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8/3/2019 V. I. Sotnikov et al- Development of global magnetohydrodynamic instabilities in Z-pinch plasmas in the presence o…
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Development of global magnetohydrodynamic instabilities in Z-pinchplasmas in the presence of nonideal effects
V. I. Sotnikova) and B. S. BauerUniversity of Nevada, Reno, Nevada 89557
J. N. LeboeufUniversity of California, Los Angeles, California 90095
P. Hellinger, P. Travnıcek, and V. Fiala Institute of Atmospheric Physics, 141 31 Praha 4, Czech Republic
Received 5 August 2003; accepted 3 February 2004; published online 14 April 2004
The development of global magnetohydrodynamic MHD instabilities in Z-pinch plasmas has been
studied with a three-dimensional hybrid simulation model. Plasma equilibria without and with axial
sheared flow, and with different values of the parameter H , which appears as a coefficient before
the Hall term in dimensionless nonideal MHD equations, have been considered. Increasing the
parameter H leads to larger simulation growth rates for both m0 sausage and m1 kink modes.
The hybrid simulations do however show that axial sheared flow severely curtails the linear and
nonlinear development of both sausage and kink instabilities. In these respects, the hybrid
simulations are in qualitative agreement with linear Hall MHD results. Moreover, in the nonlinear
stage, long wavelength modes dominate the excited wave spectrum when the parameter H is small.
For the larger value of the parameter H , small-scale structures do however develop nonlinearly inthe excited wave spectrum at late times. © 2004 American Institute of Physics.
DOI: 10.1063/1.1691452
I. INTRODUCTION
It is now well recognized1– 5 that such nonideal magne-
tohydrodynamic MHD effects as sheared flow, the Hall
term and finite Larmor radius FLR can strongly influence
the development of global MHD instabilities see also Refs.
6 – 8.
In a previous paper9 the linear stage of instability devel-
opment was investigated and the linearized system of equa-
tions based on the Hall fluid MHD model was solved nu-
merically for the m0 sausage mode. The main results of
that study are as follows. The flow shear can considerably
suppress the instability development. However, the Hall
term, with even relatively small parameter H c /( pi r 0)
0.1, where r 0 is the radius of the metal cylinder, c is the
speed of light, and pi is the ion plasma frequency, can lead
to a considerable increase in the growth rate, especially in
the short wavelength region.
Another approach to the stability analysis is based on the
Vlasov fluid model. In the simplified version of this model
ions are treated via the linearized Vlasov equation and elec-trons are added as a cold background.10 Using this hybrid
approach the stability of the azimuthal mode number m0
sausage and m1 kink modes in the collisionless, large ion
Larmor radius regime was evaluated. It was shown that large
Larmor radius effects did not lead to a significant suppres-
sion of the sausage and kink modes. A related two-
dimensional and nonlinear hybrid model in r – coordinates
which includes ion collisions has been applied to a kinetic
description of ions in aluminum wire-array precursor
plasmas.11 The emphasis in that work is on modeling wire-
array implosion and precursor development, not pinch stabil-
ity studies.
The evolution of the Rayleigh–Taylor instability in a
low beta, two-dimensional plasma was also studied with the
use of a hybrid code and a nonideal MHD code.12 In that
paper, differences between the conventional MHD and non-ideal MHD results are discussed. In the conventional MHD
regime, the usual behavior of the Rayleigh–Taylor instability
is observed. In the weak nonideal MHD regime, long wave-
length modes, reminiscent of the Kelvin–Helmholtz instabil-
ity, dominate nonlinearly but very short wavelength filaments
develop at the boundary interface. In the strong nonideal
MHD regime, small-scale structures dominate and the
boundary layer relaxes via a diffusion-like process rather
than through a large-scale nonlinear mixing process.
