PFC/JA-87-10
MHD CLUMP INSTABILITY
by
D.J. Tetreault
Massachusetts Institute of Technology
Cambridge,
MA
02139
January 1987
Submitted to The Physics of Fluids
ABSTRACT
The theory of MHD clump instability in a plasma with magnetic shear is
MHD clump fluctuations are produced when the mean magnetic field
presented.
shear
is turbulently
clump magnetic
clump
decay
growth
time
mixed.
Nonlinear
island structures
due
to
magnetic
is
on
the
order
instability
resonantly overlap,
field
of
line
the
results
as
the
the mixing overcomes
stochasticity.
Lyapunov
when,
time.
The
The
instability
renormalized
dynamical equation describing the MHD clump instability is derived from one
fluid
MHD equitions
and
conserves
the
dynamical
invariants
The renormalized equation is a nonlinear,
equations.
of
turbulent
the exact
version of
the Newcomb equation of linear MHD stability theory and can be cast into the
form
of
a
instability
nonlinear
MHD
is calculated.
Rayleigh-Taylor
interchange
energy
principle.
The
growth
The instability is a nonlinear
rate
the
analogue of the
instability in a magnetized fluid and,
1
of
in the
fully
stochastic
case,
of
the
tearing
mode
instability.
instability is a dynamical route to the Taylor state.
2
MHD
clump
I.
INTRODUCTION
called
fluctuations
been suggested that nonlinear
It has
will
clumps
occur in magnetohydrodynamic (MHD) plasma with magnetic shear.'
Clumps were
first discussed in studies of Vlasov turbulence,
where the term was used to
describe
turbulent
resonant
fluctuations
density is conserved
density
is
turbulently
density.
different
in a Vlasov
transported
In
by
the
The clump fluctuations arise
space density gradients .
space
produced
the
to
plasma:
a
new
region
corresponding
MHD
case
and,
in the phase space density.
Green-Kruskal
MHD
(BGK)
of
corresponding role is
the
phase
space
of
shear,
Clumps are localized,
were shown to be holes
played
5
In an
in the
by a hole
The Vlasov hole fluctuation
structure
Its self-consistent
nonlinear.
(trapped) orbits and,
space
of phase
In isolation, the Vlasov hole is a Bernstein-
current density, i.e., by the magnetic island.
contains closed
therefore, cannot be obtained from linear perturbation
The linear perturbation expansions also fail in the MHD case where
theory.
resonantly trapped magnetic
Hole
phase
mode and is the analogue of the modon in fluids.
plasma with shear,
is fundamentally
because the
with magnetic
in the Vlasov plasma,
4
phase
an element
clumps arise because of the conservation of energy.'
non-wave-like-structures
mixing of
properties
stability
in
are
lines
also
amplitude
close
to
For
nonlinear.
in
form
linearly
magnetic
example,
stable,
islands.
Vlasov
current
hole
driven
turbulence
can
plasma.
A turbulent Vlasov plasma is composed of a random collection of
6
grow
field
these colliding, growing holes called clumps.
Vlasov
clump
instability
instability describes
interaction
turbulence
many
the
subject
of
this
The MHD analogue of the
paper.
The
MHD
clump
the turbulence that results from the strong resonant
of magnetic
has
is
0-12
islands
features
of
at
high magnetic
fluid and
3
Vlasov
Reynolds
numbers.
plasma turbulence,
The
and
thereby provides an interesting bridge between the two types of turbulence.
MHD
clumps
are
surfaces
in
magnetic
field shear
the
plasma.
mixing rate exceeds
stochasticity
resonant
The
below
the
linear
growth rate is
time.
at
island
stability
amplitude
For
large
Instability
mode
dependent,
overlap.
analogue
of
the
replaces
the
longitudinal
mean
when
field
The
the
line
stability
(Kruskal-Shafranov
limit).
of
The
the Lyapunov
the Lyapunov time is long and the growth rate is
driven
field line
the
results
and is on the order
amplitudes,
magnetic
rational
when
due to the magnetic
(resonance)
boundary
region of stochasticity is large.
tearing
mode
produced
analogue of that for the Rayleigh-Taylor
instability.
the
are
near
above a critical but small fluctuation amplitude,
For low amplitudes,
a nonlinear
fluctuations
clump decay rate
generated
the
localized
is turbulently mixed.
boundary is nonlinear and,
is
fluctuations
by
an
diffusion.
the
Then,
Lyapunov
resistivity
The instability
condition
wave length of
time
is
short
the growth rate resembles
anomalous
Kruskal-Shafranov
interchange
threshold
where
the
due
the
to
is
(mixing)
and
that of
stochastic
a nonlinear
Lyapunov
fluctuation.
the
length
The turbulent
mixing during MHD clump instability minimizes the energy subject to magnetic
helicity conservation.
Steady state MHD clump turbulence- is
described
by
the Taylor state.
The results
models,
here
are developed
from intuitive
as well as derived from a renormalized
MHD equations.
turbulence
theories
nonlinear,
appear
presented
perturbation
and
physical
theory of the
Though most fully developed for the nonlinear description of
in simplified
one
dimensional
can nevertheless be arcane.
three dimensional
to only enhance this
vector
plasma,','"
renormalized
plasma
Applying the theories to the coupled,
equations
reputation.
We
describing MHD fluids
have,
therefore,
would
applied the
theory in a way that stresses its physical and conceptual features.
made
possible
by earlier
work on one
dimensional
Vlasov
This is
plasma where
an
integrated theoretical and computer simulation effort led to a detailed, but
intuitive and tractable model of Vlasov turbulence.
structures
coherent
(the
holes)
their
and
integration
fully turbulent state of incoherent fluctuations (the clumps).
results
clump
of the model
model
of
describes the plasma from the complementary viewpoints
clump/hole model,
localized,
This model, the
agree well
extends
the
conceptual
In particular,
antecedent work.
with the
and
computer
a
The detailed
simulations.
mathematical
into
features
The
of
MHD
this
we expand upon the preliminary work of Ref.
1 with calculations based on Ref. 8.
Much work has been done by other investigators on turbulent relaxation
in
MHD
magnetic
fluids.14-26
field
conservation
action,' a,19
These
lines,
include
fluctuations,
and
quasilinear
final
energy. 1,24-2s Except for Ref.
(time
independent)
stochastic
have assumed a given spectrum
modes).
However,
stochastic
and
models
coherent 2 0
of
dynamics
on
decay
selective
1',1
laws,'''17'1
nonlinear
studies
force-free
constraints
of
and
(Taylor)
turbulent
interacting
states
26, where the self-consistent
fields
was
considered,
of fluctuations
diffusion
(e.g.,
in the clump model of MHD turbulence,
these
of
of
of
MHD
dynamo
magnetic
minimum
generation
of
investigations
Alfven waves,
tearing
the self-consistent
generation of fluctuations is treated on an equal footing with the nonlinear
conservation laws.
One is,
previous investigations.
therefore,
led down a path different
turbulence in sheared magnetic fields.
of
magnetic
the
The result is a model that unifies many of their
features and reveals the MHD clump fluctuation
spectrum
from
islands,
a
as a new constituent of MHD
Rather than having to assume a given
priori,
5
the
theory
determines
the
fluctuations self consistently from the turbulence itself via Ampere's law
In addition,
and the conservation laws.
variational
unlike a statistical mechanical or
final states and the rates at which they are attained.
for this--unlike a
nonlinear
and,
27
One pays a price
the renormalized theory is manifestly
quasilinear model,
therefore,
nonlinear
determines
the theory self-consistently
calculation,
necessarily
more
complicated
However,
we believe that the MHD clump theory,
picture,
has a strong intuitive
appeal
besides
and
approximate.
providing a unified
that more than compensates
for its
additional complexity and approximations.
paper
This
describes
MHD clump fluctuations
paper ,
helicity are treated in a subsequent
results is
A physical
IA.
preview
presented in Sec.
IC below.
The
27
but
a brief
preview
of the
paper is organized
present
as
The current hole and its magnetic island structure are derived in
follows.
decay
instability to
Steady state MHD clump turbulence and the conservation of magnetic
growth.
Sec.
and their
is
presented
in
of magnetic
dynamics.
discussion
Sec.
of resonantly interacting hole
IB.
We
conclude
the
and
its
helicity conservation
growth and
with
Introduction
for
consequences
the
clump
The clump fields and their conservation laws are derived from the
one fluid MHD equations in Part II.
the clump field equations
One and two
are presented
in Part
point renormalizations
III.
point equation is cast into the form of a nonlinear
and solved for
the
instability growth rate.
growth rate is analogous
magnetized fluid.
to that
In Part VI,
In Part
In Part
IV,
Newcomb-like
V,
for the Rayleigh-Taylor
of
the two
equation
we show that
the
instability in a
we cast the clump growth rate into the
form
of a nonlinear MHD energy principle and show in Part VII that the stability
boundary is a non-linear analogue of that
for the
linear
kink mode.
Part
VIII compares the growth rate to that for Vlasov phase space density holes.
6
A.
Current Density Hole
The fundamental
current
density,
i.e.,
current density.
which
become
nonlinear
a
structure
fluctuation
in the theory
6Jz<O where
JZ
is
a
is the
The current hole produces self-consistent
"trapped"
about
spacial
resonances
surfaces) in a sheared magnetic field.
hole
in
the
longitudinal
magnetic fields
(so-called mode
rational
The hole is localized near the mode
rational surface where the restoring force to field line bending is minimal.
The
trajectory
dimensional
to the
vortex.
of
these
perturbed
field
lines
vortex or magnetic island structure in the
current.
A
saturated
A Kadomstev bubble
is an extreme example,
tearing mode
2-29
is
forms
plane
a
two
perpendicular
an example
of such
a
formed by a vacuum region inside the plasma,
3,
where the hole depth is maximum,
density is zero in the hole.
is the
magnetic
This self-consistent
analogue of the trapped
particle
i.e.,
the current
island/vortex structure
phase space
vortex of the Vlasov
hole 4 and of the modon in fluids. 5
Consider MHD force balance
J x
B
(J and B are current
self-consistent
equilibrium
resonant
(1)
density and magnetic field)
island
solution
Boy (x)?,
law for the
J
ji
to
(2)
structure
(1)
and
of
(2)
the
in
current
a
where
transverse direction,
Boz
i.e.,
so-called tokamak ordering.
is
assumed
6B = 6BI.
hole
sheared
simplicity, we use slab geometry and write B = Bo
S=
and Ampere's
field
V x B
The
0
constant
+
follows
magnetic
6B, Bo
and
6B
=
B1
is
from
field.
+
For
2
, and
in
the
BOZ
only
the
For Boz >> Bo
0 , this is a field with
Taking the 2 component
7
of the curl of (1)
and
using V.B = V.J = 0,
(1)
reduces
t020
B.* Jz . 0
(3)
that the Jz is constant
states
reduced
(3)
current
field lines (J
equivalent
pB).
=
and setting ye=1
= 0,
of JxB
i.e.,
that
J flows
for convenience,
(3)
Jz and B in
It is the
along magnetic
= Vx($g) defining the poloidal
With B,
in (2)
along magnetic field lines.
flux function
are coupled self-
consistently through Ampere's law:
Expressing (3)
in terms of the model field gives
J + B (x)
o3zJ
z
y
B
-
ay Jz
-
+ SB
-J
x axz
=
0
where for simplicity we have retained only 6BX in 6B.
island
structure
z=constant.
can
is
determined
The solution,
= 3B Y/3x.
B'
9y
with a solution of J
unperturbed
line
field
J
) where
in
the
plane
= o + 6p.
This
(x)
oy = B' x
cY
z + 3D3Xz(6
(6)
J(6p - By x 2 /2).
=
trajectory
(closed) when B
by Ax=(26ip/B
trajectory
by assuming a weak shear so that B
resonance by the nonlinear
are bound
z(
The two dimensional
Then, the structure equation is
x-
oy
line
is Jz =
therefore,
be made more explicit
where B
B
by the
(5)
'
go
is
The island
perturbed
and trapped
line bending term 6Bx in (5).
2
x /2
-
6i < 0, i.e.,
is formed as
near
the
the
The trajectories
the island width is given
)/2
An analogous
trapped structure occurs in a Vlasov plasma described
by
the Vlasov equation
v
- e
0+
(7)
In steady state, unperturbed particle trajectories x-vt become perturbed and
8
resonantly "turned around" or trapped
by the
potential $.
island structure is determined by contours of constant
trajectories
i.e.,
the
determines
f=f($+mv 2 /2e).
are bound when the total energy is negative,
island
(trapping) width is Av
the
self-consistent
contours of width
=
required
mV
2/2
+
The
e$
<
to
trap
the
island
Av,
a-2
6$ = -4we
fdv6f
(8)
ax
One divides
6f
0,
Poisson's equation
(-2e$/m)1/2.
potential
The phase space
? describing the island structure and fc
into two parts:
describing the nonresonant
particles passing outside the island. 4
Then, we
write (8) as
2
6$ + 4ne Jdvfc
=
dvf
-
(9)
ax
We Fourier
transform
(7)
and evaluate fc at the
phase speed of the
hole.
Then, in terms of the mean distribution gradient, fc is given by
c
k
This
nonresonant
resonant
island
e
(v>Av)
o
-
particle
potential.
shielding function, Ek,
(9),
1
r6k v f 3v
=
60
distribution
This leads
to the
tends
-(10)
to
definition
shield
of a
out
the
dielectric
from the Fourier transform of the left hand side of
i.e.,
af
2
k
where P means
1-
principal
PIdvvalue.
(11)
The Fourier
9
transform
of
(9)
can
then
be
written as
k
where
=
(4re/k
2
(12)
-
k
)fdvfk is
the
hole or resonant
part
of the
potential.
Av and Av2 =- 2e$/m, (12) then yields an approximate
Since the hole width is
relation between the hole depth and resonance width
vk (kA
2
(13)
t
where
vt is the thermal speed and
AD is the Debye length.
Equation
(13)
gives the hole depth required to trap the particle trajectories into a bound
hole structure of width Av and k'.
Ek > 0.
the
same
hole solutions result for
calculation using maximal entropy arguments yields
A more rigorous
essentially
Localized,
hole
solution
as
this
physical
balance
of
forces
arguments.4
Imposition of self
current
hole.
This
consistency
"modon"
(4)
solution
on
(3)
follows
similarly yields
from
divide Jz into currents flowing inside
(resonant)
the
small
island resonance
as in
(9).
