PFC/JA-87-11
STEADY STATE MHD CLUMP TURBULENCE
by
David J. Tetreault
Massachusetts Institute of Technology
Cambridge,
MA
02139
April 1987
Submitted to The Physics of Fluids
ABSTRACT
The
turbulent
investigated.
steady
state
of
the
clump
MHD
instability
Magnetic helicity conservation plays a decisive role
steady state;
The helicity
invariant
constrains
the
turbulent
is
in the
mixing of
the mean magnetic shear driving the instability and modifies the instability
growth rate.
The steady state
is determined
by the balance
between
this
helicity conserving growth by turbulent mixing and clump decay by field line
dynamical
stochasticity.
The
magnetic
satisfy
field
J0
0
balance
UB ,
occurs
where
1
yj
when
depends
the
on
mean
the
current
mean
and:
square
fluctuation level.
clump unstable.
instability
Above this critical point
This self-consistent
is
a
dynamical route
turbulent
dynamo
to the force
the plasma is MHD
generation of fields during MHD clump
action.
free,
> p Bo),
(J
MHD
clump
Taylor state.
instability
For the
is
a
steady state to
exist, p must exceed a threshold on the order of that required for Boz field
Only
reversal.
From the
turbulence.
spectrum
current
these Taylor states
(6Brms/B)
as
transition
p3.
is
The
where
lic
correspond
p threshold condition,
calculated
onset
of
=
J.B/B
and shown
the
2
steady
is
the
to steady
the
state MHD
clump
steady state fluctuation
to increase
state
with mean
corresponds
critical
to
point.
driving
a
phase
Fluctuation
intermittency is discussed.
I.
INTRODUCTION
This is the second paper in a series of three
instability.
In
the
first
paper
(Ref.
1),
fluctuations and their instability to growth.
in an MHD plasma when the mean magnetic
papers on the MHD clump
we
described
MHD
clump
The fluctuations are produced
field shear
is turbulently mixed.
The turbulence transports a magnetized fluid element to a new region in the
plasma where the mean
point of origin.
energy density differs
from
that
of
the
element's
The fluctuations are localized at the shear resonances of
the plasma where the decay effect of shear Alfven wave emission is minimal.
In isolation, the fluctuation is a hole (6Jz<O) in the longitudinal current
density
Jz.
As
the
holes
resonantly
interact,
their
magnetic
island
structures become disrupted by magnetic field line stochasticity. Energy in
the localized
magnetic structures
become
propagate down the stochastic field lines.
new fluctuations
are regenerated
dissipated
This decay
by the turbulent
2
as
shear
Alfven
waves
can be overcome
mixing.
The net
as
growth
rate of the mean square fluctuation level is of the form
Y=
where
R
is the
(R
mixing rate
and
Net growth
stochastic fields.
(1)
1)
-
is
T
the
Lyapunov
(decay)
time
(instability) occurs when R>1.
of
the
Equation
(1)
can be cast into a perhaps more familiar form by recalling from Ref.
the mixing rate is nonlinear and,
therefore,
1 that
evolves with the growth of the
In particular,
fluctuation level.
R
0 (1 + YT) 1
=
(2)
where
(3)
AIX
cd
Here,
xd is the
turbulent resonance
(island)
resonance width of an isolated island.
field line diffusion coefficient
xd scales as the
(see (92))
Ax for the case of a single resonance.
width which
generalizes
the
cube root of
the
and reduces to the island width
Ac is a nonlinear
version of
stability parameter (A ) of linear tearing mode theory and 2,
energy available for nonlinear clump growth.
With
the
gives the free
(2) and (3),
(1) can be
rewritten as
(Y + T 1 )2
(4)
is
analogous
to the
instability
("mixing")
(inverse Lyapunov time,
line
bending
(shear
(4)
2
growth rate
in
T1)
for
magnetized
the Rayleigh-Taylor
fluids.
3,
The
large
and (4)
amplitude,
stochastic
decay
plays the role of the restoring force to field
Alfven emission
rate) and the
magnetic
term (R) plays the role of the density gradient of light
For
interchange
fully stochastic
takes the form
3
fields,
and
shear
driving
heavy fluid.
the Lyapunov time
is short,
Y
=
lines.
The factor
in
Xd
1)
-
(5)
Xd2/T is the spacial diffusion coefficient of the stochastic field
where D
mode
D (3
-
the
in (5)
(A'D/x
c d
so-called
resembles
the growth rate of a tearing
regimes
Rutherford
but
by
driven
a
turbulent
resistivity, D.
It is an anomalous field line reconnection rate due to the
stochasticity.
The
fluctuations
factor
(R-1)
by mixing even
as
the
describes
existing
regeneration
net
fluctuations
stochastically
of
decay
("-1").
The
nonlinear
interaction
(anomalous
numbers Rm.
Rm
+
theory
While,
of
Ref.
1
describes
of magnetic
reconnection)
in the presence
the
islands
of weak collisional
strong
resonant
high Reynolds
at
dissipation
(i.e.,
-), the theory conserves the total energy and cross helicity, magnetic
This flaw is remedied in this paper.
is neglected.
helicity conservation
Global conservation of magnetic helicity constrains the dynamical
the mean shear and,
in
effect of this constraint below and find that 6A
If we
define
number
k,
A' as
then,
for
the average
value of Ak
with
clumps
island
for
widths
We calculate the
(5).
R in the growth rate
therefore,
(3)
mixing of
is given by (116).
unstable
clumps
of
wave
R is approximatley
Ax<k~,
given by
2
S-x(6)
oz
d
where Joz is the mean current
density.
The expression
the correction due to magnetic helicity conservation.
intuitively
(see
next
section)
as
a
helicity
the
form
Eoz
=
DJoz
used
in
Ref.
4
here is
It can be understood
modification
to
the
mean
Rather than a nonlinear Ohm's law
electric field (Eoz) driving the clumps.
of
in brackets
1,
magnetic
helicity
conservation
constrains
Eoz
to
be
consequence of (6)
- 1 in (5)),
of
the
is that,
form
A'
that the mean
wide
2
D(Ax
V 2 j oz.
An
important
(A
Joz satisfies
_
p2
-
-
for driven steady state MHD clump turbulence
V2 oz + 12
where
Eoz
xd /X2
In the general
for kAx<1.
vector current
stochastic
oz
spectrum
density go
where
p
case,
we show in Section V
also satisfies
would
be
(7)
so that,
independent
of
for
a
position,
This relation is known elswhere as the Taylor
. 10o in the steady state.
=0
state 6 , but here plays the role of the stability boundary for the MHD clump
instability.
The
its
derivation of
consequences
for
the magnetic helicity conserving source
steady
objectives of this paper.
are
presented
Introduction
developed
in
MHD
II
and
III.
a
brief
1
and of the importance
in Ref.
of
review
magnetic
(Sec.
that
theory has
an enlightening relationship
is discussed in Sec.
helicity.
IB.
as well as onset conditions,
scalings
for
turbulent
IA)
The
steady
transition to MHD clump turbulence
fluid flow is discussed in Sec.
ID.
we
of
phase
space
density
Vlasov plasma turbulence
interesting analogy
hole
are
between
first
the
main
the
continue
dynamical
of the
conservation
helicity
conserving
discussion
equations
laws,
MHD
dynamo models
of
this
in
clump
7
the Taylor
,
and
state
mixing length relations and amplitude
state
is
and its
presented
in
Sec.
similiarity to plane
IC.
The
Poisuille
Part IV deals with the possibility of
current hole intermittency in MHD clump turbulence.
and
are
from the MHD equations
to turbulent
A physical
equation,
the
turbulence
However,
particular
this
clump
The detailed derivations
Parts
with
state
term R and
intermittency
discussed.8~'I
in
5
(respectively)
In the Appendix,
phase transitions
instability.
Similarities with modon
and the onset
we
fluid
arid
discuss an
of MHD clump
A. Statistical Dynamics and Conservation Laws
The
dynamical
necessity,
equations
statistical.
The
describing
two
MHD
point
clump
fluctuation
plays an essential role in the dynamical model.
follows
from the conservation
of
turbulence
are,
correlation
of
function
This correlation function
energy and satisfies
an
equation of
the
form'
[,+ T(1,2)]
A detailed derivation of (8)
briefly outlined
in Sec.
C(1,2)
=
S(1,2)
(8)
was carried out in Ref.
III below).
T(1,2)
is,
1.
(The results
in the simplest
are
case,
diffusion operator describing the resonant interaction between islands.
particular,
stochastic
(cascade)
rate
describes
magnetic
of
fluctuation
lines
or,
quantity
S(1,2)
mixing of
energy
and
divergence
equivalently,
is on
The growth of
the
into turbulent
(8)--the
The
the
order
the
of
mode
coupling
Note
T~'.
decay or
fluctuations
that
the
use
of
fluctuations.
arise from S(1,2).
term
the
The growth rate
R deriving from
Direct
S(1,2),
Interaction
T(1,2)
describing mode coupling via a turbulent
turbulence
also yields
taken as a prescribed forcing function.
of
fluid turbulence,
determined
(1)
of
The
of
the
equilibrium
follows from the
and the
equations
term from
"-1"
(DIA)
form
viscosity,
(8)
2,'3
in
,
with
but with S(1,2)
Here, S(1,2) is more akin to mixing
i.e.,
self-consistently
6
(mean)
Approximation
of fluid
length models
"falling-apart"
It converts ordered
models
is
the
is the source of fluctuations resulting from the turbulent
the mean magnetic shear.
solution of
C(1,2),
neighboring
of
T(1,2) is a turbulent dissipation
instability1 2 -- describes
orbit
fluctuations.
T(1,2).
field
exponential
In
divergence of field line orbits--sometimes referred to as orbit
stochastic
energy
the
of energy to high wave numbers.
exponential
the
it
a
the
by
mean-square
turbulent
mixing
clump
of
energy,
the
mean
shear.
field
The
clump fluctuations
by
determined
model
stresses
this
self-consistent
of
production
the
produced by S(1,2) rather than the mode coupling spectra
(8)
of
We think
alone.
T(1,2)
as
conceptually
a marriage
between the DIA and mixing length models.
T(1,2)
The
perturbation
S(1,2)
and
theory and,
therfore,
the exact MHD equations.
the dynamical
treatment
of
viscosity,
the
treated
Here,
the
the
essential
mode
Ref.
by T(1,2)
we
show
coupling.
1,
In
are total
we
a
showed
that
proper
is analogous
length-mode
evolution
S(1,2)
coupling
of
renormalized
terms in
S(1,2)
space
equation
fluctuations
to that
the
conservation
such
subject
as
to
(8)
of
are
of
mixing
of
the
of magnetic
turbulence
the
is
conserved.
momentum.' 5
the
invariants
is
where
Vlasov
exact
follows
which
dynamical
coupling
conservation
Vlasov
and
and magnetic
mode
turbulent
of
proper
resistivity
helicity
conservation
density
ensures
the
a
for
helicity,
cross
of
of
nonlinear
maintains the
situation
phase
and
particular,
cross
the
treatment
The
while
a
the nonlinear
absence
energy,
by
equation,
from
This is necessary for the
in
the
in a way that energy
that
maintains
and,
physics
described
T(1,2)
derived
only approximate
magnetic shear
helicity.
are
However, the T(1,2) and S(1,2) terms must conserve
invariants
In
helicity.''
(8)
in
invariants of the exact equations.
of
preservation
terms
A
mixing
nonlinear
a
general
feature of clump models of turbulence.
Collisional resistivity and viscosity also contribute to T(1,2) in (8).
Their presence in MHD allows for changes in magnetic field topology that the
Ohm's law
E + V
of
ideal
Because
MHD prohibits
of
(9),
magnetic
(E
x
B = 0
and
(9)
V are
electric
are
field lines
7
field and
frozen
fluid
into the fluid
velocity).
flow,
so
magnetic flux surfaces are preserved.
lines can "slip" from the flow,
statistical
sense,
this also
With collisional
break and reconnect."
can
occur
in the
dissipation,
field
In a course-grained
presence
of
stochasticity.
The stochastic bending and twisting of a field line down to finer and finer
scale
lengths
will,
when
taken
appear as a dissipation.
below
the
scale
of
the
course
graining,
This is the meaning of the inviscid part of T(1,2)
The situation is similar to that of Vlasov turbulence.
in (8).
There,
the
exact Vlasov equation is time reversible and prohibits the breaking of orbit
trajectories
However,
(contours
of
statistical
dissipation
(e.g.,
as
constant
phase
course
graining
in
Quasilinear
the
space
density
introduces
do
not
break).
irreversibility
Theory).
This
and
irreversible
mixing of phase space contours carrying different density reduces the mean,
course-grained
phase space density.
A clear example
is given in Fig.
5 of
Ref. 18 where a time sequence of phase space density contours is shown.
the
figure,
causes
a
the
finite
course
spacial
graining.
grid used
As
the
to solve
phase
space
for
the
islands
contours
interact,
In
also
their
contours
break as
they become mixed and twisted
down to scales
less
that
the
size.
in
similarly
of
grid
Magnetic
flux
contours
MHD
can
be
than
dissipated.
Since the energy in MHD is dissipated
helicity'',
we view
(8)
as
turbulent mixing subject
magnetic
helicity.
averaged sense.
dissipation
to the constancy
Magnetic
helicity
E.B
=
we
the
energy
of the more
is
conserved
0
"invariant"
"rugged"
only
in
by
invariant,
the
volume
(10)
is trivially satisfied because,
However,
of
rate than magnetic
Recall that magnetic helicity is conserved in ideal MHD if
Jdx
(10)
the
at a faster
take
the
view
that,
from (9),
because
8
E.B
of
=
0 on each flux surface.
the
course-graining
or
collisional
dissipation,
only
the
flux
surrounding the plasma is preserved.
associated
with the
analogous
to
that
total
of
plasma
Vlasov
surface
Therefore,
volume
is
turbulence
at
the
conducting
shell
only the magnetic helicity
invariant.
where
momentum
The
situation
can
be
is
exchanged
locally but total momentum must be preserved globally,
fdx f dv vf = 0
'
where
the
integrals
cylindrical
longitudinal
MHD
are taken over
plasma
field,
(11)
with
the
sheared
Boz >> B0 (r).
total
plasma
poloidal
field,
Then,
volume.
Consider
Boe(r),
and
the helicity constraint
a
strong
(10)
for
the mean field becomes
fdx Eoz = 0
(12)
which, when used in Faraday's law for B0 8 (r), yields a global constraint on
the turbulent mixing of B06(r),
' f dr r B0 0 (r) = 0
(13)
where again, the integrals are taken over the plasma volume.
is the MHD clump analogue
(13)
of
(11)
the
for
The constraint
The
Vlasov case.
use of
global helicity invariance has been previously proposed by Taylor.6
The two
for
the
point energy conservation equation
fluctuation
energy.