In a recent paper13 nonideal MHD plasma regimes in the
study of dynamic Z pinches are discussed and conditions for
nonideal MHD dynamics including fluid viscosity, resistivity,
and Hall current dynamics are reviewed. In particular, thephysics of wire initiation and breakdown and liner implosion
are considered. This work emphasizes the need to go beyond
ideal MHD, as is also attempted here, for experimental rel-
evance.
In the present paper, the development of Z-pinch insta-
bilities in the presence of axial sheared flow, the Hall term
and finite Larmor radius effects FLR has been investigated
via three-dimensional 3D hybrid particle ions, fluid elec-
trons simulations. The aim is to see how the combined in-
fluence of sheared axial flow and the Hall term affects theaElectronic mail: [email protected]
PHYSICS OF PLASMAS VOLUME 11, NUMBER 5 MAY 2004
18971070-664X/2004/11(5)/1897/11/$22.00 © 2004 American Institute of Physics
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development of the sausage and kink instabilities. Sheared
axial flow leads to the rapid appearance of short wavelengths
in the spectrum of the excited MHD modes. This dictates the
necessity of incorporating nonideal MHD effects into the
model. Inclusion of the sheared axial flow and the Hall term
allows to take the Z-pinch system away from the region in
parameter space where ideal MHD is applicable and where
the Z pinch is in its most unstable configuration. This bringsthe Z pinch to a regime in operational space where nonideal
effects such as the Hall term, finite Larmor radius effects,
and sheared axial flow tend to govern stability.
The 3D version14 of the hybrid code based on the Cur-
rent Advanced Method and Cycling Leapfrog CAM–CL
algorithm15 is used. In this code electrons are considered as a
massless fluid, and the ions are treated as particles. The hy-
brid simulations have been carried out without and with axial
sheared flow and in two regimes with respect to the param-
eter H . In the first regime the parameter H is taken to be
very small at H 0.02. In the second regime it is taken to be
five times larger at H 0.1. The set of parameters chosen for
the hybrid simulations is very close to the one used in Ref. 9
and this then makes it possible to compare linear Hall MHD
theory and hybrid simulation results.
The organization of this paper is as follows. In Sec. II,
the equations used in the hybrid simulation model are pre-
sented. In Sec. III, simulation results are reported for insta-
bility development without and with axial sheared flow when
the parameter H is small and set at H 0.02. In Sec. IV,simulation results for the case when the parameter H is
larger and set at H 0.1 are presented with and without
axial sheared flow. In the last section the results obtained in
the simulations are discussed and summarized.
II. HYBRID MODEL AND NUMERICS
We use the 3D version14 of the hybrid simulation model
based on the CAM–CL algorithm.15 In this model, the
plasma is described by a combination of kinetic ions and
fluid electrons. The relevant equations are
FIG. 1. Growth rates as a function of scaled axial wave number with parameter H 0.02 and in the absence of axial sheared flow for a the m0 sausage
instability and b the m1 kink instability. The same growth rates for the calculation with increased simulation box size to capture the ideal MHD behavior
of the growth rates in the long wavelength limit for c the m0 sausage instability and d the m1 kink instability.
1898 Phys. Plasmas, Vol. 11, No. 5, May 2004 Sotnikov et al.
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dr
dt v i , 1
d v i
dt
q i
m i
E v i B , 2
B
t
E , 3
B 0 J , 4
n em e
du e
dt en e E J
e B p e , 5
p enekT e . 6
The electrons are taken to be a massless, charge-neutralizing
fluid m e0 in Eq. 5 and they furthermore obey an iso-
thermal equation of state Eq. 6. The ions on the other
hand are treated by the particle in cell PIC scheme and full
ion kinetics are therefore resolved, with their equation of
state self-consistently determined by ion kinetics.
The electric field can then be written as
E J
i B
ene
B B
0ene
p e
en e
. 7
Therefore the equation for the magnetic field becomes B
t
J i B
en e
B B
0en e
. 8
This set of equations is sufficient to investigate development
of global MHD instabilities in Z-pinch plasmas with non-
ideal MHD effects such as the Hall term, FLR and axial
sheared flow, included into the model. Throughout, sub-
scripts e and i indicate electrons and ions, respectively.