For
(4)
the
with JZ=JZ(p)'
and outside
island width,
bound
we
(nonresonant)
the nonresonant
current can be obtained from the linearized version of (3):
6JNB
k
where k.Bo
response of
= kzBoz + ky Boy(x).
tearing mode
that the field lines
(14~)
6B k
a
k.B 0x
Joz
are
(14)
Eq.
theory.2
The
particularly
rational surface where k.B=0 - 0.
is the
k.B
0
usual nonresonant
singularity
susceptible
to
in
(14)
bending at
current
implies
the mode
The island structure will, therefore, be
localized about this resonant surface at x5
into (4) gives
10
= -
kzBoz/kyBoy.
Inserting (14)
_i - k2 +kJ
y
k.B-0
x2
- 6J
=
k =
(15)
k
The well known Newcomb equation of linear stability theory follows from (15)
6J
with
=
0
resonance.
and
describes
Integrating (15)
Ak =-12kyH
the
currents
flowing
outside
the
island
over these currents gives
y-
Pf dx
6k (x
k.B
-.-
x
oz
(16)
o
Ak is the usual stability parameter of linear tearing mode theory,
-
1=
1
dx
32
s
(q
)
(17)
ak
ax
or, more conventionally2'
64 (1)
6p(2)
where
(1,2)
refer
to
singularity at x=x .
the
k.B0
positions
(infinitesimally)
on
side
of
the
singularity
in
(14).
In
linear
tearing
mode
thleory,
In the nonlinear
the
clump
an analogous role is played by the nonlinear diffusion of stochastic
field lines,
i.e., resistivity due to the turbulence.
Nonlinearly,
(4)
and
(5)
give the self-consistent
note that the nonlinear term on the left hand side of
k.Bo
either
The jump in the logarithmic derivative of 6p is due to
singularity is resolved by collsional resistivity.
model,
)
(
(2)
6
( )
singularity.
stochasticity,
_ For
example,
in
26
the
hole solution.
(5)
presence
We
will resolve the
of
field
line
the nonlinear term gets replaced by a diffusion operator with
the effect of replacing k.B0 in (14)
and (15) with k.Bo + id where 6>0.
We
can then integrate (15) over all space and, using (17), obtain
(Ak + 2k
+
iA) 6$(x
5)
where
11
-
dx6JR
(19)
rk
3J
y
oz
(20)
s
y OY
A is an effective imaginary part of the stability parameter
The factor
A'.
Since the integral in (19) only contributes in the resonant region of width
Ax, (19) implies that Ampere's law can be schematically written as Vi $
M6 JR where M~1 - [Ak +
2
ky
+
AIAx.
M is an effective permeability shielding
the island and is the analog of the dielectric
shielding comes from nonresonant
Ak+ 2IkyI>0,
the nonresonant
currents
Using Ax6JR to approximate
the resonance
width
estimate
6JR
from
fluctuation (z
=
=
(19)
shielding
(6JOZ
)1/2
Here,
the
depth
part
the resonant
the right
from
and obtain the
(11).
hand side of
solution
(-6JR)
of
to
a
(6),
is the
Vlasov hole,
(19)
and
we
can
force-free
hole
z ~ Joz) of small (I6Jz(Joz), but finite amplitude,
(21)
MHD analogue of the Vlasov
it
For
of the
6J z ~ - A'AxJ oz
(21)
the
flowing outside the island.
currents reinforce
island field.
Ax
-
relates
hole relation
(13).
As
with the
the hole depth required to self-consistently
trap
the field lines about the resonance.
As in the BGK solution of the Vlasov case, the current hole (21)
of many possible solutions to the coupled equations
nature
of
analysis.
these
However,
by
the
require
a
detailed
and (4).
pseudo
The full
potential
we take the position where that in the turbulent state,
many details of isolated,
the turbulence.
would
solutions
(3)
is one
coherent
Therefore,
approximate
hole structure will be "washed out"
any bound structures tending to form will do so
balance
of
conditions and detailed behavior
forces
just
described.
of flux contours near
separatrix are approximated by the parameter Ak.
(i.e., the use of ek in (13))
by
The
boundary
and outside of the
An analogous approximation
has proven to be very successful in the Vlasov
12
-
4.
case.
B.
Growth and Decay of Current Holes - Mixing and Stochasticity
We are concerned
at
large
primarily with the inviscid dynamics of current holes
Reynolds
Neglecting
numbers.
the
effects
dissipation
of
resistivity and viscosity, a localized magnetized fluid element--such as an
MHD
hole
addition
to
convection,
waves away from the
Another
IIIC.
radiation
hole.
its
dissipate
fluctuation--can
dissipation
hole.
This
channel--and
of shear-Alfven
the
waves
in
energy
can occur
by
the radiation
effect of pressure
one
of
down
magnetic
ways.
several
is
importance
here--is
field
of
considered
lines
In
3'
sound
in Sec.
through
away
The rate at which the energy is dissipated is VAk.B/B,
the
the
from
where VA is
the Alfven speed and
k is the wave number
of the fluctuation.
Since the
holes
near
surfaces
=
are
localized
dissipation is minimal.
resonance
eliminates
This equilibrium
overlap.
Then,
mode
Indeed,
rational
be disrupted,
even
for
or
"torn-up"
stochastic
magnetic
calculated in detail
as
Alfven
field
lines
in Sec.
III
0,
below,
amplitude
in the previous
propagate
waves
away
a finite
the
island.
if island resonances
however,
the island structure discussed
dissipated
k.B
the trapping of the field lines near the
the dissipation
can
where
from
but
the
down
section
the
hole.
strongly
This
resulting
decay
here
can be understood
is
is
by the
following physical considerations.
The
diss-ipation
approximately
(r'
rate
VAkyxdBoy'/B,
)
for
where Boy' =
a
width of the stochastic region.
plays
role
of
resonance
coherent island regime.
width
that
(In
the
the turbulent
island
width
For strongly overlapping islands,
13
island
amplitude
is
DBoy/3x is the shear strength and
xd is the
the
finite
xd
regime,
plays
-
Ax).
in
xd
the
Hole
energy is dissipated as an Alfven wave
propagates
down a stochastic field
line that random walks radially a distance xd for each longitudinal distance
of zTVA -(kyxdBoy/B)~'.
For diffusing field lines,
is the field line diffusion coefficient.
Therefore,
of
Dm
infinitesimally
expression
small
finite
sized
becomes
islands,
the
an
neighboring
for
0
resonance
0
/B
+ izo')
Di.
Its
given
D/xd where D=VADm .
in
(22)
as in Ref.
solution
apart
neighboring
The
This
is
and
(15)
26.
shows
becomes
at
(22)
island
radially
by
xd after
Two
a length
Here,
D is a turbulent
stochastic
field lines
will
T can
Lyapunov time.
also be written
resistivity which,
along
The time T is the time that two
remain
correlated,
i.e.,
diffuse
The nonlinear time T is a turbulent skin or resistive time.
dissipation can be opposed
the
origin
of
interactions
resonant
fluctuations (clumps)
by the
the
instability.
of
the
finite
production of new fluctuations.
The
instability
amplitude
occurs
islands
the
new
by turbulently mixing the average magnetic shear.
The
The energy
plays the role here that
the
production
of
turbulence
Vlasov
as
create
clumps are produced because the energy is conserved in inviscid
magnetic
known
Equation
that,
-t1-VA/zO, is the inverse
becomes nonzero at island overlap.
together.
well
(22)
In terms of the radial diffusion of the field lines,
with Dm,
the
zo is referred to as the Lyapunov length or Kolmogoroph
The dissipation rate,
entropy".
as T~
by
and the field lines become stochastic.26
field lines will diverge
The length
z-z0 .
equation
k.B
(k.B
Dm becomes nonzero
overlap,
is
x 2 d~Dmzo. In the limit
<6B 2 >k ur6(k.B /B)
-o
dk
(21r)
=
broadened to, approximately,
then
width,
= 2 Dmz, where Dm
1,
D
For
island
2
(6x)
clumps.2
fluid of given
The
phase space
density plays in the
transports
an
energy density to a new region which,
14
(ideal) MHD.
element
of
because of
the
magnetic shear,
has
a different
energy
density.
shear results from a longitudinal mean current
mean current
growth,
density is the
i.e.,
instability,
fluctuations
by
is
turbulent
stochastic decay,
density in the
energy source
achieved
mixing
at
for the
by the
a
the
faster
plasma,
characteristic
(R)
than
1
-
(R
the
new
that
of
the
1)
-
scale
time
(23)
for
of the Rayleigh-Taylor
fluid. 2 93 1
Net
of
growth
is
the
Lyapunov
T.
time
creation of MHD fluctuations by turbulent mixing is the nonlinear,
analogue
the
instability.
creation
rate
magnetic
thus yielding a net growth rate of the form
Y=
The
free
Since
The
(interchange)
stochastic
"-1"
decay
instability
term
in
(23)
in
is
This
turbulent
a magnetized
the
turbulent
analogue of the line bending (restoring) force of the Rayleigh-Taylor model.
The factor R/T is the clump analogue of the mixing rate of light
fluid.
and heavy
The analogy is discussed in detail in Sec. IIIF below.
The instability is fundamentally nonlinear and three dimensional,
though an individual
the
island structure is two dimensional.
This is because
interaction of island resonances with incomensurate field line
(sometimes called field line "helicities") is three dimensional,
Alfven waves propagate down magnetic field lines.
islands,
this resonant
interaction
is
numbers.
The
amplitude.
strength
of
Nonlinearity
the
is
important
Constrained nonlinearly
fluctuations.
shear.
fluctuation,
once
In
addition,
produced,
the
increases
for
1 5
the
(clumps)
nonlinearity
to self-organize
shear
causing significant
with
tends
fluctuation
production
by energy conservation,
(turbulent) transport creates new fluctuations
magnetic
i.e.,
coupling of energy to high wave
interaction
also
pitches
For strongly overlapping
very inelastic,
magnetic field line stochasticity and mode
even
of
the
stochastic
by the mixing of the
to
cause
a
clump
into a localized island
(hole)
Instability
structures.
nonlinear effects
of
the
from
results
These strong nonlinear
and decay.
clump production
these
between
competition
features of the instability imply that the turbulence cannot be described by
a perturbative nonlinear model in which linear theory provides the lowest
The stability analysis relies fundamentally on the
order approximation.
existence of two-dimensional magnetic island equilibria of finite amplitude
These features are reminiscent of
and their three dimensional interactions.
the transition to turbulence in plane Poisuille fluid flow32 and in current
driven
Vlasov
plasma6~9.
As
there,
we
find
turbulence is strongly nonlinear and subcritical.
of the MHD turbulence is described
the
that
transition
to
The onset and evolution
by a fully nonlinear set of equations,
rather than the linearized MHD energy principle or the Newcomb equation for
linear
disturbances.
In the
description
of turbulence
in
fluids,
inadequacy of the Orr-Sommerfeld equation for linear fluctuations
known.
12-1
Newcomb
is well
The MHD clump theory extends the linear energy principle
equation
to
the
nonlinear,
turbulent
regime.
the
Subcritical
turbulence has been apparently observed in computer simulations by Waltz.
and
MHD
4
The depth of the current density hole plays a role similar to that of
"Magnetic vorticity" (J)
vorticity in fluid turbulence.
determines
the magnetic field through Ampere's
law
(2).
self-consistently
Because of
the
existence of the finite amplitude island equilibrium, the magnetic vorticity
fluctuations tend to organize into the localized, resonant island structures
Though
of Sec.
IA above-.
current
density can occur,
negative magnetic
(21).
To
see
both positive
vorticities,
this
is the
it
current
consider
Faraday's law is
16
density
holes,
in the
i.e.,
the
These correspond to A'>0 in
that can grow.
explicitly,
magnetic vorticity, 6J.
and negative fluctuations
an
isolated
concentration
of
6$ = - D 6J
D
where
is
fluctuations
resistivity
turbulent
a
(i.e.,
(24)
other islands).
hand side of (24),
modeling
Growth,
occurs for current holes,
the
a positive
6J<O.
background
stochastic
value to the right
Using the island width,
and (3) for 6J, (8) yields a growth rate, AH = alndw/3t, resembling that for
the tearing mode,20
Y
~
H
D
(25)
Ax
Growth occurs for A'>O (free energy available)
overlapping
resonances).
For
A'>O,
and D*O
the nonresonant
(stochasticity from
current
6
JNR
flowing
and
outside the island reinforces the resonant island current 6JR (see (21))
the island grows.
of two coherent
Though the instability may be precipitated by the overlap
islands
(e.g., two tearing modes),
("collisions") between the islands lead
will quickly develop as interactions
to mode
coupling
(hole decay)
resulting turbulence will
be
and
mixing
composed of
magnetic vorticity concentrations
a truly turbulent state
(generation
of new
incoherent,
strongly
which we call MHD clumps.
holes).
The
interacting
Rather than a
coherent island structure, an MHD clump fluctuation is a flux tube or bundle
of correlated magnetic field lines with finite lifetime on the order of
The instability or turbulence
growth
of
these
clump
is modeled as the creation,
fluctuations.
In
this
turbulent
interaction,
regime,
(25)
T.
and
is
replaced by (23).
The
derivation of (23 ) is the main objective
of this
paper.
We find
that the clump regeneration factor R in ( 23) can be written approximately as
(see (112)
and (114))
R
kd 2 (1 + Yt) -1
IAkxdI
17
(26)
The Akxd factor in the numerator of (26)
for
the instability.
clumps.
instability derives
The
tearing instability.
I4xd-
The
is the magnetic shear driving force
2
its
free
energy as
factor in (26) gives the shielding of the
The clump fields that randomly mix the magnetic shear are shielded
by the nonresonant currents flowing outside the islands.
imaginary
The
part because of the stochastic
(1+YT)4'
factor
in
(26)
occurs
because
the
For YT>1,
appreciable stochastic decay,
and
Y2 _
R
analogue
YT<1,
R(1+YT)~'.
=
of
the
the
effect
and (23)
In
this
fields
causing the
growth can occur before any
(26)
regime,
give
the
instability
interchange
(mixing)
is
a
instability.
When
instability
but driven by an anomalous
resistivity
significant,
and
field diffusion.
(and flow)
In this limit,
(23)
give
Y
-
D/xd.
~
DD
k (R
xd
-
(28)
1)
Except for the factor in parenthesis,
(28)
resembles
growth rate of the tearing mode in the Rutherford regime.20 Here,
the Spitzer resistivity of the Rutherford model,
width xd replaces
accounts
nonlinear
the
is
stochasticity
due to stochastic magnetic
since T~1
clump
resonance.
