Statistical mechanics and numerical modelling of geophysical fluid dynamics - Chapter 1: Introduction

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Statistical mechanics and numerical modelling of geophysical fluid dynamics

Dubinkina, S.B.

Publication date

2010

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Dubinkina, S. B. (2010). Statistical mechanics and numerical modelling of geophysical fluid

dynamics.

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Chapter 1

Introduction

In numeri al simulations of limate variability, the ow of atmosphere and o eansisknowntobe haoti ,andthereforethequantityofparti ularinterest isnotasinglesolutiontraje tory,butratheranensembleofpossiblemotionsof thesystemgiven ertaininformationaboutthebehavior ofthissystem. Then statisti alme hani sbe omesuseful,sin eitdealswiththeprobabilityofa sys-tembeinginsomestateunderknown onstraints. Toderiveastatisti altheory for a system, the system should be volume preserving and possess onserved quantities. Ergodi ityisusuallytakenas anassumptionto relatedynami s to statisti al theory, and thevalidityof this assumptionmust be onrmed with numeri al experiments. Although a ontinuous model has onservation laws, it isnot learthat after implementinga numeri al method, the orresponding dis retemodelwillalsohaveanequivalen eofthem. Thisentirelydependson propertiesof a numeri al method. Thereforethe statisti al resultsof the dis- retemodelmaydepend(andtheya tuallydo)onthenumeri almethodused. And in an ample range of available numeri al methods, it therefore be omes even more ru ial and pronoun ed to understand how the numeri al method inuen esthestatisti alme hani sof asimulation.

Arakawa'ss hemes for 2D in ompressible ow area onvenient subje t to investigatetheimportan eof onservation be ausetheysatisfyoneor bothof a pair of onservation laws, and they are onstru ted from identi al dis rete operators, dieringonly in the order of appli ation of these. If the operators would ommuteinthedis rete aseastheydointhe ontinuousthes hemes would be identi al. Nonetheless we observe ompletely dierent equilibrium behavior for these methods, see Chapter 2, and for the introdu tion to the statisti al me hani sseeSe tion1.3.

Symple ti or Poisson dis retizations for Hamiltonian partial dierential equations are onstru ted in su h a way that the semi-dis rete systems are again Hamiltonian and possess an equivalen e of onserved quantities of the original ontinuousHamiltoniansystems. Be auseHamiltonianstru tureplays animportantroleinstatisti s,wemightexpe tthesemethodstoperformbetter thanstandardmethodsintermsofstatisti alperforman emeasures. This mo-tivatedus toderiveinthethesisstatisti altheoriesforaHamiltonian parti le-meshmethodin the aseofquasigeostrophi owwithtopographi for ing,see

Se tions 1.1and1.2.

Anotherattra tiveemploymentofstatisti al me hani sis themodelling of sub-grids ale losureof invis id uidequations, sin esu h ows,whi hmodel essential omponentsof atmosphereand o ean, support a vorti ity as ade to ever ner s ales. Any nitedis retization ofthese equationsmustbe ome un-derresolved,and we an useequilibrium statisti al me hani stheory to repre-sent the subgrid motions. In Chapter 4, we show for a point vortex ow on a dis that this an be parameterized byusing a mathemati al thermostat,a te hniquefrommole ulardynami swhi hallowstomodela systemin onta t withareservoirofnes alevorti ity. InSe tion1.3,weexplainstatisti al me- hani sin general, on ept ofmathemati al thermostat, and dene statisti al temperature.

1.1

Hamiltonian systems and geometric

integra-tion

The ontinuum models studied in this thesis have a ommon mathemati al stru ture: thatofHamiltoniansystems. Hamiltonianstru tureisalsothepoint of departureforstatisti al me hani s. WethereforereviewHamiltonian stru -tureforordinaryandpartialdierentialequationsinthisse tion,andwedis uss geometri integratorsnumeri almethodsthatpreserveHamiltonianstru ture. A geometri integrator isa numeri almethod whi h preservessome stru tural properties of a given problem, e.g. symple ti ity, rst integrals, symmetries. Hamiltonian systems in turn are ri h in su h properties. Thereforethis moti-vatesinterestingeometri integrationofHamiltoniansystems.

1.1.1

Hamiltonian systems

Inthefollowingse tionwe onsider Hamilton'sprin iplefornite-dimensional me hani alsystemsfollowingthederivationsdes ribedin[3 ,42 ℄. Inthe ontext ofgeophysi aluiddynami ssee also[57 ,73,74,82℄.

The canonical equation

Consider a me hani al system with

d

degrees of freedom. Its state an be spe iedbythegeneralized oordinates

q

= (q

1

, . . . , q

d

)

T

asfun tionsoftime

t

. Thedynami softhesystemaredeterminedbytheLagrangian

L

,thedieren e between the kineti and potential energies. The Lagrangian

L = L(q, ˙q)

is a fun tion of the oordinates

q

and thevelo ities

˙q

, where the dots on

q

stand for dierentiation with respe t to time. The evolution of the system may be determinedfrom Hamilton'sprin iple[42℄

δ

Z

t

0

0

1.1. Hamiltonian systems and geometric integration

3

forarbitraryvariations

δq

k

thatvanishat

t = 0

and

t = t

0

. Fromthisvariational problem(1.1)followsLagrange'sequationsofthesystem

d

dt

µ ∂L

∂ ˙q

¶

=

∂L

∂q

.

(1.2)

Denethe onjugatemomenta as

p

k

=

∂L

∂ ˙

q

k

(q, ˙q),

k = 1, . . . , d

(1.3)

andthe

Hamiltonian

H

_{≡ p}

T

_˙q

− L(q, ˙q)

asafun tionof

q

and

p

. ThisistheLegendretransformationof

L

withrespe t to

˙q

. Undertheassumptionthat

˙q

maybeexpressedasa fun tionof

q

and

p

via (1.3),we an hangevariablesto

(q, p)

.

Hamilton'sprin ipleintermsofvariables

(q, p)

leadsto

Hamilton’s canonical

equations

˙

q

k

=

∂H

∂p

k

(q, p),

p

˙

k

=

−

∂H

∂q

k

(q, p),

k = 1, . . . , d.

(1.4)

The anoni alequation(1.4) ontinuestoholdif

L(q, ˙q, t)

,andhen e

H(q, p, t)

, ontainsanexpli ittime-dependen e.

Itisusefulto ombineallthedependentvariablesina

2d

-dimensionalve tor

y

= (q, p)

. Then(1.4)takesa simpleform

˙y = J

∇

y

H,

(1.5)

where

J = (J

ij

)

isthe

2d

× 2d

skew-symmetri matrix

J =

Ã

0

1

−1 0

!

.

(1.6)

Here

1

and

0

representtheunitandzero

d

× d

matri es,respe tively.

The Poisson bracket

An alternativeform for a Hamiltonian systemis the

Poisson bracket

, a skew-symmetri , bilinear form. For two fun tions

F (y)

and

G(y)

this is dened as

{F, G} = ∇

y

F (y)

T

J

∇

y

G(y).

(1.7)

The Poisson bra ket of a fun tion

F (y)

with the Hamiltonian

H(y)

is a on-venientnotationforexpressingthetime-derivativeof

F

alonga solutiontothe Hamiltoniansystem(1.5)

d

dt

F (y(t)) =

{F, H} = ∇

y

F

T

_J

∇

y

H =

∇

y

F

T

dy

dt

.

Givena Poissonbra ketandaHamiltonian,thedynami alequations(1.5) an beba kedout

˙y

k

=

{y

k

, H

}, k = 1, . . . , 2d.

GeneralizedHamiltonian systemsareobtained byallowing

J

to bea skew-symmetri operator, dependent on the oordinates

y

, i.e.

J = J(y)

. The Poisson bra ket still takesthe form(1.7), but isrequiredto satisfyskew sym-metry

{F, G} = −{G, F }

andthe

Jacobi identity

{E, {F, G}} + {F, {G, E}} + {G, {E, F }} = 0, ∀E, F, G.