The
T(1,2)
term
fluctuation energy in and out of the volume.
fluctuation
which
energy
is
inside
produced
the
volume.
are
the
mean
field
is
dissipated
is a Poynting
is
the
Poynting
theorem
flux
of
The S(1,2) term is the rate at
the
volume
by
the
turbulent
S(1,2) is the rate at which the energy
mixing of the mean field shear, i.e.,
in
(8)
into
turbulent
fluctuations
inside
the
In the simplest case, S(1,2) is equal to Eoz Joz, where Eoz and Joz
mean
longitudinal
electric
field
and
current
density.
The
conservation of magnetic helicity constraining S(1,2), therefore, constrains
9
the electric field profile Eoz (x).
In this paper, we show that the portion
of the mean field Ohm's law due to the turbulence is
- FBoz
EJ o=
where D and F are diffusion
(drag)
coefficients.
(14)
Inserting
(turbulent resistivity) and dynamical friction
Only the
into Faraday's
D term
(14)
in
Law and
using
was
(4)
considered
yields
a
in
time
Ref.
1.
evolution
equation for the mean magnetic field Boy that is of the Fokker-Planck type.
The diffusion term of this Fokker-Planck equation, coming from the D term of
(14),
describes
the
random motion
of
coming from the F term in
equation,
(correlated)
Because
motion.
of
the
holes.
The
second term
of
the
describes their self-consistent
(14),
helicity conservation,
magnetic
the
two
terms are connected--in the limit of zero resonance width, the D and F terms
in (14) cancel.
The resonant interactions between the holes lead to no net
transport of the mean field
as "collisions" between
these interactions
The
collisions.
island)
collision
Fokker-Planck
situation
is
operator
for
momentum
to
conservation,
distribution. "
We
it
holes,
hole-hole
i.e.,
to
analogous
identical
to think of
is useful
the
show
no
below
net
transport
in
particles
that
net
of
transport
(island-
vanishing
collisions between like particles lead,
dimensional Vlasov plasma:
of
In this regard
B.
the
mean
from
of
the
a
one
because
particle
hole
random
collisions occurs in (14) at second order in the resonance width:
E
oz
=
-
D(x)2 V2J
.Loz
where Ax2
is the mean-square
squared).
Using Ampere's law,
fourth
order
diffusion
step size
equation
for
cancellation of lowest order
center
plasma
like-like
(on the order
insertion of (15)
similar
where
(15)
magnetic
particle
collisions
10
of
the
island width
into Ampere's law yields a
field
line
diffusion.
A
fluxes occurs in the guiding
cause
transport
(fourth
order
diffusion) at second order in the gyro radius.1'
The constraint of helicity
20
conservation on Ohm's law has also been considered by Boozer.
D.
Turbulent Dynamo
The D and F terms of the clump model Ohm's law (14)
as the a and a coefficients of dynamo theory7.
electric
field is
calculated
from the
can be interpreted
In dynamo theory,
fluctuating
flow
the mean
velocity,
6V,
and
magnetic fields in Ohm's law,
=-<6V
E
x
(16)
6B>
where <> denotes an ensemble average.
Frequently, the view taken in dynamo
theory is kinematic rather than dynamic (i.e., self-consistent) in that the
6V's
are
prescribed
Faraday's law.
and
the
6B's
are
from
these
flows
using
The result is that (16) can be written as
E
=a 0J + aB
(17)
While a depends on the mean square flow,
derivatives.
derived
Consequently,
if
the
(17),
only occurs
field
lack reflexional
a depends on the flow field and its
the so-called
In the
due to a non-zero
of
properties
statistical
symmetry. 7
a-effect,
this
clump model,
a in
flow
background
we interpret
(17)
dynamically, i.e., as the self-consistent, helicity conserving Fokker-Planck
process ( 1 4 ).
the
sum of
structures,
We consider a "test particle" picture where the flow,
two terms:
term of (14)
is proportional
any 6B.
presence
of,
self-consistent
is
island
and dVc, the response that is phase coherent with the fields of
With force
background islands.
first
6V due to the
6V,
balance
used to
evaluate
in
6V
comes from 6Vc and the second term from 6V.
to the mean square 6B and would,
The F term,
however,
is nonzero
self-consistently in the island structure.
11
therefore,
because the fields
be
(16),
the
The D term
present for
are correlated
This self-consistency causes the
lowest order cancellation between the D and F terms of
the
magnetic
helicity
conserving
demonstrate this cancellation
nonvanishing of the
form
for
F term in
(15),
the
(14),
"net"
can also
the
clump model
holes
(As
as required
by the formation
discussed in Sec.
in Sec.
be traced to the
prevalence of the current holes (over the "anti-holes").
of reflextion symmetry,
a-effect.
steady state turbulence
(14)
thus yielding
Thus,
by a turbulent dynamo,
and
preferential
IB of Ref.
1,
II.
The
statistical
the breaking
is achieved
growth of
"anti-hole"
We
the
in
current
fluctuations,
6J>O,
decay).
Comparison
copper
with the homopolar
disc rotates about
disc dynamo
its axis with angular
enlightening. 7
is
velocity Q,
A
solid
and a current
path between its rim and axle is made possible by a wire twisted in a loop
around the axle
(see Fig.
1.1 of Ref.
7).
Rotation of the disc causes
an
electromotive force MQI which drives a current I in the loop given by
L
where
M
is
inductance
L+
dt
the
and
RI = MQI
loop/rim
mutual
resistence
of
unstable to growth of current
flux induced
energy
across
exceeds
Ultimately,
the
(18)
inductance,
the
complete
L
and
circuit.
disc.).
Growth occurs
rate,
i.e.,
when
when
the disc rotation slows to the critical
steady state is achieved.
frequency
corresponds
to
0,
the
i.e.,
the
a
>
are
the
system
selfcan
be
is the magnetic
source
R/M
C
of
in
= R/M,
free
(18).
and a
The situation is similar to that of the MHD clump
and
the
resistance
magnetic clump fluctuations
stochastic decay,
(MI
value
instability where the mixing rate of the mean shear
driving
R
The
and magnetic fluctuations
dissipation
the
and
R>1.
plays
stochastic
decay
R
disc/loop
(see (1))
in
the
the role of
(turbulent
the
resistivity)
circuit.
Growth
of
occurs when turbulent mixing overcomes
Quasilinear relaxation of the shear
12
gradiehts
lowers R to the critical,
steady state value Rc
=
Multiplying
1.
(20)
by
I/R gives
(
(19)
+ R) 2 = 2 ( ),I2
dt
L
is analogous
The resistive
T(1,2) -
1.
to
(8)
for the
clump magnetic energy
As
The driving term on the right hand side of
preferred
in
2
(C-<6B
> in
(8)).
decay rate R/L corresponds to the clump stochastic decay rate
to the clump source term S of (8).
(15).
(19)
L
the
disc
Note that
dynamo where
we
sign for forcing is provided
must
by
Eoz
S
-
(19)
is analogous
<6B 2
Eoz
~
D
0
>
0
have
~
V
-
for
z
> 0
>-C
from
growth,
inside
a
the
plasma.
The MHD clump instability is a turbulent dynamo,
type.
Typically,
However,
effect
of the usual
dynamo models sustain or increase the mean magnetic field
at the expense of currents flowing across the field.
term in (14),
but not
Were it
not for the D
the F term would cause an increase in the mean magnetic field.
the mean field does not increase
is balanced
by the stochastic
because,
to lowest order,
diffusion of
the
balance is the result of magnetic helicity conservation.
field
the a-
lines.
This
To next order (in
the island width), the mean magnetic field decays according to the turbulent
mixing process (15).
The energy lost from the mean field goes,
energy conservation,
into the
degrading
where V2 J
of the
mean field occurs
<0 in (15).
of
in the
the
clump
interior
In the exterior region where
support the mean field and,
a-effect.
creation
The net effect
therefore,
of (15)
13
confined
>0,
(11)
This
plasma
tends to
acts in the spirit of a traditional
is to expel
plasma.
fluctuations.
of a
V
because of
mean
poloidal
flux from
the
C.
Steady State Turbulence - The Taylor State
The constraint
on
steady
of magnetic helicity conservation
state MHD
clump
quasilinear relaxation
instability
source
the
R
toward
zero.
The
crucial
instability
net
effect
proceeds,
the
will drive the
a-effect
(15)
will,
This will result in decaying turbulence where the energy
decays by mode coupling
(i.e.,
the "-1"
to finer and finer scale lengths.
the mean field gradients
is the critical
As
of the mean magnetic field gradients
term
therefore, vanish.
turbulence.
has a
and,
stochastic decay term in (5))
However,
therefore,
this decay can
and a steady state turbulence
be overcome
the a-effect are maintained.
= 1 noted above.
value Rc
down
The turbulence
(dynamo action)
is
is possible.
then
if
This
driven,
Of interest then
is the structure of the driven clump fluctuations rather than the detailed
features of the
The
broad spectra produced
steady state
occurs when R
=
clump
fluctuation
in the case of decaying turbulence.
level
follows
from
(1)
and
(6),
and
1, i.e.,
7 2J
V
+ P2 J
Loz
oz
=0
(20)
1
(21)
where
12-
(k2
AIX
+*
2d2
For low mode number,
A'/xd
small
2
amplitude
holes
Then,
case.
in the fully stochastic
koAx
<
1
and
multiplying
(21)
(20)
gives
by
Dxd,
y2
and
using (15) gives
E
oz
where Ec
=
D 6Jz
-
+ Ec = 0
DA'
(22)
z
Xd
z is the force turbulently dissipating
holes and
114
the
-6R
is the
root mean
structure
holes,
(see
(22)
current
hole
depth necessary
1).
Since
EOZ is the
Ref.
(dynamo)
can also
(23)
square
is a statement
steady state
(20)
AXd Jz
of mean force
to
mean
form
balance.
where Jo is the zeroth order
coordinates,
Bessel function.
island
creating
In addition to
fluctuation
be thought of as a global equilibrium
In cylindrical
trapped
field force
condition on the mean-square
density JOz.
a
being a
level
(p),
condition on the mean
(20)
gives Joz
~ JO(ir)
A more enlightening view of
this is obtained by considering poloidal
as well as toroidal currents.
approximate
is
calculation
equilibrium relation
for
this
between
case
the
carried
the mean current
out
profile
in
Sec.
An
V.
The
and the mean-square
fluctuation level is found to be a vector generalization of (20):
+
V2 J
Because
A'
and
xd
2 J
are
= 0
(24)
spectral
averaged
insensitive to position inside a broad,
quantities,
p
is
relatively
fully stochastic spectrum.
end of Sec. V for further discussion of this point).
Therefore,
(See the
we set
P
=
constant, and with V.J=0, (24) has for a solution,
J
=
yB ,
(25)
yielding again a force-free state (J
x
Bo = 0).
The global
force
balance
relation (25), the so-called Taylor state, has been known previously as the
MHD
state
of
minimum
mean
energy
with
a
given
constant
mean
magnetic
helicity 6 .
That the solution (25) should result from steady state MHD clump
turbulence
is not
energy
turbulent
via
surprising,
mixing
helicity conservation.
(25)
is a
prescribed
state is turbulent.
since
subject
Note that,
function
the
clump dynamics
to the
constraint
in the clump theory,
(21)
of
the
turbulence
minimize
of
global
the mean
magnetic
the parameter
level.
The
y in
steady
Equation (25) prescribes the mean profiles in terms
15
of
the
mean-square
fluctuation
(p).
level
The
fluctuations
generating
the
dynamo action are created self-consistently by the balance between turbulent
mixing (J 0 ) and the decay (1iB 0 ) caused by field line stochasticity.
Because
depends
i
on
length to the shear.
the mixing length
This relation follows
xd,
(20)
relates
if we multiply
the
(20)
mixing
by x
and
use (23):
R _
6J
2
x2
2
d
z
L
(26)
oz
Eq. (26) is actually an equivalent form of the MHD clump steady state mixing
length relation
<6B 2>
x
-DTx
d Joz VI
(27)
oz
(27)
follows from the steady state integration
for
S(1,2)
-
EozJoz
and
xd~(64/Joz) 1/2, and 6Bx
reduces
to
(26).
T(1,2)
3/Dy
=
Equations
~
-
helicity conservation.
diffusive
mixing
relation
However,
because
of
Using
36BX/3y
and
from
here,
(27)
differ
term in
of
the
helicity
(13)
the
hole
-
-6BX/Ay,
the
usual
(15)
width,
(27)
form
of
the mixing process is constrained by
This can be seen by integrating (8)
with only the
length
D/xd.
with the use of
6*/Ay and 6Jz =-
(26)
mixing length relations because,
~
of (8)
retained
standard
form:
conservation,
in
S.
we obtain
Then,
DTJoz 2
6B2
each
in steady state,
term
in
~
oz 2
d
(14)
a
must
be
retained--leading to the use of (15) in S(1,2) and the result (27).
Not all V values
(21)
(21)
are
consistent
with the steady state.
Solving
for xg gives
1
xd
2
2
4
,24A'
k )1
(2
28)
2A'k0
There are no real
For
reasonable
solutions for
values
of
A' and
xd
(and,
ko,
16
therefore,
this means
D)
unless
p will
be
2
an
> 2A'k0 .
order
one
quantity.
2
At the threshold
P =2A'k,, and koxd
= 1.
Above
the threshold,
ko xd > 1 and
xd
Note that
2 /Ak2
P's on the order
of one are
for a BOZ field-reversed state,
(25)
reverses
states (p),
clump
(29)
0
d
sign when
i.e.,
comparable to the
Lt values required
the solution Boz =
r>2.4.
Therefore,
of
all
1
the
~ JO(Wr) to
-oz
possible
Taylor
the ones with reversed BOZ field correspond to steady state MHD
turbulence.
Smaller
y's
apparently
correspond
to
MHD
clump
instability.
At the threshold of the clump steady state,
the mixing length relation
(27) reduces to
6B
- xd oz ko xd
(30)
(30) follows from (27) by using (23), (20) for V Joz,
and p 2 ~A'koxd
at
the
threshold
xdJozkoxd, (30)
trapped
(koxd=1).
Since
(23)
can
also
be
written
- A,/xd
as
6Bx
-
is just the fluctuation level necessary for the formation of
island structures.
Since
= 1
koxd
at
the threshold,
(30)
is,
at
threshold, just the standard mixing length relation
6B
A
similar
result
i .e.,
can
Vlasov
holes
characterized
the mixing length level
as the fluctuation
by the turbulence.
