0wfswjfxpo6odfsubjouz2vboujgjdbujpopg ...qidi/turbulence19/lecture1.pdf3.3 mathematical formulation...

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0WFSWJFX PO 6ODFSUBJOUZ 2VBOUJGJDBUJPO PG$PNQMFY 5VSCVMFOU TZTUFNT

Andrew J. Majda

Courant Institute of Mathematical Sciences

Fall 2019 Advanced Topics in Applied Math

"OESFX +� .BKEB $*.4 5VSCVMFOU %ZOBNJDBM 4ZTUFNT 4FQU� ��

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5VSCVMFOU %ZOBNJDBM 4ZTUFNT

Turbulent dynamical systems are ubiquitous complex systems ingeoscience and engineering and are characterized by a large dimensionalphase space and a large dimension of strong instabilities which transferenergy throughout the system between different spatiotemporal scales.

B Red spot on Jupiter C MJO D QG


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(SBOE $IBMMFOHF JO &YUSFNFMZ $PNQMFY 4ZTUFNImportant societal impacts: predicting long range weather forecasting(intraseasonal to interannual) and short term (decadal) climate change.

Turbulent dynamical system: huge phase space and large dimension ofinstabilities.

Other examples: engineering turbulence, neural science, material science.

Need statistical, stochastic, thinking combined with nonlinear dynamics ideas.

Central Applied Math/Science Issues:� Accurate prediction and representation of suitable statistical mechanism for

observations from nature.� Model error: lack of physical understanding and inadequate resolution due to

curse of ensemble size, computational overload in generating even smallnumber of ensemble members is overwhelming.

� Uncertainty quantification (UQ) accurate bounds for 1) and 2).� Low order models which achieve 1 ) and 3) while coping with 2) in an optimal

fashion.� Information-theoretical framework for measuring imperfect model error in

complex system.


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Modern Applied Math ParadigmModern Applied Math Paradigm

Rigorous Math Theory

Qualitative orQuantitativeModels

Novel NumericalAlgorithm

Crucial Improved Understanding ofComplex System

3 / 51"OESFX +� .BKEB $*.4 5VSCVMFOU %ZOBNJDBM 4ZTUFNT 4FQU� ��

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$PVSTF 0VUMJOF GPS 1BSU *�

Barotropic geophysical flows and two-dimensional fluid flows

The response to large-scale forcing

Nonlinear stability of steady geophysical flows

Introduction to information theory and empirical statistical theory

Equilibrium statistical mechanics for systems of ODEs

Statistical mechanics for the truncated quasi-geostrophic equations


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.BKPS $PVSTF 3FGFSFODFT�

Majda & Wang, Nonlinear Dynamics andStatistical Theories for Basic GeophysicalFlow (majorly Chapter 1, 2, 4, 6, 7, 8),Cambridge Press 2006

Majda, Introduction to TurbulentDynamical Systems for Complex Systems(85 pages), Frontiers in AppliedDynamical Systems, Springer 2016

Papers Available from Prof. Majda’swebsite


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#PPL $POUFOUT� $IBQUFS � � �

Contents

Preface page xi

1 Barotropic geophysical flows and two-dimensional fluid flows:elementary introduction 1

1.1 Introduction 11.2 Some special exact solutions 81.3 Conserved quantities 331.4 Barotropic geophysical flows in a channel domain – an important

physical model 441.5 Variational derivatives and an optimization principle for

elementary geophysical solutions 501.6 More equations for geophysical flows 52

References 58

2 The response to large-scale forcing 592.1 Introduction 592.2 Non-linear stability with Kolmogorov forcing 622.3 Stability of flows with generalized Kolmogorov forcing 76

References 79

3 The selective decay principle for basic geophysical flows 803.1 Introduction 803.2 Selective decay states and their invariance 823.3 Mathematical formulation of the selective decay principle 843.4 Energy–enstrophy decay 863.5 Bounds on the Dirichlet quotient, !"t# 883.6 Rigorous theory for selective decay 903.7 Numerical experiments demonstrating facets of selective decay 95

References 102

v


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#BSPUSPQJD HFPQIZTJDBM GMPXT BOE UXP�EJNFOTJPOBMGMVJE GMPXT� FMFNFOUBSZ JOUSPEVDUJPO

Di Qi, and Andrew J. Majda

Courant Institute of Mathematical Sciences

Fall 2019 Advanced Topics in Applied Math

%J 2J BOE "OESFX +� .BKEB $*.4 5VSCVMFOU %ZOBNJDBM 4ZTUFNT 4FQU� ��

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-FDUVSF � � �� &MFNFOUBSZ *OUSPEVDUJPO

