JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 97, NO. A4, PAGES 4265-4273, APRIL 1, 1992 On the Thermodynamicsof Diamagnetic Plasma Expansions GALEN R. GISLER AND T. G. ONSAGER • SpacePlasma Physics Group, Los Alamos National Laboratory, Los Alamos, New Mexico Particle heating in a diamagnetic plasma expansion is studied by means of well-diagnosed simulation with an axisymmetric particle-in-cell code.Moments of the particle distribution function are obtained for spatially distinct subsets of the particles to examine temperature anddensity histories for different regions of the expanding plasma. The simulation is followed through oneexpansion-contraction cycle. While adiabatic behavior is observed during much of the cycle, significant deviations from the adiabatic result in strong particle heating. Anomalous ion heating occurs throughout the plasma during the expansion phase. This is manifested earliest in the ion parallel temperature, whichincreases first in the expanding plasma's outerreaches andlastin thecenter. This heating originates from the entropy that is generated at the barrel endsof the plasma as the initial expansion alongthe ambientfield is essentially free. Later, after a diamagnetic cavity is formed,ionswithin the cavity are reflected by the magnetic mirrors at the necks andtransport some of the generated entropy backinto the center of the cavity. The ion heatingthat occurscan easily raise the bulk plasmatemperature by an order of magnitudeover the initial adiabat. FUNDAMENTAL CONSIDERATIONS The expansion of a finite quantity of hot plasma into a weak uniform magnetic field in vacuum is among the sim- plest of all possible systemsinvolving unbounded plasmas, yet even this systemis at presentpoorly understood.As an example, we consider the thermodynamical behaviorof such a system. The free expansion of an idealgasinto vacuum is well understood; it is dealt with in textbooks for undergraduates in thermodynamics. Make the gas a plasma, and the expan- sion into a field-free vacuum is no different: if the electron temperature is not very much lessthan the ion temperature, the electrons try to race aheadof the ionsbut are reined in by the electrostatic field that results. The overall expansio n speed is then very nearly the ion thermal speed, as in the case of a neutral gas. Now load the vacuum with a uniform magnetic field. If the energy density in the magnetic field is very much larger than the kinetic energy density in the plasma, then the expansion is free only along the direction of the field. Across the field the ions and electrons can only move as far as a gyroradius, and sincethe particle gyroradii are different, this resultsin a radial electric field at the plasma boundary. This electric field, together with the axial magnetic field, produces a rotation of the plasma edge. If there is a gradient in the electric field, then there is a shearin the rotation velocity. In addition, the deceleration of the plasma boundary drives an interchangeinstability that contributes to certain structures observedin expanding plasmas[Galvez et al., 1988]. While this behavior is physically rich in plasma and interesting, speaking thermodynamically this is not a difficult problem; the expansion is free alongthe field and doesnot occur at all across it. Complicationsarise when the energy density of the uni- form magnetic field is smallerthan the plasmakinetic energy 1Nowat Institute for the Study of Earth, Oceans, and Space, University of New Hampshire, Durham. Copyright 1992 by the American Geophysical Union. Paper number 91JA03024. 0148-0227/92/91 JA-03024505.00 density [Winske, 1989]. In that case, one might suppose that the expansion begins in spherical symmetry, as in the field-free case. But as the plasma expands, its kinetic energy density can only decrease, while the field energy density remains nearly constant outside the expanding plasma. Thus the dominance of particle kinetic energy, no matter how great initially, is only transitory. Eventually, the field must win, and the plasma must adopt a geometry conforming to the overall field direction. Thus the case of expansion into a weak magnetic field might start out looking like expansion into no field and end up looking like expansion into a strong field. But it turns out to be not even so simple as that. In both the field-free case and the strong-field case the expansionis free, that is unconstrained, in at least one direction. One might think, then, that in the weak-field case the expansion would remain forever free in the axial direction, the direction along the uniform ambient field. This assumption has been shown to be of some predictive value in describing the dynamicsof plasma expansions [Gisler and Lemons, 1989]. It cannot, however, be rigorously true. At some time after the beginning of the expansion the magneticfield within the expandingplasma is reduced com- pared to the ambient field, becausethe plasma is diamag- netic: it pushes the field away as it expands. At great distances along the axis, of course, the field has its ambient value, so that particles moving in the axial direction will