0
(27)
resembles that of the tearing mode,
and (26)
has a real and
2
Rayleigh-Taylor
of
A
broadening of the k.B
turbulent mixing are growing in time.
where
in the
the coherent island width
fer the net
regeneration
of
new
Ax.
the
D replaces
and the turbulent resonance
The
(-1)
fluctuations
by
factor
in
mixing even
(28)
as
existing island structures are torn up by magnetic field line stochasticity.
The (1-1) factor corrects the growth rate (25)
1.
18
calculated in Sec.
VI of Ref .
C. Magnetic Helicity Conservation
We
derive
clump
the
regeneration
factor
(26)
in
Sec.
III
from
renormalized MHD equations that are an ensemble averaged version of the one
fluid MHD equations.
and
viscosity),
In the absence of collisional dissipation
the
one
MHD
fluid
invariants:
total energy,
II).
for the smallest spacial
(Except
equations
cross helicity,
admit
(resistivity
three
dynamical
and magnetic helicity
(dissipation)
scales,
(see Sec.
these will
approximate invariants for large Reynolds number turbulence as well).
have
discussed above,
that
is responsible
an
MHD
energy
plasma
is
not
it
is
the turbulent
for the generation of clump fluctuations.
with
self-consistent
done
arbitrarily,
magnetic helicity conservation.
(and
cross
mixing of the energy
helicity)
fields,
but
the
rather
is
be
As we
invariant
However,
turbulent
mixing
of
globally
constrained
in
the
by
While we only deal explicitly with energy
conservation
in
this
conservation has important
consequences.
but are derived in detail
in a subsequent
paper ,
magnetic
helicity
These are outlined briefly below,
paper on steady state MHD clump
t ur bulence .
Mean magnetic helicity is conserved because the fields
clumps
by the mixing of the mean
clump currents
field and,
shear
derive
themselves via Ampere's law.
from Faraday's law,
generating the
self-consistntly
from
the
As a result, the mean magnetic
the mean longitudinal
electric
field follow
from a mean nonlinear Ohm's law of the form
- FB
oz = DJoz
oz
(29)
E
The D term in (29)
and would be
describes the turbulent diffusion of magnetic field lines
present for an arbitrary spectrum of stochastic fields.
term reflects the self consistency of the fields,
source of the stochastic
mixing fields
19
is
in
i.e.,
fact
the
the fact
The F
that
individual
the
cLumps
themselves.
In
conservation
the
limit
is ensured
of
zero
island
width,
magnetic
by the cancellation of the D and F terms
This situation is analogous
to the vanishing of the net transport
like-like
one
particles
in
a
dimensional
Vlasov
transport vanishes because of momentum conservation.
conserving,
resonant
interactions
("collisions")
plasma.
Here,
helicity
in
(29).
between
There,
the
we have helicity
between
islands
(current
holes) rather than between momentum conserving, shielded test particles.
next order in the island width,
E
where D-D(Axi) 2
=-
Eoz becomes
V . D . V J
= oz
D[kY 2
+ (Ax)-
2
(30)
(30)
Equation
'-.
net Eoxgo flux of guiding centers in a guiding center
of ambipolarity constraints,
of the
is reminiscent
of the
plasma where,
because
net transport across the magnetic field occurs
at second order in the gyro radius.
part
To
island structures
At island overlap, the deeply resonant
"collide"
in
an
analogous
fashion
to
the
guiding centers, with magnetic helicity conservation playing the role of the
ambipolarity
constraint.
because fdx Eo. B
= 0,
Equation
in a strong and constant
the
magnetic
helicity
(31)
= 0
longitudinal
by a perfectly
combined with Faraday's
field
conserves
i.e.,
Jdx E 0
surrounded
(30)
field Boz.
conducting wall,
For a cylindrical
the integral
law to give the equivalent
constraint
constraint
plasma
can
be
on the mear
_
f dr r B
Equation
(32)
is the
(32
= 0
analogue of momentum
conservation
of the Vlasov cas(
and the ambipolarity constraint of the guiding center case.
Because of magnetic helicity conservation,
20
the turbulent mixing expell:
poloidal
Joz
< 0),
the
edge
(where
>
j
V2
0).
(where V 2
The converse occurs at
Eoz > 0 and the flux decreases (see (30)).
plasma
plasma
a confined
Inside
instability.
flux during the
Equivalently,
from
the
helicity
an increase in Eoz inside a cylindrical plasma (where r is
constraint (31),
small) must be accompanied by a simultaneous, small decrease in Eoz near the
plasma edge (where r is large).
For a significantly large increase in Eoz
on axis (i.e., the development of large fluctuation levels), Eoz will become
negative near the
profiles
have
plasma edge.
been observed
We note that such radial electric
during
plasma disruptions
in tokamak
field
fusion
devices.
Since magnetic helicity
mean
shear
profile,
fluctuations
is
the
(-V i.Joz /k ±J)
oz
helicity
(k
2
turbulent
similarly
nonlinear Ohm's law Eoz
conservation
mixing
constrained.
DJOZ,
=
constrains
(30)
process
While
the
dynamics of the
generating
(26)
the
derives
clump
from
implies that we must multiply (26)
the
by
in order to ensure that the mixing process conserves magnetic
k2
+
(Ax)- 2 here).
Then,
the clump source term f in
(26)
becomes
a
72
x
kd
J~'2
lAxd
This
constrained
form
of
OZ)
(33)
k2
kIJoz
the
clump
mixing
rate
fluctuation level to the mean profiles being mixed.
state MHD clump turbulence
(Y
= 0 in (28)),
directly
couples
For example,
the
in steady
the turbulent mixing balances
the stochastic decay and Joz satisfies
h
(34)
o2 + w2z =0
oz
oz
where
21
d (k2 +
2
AkvX
-
We show in the subsequent
V2
1)
+
paper
2 J
(35)
x2
27
that,
(34)
becomes
= 0
(36)
in the general case of poloidal and toroidal currents.
With V.Jo = 0,
(36)
has the solution
(37)
o
J0 =
MHD clump instability drives the
therefore,
a self-consistent,
plasma toward a force
free state
dynamical route to the Taylor
and is,
state.25
This
occurs in the MHD clump model because turbulent mixing minimizes the energy
subject
to
the
constraint
of
magnetic
helicity
conservation.
The
self-
consistent generation of currents and fields during MHD clump instability is
a turbulent dynamo action.
The D and F coefficients
in (29)
correspond to
the B and a coefficients of dynamo theory.19
In the case of large,
fusion
device,
overlapping island resonances
where the mode rational
mixing lengths are large.
surfaces are widely separated),
the
Instability starts far from the Taylor state and
results in significant disruption of the current profile.
packed resonances
(as in a tokamak
For more closely-
(as in a reversed field pinch fusion device),
the initial
mixing lengths are smaller and a steady state turbulence level is possible.
in order
However,
(i.e.,
to
a
-6 for typical parameters.
changes sign when W>
Boz
provide
fully stochastic
field
a
reversed
basis
tokamaks and RFPs.
for
(Ax-xd)
steady
state
to
exist
p must exceed a threshold given approximately by P2
Ax real in (35)),
> 2Ak k
(37)
for
2
.
4
function solution to
, steady state MHD clump turbulence corresponds
Taylor
a
Since the Bessel
states.
unified
MHD
description
clump
of
instability
turbulent
appears
to
relaxation
in
The instability occurs in its growth phase as the tokamak
22
disruption and,
in its steady state turbulence
phase,
in the RFP.
Detailed
comparisons between experiments and predictions of the MHD clump theory are
presented in a second subsequent paper.
23
II.
A.
MHD EQUATIONS
Dissipative, One Fluid Equations
The one fluid MHD equations
provide
a fluid description of
consistent magnetic and velocity fields in a plasma
achieved
by
combining
momentum
balance
with
.29
the self-
Self-consistency is
Ampere's
law,
and
using
Faraday's law in conjunction with Ohm's law,
E + V
x
B
pJ,
(38)
to obtain
2
p0 (L+
0
B. VB - V1 (P + .)
V. V) V=
tP0
-tl
+ V.V) B
1
+
V 2V
(39)
2p0
= B.VV + (
)V B
(40)
where
rnp is the Spitzer resistivity,
above,
we've assumed that Bz is constant and V=6V is only transverse.
v is a collisional
Three decay laws can be derived from
(39)
and (40).
viscosity and,
as
They are for the
total energy,
' fdx
(
pV2 + B 2 /2p)
=-2v
dx w 2
- 2n p Jdx J 2
(41)
the cross helicity,
.
fdx V.B
(
+
idx w.J
-)
-at0
(42)
P0o
and the magnetic helicity,
dx A.B
where A is the vector
x
V).
=
-2
.. 2
fdx B.J
potential (B=VxA)
and W is the fluid vorticity
Because of the difference in the number of gradient
24
(43)
(w = V
operators in the
dissipation terms of
-
(41)
(43),
the magnetic helicity decays
slower rate than energy and cross helicity.
has been observed in
selective decay.16
at
a much
This feature of the dissipation
computer simulations
and is sometimes referred to as
Energy and cross helicity cascade to high wave numbers
while magnetic helicity remains approximately constant.
In order to define the Reynolds'
normalize
B to Boz,
number subsequently,
lengths to the current
Spitzer resistive time
tR
40 a 2/
=
s
p,
it
is useful to
channel radius a,
and V to a/TR.
Then,
time to the
(39)
and
(40)
become
-2
a
S ( 1
at
(
V.V _SV2)V=B.VB
v 1j.
+
where all
and (45)
V
also that:
(1)
the generalized
+
and,
VxB = J.
physical
=
(2)
= a/VA is the Alfven time),
(TH
pOa
(p
2
/v
is the viscous time).
B /2p%)
+
in (44)
(39)
in
is normalized
with J+(Poa/Boz)J in Eq.
with E+(iea/nspBoz)E in (38),
Note
(2)
Ampere's
Ohm's law becomes
Though we will frequently consider only near ideal MHD effects
therefore,
particular
(T v
pressure
to p*=(p+B2/2pj)(e/B2 oz),
law becomes VxB = J, and (3)
E
TR/Tv
(45)
The two scaling parameters
are the Lundquist number S=TR /TH
=
(144)
-l
= B.VV
quantities are dimensionless.
and the "Prandtl" number Sv
in (44)
) B
-V P*
neglect the collisional dissipation terms in (39)-(40),
normalized
form
significance
of
(44)-(45)
the
will
be
nonlinear
useful
terms
in
the
identifying
(e.g.,
as
the
anomalous
resistivities) and estimating the effects of collisional dissipation on MHD
clump dynamics.
The particular normalization will also be useful in Ref.
35
where the theory is compared to laboratory plasmas with finite S.
We
will
use
the
equation
determined from B and V.
of state
V.V
=
0 so
that
If the turbulent mixing occurs
25
p*
in
(44)
on a faster
is
time
scale
and shorter spacial
assume that mean pressure
scales
than the mean
pressure
profile,
we
can
balance will be satisfied in a quasilinear sense,
i.e.,
<B>.V<B > - V <p*>
The mean
fields.
pressure
0
=
(46)
po=<p*> responds
Of course,
in a low
quasistatically
a plasma,
the mean
magnetic
the pressure p in p* is small,
the mean current flows mainly along <B>,
In the case of Vlasov clumps,
to
i.e.,
(46)
reduces to 1
x
and
Bo
0.
=
the mean distribution is also assumed to relax
quasilinearly.
The mean distribution changes slowly compared to the mixing
rate
production.
for
width)
clump
This
occurs
if
the mixing
is less than the scale characteristic
length
(resonance
of the mean distribution.
In
the MHD clump case, this means Ax < a, where a is the radius of the current
channel.
(46),
From (45),
and V.V
=
0,
6p*
=
p* - <p*> is determined by the
equation
V 26p* =
where 6B=B-<B>.
.[6B
.B
The system
+
(44),
V<B >
SB
(45)
S- 26V.6
-
and
(47)
(47)
form a closed set for the
determination of B and V.
B.
Conservation Laws
We are interested in nonlinear MHD instability and, therefore, cases of
sizable fluctuation levels.
-
S6BAx
(see (44)
and
(45))
In such cases,
the magnetic Reynolds'
is larger than unity.
For Rm >> 1, collisional
dissipation will be a relatively
small effect in (44)
the
the
dissipation
range
Neglecting
collisional
dissipation,
small
scales
dissipation
term
in the
scale).
Ax<xc
parenthesis
of
of
(44)
and
(45),
26
number Rm
and
(xc
(45),
-
except for
(S6B)~
is
the
i.e.,
the
third
the MHD equations
admit
three
invariants: total energy,
cross helicity, and magnetic helicity.
Neglecting
surface terms, the three conservation laws can be written as
a Jdx (B2
2) = 0
S-
(48)
for mean energy,
a fdx
t
-
V.B
(49)
0
=
for mean cross helicity, and
dx
A.B
for mean magnetic helicity.
(49)
result
from
terms in (44)
the
V.V terms in (44)
(50)
As discussed in Ref. 1, the invariants (48) and
particular
and (45).
0
=
structure
For example,
and (45)
and symmetry
vanish separately upon integration
while the B.V terms,
when integrated,
and
a
where magnetic
(i.e.,
dissipation
converted into each other).
= 0.
E,
This can
i.e.,
effect
Equation
be seen directly
(50)
(i.e.,
a mode
cancel between
and
(44)
flow energy
are
results from fdxE.B=0 when E+VxB
by considering (45)
explicitly in terms
of
Faraday's law gives
a
f dx A.B
=
Since Bz is assumed constant
in (49).
-
2
dx E.B
(48),
constrain the dynamics
example,
we can set
If we further assume an ordering where Boz >> Bol, then (50)
or,
(49),
(51)
and V=6V is transverse only,
is equivalent to (31)
zero.
nonlinear
total energy is conserved because the
coupling effect)
(45)
of the
and
(50)
consider
By (49),
the
with Faraday's
simple
case where
cross helicity will remain
and fluctuating parts gives
27
law,
the
(32).
The conservation
of the B
cross
zero.
and
V fields.
helicity
Separating
laws
is
(48)
For
initially
into mean
+ S-2-
dx <6B
B >2
dx
(52)
where <> denotes an ensemble average.
The conservation law (52)
any rearrangement
mixing)
profile
will
(such
necessarily
fluctuations.