� *OUSPEVDUJPO

� 4PNF TQFDJBM FYBDU TPMVUJPOTBeta-effect and generalized Kolmogorov forcingRossby wavesTopographic effectLarge-scale shear flow

� $POTFSWFE RVBOUJUJFTConservation of energy and enstrophyVariational derivatives and an optimization principle

� .PSF FRVBUJPOT GPS HFPQIZTJDBM GMPXT


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0VUMJOF

� *OUSPEVDUJPO





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%ZOBNJDBM QSPDFTTFT JOWPMWFE JO UIF EFTDSJQUJPO PGHFPQIZTJDBM GMPXT

Physical variables:! the velocity, the pressure, the density;! in addition, the humidity in the case of atmospheric motions or the

salinity in the case of oceanic motions.

Physical processes that determine the evolution of the geophysicalflows:

! the Coriolis force due to the earth’s rotation;! the sun’s radiation;! the topographical barriers, e.g. mountain ranges in the case of

atmospheric flows and the ocean floor and the continental masses in thecase of oceanic flows;

! The dissipative energy mechanisms, for example due to eddy diffusivityor Ekman drag;

Spatial and temporal scales:! The space scales: from a few hundred meters to thousands of kilometers.! The time scales: from as short as minutes to as long as days, months, or

even years."OESFX +� .BKEB $*.4 5VSCVMFOU %ZOBNJDBM 4ZTUFNT 4FQU� ��

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The simplest set of equations that meaningfully describes the motion oftwo-dimensional geophysical flows under these circumstances is given bythe:

#BSPUSPQJD 2VBTJ�(FPTUSPQIJD &RVBUJPOT�Dq

Dt=∂q

∂t+ p ·∇q = D (∆)ψ+ F (t, t)

where

q = ω+ βy+ h (x, y) , ω = ∆ψ, p = (−∂yψ,∂xψ)T .

q: the potential vorticity; ψ: the stream function;

p = ∇⊥ψ: the horizontal velocity field; ω = ∆ψ: the relative vorticity;

βy: the beta-plane effect from the Coriolis force;

h (t): the bottom floor topography, given by ocean floor or mountainrange;

D (∆): various possible dissipation mechanisms

F (t, t): additional external forcing."OESFX +� .BKEB $*.4 5VSCVMFOU %ZOBNJDBM 4ZTUFNT 4FQU� ��

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$IPJDFT PG EJTTJQBUJPO PQFSBUPST

Newtonian (eddy) viscosity: D (∆)ψ = ν∆2ψ

Ekman drag dissipation: D (∆)ψ = −d∆ψ

Hyper-viscosity dissipation: D (∆)ψ = (−1)j dj∆jψ

Ekman drag dissipation + Hyper-viscosity:D (∆)ψ = −d∆ψ+ (−1)j dj∆

jψ

Radiative damping: D (∆)ψ = dψ

General dissipation operator: D (∆)ψ =∑l

j=0 (−1)j dj∆jψ

(FOFSBMJ[FE ,PMNPHPSPW GPSDJOH

F (t, t) =∑

|F|=Λ

FF (t) eit·F + c.c.


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5IF PSJHJO PG UIF CBSPUSPQJD RVBTJ�HFPTUSPQIJDFRVBUJPOT

Barotropic Rotational Equations:

! The slow mode of propagation corresponds to themotion of the bulk of the fluid by advection (theweather patterns in the atmosphere, evolving on atime scale of days)

! The fast mode of propagation corresponds to gravitywaves (do not contribute to the bulk motion of thefluid, evolving on a short time scale of the order ofseveral minutes)

The barotropic quasi-geostrophic equations: theresult of “filtering out” the fast gravity waves fromthe rotating barotropic equations.

Charney (1949) first formulated thequasi-geostrophic equations, and thus opened themodern era of numerical weather prediction(Charney 1949, and von Neumann, 1950).


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5XP�EJNFOTJPOBM DMBTTJDBM GMVJE GMPX FRVBUJPOT

Dω

Dt= ν∆ω+ F (t, t) , ω = ∆ψ, p = ∇⊥ψ

and in the case without dissipation we have the classical Euler equationswith forcing

Dω

Dt= F (t, t) , ω = ∆ψ, p = ∇⊥ψ

One of our objectives of this course is to compare and contrast thebarotropic quasi-geostrophic equations and the Navier–Stokes equationsto better understand the role of the beta-plane effect and the topographyon the behavior of geophysical flows.


