Statistical mechanics and numerical modelling of geophysical fluid dynamics - Chapter 3: Statistical mechanics of the Hamiltonian particle-mesh method

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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 3

Statistical mechanics of the

Hamiltonian particle-mesh

method

3.1

Introduction

Computationalappli ations inatmosphereand o ean s ien eofteninvolvethe

simulation of geophysi al uids on time intervals mu h longer than the

Lya-punove-foldingtime. Intheseinstan esthegoalofsimulationistheevaluation

ofstatisti alquantitiessu hasaveragesand orrelations. Itistherefore

impor-tant to investigate thea ura y of numeri al dis retizations in the ontext of

statisti al averages.

TheHamiltonian parti le-mesh (HPM)method was originally proposed in

the ontext of rotating shallow water ow in periodi geometry in [25 ℄ and

extendedto other physi al settingsin [16 , 17, 29 , 83 ℄. The uidis dis retized

ona niteset ofLagrangian parti lesthattransport themassof theuidand

persist during the ow evolution. The HPM method is symple ti , and one

an onstru ta ontinuumvelo ityeldinwhi hthedis reteparti levelo ities

are embedded for all time. The ontinuum velo ity eld satisies a Kelvin

ir ulation theorem, implyingmaterial onservation of potentialvorti ity and

invarian eoftheinnitefamilyofCasimirfun tionals,see[28 ℄.

For the ase of ideal uid ow in two dimensions, the HPM variant was

des ribed in [17 ℄. We will apply the HPM method to the quasigeostrophi

potential vorti ity equation des ribing a 2D barotropi ow over topography.

Here, itisthepotentialvorti ity(PV),andnot themass,that isxedonea h

parti le and adve ted in a divergen e-free velo ity eld. In this asewe show

that the parti le motion may beembedded in an area-preserving ow on the

uid label spa e. Hen e, an arbitrary fun tion of PV may be integrated by

quadratureoverlabelspa eandistherefore onserved. Ontheotherhand,this

trivial onservationisapparentlyduetothefa tthata valueofPVisassigned

toea hparti leon eandforall,anddoesnotimplyanyredu tionofthenumber

ofdegreesoffreedomoftheowevolutioninthesenseof,say,adis reteenergy

vorti ity eld, dened on a uniform grid, and numeri al simulations indi ate

thatonlyenergyandlinearfun tionalsofPVare onservedatthisma ros opi

s ale.

Consequently,one mayquestiontowhat degreethePV onservationofthe

HPM method is meaningful. Long time simulations of nonlinear dynami al

systemsaretypi allyina urateinapointwisesenseandratherare arriedout

withthegoalofsamplinganequilibriumprobabilitydistributionoverthephase

spa e under an assumption of ergodi ity. The relevant statisti al equilibrium

distribution is a fun tion of the onservation laws that restri t the dynami s.

Hen e,ifthePV onservationbyHPMistrulytrivial,thestatisti sshould

ad-here tothat ofanenergy- ir ulationequilibrium theory, whereasa meaningful

PV onservation should lead to a ri her statisti al equilibrium. Inthis

hap-ter wewill showthat thePV onservationof HPMsigni antly inuen esthe

statisti sofsimulationdataobtainedwiththeHPMmethod.

Sophisti ated statisti al equilibrium theories for ideal uids are based on

onservation ofgeneralizedenstrophiestheintegrals over thedomainof

arbi-trary fun tions of PVor, equivalently, on the area-preservation property of

thevelo ityeld. Anequilibriumtheorywithmi ro anoni altreatmentof

vor-ti ity invariants was developed independently by Lynden-Bell [47 ℄, Robert &

Sommeria [71 , 72℄, and Miller [54 ℄. An alternative approa h that treats

vor-ti ity invariants anoni allywas developedbyEllis, Haven &Turkington[22 ℄,

see Se tion1.3.3. Seealso Chavanis[13 ℄for a omparison ofthetwo. Our

nu-meri al method hasfeaturesin ommonwith themodel usedto onstru tthe

latter ontinuumtheory(spe i ally,anaturaltwo-s alestru ture),andinthis

hapter we derive analogous dis rete statisti al equilibrium models based on

both Lagrangian (Se tion 3.4) and Eulerian (Se tion 3.5) uid onsiderations

and omparethesemodelswithsimulations(Se tion3.6).

We wish to emphasize here that the obje tive of this hapter is to show

that the dis rete statisti s of the HPM method are in good agreement with

predi tions ofthemodern ontinuumtheory. Thisimpliesthatthe ontru tion

of the HPM method respe ts the dynami al onsiderations that go into the

theory, and that for the numeri al experiments in luded, the dis rete ow is

su ientlyergodi toobserve onvergen eoftheensemble averages.

Thequasigeostrophi potentialvorti ity(QG)equation[48 ,67 ,74℄des ribes

barotropi divergen e-freeowover topography

d

dt

q(x, t) = 0,

∆ψ(x, t) = q(x, t)

− h(x),

(3.1)

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

isrelatedto thestreamfun tionby

u

=

∇

⊥

_ψ

, where

∇

⊥

_{= (}

₋

∂

∂y

,

∂

∂x

)

T

. In this hapterwe onsidertheQGequationona doublyperiodi domain

The QG model des ribes a Hamiltonian PDE with Lie-Poisson stru ture

[74 ℄, implyingthe onservationofthetotalkineti energy

H = −

1

₂

Z

D

ψ

_{· (q − h) dx}

(3.2)

as wellastheinnite lassofCasimirfun tionals

C[f] =

Z

D

f (q) dx

(3.3)

for any fun tion

f

for whi h theintegral exists. Of parti ularinterest arethe PV moments:

C

r

=

Z

D

q

r

dx,

r = 1, 2, . . . ,

(3.4)

andespe iallythe ir ulation

C

1

andenstrophy

C

2

.

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}.}

(3.5)

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

q

, this fun tionis inde-pendentoftime

∂G

∂t

= 0

. Dierentiatingwithrespe tto

σ

,thefun tion

g(σ) =

∂G

∂σ

(3.6)

ispreserved. Forthe aseofapie ewiseuniformPVeld,

q(x, t)

∈ {σ

1

, . . . , σ

Λ

}

, thequantity

g

ℓ

= G(σ

ℓ+1

)

− G(σ

ℓ

)

isthemeasureofthevorti itylevelset

σ

ℓ

.

3.2

Review of continuum statistical equlibrium

theories

Givena (spatially)dis reteapproximation

q(t)

to thesolution

q(x, t)

of(3.1), obtained from a numeri al simulation,one wouldlike toanalyze thea ura y

of omputedaveragesoffun tions of thesolution. For example,thelong time

averageofafun tion

F (q(t))

ofthePVeldisdenoted

F = lim

T →∞

1

T

Z

t

0

+T

t

0

F (q(t))dt.

