ2ψ + ξ∇2τ

))

with U =√δ (1− δ) (U1 − U2), a =

√(1− δ) δ−1, ξ = (1−2δ)/

√δ(1−δ).


With 128 modes meridionally and zonally, corresponding to 256 × 256 × 2 grid points, more than125,000 state variables are needed for numerical integration of this turbulent system!

x

y

mean barotropic stream function <ψ> low latitude

0 1 2 3 4 5 60

1

2

3

4

5

6

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

x

y

mean baroclinic stream function <τ> low latitude

0 1 2 3 4 5 60

1

2

3

4

5

6

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

x

y

mean barotropic stream function <ψ> high latitude

0 1 2 3 4 5 60

1

2

3

4

5

6

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

x

y

mean baroclinic stream function <τ> high latitude

0 1 2 3 4 5 60

1

2

3

4

5

6

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06


Boussinesq equation with gravity waves and vortices72 Chapter 2. Effects of Rotation and Stratification

Summary of Boussinesq Equations

The simple Boussinesq equations are, for an inviscid fluid:

momentum equations:DvDt

+ f × v = −∇φ+ bk, (B.1)

mass conservation: ∇ · v = 0, (B.2)

buoyancy equation:DbDt

= b. (B.3)

A more general form replaces the buoyancy equation by:

thermodynamic equation:DθDt

= θ, (B.4)

salinity equation:DSDt

= S, (B.5)

equation of state: b = b(θ, S,φ). (B.6)

Energy conservation is only assured if b = b(θ, S, z).

2.4.3 Energetics of the Boussinesq system

In a uniform gravitational field but with no other forcing or dissipation, we write the simpleBoussinesq equations as

DvDt

+ 2Ω× v = bk−∇φ, ∇ · v = 0, DbDt

= 0. (2.108a,b,c)

From (2.108a) and (2.108b) the kinetic energy density evolution (cf. section 1.10) is given by

12

Dv2

Dt= bw −∇ · (φv), (2.109)

where the constant reference density ρ0 is omitted. Let us now define the potential Φ ≡ −z,so that ∇Φ = −k and

DΦDt

= ∇ · (vΦ) = −w, (2.110)

and using this and (2.108c) givesDDt(bΦ) = −wb. (2.111)

Adding (2.111) to (2.109) and expanding the material derivative gives

∂∂t

!12v2 + bΦ

"+∇ ·

#v!

12v2 + bΦ +φ

"$= 0. (2.112)

This constitutes an energy equation for the Boussinesq system, and may be compared to(1.186). (Also see problem 2.14.) The energy density (divided by ρ0) is just v2/2+ bΦ. Whatdoes the term bΦ represent? Its integral, multiplied by ρ0, is the potential energy of the

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Qualitative models

2.4 More Rich Examples of Complex Turbulent Dynamical Systems 11

2.4.1 Quantitative Models

A) The truncated turbulent Navier-Stokes equations in two or three space dimen-sions with shear and periodic or channel geometry [143].

B) Two-layer or even multi-layer stratified flows with topography and shears inperiodic, channel geometry or on the sphere [130, 171, 94]. These models in-clude more physics like baroclinic instability for transfer of heat and generalizethe one-layer model discussed in Section 2.1. There has been promising nov-el multiscale methods in two-layer models for the ocean which overcome thecurse of ensemble size for statistical dynamics and state estimation called s-tochastic superparameterization. See [110] for a survey and for the applications[65, 66, 67, 68, 64, 63] for state estimation and filtering. The numerical dynamicsof these stochastic algorithms is a fruitful and important research topic. The endof Chapter 1 of [130] contains the formal relationship of these more complexmodels to the one-layer model in Section 2.1.

C) The rotating and stratified Boussinesq equations with both gravity waves andvortices [124, 171, 94].

There are even more models with clouds and moisture which could be listed. Next isthe list of qualitative models with insight on the central issues for complex turbulentdynamical systems.

2.4.2 Qualitative Models

A) The truncated Burgers-Hopf (TBH) model: Galerkin truncation of the inviscidBurgers equation with remarkable turbulent dynamics with features predicted bysimple statistical theory [119, 125, 120]. The models mimic stochastic backscat-ter in a deterministic chaotic system [2].

B) The MMT models of dispersive wave turbulence: One-dimensional models ofwave turbulence with coherent structure, wave radiation, and direct and inverseturbulent cascades [116, 23]. Recent applications to multi-scale stochastic su-perparameterization [66], a novel multi-scale algorithm for state estimation [62],and extreme event prediction [35] are developed.