experience magnetic mirror forces, and some fraction of them will be reflected. Thus the axial expansionis by no meansfree, though it may not be as strongly constrainedas the radial expansion is. The dynamical importance of the mirror forces is nil, however, if each particle's adiabatic invariant is conserved. In that case, all particles remain forever unconfined: the perpendicular momentum is reduced in concert with the axial field, and all particles drop into the loss cone. In the more interesting case where the adiabatic invariants are broken through pitch angle scattering, anisotropy-driven instabilities, or finite gyroradius effects, a class of trapped particles is born. To thesetrappedparticles the mirror forces will be important. The expansion in the radial direction is resisted, of course, 4265
Transcript
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 97, NO. A4, PAGES 4265-4273, APRIL 1, 1992
On the Thermodynamics of Diamagnetic Plasma Expansions
GALEN R. GISLER AND T. G. ONSAGER •
Space Plasma Physics Group, Los Alamos National Laboratory, Los Alamos, New Mexico
Particle heating in a diamagnetic plasma expansion is studied by means of well-diagnosed simulation with an axisymmetric particle-in-cell code. Moments of the particle distribution function are obtained for spatially distinct subsets of the particles to examine temperature and density histories for different regions of the expanding plasma. The simulation is followed through one expansion-contraction cycle. While adiabatic behavior is observed during much of the cycle, significant deviations from the adiabatic result in strong particle heating. Anomalous ion heating occurs throughout the plasma during the expansion phase. This is manifested earliest in the ion parallel temperature, which increases first in the expanding plasma's outer reaches and last in the center. This heating originates from the entropy that is generated at the barrel ends of the plasma as the initial expansion along the ambient field is essentially free. Later, after a diamagnetic cavity is formed, ions within the cavity are reflected by the magnetic mirrors at the necks and transport some of the generated entropy back into the center of the cavity. The ion heating that occurs can easily raise the bulk plasma temperature by an order of magnitude over the initial adiabat.
FUNDAMENTAL CONSIDERATIONS
The expansion of a finite quantity of hot plasma into a weak uniform magnetic field in vacuum is among the sim- plest of all possible systems involving unbounded plasmas, yet even this system is at present poorly understood. As an example, we consider the thermodynamical behavior of such a system.
The free expansion of an ideal gas into vacuum is well understood; it is dealt with in textbooks for undergraduates in thermodynamics. Make the gas a plasma, and the expan- sion into a field-free vacuum is no different: if the electron
temperature is not very much less than the ion temperature, the electrons try to race ahead of the ions but are reined in by the electrostatic field that results. The overall expansio n speed is then very nearly the ion thermal speed, as in the case of a neutral gas.
Now load the vacuum with a uniform magnetic field. If the energy density in the magnetic field is very much larger than the kinetic energy density in the plasma, then the expansion is free only along the direction of the field. Across the field the ions and electrons can only move as far as a gyroradius, and since the particle gyroradii are different, this results in a radial electric field at the plasma boundary. This electric field, together with the axial magnetic field, produces a rotation of the plasma edge. If there is a gradient in the electric field, then there is a shear in the rotation velocity. In addition, the deceleration of the plasma boundary drives an interchange instability that contributes to certain structures observed in expanding plasmas [Galvez et al., 1988]. While this behavior is physically rich in plasma and interesting, speaking thermodynamically this is not a difficult problem; the expansion is free along the field and does not occur at all across it.
Complications arise when the energy density of the uni- form magnetic field is smaller than the plasma kinetic energy
1Now at Institute for the Study of Earth, Oceans, and Space, University of New Hampshire, Durham.
Copyright 1992 by the American Geophysical Union.
Paper number 91JA03024. 0148-0227/92/91 JA-03024505.00
density [Winske, 1989]. In that case, one might suppose that the expansion begins in spherical symmetry, as in the field-free case. But as the plasma expands, its kinetic energy density can only decrease, while the field energy density remains nearly constant outside the expanding plasma. Thus the dominance of particle kinetic energy, no matter how great initially, is only transitory. Eventually, the field must win, and the plasma must adopt a geometry conforming to the overall field direction. Thus the case of expansion into a weak magnetic field might start out looking like expansion into no field and end up looking like expansion into a strong field.