Theserequirementstranslateintoanalogouspropertiesforthematrix represen-tationof

J(y)

;respe tively,

J(y)

T

₌

−J(y)

and

J

im

∂J

jk

∂y

m

+ J

jm

∂J

ki

∂y

m

+ J

km

∂J

ij

∂y

m

= 0,

∀i, j, k,

whererepeatedindi esaresummedfrom

1

to

d

. TheJa obiidentityistrivially satised by a onstant

J

, so onlyskew-symmetry isneeded in this ase. Veri- ation or onstru tionof non onstantPoisson bra kets an bea ompli ated task[49, 64 ℄.

The spe i form (1.6) of

J

orresponding to (1.4) is alled the

canonical

form of a Hamiltoniansystem. Otherwise,the systemis

noncanonical

. If

J

is dependenton

y

,thesystemis alleda

Poisson system

.

First integrals

For an ordinary dierential equation

˙y = f (y)

, a non onstant fun tion

I(y)

is a rstintegral if

I(y(t)) =

onst. alonganysolution. Inother words, ea h solutionis onstrainedtoalevelset of

I

.

d

dt

I(y(t)) =

∇

y

I(y)

T

_{f (y) = 0,}

∀y.

IntermsofthePoissonbra ket,a rstintegralis afun tionwhosePoisson bra ketwith

H

vanishes

{I, H} = 0.

Asystemmayhavemorethenonerstintegral.Inthat asetheinitial ondition denes an interse tion of the rst integrals and the solution evolves on this interse tion. The set ofrst integrals foliate thephasespa e onstrainingthe dynami s, restri ting the behavoir of a given traje tory. When one derivesa statisti altheory,theaimisto onstru ttheleastbiaseddistributionthat still ree ts given information about the system. This given information may be

1.1. Hamiltonian systems and geometric integration

5

Duetotheskewsymmetryofthebra ket,theHamiltonianis onservedfor any(autonomous)Hamiltoniansystem

dH

dt

=

{H, H} = ∇

y

H

T

_J

∇

y

H = 0.

Inphysi alappli ationsthis often orrespondsto onservation oftotalenergy. Other rst integrals in Hamiltonian systems arise due to ontinuous sym-metriesintheHamiltonian,viaNoether'stheorem[64 ℄. Thatis,ifone annd a one parameter, ontinuous hange of variables that leaves the Hamiltonian invariant,thenNoether's theoremidentiesa orrespondingrstintegral.

Fornon anoni alHamiltoniansystems,therearesometimesfun tions

C(y)

, alled

Casimirs

, that vanish identi allyin thePoisson bra ket with any other fun tion

{F, C} = 0, ∀F = F (y)

⇔

J

∇

y

C = 0.

Casimirsareobviouslyrstintegrals,relatedto singularityof

J

. Possessionof CasimirsisapropertyofthePoissonbra ket,theydonotdependonaparti ular hoi eoftheHamiltonian. For anoni alsystemstherearenoCasimirs.

Symplectic structure

Consider anordinarydierentialequation

˙y = f (y)

withthephasespa e

R

2d

. Denetheowovertime

t

asamapping

φ

t

thatadvan esthesolutionbytime

t

, i.e.

φ

t

(y

0

_{) = y(t, y}

0

₎

,where

y(t, y

0

₎

isthesolutionofthesystem orresponding totheinitialvalue

y(0) = y

0

. Asmoothmap

φ

t

onthephasespa e

R

2d

is alled a

symplectic map

withrespe ttothe( onstantandinvertible)stru turematrix

J

ifitsJa obian

φ

′

t

(y)

satises

φ

′

t

(y)

T

J

−1

φ

′

t

(y) = J

−1

,

(1.8)

forall

y

in thedomain ofdenitionof

φ

t

.

Theorem (Poincar´

e, 1899).

Theowmap

φ

t

ofaHamiltoniansystem(1.5)

issymple ti .

Symple ti mappingsarevolumepreserving. Takingthedeterminantofboth sidesof(1.8)wehave

|φ

′

t

(y)

|

2



J

−1





=



J

−1





.

Therefore

|φ

′

t

(y)

|

is either

+1

or

−1

. Sin e at

t = 0

|φ

′

0

(y)

| = 1

, a ontinuity argument showsthat

|φ

′

t

(y)

| = 1

forany

t

,i.e. thevolume ispreservedunder thesymple ti mapping.

Theowmap

φ

t

(y)

ofaPoissonsystemisa

Poisson map

,whi h satises

1.1.2

Geometric integration

Geometri integrators arenumeri al methods that preservegeometri proper-ties oftheowofadierentialequation,e.g. symple ti orPoisson integrators forHamiltoniansystems,methodspreservingrstintegrals,et . Preservationof rstintegralsplaysanimportantroleinstatisti alme hani s,be auseoneofthe aimsofstatisti alme hani sisto onstru taproperaverageofdesired quanti-tiesofasystemundersu h onstraints. Symple ti itygivesvolumepreservation oftheHamiltonianphaseow,andinaddition,a numeri alsolutiontraje tory onashorttimes alestays losetotheexa tsolutionoftheHamiltoniansystem. On ergodi time intervals,the solution driftsover multiple traje tories, while preservingthephasedensityofsolutions.

Inthefollowingse tionwe onsiderpro esseswhi hevolve ontinuouslyin time only, that is havingno ontinuous spatial stru ture. Therefore they an be des ribed bysystems of ordinary dierential equations. In this se tion we followthederivationsdes ribedin [35,45 ,79 ℄.

Symplectic integrators

Sin esymple ti ityisa hara teristi propertyofHamiltoniansystems a ord-ingtotheoremofPoin aré,itistemptingtosear hfornumeri almethodsthat sharethis property.

Aone-step numeri almethod

Φ

h

: y

n+1

_{= Φ}

h

(y

n

)

,

t

n+1

= t

n

+ h

is alled

symplectic

ifthemap

y

n

_{→ y}

n+1

issymple ti wheneverthemethodisapplied to asmoothHamiltoniansystem.

Here aresome examples ofsymple ti methods: thesymple ti Euler rule, the impli it midpoint rule, the Störmer-Verlet s heme, the Gauss ollo ation methodsandotherRunge-Kuttamethodssatisfyingthesymple ti ity ondition onthe oe ients[77 ℄.

Somesymple ti integratorspreserverstintegralsofasystem. Forexample, impli itmidpointpreservesanyquadrati rstintegraloftheform

I =

1

2

y

T

_Ay+

b

T

_y

for a onstant symmetri matrix

A

and an ordinary dierential equation

˙y = f (y)

. Thesymple ti Euler methodand thegeneralized leapfrogmethod

preserve any rst integral of the form

I = q

T

_Ap

for the anoni al variables

(q, p)

ofa Hamiltoniansystem.

Symple ti methodsdonot,ingeneral, onservetheHamiltonian

H

exa tly. Inspiteofthistheydo onserveitapproximately. Thisgood onservationof

H

insymple ti integrationisrelatedtotheexisten eandexa t onservationofa perturbedHamiltonianasshownbyba kwarderroranalysis.

Backward error analysis

The origin of ba kward error analysis dates ba k to [87℄ in numeri al linear algebra. For thestudy of numeri aldierential equations,its importan e was

1.1. Hamiltonian systems and geometric integration

7

numeri al one-step map is interpreted as the ow of a modied dierential equation,whi his onstru tedasanasymptoti series.

Consider an ordinary dierential equation

˙y = f (y)

with exa t mapping

φ

t,f

,andaone-stepnumeri almethod

Φ

h

(y)

:

y

n+1

_{= Φ}

h

(y

n

)

,

t

n+1

= t

n

+h

. A

forward, or common, error analysis

onsistsofthestudyoftheerror

φ

h,f

(y

n

₎

−

Φ

h

(y

n

)

, i.e. the dieren e between the exa t solution and the numeri ally omputed approximation. The idea of

backward error analysis

is to derive a

modified differential equation

˙y = ˜

f

i

(y, h),

(1.10)

with

˜

f

i

(y, h) = f (y) + hf

1

(y) + h

2

f

2

(y) +

· · · + h

i

f

i

(y),

(1.11)

where the expressionfor

f

˜

i

(y, h)

depends onthe numeri almethod

Φ

h

of the original system

˙y = f (y)

byusing theTaylor expansion of theow map

φ

t, ˜

f

i

with

t = h

at

y

= y

n

and omparingitsTaylorexpansiontermswiththeterms of thenumeri almethod

Φ

h

inpowersof

h

. Ingeneralfora one-stepmethod, theseries(1.11)divergesas

i

→ ∞

butmaybeoptimallytrun atedasafun tion of

h

.