Note
are allowed.
turbulent
produce
that
those self-consistent
(32),
as
(52).
energy,
rearrangements
that
Equation (32)
the right hand side of
fluctuation
arbitrary
motions
of the mean
satisfy
(50),
are
or
states that
magnetic
i.e.,
not
field
MHD
clump
allowed.
Only
equivalently
(31)
or
constrains the source of fluctuations on
Equivalently,
we note that Faraday's law
is
linear, so that the right hand side of (52) can be rewritten in terms of the
mean electric field Eoz
=
<Ez> to give
+ S2 62>
dx <6B
The equivalent
=
2 Jdx E
J
mean magnetic helicity constraint
(53)
on
Eoz
in
(53)
is
then
the energy is dissipated.
The
(31).
Note that,
in a course grained sense,
turbulent mixing converts the large scale
(mean)
shear profile into smaller
scale clump fluctuations.
Since the magnetic helicity is conserved
the mixing,
dissipated
helicity.
the energy is
This
is reminiscent
presence of collisional
in Sec.
III that,
of
dissipation
the
subject
to
selective
(see Sec.
the
decay
of the
that
in the
occurs
Indeed,
the nonlinear
and (45)
have the effect of anomalous resistivity and
be seen
approximately
here
conservation
IIA above).
during the turbulent mixing,
by writing Rm~S(6B/Boz)
during
terms in
viscosity.
(ax/a)
we show
This
(44)
can
in dimensional
units as
Rm ~ [VA
The bracket in (54)
x(6B/Boz )/nsp
(54)
is an anomalous resistivity due to magnetic fluctuations
SB with transverse correlation lengths Ax.
28
In the renormalized turbulence
equations
derived
diffusion
coefficient,
lines.
Sec.
III,
D=VADm ,
the
bracket
for
diffusing
in
(54)
gets
stochastic
replaced
by a
magnetic
field
It is also enlightening to write (54) as
Rm
The
in
first
sp
factor
resonant layer .
in
oz
brackets
nonlinearly
(Lyapunov)
is
the
The second factor
time in the resonant layer.
respectively,
(55)
BA
[Ax)
-
then
Rm
in the
=
collisional
resistive
is the inverse of the
time
in
the
perturbed Alfven
If we denote these nonlinear times as TR and TH
TR/TH,
resonant
i.e.,
the
layer.
time is -t = TR(ri~p/D)
The
TR/Rm
=
Lundquist
dimensional
TH,
=
number
(S)
nonlinear
defined
resistive
i.e., the Alfven time defined
for the perturbed field in the resonant layer.
C.
Clump Fields
From the above,
investigated
with
Since
helicity.
a
we conclude
dynamical
these
that
model
constraints
properties of the nonlinear
MHD clump
that
are
terms in
taken in treating those terms.
due
(44)
fluctuations
conserves
to the
and
In particular,
energy
and
structure
(45),
special
can only be
magnetic
and symmetry
care must
any renormalization
be
method
used to approximate the stochastic or turbulent portion of these terms must
preserve
their
satisfied.
symmetry
(44)
so
that
the
conservation
One way to facilitate this is to rewrite
of new field variables
terms,
properties
and (45)
(
(
at
L
=
B
- S~1V and N =
B
(44)
+ S~'V.
and (45)
laws
in terms
Neglecting
can be written for unit Prandtl number (S= = 1)
are
6p*
as
- SL.V - V ) N = 0
(56)
+ SN.V
(57)
-
)
J.-
L= 0
29
where
and
<L>.V<N>
are
of
because
absent
as we show in
because the symmetry is best displayed for this case.
not
generally appropriate
driving
terms in (56)
for this is that the nonlinear terms dominate the
Since V.L =
the instability.
collisional
of
where
turbulence,
a
(49)
(31)
or (32),
can
average
the
homogeneous
with an
identified
be
of
case
ensemble
(56)
Vlasov
conservation
In
equation.
of
<N 2 > and
electron and ion
the conserved
turbulent
decay.
in describing clump dynamics
(57)
and
fluid in a Vlasov
absence
the
<L 2 > is
this
of
analogous
phase space
quantities
Here,
are
collisional
to
densities
mixed to
to
cascade
high
the
is "incompressible"
is their
i.e.,
to
dissipation,
the
plasma,
conservation
finer
wave
2
(i.e.,
,3
As
scales
during
occurs
in the
spacial
numbers
(mean
of
RM-+O),
the flow
and a magnetized fluid element retains
(the smaller the
its energy density for a finite time
longer the lifetime).
is still
in a Vlasov plasma.
With the neglect of collisional dissipation
and (57)
to (48)
<L>/3t = 3<§>/3t.
to the flow of phase space
resemblence
in (56)
of the
structure
conservative
In
dissipation.
since 3<N>/3t =
The advantage of
energy.
collisional terms during
The relevant magnetic helicity constraint
respectively.
and
there,
the
the reason
2
2
the conservation of <(N 2 +L 2 )> and <(N -L )> is equivalent
average,
square)
spacial
= 0,
V
ensure that <N2 > and <L 2> are invariants in
and (57)
nonlinear terms in (56)
absence
V.N
resulting
Again,
large Rm clump turbulence.
dissipation for
collisional
is
and effective measure of
an approximate
do provide
(57)
and
and
(56)
While S =1
the
plasmas,
interesting
all
for
in
S,=1
we have taken
Additionally,
clump production.
(57)
the
pressure
III, the 6p* effects are small compared to the magnetic shear
terms for
the
mean
We have suppressed the ap* terms here because,
balance (46) .
Sec.
terms
<N>.V<L>
During this time,
30
volume
element,
the
the turbulence randomly transports
the
As in the Vlasov
are different.
the <N 2> and <L2> energy densities
shear,
magnetic
nonzero
for
where,
of space
region
to a new
fluid element
plasma, this mixing process is the origin of the clump fluctuations.
(56)
While
deduce them
interesting to
IB
Sec.
in
that,
governing MHD
effects
to
addition
is
it
MHD equations,
the
from
physical
from the
from
We recall
evolution.
directly
follow
(57)
and
clump
a
convection,
localized clump fluctuation can dissipate its energy through the emission of
shear
Alfven
We,
fluctuations
have
therefore,
two
fluctuations
tend to exist in mutually exclusive
clump is resonant
The two
the
surface,
the mode
region away from
in the nonresonant
While
regions of space.
mode rational
near the
and localized
consider:
to
fluctuations.
(clump)
and non-wave-like
(Alfven waves)
wave-like
the
waves.
shear Alfven wave
propagates
rational surface.
The two fluctuations represent two degrees of freedom or
The Alfven wave magnetic fluctuation is given, in
excitation in the plasma.
by 6B
dimensional units,
this
expression
clump
localized
is
valid
even
by
obtained
component from the total fluctuation 6B
.
6
subtracting
of the
(11)).
the wave has two polarizations,
we define N = B + /pV
backward
Alfven
convection,
Consider
wave
has
the
been
dynamical
subtracted
collisional dissipation,
and the
31
wave
field fluctuations
is
equation
out,
the
response
and L = B - /pV
for
N
propagation
part
in the MHD case,
Since,
(see (10),
units.
this
(nonresonant)
through the dielectric constant
dimensional
out
The
(resonant) non-wave-like
the wave-like
fc determines
waves.
we wrote the total fluctuation
There,
Sfc + f where T denotes the localized
of the fluctuation and
Alfven
This subtraction and distinction
portions
and non-wave-like
analogous to the one made in (8)-(9).
as 6f =
amplitude
finite
for
is
fluctuation
between wave-like
Note that
v/p6V, where p is the mass density.29
=
N.
field
Since
decays
in
the
by
of forward Alfven
waves.
For unit Prandtl number,
V.) a N
(
we have,
in dimensional units,
2
1 B.N +
=
The equation for L is similar,
(58)
except that Alfven wave emission
is due to
the backward wave:
(
+
a~
1
V.V)L =-
A rewriting of these
B.VL +
two equations
2
nV L
(59)
expresses
their symmetry
and coupling
features:
+
N.V
+ (p-1/2
(p)
a -_ (p)p-1/2
These
equations,
when
fields N and L have
physical
n2
nV?)L
-
n2
L.V -
0
(60)
=
0
(61)
=)N
nondimensionalized,
historically been
significance
=
are
just
(56)
known as Elasser
and
(57).
The
variables."
Their
here lies in their identification with the
clump or
resonant portion of the field.
Besides providing a natural
basis for clump analysis,
N also define the correct variables
case
(see Sec.
4-3
of
Ref.
depend on the mean-square
31).
flow
the fields L and
for analogy with the unmagnetized fluid
Turbulent
fluctuation
transport
in the
pure
depend on <L 2> and (N2> in the corresponding MHD case.
of the B and V fields in the turbulent transport
coefficients
fluid
case
This equal
"frozen-in"
symmetry can be "broken"
so that
subtraction
equations
of
(56)
(44)
and
and (57)
(45)
case of
will not
expressible
footing
Physically,
B is transported with the flow.
by collisional dissipation.
0 but v = 0 as in the classic
For example,
the tearing mode,
lead to the
particularly
This
for nsp *
the addition
in terms of L and N alone.
32
will
processes can be traced to
the symmetry of the nonlinear terms in the ideal MHD equations.
the fields are
which
and
symmetric
While the N
and L dynamics are approximately conservative for Sv
down to Rm
"cools"
turbulence
form
conservative
clump-like,
1,
-
symmetry will
the
of
equations
broken
be
and
(56)
if the
1 and Rm >> 1,
-
(57)
the
and
not
will
be
preserved.
to note that the symmetry of the nonlinear terms in
It is interesting
(56)
and
(57)--specifically
the
absence
of
and
SN.V6N
similar situation also occurs in the Vlasov clump case.'
Poisson's
equation,
momentum.
As a consequence,
electron,
or ion-ion) cancel
self-consistent
and
take
fields
fields
conserve
(electron-
Net transport arises only
between themselves.
The
analogous situation
since they are self-consistent to start with (i.e.,
and (57),
(57)
and ion
A
because of
There,
contributions from like-like
As a result, the nonlinear terms in
Ampere's law has already been imposed).
(56)
electron
between like and unlike fields.
from interactions
occurs in (56)
the
terms.
6L.V6L
on
symmetrical
a
helicity and leading only to
conserving
form--thus
like-unlike
energy
interactions
(nonlinear)
and
between
the fields.
As
in
the
appropriate
case,
the
quantity to describe
point equations
elements
Vlasov
and,
two
point
clump dynamics.
therefore,
would
plasma would
the
clump
lifetime
predict
diffuse
that
field lines
at
separation,
and are
all
thus
localized,
radial
fluid
However,
an infinite lifetime,
neighboring field lines feel
correlated
forever.
33
Such
the
is
The reason is that one
of
Consequently,
independently.
infinitely small scale will have
together,
,3
When renormalized,
spacial extent would decay at the same rate.
of zero
2
function
cannot describe the correlated motion of neighboring
magnetized fluid elements in an MHD clump.
equations
correlation
fluid
correlated
the one point
positions
elements
in the
of
a fluid element
since,
of
in the limit
the same forces,
an infinite
any
diffuse
lifetime is
merely a statement that the total energy,
For a
clump of finite spacial
extent,
i.e .,
<N 2 > and <L 2>,
neighboring field lines
different rates and diverge apart exponentially.
cascades to high wave numbers.
is conserved.
at
The clump decays as energy
As discussed in Sec.
time for decay is the perturbed Alfven
diffuse
IB, the characteristic
time--the time for
Alfven waves
to
propagate down the stochastic field lines away from the clump.
It is perhaps appropriate here to comment on the reduced MHD (Strauss)
equations36
for
the
poloidal
flux
and the
t
These follow from the inverse curl of (40)
vorticity
and the
U=-.(VxV)
2 component
=-V2.
of the curl
of (4):
at
- -
( +1.)*
p(
+
3pzZ
=
z+(
V.V) U
resistive
20),
it
study
MHD fluctuations
appears that
of
production
nonlinear
is the
B.VJ
=
Though the Strauss equations
aZ
have
3
2
vV U
+
(64)
proven extremely useful
such as
tearing modes
(see,
k, (63)
equations
clump
First,
essence
fluctuations.
(64)
for
of
"phase
and
p
appropriate conservative form.
U
space
the
cascade).
equations
natural
for
way.
B
and
As
we
V
display the
In addition,
properties of nonlinear
and
(64).
Because
the
cannot
Ref.
for
of
the
clump
density" and its resonant
While U does cascade to
be
combined
into
such
an
The problem can be traced to the fact that
is not mixed during turbulent decay but,
(inverse
example,
appropriate
conservation
and
for
they are not the most
cascade under turbulent decay to high wave numbers.
high
in the study of
have
rather,
shown,
flows to long scale lengths
however,
conservative
the essential
p
the
MHD
resonant
full
unreduced
clump dynamics
localization
and
MHD
in
a
decay
clump fluctuations are not directly evident in (63)
Alfven
wave
34
response
has
not
been
explicitly
(subtracted
distinguished
appear
propagators
clump
fluctuation
waves .
(57)
as
can
this
and
However,
decay resonantly
It is the localization of
displays
(58)
in
to be nonresonant.
that minimizes this decay.
and
out
the
by the
clumps
(59)),
as
we
the
fluid
element
have
seen,
an MHD
shear
Alfven
emission
near
mode
of
rational
surfaces
The resonant property of the propagators in (56)
clump
localization
directly
and
leads
to
the
exponential increase in the separation between neighboring field lines i.e.,
to the resonant
decay of the clumps.
Of course, the Strauss equations have
the advantage of being scalar equations that do not involve the pressure p*.
However,
we shall see that the two point correlation equations we will need
to describe the
and <N(1).N(2)>.
terms in (44)
clumps
involve only scalar
We will also show that,
and (45)
can be neglected.
35
quantities
for the
such as (L(1).L(2)>
clump problem,
the
Sp*
RENORMALIZED EQUATIONS
III.
of
The renormalization
carried
out
focus only on
those terms
in the
magnetic
helicity
in
perturbation
friction" term2 7 that
the
3
, to numerous terms,
expansion
that
have
an
These are a Markovian diffusion term and a
identifiable, physical meaning.
Fokker-Planck "dynamical
perturbation
renormalized
straightforward
a
as in the case of Vlasov turbulence 3''
procedure leads,
we
While
below.
is
(57)
and
(56)
equations
field
clump
the
model.
This
is
ensures
not
to
the conservation
say
that
the
of
other
contributions to the renormalization are necessarily smaller in magnitude or
less
important.