The algorithm for solving this set of equations is de-
scribed in detail in Refs. 14 and 15. It is sufficient to say here
that in this code the particle ions are advanced by a leapfrog
scheme that requires the fields to be known at one-half time
FIG. 2. Color 3D density plots in the absence of axial sheared flow and with parameter H 0.02 at scaled times a t 0, b t 130, and c t 200.
1899Phys. Plasmas, Vol. 11, No. 5, May 2004 Development of global MHD instabilities in Z-pinch . . .
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step ahead of the particle velocity. This is effected by ad-
vancing the current density to this time step with only one
computational pass through the particle data at each time
step. Two interlaced grids are used, one with nodes at cell
centers for electric fields and another with nodes at cell ver-
tices for all other fields. The particle contribution to the cur-
rent density at the relevant nodes is evaluated with bilinear
weighting. The magnetic field is advanced in time with themodified midpoint trapezoidal method, which makes time
substepping for the field advance possible.
In the hybrid simulations, the magnetic field is scaled to
B 0 and the density to n0 . The units of space, time, and
velocity are collisionless skin depth c / pi , inverse of the ion
cyclotron frequency 1/ i , and Alfven speed v A , respec-
tively. These quantities are also defined through B 0 and n 0 .
The fields and particle moments are determined on a 3D grid
with ( N x L x / x50)( N y L y / y50)( N z L z / z
100) points or cells. There is a maximum of 128 particles
per cell for a scaled peak density of n 01 and the total
number of particles is 2 516 400 for the simulation box size
used. In the cases of interest here, the initial particle distri-
bution is not uniform and follows the density profile set by
the type of pinch equilibrium used, e.g., Bennett.
The simulation box is taken to be periodic in the axial z
direction. The simulation resolves only the grid points inside
a cylinder aligned with the z axis and centered in the middle
of the box with radius r 0 . The electric field outside that
cylinder is set to zero, and particles that cross the cylinderboundary are reflected back. The time step for the particle
advance is dt 0.025/ i , while the magnetic field B is ad-
vanced with a smaller time step, dt Bdt /10.
Simulation studies were carried out for two cylinder ra-
dii. In the first case the radius was set to r 050c / pi ,
through r 0 L x( N x x L y)/2 with N x50 and grid spac-
ing x y z2c / pi , which corresponds to a value of
H (c / pi )/ r 00.02. In the second case the radius was
equal to r 010c / pi , through r 0( N x x)/2 with N x50
and grid spacing x y z0.4c / pi , translating into
the parameter value of H 0.1.
The maximum axial wave number k zmax included in the
FIG. 3. Excited wave spectrum as a function of scaled axial wave number
vertical axis and azimuthal mode number horizontal axis in the absence
of axial sheared flow and with parameter H 0.02 at scaled times a t
130 and b t 200.
FIG. 4. Growth rates as a function of scaled axial wave number with pa-
rameter H 0.02 and with axial sheared flow V 0 z3 for a the m0
sausage instability and b the m1 kink instability.
1900 Phys. Plasmas, Vol. 11, No. 5, May 2004 Sotnikov et al.
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simulations is, in grid units, (2 )/ ( N z z)( N z /4)
( /2)/ z, which translates into k zmax(c / pi)0.78 and
3.92 for parameters H of 0.02 and 0.1, respectively. With
azimuthal mode number m, a change of coordinates from
Cartesian to cylindrical yields k m / r (k x) 2(k y)21/2.