Consider now a Hamiltoniansystem (1.5)with a smoothHamiltonian

H :

R

2d

→ R

. If a symple ti method

Φ

h

(y)

is applied to it, then the modied ve tor eld

f

˜

i

(h)

in the equation (1.10) is also Hamiltonian with a modied Hamiltonianfun tion

H

˜

i

(h)

. Morepre isely,

˜

f

i

(y, h) = J

∇

y

H

˜

i

(y, h),

andthemodiedHamiltonian

H

˜

i

(y, h)

is losetotheoriginallygiven Hamilto-nian

H(y)

,i.e.

| ˜

H

i

(y, h)

− H(y)| = O(h

p

),

where

p

≥ 1

istheorderofthesymple ti method

Φ

h

[4 ℄.

For the Hamiltonian system(1.5), themodied Hamiltonian

H

˜

i

(y, h)

an bewrittenas

˜

H

i

(y, h) = H(y) + hH

1

(y) + h

2

H

2

(y) +

· · · + h

i

H

i

(y),

where theset

{H

i

}

depends onthesymple ti method

Φ

h

as it was des ribed above,andwhereea h

H

i

isa Hamiltonianofsome Hamiltoniansystem.

Ba kwarderror analysisshowstheadvantages of usingsymple ti integra-torsforHamiltoniansystems. Itwasshownin[35 ,45 ,79 ℄thatforasymple ti integratorappliedtoanautonomousHamiltoniansystem,modiedautonomous Hamiltonianproblemsexistsothatthenumeri alsolutionoftheoriginal prob-lem is the exa t solutionof the modied problem. On the ontrary, when a nonsymple ti integratorisusedthemodiedsystemisnotingeneral Hamilto-niananymore. ForgeneralHamiltonians,in[33 ℄itwasproventhatasymple ti method

Φ

h

annot exa tly onserveenergy. Nevertheless,thesymple ti

inte-Hamiltonian system with analyti

H : K

→ R

(where

K

⊂ R

2d

is an open subset), and apply a symple ti numeri almethod

Φ

h

(y)

withstep size

h

. If thenumeri alsolutionstaysina ompa tsubsetof

K

,thenthereexistsa

ϑ > 0

su hthat

˜

H

i

(y

n

, h)

=

H

˜

i

(y

0

, h) + O(e

−ϑ/2h

),

H(y

n

, h)

= H(y

0

, h) + O(h

p

)

over exponentiallylong time intervals

nh

≤ e

ϑ/2h

. Intypi alappli ations, the Hamiltonianos illatesarounditsinitialvaluewithboundedamplitude

O(h

p

₎

. In[70 ℄,itisproventhatwhenevertheowmapofagivendierential equa-tionpossesssome geometri propertiessu hasexisten eofrstintegrals, time reversibility, preservation of volume, symple ti ness, and the numeri al dis- retizationpreservesthesepropertiesexa tly,thentheowmapofthemodied dierentialequationwillalsosatisfythesegeometri properties.

Poisson integrators

Poissonintegrators generalizeofsymple ti integratorsto Poissonsystems. Anumeri alone-step method

y

n+1

_{= Φ}

h

(y

n

)

,

t

n+1

= t

n

+ h

, isa

Poisson

integrator

for the stru ture matrix

J(y)

, if the transformation

y

n

→ y

n+1

respe ts theCasimirsand ifitis a Poisson map (1.9)whenever themethod is applied tothePoisson system.

A ordingtoba kwarderroranalysis,ifaPoissonintegrator

Φ

h

isappliedto thePoissonsystem,thenthemodiedequationislo allyaPoissonsystem[35 ℄. There is no general te hniquefor onstru ting Runge-Kutta type Poisson methods. Themostgenerallyappli ablealternativeissplittingmethods,e.g.[50 ℄.

1.2

Hamiltonian fluid dynamics

Thestudyofthedynami sofuidsisoneofthemostattra tiveareasinapplied mathemati s. Thefa tthatuiddynami sisanattra tiveresear hareaisdue tomanyreasons. Perhapsthemostimportantoneistheintrodu tionofe ient highresolutionnumeri alsimulationsintouiddynami sasaresear htool. The signi an e of thistoolis espe ially pronoun edin ase of omplexbehaviour ofa system.

1.2.1

Hamiltonian PDEs

The dynami s of uids has both propagation in time and a spatial stru ture and,hen e, annotbedes ribedbyordinarydierentialequations(ODEs) any-more but by partial dierentialequations (PDEs). Many PDEs that arise in physi s an beviewedas innite-dimensionalHamiltoniansystems. (Problems des ribed by ordinary dierential equations are nite-dimensional.) The

nu-1.2. Hamiltonian fluid dynamics

9

PDEsismu hlessexploredthanthatofODEs,sin ethesolutionbehaviourof PDEsismu hmore omplex. Nevertheless,thereexistsanumberofpapersfor HamiltonianPDEs[12 ,14 ,17 ,24 ,26 ,27 ,31 , 32 ,51 ,83 ℄.

Thenite-dimensionalHamiltoniansystem onsistsofatriple(

K,

{·, ·}, H

), where the phasespa e

K

⊂ R

2d

is anopen subset,

H : K

→ R

is the Hamil-tonian fun tion, and

{·, ·}

is a Poisson bra ket with stru ture matrix

J(y)

, see (1.7). When the phase spa e is innite-dimensional, we write the triple as

(

K, {·, ·}, H)

,andthePoissonoperator as

J

,a ordingto [64 ℄. Typi ally

K

onsistsofsetsofsmoothfun tionsonanite-dimensionalspa e

Y

. Anelement in

K

isdenoted by

u(y)

,

y

∈ Y

. TheHamiltonian

H : K → R

is a fun tional onthisspa e,andthebra ket anbewrittenas

{F, G}[u] =

Z

Y

δ

_F

δu

J (u)

δ

_G

δu

dy,

where

δ

F/δu

isthevariationalderivativedenedby

lim

ǫ→0

F[u + ǫδu] − F[u]

ǫ

≡

Z

Y

δ

_F

δu

δu dy

forappropriate

δu

.

J (u)

is,ingeneral,adierentialoperator, alledthe

Poisson

operator

.

Motivated by thesu ess of symple ti integrators, a reasonable approa h to HamiltonianPDEsisto tryto dis retizein spa ewhilepreservingthe sym-ple ti or Poisson stru ture. For anoni al stru ture, it is a simple matter to dis retizetheHamiltonian fun tionalwith anydesiredquadrature. Theresult is a Hamiltonian ODE to whi h symple ti integrators may be applied. For Poisson systems, it is a signi ant hallenge to derive a dis rete bra ket that preservestheJa obiidentity. IfthereareCasimirs,there shouldbesome rem-nantofthese. Foruids,thereisaninnitefamily,andonlyanitenumberof independent integrals an survive, ifthey onstrain thedis rete, nite dimen-sional phasespa e.

Onthe other hand,sin e Lagrangian uid dynami s is anoni al, one an approximatethePDEsolutionwithasetofmovingparti lesintera tingthrough anappropriatepotentialenergyfun tion,andaHamiltoniansemi-dis retization will be obtained for any quadrature of the Hamiltonian. The set of nite-dimensionalHamiltonianODEsisthenintegratedintimeusingasuitable sym-ple ti orPoissonintegrator.

Unfortunately, inthe aseof aPoisson PDE,unlike anoni alHamiltonian PDEs, it is not possible to establish a ommon generi approa h. For ea h parti ular problemone hastodevelopa proper wayof redu ingthePDEtoa systemofODEs. Wewill onsiderthequasigeostrophi potentialvorti ity equa-tion (a Hamiltonian PDE with Poisson stru ture), for whi h we will des ribe

1.2.2

Eulerian and Lagrangian descriptions

Geophysi al uiddynami sisthestudyofuidmotionintheatmosphereand theo ean. Therststepinthisstudyisa hoi eofframework: Eulerianor La-grangian. The

Eulerian description

is ommonlyusedinliterature,e.g.[52 ,67 ℄, ittreats motionoftheuidasaeld inwhi h thevelo ityistobedetermined at allpositions and times. The

Lagrangian description

regards the uid as a ontinuouseldof parti les,whosepositionsareto bedetermined[74℄.