If thedis retedynami s isergodi withrespe t toa uniqueinvariant measure

p(q)

onthephasespa e,thenthelongtimeaverageisequivalenttotheensemble averagewithrespe tto

p

,

hF (q)i =

Z

where theintegral is over the (fun tion) spa e of PV elds, and it su es to

derivetheinvariantmeasureasso iatedwiththenumeri almethod,andanalyze

thiswith respe ttowhatisknownaboutthe ontinuummodel.

Inaneortto hara terize thelong timemeanbehavior ofidealuidsand

explaintheirtenden ytoorganizeintolarge-s ale oherentstru tures,anumber

of authorshaveapplied ideasfrom statisti al me hani s. Thepioneeringwork

was that ofOnsager [65℄, whi haddressed the statisti alme hani sof a nite

pointvortexmodel. Heobservedthatforaboundeddomain,theavailablephase

spa emusteventuallyde reaseas afun tionofin reasingsystemenergy,

lead-ingtonegativetemperatureregimesinthemi ro anoni alstatisti alensemble.

Healsopredi tedthatinaheterogeneous systemoflike-signed and

oppositely-signedvorti esofvaryingstrengths,thelargevorti eswouldtendto lusterso

as toa hievemaximumdisorder withminimumdegrees offreedom,like-signed

vorti es at negativetemperatures andvi e-versa. Thesepredi tions were

on-rmednumeri allyin[9℄,andthisproblemundertheinuen eofmathemati al

thermostat willbe onsidered inChapter4.

Statisti alme hani stheoriesbasedona Fourier-spe traltrun ationofthe

Euler equations were proposed by [11 , 40, 76 ℄, see also Se tion 1.3.3. The

spe traltrun ationpreserves ir ulation

C

1

andthequadrati fun tions

E

and

C

2

only. Consequently,atreatmentbasedon onstrainedmaximumentropy(for anintrodu tion,see [48 ℄)yieldsGibbs-likedistributions

p(q)

over thevorti ity eld, with Gaussian distribution of lo al vorti ity u tuations. The

energy-enstrophy theory predi ts a linear relation between ensemble average stream

fun tionandpotentialvorti ity:

hq(x)i = µhψ(x)i,

(3.7)

for a s alar

µ

depending on the observed energy and enstrophy, where the ensemble average is dened with respe t to integration over an appropriate

fun tionspa einthis ase.

Given that it is not just the enstrophy

C

2

but any fun tional

C[f]

that is invariant under the ontinuum ow of (3.1), it is natural to ask what ee t

the more general onservation laws haveon statisti s. Abramov & Majda [1 ℄

investigatedthestatisti alsigni an eofthehigherPVmoments

C

r

,

r > 2

, nu-meri allyusingthePoissondis retizationofZeitlin[50 ,88 ℄,whi h onserves

M

Casimirsofan

M

× M

-modetrun ation,see Se tion1.2. Computingthelong timeaveragedPVandstreamfun tionelds,theyobservedin reasing

dis rep-an y relativeto the linearmean eld theory (3.7), as a fun tionof in reasing

skewness

C

3

oftheinitial ondition,thusprovingthestatisti alrelevan eofthis quantity.

Statisti alequilibriumtheoriesin orporatingthefullfamilyofCasimirs

im-pliedbythepreservationofarea(3.5)wereindependentlyproposedby

Lynden-Bell[47 ℄inthe ontextofastrophysi s,andMiller[54 ,55 ℄ andRobert&

Som-meria[71 ,72 ℄, seeSe tion1.3.3. These originaltheoriesused ami ro anoni al

treatment of the Casimirs

C[f]

. That is, the equilibrium distribution is de-rivedbyminimizing entropy under onstraintsofenergyand theentire family

of Casimirs. Morere ently, Ellis,Haven&Turkington[22 ℄ proposed an

alter-nativetheoryfeaturing anoni altreatmentoftheCasimirs,inwhi hthepoint

statisti sofPV isdes ribedbya priordistribution,see Se tion1.3.3. Inallof

thesepapers,a oarse-grainpotentialvorti ityeldisdes ribedbyaprobability

densityoverthe lassofne-grainPVdistributionsatea hpointinthedomain.

InChapter2 weanalyzedenergyandenstrophy onservingnitedieren e

methods forthe QG model under topographi for ing,and observed that the

dis rete time-averaged mean elds

q

and

ψ

obtained depend heavily on the onservation properties of the dis retizations used. For a dis retization that

onservesenergyonly,thepredi tedmeaneldisuniformlyzerovelo ity

hΨi =

0

. Given that theonly dynami ally onserved quantities ofthe HPMmethod are energy and total ir ulation (see below), any departure from the trivial

mean eld is an indi ation of the statisti al relevan e of the other onserved

quantities,namelytheareameasure (3.5).

3.3

Hamiltonian particle-mesh method

TheHamiltonianparti le-mesh(HPM)methodisanumeri aldis retizationof

invis iduiddynami sthatretainsHamiltonianstru ture. Themethodmakes

useofaLagrangian uiddes ription,toadve tuidparti leswhile onserving

energy, andanEulerian gridforevaluating derivativesusingnite dieren es.

Themethodwasadaptedfor2Din ompressibleowin[17 ℄.

3.3.1

HPM description

ThePVeldisdis retizedbyintrodu ingasetof

K

dis reteparti leswithxed potentialvorti ity

Q

k

,

k = 1, . . . , K

. Theparti les havetime-dependent posi-tion

X

k

(t)

∈ D

, and areadve ted in a divergen e-freevelo ity eld a ording to

d

dt

X

k

=

∇

⊥

_{Ψ(x, t)}

¯

¯

_x

₌

_X

k

(t)

where thestreamfun tion

Ψ

is des ribedbelow.

Wealsomakeuseofa uniform

M

× M

gridon

D

,withgridspa ing

∆x =

∆y = 2π/M

, and denote gridpoints by

x

i

. Given a dis rete stream fun tion

Ψ

i

(t)

onthegrid,we onstru ta ontinuouseld via

Ψ(x, t) =

X

i

Ψ

i

(t)φ

i

(x),

(3.8)

where

φ

i

(x) = φ( x

−

x

i

∆x

)

is a ompa tly supported basis fun tion satisfying symmetry,normalizationandpartitionofunityproperties, respe tively:

φ(x) = φ(

_−x),

Z

D

φ(x) dx = 1,

X

i

φ

i

(x) = 1,

∀x ∈ D.

(3.9)

Inourimplementationweusethetensorprodu tofnormalized ubi B-splines

φ(x) = φ

0

(x) φ

0

(y)

,where

φ

0

(r) =











2

3

− |r|

2

+

1

2

|r|

3

,

|r| ≤ 1,

1

6

(2

− |r|)

3

,

1 <

|r| ≤ 2,

0,

otherwise.