But it turns out to be not even so simple as that. In both the field-free case and the strong-field case the expansion is free, that is unconstrained, in at least one direction. One might think, then, that in the weak-field case the expansion would remain forever free in the axial direction, the direction along the uniform ambient field. This assumption has been shown to be of some predictive value in describing the dynamics of plasma expansions [Gisler and Lemons, 1989]. It cannot, however, be rigorously true.
At some time after the beginning of the expansion the magnetic field within the expanding plasma is reduced com- pared to the ambient field, because the plasma is diamag- netic: it pushes the field away as it expands. At great distances along the axis, of course, the field has its ambient value, so that particles moving in the axial direction will experience magnetic mirror forces, and some fraction of them will be reflected. Thus the axial expansion is by no means free, though it may not be as strongly constrained as the radial expansion is.
The dynamical importance of the mirror forces is nil, however, if each particle's adiabatic invariant is conserved. In that case, all particles remain forever unconfined: the perpendicular momentum is reduced in concert with the axial field, and all particles drop into the loss cone. In the more interesting case where the adiabatic invariants are broken through pitch angle scattering, anisotropy-driven instabilities, or finite gyroradius effects, a class of trapped particles is born. To these trapped particles the mirror forces will be important.
The expansion in the radial direction is resisted, of course,
4265
4266 GISLER AND ONSAGER: THERMODYNAMICS OF PLASMA EXPANSIONS
by the ambient field. As the expansion proceeds, there will eventually come a time when the ambient field energy density exceeds the energy density in the plasma. Then the radial expansion is decelerated, halted, and eventually re- versed. The cavity may continue to undergo some strongly damped radial oscillations.
What interests us in the expansion of a hot plasma into a weak magnetic field is the exchange and partition of energy between the various components of the system: the field, the ions, and the electrons. As the plasma expands radially, work is done against the magnetic field, and some work is also done against converging magnetic mirrors as the plasma expands axially. If the plasma is collisionless, then the various species partake unequally of the work, and the plasma departs from thermodynamic equilibrium.
The electrons have speeds much higher than the expan- sion speed, and they therefore communicate throughout the entire expanding plasma. They might therefore be expected to be isothermal throughout. They also undergo pitch angle scattering in the fluctuating fields near the edge of the expanding plasma, so they might also be expected to be isotropic throughout.
The ions, on the other hand, do not communicate through- out the plasma, and their population evolution is expected to be different in the different regions of the plasma. In the equatorial region, ions give up perpendicular energy as they push the field out radially, while ions near the poles give up parallel energy as they work against the mirrors. Then, of course, some ions are lost along the axis, namely those with small pitch angles.
All of these effects taken together result in a very compli- cated picture, and it is not at all clear, from first principles, what the thermal history of a plasma expansion should be. We therefore use particle-in-cell simulations to illuminate the evolution of thermodynamic quantities in diamagnetic plasma expansions. These simulations illustrate the interplay of energy among the ions, electrons, background magnetic field, and generated electrostatic and electromagnetic waves. Qualitatively, the simulations behave much as the discussion in the preceding paragraphs indicates. There are interesting details, however, which give rise to strikingly nonadiabatic behavior.
Our simulations are axisymmetric, which means that we suppress the development of azimuthal structures. The instabilities described by Winske [1989] and others do not occur in our simulations. These instabilities have mainly been simulated in the radius-azimuth plane, that is, ignoring the structure along the field. There have been attempts to do full three-dimensional simulations of diamagnetic plasma expansions, and the results of these have so far been difficult to interpret (M. E. Jones, private communication, 1990). Our task here was to understand the zeroth-order dynamics of these expansions, including the axial but not the azimuthal structure, partly as an aid to isolating and understanding some of the physics that expensive three-dimensional simu- lations will present.
The azimuthal instabilities that we suppress may also alter the axial dynamics that we attempt to simulate. For exam- ple, an interchange instability produces irregularities in the surface of the diamagnetic cavity. Some ions may escape through these irregularities (through breaking of the adia- batic invariant) before the cavity undergoes recompression. This escape of ions through the surface, and diffusion of field into the cavity, will tend to damp the radial oscillations of the
cavity. We already know from these runs that the escape of ions along the field is a strong source of damping for the bounces of the cavity, and it is clear that azimuthal irregu- larities may introduce additional damping. We await full three-dimensional simulations for a proper treatment of the problem.