"amorphous"
(31)
for
There,
6f-(Av/vth)fo.
fluctuations
x Az
occurs
trapping width Av.1
same order
-
level
Therefore,
"just barely"
necessary for
in
steady
self-organize
6f
-
by
a
velocity
Av3f/3v
trapped hole
state
before
formation,
t ur bulence ,
they become
In steady state,
the
dissipated
The turbulence is thus composed of colliding,
hole structures:
is the
growing,
clumps.
the diffusion
coefficient
17
Dx /T can also
be written
as D-Yn/k
~ 1~1 is
,
since k
x-1 is the typical radial wave number
-
the characteristic mixing rate.
helicity
conservation,
coefficient,
Ynz/k
Were it
would
be
D , for the mean field flux,
-
R/T
not for the constraint of
the
0.
and Ynj
cross
field
However,
D
diffusion
< D as
can be
seen by combining (15) with Faraday's law:
a990
=
2
a2
D(Ax)
-
3x
Therefore,
o
(32)
ax
D1 -D(Ax/a) 2
current channel.
D2
2
ni/k )(Ax/a) 2
where
a
is
the
radius
of
the
An analogous reduction in cross field transport occurs in
a guiding center plasma of identical particles."
from the MHD equations in Sec.
II and Sec.
III.
Equation (32)
is derived
A quasilinear equation of
the form (32) has also been derived for an assumed spectrum of tearing modes
by Strauss and Bhattacharjee.22,23
D.
Transition to Turbulence
Of course, the steady state condition (24) also describes the threshold
condition
for
the
onset
of
the
instability.
Recalling
(5),
it
is
enlightening to write (24) for arbitrary I,
V J
+ R
j
0=0
(33)
with a solution corresponding to (25)
Therefore,
for
of
(3 4)
R/2
a given
y,
(i.e.,
amplitude),
instability
occurs
when
the
mean driving current density exceeds the critical value
4
For
analysis
additional
of
this
dissipation
=
(35)
B0
instability
effects
such
threshold,
as
18
it
collisional
is
useful
resistivity
to
include
(nsp)
and
viscosity
(v).
Assuming
a
unit
dissipations change T(1,2)
of (D
+
(1
nfp) X2
The growth rate
(5)
magnetic
-
T
Prandtl
in (8)
D/x
R' )t', where Rm =
+
number,
to the net
the
dissipation rate
D/nsp is the Reynolds
(~ -
1 - -)
(36)
R
Instability occurs when the mixing overcomes
both collisional
dissipation.
term
While
we
think
of
coupling rate due to "hole-hole
a turbulent
effective
viscous
number.
then becomes
L
Ta xd
it as
additional
the
"-1"
collisions",
eddy viscosity
damping rate is
as
in
it
fluid
in
is also
here
as
a mode
useful to think of
13
12
turbulence .
+ R~')T~'.
m
(1
(36)
and turbulent
Then,
the
With these modifications,
the threshold condition becomes JO>Jc where
c=
(37)
1)1/2B
+
Thus, the instability condition on the Reynolds number is
>
Rm
2
J
o
(38)
z
where
we
have
Instability
considered
(MHD
current profile
the
right
hand
longitudinal
occurs
side
of
the
(38).
Since,
of
<
1,
current .
amplitude
the critical
koAx
the
(P)
and
value given
p decreases
by
with
smaller Reynolds numbers require larger amplitudes for
The threshold is evidently nonlinear.
instability threshold
different
for
part
for a given
if the Reynolds number exceeds
the onset of turbulence.
located at
the
clump regeneration)
increasing amplitude,
Consider
only
mode rational
in the
surfaces.
case of two large
When
the
islands
island resonances
begin to overlap,
the region of initial stochasticity will be small compared
to
widths.
the
island
The
effect
19
of
this
intermittent
region
of
stochasticity
where
ps
width.
and,
is to replace
denotes
the
the
percentage
A further modification
therefore,
two current
of
(36)
in
"-1"
and
(38)
stochasticity
compared
is due to the fact that
holes--attract
with a factor
to
the
parallel
each other.
ps,
island
currents--
This tendency
for
island coalescence will tend to inhibit the mode coupling decay rate of the
islands.
In order to account for this tendency, we further multiply the "-
1" in (36)
and
rc
and (38)
factors
by the factor rc, where rc< 1 .
also
occur
in
Vlasov
hole
The phenomenological
turbulence.
There,
p5
hole
intermi-ttency reduces the decay rate due to hole-hole collisions, while the
attraction
between
holes
collisions)
to recombine
instability
is
causes
hole
fragments
into new holes.
the analogue
of the
(produced
The magnetic
Jeans
instability
from
island
hole-hole
coalescence
for Vlasov holes. I8
The net result here is that, for psrc<<l, the Joz/1Bz term dominates in the
denominator
of
(38).
Since
p is given
by
(21),
and a critical
amplitude
(island width) is required for island overlap, the stability boundary is of
the form depicted
in Fig.
1,
a form reminscent
As discussed in the Appendix,
Fig.
of plane
Poisuille
flow.2s
1 can be viewed as a phase diagram
for
the "phase transition" to steady state MHD clump turbulence.
In a toroidal plasma,
on
the
safety
factor
q(a)
the stability condition JO<Jc sets a lower bound
- aBz(a)/RBe(a),
where
(a,R)
are
the
(minor,
major) radii of the plasma and (BZ,Be) are the (toroidal, poloidal) magnetic
This can be seen by integrating
fields.
(35)
over the plasma
(minor)
cross
section to obtain
qc(a)
where
is p averaged over the
q(a) > qc(a).
<
y'
=
(39)
2/R5
plasma cross
section.
Stability occurs
if
An alternate view of the instability threshold follows from y
2/Rq(a).
Since
c
- q(a)~1
20
increases
with
driving
current,
instability results when too much current is driven for a given Taylor state
(1).
number
magnetic
instability
exceeded.
overlapping
islands
frequently
("disruption")
In
fusion
in a tokamak
For example,
can
the
reversed
islands
have
comparable to pc (i.e.,
be
field
smaller
large
device,
develop.
expected
pinch
initial
if
overlap,
current
device,
amplitudes
R comparable to one).
Taylor state and thus relatively quiescent.
Upon
this
fusion
amplitude,
so
by
contrast,
that
(28)
strong
threshold
The plasma will
With 5 - pc,
low mode
will
is
the
be
be near the
implies that
an increase in driving current will just push the clump turbulence to higher
fluctuation levels.
21
FIELD SELF-CONSISTENCY
II.
As discussed in Sec.
(14)
I,
the Fokker-Planck structure of the mean field
results from the self-consistent
generation of the clump fluctuations.
It is this self-consistent structure that ensures the global conservation of
magnetic
helicity.
illuminating
We
case
time stationary.
show
this
steady
MHD
sheared field
9
the
fluctuation
A complimentary
derivation
from
2 component
momentum
of
"tokamak
B
.
the
simplified
spectrum
time
but
that
is
dependent
MHD
the equation
for
III.
the
BOy(x), Boz = constant),
or,
considering
stochastic
by taking
state
by
of a spacially
equations is presented in Sec.
We begin
here
of
the
balance.
Using
ordering"
(B=Bo + 6B,
curl
slab
of
geometry
Be
and
the
= B 0 +
model
'
B
§io
we obtain
VJZ
(40)
0
=
more explicitly,
3J
z+
B'
x DJ
3z B oz
oz
6B
+
y
Boz
oz
DJ
x
= 0
(41)
where we have retained only the 6BX component
of 6B for simplicity.
we are considering self-consistently generated fields,
we must
Since
couple
(41)
to Ampere's law
2
where
B1
-
x(*Y)
self-consistency,
6Jz = 6fz+
(42),
while
particular,
consistently
defines
the
flux function
fluctuation
describes
by turbulent
the
the
response
resonant
to
clump
these
Because
of
in ip generated
fields
fluctuations
mixing at the mode rational
22
$.
the
as a sum of two parts,
describes the source of fluctuations
6JO
z describes
jz
poloidal
we write the current
j
*'j
(42)
Jz
via
(41).
generated
surfaces,
via
In
self-
while
6
Jcz
describes
the nonresonant
currents flowing in response to the clumps.
decomposition is similar to that for an isolated current
IA of Ref.
Sec.
currents
1 where the coherent
flowing
structure
of
(respectively)
an isolated
useful to treat
(41)
hole.
inside
and
In the
case
hole discussed in
of Jz and 6JCz describe
outside
the
magnetic
of stochastic
the
island
fields,
it
is
as a Vlasov equation for field line trajectories where
27
z plays the role of time.
calculated as
analogues
The
For weak fields,
in the Quasilinear
theory.
the response
Neglecting
~z,
6Jcz can then
be
one obtains from
the fluctuating part of (41)
dJc = j
k
k. B
k-
where
6Jk is
current
the
- -0
Fourier
transform
response of linear
(43)
ax Joz
of
Equation
6 J1c.
tearing mode theory.
2
,
5
(43)
is
Substitution
the
usual
into the
ensemble averaged version of (41),
= -
a
3zo
xk
k Y Im
<6$
k
6J > B(44)
k
'z
gives the diffusion equation
Jo z-
Tx Ooz
(45)
r6(k .B )
(146)
Here,
<6B2 >
DmM
Xk
2
k
B
-
is
the
magnetic
usual
2k
oz
diffusion
fields.28
29
coefficient
Inclusion
of
of
stochastic
JZ
in
(44)
instability
gives
the
models
of
Fokker-Planck
equation
wz
oh
a
Dm -J
m T
o
x
where
23
Fm J
(ere
Fm =
k
6 JCz
Since Jz and
in (47)
Im (6$
~J > (B
k
Yk
k
J
(48)
oz oz
are related self-consistently through (42),
are also related.
Fm and Dm
This relationship and its consequences for global
transport are best seen by considering the nonlinear expressions for
Fm.
For stochastic 6B , the nonlinear term in (41)
diffusion operator in the same way that
(41)
Dm and
can be approximated by a
yields
(45).
6Jz,
therefore,
follows from
B1'6
3z
+
oy x
Bo
Dm
T
ax
y
oz
x
)
c
J(9
Boz az
x oz
z
so that
k-
6Jk
-
-
(x) B
k
(50)
ax oz
oz
where
gk(x)
dz exp iz k.B0/Boz -
=
(z/z0)3
(51)
1/3
(52)
0
-
is a broadened resonance function and
(ky Bgy/Boz)
2;
2
D
is the distance along z for a field line to diffuse a distance <6y2>1/2k 1.
Insertion of (50) into (44) then gives the nonlinear diffusion coefficient
<6B2
Dm
k
that is now to be used in both (47)
gk+w'f
(k.B0 /Boz) and (53)
The
physics
of
the
(53)
xk g (x)
Boz
and (50).
Note that,
for weak fields,
reduces to (46).
nonlinear
diffusion
amplitudes, field line diffusion occurs when
24
is easily
seen.
For
finite
kz + k
BI
x/B
z y oy
In terms of the position x5
kzBoz/ky B
-
=
z701
oz 1
(54)
of the mode rational surface of
is
mode k, (54)
Ix
where xm = (k B
-
xsI < xm
z/Boz)
x=
m
(55)
i.e.,
(B Dm/ 3 k BI )1/3
oz
y oy
(56)
In the stochastic case, xm plays the role of the island width
Ax = (6
so
/Joz 1/2
(55)
is
the
overlap
of
two neighboring
2
zo/Boz
where
condition
OBres
integral in (53).
is
x
equivalently,
the
island overlap.
resonances.
resonant
To see this,
From
portion
(53)
of
Using now the definition (52)
6
expressed in terms of
or,
for
(57)
m
and
6BX
for
consider
(51),
Dm~<6B>2res
contributing
zo,
the
to
Dm in (56)
the
can
be
Bres so that (56) becomes
- (6B
(57).
res
/k BI )1/2
(58)
y oy
Therefore,
at
island overlap,
Dm becomes
and a field line random walks radially as one moves along
z,
i.e.,
nonzero
(6x) 2
=
2Dmz.
The nonlinear field line diffusion
destroys
finite
amplitude magnetic
island structures and causes global transport of the mean fields.
island structures
are disrupted
different rates.
As in Sec.
seen by deriving from (51)
(49),
because neighboring field lines
IIIB of Ref.
1
(see also Ref.
27),
Localized
diffuse
this can be
the two point correlation equation analogous
i.e.,
25
at
to
B'
az
+
x
Boz--ay-
in the simplified case of
separations
X
D
a x-
-
<6J (1)6J (2)> = 0
z
)
3JOZ/Dx = 0.
Here,
x_=x 1 -x 2 ,
between field line trajectories and Dm = D
(59)
y-=y 1 -y 2 are
+ D
- D
the
-
is the relative diffusion coefficient where
DM
12
=
<6B 2
xkk
k Bz
k-
For
small
cos k y
k
(60)
Boz
separations,
two field lines feel
approximately
the same forces
and tend to diffuse together.
Then, we can write Dm = Dm k2 y2 where
(93)
the
of
fields.
Ref.
1),
ko
defines
typical
scale
length of
the
(as in
stochastic
Using this Dm, the characteristics of (59) imply (for z_ - 0) that
z/L
<y_(z)> = (y2-2xyL /L
-c
s
+ 2x L /L )ez
-c
(61)
s
where L,1 = Bo /Boz is the inverse shear length and
L-2 Dm )-1/
Lc = (12)-1/3 zo = (4k
is the
z-exponentiation
length or
3
Kolmogoroph
field lines, initially separated by x_,
(62)
entropy.2
From
(61),
two
y_ will diverge by ko- in y- after a
distance traversed in z of
zc - L
in
2
2x
3k-2
/L
222
(63)
?_
-2_7LIL +2xL /L
c s
-c
s
where
y
referred
=
y.-x+z_/L,.
to
structures
It is this orbit exponentiation process,
as stochastic
apart.
orbit
This spacial
instability,
destruction of the
their destruction in time by the Alven speed.
islands
are
disrupted
stochastic field lines.
as
that
Alfven
waves
coherent
island
islands is connected to
As described in Ref.
propagate
The Lyapunov time of Ref.
26
tears
frequently
down
1 is T
=
the
1,
the
spacially
Lc/VA (see
(62)
above
and
(95)
of
Ref.
1),
where the
D-DmVA in dimensional units (see (53)
in the temporal case with flows,
diffusion
coefficient
above and (73)
the 6B fields
of
of Ref. 1).
in (53)
Ref.
1
is
Note that,
get replaced by the
clump magnetic fields 6L and 6N.
The nonlinear version of (47) can now be obtained as follows.
substitute (50)
into (42)
to obtain
22
k
- k
+ i g (x)
(64)
J'
k B
6 k
I
-
(64)
oz
-
Equation
We first
is a resonance broadened Newcomb equation2,
by the clump currents Jk.