Any
terms
coalescing), for example,
relating
to
clump
self-energy
(island
must play a relevant role in the turbulence,
we have not been able to identify such terms.
In our
view,
but,
the simplest,
self-consistent,
energy and helicity conserving model of the turbulence can
be
from
constructed
only
diffusion
the
generated from the formal renormalization
proven
useful
in the
only a diffusion
Vlasov
and a momentum
provides
from the renrormalization
dimensional
Vlasov
phenomenologically
clump
0
the Vlasov renormalization
self-energy
corresponding
terms
basic
dynamics.
'
1
2
(Fokker-Planck
but
The
agrees
drag
terms
Such a view has also
a statistical
conserving
the
Fokker-Planck
procedure.
There,
clump self-energy,
for
computer simulations.",''
clump
case.
and
essential
model,
model retaining
friction)
features
when
of one
modified
well with the results
of
The physical meaning of additional terms in
have proven to be obscure or impenetrable.
effects
have
generated
in
been
particularly
the
renormalization
dimensional MHD case are even more complex and obscure
ignore them in this investigation.
motivated models over formal,
term
illusive.
of
and we,
the
The
The
three
therefore,
An emphasis on conceptual and physically
mathematical
36
approaches
has also
been
useful
As there,
in the understanding of fluid turbulence.
the model
developed
below focusses on the concepts of turbulent eddy diffusion and mixing.
can be
It turns out that the effects of magnetic helicity conservation
treated
from
separately
Therefore,
in
conservation
the
those of
balance
(and,
this
of
therefore,
cross
energy and
paper,
we
neglect
the "-1"
magnetic
the Fokker-Planck drag terms)
on renormalized equations of the diffusive type.
term derives from the diffusive
conservation.
helicity
helicity
and focus
This leads to
only
(28)
where
decay of the clumps and the
clump
source term R derives from the diffusive mixing of the mean magnetic shear.
Corrections
due
to magnetic
paper (Ref.
subsequent
A.
to i
27),
helicity
conservation
are
but have been reviewed in Sec.
derived
in a
IC above.
One Point Renormalization
Though
our
is
goal
a
renormalized
equation
correlation function,
the diffusion coefficients
depend
fluid element
on
one
point
for
the
in the two
propagators.
We
two
point
point equation
obtain
these
in
the
spirit of Refs. 38 and 39.
The renormalization is done at the level of the
fluid element
in position space,
transform
trajectories
space.
renormalized
allows
This
field
line
for
a more
trajectories
of
rather than
connection
transparent
Sec.
IA,
in full
as
well
as
Fourier
with the
making
the
physical meaning of neglected non-Markovian effects more clear.
Working
component
in
slab
geometry
for
of the fluctuating field,
simplicity
and
we suppress
retaining
the V
only
the
2
collision term for
simplicity and write the perturbed version of (56),
(2Consider
the
fluctuations
S<B>.V - S6L
fluctuation
6Lk(x)
6N
in
3) 6N
a
=
large
with wave numbers
37
k =
S6L
<N>
group
(kypkz).
of
(65)
turbulent
We
background
seek the ensemble
nonlinear
averaged effect of the background fluctuations in the
of
term
To do so, we calculate that part of 5N that is proportional to each
(65).
6Lk* in the nonlinear term of (65),
i.e.,
6N(x,t)=S 6N~xt)=s
(x,t,)e i'<x
0 dt' G(t,t')6L kax
Note that there is also a part of
right-side
of
(65).
However,
3 6N(x,t')
6N coming from the
that
term
(see
(66)
3<N>/3x term on the
(70))
contributes
equation for <N> rather than to the equation for AN.
to
the
The operator G(t,t')
in (66) satisfies
(
- S<B>.V - S6L
at
with G(t,t)=1.
the
phase,
x(t')=x
stochastic
G(t,t')
and,
G(t,t') = 0
magnetized
(i.e.,
Eulerian
fields,
therefore,
fluid
element
into
strictly
trajectory
we retain only the ensemble
=
set G(t,t')
<G(t,t')>
=
background
only
by setting 6N(x,t')
if
the
scale
lengths
fluctuations
are
much shorter
and
than
with
For random
in
we insert (66)
at
- S<B>.--
correlation
those
6N = S6L -3x <N>
in
We
and
This approximation is
of
times
6N(x,t),
into the nonlinear term of (65)
-ax D -)
ax
(66).
= 6N(x,t)
yields the physically appealing result of a diffusion equation.
approximations,
initial
averaged orbits
<G(t-t')>
a6N(x,t)/3x outside of the integral in (66).
valid
x(t)
variables).
Lagrangian
further make the Markovian approximation
pulling
(67)
G(t,t') is a single element propagator that converts x into
time-dependent
condition
a)
3x
of
the
but
it
With these
to obtain
(68)
where
S
D = S2J
o
is
the diffusion
dk
<L(
<6L2(x,t )> ke
dt
(2t).
coefficient.
x<G(t)e
(69)
k
According to
38
(68),
the ensemble averaged
effect
of the
turbulent
background
fluid element to diffuse.
fluctuations
is to cause
The mean field also diffuses.
a magnetized
Inverting (68) by
the use of <G(t,t')> gives the portion of 6N that is phase coherent with 6Lk
6N(x,t)C=Sf dt'<G(t,t')>e
- 6L (t')
k
0J
a (N(xt')>
(70)
(70
We again make the Markovian assumption
by setting <N(x,t')> = (N(x,t)> and
pulling
integral
a<N(x,t)>/3x
outside
of
the
in
assumption here is on a sounder footing than in (66)
(70).
The
since
Markovian
the mean
evolves more slowly and on larger scales than the fluctuation 6N.
of (70)
field
Insertion
into (56) and ensemble averaging gives
at
The turbulent
Note that the
a
<N> =
ax
(D+1)
<N>
ax
diffusion coefficient
positive
(71)
D appears as an anomalous
terms
diffusion
in the
consistent with the cascade of these quantities
equations
resistivity.
and L are
for N
(energy and cross helicity)
to small spacial scales.
The turbulent
Alfven
speed
diffusion of the mean field
to the
discussed in Sec.
IB.
diffusion
in
of spacially
stochastic
by the
field
lines
To show this, we note that
( - S<B>.V -
D
) <G(t,t')> = 0,
a result that
follows from
(67)
been
from
Assuming
derived
z
in time is related
(65).
(72)
in the same manner in which
the
model
sheared
field
(71)
of
has
just
Sec.
IB
(G(t,t')> expi(kyy+kzz) can be evaluated from (72) and inserted into (69) to
give
fdk
D
f
2 S2 <L 2 >
G
(73)
(2T)
where
39
Gk =
0
-
(74)
dt exp[iSk.gt - (t/T ) -Yt]
with
to
The
quantity
k2 B'2 S2 W-1/3
(3 ky B'
oy S D)
Y in
(74)
follows
square fluctuation level,
from the
with 6L
)0 .
i(Sk.B 0 +iT~
leads
exp ft
so that in
it
in (69).
the
is Da2 /TR = SaDm/TR = VADm where Dm is given
is useful to approximate
As with zo
dt'Y(t')
dimensional units,
field)
(the clump portion of the magnetic
For finite amplitudes,
=
slow growth of the mean-
<6L 2 (t)> = <6L2
i.e.,
anomalous resistivity D in (71)
(22)
assumed
Gk = irVSk.po),
For weak static fields,
by
(5
(75)
T=1
(see Sec.
to a nonzero D at resonance
IA),
overlap.
(74)
5B.
replacing
as a Lorentzian,
Gk
To broadens the resonance and
From
(74),
D becomes
nonzero
when
x-x
where x5
(76)
D~S
=-
|
x0
< (D/3Sk B' )1
Sy
oy
(76)
o
is the position of the rational
kzBoz/kyBoy
is just the island resonance overlap condition.
2
6B2 res to where 6Bres here denotes the clump
within a resonance width of x5 .
Equation
(76)
surface.
Equation
To see this, we write
part of the field that is
then gives D~S6Bresxo and xO
becomes
x
-
0
(6B
res
/k B' )1/2
o oy
(77)
Since (77) is on the order of the island width in the resonant modes,
(76)
becomes
Note
x-x53<Ax and, therefore, D becomes nonzero at island overlap.
that DX 00/T0
-
(AX) 2 /.
The field lines diffuse
with a step size of the
island width and with a time step of the Lyapunov time.
This suggests the
model
in
of
dimensional
colliding
units,
islands
that
D~(6Bres/Boz)
we
VAXo
40
have
alluded
where
we
to
interpret
Sec.
I.
In
xO
as
the
transverse correlation length.
Similar
calculations
as
these
can
be
carried
out
on
(57).
For
simplicity, we consider only the case of L and N coupling where (V> = 0 and
<V.B> = 0.
This is a "strong" coupling where <N 2> = <L2> = <L.N>.
consequence,
<L> also satisfies (71).
sign of S (i.e.,
the direction of Alfven wave
(56)
for N to (57)
only
the
real
Physically,
Note that DL = DN = D.
propagation)
for L does not effect the
part
Alfven
of
the
wave
propagator
emission
Gk
causes
direction of the wave propagation.
diffusion
is
The change in
in
going from
coefficients
required
clump
As a
for
D
decay--regardless
since
or
DN.
of
the
The strong coupling limit means that the
mean-square magnetic and flow fields are transported similarly in accordance
with the
respond
(approximate)
nonlinearly
frozen-in
on the
property
same
of the
scale,
time
fields.
i.e.,
The two fields
TL
=
TN=
The
To.
situation is analogous to the case of electron and ion Vlasov clumps where
The strong coupling limit has an
the electron and ion masses are equal."
additional advantage: we need only consider the energy and magnetic helicity
invariants
(48)
and
(50).
The
third
invariant
of
cross
helicity
is
satisfied identically for all time (see (49)).
B. Two Point Renormalization
The
renormalization
of
<N(x,,t).N(x2,t)> and <Ll.L2>
of the last section.
the
=
two
point
equations
for
<N1 'N2 >
<L(x 1 ,t).L(x2 ,t)> can be done in the spirit
We only sketch the procedure here,
but the reader can
find details in Sec. VII of Ref. 1.
The two point version of (56) is
- SL.Y
where
we
have
- SL 2 ' 2
again
suppressed
(78)
0
2'2
collisional
41
dissipation
terms
for
simplicity.
6(Nl.82),
Separating mean and fluctuating quantities
by NI.N
= <N1.'2) +
2
becomes
(78)
at
S<B >.V - S<B >.V ) (N .N
1i -1
-2 -2 -1 -
-
i
- S
<6Li 6(N
N2)> - 0
(79)
i-1,2
In order
to evaluate
version of (70),
the nonlinear
term in
(79),
use
the
two
point
i.e.,
c t
-ik.x.i
6Lk (t') e
dt' <G 12 (t,t')>
6(N 1.N2) = S
.V<Nl
(t')
~ ~
-
i-1,2
where G1 2 (tt')
we
(80)
.N2 (t')>
satisfies
(-sL
.
- g222) G
= 1.
with G 1 2 (t,t)
<N (t').N2(t')>
reintroducing
=
the
t~t')
Again we make the
<Nj(t).N
2
(t)> in
collisional
= 0
(81)
Markovian
approximation
(80)
into
gives
the
Substituting
(80).
dissipation
term
then
by setting
(79)
and
bivariate
diffusion equation
S
<B.
i=1,2
1
V .(D.
-.
1
+2).V
1,2
<N 1 .2>
<0
(82)
2
-l
where
D. .=S2
=o
CO
dk
f 2dt2 <L(xit)6L(x ,t)> e
(2 ) 2
- ik1
Similar calculations
can be
-ik.xik.x.
<G2(t)> e
carried out on
Markovian approximation,
we find that <G
as <N1 'N2 >, i.e., (82).
The orbit function
then
12
be evaluated as in the last section.
field of Sec.
IA and find that
42
(81)
and,
(t)> satisfies
-
-j
again
using the
the same
<G 1 2 (t)> exp ik.xj
We again
(83)
in
use the model
equation
(83)
can
sheared
dk
Dij = S2
<
(2
where Gk is given
by (74)
kG exp ik (y
and,
for the near resonance
xl=x 2 in the spectral function and set
k.(xi-x)
case,
we have set
= ky(yi-yj)
in the orbit
Note that the xx component of (84) , which we denote
function.
just (73)
(84)
y
as y1 _+y
2, i.e., D11 = D2 2
Retaining
only
the
x
=
by Di
, is
fluctuations
for
D.
of
component
the
field
simplicity, we can straightforwardly obtain the equation for
from
(71)
and
subtract
<6N 1 .6N2 > = <1'2>
the result
<N1>.(2>.
-
from
(82)
Since uncorrelated fluid elements
= 0.
coordinate x. = _x-x
2>
Assuming that <6N,.6N2 > depends
2
2
equation
to obtain the
independently, we can equivalently deduce that <N1 >.(N
with D12 = D21
D(N1 >.<N
>/3t
for
diffuse
satisfies (37), but
only on the relative
, an approximation valid for scale sizes less than the
mean shear length, the equation for <6N,.6N 2 > can be written as
<B_>.V
-
=2D
where D_ = D
equation
similar way.
at
for
vanishes.
Also,
emission)
terms
term
3x
12
(N
' 2
2
dk2
<6N .6N2
3<N I
(85)
with 6L as the x component of 6L,
G, exp ik
2(xk=)>
(86)
y_
(21T)
<6L,.6L 2 )>
corresponding
to
(85)
can
be
obtained
in
a
Note that we have evaluated D12 on the right hand side of (85)
x_-0 since,
source
(D_+2)
D22 - D12 - D2 1 and,
+
D12
The
-
for
IxI>xd,
D1 2 +0
(see
(84))
and
the
clump source
since the linear and nonlinear shear damping
vanish on the
left-hand
on the right-hand-side
of
when x_=0.
43
side
(85)
of
(85)
as
x_+O,
makes is largest
term
(Alfven wave
the
clump
contribution
The correlated and relative diffusion coefficients D1 2 and D_ have the
following meaning.
As x1 +x2 , two points in a magnetized fluid element feel
the same stochastic forces and,
nonlinear
diffusion terms
helicity
are
again
In the case of nonzero shear,
diffusive mixing
i.e.,
the
conserved.
The
(rearrangement)
of
clumps.
This
is the
= 0,
coupling limit where <V.B>
relation (52).
