# Reduced MHD “favors” straight field line coordinates for mode

Document Sample

Reduced MHD “favors” straight field line coordinates for mode description. Reason: In straight field line coordinates, the safety factor q has no poloidal dependence on a given flux surface. Particle guiding center theory “favors” orthogonal magnetic coordinates. Reason: Guiding center equations of motion have a convenient Hamiltonian structure in orthogonal magnetic coordinates.

Representation of the unperturbed axisymmetric magnetic field in orthogonal coordinates:
A = A! "! + A# "# B = B! "! + B# "#

! - toroidal angle # - poloidal angle \$ - flux coordinate
Note that there is no "\$ - component in A or B in orthogonal coordinates.

Littlejohn Lagrangian in orthogonal magnetic coordinates:
e" Mc B! % e" Mc B( % ! Mc Mv||2 ! ! L = \$ A! + v|| ' ! + \$ A( + v|| ' ( + µ) * µ B * c# e B& c# e B& e 2

Dynamical variables:

! - toroidal angle ( - poloidal angle + - flux coordinate

v" - parallel velocity ) - gyroangle µ - magnetic moment

Hamiltonian form:

! ! ! L = P! ! + P"" + P## \$ H (P! ; P" ; P# ;" ) P! % P" % e& Mc B! ) A! + v|| + c( e B* '

e& Mc B" ) A" + v|| + c( e B* ' Mc P# % µ e Mv||2 H % µB + 2
Transformation from straight field line coordinates r; " ;! to orthogonal coordinates

( r;";! ) :
r=r

(

)

! =!
r / r -, 2 ˆ " = " \$ 1 + -, + . dr 4 sin" % " \$ 5 sin " r 3 0 R0 0

-, % Shafranov shift

Mode representation (single poloidal component) in straight field line coordinates:
! SAW = ! nm (r )exp("i# t + in\$ " im% )

Mode representation in orthogonal coordinates:

ˆ ! SAW = !nm (r)exp("i# t + in\$ " im% + im& sin % )