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PPPL-1733

1981

UC-20g

HAMILTONIAN FIELD DESCRIPTION OF T\~O-D IMENS tONAL VORTEX FLU IDS AND GUIDING CENTER PLASMAS

, ,

, ,

BY ,

~

..; "

PI J. ",10RR I SON,

PLASMA 'PHYSICS' LABORATORY

PRINCETON UNIVERSITY, P R IN CE TON, NEW J E R S E Y This work supported by the U.S. Department of Ener~y Contract No. DE-AC02-76-CHO-3073. Reproduction, translation, pubJ icailon, use and disposal, in whole or in part:, by or for the United States Government is permitted,.

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"

Hamiltonian Field Description of Two-Dimensional Vortex Fluids and Guiding Center Plasmas

by

/

Philip J. Morrison Plasma Physics Laboratory, Princeton University Princeton, New Jersey 08544

Abstract

The equations that describe the motion of two-dimensional vortex fluids

I

and guiding center plasmas are shown to possess underlying field Hamiltonian structure.

A Poisson bracket which is gi ven . in terms of the vorticity,

the

physical although noncanonical dynamical variable, casts these equations into Heisenberg form.

The Hamiltonian density is the kinetic energy density of the

fluid.

The well-known conserved quantities are seen to be in involution with

respect

to

this

Fourier-Dirac

Poisson

series

bracket.

transforms

usual canonical equations for

the

Expanding the field

vorticity

description

discrete vortex motion.

in

terms

given here

of

a

into the

A Clebsch potential

representation of the vorticity transforms the noncanonical field description into a canonical description.

-2-

I.

Introduction

This paper is concerned with the Hamiltonian field formulation of the equations fluid.

which

describe

the

advection

of

vorticity in a

two-dimensional

These equations have received a great deal of attention in the past

thirty years and are believed to model the large scale motions which occur in atmospheres

and

They

oceans.

transport perpendicular to a

have

also

arisen

in

uniform magnetic field,

the

study

of

plasma

the so-called guiding

center plasma. 1 ,2 (For recent reviews see Refs. 3 and 4.) It has been known for some time that a system of discrete vortex charge)

filaments

possesses

a

Hamiltonian

description. 5

(or

The equations of

motion are

dx.

1.

ki

dt

where k i YiO

dy.

aH

1.

=--

ki

ay.1.

dt

aH ax.1.

=

(1)

is the circulation of the i th vortex which has coordinates xi and

The Hamiltonian, H, is the interaction energy and for an unbounded fluid

has the form

H =

-1

L

2TI i>j

k.k. In R. of 1. J 1.J

The variables xi and Yi are canonically conjugate.

The

formulation

we

describe here is a field formulation which

possesses this underlying discrete dynamics. In Sec. Hamiltonian

II we briefly review some aspects of finite degree of freedom dynamics.

The

emphasis

properties of the Poisson bracket.

here

is

placed on

the

Lie

algebraic

This is used as a framework in which to

-3-

explain the "constructive" approach to Hamiltonian dynamics.

Such an approach

frees one from the prejudice that a system need be in canonical variables to This section is then concluded by the extension of these

be Hamiltonian.

notions to infinite degree of freedom or Hamiltonian density systems. III

we

present

a

Poisson bracket that

renders

the

In Sec.

vortex equations

into

Heisenberg form.

This formulation is novel in that it is noncanonical.

the

this

remainder

constants

of

motion

section for

In Sec.

truncation. which,

of

this

we

discuss

system,

involutivity

Fourier

space

IV we expand the vorticity in a

of

canonial

form.

potentials Finally,

we

which

also

obtain

a

bring

spectral

conjugate pairs are canonically conjugate.

the

and

Fourier-Dirac series

canonical discrete vortex description of the Introduction. Clebsch

well-known

representation

upon substitution into the Poisson bracket of Sec.

introduce

the

In

III,

yields the

Following this we

Poisson

description

bracket where

into

complex

A quartic interaction Hamiltonian

is obtained.

II.

Constructive Hamiltonian Dynamics

The standard approach 6 to a Hamiltonian description is via a Lagrangian description.

One constructs a Lagrangian on physical bases and through the

Legendre transformation \1

following

2N

(where

(assuming convexity) obtains the Hamiltonian and the

N is

the

number

of

degrees

of

freedom)

first

order

ordinary differential equations:

k=1,2, ••• N

Here the Poisson bracket has the form

(2 )

-4-

N

k _ .2L k ) =.2L L (.2L i aqk aPk aPk aqk

[f, g)

k=1

Jij

k

az

az

( 3)

j

The last equality of Eq. (4) follows from the substitutions,

for i

f'

zi

k

=

for i

Pk

=

1,2 ••• N

N + k

=

(N + 1 ) •••

2N

and

(J

ij

)

=

:N) C -I

(4 )

N

where IN is the N x N unit matrix. convention here and henceforth. tensor

or

the

cosymplectic

We assume the repeated index summation

The quantity Jij is known as the Poisson

form.