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" MJTU PG UIF UPQJDT UIBU XF XJMM TUVEZ JO TVCTFRVFOUMFDUVSFT

J Exact solutions showing interesting physics

JJ Conserved quantities

JJJ Response to large-scale forcing

JW Large- and small-scale interaction via topographic stress.

W Non-linear stability of certain steady geophysical flows

WJ Equilibrium statistics mechanics and statistical theories forlarge coherent structures

WJJ Dynamic modeling for geophysical flows

WJJJ How information theory can be used to quantifypredictability for ensemble predictions in geophysical flows


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0VUMJOF

� *OUSPEVDUJPO





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4PNF TQFDJBM FYBDU TPMVUJPOT

In general it is far from easy to find exact solutions of the barotropicquasi-geostrophic equations due to the non-linear character through thenonlinear advection term in

Dq

Dt=∂q

∂t+ p ·∇q, p ·∇q = ∇⊥ψ ·∇q = J (ψ, q) .

The vanishing of the Jacobian determinant J (ψ, q) certainly happens if thepotential vorticity q and the stream function ψ are functionally dependent,i.e. assuming

F (ψ) = q = ∆ψ+ βy+ h.


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3FEVDFE MJOFBS TZTUFN GPS UIF TUSFBN GVODUJPO ψ

Here we concentrate on finding special exact solutions with both forcingand dissipation with the stronger ansatz: we assume that q and ψ arelinearly dependent

q = µψ = ∆ψ+ βy+ h

The the solution of the barotropic quasi-geostrophic equations under thelinear dependence assumption

µ∂ψ

∂t= D (∆)ψ+ F (t, t) ,

µψ = ∆ψ+ βy+ h (t) .


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&YBDU TPMVUJPOT GPS UIF &VMFS FRVBUJPOT

In the absence of physical effects from beta-plane, topography, dissipation,and external forcing,

µ∂ψ

∂t= 0, µψ = ∆ψ.

so that the stream function ψ is steady and corresponds to aneigenfunction of the Laplacian with eigenvalue µ.

The most general form of the associated eigenfunction ψj with eigenvalueµ = −Λj = − |F|2 is

ψj =∑

|F|2=Λj

AFeiF·t + c.c.


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Example 1: Ground stateeigenmode flow |F|2 = Λ1 = 1

ψ =a bBM (x) + b bBM (y)

+ c +Qb (x) + c +Qb (y)

14 Barotropic geophysical flows and two-dimensional fluid flows

y

x

v = (0, ν(x))

ψ

ψ

x

ψψ

ψ = ψ(x)

stre

amlin

e

stre

amlin

e

stre

amlin

e

Figure 1.1 Example of a simple shear flow with stream function ! = A sin"x#and velocity field v = "0$A cos"x##. Notice that the fluid velocity is parallel tothe streamlines.

–1 –0.5 0 0.5 1–1

–0.5

0

0.5

1

x

y

–1

0–0.5

0.51

–1–0.5

00.5

1–2

–1

0

1

2

x

y

ω

(a) (b)

Figure 1.2 2-D array of swirling eddies with stream function !"x$y# = sin"x#+sin"y# (x and y both normalized by 2%). (a) Plot of the streamlines: counter-clockwise rotation (—); clockwise rotation (- -). (b) Plot of the vorticity &.'JHVSF� 2-D array of swirling eddies

Example 2: Taylor vortices and flowwith symmetries |F|2 = Λ2 = 2

ψ =a bBM (x) bBM (y) + b bBM (x) +Qb (y)+ c +Qb (x) bBM (y) + d +Qb (x) +Qb (y) .


–1

–0.5

0

0.5

1

y

–1

–0.5

0

0.5

1

–1–0.5

00.5

1

x–1–0.5

00.5

1

y

ω

–1 –0.5 0 0.5 1x

(a) (b)

Figure 1.4 Taylor vortices with stream function !"x#y$ = sin"x$ sin"y$ (x and yboth normalized by 2%). (a) Plot of the streamlines: counter-clockwise rotation(—); clockwise rotation (- -). (b) Plot of the vorticity &.

across the lines given by C = 'x"!"x$ = 0(, which include the lines x = % andy = %. In particular, for the lines x = % and y = % it follows that

!"% −x#y$ = −!"% +x#y$# !"x#% −y$ = −!"x#% +y$)

Clearly, the Taylor vortices satisfy the no-penetration boundary condition on thelevel set C = 'x"!"x$ = 0(

v · n∣∣C

= * ⊥ ! · *!