C) Conceptual dynamical models for turbulence: There are low-dimensional modelscapturing key features of complex turbulent systems such as non-Gaussian in-termittency through energy conserving dyad interactions between the mean andfluctuations in a short self-contained paper [115]. Applications as a test model fornon-Gaussian multi-scale filtering algorithms for state estimation and prediction[91] will be discussed in Section 5.4.

It is very interesting and accessible to develop a rigorous analysis of these modelsand also the above algorithms.


MMT model

Need Statistically Accurate Inexpensive ForecastModels to Beat the Curse of Ensemble Size for

Prediction, State Estimation and UQ

The MMT equation

The MMT equation (Majda, McLaughlin and Tabak, 1997; Cai and M.M.T., Phys. D2001)

iut = |@x | 12 u + |u|2u iAu + F .

Here we consider the case with the focusing nonlinearity, = 1, which inducesspatially coherent ’solitonic’ excitations at random spatial locations.

I The instability of collapsing solitons radiate energy to large scales producingdirect and inverse turbulent cascades.

I In geophysical applications energy oftern flows from small scales to large scales(inverse cascade) creating a challenge for reduced modelling.

I Fractional dispersion are crucial with completely different behavior from NLSequation!

39 / 51


Visualization of | (x , t)| from simulation with F0 = 0.0163; darker colors indicatehigher amplitudes. Here the number of Fourier modes are 642 4000.

From Cai etal, Physica D 2001.40 / 51


Conceptual dynamical models for turbulenceA hallmark of turbulence is that the large scales can destabilize the smaller scales in fluctuationsintermittently and this increased small scale energy can impact the large scales (Majda & Lee,PNAS, 2014)

du

dt= −d u + γ

K∑

k=1

(u′k)2 − αu3 + F ,

du′kdt

= −dk u′k − γuu′k + σk Wk , 1 ≤ k ≤ K .

Negative large scale damping d = −0.1

Note that u is exactly the critical value of neutral stabilityfrom [5] for the conceptual model. The linear stability matrixat these critical points for [7] has the form

d 3↵ u2

2!! 0

[10]

so these critical points are stable (unstable) if and only if

d 3↵ u2 < 0 (> 0).

To develop guidelines in choosing parameters for the nu-merical experiments for K 2 with the conceptual model in[4], we consider the phase plane analysis in two scenarios withpositive and negative large scale damping. In both cases, theparameter = 1.5 and d = d1 1 are fixed below while for

positive large scale damping, d = 0.01 and ↵ = 0

negative large scale damping, d = 0.1 and ↵ = 0.05.[11]

First consider positive large scale damping; the two critical

points (u, ±u0CR) occur for F < F CR =

dd

= 0.0067

and are both stable by the criterion in [10] while the crit-

ical point (F

d, 0) along the u-axis is unstable to u0 pertur-

bation provided F < F CR. Since the energy is a Lyapunovfunction for [7], trajectories o↵ the u-axis converge to ei-ther of the critical points (u, ±u0

CR) with u the marginallystable value; thus we can expect more turbulent behavior inthe conceptual stochastic models with K 2 as the forcingF increases in magnitude through negative values, F withF F CR = 0.0067. A similar scenario occurs for the casewith negative damping in [7] for F 0.0545 with a singlecritical point along the u-axis which is unstable to pertur-bations in u0 with two critical points (u, ±u0

CR), u0CR 6= 0,

which are also unstable because d 3↵ u2 > 0; in this casewith all three equilibrium points unstable, trajectories o↵ theu-axis necessarily converge to periodic orbits encircling thecritical points (u, ±u0

CR) and frequently visit values of u withinstability in the u0 dynamics. We also anticipate di↵erent be-havior for F > 0.0545 since a stable critical point appearsat u = 0.8329 for this and larger values of F . See the tablesin the supplementary material.

Numerical Experiments for K=5 in the Conceptual

ModelHere we use simple numerical experiments to demonstratethat the six dimensional conceptual model in [4] with K = 5has all the statistical features listed in (A)-(D) including in-

termittency of the small scales. The parameters, d,↵, and = 1.5 have already been discussed in [11]. The dampingcoecients dk are a mixture of uniform and scale selectivedamping with dk = 1 + 0.02k2 for k = 1, 2, ..., 5 so that thesmaller scales are damped more rapidly; the noise level set byk for the k-th mode is determined by

2k

dk=

0.004

(1 + k)5/3, k = 1, 2, ..., 5 [12]

so that a -5/3 spectrum is calibrated to occur for these modesprovided u 0 in the equations for u0

k [8]. This specifiesall parameters in the conceptual model for turbulence usedhere. For all numerical simulations below and in the supple-mentary material, the Euler-Maruyama method is used witha time step t = 5 103 and the system is integrated fora long time T = 2 105 with the first t = 2 103 time dataignored for post processing the climatological statistics. In

all simulations the initial value is u = 1.5 with u0k = 0 for

k = 1, 2, ..., 5.