30
that is driven
Let us define the clump or resonant part of the
flux function as
Wk = Xm
then,
integration of (64)
k
k
J
jk
dx
(65)
gives
=
k
(AI+21kIJ)x
(
+
(66 )
where
A
is
the
ik y/Bo
= - 21k
resonance
denominator,
Pfdx g (x)6$ (x)
broadened
tearing
as discussed in Ref.
1,
mode
in (44)
stability
parameter.2
The
describes the shielding of the clump
field by the nonresonant currents 6Jc.
relates Fm to Dm.
Joz
Using (66) and (50),
The self-consistency relation
(66)
the 6Jk contribution to the bracket
becomes
k 2<*fdx'J*(x
(x)
oz
-
k
JA +2|k
while the jk contribution is
27
(68)
2x
k 2
y ~~
oz jA +2|k
2
(69)
x
k y
m
Passing now to the Fourier integral limit and noting that
<~(1)J(2)>k = 2Regk(1)<J(1)J(2)>k
-
(70)
y
-
(44) becomes
-J
3z
-z 2x
ax
OZ
m
B
f
oz. (27) 2 k2
y
3J
(x)
+2 1-2
dx'
Reg (x')Reg (x)
k
kX
-
3J
>
-
J<
k
(x')
(71)
a(x')>k
This can be cast in the form of the Fokker-Planck equation
(47)
where, using
(65) and (70),
2
12
Dm
B
dk2
(21)
-2
y
+2lk
|A
k2
2
g k(x)
(72)
X
x1
and
Fm
Recalling (66),
<65~J >
dk
dk
k
(27) 2 y 1,+21k
Boz
we see that
(78)
is just
112x
(53).
ImA'
k
(73)
Rather
than the
diffusion
equation (45) for Joz in an arbitrary (non--self-consistent) field 6Bx, (47)
describes
fields
the
(66).
evolution
Note
that
of
Joz
Fm*0
due
to
because
self-consistent,
the
fields
are
shielded
island
correlated
self-
consistently by Ampere's law (42).
This correlation connects Fm with Dm and
causes the right hand side of (47)
to vanish to lowest order in the
width.
To see this, we note that as xm-z-'+O in (51),
and the two terms in the
bracket in
(71)
cancel.
island
Regk(x)+76(k.Bo/Bz),
There is no net
radial
transport of the field lines.
For finite island widths,
the two resonance functions
28
in
(71)
overlap
and cancellation
does
not
occur.
expansion of the Fokker-Planck
To show
this,
we
collision operator
follow
in Vlasov
the resonance
turbulence."
We assume that the correlation function is factorable:
<( ~(x)> k
Then, (71)
a(x) b(k )
(74)
can be written as
TZ Joz = L
ax
dx' K(x'-x,
2+x)H(x,x')
(75)
where
K(x'-x,
)
-
2x
m
Boz2
=
2
Reg (x)Reg (x')
d
(2i)2
+2 k
b(k )
y
2
(76)
and
H(x,x')
Recalling
therefore,
=
aiJo (xI)
oJ (x)
Z
- a(x) - ax
a(x')
(51), we note that
function of Ax
We,
=
(77)
axax
K-0 unless
Ix-x'I <xm.
Moreover,
K is an even
x' - x and has a weak (nonresonant) dependence on (x'+x)/2.
expand K as
K(Ax, x
-)
=
K(Ax,x) +
K(Ax,x)
2~
(78)
3ax
and H as
S(xAx) 2
H(x,x')
=
2
iii+
a
x
aa(x) aJz) a(a2
-
a7x_
Fax
J0 (x)
ax)
ax 2
a
(79)
79
..
Substituting (78) and (79) into (75) gives
2
J
Tzo
Because
of
effectively
recalling
2(
2
fdAx (Ax
K(Ax,x)
[a(x)
a2 Jozx
2
x2
the
ax
resonance
(74)
Ax2/2
and
functions
with
the
(76),
in
the
mean-square
the
29
K,
first
(80)
a
x
replaces
(72),
3(x) ajo (x)
-
dAx
integral
island
term
in
width.
in
(80)
(80)
Then,
is
-(3 2 /aX 2)Dm(Ax) 2 (a2
.z.
2)
insensitive to
For Dm relatively
broad, fully stochastic spectrum, the last
term in
(80)
can
a
x inside
be neglected.
Therefore, (80) becomes:
22
a ja2
az
oZ
m(,)
2j(1
2ax 2
2
oz
The resonant interaction of finite amplitude islands,
net
diffusive
Since
Alfven
transport
waves
will
of
the mean field,
carry
energy
albeit
away
therefore,
fourth order
along the
cause
diffusion.
stochastic
magnetic
field lines at the Alfven speed, the effective global transport rate in time
due to the stochasticity
follows
by multiplying
(81)
by
VA.
Then,
(81)
gives
2
)2 a2
oz
(82)
where, as we have discussed above, VADm + D.
Assuming that Dm is relatively
at
insensitive to x,
oz
result as
2
we use the mean field
the time evolution
alternative
2
equation
(32)
for
part of (42),
and thus
the mean flux,
$e.
(82)
In Sec.
yields
III,
an
derivation from the time dependent MHD equations yields the same
(32).
Generalizing
(32)
to
cylindrical
coordinates
and
noting
that Eoz= -3$ 0/Dt from Faraday's law, (32) gives
2
E
Eoz
-V
.D(Ax)
gx)*
.
ensured by the Fokker-Planck coefficients
width,
the right
hand sides
field line transport and (12)
second order in the
of
(83)
0.
oZ
satisfies
Magnetic helicity is conserved since (83)
island
v
(12).
D and F.
(14)
and
Note that this is
To zeroth order in the
(47)
vanish.
There
is no
is identically satisfied.
Transport occurs at
thus leading to the
helicity conserving
island width,
form (83).
30
III. MAGNETIC HELICITY CONSERVATION
Here
consider
we
For this purpose, we need the explicit time
growing MHD clump fluctuations.
Time dependent equations for MHD clump turbulence were
dependent equations.
derived
equations,
that subtraction of the Alfven wave field from the total
1
clump or resonant part of the field L=B-S~'V, N=B+S~ V.
units here
We showed
field B yields the
[We use dimensional
1), where S is the Lundquist number
IIA of Ref.
(see Sec.
MHD
vector
full
the
neglected.
conservation
helicity
with magnetic
but
were obtained from
The equations
1.
in Ref.
on
conservation
helicity
magnetic
of
effect
the
(S
=
times for
R /TH where TR and TH are, respectively, the resistive and Alfven
the current
channel
For
radius)].
example,
neglecting
pressure
(shown
in
Ref. 1 to be small for the clumps), the N equation is
(
at
- S<B>.V -
S6L.V
V ) N = 0
-
(84)
I-
The <B>.V term describes Alfven wave emission and,
clumps
near
renormalized,
mode
The
surfaces.
rational
localizes the
therefore,
nonlinear
term,
6L.V
II above.
becomes a diffusion operator as in (49) of Sec.
when
The
diffusion coefficient is a turbulent or anomalous resistivity to be added to
the
collisional
resistivity
term in
V
(the
(84)).
The
mean
field
<N>
satisfies
(N > =
ty
ax
(D
+
1)
-
ax
y
where D is the turbulent resistivity in terms
generates
clump
of the fluctuations
fluctuations
satisfying
31
<6N(xj,t).6N(x 2 ,t)>
<6L2 x >k
of the mean
the turbulent mixing (85)
Because of the conservation of energy,
shear
(85)
<N >
u
<6N1.6N2
[~S<B_>.V_
where D_=D
1
a(D +2)
12
2<N >
D< 2
x
*ax
1
2
a
-
+D2 2 -D 12 -D 2 1 =2 (D-D
S2 f
1(22
D
<6N .
-
-S<a>7
12
2
2
(86)
) with
2dk
<6L 2 >
G
xk
k
exp ik y
(87)
(87
The broadened resonance function is
Gk =
isk.B0t - (t/T 0 )3-Yt
dt exp
where Y is the clump growth rate and To
the
two
point
operator
on
the
=
(88)
2
ky
B?Oy 2
(1/3
left-hand-side
of
Inverting
2D)1/3.
(92)
gives
the
clump
fluctuation level (see (97) of Ref. 1):
= 2T_ D 12(B
<6N 1 .Y2>
'2
)
(89)
where T_ = TCZ (1 + YT)~', with
3kTcZ
2
2
0
7_-27_x
-oy STB
= T Zn
2
2 2 2
+2S B' x T
(90)
oy
and
T
=
as the Lyapunov time.
(91)
is
In
that
earlier,
(97),
7
(91)
y. - xBz_.
=
The resonance
width
in
-
xd
so
To = (4k 2S2 B, 2 D)
0
OY
(12)
two
field
lines
these results
spacially stochastic
dimensionalized
(D/Sk
are
only
of the time
case of
units).
B' )1/3
We
Sec.
define
(92)
correlated
dependent
II
by the
the
32
clump
if
Ix_.<xd.
As
discussed
evolution are related to the
Alfven
"flux"
speed
(i.e.,
function
by
1
=i
S
+ S1
in
,
where Bl-Vx(2*) and V =Vx(2$)
velocity stream function,
4.
define the poloidal flux function,
The
Fourier
transform
of
(89)
4, and the
can
be
then
written as
(
- k 2) <6 (1)6s(2)>
ax 2
where i(c,
Ref.
1,
y
= -
2DiE
(x_,k)(B' )2
oy
ci-'
k
(x-,k) is the Fourier transform of (90).
(93)
can be cast in a form reminiscent
linear MHD stability theory.
33
(93)
As shown in Sec. IVA of
of the Newcomb equation
of
We would like to include magnetic helicity conserving terms in the time
dependent equation (86)
we need to distinguish between the phase coherent
As there,
II.
as we did in the spacially stochastic case of Sec.
parts
field
has
already been distinguished by the use of the N and L field variables).
We
incoherent
write 6N =
of
field
(the
Alfven
wave
part
of
the
N, where 6Nc, the part of 6N phase coherent with 6L,
+
6 Nc
the
and phase
given by (70) of Ref. 1 and produces the diffusion equation (85).
resonant
clump
fluctuation
produced
However,
N will
be a random,
we will only need its
contribution
of
R,
(85)
the
self-consistently
as
the
6L
fields
Because the mixing occurs at the overlap of
turbulently mix the mean shear.
resonances,
Note that
field N represents
The
any fluctuations 6L.
be present for
6Nc would
is
incoherent
function
correlation
function.
(neglecting
becomes
of
the
field
phases.
With the additional
dissipation
collisional
for
simplicity)
<N>
L
at y
L
superscript
coefficients
on
<N
TQX
> - FL (N>
yy
D
here
to
the
distinguish
of
(86)
and
The additional
but with FN--S<(xLLy>/<Ly>.
modify the right-hand-side
<6L 2>
We introduce
driven
<6N2 >
A corresponding equation to (94)
that we will consider below.
holds for 3<L y>/at,
(94)
is a Fokker-Planck "drag" coefficient.
where FL - S<LxNy>/<Ny>
the
D-L
ax
so that,
when integrated,
term FL will
(90)
will
be
replaced by
<6N
.
2>
=
2
<6
.6 >
=
2TN
L
D
2
B
+ FL
Bgy
(95)
Similarly,
Strictly speaking,
sides
of
the
two
D
2
B1
-
F
2
the R and L terms also produce
point
equations
such
34
as
(96)
BI
(86).
F terms on the left-handHowever,
the
two
point
propagators
clump
are less
source
term.
sensitive
While
to magnetic
the
global
effects the mixing of the mean field,
mixing of the
fluctuating
(95)
to be given by
and (96)
part
conservation
it
(90)
Adding (95)
of
than
helicity
the
directly
has much less effect on the local
of the field.
as before.
the strong N/L coupling limit where
Sec. IIIB of Ref. 1).
helicity constraints
<N.L>-0
and (96)
We,
therefore,
take
For
simplicity,
we consider
so that
TNz
TL
-
tcz in
Tc,
(see
then gives the equation for the
total energy correlation,
<6B
. 6B2 + S-2
1 * 6-2> = 2 Tlt
E
J
(97)
where
E
With F = <V
(98)
B>z/Boz,
x
(98)
z
Joz -Vb>
= D
form (14).
to the Fokker-Planck
corresponds
We
can also write F=S<NxL>Z/Boz. F gives the self-consistent, correlated motion
of the clump part of the magnetic field.
into Faraday's law gives a Fokker-Planck equation for
Insertion of (98)
.
the mean field Boy with F-<VxB> /B
D
and
F
terms
conservation
to
the
in this
of
the
by
dynamical
mean
Q -
by
demanded
distribution
There,
in a Vlasov plasma.1-
equation of the form 3f
given
is
constraining the dynamics of Boy.
dynamics
particles
equation
Lowest order cancellation between the
0
/3t
a3Q/3v,
D(afo/3v)-Ffo.
friction
D
coefficients,
global
0
)
of
clumps
or
discrete
a Fokker-Planck
also satisfies
fo
helicity
situation is analogous
The
(f
magnetic
where Q is a (x,v) phase space current
and
are
F
e.g.,
velocity
space
F-<Et'>/fo where
E
diffusion
is
the
electric
field from Poisson's equation and f is the clump or discrete particle
of f.
Because of global momentum conservation,
(Q-0) to lowest
order
in
the
resonance
35
width
and
part
these D and F terms cancel
Av.
To
next
order ,
Q--
D(Av) 2 9 3 f
3
/v
between like
known effect of collisions
is the well
This
, thus giving a fourth order diffusion process for fC.
In the MHD clump model,
conservation
(98)
the
playing
the role of
plays
constraining
role
Q,
(or clumps).
particles
helicity
with magnetic
to
analogous
momentum
conservation.
Unlike the stochastic,
have
not
been
able
to
renormalization
techniques
conservation
Eoz.
equation
(41)
in
The
and
but
static magnetic field
evaluate
and
show
static
yields
the
case of Sec.
the
correlation
that
it
case
ensures
involves
helicity
the
<(xL>
conserving
with
magnetic
rather
II,
our
helicity
simple
results
we
scalar
(71)
in
straightforward way.
The dynamic case with the full vector MHD equations is
much
to
more
difficult
treat.
Preserving
correlation is particularly difficult.
the
properties
of
the
Two points about the calculation are
Unlike kinetic dynamo models,' the evaluation of
worth mentioning, however.