IntheEuleriandes ription,theindependentvariablesarethespa e oordi-nates

x

= (x, y, z)

∈ D

and the time

t

. Thedependent variables in lude the velo ity

v(x, t)

andthemassdensity

ρ(x, t)

Thenon anoni alPoisson bra ket foridealuidowinEulerian variables, a ording to[58 ℄,is

{F, G} = −

Z

D

dx

· δF

δρ

∇

x

·

δG

δv

−

δG

δρ

∇

x

·

δF

δv

+

µ ∇

x

× v

ρ

¶

·

µ δG

_δv

×

δF

_δv

¶¸

withHamiltonian

H[v, ρ] =

Z

D

dx

· v · v

2ρ

+ ρE(ρ)

¸

.

Here

x

is anindependentvariable.

ThePoisson bra ket has aninnite lassof potentialvorti ityCasimirs of theform

C[ρ] =

Z

D

dx ρf

µ ∇

x

× v

ρ

¶

foranarbitraryfun tion

f

.

In the Lagrangian des ription, ea h uid parti le is assigned a label

a

=

(a, b, c)

∈ A

. Forexample,thelabelsmaybedenedasthepositionsofparti les at the initial time. The independent variables are set of

a

, whi h are xed for ea h parti le, and the time

t

. The dependent variables are the position oordinates

x(a, t)

. Thevelo ityofa parti leisgivenby

v

=

µ ∂x

∂t

,

∂y

∂t

,

∂z

∂t

¶

.

Themassdensity

ρ

isdenedvia Ja obianmatrix

|

∂

x

∂

a |

as

ρ = ρ

0









∂x

∂a









−1

,

where

ρ

0

= ρ

0

(a)

doesnotdepend ontime

t

.

Taking derivativesoftheexpressionaboveleadsto the ontinuityequation

dρ

1.2. Hamiltonian fluid dynamics

11

ThePoissonbra ketforidealuidowin Lagrangianvariablesis anoni al

{F, G} =

Z

A

da

· δF

δx

·

δG

δv

−

δG

δx

·

δF

δv

¸

withHamiltonian

H[x, v] =

Z

A

da

"

v

_{· v}

2ρ

0

+ ρ

0

E

Ã

ρ

0









∂x

∂a









−1

!#

.

Here

x

is adependentvariable.

The quasigeostrophic model

The two-dimensional quasigeostrophi potential vorti ity (QG) equation [48 , 67 ,74 ℄des ribesdivergen e-freeowovertopographyby

d

dt

q = 0,

∆ψ(x, t) = q(x, t)

− h(x),

(1.12)

where

q

isthepotentialvorti ity(PV)eld,

ψ

isthestreamfun tion,and

h

is thetopographyoftheearth. TheLapla ian operatoris denotedby

∆

andthe material derivativeby

d

dt

=

∂

∂t

+ u

· ∇

. Here, thedivergen e-freevelo ityeld

u

isrelatedtothestreamfun tionby

u

=

∇

⊥

_ψ

,where

∇

⊥

_{= (}

−

∂y

∂

,

∂

∂x

)

T

. We onsider theQG equationona doublyperiodi domain

x

= (x, y)

∈ D ≡ [0, 2π) × [0, 2π).

Dene the operator

J (q, ψ) = q

x

ψ

y

− q

y

ψ

x

. The QG model des ribes a HamiltonianPDEwithPoissonstru ture[57 ℄,

{F, G} =

Z

D

q

_J

µ δF

δq

,

δ

G

δq

¶

dx,

implyingthe onservationoftheHamiltonianortotalkineti energy

H = E = −

1

₂

Z

D

ψ

_{· (q − h) dx}

as wellastheinnite lassofCasimirfun tionals

C[f] =

Z

D

f (q) dx

for any fun tion

f

for whi h the integral exists. Of spe i interest are the momentsofPV

C

r

=

Z

D

q

r

dx,

r = 0, 1, 2, . . . ,

(1.13)

PreservationoftheCasimirfun tionalsfollowsfromarea-preservationunder the divergen e-free velo ity eld [55 ℄. Dene a fun tion

G(σ, t)

denoting the measure ofthatpartofthedomain

D

forwhi h thevorti ityislessthan

σ

:

G(σ, t) = meas

_{{x ∈ D | q(x, t) < σ}.}

We note that due to the divergen e-free adve tion of

q

, this fun tionis inde-pendentoftime,

∂G

∂t

= 0

. Dierentiatingwithrespe t to

σ

,thefun tion

g(σ) =

∂G

∂σ

(1.14)

ispreserved. Forthe aseofapie ewiseuniformPVeld,

q(x, t)

∈ {σ

1

, . . . , σ

L

}

, thisquantity

g

ℓ

= G(σ

ℓ+1

)

− G(σ

ℓ

)

isthemeasure ofthevorti itylevelset

σ

ℓ

.

1.2.3

Numerical methods

In the following se tion we des ribe several numeri al methods to solve the quasigeostrophi model(1.12).

The Zeitlin method

Normalspe tralmethods fortheQG equationpreservetheenergyand enstro-phy at most. However, the Zeitlin method [88 ℄ is a spe tral method whi h preservesa Poisson stru ture, theHamiltonian and

2M

Casimirsin a

(2M +

1)

× (2M + 1)

modetrun ation.

Therstequationin(1.12)istransformedthroughtwo-dimensionalFourier seriestakingtheform ofaninnitesystemofODEs

dˆ

q

k

dt

=

∞

X

k

′

1

,k

′

2

=−∞

k

′

6=0

k

_{× k}

′

|k

′

|

2

q

ˆ

k+k

′

(ˆ

q

−k

′

− ˆh

−k

′

).

(1.15)

Here

q

ˆ

k

denotes the spe tral oe ient asso iated with the two-dimensional wave ve tor

k

, whose omponents are integers. The skew-symmetri s alar produ t

k

× k

′

is

k

1

k

′

2

− k

2

k

′

1

, andthe norm

|k|

is

pk

2

1

+ k

2

2

. Sin e

q

is real,

ˆ

q

∗

k

= ˆ

q

−k

.

Zeitlin proposed the sine-bra ket trun ation of the equations. The nite-dimensional setofequationsfortheFourier oe ientsisthengivenby

dˆ

q

k

dt

=

M

X

k

′

1

,k

′

2

=−M

k

′

6=0

1

ǫ

sin(ǫk

× k

′

)

|k

′

|

2

q

ˆ

k+k

′

(ˆ

q

−k

′

− ˆh

−k

′

),

ǫ =

2π

2M + 1

,

(1.16)

where all indi es are redu ed modulo

2M + 1

to the periodi latti e

−M ≤

k

1

, k

2

≤ M

. The summation o urs on the

(2M + 1)

× (2M + 1)

domain of the Fourier oe ients. For

M

→ ∞

and given

k

and

k

′

,

ǫ

−1

_sin(ǫk

× k

′

) =

k

_{× k}

′

+ O(ǫ

2

₎

1.2. Hamiltonian fluid dynamics

13

Thistrun ationpossessesaHamiltonianstru turewithsymple ti operator

J

kk

′

=

−ǫ

−1

sin(ǫk

× k

′

)ˆ

q

_k+k

′

,

andHamiltonian

H = E =

1

2

X

k6=0

|k|

2

| ˆ

ψ

k

|

2

=

1

2

X

k6=0

|ˆq

k

− ˆh

k

|

2

|k|

2

.

(1.17)

Thesymple ti matrixisskew-symmetri andsatisestheJa obiidentity. The sine-bra ket trun ation (1.16)preservesthe Hamiltonian(1.17) and

2M

inde-pendentCasimirinvariants orrespondingtotherst

2M

momentsofpotential vorti ity. If in the Zeitlin method thePoisson dis retization is integrated us-ingthePoissonsplittingofM La hlan[50 ℄,thenthesequantities arepreserved by the splitting (the energy is only preserved approximately, in the sense of ba kwarderroranalysis[35 ℄).

Unfortunately, theZeitlin method islimitedto 2Din ompressible owson periodi geometry.