In addition, azimuthal modes of certain microinstabilities may contribute to the exchange of energy between the particles and fields, between electrons and ions, or between perpendicular and parallel ion temperatures. We obviously can say nothing about these; we observe and document certain exchanges in this axial-plane simulation, and other exchanges may be observed in azimuthal-plane simulations, while the full three-dimensional picture may be richer still. But we believe that both the axial-plane and azimuthal-plane simulations, being simpler than three-dimensional simula- tions, are necessary to the understanding of the full picture.
We proceed to describe the characteristics of the compu- tational model and then detail our procedure for extracting the thermodynamic quantities of interest from the simulation run. We present our results in a series of figures document- ing the thermodynamic quantities in different regions of the expanding plasma, and finally we offer our interpretations and conclusions.
SIMULATIONS
Initially, the ambient magnetic field is everywhere uniform and straight. A finite quantity of plasma is placed in a small region at the center of a cylindrical computing volume whose axis is along the magnetic field. The rest of the computing volume is vacuum. The plasma is given an initial tempera- ture such that the thermal energy density greatly exceeds the energy density of the magnetic field.
As expected, the thermal expansion of the plasma begins in spherical symmetry (see Figure 1). Before long, however, the axial extent of the plasma is noticeably greater than its radial extent, and the overall shape is that of a barrel. As the calculation proceeds, more and more plasma is lost out the ends of the barrel. Eventually, the radial expansion is halted; the kinetic energy in the radial expansion has done work in pushing the field out of the way, and that energy is eventu- ally spent. The field is somewhat compressed at the edge of the field-free cavity, and the steep field gradient there now pushes back on the plasma. The barrel-shaped plasma is then radially compressed to about its initial radius. The field within the plasma is increased during the compression, and there is an overshoot, so another radial expansion follows. The bounce may not occur in real three-dimensional expan- sions because of additional dissipation due to nonaxisym- metric effects.
Particles at small pitch angles leak out after doing some work against the field and end up in regions of space where the field has been only mildly disturbed. The recovery of the field in the cavity strongly affects those particles that have remained within the cavity but has little effect on the particles that have migrated into weakly disturbed fields. Those escaped particles do not regain the energy they have spent in pushing on the field. This energy is returned to the particles left behind.
Put another way, because of the leakage of plasma along the field, the kinetic energy that is spent in pushing the field away is returned to fewer particles during the collapse. This has the result that the plasma remaining within the cavity
GISLER AND ONSAGER: THERMODYNAMICS OF PLASMA EXPANSIONS 4267
Fig. 1. The configuration space of the electrons at four times during a typical diamagnetic plasma expansion simulation. The four panels show the initial conditions, the quasi-spherical phase, the barrel shape at maximum expansion, and finally the contraction. The units of space and time are 2 km and 6.6/zs, respectively. The center of the expansion is at (z = 164, r = 0).
cycles to an adiabat of higher entropy than the original one and, in some cases, can actually attain a temperature greater than the original temperature.
In the remainder of this work we devote our attention to
the description of a single very well diagnosed simulation that we have followed through an expansion and recompres- sion. The qualitative remarks we make apply to a large number of less well diagnosed simulations, over a large range of parameter space, that we have done over the last three years. Some of these simulations have been reported by Gisler [1989] and Gisler and Lemons [1989].
The parameters of the simulation described in this work follow in the next few paragraphs. The numbers are chosen purely for computational convenience, and this simulation does not resemble any actual event. Nevertheless, the thermodynamic behavior is broadly characteristic of all the simulations we have performed, and it may be expected to resemble reality. Scaling to a real event can be accomplished by using the model relations given by Gisler and Lemons [1989], which were shown to be successful in predicting quantities of interest for the AMPTE chemical release experiments [Gurnett et al., 1986a, b; Bernhardt et al., 1987].
The computational volume is a cylinder of length 656 km and radius 328 km. This is divided into 524 axial by 262 radial computational cells. The hot plasma at the beginning of the calculation uniformly fills a cylinder of initial length and radius 49 km.
At the beginning of the calculation the ions and electrons have equal density, ne = ni = 20 cm -3 and equal temperature, T e = T i = 274 keV. The electron mass is 1/300 times the ion mass, and both are treated, explicitly, as particles in the simulation. The high temperature and high electron mass were chosen to minimize the computer time
while preserving the important physics. Other runs at differ- ent temperatures and electron/ion mass ratios show similar qualitative behavior.