F in the clump model requires self-consistent
flow fields V.
vector
rather than arbitrarily given
From momentum balance one obtains
vSL
t dt' J [x(t'),t']
=2
x
B[x(t'),t']
(99)
so that
<VSC x B>
where the subscript t'
with
the
obtained
so-calfed
B
=
-S
tdt'<(B.B) Jz >
means evaluation of time and orbits at t-t'.
term
of
kinetic
from the time integration
(100) yields a contribution to Eoz
<6Vx6B>z
(100)
of
dynamo
theory
(the
Faraday's law in a
a
term
given V
Along
being
field),
f
tdt' <(6V 2 + S2 6B2 )z >
(101)
The coherent part of this response, coming from Jz=Joz, gives the result Eoz
36
=
DJOZ-
to the incoherent
The response due to Jzjz contributes
part of -Eoz, i.e., the F-<V
x
or
clump
This contribution depends on
B> term in (98).
the three point correlation <6B26JZ> and is sensitive to the distribution of
fluctuation amplitudes.
is equally populated with
If the turbulence
6Jz>o
and 6Jz<O fluctuations,
this contribution to the a-effect will vanish.
If,
however,
are more
the
holes
(6Jz<O)
prevalent
because
of
reflextion symmetry of the spectrum will be broken.'
nonzero,
and the B term in
will
However,
(17)).
at
growth,
F and a will
a being negative and F--a being positive.
is countered by the diffusion of the field lines
their
this effect
Of course,
(i.e.,
then be
the D term in (14)
the edge of a confined
be small and Eoz can expected to become negative
as
plasma,
holes
(bubbles)
intermittently develop.
Though interesting, these implications of (101)
not
(101)
satisfactory
helicity.
since
does
not
ensure
conservation
Clearly, additional contributions to <6V
we have not
been
able
to identify or
calculate
x
Joz
of
are
magnetic
6B> are warranted, but
them from
the
vector
MHD
equations.
What
conserving
equations.
flux
we
have
Eoz
been
from
i,
to
do
a renormalized
The Strauss
function,
able
equations2 6
is
to
version
of the
are scalar
the stream function,
and
calculate
the
reduced
equations
$,
where
for
helicity
net,
MHD
(Strauss)
the
V=Vxof.
poloidal
They
are,
neglecting collisional dissipation,
-
at
S-B.V$
S-2
where U = -V2
aU
(102)
B.VJ z
is the vorticity.
obtained from these equations
(109)
A helicity conserving form for Eoz can be
since
the ensemble
written as
37
average
of
(102)
can
be
a
(104)
af<> - V.<6B 16$>
and,
with
Eoz--3<$>/3t,
constraint
(12).
will
The renormalization
<6B16$>
to be the current
Strauss
equations
obtain
automatically
are
only
valid
for
to
(14)
and
diffusion
"tokamak
However,
mainly concerned with in this paper.
leading
magnetic
merely determines
of a fourth order
is only valid in this limit.
argument
satisfy the
the
structure
process.
ordering",
of
Since the
the
result
we
this is the limit that we are
The result agrees
the
helicity
stochastic
with the
magnetic
field
physical
transport
calculation of Sec. II.
Since we are interested in the self-consistent evaluation of (104),
we
will need the response 6$c that is coherent with 6B1 as well as the part 6B
that is coherent with
6$.
(Kinetic or
quasilinear
only one of the responses, usually 6B ).
dynamo models focus on
For this purpose, we rewrite (104)
as
S< > - V.<6B 6$c> Next,
>
(105)
we again distinguish between the wave-like
(clump)
parts
dimensionless
of the
field.
units)
is
written
as
the right-hand-side
brought
where
term is small near
for
to the
6
$c.
(6 Bw
left-hand
side
of
of
and,
and
For 6Jw
z - S_'
6U, the equation for 6Uc is
38
6V in
(102)
and
can
then
be
clump
part
of
the
therefore,
when written
(103),
1
(103)
has
in terms
the
subtracting out the forward or backward Alfven wave as in Sec.
1.
S
=±
(and therefore 6$c)
the resonance
The 6Jwz term,
of
is the
J
The last term here gives 6Uc
neglected as a source
be
of
B.VJ z + B.6Jwz + 6B.VJO,
-- zz
The first
seek.
and non-wave-like
- ±S16U in terms of the fields
6 Jw
z
The fluctuating part
current density.
(Alfven)
The Alfven wave response
(103).
can
V 6
that we
can
be
of 6U,
effect
of
IIC of Ref.
-2 8
- S L.V) 6U
(.
S
c
- 6B.VJOZ
(106)
The renormalization converts the 6L.V term into a diffusion operator
Sec. II above.
Therefore,
as in
6$c follows from
2k
-
S
iSk.B +[Y+(T
(107)
L -1
) ]
where we've approximated the inverted propagator as a Lorentzian and set 6B
-k since we are only interested in the transport due to the clump part of
the
field.
fashion.
The
governing
equation
For the fully stochastic
the static version of (103),
6J=
for
case
6d
0
can
(Yt<<1),
i.e., (40).
be
obtained
6Jz can
in
be obtained
08)
where 6B is that part of 6B that is driven by the clump part of 6V,
is the Fourier transform of
6
from
Therefore,
6B k'oz
k.B
=
similar
Jc.
z. We obtain 69 from (102),
or,
and 6JO
equivalently,
its curl
+ V.V) B,
where we write
response,
B.V6V as
B.V6Vw
S6B , (109)
6Vw
(
B.VV
(109)
+ B.VV.
-- SL.V)
6Bk
-
Alfven wave
becomes
dB - B.VV
(110)
Renormalization and time inversion of (110)
-
Taking the forward
ik.B Vk
L -1
iSk.B +[Y+(T
then give
(111)
)
so that
.v
5
6 Jk
-
-k -
J
oz
(112)
iSk.B +[Y+(Tr)
-0
]
0
39
Since V2(*,$) -- (Jz,
U), the equations (107) and (112) give
V22
2
)(~S
-k
2
2
x-
(
y
where we have set rL
Note
that,
(111)
and (112)),
if we
B§ ).VJ(13
oz(
2 k
(~k'
k) -6f
iSk.B +(Y+T
-
-1
)
since, in the strong N/L coupling limit,
TO
had subtracted
out
the
backward
S would change sign in (113).
Alfven wave
However,
L TN.
(SL+-SN
in
this change would
not matter since we will only need the real part of the inverse propagator.
Of course,
this occurs
leads to clump decay.
in
(113)
with
because Alfven wave emission of either polarization
For simplicity,
k-(Axk-
2
we approximate the Laplacian operator
+ky2, where Axk is the resonance width for mode k.
Then, inverting (113) and substituting 6$c and
a<>
= V.D.V J
D - S2
fdk
J (2Sr 2
6 *c
into (105) gives
(114)
where
~(2,r)--
=
This is just
(kyAx<<1),
k1
(32)
-
since,
for
2
+ S-2-Y->
-- k G k k 1
--
field
line steps
D-D(Ax) 2 .
Ax so that
Note
(115)
mainly in the
also that
x direction
in the
strong N/L
limit, the total energy correlation function in (115) is the same as <6L6L>
The magnetic helicity
or <6N6N>k.
The form of (114)
is conserved
since
(114)
has been shown by Boozer to follow from general
properties of the energy and magnetic helicity invariants.
Strauss
has
obtained
an
equation
similar
(given) quasi-linear spectrum of tearing modes.
and
their
evolution
Equation (114)
because
MHD
fluctuations
ensures
are
very different
from
differs from the nonresonant,
clumps
whose
are
nonlinear,
dynamics
to
22
(114)
40
the
transport
20
for
an
assumed
However, the fluctuations
the
clumps
considered
here.
unrenormalized model of Strauss
self-consistently
conserve
(12).
magnetic
generated
helicity
in
resonant
both
the
growing
and
(Y>O)
effect
on
MHD
steady
clump
(Y=O)
states.
evolution.
For
These
example,
features
clump
the
overcome
nonlinear
lines,
the
i.e.,
turbulent
decay
in (5).
R>1
mixing
rate
rate
due
(114)
to
If VJJoz vanished,
profound
does
In the case of
must
the
a
turbulence
approach the Taylor State by the vanishing of V J
fluctuations,
have
be
large
stochastic
clump
enough
magnetic
no new fluctuations
not
to
field
would
be
produced and any existing fluctuations (including, apparently, those of Ref.
22)
would decay away due to the field line stochasticity.
level
The turbulence
would decrease to zero in a time on the order of the Lyapunov time.
Of course, the mixing and the continuous generation of clump fluctuations is
maintained in the clump model
by the maintenance
of the Joz profile, e.g.,
with an applied Eoz.
The
use of
Eoz=DJoz in the
the
helicity conserving
form
clump source term 2EozJoz still
-D(Ax) 2
E
produces
2
z instead
(3)
and
(5),
of
but
with Ac now given by (see (112) of Ref. 1).
1 dk
f -f y
1
c
-2
ReA'+22k I
k
y
o (ReA +2Ik yI) +X
where Ak is to be evaluated at
k.B 0
helicity conserving form of Ac which,
= 0.
J
k2 A(k ) -
(116)
k
Equation
when used in
(3)
(116)
and
helicity conserving growth rate for MHD clump fluctuations.
41
ozis the magnetic
(5),
gives the
IV.
Rather than
clump model
are
STEADY STATE CLUMP SPECTRUM
being assumed or given,
determined self-consistently
the mean shear.
The mean-square
the
spectrum
equation
conservation (DB' = DJ
oy
limit.
+ -
fluctuations
from the turbulent
fluctuation
calculated for the case of driven,
modify
the nonlinear
)
of
(D/k .1.
2)J)
IOZ
mixing of
amplitude or spectrum
steady state turbulence
(108
in the
Ref.
1 for
can be
as follows.
magnetic
We
helicity
and assume the strong N/L coupling
Then, expressing DL in terms of the spectrum, we have
2
<a
(0) >
k
-27B
=
A'
k
-
where A(ky)
oy
16(k.B )A(k )
-o
y
+ 21k I3
y
J
oz
d
(
dk'
J
oz
-
k 2 <6i2 (0)>
y ) 2k
k'
-1
1
iSk'.B +T0
2
(27T)
(117)
is given by (110) of Ref 1 and we have again approximated ReGk
by a Lorentzian.
Setting (3)
equal to unity and using (116),
the solution
to the integral equation (117) is
J"
<6 2(0)>k=M(x)6(k.Bo)
(-
ReA'+2|kjI
)
Y
lIA+2|k II
) [A(k
oz
where M(x)
is arbitrary
if
M =
1
and
is
zero
(118)
otherwise.
(118)
is
the
steady state clump spectrum.
The
square
fluctuations
brackets
in
are
(118)
driven
by
is
nonresonant
the
A
fluctuations (as for the tearing mode).
and
Jz.
free
The (JIOZ
constraint of magnetic helicity conservation.
localizes
for
field
minimum).
In
bending
reality,
is
the
broadened resonance function,
obtain 7.
in (90),
maximum
6(k.B
0
)
(decay
factor
energy
in
ReA'k
source
for
the
OZ) factor is due to the
The delta function in
the fluctuations at the mode rational
line
The
(118)
surface where the tendency
by
Alfven
wave
singularity
should
be
emission
replaced
is
by
a
since the unperturbed field line orbit used to
and producing the 6(k.B 0 ) in (117)
42
(see (109)
of Ref.
1)
should be the nonlinear trajectory.
The A(k ), coming from an
x. integral
is a measure of the clump energy.
of Tc,
The MHD clump spectrum
(1118)
is similar
to that
The steady state, mean-square electric potential
for Vlasov
spectrum for Vlasov clumps
has been calculated without momentum constraints in Ref. 32.
result
by
source
term,
the
factor
the
(-ImEi/ImEe)
electron
nonlinearity) proportional
a
to ensure
Vlasov
clump
clumps.
momentum
spectrum
is
Modifying that
conserving
clump
(neglecting
ion
to
e()2
ImE(
()
Imek
k2
k
k
(119)
k lEki
where
e
is the electron dielectric function
for mode
k.
(118) and
can be cast into an even more similar form by using the model for A
in (26) of Ref. 2.
(-Jo
oz) ReA1
Note
that momentum
requires
ImEi
With ReAk
(k.B
in (119) resembles the Ime
conservation
ImEe <
0 for Vlasov clump
difference
factor
(119)
in
(118).
because it
equation.
between
(118)
3f /3v driving terms in (119).
the clump source
instability.
case,
to an
(118)
by
Gk and
produces,
because
term and thus
Similarly,
magnetic
< 0 for MHD clump instability.
(119)
is
the
A delta function localization
The
delta
function
resonance
factor
does not
appear
in
gets integrated over by the velocity integral in Poisson's
The lack of a corresponding integral
delta function in
(118).
amplitude
This difference
dependent
integrating
of 6(k.B
0
r factor is just what is left
in (3).
and
given
the free energy driving term
-
constrains
helicity conservation requires JOZ" ReA
main
~J
1)2
(119)
As a result,
)
over
in
also leads,
instability
k
to
(118),
in Ampere's law leaves
unlike in the Vlasov
threshold,
obtain
the
the (Y+T~')~'
i.e.,
diffusion
43
multiplying
coefficient
factor in (2).
in Gk when k.BO=0 and what gives R
the MHD clump turbulence
the
-
Tr
is strongly resonant.
This
-
Xd 1
The
2
MHD energy spectrum <6B 2 +S
rational surfaces,
spectrum
<E 2 >kw
One
peaks.
resonant
charge
i.e.,
for
must
density
one
of
due to clumps will peak only at the mode
k. B=0.
k's where
look
structure
6V2 >>
1
dimensional
to
the
spectra
in
Vlasov
clumps
space
phase
Vlasov clump
By contrast, the electric field
density
fluctuations.
Poisson's
does
reveal
correlation
The
equation
not
electric
integrate
such
for
the
field
and
over
these
resonances.
Multiplying
(118)
by Gk and integrating over
k gives the steady state
diffusion coefficient
2
D
t0 S2 M(x)(-
=
Though M(x)
can
be
2ReA
A(k )
y
y
dk
2
(27)
arbitrary,
D(x)
limits on M(x).
J"1
O)
ozky
oz
the
2
physical
must be a smooth,
well
+,21k
parameters
of
the
compared
overlapping
to
T0
.
will
This
resonances,
roughly equal amplitude modes.
i.e.,
a
also limited.
behaved function of x so that
(S
ky
the diffusion
Further,
J
Clearly,
wide
occur
for
a
fully
stochastic
The amplitudes
Ax g)~ , where
constraint Tac
<
of
spectrum
of
Axsp is
is equivalent
the
spacial
to Ax<Axsp,
width of
the
D are
amplitude.
A
Also, since Te
, the Tac<To condition limits D to even smaller values.
Boy
spectrum
of M and
sets an upper limit on the
current density hole cannot be deeper than a vacuum bubble.
D~
set
D(x) will thus be relatively independent of
x in the unstable region and zero outside.
-
plasma
approximation demands that the fluctuation auto-correlation time
(Tac ) be short
strongly
(120)
2
A+21ky
the island overlap criterion is smoothly satisfied.
(Markovian)
|
1ky
Since Tac~
spectrum,
the
where Ax is the island width.