Thedis retestreamfun tion

Ψ

i

(t)

isobtainedbysolvingaPoissonequation onthegrid. Givenadis retegrid-basedPV eld

q

i

(t)

wesolve

X

j

∆

ij

Ψ

j

= q

i

− h

i

(3.10)

where

h

i

= h(x

i

)

is the topography fun tion sampled at gridpoints and

∆

ij

is an appropriate dis retization of the Lapla ian. In our implementation, we

usea spe tralapproximationandFFTs,but anitedieren eformulamaybe

su ient.

Finally, thePV eld on the gridis approximated from the parti les using

therelation

q

i

(t) =

X

k

Q

k

φ

i

(X

k

(t)) .

(3.11)

In[28 ℄ itisshownthat theaboveformulasamplestheexa t solutionofa

on-tinuity equationof theform

q

t

+

∇ · (q ˆ

u) = 0

with density fun tion

q(x, t) =

P

k

Q

k

φ(x

− X

k

)

andauxiliary velo ityeld

u(x, t)

ˆ

appropriatelydened. In thepresent ase,althoughtheparti levelo ityeldisgivenby

∇

⊥

_{Ψ(x, t)}

and

isthereforedivergen e-free,thiswill onlyholdinanapproximate senseforthe

auxiliary velo ityeld

u(x, t)

ˆ

.

Inthe present ontext of vortex dynami s, the HPM method is related to

the lassi al point vortex ow (see [15 ℄ and referen es therein). The singular

point vorti es have been regularized by onvolution with the basis fun tions

φ

. TheEuleriangridredu esthe omplexityofvortex-vortexintera tionsfrom

O(K

2

₎

to

O(K ln K)

(using FFT). The onstru tionof themethod preserves the Hamiltonian stru tureof the point vortex ow. However,as notedin the

introdu tion, the HPM method was originally in the setting of ompressible

owandisinthissenseappli abletomoregeneraluidsthanthepointvortex

3.3.2

Properties of the discretization

By onstru tion,thenumeri almethoddes ribedabovedenesa Hamiltonian

system. TheHamiltonianis

H(X) =

−

1

₂

X

i

Ψ

i

(q

i

− h

i

) ∆x

2

=

−

1

₂

X

i,j

"Ã

X

k

Q

k

φ

i

(X

k

)

!

− h

i

#

(∆

−1

)

i,j

"Ã

X

ℓ

Q

ℓ

φ

j

(X

ℓ

)

!

− h

j

#

∆x

2

.

(3.12)

Introdu ingphasespa e oordinates

X

= (X

1

, . . . , X

K

, Y

1

, . . . , Y

K

)

T

and

sym-ple ti two-formstru turematrix

B =

"

0

_{−diag Q}

diag Q

0

#

,

theequationsofparti lemotionaredes ribedby

B

dX

dt

=

∇H(X).

TheHamiltonianisarstintegralofthedynami sandapproximatesthetotal

kineti energy. Additionally, the phase ow is symple ti and onsequently

volume-preservingon

R

2K

.

Tointegratethenumeri aldis retizationintimeweusetheimpli itmidpoint

rule:

B

X

n+1

− X

n

∆t

=

∇H

µ X

n+1

+ X

n

2

¶

.

The numeri al map is symple ti for this problem, implying that volume is

preserved in the

2K

-dimensional phase spa e of parti le positions. Also the energy is well-preserved, with u tuations bounded by a term of

O(∆t

2

₎

for

long times, onsistentwiththeoryreportedin [35 ,45℄,see alsoSe tion 1.1.

Sin etheparti lePV values

Q

k

,

k = 1, . . . , K

arexed forthedurationof the omputation, PV is onserved along parti le paths for any motion of the

parti les. However,sin ethe

Q

k

playtheroleofparametersinthespe i ation of HPM, their onservation does notimply a redu tion in degrees of freedom

of thedynami sin theway exa t onservation of

H

does. Ontheotherhand, the motion of the parti les is not arbitrary, but area-preserving in the sense

des ribed next. The ombination of material onservation of PV in an

area-preserving ow is the essential feature of thene s ale motion of ideal uids

that entersinto themodernstatisti alme hani stheories.

Given an arbitrary ontinuous motion of the parti les

X

k

(t)

, equations (3.11),(3.10)and(3.8)denea ontinuumapproximatestreamfun tion

Ψ(x, t)

, withvelo ityeld

Letusdenelabel oordinates

a

= (a, b)

on

D

andtheLagrangianow

χ(a, t) :

D × R → D

indu edby

Ψ(x, t)

:

∂

∂t

χ(a, t) = U (χ(a, t), t),

(3.13)

Sin e

∇ · U ≡ 0

,theLagrangianow

χ

isarea-preservingon

D

. Thatis,

| det

∂χ

_∂a

| ≡ 1.

This propertyis retained undertemporalsemi-dis retization with theimpli it

midpointrule. Thatis, themapping

χ

n

_(a)

_{7→ χ}

n+1

_(a)

isarea-preserving.

Ontheotherhand,forthenumeri almethod, theparti lemotion

X

k

(t)

is justgivenby

X

k

(t) = χ(X

k

(0), t),

i.e. theparti lemotionisembeddedinitsownLagrangianow. Therefore,the

dis reteparti lemotionisarea-preservinginthesensethatit anbeembedded

in anarea-preservingow.

Typi ally, weinitializethe parti lesona uniformgrid

1

withspa ing

∆a =

∆x/κ

forsomepositiveinteger

κ

. Let

a

k

∈ D

denotetheinitialpositionofthe

k

thparti le. Let

A

k

denotethesetoflabelsinthegrid ell enteredat

a

k

:

A

k

=

{a ∈ D : |a − a

k

|

∞

<

∆a

2

}.

Then

D = ∪

k

A

k

. Denea pie ewise onstantinitialvorti ityeldthrough

Q

0

(a) =

X

k

Q

k

1

k

(a),

(3.14)

where

1

k

isthe hara teristi fun tionon

A

k

. Thisvorti ityeldistransported bytheow

χ(a, t)

via

Q(χ(a, t), t) = Q

0

(a).

(3.15)

Givenanyfun tion

f (Q)

,wehave

Z

D

f (Q(x, t)) dx =

Z

D

f (Q(χ(a, t), t))

¯

¯

det

∂χ

∂a

¯

¯

da

=

Z

D

f (Q

0

(a))

¯

¯

det

∂χ

∂a

¯

¯

da

=

X

k

f (Q

k

)

Z

A

k

¯

¯

det

∂χ

∂a

¯

¯

da

=

X

k

f (Q

k

)

|A

k

|

= ∆a

2

X

k

f (Q

k

),

(3.16)

1

_{For an arbitrary initial particle configuration, the subsequent quadrature could be carried}

whi h is onstant. Inparti ular, thearea asso iatedwith anyparti ular level

set ofPVis onserved.

In this sense wesee that the ne s ale parti le ow trivially onserves all

Casimirs,andin parti ularthepolynomialsfun tions

C

r

= ∆a

2

X

k

Q

r

k

,

r = 1, 2, . . . .