Arakawa’s scheme

TheZeitlinmethodistheonlyknowndis retizationwithPoissonstru turefor Eulerian uid models. For more general uid problems ( ompressible, non-periodi boundary onditions, et .) no Poisson dis retizations are available. How anwepreserveatleastsomequantities?Awellknowns hemeisArakawa's s heme[2℄whi hpreserveslinearandquadrati invariants.

For astartwerewritetherstequationin (1.12)as

q

t

=

J (q, ψ),

(1.18)

where theoperator

J

isdened by

J (q, ψ) = q

x

ψ

y

− q

y

ψ

x

.

Arakawa'sidea onsistsofseveralsteps. Firstofall,heuses entraldieren es for

x

-and

y

-derivatives. Thenherewritesthe ontinuous

J

inthreeequivalent formsbasedonthefa tthatthederivativeswithrespe tto

x

and

y

ommutein the ontinuous ase,namely

J (q, ψ) = ∂

x

(qψ

y

)

− ∂

y

(qψ

x

) = ∂

y

(q

x

ψ)

− ∂

x

(q

y

ψ)

. After dis retizing these three equivalent forms of

J

and taking theiraverage, one gets four dis rete non-equivalentright-hand sidesof (1.18), therefore four dis retizations. Non-equivalen eofthedis reteright-handsidesisexplainedby the fa t that theprodu t rule does nothold anymore in the dis rete ase. It an beshownthat one dis retization does not onserveanythingand, in fa t, is unstable;se ond one onservesonlyenergy; thirdoneonly enstrophy; and fourthone,whi hisanaverageofthepreviousthree, onservesbothenergyand enstrophy. Itisworthmentioningthatallthesedis retizationsarealsovolume preserving in the sense of the Liouville property, see Se tion (1.3). This is a ne essary ingredientforastatisti altheory.

The Hamiltonian particle-mesh method

The Hamiltonian parti le-mesh (HPM) method approximates the solution of an ideal uid ow using a set of moving parti les that intera t through an appropriate potential energy fun tion. The HPM method was originally de-signed for rotating shallow water owwith periodi boundary onditions [25 ℄ andextendedtootherphysi alsettingsrotatingtwo-layershallowwatermodel withrigid-lid onstraint,barotropi model,non-hydrostati verti alsli emodel, in [16 ,17,28,83℄.

HPM isbased onthe Lagrangian formulation of uiddynami s, anda set of moving point masses ombined with a xed Eulerian grid. The spatially trun atedequationsare anoni alHamiltonianandsatisfyaKelvin ir ulation theorem[25 ℄. Implementationofasplittingmethodintimetothesemi-dis rete system of Hamiltonian ODEs gives symple ti ity. Hen e the HPM method is symple ti , and one an onstru t a ontinuum velo ity eld in whi h the dis rete parti le velo ities areembeddedfor alltime. Energy is preserved ap-proximatelyinthesenseofba kwarderroranalysis,i.e. goodlong-timeenergy onservation. Convergen eofthemethodwas onsideredin[56 ℄.

Theideaofxingthepotentialvorti ityratherthanthemasstoa parti le wasoriginallyproposedinthis ontextfortwo-dimensionaladve tionunder in- ompressibleoweldsin[17 ℄. Theresultisaregularizedpointvortexmethod. Therethepotentialvorti ityofaquasigeostrophi modelissimplyadve tedin a divergen e-free velo ity eld, obtained by re onstru ting the PV eld on a uniformgrid. Theparti lemotion anbeembeddedin anarea preservingow ontheuidlabelspa e. Hen e,theCasimirs(1.13)are trivially onserved ifa valueofpotentialvorti ity issimplyassignedtoea hparti leon e andforall. The semi-dis rete system is still Hamiltonian, and an be integrated in time usinga symple ti integrator. Sin etheHPMmethod issymple ti ,thephase owisvolumepreservinginthesenseoftheLiouvilleproperty,seeSe tion(1.3), whi hisa ne essaryingredientforastatisti altheory.

TheHPMmethodforthequasigeostrophi modelisexplainedinmoredetail in Chapter3.

1.3

Statistical mechanics of fluids

Statisti alme hani sisapowerfultoolforunderstanding omplexphysi al sys-tems, e.g. [23,30 ℄. There aredierent purposesof statisti al me hani s. One may onsider a systemfor a very long time. Then the quantity of statisti al interest is the average behaviour of a system rather than the behaviour of a system at a ertain time. Anexample is a tra outside of your window. A statisti al quantityof interestmaybetheaveragespeedof ea h ar. Another purposeofstatisti al me hani sisto designmethodsto handlesystems whi h arein ompletelyknown. For example,ifwedonotknowtheinitial onditions of a system,and wewantto know itsmostlikelybehaviour. Thisinvolvesan

1.3. Statistical mechanics of fluids

15

Statisti alme hani sisbroadlyemployedinnumeri alsimulationsof mole -ular dynami s and limate variability. Here either a systemis very big (

10

23

mole ules)orlongtimesimulationsare haoti ,orboth. Moreover,dierent nu-meri al dis retizationsofadynami alsystemhavedierentdis retedynami s. Thereforethestatisti alresults analsobedistin t.

Themainpremiseoftheequilibriumstatisti altheoriesofeither ontinuous or dis rete dynami s is ergodi ity relating the equilibrium distribution to the dynami s. Ne essaryingredientsare onservationlawsandvolumepreservation ofthephaseow. Therefore,toderiveastatisti altheoryofanumeri almethod thenumeri almethodhastopossess onservationlaws,and thedis retephase owshould bevolumepreserving. Whenoneintegratesa Hamiltoniansystem with a symple ti integrator, there is an automati onservation of dis rete analoguesoftheexa t onstantsofmotionand,be auseofsymple ti ity,there isvolumepreservation. Ergodi ityisdi ulttoshowfornontrivialsystems,so thisisusuallytakenasanassumptionor'approximation'whi hmustbeveried withnumeri s.

Theaimofthefollowingse tionistoexplainstatisti alme hani s,togivea denitionofdierentensemblesandpurposeofea h,andtodis ussthe on ept ofergodi ity, entropyandtemperaturein statisti alsense.

Statistical equilibrium

Consider anordinarydierentialequation

˙y = f (y)

withthephasespa e

K

⊂

R

2d

,thenaprobabilitydistributionfun tion

ρ(y, t)

,

ρ : K

×R → R

,forexample, over a set of un ertain initial onditions,is transported bythe ow a ording to

∂

∂t

ρ +

∇

y

· ρf = 0.

Now onsider aHamiltoniansystem(1.5).

Theorem (Liouville, 1838).

The phaseow ofa Hamiltonian system(1.5)

isvolumepreserving[3℄.

For onstant

J

this followsfrom theskew-symmetry of

J

, or equivalently, thedivergen e-freenatureofthe anoni alphaseow

y

. Onesaysthattheow hasthe

Liouville property

,ifthisowsatisesLiouville'stheorem,i.e. theow is volume preserving, i.e. the ow is divergen e-free. The Liouville property is a ne essary ingredient fora statisti al theory, sin efrom itfollowsthat the probability measure is transported under the divergen e-free ow. Therefore before deriving a statisti al theory, one has to prove that the phase ow is divergen e-free.

Thetransportequationinthe aseofa Hamiltoniansystemsimpliesto

∂

∂t

ρ + J

∇

y

H

· ∇

y

ρ = 0,

A steadystateof theLiouville equation forthe Hamiltonianow,

∂ρ

∂t

= 0

, isthen

J

∇

y

H

· ∇

y

ρ = 0,

whi hisoftenreferedas an

equilibrium

probabilitydistributionfun tion. Note that anyfun tion

ρ(y) = ρ(H(y))

oftheHamiltonianisanequilibrium proba-bilitydistribution,andtondtheproperprobability,whi h orrespondstothe dynami sdes ribedbytheODE,isthetopi ofergodi theory,seee.g.[68 ,85 ℄. To bemorepre ise, letusdene thetimeaverageofafun tion

F (y(t))

as

F

≡ lim

T →∞

1

T

Z

T

0

F (y(t))dt,

providedthatthelimitexists.

Theensembleaverageof

F

isdened as

hF i ≡

Z

K

F (y)ρ(y) dy

_≡

Z

K

F ν(dy)

fora propermeasure

ν

su h that

ν > 0

and

R

K

ν(dy) = 1

.