For the sake of comparison, we take m max at r r 0 , so that
mmax
r 0(k xmax
)2
(k ymax
)2
1/2
. With k xmax
/(2 x) andk y
max /(2 y), the maximum m, defined at r r 0 , is m max
55 in both cases. Furthermore, we have tested the numeri-
cal scheme to estimate its residual diffusion. The tests show
that numerical diffusion is important only for time scales that
significantly exceed the duration of our simulations. Numeri-
cal diffusion can then be considered to be negligible less
than 1%. As a consequence, the wave vectors that are well
resolved in our simulations do indeed range from k zmin
2 / L z to k zmax /(2 z), so that k z
mink k zmax . Therefore,
the maximum wave numbers and azimuthal mode number
provide adequate resolution.
With the two different cylinder radii routinely used,
simulations have been carried out both without and with
axial sheared flow. In the simulations we initially set electron
and ion temperatures so that i e0.5, with i ,e
n0kT i,e /( B 02 /20). Taking into account that c / pi
(2/ i) i , where i is the ion Larmor radius, we therefore
have c / pi2 i for i0.5. This implies that we are still in
the regime when Hall MHD is valid strictly speaking itshould be c / pi i). In addition to the parameter H which
serves as a measure of the importance of nonideal effects
such as the Hall term, we can also introduce the parameter
FLR i / r 0 , which represents the influence of finite Larmor
radius effects on the system. With such a choice of param-
eters ( FLR0.01 and FLR0.05 for the two cylinder radii
considered, namely r 050c / pi and r 010c / pi ), the in-
fluence of finite Larmor radius effects is therefore expected
to be small. Moreover, the ratio H / FLR2/ i remains
large for plasmas with i1. This means that FLR effects
are not so important in such plasmas.
The code is initialized with Bennett equilibrium profiles.
FIG. 5. Color 3D density plots in the presence of axial sheared flow and with parameter H 0.02 at scaled times a t 0, b t 130, and c t 200.
1901Phys. Plasmas, Vol. 11, No. 5, May 2004 Development of global MHD instabilities in Z-pinch . . .
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The initial density and magnetic field are set to
nn0
1r 2 / a 22, 9
B
B 0
r / a
1r 2 / a2 , 10
where r is the radial distance from the cylinder axis and the
pinch radius ar 0 /3. The radial profile of the axial velocity
is taken to be of the form V 0 z3V A(1r 2 / r 02), so that the
axial velocity shear is linearly proportional to the radius.
In what follows, results of the hybrid simulations will be
presented in terms of plots for the growth rates of m0 and
m1 modes measured in the linear phase of the simulations,
of 3D density plots, and of wave number spectra of the den-
sity perturbations as a function of axial wave number k z and
azimuthal m wave number at various instants of time in in-
stability development. Every mode in the excited wave spec-
trum has its own time necessary for the nonlinear saturation
process to start. For instance, modes with larger axial wave
numbers k z have larger linear growth rates and as a result
their saturation times are smaller. To measure the linear
growth rates in the hybrid simulations, we have therefore
chosen time slices which correspond to the linear stage of
instability development for each of the modes of interest. In
the various plots for the growth rates, we have also restricted
the maximum axial wave numbers to those already calcu-
lated from the linear Hall MHD theory in Ref. 9. Moreover,to determine the growth rate of the particular mode with
given axial k z and azimuthal m wave numbers, we first trans-
form the density to cylindrical coordinates n( t , x , y , z)
n( t ,r , , z). Now we make a Fourier transformation of the
last two coordinates and z to get nft(t ,r ,m ,k z). Finally, for
different r , m, and k z we determine the growth rate using the
linear fit of lnnft(t ) during the exponential stage of the wave
amplitude growth.
In the hybrid simulations presented next, the source of
energy for the instabilities is the axial current in the system
which supports the azimuthal magnetic field, Eq. 10, in the
Bennett pinch equilibrium. Unstable modes grow from noise
FIG. 6. Excited wave spectrum in the same representation as in Fig. 3 with
axial sheared flow and with parameter H 0.02 at scaled times a t 100
and b t 200.