However,thispropertydoesnottransfertothegriddedPVeld

q

. Thatis,the grid-based analogs

ˆ

C

r

= ∆x

2

X

i

q

i

r

,

r = 1, 2, . . .

arenot onserved ingeneral. Thesoleex eptionisthetotal ir ulation

C

ˆ

1

for whi hwehave,usingthethirdpropertyof(3.9),

ˆ

C

1

=

X

i

q

i

∆x

2

=

X

i

X

k

Q

k

φ

i

(X

k

)κ

2

∆a

2

=

X

k

Q

k

κ

2

∆a

2

,

whi h is independent of time. For arbitrary nonlinear

f (q)

, one would not expe tthequantity

P

i

f (q

i

)∆x

2

to beinvariantingeneral. InFigure 3.1weplottherelativedrift

ε

rel

[H](t) =

¯

¯

¯

¯

H(t)

_{− H(0)}

H(0)

¯

¯

¯

¯

in thequantities

H

and

C

ˆ

r

,

r = 2, . . . 4

as afun tions oftime duringa typi al simulation (theexperiment des ribedin Se tion 3.6.3, forthe ase

γ = 0

,

δ =

90

). The ir ulation ispreserved toma hine pre ision andis notshown. The energyos illationsarebounded by

ε

rel

[H](t) < 2.1

× 10

−4

,

andtheboundde reasesquadrati allywithstepsize. Thehigherordervorti ity

momentsarenotpreserved,anden ounter relativedrifts

max

t

ε

rel

[ ˆ

C

2

](t) = 0.74,

max

t

ε

rel

[ ˆ

C

3

](t) = 16.9,

max

t

ε

rel

[ ˆ

C

4

](t) = 16.5.

Clearly,thesearenot onserved.

In some ases, it is useful to onsider the bulk motion of the uid to be

pres ribedbyatimedependentstreamfun tion

Ψ(x, t)

,and onsiderthemotion ofatypi alparti leembeddedintheow. Themotionofsu haparti lesatises

a nonautonomous Hamiltoniansystem. This pointof view andits oupling to

thedynami sisstudied in[5 ℄. Thepar elHamiltonianbe omes

˜

H =

Z

0

100

200

300

400

500

600

700

800

900

1000

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

10

1

10

2

time,t

ε

r

e

l

H

C

2

C

3

C

₄

Figure

_{3.1: Relative change in energy H and higher vorticity moments ˆ}

_C

₂

_,

ˆ

C

3

, ˆ

C

4

during the simulation described in Section 3.6.3, for the case γ = 0,

δ = 90.

andthedynami sonlabelspa e(3.13)satisfy

q

0

(a)

d

dt

χ(a, t) = J

δ ˜

_H

δχ

= q

0

(a)

∇

⊥

_{Ψ(χ(a, t), t),}

∀a.

where

J =

³

_{0 1}

−

1 0

´

.

Similarly,fortheHPMdis retization,

Q

k

∂

∂t

X

k

= J

∇X

k

(t)

˜

H = Q

k

∇

⊥

Ψ(X

k

(t), t),

(3.17)

where

˜

H =

X

k

Q

k

Ψ(X

k

(t), t).

(3.18)

63

3.4

A Lagrangian statistical model based on

canonical particle distributions

DuetoitsLagrangiannature,theHPMmethodhassimilaritieswiththe

point-vortexmethodwhosestatisti alme hani swas onsideredbyOnsager[65 ℄. The

parti lemotion anbe onsideredaregularizedpointvortexmethod. Thephase

spa eoftheHPMmethodisbounded: itissimply

D

K

. In ontrasttothepoint

vortexmethod, therangeof energyfortheHPM method isalso bounded (for

nite

Q

k

,

k = 1, . . . , K

). If one onsiders possible ongurations for a given energy, as the energy level be omes large enough the available phase spa e

eventually starts to de rease. In other words, the HPM method supports a

negativetemperatureregime.

Inthis se tion we onstru ta statisti al equilibrium theory in thenatural

phase spa e of HPM parti le positions

X

k

∈ D

. However, in some ases it maybepreferabletodire tly onsiderthestatisti softhe oarse-grainvorti ity

eld

q

onthegrid(3.11),whi hallows omparisionwiththeexistingequilibrium eldtheories. Inthenextse tionwepresentanEulerianapproximatestatisti al

modelfromthispointofview. Todistinguishthetwo,werefertothetheoryin

thisse tionas theLagrangian statisti alme hani al model.

Let us onsider the statisti s of a single distinguished parti le in onta t

withthereservoirformedbyallotherparti les. Re allthatthemotionofsu h

aparti leobeysanonautonomousHamiltoniansystem(3.17)withHamiltonian

(3.18). Theenergy ontribution of parti le

k

is

Q

k

Ψ(X

k

, t)

. We expand the streamfun tionabouttheensemblemeaneld

Ψ(x, t) =

hΨ(x)i + δΨ(x, t).

Negle ting the long time ee ts of the perturbation part

δΨ

, we obtain the anoni aldistributionforadistinguishedparti le(seeSe tion1.3.1about

anon-i al sampling)

ρ

k

(x) =

1

ζ

k

exp [

−βhΨ(x)iQ

k

] ,

ζ

k

=

Z

x

∈D

exp [

−βhΨ(x)iQ

k

] dx. (3.19)

Figure3.2 omparestypi alfun tions

ρ

k

(x)

withhistogramsofpositiondata for two arbitrarily hosen parti les with

Q

k

+

= 1.098

and

Q

k

−

=

−2.165

ob-tained from HPM simulations with normally distributed

{Q

k

}

. We observe goodagreement. Dueto the hoi eof topographyin Se tion3.6 andnormally

distributed

Q

k

,thedistributions

ρ

k

areuniformin the

y

dire tion.

Theone-parti le anoni alstatisti s an beused to onstru ta meaneld

theory. For parti le

k

theone-parti le statisti sis (3.19). This quantitygives the probability that

X

k

is near

x

∈ D

. Next onsider oordinates

Ξ

=

(ξ

1

, ξ

2

, . . . , ξ

K

)

∈ D

K

ontheparti lephasespa e,andtheprodu tdistribution

ρ(Ξ) =

Y

k

0

2

4

6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

X

k

−

ρ

k

−

0

2

4

6

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

X

k

+

ρ

k

+

Figure

_{3.2: Histograms of x-component of position for two distinct particles}

(dash line), compared with the predicted canonical distribution (solid line).

whi h governs the probability of parti le ongurations under the modelling

assumptionthat theparti lepositionsareindependent.

To ea h

Ξ

∈ D

K

,isanasso iatedgrid-basedPVeld

q(Ξ)

with

q

i

(Ξ) =

X

k

Q

k

φ

i

(ξ

k

).