2000 2200 2400 2600 2800 3000−2

−1

0

1

u

2000 2200 2400 2600 2800 3000−1.5

−1

−0.5

0

u

−1 −0.5 0

10−2

100

u

2000 2200 2400 2600 2800 3000−1

0

1

u′ 1

−0.5 0 0.5

10−4

10−2

100

u′

1

2000 2200 2400 2600 2800 3000

−0.5

0

0.5

u′ 2

−0.5 0 0.5

10−4

10−2

100

u′

2

2000 2200 2400 2600 2800 3000−0.5

0

0.5

u′ 3

−0.5 0 0.5

10−4

10−2

100

u′

3

2000 2200 2400 2600 2800 3000−0.5

0

0.5

u′ 4

−0.5 0 0.5

10−4

10−2

100

u′

4

−0.2 −0.1 0 0.1 0.210

−4

10−2

100

u′

5

2000 2200 2400 2600 2800 3000−0.2

0

0.2

time

u′ 5

−2 −1.5 −1 −0.5 0 0.5

10−4

10−2

100

u

Fig. 1. Negative large scale damping : time series (left column) and pdfs (right

column) of the turbulent signal u, u and u0k, k = 1, 2, ..., 5 with F = 0.055.

Note the logarithmic scale of pdfs in the y-axis. Dashed lines are Gaussian distribu-

tions with the same mean and variance.

First we consider the case with large scale instability foru with negative damping, d = 0.1 and ↵ = 0.05 with theforcing value F = 0.055 motivated by the phase portraitanalysis above. Fig. 1 depicts the pdfs for the total turbulentfield u, the large-scale mean u, and the turbulent fluctuationsu0

k, k = 1, 2, ..., 5 as well as a sample of the time series of eachvariable in the conceptual model; the pdfs are plotted with alogarithmic vertical coordinate in order to highlight fat tails ofintermittency while the Gaussian distribution with the samevariance is the parabola in the figure. The pdf for the over-all turbulent field u in [2] is nearly Gaussian while the pdfsfor the mean u and the largest scale fluctuating mode, u0

1,are both slightly sub-Gaussian. The variable u0

2 has a Gaus-sian tail while the variables u0

3, u04, u

05 all have significant fat

tails which are a hallmark of intermittency; the time seriesfor u0

3, u04, u

05 in Fig. 1 clearly display highly intermittent be-

havior of extreme values with the amplitude of u03 occasionally

spiking to the typical amplitude of u01 even through the clima-

tological variance of u03 is nearly eight times smaller than that

for u01 (see Table 2 of supplementary material). The climato-

logical mean value for u is 0.6733 = hui and hui is very close

to the marginal stability value u = 0.6667 = d

motivated

from [7] while the standard deviation of u is 0.1993 indicating

Footline Author PNAS Issue Date Volume Issue Number 3

Positive large scale damping d = 0.01

that the instability mechanism elucidated in [5] is operatingon all modes and creating intermittency. The total energy ofthe mean flow u exceeds that of the fluctuations, u0

k. Thevariables u0

k, k = 1, 2, ..., 5 have essentially zero means withvariances 0.0446, 0.0174, 0.0049, 0.0014, and 0.0005 respec-tively with the correlation time for u approximately 34 whilethose for u0

k, k = 1, 2, ..., 5 are decreasing with k and approxi-mately 29,16,6,4, and 3 respectively. These are all the featuresof anisotropic turbulence required from (A)-(D) and demon-strated in the conceptual dynamical models; furthermore allof these conditions occur in a robust fashion for F increasingin magnitude with F 0.055 and 0.055 |F | 0.1. Allof the detailed data discussed above can be found in Tables1-3 of the supplementary material. There is an evident rolefor the unstable damping of the large scales, d = 0.1 to in-crease the variance of u with its mean near the marginallycritical value u so that the instability mechanism from [5]operates vigorously in the model and creates more variancein u0

k, k = 1, 2, ..., 5. Thus, we expect the system with stabledamping and the same values of F with F = 0.055 to haveless variance.