FIG. 7. Growth rates as a function of scaled axial wave number in the
absence of axial sheared flow and with term parameter H 0.1 for a the
m0 sausage instability and b the m1 kink instability.
1902 Phys. Plasmas, Vol. 11, No. 5, May 2004 Sotnikov et al.
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which in the hybrid simulations is mostly due to random
seeding of the initial Maxwellian velocity distribution of the
particle ions. Along with the development of sausage and
kink instabilities in the system, magnetic energy is decreas-
ing and kinetic energy is growing. Throughout, the total en-
ergy conserves to an accuracy 4%.
III. HYBRID SIMULATION RESULTS FOR SMALLPARAMETER H Ä0.02
In this section results of 3D hybrid simulations of
Z-pinch instabilities with and without axial sheared flow and
with the parameter H measuring the importance of nonideal
effects such as the Hall term set to H 0.02 are presented.
A. Instability development in the absence of axialsheared flow
For the simulations with small H , the growth rates of
the sausage (m0) and kink (m1) instabilities measured
in the linear phase of the simulations are plotted in Figs. 1a
and 1b. They are displayed as a function of the wave num-
ber along the axial direction k z , normalized to the ion skin
depth c / pi . The growth rates are normalized to V Ti / r 0 ,
where V Ti corresponds to the spatially uniform ion thermal
speed and ion temperature at t 0 and r 0 is the radius of
the cylinder.
The growth rate for the sausage instability presented in
Fig. 1a is in fact in reasonably good agreement with the
linear Hall MHD calculations with parameter H set at H
0.01, presented in Fig. 3 of Ref. 9. The growth rate of the
m1 mode is slightly larger than that of the m0 mode.
To check the growth rate behavior in the long axial
wavelength limit, where according to ideal MHD the growth
rate should go to zero, we have carried out simulations for a
box size with twice the length in the z direction and therefore
twice the resolution. The fields and particle moments are
now determined on a 3D grid with ( N x L x / x100)
( N y L y / y100)( N z L z / z200) points or cells.
There is as before a maximum of 128 particles per cell for ascaled peak density of n 01 and the total number of par-
ticles is now 20 131 200 8 times more particles than usual.
Simulation results for the larger box size are presented in
Figs. 1c and 1d. It is now clearly seen that the growth rate
is going to zero when the axial wave number goes to zero.
Simulations with the larger box size were only carried out for
this case to demonstrate that the code can correctly capture
the growth rate behavior in the long wavelength limit.
Figure 2 contains 3D plots of plasma density inside the
cylinder at scaled times t 0, t 130, and t 200, where t is
in 1/ i units. Figure 2 indicates that at earlier moments in
time sausage and kink modes coexist, but that kink modes
FIG. 8. Color 3D density plots in the absence of axial sheared flow and with parameter H 0.1 at scaled times a t 0, b t 110, and c t 180.
1903Phys. Plasmas, Vol. 11, No. 5, May 2004 Development of global MHD instabilities in Z-pinch . . .
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take over later on in time as can be seen from Fig. 2c. This
corroborates the results of Figs. 1a and 1b for the respec-
tive growth rates of the sausage and kink modes, the latter
being larger.
Wave number spectra are presented in Figs. 3a and
3b, at scaled times t 130 and t 200, respectively, as a
function of axial wave number k z , measured in the units of
pi / c and plotted along the vertical axis and of azimuthal
wave number m along the horizontal axis. These plots show
how the density perturbations are distributed among the dif-ferent axial and azimuthal wave numbers in the excited wave
spectrum. The darker regions correspond to higher density
amplitudes in the excited wave spectrum. Figure 3 shows
that the sausage m0 and kink m1 modes appear at ear-
lier moments in time (t 130), as has already been seen in
the structures of Fig. 2. This is also in agreement with Fig. 1,
where the growth rates of two most rapidly growing modes
are presented. At late times ( t 200) modes with larger azi-
muthal wave numbers m appear. Moreover, Fig. 3 indicates
that in the case of small H 0.02 the excited wave spectrum
is concentrated in the region of long axial wavelengths
small k z) in the nonlinear stage.