(85)
and D.+2D.
meaning of
is just
produce
(85)
and,
conserve
means
total
that
fluctuations
in
the
the
<6N
2
>,
strong N/L
production
the energy fluctuation
yields the same result.
different forces
feel
(N>2 must
<L2 >, the
energy and cross
<N 2 >
of
that
In this limit, the equation for <6L 1 .6L2 >, being the same as
but with S+-S,
elements
conservation
For the
dissipation,
generated in the renormalization
energy and cross helicity.
D.+O.
holds for
Since an analogous result
(82).
(N2 > is conserved in
i.e.,
in the absence of collisional
case this means,
shearless
diffuse together,
thus,
net
The
effect
independently,
and thus diffuse
of
D_
is
elements of different spacial scales.
to
xI-x 21, two fluid
For large
diffusively
In this way,
D
i.e.,
mix magnetic
12+0
fluid
energy cascades to small
spacial scales (high wave numbers).
Along with the
the
conservation
renormalization
preserves
of magnetic helicity
all
original one fluid MHD equations.
ensures
us
equations
that,
while
we have
One
dynamical
invariants
Since
turbulent
the
and
could
some terms
go so
far
is
the
the
dynamical
in Ref.
invariants
This is of crucial importance.
have
been
neglected,
the essential
derived maintain
equations.
renormalization
of
imposed
as
to
most
say
that
important
physics
the
of
27,
the
First, it
the renormalized
of the original
preservation
requirement
of
the
of
the
x_-0,
the
procedure.
two
point
collisional
correlation
diffusion
function
terms
44
on
is peaked
the
about
left-hand-side
of
(85)
cause
decay of
the
corresponding
There,
(2 4 ).
terms produce
=
in
e.g.,
(68)
and
in
the
clump
localized
point
two
the diffusion
Physically,
(as for the tearing mode).
decay
a
the
- 6Jz > 0 for a hole fluctuation so that the diffusion
growth
cause
operators
elements
V2
to
contrast
in
point equations,
in the one
diffusion terms
is
This
fluctuations.
localized
equations
because
orbit
stochastic
undergo
fluctuation
neighboring
Correlations are destroyed
instability, and thereby diffuse apart radially.
by this effect.
C.
Effect of Pressure
Because the p* pressure terms conserve energy by themselves when V.V=0,
we have
and
been able to treat the renormalization
properties
conservation
of the nonlinear N.V and L.V terms separately from the pressure terms.
now
consider
the
pressure
terms
negligible compared to the current
does not effect the mean magnetic
has
no
(46),
their
(magnetic shear)
it
is (B> that determines <p>.
equations.
effect on the correlation function
means that the corresponding
equation
for
(85)
is
the
Because of (46),
<p>
While the remaining
from V.V=0,
6p* does
fdx<V.V6p*>
vanishing of
The
balance.
because of mean
Indeed,
field profile.
contribute to the total energy
function
on
driven clump effects we
6p* terms do have an effect, the effect is small since,
not
effect
The main reason for the neglect of <p> is that <p>
have already considered.
pressure balance
that
and show
We
6p* contributions to the two point correlation
energy will
vanish
as
x-+0.
Since
the
shear
driving (mixing) term on the right-hand-side of (85) is finite and indeed at
its largest value at
negligible.
part
of
6p*
x_=0, the effect of the 6p*
It is useful to see this in
in
(47)
(consideration
of
45
detail
the
terms are,
by comparison,
by considering the linear
nonlinear
leads
to the
same
conclusion).
Using V.<BX>=O and V.6B=O, this part of 6p* satisfies
V 2 p* = 2 B'
oy
1
as above,
where,
correlation
is the
6B
6V
between
(87)
6B
ay
x component
and
6B
(strong
of
coupling
neglect
the
makes
the
again
If we
6B.
limit),
5p*
following contribution to the 6N correlation function equation (85):
a1
Since
6p*
- 6B
<6p* 6B2>
from
1
-
2
(87)
-
and
ax2
<66B >
2-
<6B6B>
a
is
function
so that
- can be evaluated from (87)
a<6p*6B2
(88)
(88)
of
x_,
the
term
gives a contribution
to (85) of
x
-4 B'
oy
dx'
ay_
<6B(1)6B(2)>
(89)
0
In wave
number
90(2)].
The additional source term to (89) vanishes as x_+O, thus making it
space,
this
6p*
contribution
is proportional
to
k.[-E(1)-
negligible to the shear mixing term already present on the right-hand-side
of
(85).
The
terms
corresponding to (89)
sign.
are
in
appears
ratio
in the
of
x_/xd
for
small
.<6Li.5L2 > equation,
but
A
term
with a
plus
x-.
CALCULATION OF THE GROWTH RATE
IV.
A.
Nonlinear Newcomb Equation
and
will
fluctuations
determine
in the
of the
determined
the
effect
vicinity of
fluctuations
by the resonance
of
a mode
away
turbulent
rational
mixing
surface
broadened
the growth rate Y in the
and
a
decay
of
xs=x 1 .
1x 5-x>Ax,
from the resonance,
The
will
be
version of the Newcomb equation.
analogy with linear tearing mode theory,
relates
fx1 -
The nonlinear equation will dominate in the resonant region
equation.
behavior
can
turbulent version of the Newcomb
be inverted in time to yield a nonlinear,
x2 1<Ax
level
for the mean-square fluctuation
(85)
- The dynamical equation
a matching
inversion of
In
of the two solutions
(85)
to
A' of the Newcomb
equation.
the case of time stationary turbulence where the mean-
Consider first
square
fluctuation
have
fluctuations
level
is
scale sizes
not
growing
less
than the
in
Assuming
time.
mean
shear
length,
that
the
the
time
along the stochastic two point orbits can be written as
inversion of (85)
T_(x_,y_,z_) B, 2
<6N .6N2 > = 2D
(90)
where
-t(x)
and we
have
localized
(Boy),
S2
D
=
f
noted
d
2
2 <6L 2 >kGk
that
3<N>/3x
clump fluctuations
a result consistent
scale lengths.
t
dt'
=
<exp ikyy_(t')>
Boy
S.
by the large
are driven
with the cascade
The meaning of the -r_ factor
the mixing process
generating the clumps.
from the diffusive
(D)
transport
of differing energy density,
Equation
(91)
(90)
means
scale magnetic
47
the
shear
of energy from large to small
in (90)
can be understood from
Since a clump fluctuatin
of a magnetized fluid element
a larger
that
fluctuation
results
as
arises
to a region
the
element
diffuses farther and farther
away from its density of origin.
is not susceptible to decay during the transport,
inexorably larger with time,
However,
time t.
finite
size
because
has
a
i.e.,
T. in (90)
of stochastic
finite
lifetime
the fluctuation would get
would be equal to the elapsed
decay processes,
(T_),
If the clump
so
that
a fluctuation
the
magnitude
of
of
the
fluctuation is limited to the value given by (90).
The clump lifetime is determined by the operators on the left-hand-side
of
numbers by a relative diffusion
In principle,
correlation.
clump but,
the
approximate
operators
These
(85).
energy
cascade
(mode coupling) processes
cascade involves
the
because of complexities
to
wave
high
in the two point
all scale lengths in the
in an analytical
evaluation
of
we
T_,
will be forced to consider only the turbulent dissipation of scales smaller
than the
While the
( cl)~1.
for
%
=
less
scales
reasonable result
find that the
decay occurs
than
for the
the
clump
characteristic
we will only need an integral of T_.
case
have
neighboring
time
at
a rate
Tcl expression we calculate below is only strictly
size,
it
yielded a
clump
since,
Moreover,
a
the
as we shall see,
similar approximations
lifetime in
gives
nevertheless
clump decay time.
precise analytical expression for T_ is not crucial,
clump
as
clump diverge apart exponentially with
field lines in the
valid
We will
clump size.
good agreement
for the Vlasov
with
computer
simulations .6
We expand the randomly fluctuating
(t)]
of
(90)
in cumulants
<exp iky_> = exp(-1/2 k 2 <y
y
from
the two
point
orbit
40,2
2
-
>).
and,
for
a normally distributed
y_,
obtain
The time evolution of <y2> can be calculated
characteristics
Using the model sheared field of Sec.
the characteristic equations imply that
i
(stochastic) orbit function exp[iky_
of the
left
hand side
IA, <B_.>.V_.= B' x_ 3/y
of
for
(85).
z =z2 '
(92)
<D_>
(t)> = 2S 2B
<y
toy
(92)
can be solved approximately by expanding D_
= 2(D-D
12
) for small y_ so
2 2
<D_>=2k2D<y2>,
where k. is defined by
2
that
32
k
=-(
)
(93)
y-=O
2D (2
o
Because of the cascade of energy to high k,
one might worry that this small
kyy_ expansion will quickly become invalid during the cascade.
instability growth rate
is on the order of
valid during most of the
Using this <D_> in (92),
= y_
y_(O)
first
e-folding
T-1,
However, the
so that the expansion
growth time of
the
is
instability.
the time asymptotic solution with initial condition
is
<y (t)>--= (y2-2y
SB
x
oy-0
T
+
2S 2B
x -r )e t/T(94)
y-
where
-1
T = (4k
is
the
apart
2 2 t2
-S2B
D) 3
characteristic
exponentially.
high wave numbers,
time
(12)
for
This divgerence
While
dependence can be recovered by
(91),
the
=
1/2 (xj
time
(-cl)>/2 ~ 1.
o
(95)
+
x 2 ).
integral
Using (93),
line
the cascade
since
(94)
(94)
has
been
calculated
replacing y_ in (94) with
(91)
is the typical
will
converge
Tcl is given by
49
orbits
of
to
diverge
the energy to
is determined
In dimensional units,
Since k.
in
field
causes
not a surprising result,
Lyapunov scale.
where x+
T
neighboring
diffusive dynamics of D_ in (85).
is the
3
by the
-C= zo/VA where zo
for
_
k
in the
for
t>Tci,
=
z_=O,
the
z_
y_ - xz- B
k integral
where
of
k2<y2
~y
7
'c ' ' -' ~
3
*ln
~i
~ 2
2
k9 (Y _-27_SB
0
for argin>1
and is zero otherwise.
T_~T cl
are growing during the instability,
the
(85)
correlation
is effectively
function.
96)
2 2 2
Since the time integral
the order of Tcl, we conclude that
hand size of
2
x_+2S B' x T
0Y
O y-
in
the two
(Y+-rc
Therefore,
point
),
1
the
(90).
When
(91)
is on
the fluctuations
propagator
where Y is the
time
in
inversion
on the leftgrowth rate of
of
(85)
can
be
generally written as
<6N 1 .6N2 > = 2DTcl B
The
equation
for
<6I1.6L2>
'2
-1
(1 + Y Tct)
follows
by
(97)
making
D=DL+DN on the right hand side of (85) (see (56),
With Y=0,
The mean
(97)
clump fluctuations.
shear
(B'
oy ) is mixed to
As discussed above,
along with the
xd = (4D/Sk0 B
rkOSB oyxd=l.
factor D,
nonlinear
determines
1
the mixing length.
point
generalization
of the one
point
Two field line trajectories are in resonance
Ix_|<xd'
version of the Newcomb
3<N (1) 3% (2)
x
decay or lifetime
Using (94),
We define the clump "flux" functions T+*
-. 2
does not mean that the
(98)
(feel the same stochastic forces) when
from (51).
and
scale
)1/3
xd is the two
resonance width xo (see (76)).
generate the smaller
Tcl describes the
The mean-square mixing length is x2 = Dic.
The resonant,
S+-S
(57) and (68)).
Tcl in (97)
clumps are driven by the small scales.
so that
replacements
is in the form of a standard mixing length relation.
(large scale)
of the clumps and,
the
2
i+ (1) 3T + (2)
T e
Therefore,
50
1
ay
y
equation
S~_'
is
derived
so that
<6N .6 2 k
=
(-
+ k ) <6T+(1)
and the Fourier transform of (97)
can be written as
<6T (1)6+
- k )
(!
(100)
6T+(2)>
-
=
2DB '2
(101)
-,k)
c+i+
oy
k
kwhere
Tl
Recalling
(73)
S 2ky2<6T2>
(Y+')
(100)
adz-e
-x,k)
and
Z
(74),
can
D-S2 <6L 2 >k(Y+To~
write
Substituting this into
and rearranging produces the Newcomb-like equation
Fluctuations
mixed
(D)
S2
created as
forces
+(1)6T
(2)>
the mean magnetic
(Sk.B-T('),
the instability
shear
=
0
for
a
large
enough
(103)
) is
(B
This mixing is opposed
but
(Y>0) of the mean-square
IIIF that
<T
)2
near the resonant surface.
bending
growth
are
(BI
22
2
Sec.
we
(102)
T(10)cl
near the mode rational surface.
k 2+
line
dy_e
turbulently
by stochastic
mixing rate,
fluctuation level is possible.
is a nonlinear,
turbulent
net
We show in
version of
the
Rayleigh-Taylor instability in a magnetized fluid.
Near the mode rational surface
by the
nonlinear
Newcomb equation
(Ixl<xd),
(101).
the instability is described
Close to
the resonance,
decay by stochastic line bending (shear Alfven wave emission)
the limit,
a clump of infinitesimally
conservation,
have an infinite life time
fluctuations on the right
hand
time
In
and
rational
small
thus
diverge.
surface,
shear
side of
the
Alfven
(t
(101)
51
will,
is small.
because
In
of energy
The mean-square source of
0 1 +o).
nonresonant
wave
scale
clump
will
increase secularly
region
emission
away
from
(propagation)
the
with
mode
dominates.
Energy is carried away from the clumps
yielding
a
short
clump
life-time
(Icl~Ic1+0).
The
clump source
as we show in the next section,
then goes to zero and,
(101)
along stochastic field lines,
to the two point Newcomb equation with broadened resonance
2
k
_-
X_2
Eq.
(104)
position
k J'
+
y
<6 (1)6(2))>
26
=0
mode
and multiplying by 6(2) and ensemble averaging,
since
source
x 1 -x, 2 =x-
rational
singularity
term
(104)
((15) with 6JR = 0)
= -
This zo
surface.
in
for
(101)
exists
turbulent
broadening resolves
because
However,
to all
mixing
orders
in
resonance region.
52
(101),
where
at
(2)
is
kzBoz/ky BY where k.B(2) = 0.
Note that we have also included the zo broadening in (104)
2laX
goes over
o
y OY-
at the mode rational surface, i.e., at x 2
a
term in
k
k B? x +iz
follows by evaluating the Newcomb equation
(1)
(101)
thus
of
in
2
/axi1
=
the singularity at the
energy
the
and set
conservation,
field
therefore,
amplitude.
dominates
the
The
in the
B.