TheensembleaveragePV eldis

hq

i

i =

Z

q

i

(Ξ)ρ(Ξ) dΞ,

whi h anbesimpliedasfollows

hq

i

i =

Z

D

K

X

k

Q

k

φ

i

(ξ

k

)

Y

ℓ

ρ

ℓ

(ξ

ℓ

) dξ

1

· · · dξ

K

=

X

k

Q

k

Z

D

φ

i

(x)ρ

k

(x) dx

=

X

k

Q

k

R

D

φ

i

(x) exp [

−βhψ(x)iQ

k

] dx

R

D

exp [

−βhψ(x)iQ

k

] dx

=

X

k

Q

k

hφ

i

i

k

where

h·i

k

istheensembleaverage inthemeasure(3.19).

Ifwe onsider a pie ewise onstantvorti itydistribution with

K

ℓ

parti les withPV

σ

ℓ

,therelationsabove anbeexpressedas

hq

i

i =

X

ℓ

σ

ℓ

K

ℓ

R

D

φ

i

(x) exp [

−βhψ(x)iσ

ℓ

] dx

R

D

exp [

−βhψ(x)iσ

ℓ

] dx

=

X

ℓ

σ

ℓ

p

i,ℓ

(3.20)

where

p

i,ℓ

= K

ℓ

hφ

i

i

ℓ

.

(3.21)

Thisquantityistheproportionoftimethataparti lewithPV

σ

ℓ

spendswithin thesupportofgridpoint

x

i

,weightedbythekernelfun tion,timesthenumber ofsu hparti les.

To omputethemeanstate,weapproximatetheintegralin(3.20)by

quadra-tureatthegridpoints

hq

i

i ≈

X

ℓ

σ

ℓ

K

ℓ

P

j

φ

i

(x

j

) exp [

−βhψ

j

iσ

ℓ

]

P

j

exp [

−βhψ

j

iσ

ℓ

]

.

Thisrelationissolvedtogetherwith

∆

ij

hψ

j

i = hq

i

i − h

i

,

andthe onstraintrelation

H(

_{hqi) = H}

0

,

whi hspe iesthevalueof

β

.

OurLagrangianstatisti al theoryfortheHPMmethodisanalogous tothe

anoni al theory of Ellis, Haven & Turkington [22 ℄, inasmu h as the energy

is treated mi ro anoni ally through the spe i ation of

β

, and the ne-s ale vorti ity onservationistreated anoni ally.

3.5

Eulerian statistical model for HPM

The ontinuum statisti al me hani s theories of Miller [54 , 55 ℄ and Robert

[71 , 72 ℄ an be onstru ted using a two-level dis retization of the ontinuum

vorti ityeld. Themi ros opi ongurationspa e onsistsofpermutationsof

a pie ewiseuniform vorti ity eld,assuming onstant values on ea h ell ofa

ne mesh. The ma ros opi vorti ity eld is the lo al average of the

mi ro-s opi eld on anembedding oarse mesh. The ontinuum theory isobtained

by rstletting the ne mesh size tend to zeroforxed oarse gridmesh size,

andsubsequentlytakingthe ontinuumlimitofthe oarsemesh.

Asimilar approa hnegle tingthe ontinuumlimits an beused to

on-stru t a dis rete statisti almodelfor theHPM method. Keepingin mind the

interpolating ontinuum ow (3.15), wedene

p

ℓ

i

to bethe probability of

ob-serving

Q(x

i

, t) = σ

ℓ

neargridpoint

x

i

. Then

p

ℓ

i

hastheproperties

X

i

p

ℓ

i

∆x

2

= g

ℓ

,

X

ℓ

p

ℓ

i

= 1.

(3.22)

It is natural to asso iate

p

ℓ

i

with the hara teristi fun tions at the grid

points,smoothedbytheHPMbasisfun tions

φ

. Denoteby

K

ℓ

theindexsetof parti leswithvorti itylevel

σ

ℓ

(

ℓ = 1, . . . , Λ

, forsome

Λ

≤ K

):

K

ℓ

=

{1 ≤ k ≤ K : Q

k

= σ

ℓ

}.

Dene thefun tion

ϕ

ℓ

i

=

X

k∈K

ℓ

φ

i

(X

k

)

1

κ

2

.

If the parti les are initialized on a uniform grid of spa ing

∆a = ∆x/κ

, for

κ

_{≥ 1}

an integer, then

ϕ

ℓ

i

hasthe required properties (3.22). To onstru ta

Miller/RoberttheoryfortheHPMmethod,wewouldinitializetheparti leson

su hauniformgrid,and onsiderpermutationsofthe

Q

k

as anapproximation ofthe ongurationspa e.

The motionof parti les in the HPM method onservesenergy, as pointed

out in theSe tion 3.3. It istherefore ne essary to further restri tthe sample

spa e to those permutations of PV that preserve the initial energy to within

some toleran e. Denethe oarsegrain meanpotentialvorti ityby

hq

i

i =

X

ℓ

p

ℓ

i

σ

ℓ

,

(3.23)

the oarsegrainmeanstreamfun tionby

∆

_{hΨi = hqi − h,}

(3.24)

andtheenergyofthemeaneld by

H(

hqi) = −

1

₂

hΨi

T

₍

hqi − h) ∆x

2

_.

Substituting

ϕ

ℓ

i

for

p

ℓ

i

in (3.23), the above denitions are onsistent with the

( oarse-grain) grid quantities

q

,

Ψ

and

H

given in (3.11), (3.10), and (3.12), respe tively.

Ami ro anoni alstatisti almodelanalogoustotheMiller/Robertapproa h

pro eedsat thispointbyintrodu ingtheShannon informationentropy

S[p] =

₋

X

i,ℓ

p

ℓ

i

ln p

ℓ

i

,

(3.25)

andmaximizingthisfun tionwithrespe tto

p

ℓ

i

subje tto ontraintsofobserved

values ofenergy,

H(

hqi) = H

0

,andthe onditions(3.22).

Insteadwetakehere the alternativeapproa h proposed byEllis, Haven&

Turkington[22 ℄,whi hassumesa anoni alensemblewithrespe ttothehigher

order Casimirs, as determinedby a priordistribution over pointwise vorti ity,

in ombination with a mi ro anoni al distribution with respe t to

H

and

C

1

. This is onsistentwith observationsof invis iduids, where

H

and

C

1

depend

onlyonthelarges alevorti itywhereasthe

C

r

,

r > 1

dependonthenes ale detailed vorti ity and the lengths ale ofaveraging. To that end wedrop the

requirementthat

p

ℓ

i

satisfytherst onditionof(3.22).