B. Instability development in the presence of axialsheared flow
For the simulations with axial sheared flow and with
small H , the growth rates of the sausage ( m0) and kink
(m1) instabilities as a function of scaled axial wave vector
are plotted in Figs. 4a and 4b. When compared to those of
Fig. 1 we see that the simulation growth rates of both m
0 and m1 modes are substantially reduced in the pres-
ence of axial sheared flow, particularly for k z0.5. The
growth rate for the sausage instability presented in Fig. 4a
is again in reasonably good agreement with the linear Hall
MHD calculations with H 0.01 and with axial sheared
flow presented in Fig. 3 of Ref. 9. To elaborate, in the long
wavelength limit the simulation growth rate of the m0
mode is in good agreement with the linear theory results
presented in Fig. 3 of Ref. 9, while it is slightly larger in the
short wavelength limit.
Figures 5a, 5b, and 5c are 3D plots of plasma den-
sity inside the cylinder at scaled times t 0, t 130, and t
200. These plots, when compared to those of Fig. 2,
clearly show that instability development in both the linearand nonlinear phases for sausage and kink modes alike is
strongly suppressed by the axial sheared flow.
Wave number spectra at scaled times t 130 and t
200 are presented in Figs. 6a and 6b, again as a func-
tion of scaled axial wave number on the vertical axis and
azimuthal mode number on the horizontal axis. Even more
than in the case without flow shear of Fig. 3, the long axial
wavelength amplitudes and the low azimuthal mode numbers
dominate the excited wave spectrum, although at time t
200 harmonics with high m numbers have not developed
yet, because their growth is now suppressed by the flow
shear.
IV. HYBRID SIMULATION RESULTS FOR LARGERPARAMETER H Ä0.1
In this section results of 3D hybrid simulations for the
case when the parameter H is five times larger than in the
simulations described in Sec. III, are presented.
A. Instability development in the absence of axialsheared flow
For the simulations without axial sheared flow but with
large H parameter at H 0.1, the growth rates of the sau-
sage (m0) and kink ( m1) instabilities are presented in
Figs. 7a and 7b. As can be seen from the plots, the simu-lation growth rates of both m0 and m1 modes are con-
siderably larger for all k z values except those discussed in
Sec. III A, where the parameter H was much smaller at H
0.02.
Figures 8a, 8b, and 8c are 3D plots of plasma den-
sity inside the cylinder at scaled times t 0, t 60, and t
100. These plots clearly show that instability development
for both sausage and kink modes is now taking place much
faster in time and therefore with much larger growth rates, as
attested by the very distorted plasma density patterns of Fig.
8c. These results confirm those of the linear Hall MHD
theory presented in Ref. 9 which predict that an increase in
FIG. 9. Excited wave spectrum in the same representation as in Figs. 3 and
6 in the absence of axial sheared flow and with parameter H 0.1 at scaled
times a t 100 and b t 160.
1904 Phys. Plasmas, Vol. 11, No. 5, May 2004 Sotnikov et al.
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the parameter H leads to higher growth rates for sausage
modes. Figure 8b also reveals the dominance of short scale
perturbations in the excited axial wave spectrum, in contrast
with Fig. 2c, where large scale axial perturbations are the
most pronounced in the case of smaller H parameter.
Spectral analysis of the excited wave spectrum at scaled
times t 60 and t 100 is presented in Figs. 9a and 9b.
These figures also show that at these instants in time the
wave spectra have spread into the regions of smaller axialwavelengths along the vertical axis and of large azimuthal
wave numbers along the horizontal axis. This is different
from the excited spectrum behavior in the case of smaller
H 0.02 presented in Fig. 3, where the wave spectra mostly
consist of long axial wavelengths and low azimuthal wave
numbers at late times.