Growth Rate
Since D in
depends on <6Y 2 (0)>k
(86)),
(see
denotes the region about this singular surface,
X2 =X+~x-,
we
can solve
(100)
Let x+ = (x 1 +x 2 )/2 be located at the mode rational surface.
for <ST 2>.
E
(101)
then,
If
using xl=x++x_
and
we have
= <[6T'i(x +x_)5'(x-x_)
15X<6T(1)6T(2) >1
E
-
S6(x
+x_)
6T'(x+x)-
-
2 <[6 'I(x++E)6(x +- ) - 6 (x++E) 6T''(x +-e)]>
=
=2 <6T1(1)6Y(2) - 6T(1)6T'(2)>
<I-62;(21)
= 2
616(22)>
= 2 6' <6T(1) 5'(2)>
(105)
where 6' denotes the jump (discontinuity) in (101), and (1) and (2) in (105)
denote x++E and x+-E respectively.
Therefore,
2
P
dx-
D2 2 <6'(1)6i(2)>
= - 26'
23k
<6T(1)6Y(2)>k
singularity
Y6~k .B
reduces
when
6$
Therefore,
-
to
YT<1.
k B yX6$
the
To
~
Newcomb
see
Since we are interested
approximation
given by (57) .
is a
good
<6$(1)6p(2)>k
we
from
note
T~'(6/S)(x/xd), so
primarily
one.
6i
6'-6W except
in the
53
-
outside
Faraday's
(YT)~'
when
YT is
law
equation
for
the
that
(x/ xd ) (6/S).
very large.
stochastic regime where YT<l,
The Newcomb
Since
(106)
0)>
solution
this,
away from the singularity,
<6y-2
<6(1)6(2)>k
this
is
<6$(1)65 (2)> = 2
'1 ) -
$
2 A' <6 (1)6p (2)>
the
discontinuity in
parameter.
is just
(104)
Therefore,
A'k,
the
resonant
hole turbulence,
this
hole solution to the
IA,
hole,
IB where 6Jz=
the matching
-
procedure
2
<
in the turbulent case,
by setting
5'
A'k
describes the growth
just matches
Near
(for each
k)
the
the resonance,
the
0) so that,
6' and A'.
The results
i.e.,
(106)
one
and
(107)
the matching of solutions is also obtained
is
and 6'k
<6Y
(0)>k
+
given
by
=
, (107)
A
gives
2D LB 2
oy yI
A +21k
the
dx-
-
resonance
T (x-,k)exp[-Iky Ix]
c9.
broadened
version
of
An approximate way to evaluate the integral in (108)
the integrand is nonzero in the range ky_-koy_-1/2
can replace
YT.,
exp(-kyx_)-1
in
first
see
for an isolated coherent
in the
(108).
denominator
Then,
of
(102)
using (96)
and
(102),
1TA(k )
dx_
(x
ci-
k) =
A( y
ISBy I
the
<6y2
i.e.,
2
so that we
(see Ref.
we do the
by completing the square in the denominator of (96)
and
is to note that
and x_/xd-1/
with -2YT
(108)
(16)
superscript L on DL is a reminder that DL depends on <6L 2 (0)>k,
(0)>k.
(i.e.,
= A'.
Using (106)
where
we match the inner
2,
(101)
stability
as in standard tearing mode theory,
equates the logarithmic derivatives
mean that,
tearing mode
dependence as the tearing mode
26$/x
is done
Since
Newcomb solution.
current hole has the same radial
Sec.
usual
as in the tearing mode theory
and outer solutions by setting 6'k = A'k.
of current
(107)
,
k
x_
8),
and
integral
and get
(1+2Y-t)
where
54
6(k +k x+B )
Z y + oy
(109)
A(k) = 2
[1 - J
(/6 k/k )]
(J
is the Bessel function).
magnetic
energy
(see
function Gk with
(90)).
a
D
N
r
= Y +
t0
1
f
=
L
D
Recalling
(73)
is proportional
and
(74)),
(see
I
L
replacing
to the
clump
the resonance
we can construct DN from the
k0oSB
N
A'
c
(with To given by (75)),
dk
ReA'k+21k
2 ky
y
o (ReA'
c
is an effective
(110)
to obtain
1+2Yt
where
Equation
Lorentzian
left-hand side of (108)
(110)
00
k2
(inverse)
I
I)
+2 1 k
and
k
2
)
A(k
(112)
+A
A'k averaged over the clump spectrum.
A'k is (ky,-kyByx+) where A is given by (20).
can be derived for <6_> in terms of DN.
Here,
k in
An equation similar to (101)
The equation can be integrated and
used to construct DL:
DL =
N
SBY
oyN
1+2YT
As in the
Vlasov clump case
diffusion coefficients
k0
o
A'r
c
(113)
(see Ref.
yield a
equations
(111)-and
(113)
the
coupled equations
quadratic equation
the situation here by taking the N/L
cross helicity, we can set
8),
for
TN = TL = T0 and TN =
TL
= T.
the
can simplify
for
zero
Then, the two
each yield the same equation for the total energy
with <6L 6L> replaced
by
(12) 1
and (113)
2t, so (111)
We
strong coupling limit where,
correlation function, i.e., for DL = DN = DE where DE
[-
Y.
for
<6B6B
+
S- 2 6V6V>.
finally give
55
is given by (73),
Further , we note
that
but
to
-
where,
ISB0y
since
k xd T
(11 4a)
2
(1 +YT
1,
2
R =A,
(11 4b)
C,'d
Note that the
B
factors
on the right-hand-side
of
(101)--ostensibly
the
clump driving terms--divide out in the derivation of (11L4) in favor of A'k
J., as the driving term.
Tci~(Boy)1 in (109).
One of the B,
Because of (73)
factors
and (74),
= 0 removes the other factor of BO.
divides out
because
fdx.
the evaluation of D at k.Bo
The free energy source for instability
is, therefore, the same as for the tearing mode.
In the
before
limit
field
hydrodynamic
the
YT>1,
line
Lyapunov time
stochasticity
limit,
(114a)
has
becomes
Y2
is long and
any
appreciable
R/T2 which,
=
growth can
effect.
with
occur
In
(114b),
this
can
be
approximately written as
2
ReAl
k
x
(115)
IA xd 1
If
we
now
(6Bes/k B
recall
T~
that
2
=koBoySxd
a.nd
write
(as
in
(77))
for the island width in the resonant modes,
)/2
xd
=
(115) can
be
rewritten in dimensional units as
ReA
_2
TH
k 2
<6B2 >
x res
Axd
(116)
d
Except for the clump shielding factor
IAxd
the growth rate calculated in Ref.
However,
21.
for a tearing mode
in an
(70)
growth of resonant
describes
the
self-consistently
assumed static
generated
and
,
(116)
is the same form
background of
as
as
rather than the growth rate
stochastic
clump fluctuations.
shielded
56
2
other
The
growing
fields,
clumps
are
(background)
clumps turbulently mix the mean shear.
In t-he
limit
large
where
amplitude
overlap,
islands
region is large and the Lyapunov time is short.
This
the
YT<1
stochastic
limit of
(114)
yields a growth rate of
Y - D
where,
the
xd (
1)
-
for large amplitude islands,
parenthesis of
(117).
Here,
(117)
A' is an effective
taken over the wave numbers of unstable
(112).
Without the parenthesis,
amplitude,
tearing
Replacing
turbulent
quantities,
Spitzer
i.e.,
value
brings
(117)
regime.
mode
rgp,
island
with
coherent
an
xd
square
fluctuation
hence,
with
the
value of
dominant
Rutherford
corresponding
A'k
A'k in
Ax,
and
the
Rutherford result.
The
(-1)
Rutherford
will
the
grow
at
interaction
island decay.
if
result
into
the
of
factor
new
in
turbulent
rate
between
island
collisional
the
(25),
islands
Net growth (117)
creation
regime.20
coherent
with
island
level results
i.e.,
so-called
their
stochasticity resulting from the resonant
cause mode coupling and,
average
D
island
individual
the
with
A'k,
clumps,
A'xd in front of
-
resembles the growth rate of a finite
in
we obtain the
the
Though
(117)
parameters
A'
used (2f)~'
we have
the
will
of the mean-
fluctuations
by
mixing (4) occurs at a rate exceeding the decay rate due to stochasticity (1),
i.e.,
R>1 in (117).
In
the- stochastic
regime,
reconnection of field lines.
the
instability
describes
exponentially.
the
Lyapunov
time
for
stochastic
field
(-T'
lines
-
to
D/x2) or,
diverge
This stochastic transport of field lines across the sheared-
fields of the plasma randomly mixes
- .
nonlinear
The characteristic reconnection time is -t--the
nonlinear or turbulent resistive time in the resonant layer
equivalently,
the
Because of
elements of magnetized
energy conservation,
57
this
random
mixing
fluid at a rate
process
creates
clump fluctuations at the same rate,
The
speed
at
resonance--a
which
field
and
xd
is
approximately
on
the
given
by
field.
to
Since,
order
are
in the
randomly
space
transported
of
the
island
where
width,
VA
is
across
literature 4 '
plasma
in dimensional units,
VA(6B/Boz),
longitudinal field BOz.
perturbed
lines
speed referred
"merging rate"--is Xd/T.
and is the source of the instability.
as
the
~ VAkyBoy xd/Boz,
~
the
the
the
merging
Alfven
rate
speed
in
is
the
The merging rate is the Alfven speed in the local
Again,
the
Alfven
speed appears
dissipation of a clump is due to the
because
the
nonlinear
propagation of Alfven waves
down the
stochastic field lines away from the clump.
Evaluation of (113)
for Y requires k'
which,
from. (86),
(93),
and D
=
2 (D-D1 2 ) is given by
k
2
f
1fdk dk
=
y ~2 k~ <6Vi2(O)>k =1
(27r)
y
k
z
(118)
where
S
I
n
Since
i
solved
=
Jdk
-
depends on
ko
simultaneously
ko is ReA'k.
(see
(114)
in order
+
and
to
coupled
evaluation.
toroidal,
There,
kn+2A(k )
y
y
(112)),
obtain
(114)
the
Moreover,
(119)
and
growth
2IkyI) that is positive
For
confined
a
equations
rough
they are integrals
are
estimate
laboratory
plasma
non-trivial
of
these
typical
rate.
ReA'k>O for low mode number modes.28
58
to
and
we expect
require
a tokamak
In particular,
large
to determine
over ReA'k.
we
be
Instability
profile,
quantities,
of
have
and sufficiently
While ReA'k is rather sensitive to current
integral
(118)
the only scale length in (119)
less sensitivity in ko and i since
the
+
Y (ReA' k +21k 1)2+x 2
requires a value of (ReA'
to ensure that R>1.
I
Re A'+2|k
Still,
numerical
consider
fusion
ReA'k
a
device.
-
a-
in
dimensional
Equations
units
(118),
fluctuations
order
of
requires
where
(119),
a
inverse
(ReA'k
+
the
and (114),
will have low
the
is
2Iky )>O
radius
therefore,
poloidal
Lyapunov
minor
rather
-~-.
Note
than
ReA' >O,
bending restoring force term 2Iky l in (16)
the nonlinear
part of ReA
T
term in (1114),
i.e.,
the
the
plasma
column.
imply that unstable MHD clump
mode numbers
time
of
and
growth rates
that
since
on the
instability
the
linear
(R>1)
line
is already taken into account
(-1)
term in (117).
appears in the clump growth factor i of (117).
59
in
Only the J V
V.
The net
growth
of
RAYLEIGH-TAYLOR ANALOGY
fluctuations
(113)
by mixing
analogy with the
Rayleigh-Taylor
interchange
inverted
gradient.
In
density
instability
clump
known
is
instability
as
is
the
a
a
be obtained
instability for
magnetized
plasma
Kruskal-Schwarzschild
nonlinear,
can
turbulent
a fluid with
shear,
with
3
instability.
analogue
of
by
'
the
the
The
MHD
Kruskal-
Schwazschild instability.
Consider
unmagnetized
Rayleigh-Taylor
the
fluid.
instability
In dimensional
units,
for
an
incompressible,
the fluid evolves
according
to
mass conservation
a + V.Vp =
0
(120)
and momentum balance
P(a + V.V) V
=
pgR - Vp
-
(121)
where p is the pressure and we have assumed that acceleration due to gravity
points
in the
negative
2
direction.
Linearizing
(120)
and
(121),
one
obtains
aD
where
6
rate,
and
shear,
2 a
x
(p Y ) L
ax
o
6V
k
-
k
2
y
(p
0
Y2
2 _
N )
ax
Vk is the Fourier transform of the 2 component of 6V,
p0
is the mean
one couples
(120)
and (121)
a =
and (122)
density.
V
x
(V
In the
+
(k.B)2]
Y is the growth
of magnetized
fluid with
to the Faraday/Ohm law
x B)
(123)
becomes
1
case
(122)
k(2)
-
6V
60
p
k
Instability occurs when the forcing
mixing
of
fluid
potential
elements.
If
For
energy.
tension,
k. B*0,
When 3p 0 /3x
> 0,
energy is decreased
the
elements is perpendicular to Bo,
(124)
Vk
by the density gradient term overcomes
the stabilizing effect of line bending.
on top of light fluid,
6
2 - g
Q
+
wave
number
of
i.e., heavy fluid
upon interchange
the
frozen-in
or
fluid
the interchange will not alter the magnetic
the mixing
bends
field lines
and increases
and thus tends to stabilize the instability.
magnetic
Stabilization occurs
as energy is carried away from the fluctuation in the form of linear shearAlfven waves.
Extrapolating
(124)
to
the
nonlinear,
turbulent
governing equation for the MHD clump instability.
the nonlinear
mean-square
fluctuation.
The
fluctuation,
conservation
of mass
conservation of energy density.
one
has
an
equation
y1
k 2<61P(1S(
yk
2
Vxf,
=
)>
=
bending term k.B
the
we
=
stochastic regime.
-
iB0 .V in
than
the
we must deal with
linear,
one
point
is replaced with the
functiOn
+
<6V(1)6V(2)
Vx2p, and defining the stream function
a
nonlinear
+ po6$(1)6$(2)]>k.
(124)
the
yields
instead of an equation for 5Vk,
correlation
have
(1)6*(2)
2<[6
0
will
First,
density (120)
Therefore,
Recalling that B1
po I6B(1)6B(2)>k.