Ergodicity

implies thatthelong time average isequivalentto theensemble average

F =

hF i

(1.19)

withtheprobabilitymeasure

ν

ora reasonableapproximationtoit. Giventhe probability measure

ν

it is a hallenging task to prove theergodi ity, and we will takeitasanassumption.

1.3.1

Statistical ensembles

Let usintrodu eanimportantideaof mi rostateand ma rostate ofa system. Consideramodelinwhi huidmotionisdes ribedbyasetofmovingparti les. Thena

microstate

ofsu hsystem anbedes ribedbypositionsoftheseparti les,

anda

macrostate

is, forexample,theobservableenergywhi hmay orrespond

to alargenumberofmi rostates.

Belowwewillexplainthisinmoredetailfollowingthederivationsdes ribed in [10 ℄.

The microcanonical ensemble

Consideradis retespa e

K

withasinglema rostategivenbytheenergy

H = E

. Let

y

∈ K

. Thesubset

D(E) =

{y ∈ K : H(y) = E}

onsistsofdis retestates

y

withthesameenergy

E

.

Dene

Ω(E)

to be the total number of states

y

∈ K

with the energy

E

. The mi ro anoni alensembleis the set of all

y

having

H(y) = E

. Assuming

1.3. Statistical mechanics of fluids

17

all su h states are equally likelywe dene the

microcanonical density

for the dis retespa e

Prob

_{{y | H = E} =}







0,

H(y)

6= E

1/Ω(E),

H(y) = E.

Themi ro anoni alentropyisdened as

S(E) = ln Ω(E).

ConsidernowajointHamiltoniansystem

AB

,whi h onsistsoftwosystems

A

and

B

,withtotalenergy

E

su hthat,the ouplingelementallowsex hange of the energy between systems

A

and

B

and adds neither new states to the systemnornewtermsintotheHamiltonian.

A mi rostateof system

A

is dened as

y

A

. If system

A

isin a mi rostate

y

_A

with orrespondingenergy

E

A

,thensin ethereistotalenergy onservation, system

B

shouldbeinastatewiththeenergy

E

− E

A

. Theprobabilityofthe mi rostate

y

A

is

Prob

_{y

A

| H = E} =

Ω

B

(E

− H

A

(y

A

))

N (E)

,

(1.20)

where

N (E)

isanormalization onstantsu h that

X

y

A

∈K

A

Prob

_{y

A

| H = E} = 1

⇔

N (E) =

X

y

A

∈K

A

Ω

B

(E

− H

A

(y

A

)).

This is the mi ro anoni al probability of a mi rostate. The mi ro anoni al probabilityofthema rostate

H

A

= E

A

istheenergysplit

Prob

{H

A

= E

A

| H = E} =

Ω

A

(E

A

)Ω

B

(E

− E

A

)

N (E)

withthenormalization onstant

N (E)

.

Themostprobable ma rostate anbefoundbymaximizing thenumber of stateswiththeenergysplitoverallpossiblestatesofsystem

A

withtheenergy

E

A

max

E

A

[Ω

A

(E

A

)Ω

B

(E

− E

A

)] ,

whi hin termsofentropiesgives

max

E

A

[S

A

(E

A

) + S

B

(E

− E

A

)] .

Fromthelastexpressionitis learthatthemostprobablestateisthemaximizer ofthetotalentropy. Whenthejointsystem

AB

isverylarge,thenitispossible

the maximizer of the total entropy an be found analyti ally: the maximum o urs atsome

E

∗

A

where

S

A

′

(E

∗

A

) = S

B

′

(E

− E

∗

A

),

and the prime denotes thederivative with respe tto theargument. This has motivatedthedenition ofthe

microcanonical statistical temperature

S

′

(E) =

1

T

.

Note that

T

A

=

T

B

at the mostlikelyma rostate. Inthe literatureone often usesthe

inverse statistical temperature

β = 1/

T

.

Now we show how to derive the mi ro anoni al density for a ontinuous phasespa e. Therestofthemi ro anoni alstatisti altheoryfora ontinuous phasespa efollowsautomati allywithsumsrepla edbyintegrals.

Iftheenergyistheonly onservedquantityofthesystem,thenwe onsider thesubspa e

D(E, dE) =

{y ∈ A : H(y) ∈ [E, E + dE]},

(1.21)

with orrespondingdensity

ρ(y) =







0,

H(X) /

_{∈ [E, E + dE]}

1/vol

{D}, H(X) ∈ [E, E + dE].

(1.22)

The density (1.22) is a stationary density, sin e for xed

(E, dE)

it depends onlyonautonomous

H

. Ifwetakethelimit

dE

→ 0

,thedensity

ρ

ispresented onlyonthesurfa e

H = E

. Thenthemi ro anoni aldensityforthe ontinuous phasespa e anbewrittendownin termsofDira deltafun tions

ρ(y) =

1

Ω(E)

δ(H

− E)

with

Ω(E) =

Z

K

δ(H

− E) dy,

where

Ω(E)

isthemeasureofthesurfa e

H = E

.

We want to underlinethat the mi ro anoni alstatisti al me hani s is de-rivedassuming onservationofsome quantities,e.g. energy. Thereforeall pos-sible stateswith these onstantquantities,i.e. the ma rostate, determinethe mi ro anoni alensemble.

The canonical ensemble

Consider again thejoint system

AB

, but withthe size of thesystem

B

mu h largerthenthesizeofthesystem

A

,thesizeofthesystem

A

mayormaynotbe large omparedtounity. Inthis asethesystem

B

is alledanenergyreservoir forthesystem

A

.

Nowwewouldliketoderiveanequivalen eof(1.20)forthedes ribedsystem. First,letuswritedown(1.20)in termsofentropy:

1.3. Statistical mechanics of fluids

19

Sin ethefun tion

S

isa slowlyvaryingfun tionin anamplerangeofpossible mi rostates ofthelargesystem

B

,in ontrastto

Ω

whi h isnot, we an write downtheTaylorexpansionof

S

B

(E

− E

A

)

,andwetrun atetheseriesafterthe rstterm

S

B

(E

− E

A

)

∼ S

B

(E) + S

B

′

(E)(E

− E

A

− E).

Absorbing

S

B

(E)

into the onstantof normalizationandnoti ing that

S

′

B

(E)

istheinversetemperature

β

B

,weobtain

Prob

{y

A

| H = E} ∼ exp(−β

B

E

A

).

Thismotivatesthedenition ofthe

canonical probability density

ofasystemin onta twithanenergyreservoirin aseofa dis retephasespa e

Prob

_{{y | β} =}

1

N (β)

exp(

−βH(y))

and

N (β) =

X

y∈K

exp(

_{−βH(y)). (1.23)}

For a ontinuousphasespa e

K

, the sumin (1.23) isrepla ed bytheintegral over

K

.

Sampling

Nowthatwehavedenedthestatisti alensemblesandtheprobabilitydensities asso iatedwiththem,aquestionariseshowtoensuresamplingofthese ensem-bles. If we onsider a system with onserved energy, and having ergodi ity, we an samplea mi ro anoni aldistribution of onstant energybysimulating the dynami sof thesystemfor a longtime. Then theensemble originatedby the time series of the dynami s is equivalent to the mi ro anoni al ensemble asso iatedwiththe onstantenergy.

Thereareseveralapproa hestoensuresamplingofa anoni aldistribution. Theseapproa hesworkinsu hawaythatiteithermodiesthedynami al sys-temorintrodu esasto hasti perturbation. Themostknown lassi almethod for sampling a anoni al distribution is the Metropolis algorithm [53℄. It is basedonarandom hoi eofa stateanda eptan eofthisstatedependingon theprobabilitywhi hshouldbesampled. Apopularmethodologyinmole ular dynami s to samplea anoni al distribution isa mathemati al thermostata tooltomodelthesysteminthermalequilibriumwithareservoir. The thermo-statis responsiblefortheenergyex hange betweenthesystemandtheenergy reservoirsu hthatthesystemstaysata giventemperature,whi hfor es sam-pling of the anoni al equilibrium distribution. Here are several thermostat te hniques. The lassi alonesareLangevindynami s[80℄,whi hisasto hasti thermostat,anddeterministi thermostatssu hastheNosémethod[61 ,62 ℄and theNosé-Hoover method [37 ,62 ℄. InLangevindynami s the ombinationofa damping for e anda sto hasti termmaintainsthesystem ata given temper-ature. Be ause of thepresen e of damping and theintrodu tion of a random for ing, the dynami s are not any longer Hamiltonian. The Nosé and Nosé-Hoover methods preserve the Hamiltonian stru ture and a hieve sampling by

1.3.2

Information theory

Wehave onsideredthemaximumentropyprin iplebasedonthemaximization of the number of states of a system and repla ing this number by the total entropy. There is an alternative approa h to the maximum entropy prin iple based on Shannon entropy of information theory, whi h is not derived from physi alprin iples.