It

is

not

difficult

to

show

that

transforms as a contravariant tensor under a change of coordinates. transformations which preserve its form,

it

'!hose

and hence the form of the Eqs.

(2),

the equations of motion, are canonical. The

constructive

approach

differs

from

the above

in that

one

is

not

concerned with any underlying action principle nor with (initially at least) the necessity of canonical variables. properties of the Poisson bracket. of Eqs.

(2) with Eq.

A system need not have the canonical form

(3) to be Hamiltonian.

introduce a few mathematical concepts. bracket

acts

are

The emphasis is placed on the algebraic

differentiable

To make the idea more precise we

'!he quantities on which the Poisson

functions

defined

on

phase

space.

The

collection of all such functions is a vector space (call it n) under addition and scaler multiplication. maps

n

x

n

to

n.

The Poisson bracket is a bilinear function which

Also note that the Poisson bracket possesses the following

-5-

two important properties: Jacobi identity, Le.,

e:

vector space

A

Q.

Property (i)

(i)

=-

[f,g]

[f, [g,h]]

for every f,g e: Q' and (ii) the

[g,f]

+ [g, [h,f]] + [h, [f,g]] = 0 for every f,g,h

together with

such a

bracket defines

a

Lie algebra.

requires that the Poisson tensor be antisymmetric and property

(ii) requires the following: ()

o

One

can

show

that

sijk

transforms

contravariantlYi

hence

identically in one coordinate frame it does so in all. is coordinate independent. the converse outlook:

(5)

if

it

vanishes

Similarly anti symmetry

The covariance of properties (i) and (ii) suggests

if a system of equations possesses the form

·i

(6)

i, j = 1,2 ••• 2N

Z

where Jij is antisymmetric and fulfills the Jacobi requirement, but is not of the form of Eg. theorem

due

(locally)

(4), then it is Hamiltonian.

to

Darboux

canonical

(1882)

coordinates

which can

be

This outlook is justified by a

states

that

assuming

constructed.

theorem may be found in Refs. 7, 8, and 9.)

(The

det(Jij) proof

of

*

0

this

Hence in order for a system to be

D

Hamiltonian it is only necessary for it be representable in Heisenberg form with a

Poisson bracket that makes Q into a

Lie algebra.

The constructive

approach simply amounts to constructing Poisson brackets with the appropriate properties. The rigorous systems

requires

generalization of the above ideas to infinite dimensional the

language

of

functional

geometry of infinite dimensional manifolds.

analysis

and the

differential

(See Ref. 7, Ch. V and Refs.

10 -

-6-·

This of course is not our purpose here~ rather we simply parallel the

15.) above.

The

Poisson bracket for

following form:

set of field equations usually has the

6

N

=

[F,G]

a

I

J

(7)

k=1

where the integration is taken over a fixed volume.

The quantities on which

the bracket acts are now functionals, such as the integral of the Hamiltonian density [e.g., Eg.(13)].

dF d£

where

(nk

+

£W)

I

£=0

The functional derivative is defined by

J -of on

k

WdT

of -

n

k

the bra-ket notation is used to indicate

J fg dT.

In terms of this notation Eg.

the

inner product



(7) becomes

(8)

[F ,G]

where the 2N quantities nk and TIk are as previously the 2N indexed quantities ui •

The canonical cosymplectic density has the form

In noncanonical variables the quantity (Oij) may depend upon the variables u i , and

further

variables.

it

may

contain

derivatives

In general anti symmetry of Eg.

with

respect

to

the

independent

(8) requires that the (Oij) be an

< .

anti-self-adjoint operator. analogous to Eg.

The Jacobi identity places further restrictions,

(5), on this quantity.

We defer a discussion of this to the

-7-

Appendix where the Jacobi identity for the bracket we present· [Eq. proved.

(15)]

is

The extension of the Darboux theorem to infinite dimensions has been

proved by J. Marsden. 14

For a discussion pertinent here see Ref. 15.

G

III.

Noncanonical Poisson Bracket

The equations under consideration are the following:

= -

(9)

V.v = 0

(10)

Here we use the usual Euclidian coordinate system with uniformity in the z direction where x

(which

= (x,y),

vez = O. and v to

has is

unit

the

vector z). The

vorticity

and v is

quantity the

w(~,t)

flow

= ze'iJ X ~(~,t),

velocity

such

that

(For the guiding center plasma w corresponds to the charge density the E

x

B drift

velocity.)