If we make the reasonable assumption that the unstable spectrum encompasses
a sizeable fraction of the current channel radius,
The
k y dependence
of
the
clump
spectrum
44
is,
we must have Ax < a.
in
general,
nontrivial.
For example,
and A(k y).
the dependence of Ak is sensitive to the mean current
Moreover,
the
k.
factor
necessary
in the
must be obtained from the selfconsistent solution
Ref. (1).
k
evaluation of
of (114b
) and
A(ky)
) of
(118
Because of these complexities, we have not determined a general
dependence
of
the
steady
state
spectrum.
hyperbolic tangent model current
profile of
fdx<6T2(x)>k
ky>k0 , scales
scales
profile
as
-
A(ky)
2
2
which,
(1-k /k ).
y
0
for
Again,
the
large
t
Eq.
k
localization of the clumps to y_ scales koy- l.
45
However,
(27)
as
of
k
if
Ref.
but,
dependence
we
2,
for
use
the
we obtain
k
<
reflects
ko
the
V.
CLUMP DERIVATION OF Jo=
a
In Sec. IC, we obtained the Taylor state Jo
0 =iio as a solution to (24).
There,
we
suggested
(24)
as
While we have derived (20)
generalization
would derive
to
(24)
clump dynamics.
and,
(20),
in Sec.
vector
III,
by including both
current
and (24)
we would like
to
and
of
(20).
appears to be a reasonable
derive
Bz(x)
generalization
(24)
also.
By(x) shear
Ideally one
(86)
in
for
We would then proceed to derive the generalization of
upon integration,
However,
the
we have
not
set the
been
generlized i
able
to
derive
(93),
equal to one and obtain
(24)
in
this
way.
the
(24).
While
the
assumption of tokamak ordering greatly simplified the calculation of R from
(86),
the
general
calculation
is much
more
difficult.
For
example,
the
Bz (x)*o effects will contribute to both the mixing term for self-consistent
clump generation
B
and
and to the propagators
for
clump decay.
These additional
vector contributions must be evaluated in a way that the self-consistency
conservation
properties
Alternatively,
one might
spirit of Ref.
22.
of
the
imagine
However,
two
point
trying a
quasilinear
such calculations
consider only nonresonant diffusion effects.
propagators
in the diffusion coefficients
equations
are
maintained.
calculation
in
the
are not fully nonlinear
and
For example,
in Ref.
we note that the
22 are of
the form Y /W.
We recognize this as the well known Y/w2 structure of non-resonant diffusion
in quasilinear theory (here, wk
clump fluctuations,
resonance,
(86)
since,
For
at the
the turbulent mixing makes its largest contribution and the decay
quasilinear
of
k'Bo is the linear Alfven frequency).
the resonant contribution to D dominates,
by Alfven wave emission
diffusion
=
is minimal.
The nonlinear terms
approach are also crucial
coefficients in the nonlinear
would,
without their
resonance
neglected
to clump dynamics.
For example,
mixing term on the right
broadening factors,
in the
the
hand side
diverge
in the
(ideal) solution for the clump growth rate.
renormalization
resolves
this
k.B0 =0
amplitude dependent threshold (R=1)
growth of the instability.
I
parameter
matter
how small
terms
ordering
or
amplitude
are
also
they
Therefore,
small
amplitudes
analytical
calculation,
difficult
in
the
and
leads
to
the
and, for YT>l, the Y 2 hydrodynamic-like
the
in magnitude,
localization.
singularity
by the
It is the amplitude dependence of this threshold
determines
The nonlinear
(21)).
clump
that
The broadening provided
'dependence
important
will
while
for
of
(20)
and
since,
no
the resonances
of
the
assumption
is
simplifying
enough
to
a rigorous
calculation
appears
to be
general
case
where
(see
clump decay
dominate near
either
p
neither
of
allow
tokamak
tractable
prohibitively
assumption
is
made.
Unfortunately, such is the case for a rigorous derivation of (24).
Assuming some reasonable symmetry properties for the steady state clump
spectrum
and
calculation
and,
derivation
of
rectilinear,
vector
diffusion
when
(114)
slab
equation
dimensionalized
coefficients,
prohibitive,
is
valid
variables
any
For
our
final
coordinate
where
magnetic
derivation
argument
however.
However,
in
hybrid
physical
possible,
geometry.
a
based on
and analogy
simplicity,
result
system.
fields
We
are
with
we
(137)
also
direct
the
work
in
will
be
work
with
normalized
to
a
the
spacially averaged mean magnetic field go and lengths to the current channel
radius a (see Sec. IIA of Ref. 1 for details).
We begin with the generalization of
Eoz Joz to E*.
magnetic
= EozJoz
+
field B=VxA is now
EoJ oy.
given
the mean-square
In the
in terms
presence
of the
clump source
of Joz and Joy,
poloidal
=
(Az
p)
term
the
and
toroidal (Ay =
T) flux functions.
As with Eoz, the mean electric field Eoy
will
Fokker-Planck
(14),
have
the
discussed in the Introduction,
form
i.e.,
Eoy
=
DJOy
-
F
Boy.
As
this structure is a consequence of the self-
47
consistent nature of the clumps,
of the clump field.
the F term being due to the incoherent part
Because of magnetic helicity conservation,
the D and F
terms cancel to lowest order in the island width and lead to a fourth order
diffusion process of the form (114) or, more simply, (15).
(114)
for
E0 z =-3<$P >/3t
perturbation
6T,
i.e.,
analogous
fashion,
replaced
by
Eoy
6
the
'T
is due
so
to the
6Tp
that
=-3< T>/at
driven
contribution from poloidal
poloidal
D+DP
will
be of
coefficient
and toroidal
of
the
poloidal
(25)).
and toroidal
H is the
diffusion process.
(10)
or (114))
current
rather than (12)
the
DT.
clump
Therefore,
form
This
(114),
but
assumes
are
flux
in
the
an
with
that
D
the
independent,
contributions are additive.
Note
possibility that the mean parts
functions
are
correlated
for the fourth order
(i.e.,
as
in
here is required
Therefore,
and
poloidal)
in the general
preserves the magnetic helicity.
(Note that for
form here reduces
and the helicity is preserved via (12)
simplicity that,
(toroidal
since,
the full vector
that D is diagonal,
D.
flux
The factor B /B
tokamak ordering,
for
preclude
(114).
of
These Eoy and Eoz components combine to give an Eo of the form BO/B2
V.H, where
case,
does not
component
flux fluctuations
and produces a model where their diffusive
that this assumption
in
Recall that D in
in the
steady state,
to E
V.H
=
as before).
the turbulence
(see (83)
We also assume
is
isotropic
so
with the RR and g? components equal and each denoted by
with
J,,
.B /B0 , the
=
helicity
conserving,
two
point
source term for poloidally and toroidally shear driven clumps is
J11
B
-2
(6
2
12
where 5P is given by (115)
1
al
oz
BT~ V
12
2
1
1
(121)
oy
but with Gk replaced
by BoGk
(similarly for DT).
This additional BO factor in the D's provides for the correct magnetic field
normalization
of
Gk
in
the
general
case
(see
(29)).
Note
that
in
the
tokamak ordered case,
the
first
term
of
B0 = 2 Boz = :
(121)
gives
in dimensionless units,
the
previous
result
so
BP = DP and
calculated
from
the
Strauss equations.
replaces the right-hand-side of (97).
Equation (121)
Integration along
the two point orbits gives
-2 (-r 11p
-2
B
where
tj
surfaces
and TT
(ck
2 JZ
5P
12
1
oz
+ T
c
D12 V1 Joy
Since
the
essential
flux surfaces comes from their y_ separation,
<B->.V_
= B
evaluate <y2(t)> and
thus
put
given by (90).
essential
Bo z x_
TT
x_
D/3y_
.
The
T
correlation
so we,
between
therefore,
toroidal
poloidal
lifetime
of Ref.
flux surfaces
set expik.r_
propagator
point
in the two
clump
between
poloidal
we set expik.r_ = exp ikyy_
Its x_ integral is given by (109)
correlation
dependence,
(122)
are the clump lifetimes in the poloidal and toroidal flux
respectively.
DP12 and
2
TZ T
1.
T
used to
is
then
Similarly the
comes
= expik z_ in
T
in
from
their
z_
and put <B_>.V_
V/z_ in the two point propagator used to evaluate <zf(t)> and thus
Then, the
x
integral of the clump toroidal lifetime TT
(109) of Ref. 1, but with the replacements Boy
6(ky + kz Boz X+/Boy),
and A(k )
+
+
is given by
Bz,
x+/B
oz, 6(k z + ky Boy+oz
A(kz).
The clump correlation function now contains contributions
from both 6T
and
6
TT'
We again assume that the poloidal and toroidal clump fluctuations
can be treated independently so that the fluctuation correlation <5llp 6 TT> is
negligible.
For example, in the strong N/L coupling limit, this means that
< 6 N (1)
6N ( 2 )>
<5 p(1)
5T p (2)>
=
k2 <[6' (1)
<[6$P(1)
6T (2)>
6 p(2)
+
S-2
transformed nonlinear Newcomb equation (93)
49
-
k
5$p(1)
<6T
T(2)]>
6$ p(2)]>.
now becomes
The
where
Fourier
- k )
(
y
a2
=
where TP,
<6.F (1)6T (2)>
p
p
2 0
(T
+ (-
k,
- k )<6T (1)6T (2)>
3 2
F V Jo Z
+
z
T
Ct
V
Joy)
IVB of Ref.
1,
1<62 (0).>
'
A(k
S
V2
+ k
)B 6(k
z
B?
x /B0 z)
joz
J
y oy
J
2
+ A(k ) BT 6(k
z
6'(p)
equate
the
is kthe
y
discontinuity
discontinuities
case,
6' to
z OZ
B'
x /B ) V
+
y
JYiO
<6T p (1)
6' p (2)> k
k
+
in
that
the Newcomb equation
of
can
the
be written
poloidal and toroidal fluctuations are uncorrelated
x
x
k
= k2
y
6
(1)6$
p
(2)>
p
z
As in
1,
we
In
the
of the radial
We again assume that
(<64 p6ST>
T
Ref.
solution.
in terms
+ k2 <6$ (1)6$
k<
(124)
Newcomb
component of the magnetic field 6B =ad4 /ay - a6$T/;z.
<6B (1)6B (2)>
as
T (O)>k
z
k
0V
where
(123)
<62
6 (T)+21kz
-
y p k
0
If we integrate
we obtain
[6(p)+2Ik
general
(123)
is the Fourier transform of Tct and we have set D1= D since the
dominant x_ dependence comes from the Tct factors.
in Sec.
k
T
T
=
(2)>
k
0) so that
(125)
Then,
<6B
;- x (1)6B x(2)> k =yk
6 (T)<6T(1 )M6T(
+
But,
6k (p)<5
()<p (1)6$ p (2)>k
2
)>
(126)
it is also true that
ax
S6BI(1)
<6B (1) 6B (2)>
x
x
k<
6B(
. 6B
50
1
)
6B'(2)'
6B (2)6B (1)6(2)>
X
(127)
=
k
(6B (1)6B (2)>
<
x
x
(128)
k(18
Therefore, the discontinuity of the general Newcomb solution can be written
in terms of the 6' factors as
k2<6
A = 6(p)
2
2
2
k <62 (0)> +k 2z6p (0)>
k
-
k <62 (0)>k+k
clump fluctuation are
pk
y
poloidal
the
(129)
k
k(
-
that
<6 2(0)>
k
zT
and toroidal
a resonant
of
numbers
This seems to be
a
isotropic spectrum considered
Equation (129) then simplifies to
A'
(130)
wave
comparable in the steady state.
reasonable assumption for the fully developed,
here.
k
2 0)>
k2
assume
T
k z
p
y
+
We now
(0)>
=
6 ,(p)<6$P2 (0)>k+6 <(T)<6
(0)>k
k
T
kk
k
<6 p (0)>
+ <6c T (0)>
and a matching
the
of
then
solutions
(130)
allows
the
left-hand-side
of
(124) to be written as
-(A
Since
k
+2 1k
1) <62 (0)> - (A
k
p
and poloidal
the toroidal
break (124)
yY
+2|kz|) <62 (0)>
<5YT<zI)k
fluctuations
are
assumed
(131)
uncorrelated,
we
into its toroidal and poloidal components,
I
(k2
ReA' +21k 2
<6T 2 (0)>
A
SBo
k
(ReA+2
y2
)+
kBJ
2I6(k+k BIx+/B
z yoy+ oz
J oz
JOZ
and
IT
<6Y 2 (0)>
k
T
r
where
A
=
Im
-
Xo
SB
.We
V2J
ReA'+2k
l
(ReA +2k
+k BI x /B )
B6(k
DI
y z oz + oy )
2 2
+c df)
now form the diffusion coefficients
51
oy
(133)
oy
by multiplying
2 22
2
(132)
by
kz2 is
the generalization of k
(132)
and (133)
S2BOG kY/kd
2
Again,
III.
in Sec.
=
A
-2
(LX)-
+k2+
+
k
+
the delta functions in
function and give the Lyapunov time to.
will collapse the G
of Ref. 1,
Thus, similar to (111)
V2
ki
=
2
2
and (133) by S2BoG kz/kd where kd
(ST
-
B
0
-(ST
I oz(134)
J
oz
I oy
y_135
T
k B ) DI(135)
J
oy
IT= (S
B
D
B ) BPI
k
000
kB9
_
0 00
0
where
ReA +2Ik I
dk
I irk
as in (116),
and,
k
A(k
2
o (ReA +2Ik i) +X kd
2
is evaluated at k.B0 =0.
A
(136)
)
Of course,
is just 1=
(134)
for the poloidal driving term whereas (135) is the toroidal analogue.
zero,
Provided the D's are not
(134)
and
(135)
can be expressed
in
vector form as
2
+ P2 J = 0
-0
1-0
(137)
where
2 =
1
k
(ST
(138)
B )I
0
In
the
case
=
xd so that
T0=SkoyJoz
case,
Boz
T~=
Sk0 J
xd
constant
y
1
Boy,
2
~Ak xd and (137)
so that
relatively flat spectrum,
(137)
>>
p2=xd/I.
then
J
/Bo=Joz,
reduces to (21).
Then,
for
k0 B0 =koy,
In the general
a smooth,
p is constant and, with V'Jo = 0,
and
broad
and
the solution to
and Ampere's law is the force-free Taylor state
Jo
(139)
= 4i
52
where
(1140)
d /1
=
y2d 2
We can redefine I to make it dimensionless and of order unity by writing
I
= A
d
for the
where A' is a mean value of A
k
(141)
k2 + (Ax )-21
0I
o0
unstable modes
ko) and Ax0 is the island width for mode k0 .