Throughout the computational volume, both in the vac- uum outside the initial plug of hot plasma and within the plug, there is a uniform axial magnetic field of strength B z = 86 nT. The plasma has an initial ratio of particle thermal kinetic energy density to magnetic field energy density,/3 = 300, and a ratio of electron plasma frequency to electron
cyclotron frequency of tOpe/tO•e = 17. The analytic "free- axial" model of Gisler and Lemons [1989] predicts a maxi- mum radial extent of 228 km for the plasma with these parameters, which compares very well with the maximum radial extent in the simulation of 220 km. The radial expan- sion of the plasma is very well accommodated within the computational volume.
The code units in which we will be plotting various quantities are as follows: unit of length, 2 km; unit of time, 6.6 •s; unit of temperature, 6200 keV; unit of number density, 7 cm -3' and unit of magnetic field, 860 nT.
The simulation was done with the Los Alamos particle-in- cell plasma simulation code ISIS [Gisler et al., 1984], in fully electromagnetic, explicit mode, in two spatial dimensions (r, z), with all three momentum components and all six electro- magnetic field components. The problem was run for a physical time of 0.05 s, which was 20,000 time steps. There were 40,000 particles of each species, which amounted to 26 particles per cell in the original hot plasma, fewer after expansion. Energy conservation was monitored and found to be good to within 10 -3 . This run is similar to the plasma expansion runs reported earlier, which had a wide range of physical parameters [Gisler, 1989; Gisler and Lemons, 1989].
4268 GISLER AND ONSAGER: THERMODYNAMICS OF PLASMA EXPANSIONS
INTERNAL PLASMA MEASUREMENTS
The internal temperature and density of the expanding plasma were diagnosed by forming the appropriate moments of the distribution function of the computational particles in separate, dynamically adjusted "boxes" containing approx- imately equal masses. These analysis boxes were con- structed as follows.
First, during the simulation run, all particle quantities were periodically dumped to disk. For this run, there are 40 such particle dumps. Then we used a separate analysis code to divide the dumped particles into the boxes and calculate the appropriate moments. At each dump time we determined the total mass of particles remaining on the grid. Then we sorted the particles in z and divided them into 6 equal-mass axial bins. Finally, we sorted the particles in each axial bin in r and divided those particles into 3 equal-mass radial bins. This produced a total of 18 boxes of nearly equal mass; the mass in each box differed from 1/18 the total mass by at most the mass of a single particle, and there were always at least 990 particles of each species per box. Within each of the 18 analysis boxes we formed moments of the distribution func- tion corresponding to density, center of mass, average velocity in each of the three components, and temperature both parallel and perpendicular to the ambient magnetic field.
To improve the statistics, we then combined the data from the nine boxes below the midplane of the calculation (z < 164 in Figure 1) with the data from the nine boxes above the midplane after first checking that there were no gross asym- metries. This was done by reflection in z; that is, the z coordinates and velocities were reflected about 164 and 0, respectively, while the other components were unchanged. The evolution of the resulting nine boxes and their mass- weighted centers in configuration space is shown in Figure 2, for the same times as the configuration space particle plots in Figure 1. In the maximally expanded configuration in Figure 2 we have labeled the boxes with the numerals 1-9. We will
use these labels in the subsequent discussion to refer to the individual boxes. Thus box 1 is the central box, while box 7 is at the equatorial edge and box 3 is at the polar mirror.
Because the boxes are dynamically redetermined by the postsimulation analysis at every dump, the boxes can be said to be roughly comoving with the plasma, up to a point. Thus the box positions are always moving with respect to the computational grid, but they maintain a relatively fixed position with respect to the plasma cloud as it expands and recontracts. Of course, the box borders can never cross the boundaries of the computational box, while the expanding plasma obviously does (but only along the axis). Thus boxes 3, 6, and 9 have their upper z boundary at the edge of the computational box at all but the earliest times. Also, the boxes do not contain the same particles at all times: particles can and do move freely between boxes.