"*!"

∣∣∣∣C

= 0)

In addition, the vorticity & = −+2! also vanishes on the set C. This implies thatthe Taylor vortices also satisfy the free-slip boundary condition

,

,n"v · -$

∣∣∣∣C

= 0#

where - is the tangent vector to C.Finally, we also remark that additional flows satisfying the symmetry conditions

are also obtained when the stream function ! is of the form

! =%∑

k1#k2=−%ak1k2

sin"k1x$ sin"k2y$)

These flows with symmetries include as a particular case the Taylor vortices andfor that reason they are called generalized Taylor vortices.

'JHVSF� Taylor vortices


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&YBDU TPMVUJPOT XJUI CFUB�FGGFDU BOE GPSDJOHThe small and large scale of stream functions ψ = −V0y+ψ′ (x, y, t)

µ∂ψ′

∂t= D (∆)ψ′ + F (t, t) ,

µψ′ = ∆ψ′, V0 = −β/µ.

The generalized Kolmogorov forcing and corresponding stream function

F (t, t) =∑

|F|2=Λ

FF (t) eit·F, ψ′ =

∑

|F|2=Λ

AF (t) eit·F + c.c.

Summarizing, the stream function ψ is given by

ψ = −V0β+∑

|F|2=Λ

AF (t) eit·F + c.c.

with the Fourier amplitude

dAFd

=D (−Λ)

−ΛAF +

FF−Λ

.


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3PTTCZ XBWFTThe potential vorticity equation for q with large-scale mean V0

∂q

∂t+ V0

∂q

∂x+ J (ψ′, q) = D (∆)ψ′ + F (t, t) .

The constant mean flow vorticity V0 is easily removed with a Galileancoordinate transform

x = x − V0t, y = y, t = t

→ ∂

∂x=∂

∂x,∂

∂y=∂

∂y,∂

∂t=∂

∂t− V0

∂

∂x.

#BSPUSPQJD CFUB�QMBOF FRVBUJPOT

∂q

∂t+ J (ψ′, q) = D (∆)ψ′ + F,

q = ∆ψ′ + βy.

ψ′ (t, t)=

∑

|F|2=Λ

AF (t) ei(F·�t+k1V0t) + c.c

=∑

|F|2=Λ

AF (t) ei(

F·�t+k1β

|F|2t)

+ c.c


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Phase speed ω(F)k1

= − β

|F|2

Group speed ∂ω(F)∂k1

=(k21−k22)β

|F|4


Since we know that the small-scale solution !′ in the previous example wasgiven by equation (1.26), we can write this solution in terms of the new coordi-nates. Keeping in mind that the mean velocity V0 = − "/# = "/$, we obtain

!′%x& t' =∑

$k$2=$

Akei%k·x+k1V0 t' +c(c

=∑

$k$2=$

Ake

i%k·x+k1"

$k$2 t' +c(c(& (1.31)

and we recognize that !′ is a traveling wave of the form eik·x− i)%k't, where ) andk satisfy the dispersion relation

)%k' = − k1"

$k$2( (1.32)

This solution represents a dispersive wave known as a Rossby wave (Pedlosky,1987). Indeed these waves travel horizontally at the speed (the so-called horizontal

phase speed) of )%k'k1

= − "

$k$2 . Thus waves with different wave numbers travel at

different speeds and hence are dispersive. See the lecture notes by Majda (2003)for an extensive discussion of dispersive waves in geophysical flows.

In the absence of dissipation and forcing it is interesting to see how the ele-mentary exact solutions of the Euler equations in Subsection 1.2.1 are modifiedby the beta-effect. They become time-dependent solutions with different phasevelocities. Indeed it is easy to see that we have the following:

1. The simple shear in x, ! = A sin%x', as presented in Example 1 of Subsection 1.2.1,becomes a Rossby wave ! = A sin%x+"t' with horizontal phase velocity = − ".

2. General ground eigenmode flows, ! = a sin%x' + b cos%x' + csin%y' + d cos%y', aspresented in Example 1, become the super-position of Rossby waves and zonal shearflows ! = a sin%x + "t' + b cos%x + "t' + csin%y' + d cos%y' with horizontal phasevelocity = − ".

3. Taylor vortex, ! = A sin%x' sin%y', as presented in Example 2 of Subsection 1.2.1becomes a Rossby wave ! = A sin%x+ "

2 t' sin%y' with horizontal phase velocity = − "2 .

