1856-15

2007 Summer College on Plasma Physics

Z. Yoshida

30 July - 24 August, 2007

The University of TokyoChiba, Japan

Clebsch parameterization - theory and applications

Clebsch parameterization― theory and applications

Zensho Yoshida

The University of Tokyo

parameterization of vectors

• “state” = “vector” (in a Hilbert space)

• “observation” = “parameterization”

1x

2x

3x

= ⇒observation(analysis)

some basic notions

• Parameterization of a vector depends on thechoice of “frame”, while scalars are frame-independent.

(→ covariance arguments)• “functional” : vector → scalar (→ variational principles)• Linear functional = 〈 f | (→ observable arguments; Riesz’ theorem)

• Cartesian:

• Covariant form (1-form):

• Contravariant form:

• Mixed form (Grad form):

• Clebsch form:

some different parameterizations (forms)of (3D) vector fields

zzyyxxuuu eeeu ++=

! "=j

ju #u

! "#"=j

ju $u

!"!# $%$+$=u

!"# $+$=u

frame → potential

• curl-free (irrotational) field:

• div-free (solenoidal) field:

• Poincare-de Rham-Hodge-Kodaira-Wyletheorem:

!"#=E

AB !"=

!"#$"#"= A%u

Θ=multi-valued potential = angle

application I (streamlines)

• Grad form of a symmetric vector field

→ integrable streamlines

→ integrals of hyperbolic PDE (Casimir)

( ) !!" ! #+#$#= rBB

!"

"d

r

d

z

dz

r

dr=

#$=

#$=

#$ BBB!!

"

!!

#

$

%

%&=

%

%=

rd

dz

zd

dr

'

(

'

(

Nonlinear elliptic-hyperbolic PED theory-- including Cauchy problem to generate Casimirs

!"#

=$%

&%=''%

0

,)(

B

BB p

!"#

=$%

+&%=''%

0

),2/()( 2

v

vvv p

ideal MHD ideal fluid (Euler eq.)

!"!# $+$%$=u

0},{ =!")(')()('

)(),(

!"!"!!

!!""

+=#

==

PL

PP

• double contravariant form (Boozer form) ina toridal (ρ, θ, φ ) frame

→ Hemiltonian form of general (non-integrable) streamlines

!"# $%$+$%&$=B

!!

"

!!

#

$

%

&%'=

%

&%=

()

*

*)

(

d

d

d

d

application II (canonical form)

• Hamiltonian fluid mechanics:

irrotational → zero-helicity → general???

dxuH !!"# $%

&'(

)+= )(

2

1 2

!"=u

( )

( )!"

!#

$

+%=&%=&

'(%=&=&

huH

H

t

t

2/2

)

*

*

)) u

ht

!"=!#+$ uuu )(

!"# $+$=u

( )

( )!"

!#

$

+%=&%=&

'(%=&=&

huH

H

t

t

2/2

)

*

*

)) u

( )

!"

!#

$

%&'=('=(

&%'=(=(

))

**

*

)

u

u

H

H

t

t

'

''

)'( !"! =

ht

!"=!#+$ uuu )(

local → global

}0formsclosed{}formsexact{ =!= "#" dd

the gap = topology of the domain (cohomology)

helicity (topological degree)

• Gauss’ liking number:

• helicity:

)kernelsSavart'Biot(4

1),(

),(),(

3

12

1221

221121

21

!!

!"

#$= %%

xx

xxxxK

xxxKx

&

ddCCl

CC

! "=#

dxcurlC ww1 Casimir invariant of ideal

vortex dynamics

incompleteness of the Clebsch formand its amendment

• The Clebsch form isincomplete

(1) to describe streamline chaos,

(2) to represent a field with a helicity.

• A complete form:

!"# $+$=u

j

j! "+"= #$%u

Theorem (complete Clebsch form) Let u be a continuous 1-form in Ω:

(*)

With chosing a complete 1-form , we can transform (*) into

(**)

where

!=

=n

j

j

jdxu1

u

!ddxun

j

j

j +="#

=

1

1

'u

!d

!jxjj

uu "#='

);,,(

),,,(),,(

11

111

mxxC

dyyxxuxx

n

x

a

n

n

nn

m

!

!

+

= "K

KK#

remarks

• Counting the number of fields does not suffice.

• The set is NOT alinear space.

• The set is equal to thelinear space of solenoidal fields, proving thecompleteness of the Boozer form [Z. Yoshida,Phys. Plasmas 1 (1994), 208-209].

})({ !"!"# $%$=$+$%$

}{ 2

2

1

1 !"!" #$#+#$#