=
overlap is more than satisfied in the steady state,
evaluated for
(ReA'
If we assume that island
then Ax0
-
(141)
xd and
can be rearranged and written as
2
I
d
2k xd
o2A'k
0 0
(142)
1) = (1-k xd 2
-
o d(14)
A real and positive solution for xd (and,
therefore,
only occurs when the
D)
threshold condition
(143)
y2 > 2A'k 0d
is
possible
(143)
Taylor
states
(i.e.,
to
steady
correspond
unstable.
is
plasma
the
Otherwise,
satisfied.
yi values),
the
turbulence.
clump
state
one with
Therefore,
of
p values
given
that
Notice
all
by
this
threshold behavior can be traced to the presence of the inverse squared step
size
in
brackets
conservation.
reasonable
=Py
J
Since Id-1,
parameters.
-
J 0 (pr)
to
i.e.,
(141),
in
(143)
the
of
Such y's also lead to Boz field reversal,
from
Therefore,
(141).
Much above threshold,
2
=
' k
2
helicity
magnetic
implies that y is an order one quantity for
occurs for Boz field reversed Taylor states.
i 2=2A'k 0 /Id.
constraint
steady state
since Boz
clump turbulence
At the threshold,
koxd = 1 and
koxd >>1 and
(144)
xd /d
or, in terms of the pinch parameter
(0 = lia/2 in dimensional units),
53
x
Since
xd
d
D
-
= 41
62/A'
d
as
(145)
0
(139)
<6B>1/3,
increasing current (6)
k2
with
increases
6Brms
that
implies
63
In the Taylor model where (25) is calculated as a final state, P enters
as a Lagrange multiplier
the dynamical
In the MHD clump model,
a constant.
and is, therefore,
p only becomes
to the final state is prescribed and
Such spectral
properties result from a large number of overlapping resonances.
for
required
the
diffusion
the
Inside
theory.
the MHD clump
used throughout
approximation
Markovian
the
of
application
strict
As we have
such conditions on the spectrum are
noted in the discussion following (120),
also
route
constant in space when
flat and fully stochastic.
the clump spectrum becomes broad,
principle
minimization)
(energy
variational
in a
spectrum
where these conditions are satisfied, D and p will be relatively independent
the individual
Near the edges of the spectrum,
of position.
mode structures
become important and will give a spacial dependence to D and p.
more restrictive
namely,
y
assumption
the
in
Taylor
flux
by each
notes
on each
flux
i.e.,
conducting wall
will
not
be
surrounding
turbulent
preserved.
the
depends
p
plasma
boundary.
the
in the Taylor model--the
that
by
yB
is
However,
at
surface
boundary,
the
assuming
there
is
p associated with the
The variational principle then leads to J = pB, where now,
54
in
magnetic
yi in J =
Therefore,
plasma
a
in an MHD plasma,
flux
is invariant.
the
built
position.
on
relaxation
Only
that magnetic helicity is only conserved at
only one Lagrange multiplier
assumes
the parameter
flux surface,
surface,
is also
p
one
if
There,
that during "violent",
surfaces
constancy of
the
model.
Taylor
helicity is invariant
determined
Actually,
constant.
=
makes
clump model,
the
than
state
final
the
about
prediction
model
the* Taylor
that
conclude
one might
throught,
At first
p is a
constant.
to
The constancy of y in the Taylor model can,
destruction
the
Of course,
plasma.
of
flux
surfaces
flux surfaces
during
be traced
therefore,
turbulent
relaxation
the
of
can only be "violently" destroyed over a
significant portion of the plasma crosssection if their resonances strongly
overlap.
wide,
Strong
fully
mode
stochastic
coupling
and mixing will
spectrum
will
develop.
ensue
The
and
dynamically,
resulting
stochastic
field lines will transport current throughout the plasma crosssection,
"homogenizing" the turbulence.
models
assume
the
constancy
Therefore,
of
p.
thus
both the Taylor and the MHD clump
Moreover,
assumption is essentially the same for each model.
55
a
the
justification
for
the
IV.
INTERMITTENCY
An outstanding issue in the model concerns the complete role of finite
amplitude
island
nonlinear
interaction
exponential
the
equilibria
divergence
initial state,
with
term
time.
of
i.e.,
This
lack
in
T(1,2)
tendency
toward
turbulence.
describes,
neighboring
initial
of
In
in
field line
correlations,
predictability,
because of the nonlinearity,
this
the
equation
its
simplest
trajectories.
will
however,
(8),
the
form,
the
Memory
be lost
exponentially
will
lessened
be
if,
coherent island structures tend to form.
fluctuation
self-organization,
of
the
net
With
mean-square
fluctuation decay rate T(1,2) will be reduced, and fluctuation intermittency
will
tend
to
develop.
Such
a
tendency
toward
intermittency
would
qualitatively alter the conceptual picture of the turbulence.
The intermittency issue is of importance
we can look there for some guidance.
graphically
show
the development
can
of hole
the
plasma
intermittency
in
that maximize
to
subject
the Maxwell-Boltzman
dynamical
decaying3 3
It has been proposed by Dupree
be partially understood conceptually as
develop fluctuations
and
Computer simulations of Vlasov plasma
driven 9 (i.e., unstable) turbulence.
these features
in Vlasov hole turbulence,
constraints
of
that
21
the tendency to
entropy
energy
and
(fznf)
and
of
momentum
conservation. The variational principle yields a fluctuation amplitude of
Av
6f ~
(146)
-~2
vth
where
vth,e,
fluctuation
and
are
velocity width.
the entropy,
we can
space density,
Vlasov
Av
plasma,
f2,
thermal
Rather
also obtain
velocity,
dielectric
than assuming a
(146)
function,
particular
f2 is a dynamical
invariant
56
measure
by minimizing the mean-square
subject to momentum and energy conservation.
(it
is constant
and
of
phase
While in a
along
particle
orbits),
it
will,
electric
fields
in a
in the
plasma
sense,
cascade
to the
be
dissipated
by
turbulent
to high wave numbers.
Such
decay hypothesis"
selective
a
of
(see below).
fluid turbulence
(146)
Equation
and
is similar
principle
minimization
coarse-grained
describes
a self-consistent,
bound
phase space
Using Poisson's equation for the hole
hole structure.
density
$,
potential
(146)
yields the trapping condition
Av
-
(eo/m) 1 /2
for a virialized equilibrium hole
(147)
(e/m is the charge to mass ratio).
Hole
material tends to self-attract and coalesce into new holes in much the same
way as
a
self-gravitating
An
formation.
analysis
fluid.' a
of
This
collisions
is
a
between
further
holes
impetus
shows
for
hole
a
dual
that
cascade occurs with 6f tending toward small scales and energy toward large
scales.
x and v)
of
Given that the Vlasov fluid is governed by a two dimensional
21
phase space incompressible flow,
two
dimensional
density
plays
the
Navier
role
probable
state
density,
fluctuations
structures
in
for
the
of
Stokes
turbulence.
fluid
vorticity.
distribution
can
a turbulent
be
self-organization
intermittent
of
expected
plasma.
"Vlasov vorticity" concentrations
The
Because
energy,
to
Another
is analogous to that
Vlasov
(146)
momentum
self-organize
reason
for the
phase
is
and
into
the
space
most
phase space
such
hole
development
of
is that an isolated hole can be unstable
small free
to growth for-arbitarily
this cascade
(in
energies.
These tendencies for hole
and growth imply a turbulent state that is composed of an
distribution
of
colliding,
theory deals with this turbulence,
growing
holes.
strictly speaking,
The
clump/hole
only in its extremes.
In the intermittent case of isolated coherent holes, one deals with the hole
model
and
considers
self-consistent
57
hole
structure,
dynamics,
and
-
growth.
In the opposite extreme where the packing fraction of the holes
21,34
in phase space is one half (i.e.,
deals
with the clump model
mode
coupling,
and
the distribution of 6f is Gaussian),
and
considers
of
the
growth
two point
mean-square
one
correlation functions,
fluctuation
level.'s
The
tendency for hole formation and attraction is modeled phenomenologically in
the
statistical
exponentially
equations.
diverging
simulations",
a
by
The
orbits
T(1,2)
is
term
reduced,
in
factor
on
multiplicative
describing
accord
the
by
decay
with
computer
order
of
1/3.
Interestingly, in the intermediate regime where the region of applicability
of
two
models
overlap,
the
clump
and
hole
models
agree.
For
example,
neglecting mode coupling (hole-hole collisions), the fluctuation growth rate
of the clump model
of
coming
the
agrees with that of the isolated hole model.
clump/hole
theory
quantitatively the development
statistical
However,
correlation
function
develop.
growth arguments,
Probability
in
its
inability
to
predict
of hole formation and intermittency from the
the theory can predict,
isolated hole
lies
The short
equations
describing
the
turbulence.
based on fluctuation self-organization
that hole formation
arguments
have
also
and
been
intermittency
used
to
and
will
calculate
approximately under what conditions intermittency will occur.
Intermittency
in fluid
turbulence
is
well
known.
Concentrations
of
fluid vorticity have been observed in computer simulations of decaying fluid
turbulence.
As
concentrations,
called modons,
encounters
(i.e.,
the
nonGaussian
in
with other
modon
decaying
modons,
"packing
statistics.
geophysical flows. 3
7 3
,
Vlasov
turbulence,
develop spontaneously and,
persist.
fraction")
Modons
also
The
space
decreases
develop
with
in
vorticity
except for
occupied
Their formation is important
58
the
by
time,
the modons
leading
apparently
to the
brief
to
driven,
predictability
of
such
flows.
One
route
taken
for
the
integration
of
modons
turbulence is similar to that for holes in Vlasov turbulence.
into
the
A variational
or minimization principle is invoked, based on the so-called selective decay
hypothesis.
Of all
the inviscid
dynamical
one that viscously decays the fastest
invariants,
the
(to high wave numbers)
"dissipated"
is selected to
be minimized subject to the constancy of the remaining, more slowly decaying
("rugged") invariants.
The
stability
and
The variational
interactions
principle yields modon solutions."
between
modons
are
the
subject
of
active
research.
Though
current
density
concentrations
(intermittency)
have
been
observed in simulations of decaying, homogeneous MHD turbulence "', the issue
of isolated island formation and intermittency is less clear in shear driven
MHD turbulence.
However,
intermittency.
there are suggestive
One is the existence of most
with the Vlasov case,
one might
parallels
with
Vlasov hole
probable states.
calculate such a
state
energy subject to the constancy of magnetic helicity.
In parallel
by minimizing the
Recall that it
is the
energy--being the MHD analogue of the Vlasov phase space density--that mixes
and
cascades
grained sense,
selective
to
high
wave
the energy
decay
numbers
during
is dissipated.
hypothesis
(see
turbulent
This
mixing.
is another
Montgomery's
review).
In
a
variant
coarse
of
Performing
variational principle, one obtains a result known as Woltjer's theorem,
-
VB=
1
P9
the
the
4
(148)
where W is the Lagrange multiplier for the magnetic helicity.
Coupled with
Ampere's law, (148) is just the force-free (JxB = 0), Taylor state (J = pJB).
The global or mean field part of (148)
is (25).
Ref.
fluctuation
1,
one
localized
or
resonant
As we showed explicitly
solution
consistent force-free state equations, J = VxB and JxB = 0,
59
to
the
in
self-
is the magnetic
-
island or current
hole.
here is that,
will
fluctuations
shear,
with magnetic
plasma
The suggestion
described by
These localized fluctuations
(57)
of the Vlasov holes described
The magnetic
island
coalescence
self-organization
T(1,2)
term in
(8)
into
will
Further
organization.
structures.
island
be reduced
by
(146)
analogy
(magnetic
(23)
and
and
(147).
with the
self-
decay
We,
form
the
fluctuation selfcomes
expect that coherent,
therefore,
will
structures
of
effect
vorticity concentrations
impetus for magnetic
island)
The
by this tendency for
from the growth of the current holes.
current
into
Jean's instability of Vlasov holes--would also tend to promote
gravitating,
this
instability4'--in
MHD
to self-organize
tend
current density holes.
are the analogues
in a turbulent
intermittently
out
of MHD
clump turbulence.
Without
the
intermittent
turbulence would have
formation
a symmetric
and
growth of
distribution
6Jz<0 and 6Jz>0 values would be equally likely.
breaks this symmetry.
growth of the holes (6Jz<O)
6
of
current
holes,
Jz fluctuations,
Preferential
the
i.e.,
formation and
Such symmetry breaking is
required of a turbulent dynamo, though it is the statistics of the flow (5V)
field that
assumed,
as
are broken in
conventional
in a kinetic model,
generated via Ampere's law.
broken.
MHD clump turbulence
Rather than
is self-consistently
Ampere's law determines both the hole structure
and the free energy source for
self-consistently
kinetic dynamo models.'
The
hole growth.
breaking
The
occurs
6
Jz+-6Jz
symmetry is thus
spontaneously
as
parallel
current fluctuations 6Jz <0 intermittently coalesce into self-consistent hole
structures,
and dynamically as the holes
via MHD clump instability
(recall that
(once formed)
preferentially
the 6Jz >0 fluctuations
decay).
grow
An
analogous situation occurs in the Vlasov plasma where the 6f-+-6f symmetry of
weak turbulence
is self-consistently broken
60
via
Poisson's equation.
Phase
space density holes
(6f<O)
spontaneously
coalesce
(Jean's instability)
grow (hole instability) while 6f>O fluctuations decay.
the
spontaneous
generation
of
nonperturbative,
also occurs in models of supeconductivity
theory (Higgs field).
phase
transition.
42
The
-
Symmetry breaking by
self-consistent
(Cooper
and
structures
pairs) and quantum field
There, the symmetry breaking is interpreted as a
condensation
of
holes
out
of
Gaussian
background
turbulence in a Vlasov plasma can also be thought of as a phase transition.
An interpretation of the MHD clump instability as a phase transition to the
Taylor state is discussed in the Appendix.
61
ACKNOWLEDGEMENT
I would like to thank J.B. Taylor, I. Hutchinson, A. Boozer, H. Strauss
and
T.
Chang
for
stimulating
conversations,
and
T.H.
Dupree
and
I.
Hutchinson for a critical reading of the manuscript.
This work was supported
by the
Department
of Energy,
Office of Naval
Research, National Science Foundation and the Air Force Office of Scientific
Research.
62
APPENDIX
In this Appendix,
via MHD clump
is
/B0
o'.
effect
in
a
we suggest that the relaxation
instability can
the
critical
be
For
point.
superconducting,
current
magnetization in a ferromagnet.
alternative
viewed as a
view of the instability.
phase transition where
analogy,
we
carrying
wire
At least,
to the Taylor
and
the
uc
Meissner
the
consider
state
spontaneous
the discussion below presents an
At best,
the results suggest the MHD
clump instability as an example of a dynamical model for phase transitions.
