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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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http://dx.doi.org/10.1063/1.1691452

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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.




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 . . .




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.





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.





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.





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.





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.





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.





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.





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.


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