2000 2200 2400 2600 2800 3000−2

−1

0

1

u

−1.5 −1 −0.5 0 0.5

10−4

10−2

100

u

2000 2200 2400 2600 2800 3000−1.5

−1

−0.5

0

u

−1.2 −1 −0.8 −0.6 −0.4 −0.2

10−4

10−2

100

u

2000 2200 2400 2600 2800 3000

−0.5

0

0.5

u′ 1

−0.5 0 0.5

10−4

10−2

100

u′

1

2000 2200 2400 2600 2800 3000−0.5

0

0.5

u′ 2

−0.5 0 0.5

10−4

10−2

100

u′

2

2000 2200 2400 2600 2800 3000−0.5

0

0.5

u′ 3

−0.5 0 0.5

10−4

10−2

100

u′

3

2000 2200 2400 2600 2800 3000−0.2

0

0.2

u′ 4

−0.4 −0.2 0 0.2 0.4

10−4

10−2

100

u′

4

2000 2200 2400 2600 2800 3000−0.1

0

0.1

time

u′ 5

−0.2 −0.1 0 0.1 0.210

−4

10−2

100

u′

5

Fig. 2. Positive large scale damping : time series (left column) and pdfs (right

column) of the turbulent signal u, u and u0k, k = 1, 2, ..., 5 with F = 0.080.

Note the logarithmic scale of pdfs in the y-axis. Dashed lines are Gaussian distribu-

tions with the same mean and variance.

We consider the case with positive large scale damping,d = 0.01, for F = 0.080 and in Fig. 2 we show the pdfsof all variables as well as a piece of the time series of theturbulent signal u, u, and u0

k, k = 1, 2, ..., 5. The intermit-tency of the small scale modes with less variance is evident

in Fig. 2. The mean flow variable, u, has the largest to-tal energy with climatological mean hui = 0.6853 which isvery close to the marginal critical values u = 0.6667 so theintermittent instability mechanism in [5] is operating onceagain. Both the variances and correlation times behave in asimilar fashion as for the negative large scale damping casediscussed above and as required in (A)-(D) so the conceptualmodel with positive large scale damping also is a qualitativedynamical model for anisotropic turbulence with all the fea-tures in (A)-(D). Furthermore, all of these features persist for

F with 0.055 F 0.1; the pdfs are all Gaussian withno fat tails for F with suciently small absolute value suchas F = 0.01 as shown in the supplementary material. Asexpected from our discussion of the unstable case; for fixedforcing with F 0.055, there is between a factor of two andthree less variance in all variables in the positive large scaledamping case compared with the negative large scale dampingcase. Documentation for all of the above claims is found in ex-tensive tables in the supplementary material. For both casescross correlation among the variables u, u0

k, k = 1, 2, ..., 5 arenegligible in the climatological mean state with values roughlyless than the 5% level.

In the above paragraphs, we emphasized models with K =5 to mimic the many degrees of freedom in real anisotropicturbulence and their interaction with the mean flow. Froma mathematical viewpoint, it is interesting to address the fol-lowing: what is the lowest dimensional conceptual model withintermittency and satisfying all the requirements in (A)-(D)?Versions of the conceptual model with K = 2 already exhibitintermittency in u0

2 as well as all the other features required in(A)-(D) for both positive and negative damping as shown inthe supplementary material. However, the two mode modelswith K = 1 always exhibit either sub-Gaussian or at mostGaussian behavior in u0

1 without intermittency as the noiselevel is varied in all of our numerical experiments.

Concluding DiscussionConceptual dynamical models for anisotropic turbulence havebeen introduced here which, despite their simplicity, capturekey features of vastly more complicated systems. The concep-tual dynamical models introduced here in [4] involve a largescale mean flow u and turbulent fluctuations, u0

k, 1 k K,on variety of spatial scales and involve energy conservingwave-mean flow interactions as well as suitable degeneratestochastic forcing of the fluctuations u0

k. The models havea transparent mechanism where the mean flow, u, can desta-bilize the k-th mode whenever dk + u < 0; a phase planeanalysis yields parameters and robust regimes of sucientlystrong large scale external forcing, F , where the models havea climatological mean state hui which is nearly neutrally sta-ble in the sense that d1 + hui = 0 so that fluctuations inthe mean u often introduce intermittent instability. Numer-ical experiments with a six-dimensional version of the modelsummarized here and in the supplementary material confirmthat it captures key statistical features of vastly more complexanisotropic turbulent systems. These include chaotic statisti-cal behavior of the mean flow, u, with a sub-Gaussian pdf forits fluctuations while the turbulent fluctuations, u0

k, 1 k 5,have decreasing energy and correlation times as k increaseswith nearly Gaussian pdfs for the large scale fluctuations andfat-tailed non-Gaussian pdfs for the smaller scale fluctuations;this last feature allows for intermittency of the small scale fluc-tuations where turbulent modes with small variance can haverelatively frequent large amplitude extreme events which di-rectly impact the mean flow, u. Remarkably, vastly more com-plex realistic turbulent systems often exhibit such marginal

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