B. Instability development in the presence of axialsheared flow
For the simulations with large H parameter at H 0.1
and with axial sheared flow, the growth rates of the sausage
(m0) and kink (m1) instabilities are plotted in Figs.10a and 10b. As can be seen from these figures, the
growth rates of both m0 and m1 modes are dramatically
reduced compared to those of Figs. 7a and 7b. It is inter-
esting to note that the growth rate of the m0 mode is now
larger than the growth rate of the m1 mode in the simula-
tions.
Figures 11a, 11b, and 11c are 3D plots of plasma
density inside the cylinder at scaled times t 0, t 110, and
t 180. These plots, when compared to those of Fig. 8,
clearly show that instability development for both sausage
and kink modes is now suppressed by axial sheared flow. As
mentioned before, the growth rate of the m0 mode is
larger and this explains the density structure appearing inFig. 11b. Later on in time, the m1 mode and even higher
m modes emerge as can be seen in Fig. 11c.
Wave number spectra at scaled times t 100 and t
160 are presented in Figs. 12a and 12b as a function of
scaled axial wave vector on the vertical axis and azimuthal
mode number on the horizontal axis. This spectrum mark-
edly differs from that without axial sheared flow presented in
Fig. 9 in the sense that there is much less of a spread in axial
wavelength and in azimuthal wave number even at time t
160. This may just be another reflection of the much
slower development of the instabilities because their growth
rates are now substantially suppressed by the axial sheared
flow, as shown in Fig. 10.
V. SUMMARY, DISCUSSION, AND CONCLUSIONS
In this paper we have examined the linear and nonlinear
development of instabilities of current-carrying Z-pinch plas-
mas via 3D hybrid simulations with inclusion into the model
of nonideal MHD effects connected with the Hall term
through parameter H and with axial sheared flow. The simu-
lations have been initialized with Bennett equilibrium pro-
files with two different values of the parameter H ( H
0.02 and H 0.1), without and with axial sheared flow.
The latter was chosen of the form V 0 z3V A(1r 2 / r 02) so
that flow shear is everywhere proportional to radius. For ev-
ery simulation performed, results were presented in terms of
plots for the growth rates of m0 and m1 modes mea-
sured in the linear phase of the simulations, of 3D density
plots, and of wave number spectra of the density perturba-
tions as a function of axial wave number k z and azimuthal m
wave number at various instants of time in instability devel-
opment.
The parameter H can be expressed through the ratio of the ion Larmor radius i to the cylinder radius r 0 which we
call FLR and ion plasma i as follows H (2/ i) FLR . So
only for very large plasma i1, we shall have H FLR
and effects connected with the Hall term can be neglected in
comparison with FLR effects. For our choice of i0.5, we
have H 2 FLR and the influence of the Hall term on insta-
bility development can therefore exceed that of FLR effects.
It is also important to mention that in the simulations k z i for
the long axial wavelength part of the spectrum was small
(k z i1) for both H 0.02 and H 0.1.
The results reported here show that when the parameter
H increases, the growth rates of both sausage and kink in-
FIG. 10. Growth rates as a function of scaled axial wave number with axial
sheared flow and with parameter H 0.1 for a the m0 sausage instabil-
ity and b the m1 kink instability.
1905Phys. Plasmas, Vol. 11, No. 5, May 2004 Development of global MHD instabilities in Z-pinch . . .
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stabilities increase as well. This happens in cases with and
without the flow shear. The results of this study also show
that axial sheared flow does tend to suppress sausage and
kink instabilities for both small and large H cases. This is in
agreement with ideal MHD simulations of DeSouza-
Machado and co-workers5 that demonstrate stabilization of
the Z pinch by velocity shear. Here, we show that the sup-
pression by axial sheared flow is more pronounced for largeparameter H . The growth rates obtained in the hybrid simu-
lations for the m0 mode appear to be in reasonable agree-
ment with the linear Hall MHD results presented in Ref. 9
for a similar set of initial parameters. In Ref. 9, only the
sausage (m0) instability was examined in the framework
of the Hall MHD model, although equations were derived for
arbitrary azimuthal mode number m. It was shown that even
small values of the parameter H can destabilize the m0
mode. FLR effects were not taken into account in that model.