We observe that waves with different wave numbers move at different speeds. Infact, waves with higher wave numbers move slower than waves with lower wavenumbers in general. This is a feature for geophysical waves, which is differentfrom most of the classical dispersive waves where waves with higher wave numbermove faster (Majda, 2003).

Let us verify that the Rossby waves are indeed dispersive waves from themathematical point of view. This is equivalent to checking that the determinant ofthe Hessian of the dispersive relation does not vanish except for a set of measure


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5PQPHSBQIJD FGGFDU JO TUFBEZ TUBUFT

Consider topography and beta-effect

µ∂ψ

∂t= 0, µψ = ∆ψ+ βy+ h.

V0 = −β

µ, µψ′ = ∆ψ′ + h (t) .

Assume periodic topography and stream function

h =∑

|F| =0

hFeit·F + c.c., ψ′ =

∑

|F| =0

ψ′Fe

it·F + c.c.

In this case we can solve for the Fourier coefficients and obtain the streamfunction

ψ = −V0y+ψ′ =β

µy+

∑

|F| =0

1

|F|2 + µhFe

it·F.


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EiQi�H =1

2V20 + EbK�HH b+�H2 =

β2

2µ2+

1

2

∑

|F| =0

|F|2∣∣∣hF

∣∣∣2

(|F|2 + µ

)2.

1.2 Some special exact solutions 23

E

0–1–2 µ

Figure 1.5 Plot of the total energy E = 12

∑

!k!=0

!2"#2!k!2!hk!2

!!k!2 +$#2versus $ in the

case of no beta-plane effect (and no mean flow V0). The singularities are givenby the eigenvalues of the Laplacian, $ = − 1% − 2% − 5% & & &

These are somehow robust states. In a later chapter we will see that these statesare non-linearly stable. In other words, they are typically observed states withoutmean flow.

Next we consider the case ' = 0. In this case there is a mean flow contributiongiven by V0 = − '/$, so that the total kinetic energy has contributions from thelarge-scale mean flow and the small-scale flow. From the Fourier representationof the velocity field v in equation (1.28), it follows that the total kinetic energyE is given by

Etot = 12

V 20 !2"#2 +Esmall scale (1.42)

=

⎛

⎝'2

2$2 + 12

∑

!k!=0

!k!2!hk!2

!!k!2 +$#2

⎞

⎠!2"#2&

A plot of the kinetic energy E versus $ for this case is given in Figure 1.6. Inthis case the presence of the beta-plane effect introduces through the presence ofthe mean flow an additional singularity at $ = 0, so that in this case the energy Egoes to infinity as $ approaches zero, while in the range 0 < $ < %. Again weexpect that the steady state solutions with $ > 0 will be stable. Interestingly, when' > 0 these steady states have negative mean flow velocity, V0 = − '/$ < 0,and are associated with westward mean flow. This is in marked contrast with theearlier example, where there was beta-plane effect but no topography present, and

B β = 0


µ

E

–2 –1 0

Figure 1.6 Plot of the total energy E =

⎛

⎝ !2

2"2+ 1

2

∑

!k!=0

!k!2!hk!2

#!k!2 +"$2

⎞

⎠#2%$2 ver-

sus " in the case of beta-plane effect (and non-zero mean flow V0). The singu-larities are " = 0& − 1& − 2& − 5& ' ' '

the resulting flow had an eastward mean flow. Nevertheless, this conclusion isconsistent with the observations of flows in the ocean. The flow in the oceans ismuch slower than in the atmosphere and the effects due to the topography aremore important. Indeed, in many cases the flow will be westward because of thetopographic effect.

Next we would like to discuss the effect that the topography has on the vorticity.For example, we can wonder whether a rise in the terrain will increase or decreasethe rotation of the flow? It turns out that the vorticity of the flow decreases as itclimbs up the terrain and increases as it climbs down. If we ignore the dissipationand external forcing effects, then the equations for the potential vorticity q in(1.1) reduce to

(q

(t+ v ·)q = 0& q = *+!y +h'

Thus q should be conserved along the particle trajectory X#+& t$, which satisfies

dX#+& t$

dt= v#X#+& t$& t$'

That is

q#X#+& t$& t$ = q0#X#+& 0$$'

C β = 0


0wfswjfxpo6odfsubjouz2vboujgjdbujpopg ...qidi/turbulence19/lecture1.pdf3.3 mathematical formulation...

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