Consideraninnitedimensionalphasespa e

K

(it ouldalsobenite)with an element of it denoted by

y

. Then the

Shannon entropy

, or information entropy,is

S[ρ] =

−

Z

K

ρ ln ρ dy.

(1.24)

The on ept of Shannon entropyplays the entral role in information theory, sometimesreferredasmeasureofun ertainty[81 ℄. Theprobabilitydensity fun -tion

ρ

is hosentomaximize

S

under onstraints orrespondingtoobservations on the system. These minimal assumptions lead to the distribution of least bias, i.e. the distribution whi h is most generaland still explains the obser-vations. Weshow that themi ro anoni aland anoni aldistributions arethe maximizersofShannonentropyundersuitable onstraints.

Supposethatwehaveonlyone onstraintontheprobabilitydensity,namely thenormalization onstraint

Z

K

ρ dy

_{− 1 = 0.}

(1.25)

To ensurethis onstraintwe needto in lude it in thea tion prin iple via La-grange multiplier

θ

δ

·

−

Z

K

ρ ln ρ dy

_{− θ}

µZ

K

ρ dy

_{− 1}

¶¸

= 0

forarbitraryvariations

δρ

and

δθ

. Aftertakingvariationswithrespe tto

ρ

we

have

ln ρ =

−1 − θ

with

θ

determinedby thenormalization onstraint(1.25).

Therefore

ρ =

1

vol

{K}

,

and

S

∗

_{= ln vol}

{K}

is themaximized entropy. Thus,withnofurther assumptions,theleastbiased distribution isuniform.

Considerthephasespa e(1.21)withthemi ro anoni alprobabilitydensity (1.22). Then it an be shown that this density is the maximizer of Shannon entropyunderboththenormalization onstraint(1.25)andthe onstraintthat theenergy anonlytakevalues

H

∈ [E, E +dE]

. The orrespondingmaximized Shannon entropyis

S

∗

_{= ln vol}

_{{D(E, dE)}}

.

The anoni aldistribution (1.23) is the maximizer of Shannon entropy as well, but under other onstraints,namely, under the normalization onstraint (1.25)and the onstraintofobservedmeanenergy

U

Z

K

1.3. Statistical mechanics of fluids

21

forsome xed

U

.

Thea tionprin iple statesthat

δ

·

−

Z

K

ρ ln ρ dy

_{− θ}

µZ

K

ρ dy

_{− 1}

¶

− β

µZ

K

Hρ dy

_{− U}

¶¸

= 0

forarbitrary

δρ

,

δθ

and

δβ

. Here

θ

and

β

areLagrangemultipliers orresponding to thenormalizationandthexedmeanenergy onstraintsrespe tively.

Taking variationswithrespe tto

ρ

weobtain

ρ =

1

exp(1 + θ)

exp(

−βH).

Sin e

θ

here orrespondsto the normalization onstraint, after renamingitas

N

,itbe omes learthat thisisthe anoni aldistribution (1.23).

Thereforethemi ro anoni aland anoni aldistributionsarethemaximizers of the Shannon entropy under suitable onstraints. This an be extended to a generalsystemwith more onstraintsmore information about thesystem. For example, if one onsiders a numeri al method with some quantities, say, preservationof energy, or anyother information,then thisinformation anbe usedto onstru ttheleastbiaseddensity onsistentwiththeobservations.

The ee tiveness of a density derived this way depends on the detail to whi h known information about the system is in luded. For example, the energy-enstrophystatisti altheoryforthequasigeostrophi model(1.12)based onpreservationofenergyandenstrophyisamodelandisin omplete[1℄,sin e ittakesintoa ountonlylinear( ir ulation)andquadrati (enstrophyand en-ergy) invariants,while weknow that thequasigeostrophi model preservesan innitenumberofCasimirs(1.13).

Anothermeasureofinformational ontentis

relative entropy

,alsoknownas theKullba k-Leibler'sdistan eor thedivergen e [41℄

S[ρ, Π] =

₋

Z

K

ρ ln

³

ρ

Π

´

dy,

(1.27)

where

Π

is a probability density over

K

representing an externalbias due to someadditionalinformation. Forexample,inthequasigeostrophi model(1.12) there areaninnitenumber ofCasimirs(1.13). Thismeansthat formallyone has to onsider an innite number of onstraints on the entropy. To nd a solution to all these onstraints might be a di ult or even impossible task. Insteadofthis,

Π

anbe hosensu hthatitree tstheCasimirs,andtherefore itgivesanexternalbiasonthespatialdistributionofPV.

It an be shownthat

S

is non-positive, and

S

is zeroonly if

ρ

≡ Π

every-where. Thisexplainsthetermdistan efor(1.27). Thedensity

Π

isoften alled a

prior distribution

andin atypi alappli ation

Π

isgivenand

S

ismaximized overpossible hoi esof

ρ

.

1.3.3

Statistical theories for quasigeostrophic flow

One on lusion of this thesis is that numeri al methods an give ompletely dierent statisti al behaviour depending on their onservation properties. In order to dis uss statisti al theories of numeri al dis retizations of the quasi-geostrophi equation wewant to des ribe, rst, several statisti al theories for the ontinuousquasigeostrophi model. Thenthereader anmoreeasilyfollow the statisti altheories derivedfor Arakawa'ss heme inChapter 2 and forthe Hamiltonianparti le-mesh methodinChapter3.

Energy-enstrophy statistical theory

Theequilibriumstatisti alme hani altheoryfor2Didealuidswas developed by Krai hnan [40 ℄, Salmon et al. [76 ℄, andCarnevale& Frederiksen [11 ℄. It is based ona nite trun ation of the spe tral de omposition of theequations of motion. Statisti alpredi tionsareobtainedforthetrun atedsystem,andthese are extended to theinnite dimensional limit. In this se tion we go through thederivationofenergy-enstrophystatisti altheorydes ribedin[48 ℄foranite trun ationofthequasigeostrophi equation(1.12).

The Fourier spa e equation (1.15) of the quasigeostrophi model (1.12) is derivedthrough two-dimensional Fourier series. Thestandard trun ation, the Galerkin trun ationofthequasigeostrophi equation(1.12) anbeinterpreted as theFourier spa eequation(1.15)limitedtothenite

(2M + 1)

× (2M + 1)

domainoftheFourier oe ients

dˆ

q

k

dt

=

M

X

k

′

1

,k

′

2

=−M

k

′

6=0

k

_{× k}

′

|k

′

|

2

q

ˆ

k+k

′

(ˆ

q

−k

′

− ˆh

−k

′

)

(1.28)

with oe ients

q

ˆ

k

having period

(2M + 1)

in

k

. The trun ated potential vorti ity

q

M

andstreamfun tion

ψ

M

haveto satisfythePoissonequation,the se ond equationin(1.12)aswell

∆ψ

M

= q

M

− h

M

.

(1.29)

Onlylinearandquadrati onservedquantitiessurvivethetrun ation. There-forethe trun atedenergy

E

M

and thetrun ated enstrophy

Z

M

are onserved in thenite-dimensionally trun ateddynami s

E

M

=

1

2

M

X

k

′

1

,k

′

2

=−M

k

′

6=0

|ˆq

k

− ˆh

k

|

2

|k|

2

=

1

2

M

X

k

′

1

,k

′

2

=−M

k

′

6=0

|k|

2

| ˆ

ψ

k

|

2

,

(1.30)

Z

M

=

1

2

M

X

k

′

1

,k

′

2

=−M

k

′

₆₌₀

|ˆq

k

|

2

=

1

2

M

X

k

′

1

,k

′

2

=−M

k

′

₆₌₀

| − |k|

2

_ψ

_ˆ

k

+ ˆ

h

k

|

2

.