These earlier results are in agreement with nonideal MHD
simulations of Z-pinch stability performed by Sheehey and
Lindemuth.16
Results presented in this paper also indicate that in a
Z-pinch plasma with i1 in the regime where ir 0 and
c / pi i , the Hall term plays the dominant role in destabi-
lizing m0 and m1 modes, whereas the stabilizing effect
due to FLR is not so pronounced. This corroborates the re-
sults of Ref. 10, where instability of m0 and m1 modes
was studied in the presence of large FLR, but with zero
electron temperature and where stabilization due to largeFLR was not observed. This further agrees with results of
Ref. 17 where finite electron temperature was included into
the Vlasov fluid model which in fact leads to even larger
growth rates for the m0 modes than those obtained with
cold electrons.
The 3D hybrid simulations presented in this paper show
that the nonlinear development of the Z-pinch plasma col-
umn is complicated by the alternate dominance of modes
with long and short axial wavelengths and small and large
azimuthal wave numbers as time progresses away from the
linear phase of sausage and kink instabilities. This is appar-
ent in 3D plots of plasma density in the cylinder as a func-
FIG. 11. Color 3D density plots in the presence of axial sheared flow and with parameter H 0.1 at scaled times a t 0, b t 110, and c t 180.
1906 Phys. Plasmas, Vol. 11, No. 5, May 2004 Sotnikov et al.
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tion of time and in plots of excited wave spectra as a func-
tion of axial wave number and azimuthal mode number. The
nonlinear stage of development of the plasma column is af-
fected by the presence of axial sheared flow and by an in-
crease in the parameter H as much as the linear results are.
It is speeded up as the parameter H increases and slowed in
the presence of axial sheared flow.
In particular, analysis of the excited wave spectra for the
density perturbations as a function of axial wave number and
azimuthal wave number shows that axial sheared flow tendsto suppress the spread of the spectrum at late times towards
large azimuthal wave numbers. The axial sheared flow is
more effective at reducing the spectral spread for small pa-
rameter H . For the larger value of the parameter H , the
excited axial wave spectrum spreads at late times, the spread
being somewhat less pronounced in the presence of axial
sheared flow. All in all, this produces the small scale axial
and azimuthal structures which dominate the nonlinear stage
of instability development.
The appearance of the short axial and azimuthal wave-
lengths in the nonlinear spectrum of the hybrid simulationswith increase of the parameter H is consistent with the con-
jecture advanced in the Introduction that inclusion of the
Hall term into the system leads to development of small
scales in the turbulence spectrum.
ACKNOWLEDGMENTS
The authors wish to acknowledge valuable discussions
with I. Lindemuth, L. Rudakov, P. Sheehey, R. Siemon, and
F. Winterberg. We also express our gratitude to R. A. Fon-
seca and F. Tsung, respectively, from IST, Portugal and
UCLA, for their permission to use the OSIRISAnalysis sci-
entific visualization package developed in the context of laser- and beam-plasma interactions.
This work was supported by the United States Depart-
ment of Energy under Grant No. DE-FG03-01ER54617 at
the University of California at Los Angeles, Grant No. DE-
FC08-01NV14050 at the University of Nevada Reno and
Grant B 3042106/01 of the Czech Academy of Science at the
Institute of Atmospheric Physics in Prague.
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FIG. 12. Excited wave spectrum in the same representation as in Figs. 3, 6,
and 9 in the presence of axial sheared flow and with parameter H 0.1 at
times a t 100 and b t 200.
1907Phys. Plasmas, Vol. 11, No. 5, May 2004 Development of global MHD instabilities in Z-pinch . . .