For

an

unbounded

fluid v can

be

eliminated from Fq. (9) by16

v

( 11)

o l\

where ~ =

we

display

only

the

arguments

'"

necessary

to

avoid

confusion.

Here

z x 'iJK(~ I~") and K( ~ I~") is the Green function for Laplace' s equation in

two dimensions,

1 2 2 = - - In I(x-x") + (y-y") 27T

-8-

The integration in Eq.

(11) is over the entire x-y plane; dT _ dxdy.

form Eq. (10) is satisfied manifestly.

In this

Equation (9) becomes

(12)

Equations (9) and (10) are known to possess conserved densities; that is, quanti ties wi th

Eq.

which ( 1 2) •

satisfy an equation of the

form p

t

+ V.J = p, -

Clearly any function of w is conserved.

consistent

In addition, the

kinetic energy is conserved which is the natural choice for the Hamiltonian. With the density (mass) set to unity we have

~{ w} 1

=

2

J

K(~I~') w(~') w(x) dT dT'

(13)

The functional derivative of Eq. (13) is the following:

oH ow = -J

K(~I~') w(x') dT'

(14 )

We introduce the Poisson bracket 17

[F,G]

where

J

w(x)

af- ~ acr {f , g } = -dX ay

{OF ow' -

OG} ow dT

acr af ay -ax

~



( 15)

One observes that the discrete vortex Poisson

bracket is nestled inside the field Poisson bracket. how

to

regain

the

discrete

bracket

from

this

difficult to show from Eqs. (14) and (15) that

field

In Sec. IV we will see bracket.

It

is

not

-9-

-f

[w,H]

o

w MdT' • 'lw

Clearly this bracket is antisymmetric by virtue of the anti symmetry of the discrete bracket.

o

We prove the Jacobi identity in the Appendix.

We note by examination of Eg. '" involution 1 that is, if F.{W} = 1.

f

(15) that any two functionals of ware in

F. {W)dT (for i=1,2) are two such functionals 1.

where the F. are arbitrary functions of w, then 1.

Also,

substitution

of

any

such F. and H [Eg.

( 13) ]

into

Eq.

(15 )

and

1.

integration by parts yields

[F., H] 1.

a

In particular, we see

(when F.1.

w2 ) that the enstrophy commutes with the

Hamiltonian. The close relationship between this functional Hamiltonian formulation and the conventional formulation of Sec. II .is seen by Fourier expanding the

o

vorticity in a unit box with periodic boundary conditions,

(16 )

where k = (k , k ). The reality of w implies 1 2

w* = w_ • If we suppose for the k k

moment that w{x) depends upon some additional independent variable have the following for some functiona1 18 ;:

~,

then we

-10-

dF 1fil

f of ow

=

dW dx dy

From this we see for

of

ow =

1 ( 21r)

(17)

d)1

=

)1

wk upon Fourier inversion that

I

2

( 18)

k

where the F on the left hand side is treated as a functional of

W

while the

F of the right hand side is to be regarded as a function of the variables W

k

• Substituting Eqs.

(16) and (18) into Eq.

(15) yields,

[F,G]

The Hamiltonian becomes

H = 21T2

I &

and the equations of motion are

" z.

~

IR,

(~

x R,

&) W,q, ~-R, =

2

I

'Jk,,q,

R,

--

dH

( 19)

dWR,

where Jk,R,' the cosymplectic form, is

z.

(& x (21T)2

Clearly, Eqs.

Eq.

(19)

(6), of Sec.

~)

w~+&

(20)

is of the form of the finite degree of freedom equations, II except here the sum ranges to infinity.

The form Eq.

-11-

(20) is obviously antisymmetric anq it is not difficult to verify Eg. At first,

one might think that a truncation of the J

~,&

(5).

would yield a

finite Hamiltonian system which to some accuracy would mimic the original. ()

Unfortunately, the process of truncation destroys the Jacobi identity. must seek a change of variables which allows truncation.

One

canonical variables

(l

are suited for this purpose and in the next section we discuss this.

canonical Descriptions

IV.

As was noted in Sec.

the Poisson bracket for the discrete vortex

III,

picture is embedded in that for the field.

To see the connection between the

two, we expand the vorticity (distributed vorticity) as follows:

w(x) = \' k

L

i

i

o(x-x

- -i

)

(21)

where o(x) is the Dirac delta function, the k i are constants and w obtains its t dependence through the x,

Then using Eg.

-1

aF ax, =

k.L

of i ax ow

].

(22) x= (x"y,) -

D

where o

the

1],

'"

functional F on

the

left

function of the variables xi and Yi. aF/ay,. Substituting Egs. ].