The division of the plasma into analysis boxes is arbitrary, in the sense that a different division scheme yields slightly different results. It is our aim here to compare the behavior in the central region of the expanding plasma with the behavior at the radial and axial expansion edges. Before arriving at the scheme just outlined, we tried a number of different approaches: purely axial boxes, purely radial boxes, different numbers of boxes, and boxes with fixed spatial dimensions. Qualitatively, all such schemes give
164
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4'1 8'2 z 1•. 3 164 0 41 82 z 123 164
164
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0 41 8'2 z 1•.3 164 0 41 82 123 164
Fig. 2. The configuration space locations of the nine analysis boxes for the same epochs as in Figure 1. In this figure the center of the expansion is at (0, 0); data for z < 164 and z > 164 in Figure 1 have been combined by reflection. The box boundaries were deter- mined by sorting the particles into equal-mass bins in z and then subsorting on each of those bins into equal-mass bins in r. The dot in each analysis box represents the mass centroid of that box.
similar results; the schemes that do best at isolating the important differences while preserving good particle statis- tics are those with comoving boxes containing roughly equal masses of plasma.
For an overview of the dynamics we first plot in Figure 3 the time histories of the density and temperature averaged over all boxes, as well as the time history of the magnetic field at a point in the equatorial plane outside the initial plasma volume. The overall density evolution simply illus- trates the dynamics. There is a decelerated expansion fol- lowed by a contraction followed by another expansion. At maximum expansion the ions are underdense compared to the electrons because of the finite ion gyroradius: the ions occupy a greater volume. The density reached during the contraction is down from the original density by a factor of 10.
The magnetic field history illustrates the snowplow pileup of the magnetic field in advance of the plasma expansion front, followed by the diamagnetic cavity and the subsequent field recovery.
The electron temperature plot offers no qualitative sur- prises. Parallel and perpendicular temperatures are equal, and the temperature falls during the expansion and rises during the subsequent contraction. The electrons have an initial thermal velocity such as to make 10 crossings of the entire computational volume over the lifetime of the calcu- lation. Since the plasma itself is generally rather smaller than that volume, the electrons are and remain pretty well mixed, communicating between all the analysis boxes. Therefore we will not display the electron temperature for each box separately; such plots show only small differences from the overall average temperature shown here in Figure 3.
GISLER AND ONSAGER: THERMODYNAMICS OF PLASMA EXPANSIONS 4269
Fig. 3. The time history of important quantities: (top two pan- els) the overall average electron and ion density history and a characteristic magnetic field history and (bottom two panels) the overall average electron and ion temperatures. In the top panel we show the electron and ion density as a function of time averaged over all analysis boxes for the entire simulation run. The expansion to a minimum density and the following contraction are clearly seen. The second panel from the top shows the magnetic field as a function of time at a point in the simulation midplane outside the initial plasma volume. More precisely, this field probe is located at (z = 164, r - 55) in the coordinate system of Figure 1. The field at this location first increases above its initial value as the expanding plasma plows into the ambient field. As the edge of the plasma sweeps past, this probe is overtaken by the diamagnetic cavity, where the field has about 15% of the ambient value. During the contraction the field recovers its initial strength until the probe is again overtaken by the diamagnetic cavity of the next expansion. The third panel from the top shows the electron parallel and perpendicular temperatures during the simulation. The electrons remain very isotropic at all times, and the fall and rise of the temperature occur in synchrony with the density fall and rise. The ion parallel, perpendicular, and total temperatures are shown in the bottom panel. There are two phases of parallel ion reheating, one during the first expansion at t = 2000 and another during the contraction at t = 6000.
The ions, on the other hand, offer a number of small surprises and one big one. The small surprises are, first, that a large temperature anisotropy develops, first favoring T•_ and later favoring Tll; second, that the minimum in the parallel temperature occurs well before the time of maximum expansion; and, third, that there are two distinct episodes of
rising Tll, the first one just after the minimum, while the plasma is still expanding, and the second one during the contraction. The big surprise is that the parallel temperature achieved during the contraction is several times larger than the initial parallel temperature.
The ions make only a few crossings of the expanding plasma during the life of the calculation. This is inevitable, since the initial expansion speed is very close to the ion thermal speed. But this means that the ions do not commu- nicate between the analysis boxes as readily as do the electrons. We shall see that the ion temperature histories in separate regions of the expansion differ remarkably from each other.
We now examine the time histories of density and ion temperature in the analysis boxes. In Figure 4 we plot the time history of the electron and ion number densities in all nine boxes. The dynamics of the expansion and contraction are again clearly reflected in these plots. The density drops to a minimum in most boxes around t = 4000 and then
increases to a maximum around t = 7000 before decreasing again. The maximum in the density observed during contrac- tion is considerably smaller than the initial density.