The salient
self-consistent,
features
of
collective
superconductivity
interaction
are
of the
well
known.2-"
electrons
with the
The
lattice
ions in a superconductor produces an attractive force between the electrons.
For
small
force
electron
overcomes
energies
the
normal
(i.e.,
low
Coulomb
repulsion
together into so-called Cooper pairs.
to
penetrate
into
the
temperatures,
T),
and
this
binds
attractive
the
electrons
When an imposed magnetic field tries
superconductor,
it
induces
a
macroscopic
flow
of
Cooper pair current given by
(Al)
J= -K2A
where
A
is
the
vector
potential
and
K2
is
a
positive
diamagnetic current in turn self-consistently generates
magnetic field which tends to cancel the imposed field.
constant.
This
(via Ampere's law)
a
The magnetic field
satisfies
V2B = K2B
so that the
field only
superconductor.
penetrates
This expulsion of
(A2)
a
characteristic
distance
flux from a superconductor
the Meissner effect.
63
K~1
into the
is known as .
these superconductivity effects only operate below a
As implied above,
specific critical temperature
(Tc).
TC,
transition
at
T=Tc.
Above
fluctuate
with
no
preferred
The metal
the
However,
phase
currents
diamagnetic
microscopic
orientation.
a
is said to undergo
this
T<TC,
for
orientation symmetry is broken as the currents tend to line up to yield the
Such
(Al).
field
mean
nonzero
macroscopic,
"long
range
is
ordering"
strongest when the energy of the collective electron motion is minimum, i.e.
When T=O, all the current fluctuations line up in the
for the ground state.
same direction.
In such a state of vanishing thermal motion,
current flow is
zero--a well
known
property
resistance to
of superconductors.
A similar
symmetry breaking and long range ordering also occurs in a ferromagnet where
spin alignment leads to a net magnetization (M) of
[1-( T -) 2 ]
M(T) = M(O)
(A3)
c
for T<Tc and M = 0 for
critical
theoretically
point
The bracketed expression in (A3)
Tc.
measures
breaking and long range ordering of the spins below
the amount of symmetry
the
T >
T=Tc.
Such
phase
transitions
can
be
described
mean field model in which the nonzero
by the Ginzberg-Landau
magnetization of the lowest energy state is obtained by minimizing the Gibbs
free energy subject to constant M.
are supported
ordered
field
mechanically.
by
the
into
These
so-called
the
are
The conclusions of the mean field model
the
theory where
BCS
state
ground
equilibrium
has
been
thermodynamic
or
condensation
of
the
calculated
quantum
statistical
models.
They do not address the dynamical route of the phase transition.
As in
Consider now MHD clump instability in a current carrying plasma.
Section
IV,
we imagine that the instability develops from background noise
where current fluctuations
A'>0 and R>1,
current
hole
are Gaussianly distributed.
(Jz<0)
fluctuations
64
will
As we've
seen,
preferentially
for
grow--
thus breaking the 6Jz
~
A'>O, the currents
(43)
induced in response to the magnetic fields
( 4 3 ))
fluctuations,
of the
law,
hole
z symmetry of the pre-instability
JZ,
this is a paramagnetic effect
Jz fluctuation would develop
IA of Ref.
(self
1,
the
binding)
resonances)
the
(see (64)
the
magnetic
variation
in
forms
fluctuations
occurs
stochastic decay.
further.
- (67)).
field
the
lines
field
(6B
in
In isolation, such a
As discussed in Sec.
balance
line
For
From Ampere's
as the self-consistent
environment of many overlapping resonances,
field line stochasticity.
iz
into a magnetic island.
island structure
of
reinforce
phase.
(near
pitch.
trapping
the
shear
However,
in
an
an island will be dissipated by
In the MHD clump instability, net growth of hole
as
the
regeneration
of
new
This occurs when R>1, where
R is
holes
overcomes
given
by (33).
the
Here,
the stochastic decay plays the role of decay by thermal agitation (T) of the
superconductivity
case.
In
the
steady
state,
the
fluctuation
level
is
nonvanishing and, with R = 1, (33) reduces to (137), or with Ampere's law,
2 B0(0n)
V722B 0= - p2B
Unlike
(rather
(A2),
(A4) has
than
fluctuations
a minus
diamagnetic)
sustain
superconductivity
the
case,
( 4
sign--a
effect
field
by
sign
in
dynamo
action.
the self-organization
(A4) is the Taylor state (25),
i.e.,
constant
magnetic
helicity.
It
symmetry
has
broken
to form
reflects
where,
system relaxes to its lowest energy state.
been
that
the
steady
In
of 'the
As we've seen,
a
turbulent
a macroscopic
steady
but
paramagnetic
state,
analogy
fields
the state of lowest
is
a
clump
with
occurs
the
as the
the solution to
energy subject
state
turbulent
where
to
the
fluctuation
level of current holes.
A measure of the strength of the phase transition to the steady state
is
the
growth
rate of
the
fluctuations.
65
In
the
fully
stochastic
case
(YT<1),
gives
(i4)
- - 1 /2
- R
)
11
'
(A5)
which, when combined with (34), can also be written as
Y
[1
(,)2]
-
(A6)
2T
= J 0 *o/B
where p.
For
the
y>ljc,
.
The critical
plasma
is
point for the phase transition is y=ljc'
stable--no
current
holes
grow.
For
p<,c
the
symmetry is broken and current holes preferentially self-organize and grow.
This
is
analogous
to
the
of
development
long
range
order
in
the
superconducting state.
An
analogy
relating (A6)
with the
"order
parameter"
to the steady-state
background magnetic noise
level,
M
(see
(A3))
can
hole fluctuation level.
then the
fluctuation
be made
by
If <6B > is the
amplitude resulting
from MHD clump instability will be given at time t by
<6B 2 > = <6B 2> exp
dt'
Y
(A7)
0
where Y is given by (A6).
Sec.
If the plasma is driven,
as we have discussed in
I(C,D), the fluctuation level will grow and then saturate
level.
Then,
Let
the time
the current
for
the
steady
state
hole fluctuation level,
to
AB
2
develop
, will
be
at a finite
denoted
by t .
be given approximately
by
2
2
AB= <B >Yt
where AB 2 = <6B 2 (t ) >-
<6B >.
AB2 (0)
AB2
(A8)
Then, (A6) and (A8) give
-(E
)2]
(A9)
1c
for
p
<
parameters
pc
p
and
and
AB2
yic
=
0
play
otherwise.
the
roles
66
Here,
of
T
AB2 (0)
and
Tc
U
(5Bb>
tR/T.
respectively
in
The
the
superconductivity case.
p
< uc which means
For the transition (instability) to occur, we need
that
R >
1, i.e.,
hole
self-organization
and
growth
overcomes stochastic decay caused by "collisions" with other holes.
This is
analogous
of
to
magnetization
the
condition
currents
Tc
>
overcomes
T
the
where
the
competing
self-ordering
effect
of
random
the
thermal
motion (interparticle collisions).
Note that an increase in driving current
fluctuation
levels.
Because
stochastic limit),
(A9)
(A6)
is
(i.e.,
only strictly
is only valid
near the
in pc)
leads to larger
valid
for
critical
YT>1, the instability is more of the interchange type,
of
AB2
changes.
changes
in
transitions.
Of
course,
parametric
If,
With
(21),
this
new
dependence
as in Sec.
overlap of two magnetic
when
ID,
islands,
gives Fig.
1.
we
physics
also
consider the
the stability
Figure
1
is a
point
y=yc.
For
and the y1 dependence
comes
occur
(fully
Yt<1
into
in
play,
similar
superconductivity
instability onset at the
boundary
phase
is Rm
c)2
t(ii
diagram for the phase
transition caused by MHD clump instability for finite Rm.
As we have discussed here and in Ref.
1,
the clump fluctuations arise,
because of energy and magnetic helicity conservation, from the mixing of the
mean sheared fields.
As the fluctuations grow during MHD clump instability,
the mean (course grained) fields are dissipated as the mean flux is expelled
from the plasma.
This spontaneous expulsion of mean flux during
instability
analogous
is
to
the
Meissner
effect.
The
MHD clump
disruptive
instability 3 in a tokamak fusion device is a "Meissner effect" in which the
plasma
confinement
reached.
driven
The
before
phase transition
reversed
(replenished)
is lost
field
pinch
does
the
not
Taylor
reach
device,
and a steady state turbulence
67
the
state
of
lowest
energy
completion.
However,
mean
is
flux
is possible.
is
in a
maintained
At the attainment
of
the
steady
state,
the
phase
transition
reaches
the
Taylor
state
minimum energy and the mean fields are supported by the paramagnetic,
action (A4),
i.e.,
Equilibrium
transitions
plasma.
are
dynamo
(25).
thermodynamic
analogous
or
to the
Such energy minimization
transition,
of
not the dynamical
MHD clump instability
statistical
mechanical
Taylor
6
model
of
models
relaxation
of
in
phase
an
MHD
principles predict the final state of the
route taken.
is a dynamical
Taylor state in an MHD plasma.
68
route
However,
as we have
through the
seen,
the
transition to the
REFERENCES
1.
D. Tetreault, Phys. Fluids 25, 527 (1982).
2.
H. Furth, J. Killeen and M. Rosenbluth, Phys. Fluids 6, 459 (1963).
3.
G. Bateman, MHD Instabilities, (MIT Press, 1978).
4.
T.J. Boyd and J.J. Sanderson,
5.
P.
6.
J.B. Taylor,
7.
H.K.
Fluids,
8.
Oceans 5,
Atmos.
9.
Dyn.
1 (1980).
D.J.
Berman,
R.H.
Conducting
Reznik,
G.M.
and
McWilliams,
J.C.
Larichev,
V.D.
Flierl,
G.R.
Electrically
1978.
Press,
Cambridge U.
in
Generation
Field
Magnetic
Moffatt,
1139 (1974).
33,
Lett.
Rev.
Phys.
1969).
1903 (1973).
Fhys. Fluids 16,
Rutherford,
(Nelson
Plasma Dynamics,
Dupree,
and T.H.
Tetreault,
Fluids
Fbys.
28,
155
(1984).
10.
R.
Berman,
11.
W.H.
and D.
Matthaeus
Tetreault,
and D.
Dupree,
T.
of the
Annals
Montgomery,
29,
Fluids,
Phys.
N.Y.
2860 (1986).
Sci.
Acad.
357,
203 (1980).
12.
J.A.
"Statistical Descriptions
Krommes,
Plasma
Physics
II,
Galeev
A.A.
(ed.
and Plasma
R.N.
and
Physics,"
Sudan,
in Basic
North-Holland
1984), pg. 183, pg. 198.
13.
R.H.
14.
H.
Kraichnan,
Tennekes
1972),
Lumley,
and J.L.
see Sec.
497 (1959).
5,
J. Fluid Mech.
A First Course
1.2 and 2.3.
1839 (1981).
Boutros-Ghali and T.H. Dupree 25,
15.
T.
16.
J.P.
17.
T.
18.
H.L.
Berk,
19.
C.L.
Longmire and M.N.
Freidberg,
O'Neil,
Phys.
C.E.
(MIT Press
in Turbulence,
Rev.
Phys.
Mod.
Fluids 8,
Nielsen,
54,
801
(1982).
76 (1965).
and K.V.
Rosenbluth,
69
Roberts,
Phys.
Phys.
Rev.
103,
Fluids 13,
980 (1970).
507 (1956).
20.
A.H.
Boozer,
"Ohm's
Law
T.H. Dupree,
22.
H.R. Strauss,
23.
A.
24.
S.A.
Magnetic
Phys Fluids 28,
Phys.
Hameiri,
in
Transition
Academic Press 1981),
pg.
127.
Stuart,
57,
206 (1986).
Plasma
and
Turbulence
(ed.
Pipes
and
Transition
"Instability and
R.E.
(ed.
Transition and Turbulence
Lett.
Rev.
"Subcritical Transition to Turbulence
Flows",
J.T.
Princeton
2786 (1985).
Patera,
Orszag and A.T.
Shear
Fields",
277 (1982).
Fluids 25,
Phys.
ahattacharjee and E.
Planar
25.
Mean
PPPL-2153 (1984).
Physics Laboratory,
21.
for
in
Academic
Meyer,
R.E.
Meyer,
Channels,
in
pg.
1981),
Press
in
77.
26.
H.R. Strauss,
27.
M.N
Rosenbluth,
134 (1976).
Fluids 19,
Phys.
Taylor,
and G.M.
Tetreault,
Phys.
J.P.
Sagdeev,
R.Z.
Nucl.
Zaslavsky,
Fusion 6, 297 (1966).
28.
P.H.
T.H.
Diamond,
and D.
Dupree,
45,
562
Lett.
42,
Lett.
Rev.
(1980).
29.
A.B.
M.N.
Rechester,
and R.B.
Rosenbluth,
White,
Phys.
Rev.
1247 (1979).
Phys.
Ann.
3,
347 (1958).
30.
W.A. Newcomb,
31.
T.H. Dupree, private communication.
32.
T.H. Dupree,
Phys.
33.
R.H. Berman,
D.J. Tetreault,
34.
T.H. Dupree, Phys. Fluids 26, 2460 (1983).
35.
D.
36.
J.C.
37.
Topics in
Tetreault,
Phys.
Phys.
McWilliams,
334 (1972).
Fluids 15,
Fluids 26,
Enrico Fermi,
ed.
Fluids 26,
2437 (1983).
3247 (1983).
1_46,
J. Fluid Mech.
Ocean Physics,
Phys.
Dupree,
T.H.
A.R.
Northholand,
21
Osborne
1982).
70
(1984).
(Proc.
of the Int.
Sch.
of
38.
Predictability of Fluid Motions, ed. G. Holloway and B. West (AIP Conf.
Proc.
No.
106,
AIP,
1984).
39.
C.E. Leith, Phys. Fluids 27, 1388 (1984).
40.
L. Woltjer, Proc. Nat. Acad. Sci. 44, 489 (1958).
41.
A.
Ehattacharjee,
F.
Brunel,
T.
and
Tajima,
Phys.
Fluids
26,
3332
(1983).
42.
M. Tinkham, Introduction to Superconductivity, (McGraw-Hill, 1975).
43.
K.
Moriyasu,
An Elementary Primer For Gauge Theory,
(World Scientific,
1983).
44.
I.J.R.
Aitchison
and
A.J.G.
Hey,
Gauge
(Adam Hilger LTD, 1982).
71
Theories
in
Particle
Physics,
FIGURE CAPTION
1.
Reynolds
number
Rm
vs.
amplitude
instability (schematic).
72
for
transition
to
MHD
clump
A
Rm
stable
Ampabld
Amplitude
73