Plasma Physics Laboratory, Princeton University. Princeton, New Jersey 08544. Abstract. The equations that describe the motion of two-dimensional vortex ...
r~1ARCH
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
=