In all boxes except the central ones (1, 2, 4, 5) the initial decrease in density is precipitous. This is an indication that the initial expansion is essentially free. The plasma in the central boxes, on the other hand, is initially confined by the plasma in the outer boxes, so its density remains close to the initial value for the first few dumps before subsequently declining. The delay in the ion density decline is comparable to the thermal electron crossing time of the original plug. From this we conclude that the ions at the center start
moving outward in response to the electric field carried by the electron rarefaction wave that reaches the center at t -
116 (the second dump is at t = 200). In the outer radial boxes (7, 8, 9) the ions appear under-
abundant with respect to the electrons. This is because of finite gyroradius effects: since the ions make excursions across the field, they occupy a larger volume in the outer boxes than the electrons do, and hence their density is lower. Similarly in the end boxes (3, 6, 9) the electron density drops faster than the ion density at the beginning because the electrons race out along the axis and fill a bigger volume. The ion density eventually catches up, since the lead electrons leave the computational volume fairly quickly. In the inner boxes the electron and ion number densities remain so nearly equal throughout the expansion- contraction cycle that they cannot be distinguished on these plots.
In Figure 5 we show the ion temperature time histories for each box separately. These histories differ markedly from each other. Nevertheless, all boxes show the minimum of Tll well before maximum expansion, the two distinct episodes of rising Tll, and the greater Tll at contraction than at the beginning. As for the temperature anisotropy, only the axial end boxes (3, 6, 9) ever show T_• greater than Tll; in all others, Tll dominates from near the beginning. Note in particular that the time of the initial reheating in Tll differs in
4270 GISLER AND ONSAGER.' THERMODYNAMICS OF PLASMA EXPANSIONS
10
ø'øø•0 20'00 40'00 60'00 8000 ø'øø•0 20'00 40'00 60'00 8000 ø'øø•6 •6'•"'E•j-'•j•"'•-ooo time time time
Fig. 4. Separate time histories of ion and electron number densities for all nine analysis boxes. The expansion begins later in the four innermost boxes than in the outer boxes. Charge neutrality is preserved except in the outer boxes where the ions are underdense because of finite gyroradius effects.
a systematic way from box to box. This reheating occurs first in box 9 and last in box 1, with an evolution from right (axial end) to left and from top (radial edge) to bottom. The earliest
observed minimum Tii, in box 9, is at time t --- 1000, and the latest, in box 1, is at t --- 2600. This time difference is roughly commensurate with the ion acoustic crossing time across half the original plug, which is ---1439 units.
In order to compare the behavior of this expanding plasma with an idealized adiabatic expansion, we display in Figure 6 plots of log T i versus log n i in each of the nine analysis
boxes. Also plotted is the line for the three-dimensional adiabatic expansion of a monatomic ideal gas, T oc n 2/3 ' The initial condition is at right, where the adiabatic line meets the curves for parallel and perpendicular ion temperature.
During the first part of the expansion phase (moving down and to the left in these plots) the inner boxes (1, 2, 4, $) are closely adiabatic. That makes sense; the plasma in these boxes is confined by the plasma in the outer regions and must do work on the confining plasma in order to expand. The boxes at the radial edge (7, 8) expand to the time of the
0 2000 4000 6000 8000 0 20'00 40'00 60'00 8000 time time
Box 2
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1
T o.1 O.Ol
o.ool
Box 3
cccCC•c• •
0 20'00 40'00 60'00 8000 0 20'00 40'00 60'00 8000 time time
Fig. 5. Separate time histories of the ion temperature for all nine analysis boxes. All boxes show a minimum in parallel temperature well before the time of maximum expansion. The subsequent reheating occurs first in the outer boxes and last in the central box.