Given a set of parti les initialized on a uniform grid with PV values

Q

k

,

k = 1, . . . , K

,we onsidertheasso iatedpie ewise onstant ontinuumvorti ity eld as des ribedin (3.14)(3.15). To ea h vorti ity level set

σ

ℓ

,

ℓ = 1, . . . , Λ

weasso iatethefra tionalarea

Π

ℓ

=

K

ℓ

∆a

2

|D|

,

where

K

ℓ

isthenumber ofparti leswithvorti ity

σ

ℓ

and

|D|

isthetotalarea of

D

. Notethat

P

ℓ

Π

ℓ

= 1

. We take

Π

ℓ

to bethe priordistribution onPV. Givenno other informationabout theow,

Π

ℓ

is theprobabilityof observing PV value

σ

ℓ

atanarbitrarily hosenpointin

D

. Theprobabilityisuniformin spa e.

To determine theprobability distribution

p

ℓ

i

we maximize the relative

en-tropy

S[p, Π] =

₋

X

i,ℓ

p

ℓ

i

ln

p

ℓ

i

Π

ℓ

.

(3.26)

Givennootherinformationaboutthesystem,we anmaximizethisentropy

as afun tionof

p

ℓ

i

tond

p

ℓ

i

= Π

ℓ

,

whi histhepriordistributionatea hpointonthegrid, onrmingtheearlier

statement.

Instead we wish to maximize (3.26) subje tto mi ro anoni al onstraints

ontheenergy

˜

E = H(

hqi) − H

0

= 0,

(3.27)

andthe ir ulation

˜

Γ = ˆ

C

1

(

hqi) − ˆ

C

1

(q(0)) = 0,

(3.28)

as wellasthenormalization onstraints

˜

N

i

=

X

ℓ

p

ℓ

i

− 1 = 0, ∀i.

(3.29)

Introdu ing Lagrange multipliers

β

,

α

and

λ

i

, respe tively, for these on-straints,wesolve

∂S

∂p

ℓ

i

+ β

∂ ˜

E

∂p

ℓ

i

+ α

∂ ˜

Γ

∂p

ℓ

i

+

X

j

λ

j

∂ ˜

N

j

∂p

ℓ

i

= 0.

Therespe tivederivativesare

∂S

∂p

ℓ

i

=

−(ln

p

ℓ

i

Π

ℓ

+ 1),

∂ ˜

E

∂p

ℓ

i

=

X

j

∂H

∂

hq

j

i

∂

_hq

j

i

∂p

ℓ

i

=

₋

X

j

hΨ

j

iσ

ℓ

δ

ij

∆x

2

=

−hΨ

i

iσ

ℓ

∆x

2

,

∂ ˜

Γ

∂p

ℓ

i

= σ

ℓ

∆x

2

,

∂ ˜

N

j

∂p

ℓ

i

= δ

ij

.

Puttingthisalltogether,anextremeentropystatemusthave

ln p

ℓ

i

= ln Π

ℓ

− 1 − βhΨ

i

iσ

ℓ

+ ασ

ℓ

+ λ

i

,

where a onstant

∆x

2

hasbeen absorbed into

α

and

β

. Solving for

p

ℓ

i

yields

theequilibriumdistribution

p

ℓ

i

= N

i

−1

exp [(

−βhΨi

i

+ α) σ

ℓ

] Π

ℓ

,

(3.30)

where

β

and

α

an be hosen to satisfy the ontraints (3.27) and (3.28), and thepartitionfun tion

N

i

isgivenby

N

i

=

X

ℓ

exp [(

_−βhΨi

i

+ α) σ

ℓ

] Π

ℓ

.

(3.31)

The relation (3.30) an be ombined with (3.23) and (3.24) to solve for

prospe tivemeanelds. Themeanstreamfun tion

hΨi

isfoundbysolving

X

j

∆

ij

hΨ

j

i =

P

ℓ

σ

ℓ

exp [(

−βhΨ

i

i + α) σ

ℓ

] Π

ℓ

P

ℓ

exp [(

−βhΨ

i

i + α) σ

ℓ

] Π

ℓ

− h(x

i

).

(3.32)

togetherwiththe onstraints(3.27)and(3.28).

TheEHT theoryis mi ro anoni alwithrespe t to theenergyand

ir ula-tion,inthesensethattheparameters

β

and

α

are hosenasLagrangemultipliers to ensurethattheresultingmeaneld assumesdesiredvalues ofthese

quanti-ties. Itis anoni alwithrespe ttohigherorderCasimir'sin thesensethatthe

ne s alevorti ityisspe iedas adistribution.

3.6

Numerical Verification of the HPM

Statis-tical Equilibrium Theories

Inthisse tionwe omparethepredi tedmeanelds

hqi

and

hΨi

ofthedis rete equilibrium statisti al modelsfrom theprevious se tions,with longtime

method,undertheassumptionthatthesimulatedsolutionisapproximately

er-godi . It should be notedthat the probabilitydistributions (3.21) and (3.30)

predi t mu h more thanjustthemean states

hqi

and

hΨi

, so our omparison is ne essarilya limitedone. Yet froma numeri alpointofview, orre t

repre-sentationof themeanstateis a minimalrequirement,as it setsthe statisti al

ba kgroundfordynami s.

Thetheoreti almeanelds (3.20)and (3.32)basedontheLagrangian and

Eulerianstatisti almodelsare omputednumeri ally. Duetopointwise

onser-vation of PV on the parti les, and the onstru tion (3.11), the spa e of

grid-based PV elds is bounded, as is the partition fun tion (3.31). For a given

parti leeld

Q

andvaluesforthe onstraintswesolveforthemeanelds(3.20) and (3.32) plusasso iatedLagrange multipliers usinga modiedNewton

iter-ation. These elds are ompared with average elds generated by long time

simulations.

Forthenumeri alsimulationsweusethetestsetupofAbramov&Majda[1℄.

We hoose grid resolution

M = 24

. The topography is a fun tion of

x

only, spe i ally

h(x, y) = 0.2 cos x + 0.4 cos 2x,

whi hisintendedtomakedeparturesfromGaussianPVtheoryreadily

observ-able(seebelow).

The integrations were arried out using a step size of

∆t = 2/M

on the interval

t

∈ [0, t

0

+ T ]

. Thesolutions areaveraged over the time interval

t

∈

[t

0

, t

0

+T ]

,where

t

0

isthetimerequiredforde orrelationoftheinitial ondition. In all experiments we use

t

0

= 10

3

and

T = 10

4

. Longer simulations with

T = 10

6

werealsorunwithnoobservabledieren eintheresults. Theimpli it midpointrulenonlinearrelationsweresolvedtoma hine pre ision.

Allsimulationswere arriedoutwith

κ = 1

. Togetherwiththelowvalueof

M

,thisimpliesthesimulationswerehighlyunder-resolved. Thishasthedouble ee t of allowingus tostret hthe limitsofthe dis retestatisti al models, for

whi h various approximations were made, and to allow the system to sample

theavailablephasespa e(assumingergodi ity)inareasonablyshortsimulation

interval.