(1.31)

1.3. Statistical mechanics of fluids

23

An(exa t)steadystatesolutionofthetrun atedequation(1.28)is

q

M

= µψ

M

withs alar

µ

,anditisnonlinearlystablefor

µ >

−1

.

The phase ow of (1.28) satises the Liouville property. To simplify the index notationwerewrite

{ˆq

k

: k

1

, k

2

∈ [−M, M]} = {ˆq

1

, . . . , ˆ

q

Λ

}

forsome

Λ

. Dene

y

=

{Re ˆq

1

, Im ˆ

q

1

, . . . , Re ˆ

q

Λ

, Im ˆ

q

Λ

}

,

y

∈ R

2Λ

≡ R

d

, then (1.28) anbe written ina ompa tform

dy

dt

= f (y)

with

f = (f

1

, . . . , f

d

)

su hthat

f

k

doesnotdependon

y

k

(

f

k

dependson

y

k

⇔

in(1.28)

k

′

= 0

,whi h is ex luded from the sum). This immediately implies the Liouville property, sin e

div

y

f = 0

.

Having the Liouville property, we an derive a statisti al theory based on preservationof energyand enstrophy. This is alled the

energy-enstrophy

sta-tistical theory

.

We maximize the Shannon entropy

S

of (1.24) under the following on-straints

• ρ(y) ≥ 0

;

•

normalization onstraint

R

R

d

ρ(y) dy = 1

;

•

meanenergy onstraint

hE

M

i ≡

R

R

d

E

_M

(y)ρ(y) dy =

E

withxed

E

;

•

meanenstrophy onstraint

hZ

M

i ≡

R

R

d

Z

M

(y)ρ(y) dy =

Z

withxed

Z

. The variational prin iple forndingthe maximizerof

S

under the above on-straintsgives

G(y) = N

−1

exp [

_{−α (Z}

M

+ µE

M

)] ,

(1.32)

where

µ

and

α

areLagrangemultipliers orrespondingtothemeanenergyand meanenstrophy onstraints,respe tively,and

N

orrespondstonormalization. Thisisa Gibbs-likedistribution.

Thedistribution existsif

N =

Z

R

d

exp [

_{−α (Z}

M

+ µE

M

)] dy <

∞.

To guaranteenormalizationweneedto ensurethatthe oe ientsofthe qua-drati termsin (1.32) with substitutedtrun ated energy(1.30) and trun ated enstrophy(1.31), arenegative. A ordingto[48 ℄, thisimpliesthat

α > 0

and

µ >

−1.

It anbeshownthat

G

isaprodu tofGaussians,whi hallowsderivationof themeaneldequationforthenite-dimensionaldynami sdes ribedby(1.28)

ThemeanstatehastosatisfythePoissonequation(1.29)aswell,thus

∆

_hψ

M

i + h = µ hψ

M

i .

Toderivethestatisti altheoryfortheinnitedimensionalQGequation,one hasto takethe ontinuumlimitas

M

→ ∞

and he k whetherallpredi tions hold.

Miller’s equilibrium theory

Theenergy-enstrophystatisti altheorytakesintoa ountonlylinearand qua-drati invariants. Butasweknowthequasigeostrophi modelpreserveshigher Casimirsaswell. Therearestatisti alequilibriumtheoriesforidealuids,whi h arebasedon onservationof allCasimirs,derivedbyLynden-Bell [47 ℄,Robert &Sommeria[71 ,72 ℄, andMiller[54 ℄. Inthis se tionwe onsiderMiller's equi-librium theory. Itwas originallydevelopedfor theEuler equations, whi h are mathemati allyequivalenttothequasigeostrophi equationwithtrivial topog-raphy.

For derivation of Miller'stheory it is ne essary to onsider preservation of theCasimirfun tionals(1.13)astheareapreservationofPVlevels(1.14). Then theprobabilitydensityfun tionasso iatedtothePVvalue

σ

atapoint

x

∈ D

is denoted by

ρ(σ, x)

. In ompressibility impliesthe onstraint

R

+∞

−∞

dσ ρ(σ, x) =

1

, and area preservation implies the onstraint

R

D

dx ρ(σ, x) = g(σ)

. Miller dis retized

q

ona latti eof ne size

a

, assuming onstantvalues of PV

σ

ℓ

on ea h ell

ℓ

of ane mesh. Thereforethepermutationsof PVvalues

{σ

ℓ

}

form themi ros opi ongurationspa e. Thema ros opi vorti ityeldisthelo al averageofthemi ros opi eldona oarsemesh. Using ombinatorialanalysis and lettingthesize ofthene latti e

a

→ 0

,Millermaximizestheentropyfor

ρ

undertheabove onstraintstoarriveat thedistribution

G =

R

+∞

exp [

−βσhψi(x) + µ(σ)]

−∞

dσ exp [

−βσhψi(x) + µ(σ)]

,

where

hψi

is theimpli itly dened expe tation streamfun tion,

β

determines the energy and

µ(σ)

are Lagrange multipliers to ensurethe area-preservation onstraint.

Themain obsta le in setting upMiller's statisti al theoryfor a dynami al systemissolvingthenonlinearrelationsfor

hψi

.

Prior distribution

In the statisti al theories of Lynden-Bell [47 ℄, Robert & Sommeria [71 , 72 ℄, and Miller [54 ℄, vorti ity invariants are treated mi ro anoni ally in the sense that

µ(σ)

is hosenas aLagrange multiplier tosatisfy onstraintsonthearea distribution fun tion

g(σ)

. Analternativeapproa hdevelopedbyEllis,Haven

1.3. Statistical mechanics of fluids

25

energy and ir ulation onstraints mi ro anoni ally. Canoni al treatment of vorti ityinvariantsresultsin therelativeentropy,see(1.27),

S[ρ, Π] =

₋

Z

D

ρ ln

³

ρ

Π

´

dx,

with prior distribution

Π(σ)

determined with respe t to the vorti ity invari-ants. When a prior distribution is given, the statisti al equilibrium state is obtainedbymaximizingtherelativeentropy

S[ρ, Π]

atxedenergy, ir ulation andnormalization ondition.

The variational prin iple for nding the maximizer of relative entropy

S

under theabove onstraintsgives

G =

R

+∞

Π(σ) exp [(

−βhψi(x) + α)σ]

−∞

dσΠ(σ) exp [(

−βhψi(x) + α)σ]

,

where

β

and

α

areLagrangemultipliers orrespondingtothemainenergyand ir ulation onstraints,respe tively.

Note an equivalen e of Miller'sand Ellis, Haven & Turkington's distribu-tions,when

e

µ(σ)

_{= e}

ασ

_Π

,i.e.

µ(σ) = ln (e

ασ

_Π)

.

Miller's statisti al theory is based on the assumption that the ow is de-s ribed by 2D Euler equations, without for ing and dissipation. However, in geophysi al situations, theowis for ed anddissipated at smalls ales, whi h destroysthe onservationofCasimirs. Ellis,Haven&Turkington'sequilibrium theory takes are of this situation by xing the prior distribution instead of vorti ity invariants, sin e it is more easy to determine the prior distribution fromdatathantodeterminehighermomentsoftheatmospheri vorti ity a u-rately. For more omparisonbetweenMiller'sandEllis,Haven&Turkington's statisti al theoriesseeChavanis[13 ℄.

Overview of thesis

In Chapter 2, we will onsider quasigeostrophi ow with topographi for -ing. Wewill onstru tstatisti alme hani altheoriesforthedis retedynami al systems arising from three dis retizations due to Arakawa [2 ℄ whi h onserve energy,enstrophyorboth. TherefereestoSe tions1.1and1.3 ouldbehelpful forthereader.

In Chapter 3, we will onsider the Hamiltonian parti le-mesh method for quasigeostrophi owover topography. Wewill propose Lagrangianand Eule-rian statisti al models for thedis rete dynami s. Thereferees to Se tions 1.2 and1.3 ouldbehelpfulforthereader.

InChapter4,wewill onsiderthepointvortexowonadisk. Wewillpresent a losure for in ompressible ideal uid ow in the form of the mathemati al thermostat. TherefereestoSe tion 1.3 ouldbehelpfulforthereader.