$ through
for
rather
regime
must
be
dynamical
Second,
for
the linear line
generalized to the nonlinear,
[p0 Y2 +(k.B )2 ]
This means that in (124),
equation
=
j/pOY+ik.Boj 2
-1-0
+
p(Y+T-)2,
since VA .B(x)/B
finite amplitude
by
field
elements)
line
and stochastic
diffusion.
diverge
stochastically
- VAkyB
in
the
bending of
Neighboring
exponentially
form
xd/B~-T1 near the resonance.
at
of
the
the field lines is approximated
field
rate
nonlinear
61
The
lines
T_'
(and
and
frozen-in-fluid
carry
shear-Alfven
energy
waves.
away
This
stabilizing
turbulent
replaces
effect
is
gravitational
by
the
forcing
term
9p 0 /ax32W/Dx
2
, where W
=-
generalized to the turbulent,
correlation
point,
function,
nonlinear
new
fluctuations
mixing
of
the
magnetic
for
coherent
a 2W(1,2)/ x,
energy
energy,
potential
current driven case.
potential
interchange
by
shear
of
the
the coherent mixing term -
is the mean
pogx
this means
magnetic
of
Therefore,
density in the Rayleigh-Taylor model.
g
creation
turbulent
Nonlinear,
mixing.
the
countered
must
be
In the equation for the
where
of
W(1,2)
the
is
clumps.
the
two
Neglecting
helicity conservation, we can use the turbulent mixing length relation <6B2 >
-~
<2 >(By)2
(B'
)2
-
(t
~ D-
(B'
2
and
estimate
Therefore,
t
/)(B'y)2.
3 2W(1,2)/ax2
the turbulent,
W/x 2
Dx2
"ci
current driven analogue
of (124) then becomes
ax
-)2
1)
o
(
aSp ( Yo
2
a
- D<6T
22 (x-)>
ax
-1
= ky
where we have divided out the k
<5(1)6'(2)>.
(125)
Rearranging
2
ax 2
Since
T
c
quadratic
~
ci0
k
)2-
(113)
2
<6T (x_)>
A y
oy
<
0
2(x )>
k
=
0,
(126)
0
(126) is just
on
(102)
in dimensional units.
Y is thus due to the second
derivative (i.e., acceleration) in the momentum balance.
regime the acceleration
(125)
gives
ky +-1)2
dependence of
21
factor from both sides and set <6T 2(X_)>
y
+
+ YTO)~',
(B' )
(V k B' /B )2
T
-
2_ 'ck
p(Y+T
'k
goes over to diffusion
(6x
2
-
dependence on Y becomes a linear dependence as in (117).
62
order
The
time
In the stochastic
t) and the
quadratic
NONLINEAR ENERGY PRINCIPLE
VI.
The equations for the MHD clump instability can be recast into the form
of an "energy principle" similar to that of linear
The
linearized
energy
principle
of
MHD derives
MHD stability theory.29
from
the
conservation
of
energy
Jdx
where
6W is
instance,
the
in the
p0 <6V2> +
*
potential
energy
case of the
quadratic form can be
dx6W = 0
density
y2
driving
Rayleigh-Taylor
constructed from (124)
(127)
the
instability.
instability
of Sec.
IIIF,
fdx-
2 0
<p
(128)
>
k
where 6k is the fluid displacement and, in the simplified case of 36Vk/ 3 x
Y9Ek/3x
=
a
to give
fdx 6W
_
For
=
0, 6W is
6W
Instability results
(kB)2 -
=
when
E, 2 >
g
6W<O, i.e.,
(129)
when mixing of the
density
gradient
overcomes the stabilizing effect of line bending.
For the MHD clump case,
thus remove <6B
2
>
we deal with the total fluctuation energy and,
The two-point,
from 6W in (127).
clump analogue of (128)
is
fdx 6W c
fdx_<6V 1 .6V 2
-(129)
2
26B 1 .6B2 >k
where
6 W12
6W
=
to0
<6V .6
-1 62
tcz
+
S
2
B .6B >-2 DB
, -2 k
T0
oy(10
(130)
Instability (6Wc<O) results when turbulent mixing of the magnetic shear
second term in (130))
overcomes
stochastic line bending
63
(the first
(the
term in
(130)).
Insertion of (130) into (129) and integration over x_ gives
DB'
Y
or,
t
o
<c
0)>
(0)>2<< 6 2
k
upon rearrangement
- f(+2 1k
This is just (113) or (23).
|x_]
(131)
and (73),
(1-R)
-
T0
Clearly,
dx_-Tcexp[- Ik
-
and use of (108)
S1
T0
2
(132)
T0
6W0 <O means that R>1,
>1l.
64
or, when YTo<<1
VII.
Though
the
NONLINEAR KRUSKAL-SHAFRANOV CONDITION
unconstrained
mixing
calculation
of
the
MHD
instability presented here neglects magnetic helicity conservation,
clump
it does
provide an interesting physical interpretation of the instability threshold.
For this
purpose we use the Re Ak model of Eq.
wavelength
(low mode number)
modes in a general
Ak ~ (k.B/k.Bo)2/ky - (Boy/Boy) 2 /ky.
ko,
(112)
gives
A
c~
ReAk
approximately then xdReAk <
stochastic layer .
-
1,
(26)
current
28 for
long
, i.e.,
profile
Re
Since the relevant ky's are of order
ko.
The
i.e.,
the "constant
instability
koxd BOY/Boz,
Defining L.
of Ref.
condition
R
>
approximation"
i
1
is
in the
the instability threshold
condition becomes in dimensional units
1
Joz
where again we have
set
c
Boz
%=1.
As discussed
in Sec.
IB,
Lc~zo
is
the
z
stochasticity length--the length one must move in z for the stochastic field
lines
to
diffuse
sufficient
overcome
radially
production
the
spacial
by
xd.
For
instability,
of fluctuations
by mixing
destruction
the
of
(Joz
localized
(133)
B'y)
-
clumps
field lines diverge apart stochastically.
Equation (133)
the
the
instability
Shafranov
arises
if
generated
column
condition
condition). 29
the
"pressure"
Joz>kzBoz
In
a
for
linear
by
the
as
kink
to
of
poloidal
the
Bo z-field
line
tension.
neighboring
mode
the
bunching
to
is reminiscent
picture,
due
that
must occur
conventional
by Joz can overcome the resistance to this
provided
states
kink
of
(Kruskalinstability
field
lines
bending of the
plasma
In
the
MHD
clump
instability, random localized bending and bunching of the field lines occurs
nonlinearly as the mean poloidal
process is opposed
field lines.
shear
profile is turbulently mixed.
This
by the random restoring force of the stochastic magnetic
Since this restoring force
65
(and the
Alfven wave
emission
it
causes)
is minimal
stability
condition
near
the mode rational
is
replaced
by
surfaces,
kzBoz
xd-Boz I
k.B-k B
in
of the
the
linear
nonlinear
stability condition.
(clump)
A further
connection with the linear
by integrating (133)
kink instability can be obtained
over the minor cross section of a toroidal plasma.
The
instability threshold condition then is
L
< 2
q(a)
where,
(134)
q is the "safety factor"
Equation
(134)
Shafranov
and R is the major radius of the plasma.
is the turbulent,
condition,
q(a)
<
nonlinear
clump analogue of
Instability results when the
1.29
length is less than the sochasticity length.
characteristic spacial
island overlap,
field line,
clumps
size)
strength is reached.
in
z
of
Lc,
length can
regenerate
and
before
the
the
distance
stability
be traversed
instability
before
z-Rq
Lc can easily
Lc
With
(133)
radially by xd
characterizing
the shear
by xd after a distance
is
Rq>Lc.
is reached,
the
dynamical
For
the
L,>Rq,
clumps
constraint
>
Then,
kink mode.
(134)
of
be small
implies
i.e.,
a nonlinear
subcritical
to
We note that the threshold
for the Vlasov clump instability is also subcritical
66
can
For such a small xd at instability
stability boundary below the Kruskal-Schafranov limit,
the stability boundary of the linear
a
P
the initial region of stochasticity will
be larger than R.
after
along a magnetic
gets replaced with J0o
and confined to the island seperatrices.
2
as one moves
condition
results.
helicity conservation,
At island overlap,
onset,
is the spacial
Physically,
1/T.
Since the field lines diverge
connection
magnetic
if,
it
neighboring field lines inside a clump diverge
(the clump scale
traversed
instability condition l/T >
cannot regenerate
connection
Expressed this way in terms of
scales of the stochastic fields,
version of the temporal
the Kruskal-
to the corresponding
linear instability boundary."'
VIII.
COMPARISON WITH VLASOV HOLE GROWTH
It is interesting to
Consider the growth
that of phase space density holes in a Vlasov plasma.
of a current
1),
<
the
for
(24)
obtain
resulting growth rate (25)
in terms of xd,
where
k-'
0
factors
the
=
in (135),
characteristic
stochastically
driven
by
the
at
the
as YH
coming from the reconnection
instability is current driven.
rate
x
6B,
field
or
as
field
bend
lines
reconnection
tension
line
force.
is
The
The rate at which the reconnection occurs is
given by the shear Alfven wave frequency VABo/Boz
a resonant stochastic
D/x2, determine
-
T~
two
first
The
hole.
the
course-grained
This
overlap.
Joz
(135)
Reconnection occurs
growth time.
stochastic
Expressing T
(4xd)/T.
-
The
Xd)
(
scale length of
poloidal
island
function.
flux
poloidal
(in dimensional units)
A (x/Ay) IBI /Bo
Ay is the
can
Inserting D into Faraday's law,
can be rewritten
this becomes
YH
of
evolution
In the fully
as we've seen,
the background stochasticity,
be modeled as an anomalous resistivity D.
we
holes.
bath of background
hole in a stochastic
stochastic case (YT
holes to
compare the growth rate of MHD current
layer of width
xd about
-
VAk B
the .island.
xd/BOZ
T
-
in
island
Larger
fluct.uations grow faster because the width of the stochastic or reconnection
region
xd.increases with amplitude as in
growth comes
from
the shielding
(80).
function
gradient in the region outside the island.
Akxd,
The free energy source for
i.e.,
For A >0,
density
the current
there
is a
positive
discontinuity or jump in the perturbed magnetic field across the island,
the island grows.
7
The growth rate of an isolated ion hole in the Vlasov plasma
67
is
and
Y
~
-
)
(Av/Ax) (v f ) (v ft
ee
01
i1
where vi, ve are ion and electron thermal
are
Safoe/3v
by the ion hole
electric
The
electrons.
2
'
the
the second
Electrons
the
are resonantly
going faster than slower
than the
hole,
field
potential
created
drop
by
the reflecting
the
across
The
electrons.
the
to
through the
dependent
by the
is provided
0
structure is frequently referred to as a double layer.1 '''
is amplitude
it
hole.
or
due
function
dielectric
as
depth
in
is
hole
where ImEe is the imaginary or resonant part
foe ~ Imee,
proportional to Ve
of
field
of
with the
electric
by an
accelerated
is
hole
grows
hole
and exchange momentum
potential
more electrons
i.e.,
For foe > 0,
the
hole
(136)).
in
(the third factor
gradient
reflected
The
Ax.
ion
an
of
The free energy for the deceleration
in (136).
electron
growth
to regions of larger ion phase space density--hence,
decelerates
factor
width
and spatial
Av
width
velocity
the
mean
electron
and
ion
the
of
describes
(136)
Equation
distributions.
= afoi/3v, fe
velocities and fi
gradients
velocity
the
(136)
hole
potential
The growth rate
As the hole
hole trapping time Ax/Av.
growth
rate
potential
increases,
electrons
are reflected and each electron
transfers
more
since
increases
drop
resonant
more momentum to the
hole.
Despite
similarities
position
in
(135)
While the MHD
differences.
important
resonant,
the
(i.e.,
mode
hole
rational
and
(136),
there
in
amplitude
grows
surface)
in
the
Vlasov
case
is
resonant--coming
velocity trapping width (-Av) of the hole.
for
growth derives
from the nonresonant
from
a fixed
electrons
(Imc)
grows
by
The free
within
a
the free energy
Consequently,
comes from the imaginary part of the shielding function
68
at
is larger.
In the MHD case,
region.
several
the Vlasov hole
0
decelerating to different resonant velocities where fo
energy
are
free energy
in the Vlasov
case,
but
case.
from the real
However,
in each case,
generate opposing currents
plasma].
requires
(Rev ) in the
growth is due to a jump or
The opposite signs of 6ip'
the field (see Ref. 30).
the
of the shielding function
MHD
discontinuity
in
[Note that the Kadomstev bubble also grows by a discontinuity in
the field.
boundaries
part
While the Vlasov
the stochasticity
the overlap with other
stochasticity
is the
at the bubble and plasma
which force the
hole
can
grow
bubble boundary into
in isolation,
the
(i.e., the turbulent resistivity D)
hole resonances.
destruction
of
Since an additional
coherent
islands
hole
produced
effect
by the
by
of the
exponential
divergence of neighboring field lines,
net
creation of new fuctuations by mixing.
The MHD instability is, therefore, a
clump
instability
rather
relevant growth rate is (28)
than
an
growth must
MHD
isolated
rather than (25).
69
hole
be achieved
instability,
i.e.,
by the
the
ACKNOWLEDGEMENT
and
With the premature start of Ref.
1 and a necessary diversion to develop
understand
one
the
instability
of
"hydrogen atom" of clump fluctuations),
making.
Vlasov
clumps
computer simulation results
compelling
to
First,
Carreras and the ORNL group for sharing their
early on in this work.
show
the
importance
of
the
fluctuations.
nature
(the
this work has been many years in the
Several people have influenced the outcome presented here.
I would like again to thank B.
resonant
dimensional
of
mixing,
I.
Nothing has
stochastic
Hutchinson
has
been more
decay,
been
and the
a
constant
source of information on laboratory experiments as well as providing useful
criticism
helicity
for
of
ideas.
conservation
useful
has
conversations
Strauss,
R.
Critical
reading
gratefully
Along
Granetz,
of
with A.
been
particularly
also
goes
to
to
T.H.
Dupree
and
the
acknowledged.
Boozer,
manuscript
I
would
by
also
his
stimulating.
J.B.
I.
championing
Taylor,
for
of
magnetic
Acknowledgement
J.P.
Freidberg,
constant
H.
encouragement.
Hutchinson
and
T.
Dupree
is
to thank
The
Department
of
like
Energy for supporting the interim study of the basic one dimensional Vlasov
case,
without which this work could not have been possible.
This work was supported
by The
Department
Research, and The National Science Foundation.
70
of Energy,
Office of Naval
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