We onstru tinitial onditionswitha desiredpriordistributionandenergy

value. Themeanstate(3.30)isfullydenedbythesequantities. Ifthedynami s

issu ientlyergodi ,thenthetimeaveragemeanstreamfun tion

Ψ

andmean potentialvorti ity

q

should agreewiththeensembleaverages(3.24)and(3.23) Given a ontinuous priordistribution on vorti ity

Π(σ)

, wedene parti le PV values

Q

k

as follows. Thenumber ofparti lesis

K = κ

2

_M

2

. Wedis retize

therangeofvorti ity

σ

into

Λ

equal partitionsofsize

∆σ

where

σ

ℓ

= σ

0

+ ℓ∆σ,

ℓ = 1, . . . , Λ.

We hoosethenumber ofparti leswith vorti ity

σ

ℓ+1/2

= (σ

ℓ

+ σ

ℓ+1

)/2

to be

K

ℓ+1/2

=

⌊K

Z

σ

ℓ+1

σ

ℓ

Anyremainderparti lesareassignedthevaluesofthe onse utivemostprobable

levelsets.

Theparti lesare initially pla edon a uniformgrid ofspa ing

∆a = ∆x/κ

in ea hdire tion. UsingMonteCarlosimulations,thePV valuesarerandomly

permuteduntila ongurationisfoundwithin desiredtotalenergy(grid

fun -tion)

H

0

± 0.01

. Inallsimulations,thetargetenergywas

H

0

= 7

,andthetotal ir ulationwas

C

1

(Q) = 0

, onsistentwith[1 ℄. TheLagrangemultipliers

β

and

α

followfromthe onstraintsoftotalenergyand ir ulation.

3.6.1

Normally distributed PV

Fromthe lassi alenergy-enstrophytheoryofKrai hnanandothers[11,40,76 ℄

itisknownthatifthePVeldis normallydistributed,themeaneldrelation

shouldbelinearoftheform(3.7). ToverifythisfortheHPMmethod,wedraw

theparti levorti itiesfroma zero-meanGaussianpriordistribution

Q

k

∼ Π(σ) = exp

µ

−

σ

2

2θ

2

¶

.

Inthis asetheEHTtheoryyields(inthesemi-dis rete ase)

p

i

(σ) = N

i

−1

exp [

−βhΨ

i

iσ] Π(σ)

whi his ontinuousinthePV

σ

. Thisdensity anbeexa tlyintegratedtoyield thelinearmeaneld relation

hq

i

i = −βθ

2

hΨ

i

i.

We hoose

β

and

θ

tospe ifyenergy

H

0

= 7

andenstrophy

C

2

= 40

. IntheleftpanelofFigure3.3,thelo usofdatapoints

(Ψ

i

, q

i

)

isplottedfor thetime-averaged elds. Thevorti ity-streamfun tionrelationisnearlylinear

as predi ted. Dueto thenitesamplingoftheGaussian distribution,the

sim-ulationdataisnotpre iselylinear. TheEulerianstatisti almodel(3.30)yields

a more linear meaneld predi tion(dashline), but the Lagrangian statisti al

model(3.21)morepre iselytsthesimulationdata(solidline).

Duetothelinearityandisotropyof(3.7)and(3.8),themeanstreamfun tion

hψi

satisesa Helmholtzequationand isexpe tedto beindependentof

y

due tothespe ial hoi eoftopography. Intherightpanelweobservethatthemean

streamfun tionisindeedindependentof

y

.

3.6.2

Skew PV distributions

In[1 ℄,AbramovandMajdashowthatnonzerovaluesofthethirdmomentof

po-tentialvorti ity an ausesigni antdeviation from thestatisti alpredi tions

of the normally distributed PV ase. They use the Poisson dis retization of

Zeitlin[88 ℄tosolvetheQGmodel. Onan

M

× M

gridtheZeitlinmethod on-servesenergyandapproximationsoftherst

M

momentsofpotentialvorti ity

ˆ

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

¯q

¯

Ψ

0

2

4

6

0

1

2

3

4

5

6

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Figure

_{3.3: Normally distributed PV on the particles. The scatter plot of mean}

fields (left) with linear fit: points are locus (q

i

, Ψ

i

), the theoretical prediction

based on (3.21) is solid line and on (3.30)—dash line. Mean stream function

(right).

Wegenerateinitial onditions

Q

fromtheshiftedgamma-distribution[13 ℄:

Π(σ) =

1

C

2

|λ|

R

µ

1

C

2

λ

(σ + λ

−1

_);

1

C

2

λ

2

¶

,

where

R(z; a) = Γ(a)

−1

_z

a−1

_exp(

−z)

for

z

≥ 0

and

R = 0

otherwisewith the Gammafun tion

Γ

,and

γ =

C

3

C

2

3/2

= 2C

2

1/2

λ

is theskewness ofthedistribution. Wetake

C

2

= 40

and

γ = 0

,

2

,

4

and

6

to omparetheresultsof[1 ℄withtheHPMmethod.

Figures 3.4 and 3.5 gives the

(Ψ

i

, q

i

)

lo i for the time-averaged elds, for these values of

γ

. Figure 3.6illustrates theasso iatedmean streamfun tions. The solutions arereminis entof those reported in [1 ℄,but there are some

dif-feren esdue tothedetailsofthemethods.

For the ase

γ = 0

, the energy-enstrophy theory predi tsa linear relation (3.7) betweenmean PV andmean stream fun tion,as well as a layered mean

stream fun tion. These predi tions are onrmed in the upper left panels of

Figures3.4,3.5and3.6. For

γ > 0

,thereissigni antnonlinearityinthemean eld relationandvorti alstru turesobservablein themeanstreamfun tion.

Alsoshownin Figures3.4and3.5arethetheoreti almeanstatespredi ted

by the dis rete statisti al equilibrium theories in Se tions 3.4 and 3.5. The

−1

0

1

−1

−0.5

0

0.5

1

γ = 0

¯q

¯

Ψ

−1

0

1

−1

−0.5

0

0.5

1

γ = 2

¯q

¯

Ψ

−1

0

1

−1

−0.5

0

0.5

1

γ = 4

¯q

¯

Ψ

−1

0

1

−1

−0.5

0

0.5

1

γ = 6

¯q

¯

Ψ

Figure

_{3.4: Locus (q}

i

, Ψ

i

) for skewed PV distributions, γ=0, 2, 4 and 6

(points). The theoretical prediction based on (3.21) (line).

−1

0

1

−1

−0.5

0

0.5

1

γ = 0

¯q

¯

Ψ

−1

0

1

−1

−0.5

0

0.5

1

γ = 2

¯q

¯

Ψ

−1

0

1

−1

−0.5

0

0.5

1

γ = 4

¯q

¯

Ψ

−1

0

1

−1

−0.5

0

0.5

1

γ = 6

¯q

¯

Ψ

Figure

_{3.5: Locus (q}

i

, Ψ

i

) for skewed PV distributions, γ=0, 2, 4 and 6

γ = 0

0

2

4

6

0

2

4

6

γ = 2

0

2

4

6

0

2

4

6

γ = 4

0

2

4

6

0

2

4

6

γ = 6

0

2

4

6

0

2

4

6

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Figure

_{3.6: Mean stream functions (time averages) for skewed PV}

distribu-tions, γ=0, 2, 4 and 6. For nonzero skewness γ

_{6= 0 the stream function is}

two-dimensional, despite one-dimensional topography.