[F ,G)

(17) we obtain the identity

=I

1 ' k, J J

(~~ ax, ay, J J

hand side is now

to be

a

Similarly we obtain the relation for

(21) and (22) into Eg.

(15) yields

~~) ay, ax, J

regarded as

(23)

J

Further, if we substitute Eq. (21) into the Hamiltonian, Eg.

(13), we obtain

-12-

H

-1 41T

I

i,j

k. k. In R .. ~ ] ~J

Since this is singular along the diagonal i=j, we remove the self-energy of each vortex and obtain the usual result

H

-1 k.k. In R .. 2 1T i>j ~ ] ~J

I

(24)

Equations (23) and (24) reproduce the Eqs. of w in a

(1).

Hence, we see that expansion

Fourier-Dirac series is a particular way of discretizing,

which allows truncation without destroying the Hamiltonian structure.

a way We now

discuss another approach. The cosymplectic form, Eq.

(20), suggests by its linearity in w

~+&

a quadratic change of variables (i.e.,

that

w ~ ~2, where ~ is the new variable) is

needed in order to achieve canonical form.

Such a transformation [given by

Eq.

(31)] removes the nonlinearty present in the Poisson bracket and places it

in

the

Hamiltonian

[Eq.

(30)].

Enroute

to

arriving

at

this

result

we

introduce a Clebsch potential representation of the vorticity,21

w=~h_~h ax ay

This form.

(25)

ay ax

substitution transforms Clearly, Eq.

the Poisson bracket,

Fq.

(25) is not uniquely invertable.

-

x__ - 1/1

condi tion that any function 1/1, such that 1/1

x"y

X

y x

(15),

into canonical

We have the local gauge = 0, can be added to 1/1

(and likewise for X). The chain rule for functional differentiation yields

-13-

of

OW

V

=

of

0

A

of

oX

V

= -

A

(OW

z

of

0

(OW

X

VX)

A

z

X

(26)

VW)

where on the left F is now regarded as a functional of wand Xo

o

Poisson bracket for X and

The canonical

Wis

[F,G]

which upon substitution of Egso

(26) yields the bracket Ego

(15).

Clearly

W

and X satisfy

o

W=

oH

X

oX

oH

= -oW

Upon Fourier transformation Egso

(27 )

(27) become

(28)

We now introduce the field variable

~k

as follows:

*

*

~k + ~-k

W k

=1- (

27T

/2

-)

~k - ~-k

(

-) /2

(This form maintains the reality condition for variables Egso

3H

=--

W and k

X .) In terms of these k

(28) become

(29)

-14-

and the Hamiltonian has the form

H =

* * S~_,m_,~,~ ~~_ ~m ~s ~t

L

~+m=s+t

where the matrix elements S

are

~,~,~,:!:

z·(t x

~)

(30 )

z·(m x s)

1& -:!:I

I~

-

~)

zo(s x

[--------,----- + ~I

1& -

z·(m x t)

~I I~

-

~I

}

The quadratic transformation mentioned above is

iz.

L

(:!:

x

&) (31)

t=k+~

( 21T ) 2

Hence, we see the connection between Clebsch potentials and our bracket.

This

transformation allows discretization and truncation while not destroying the Hamiltonian structure.

Acknowledgments

I

would

encouragement

like of

to

thank

this

work.

J.

stimulating my interest in the

M.

Greene

Also, 2-D

I

for

would

fluid

his like

equations.

continuing support

and

to

for

thank I

H.

Segur

am grateful to C.

Oberman, J. B. Taylor, and G. Sandri for many discussions which contributed to this paper.

I would like to thank J. E. Marsden for sending me an early copy

of Ref. 15.

This work was supported by the AC02-76-CH03073.

u.s.

Department of Energy contract No. DE-

-15-

Appendix

Here we generalize the method used by P. Lax 22 for the Gardner bracket,

Q

to prove the Jacobi identity for Eq •. (15). the variable u.

" We suppose F{U} is a functional of

Recall the functional derivative is defined by

ou

We

denote G

=

+ < of OU ou



2" o G w > + < of 2 ou ou

0--

00

w OU

oG > OU

By the

-16-

In this expression the first two terms are straight forward, the last comes from any dependence the operator O.may have upon

~ de:

0

(u

+ e:w)

I e:=0

u~

i.e.,

cO w - cu

Isolating w we obtain

2

2

15 [F,G] = 15 ; 0 c{; _ c {; 0 15; + T (CF cu

cu2

au

cu2

cu

cu

aG) au

(A-1 )

where the operator T comes from removing COw/cu from w.

T is antisymmetric in

its arguments. The Jacobi identity is

s

I

0 C[F,G]> + + = 0 au

(A-2)

Inserting Eg. (A-1) into (A-2) and using the self-adjointness of

and the

=