3.6.3

PV distributions with kurtosis

Abramov&Majda[1 ℄also onje turethatthehigher-ordermoments

C

r

,

r

≥ 4

, arestatisti allyirrelevantforpredi tingthelarge-s alemeanow,basedonthe

observationthat theexperimentsagreed wellwiththe energy-enstrophymean

eldtheory(3.7)inthe ase

γ = 0

,despitethefa tthatthemoments

C

ˆ

r

,

r

≥ 4

were nonzero as arbitrarily determined by their initialization pro edure, and

onservedbythemethod.

Toinvestigatethis onje turewe hooseinitialdistributions

Q

having skew-ness

γ = 0

andnonzerokurtosis(s aledfourthmomentofPV),

δ =

C

4

C

2

2

− 3.

In this asewe generatedthe initial parti lePV eld by rst drawingthe

Q

k

from auniformdistribution andthenproje tingontothe onstraintset

{H

0

=

7, C

1

= 0, C

2

= 40, C

3

= 0, C

4

= (δ + 3)C

2

2

}

.

Figures3.7and3.8showthemeaneldrelations

(q

i

, Ψ

i

)

forin reasing

δ = 0

,

10

,

50

and

90

.

−1

0

1

−1

−0.5

0

0.5

1

δ = 0

¯q

¯

Ψ

−1

0

1

−1

−0.5

0

0.5

1

δ = 10

¯q

¯

Ψ

−1

0

1

−1

−0.5

0

0.5

1

δ = 50

¯q

¯

Ψ

−1

0

1

−1

−0.5

0

0.5

1

δ = 90

¯q

¯

Ψ

Figure

_{3.7: Locus (q}

i

, Ψ

i

) for kurtotic PV distributions, δ=0, 10, 50 and 90

(points). The theoretical prediction based on (3.21) (line).

−1

0

1

−1

−0.5

0

0.5

1

δ = 0

¯q

¯

Ψ

−1

0

1

−1

−0.5

0

0.5

1

δ = 10

¯q

¯

Ψ

−1

0

1

−1

−0.5

0

0.5

1

δ = 50

¯q

¯

Ψ

−1

0

1

−1

−0.5

0

0.5

1

δ = 90

¯q

¯

Ψ

Figure

_{3.8: Locus (q}

i

, Ψ

i

) for kurtotic PV distributions, δ=0, 10, 50 and 90

δ = 0

0

2

4

6

0

2

4

6

δ = 10

0

2

4

6

0

2

4

6

δ = 50

0

2

4

6

0

2

4

6

δ = 90

0

2

4

6

0

2

4

6

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Figure

_{3.9: Mean stream functions (time averages) for kurtotic PV}

distribu-tions, δ=0, 10, 50 and 90. For nonzero kurtosis δ

_{6= 0 the stream function is}

two-dimensional, despite one-dimensional topography.

The orresponding mean stream fun tions are shown in Figure 3.9. We

observethat nontrivial kurtosis may alsosigni antlyinuen e themeaneld

statisti s,whi h disprovesthe onje tureof[1℄.

Againweobservein Figures3.7 and3.8 thatboth(3.20) and(3.32) doan

ex ellentjobofpredi tingthemeanstates.

3.7

Conclusions

The HPMmethod, as adapted for 2Din ompressibleow, onservestotal

en-ergy by onstru tion. Ea h parti le is assigned a onstant value of potential

vorti ityat initialization,andthisdis retePV eldis onservedpoint-wise,as

theparti les evolvein thedivergen e-freeow. Inthissense,PV onservation

indu es no redu tion in degrees of freedom on the dynami s. At the oarse

s ale, the vorti ity eld on the mesh satises onservation of energy and

to-tal ir ulation,butexhibitssigni antdriftfornonlinearPV fun tionals. This

is onsistentwithwhat would beobserved ifa oarse-grainingpro edurewere

applied toarealinvis idow.

A maximum entropy theory based only on energy and ir ulation would

demonstratedinthis hapterthattheHPMmethodhasamu hri herstatisti al

me hani s,withnonlinearmeaneldrelationssimilartothoseof[1 ℄,and

onsis-tentwiththe anoni alEHTtheory[22 ℄. Inparti ular,wehavedemonstrated

thatboththethirdandfourthmomentsofPV(

C

3

and

C

4

) ansigni antly af-fe tthemeaneldrelation. Thelatterresultdisprovesa onje tureofAbramov

&Majda[1 ℄.

Wehavealsopresentedtwostatisti alme hani smodelsfortheHPMmethod,

a Lagrangian andanEulerian model. TheEulerian model isanalogousto the

EHTtheory,whi husesa anoni altreatmentofne-s alevorti ityintheform

ofapriordistribution,andenfor es onservationofenergyandtotal ir ulation

throughtheuseofLagrangemultipliers. Inthepresent ase,theprior

distribu-tion hara terizestheparti levorti ityeld,andtheenergyand ir ulationare

onservedatthegrids ale. TheLagrangianstatisti almodelis onstru tedon

thephasespa eofparti lepositions, onsideringea hparti letobeimmersedin

areservoirdenedbythemeanow. Thenes ale,parti lestatisti saregiven

by anoni alensembledistributions,andthetemperatureparameterisusedas

a Lagrange multiplier to enfor e energy onservation. Mean states omputed

with bothstatisti al models omparevery wellwith the long time simulation

data.

Although PV issimply assigned to parti les andits onservation doesnot

imply any dynami onstrainton the evolution, anappeal to the Lagrangian

statisti al modelsuggeststhat aparti le'sPVvaluedeterminesitsresponse to

themeanow,andtherebyitsresiden etimeinanyparti ularregionoftheow

domain. Viathebasisfun tions

φ

, thelo alresiden etimeistranslatedtothe grids alewherethe oarse-graindynami sisgovernedbyenergy onservation.

Theessentialingredientsof theMiller/RobertandEHT statisti altheories

are thene s alepoint-wise adve tionofPV andthe oarses aling asso iated

with the stream fun tion, under the onstraint of energy onservation. The

HPMmethodretainsthesefeaturesunderdis retization,andforthisreasonits

equilibrium statisti sare analogousto those theories. Fromthenumeri al

ex-perimentswe an on ludethattheHPMmethodisfreeofarti ialdissipation

orothererrorsthatmightdestroytheequilibriumstatisti alme hani s. Forthe

experiments ondu ted, the dis retedynami s is also su ientlyergodi that