GISLER AND ONSAGER: THERMODYNAMICS OF PLASMA EXPANSIONS 4271
1 '=' TII .r,. T_L Box7 I T0.1 -- adiabatic ._•
1
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0.001 0.001 0.01 0.1 n 1
1 1
0.01 0.01
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1 Box 1
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Box 2 1
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0.001 0.001 0.01 011 n ; 0.001 0.01 011 n ; 0.001 0.01 0.1 n 1
Box 3 1
Fig. 6. Ion temperature-density plots for all nine analysis boxes. The straight diagonal line is the adiabatic relation between temperature and density for an ideal gas with 3 degrees of freedom. The outer boxes all start out in isothermal free expansion and thereafter tend toward the adiabatic. The inner boxes start out expanding adiabatically, until the parallel reheating occurs. This reheating is seen in these plots as an excursion nearly perpendicular to the adiabatic line.
second dump without losing temperature; they are evidently in free expansion. This is because it takes time, roughly a quarter of an ion gyroperiod, for the magnetic field to arrest the expansion of the suddenly released hot plasma. There- after, the expansion is nearly adiabatic, along a line that parallels the original adiabat but at greater entropy. This
adiabat is followed more closely by Tñ than by TiI, since it is principally the radial (perpendicular) expansion that does work against the confining magnetic field. The boxes at the axial edge (3, 6, 9) also start out in free expansion, and T•_ in these boxes remains nearly constant over a huge drop in before eventually decreasing. In the "neck" boxes (3, 6), starts to decline along an adiabat at about the third dump. By this time the plasma has started to adopt the barrel-shaped configuration, or magnetic bottle, and so the plasma within the cavity is partially confined by the mirror that forms in these boxes. Further axial expansion of the cavity can then occur only at the expense of the parallel temperature, and so Tii declines adiabatically, again along an adiabat at higher entropy than the original one. The fact that the neck is a mirror region is partially substantiated by the observation that T•_ is considerably greater than Tii here.
As observed before with regard to the time history curves, all boxes show a reheating in the parallel ion temperature well before the time of minimum density. This reheating has a particularly striking form in these plots, for all except the edge boxes (7, 8, 9). It is an excursion in the (T, n) plane nearly perpendicular to the adiabatic line. This suggests that it is due to a pure change in entropy. The end of this excursion coincides with maximum expansion (minimum density), after which the contraction occurs nearly adiabat- ically.
The perpendicular ion temperature remains nearly adia- batic throughout the expansion and contraction in the inner boxes (1, 2, 4, 5), but during the second expansion, Tñ is also seen to be climbing above the initial adiabat. This may
be due to an anisotropy-driven instability. In simulations that we have followed through three expansion-contraction cycles, isotropy in the ion temperature is eventually re- stored, with both temperatures well above the initial adiabat.
To follow up on the suggestion that the Tii excursion represents a pure change in entropy, we have plotted in Figure 7 the quantity
S=log T+(y+ 1) log V (1)
for each of the analysis boxes, where T is the total temper- ature, T = (Tll + 2T•_)/3, V is the box volume, and 7 is the ratio of specific heats, taken here to be 5/3. This quantity is proportional to the entropy, disregarding an additive con- stant [see Chandrasekhar, 1939].
The initial steep rise in entropy observed in the axial end boxes (3, 6, 9) is due to the free expansion at the beginning of the calculation. This rise stops at about t = 1000, when the neck forms and seals off the cavity. Thereafter, the entropy in the end boxes actually decreases, as the flow becomes organized by the mirrors, and particles are sorted according to whether they fall in the loss cone or not. This decrease continues until the mirrors are destroyed in the contraction.
In the central boxes (1, 2, 4, 5) the entropy is nearly constant at first; then there is a sharp rise, occurring earlier in boxes 4 and 5 and later in boxes 1 and 2, after which the entropy is again nearly constant. The outer equatorial boxes (7, 8) show the initial free-expansion increase in entropy followed by an increase which is similar to but earlier than the increases in the central boxes.
INTERPRETATION
The following scenario may account for the early rise in Tll in the inner boxes. At early times, plasma at the axial end of the plug streams out in free expansion. Because this expan-
4272 GISLER AND ONSAGER: THERMODYNAMICS OF PLASMA EXPANSIONS
Fig. 7. The time history of the entropy for all nine analysis boxes. Entropy is generated in the outer boxes, which are initially in free expansion. Eventually, some of the increased entropy is shared with the inner boxes as the mirrors organize the flow.
sion has a radial component as well, it begins to bend the field lines out, forming a magnetic mirror. No plasma is as yet trapped by this mirror, however, since the mirror is formed by decreasing the field within the plasma. If all particles strictly preserved their adiabatic invariants during the mirror formation, then Tñ would decrease while Tll remained constant, so the population would remain un- trapped. As we have seen, however, though Tñ does de- crease a little faster than Tll within the body of the plasma, isotropy is fairly well maintained (until Tll starts to increase), presumably through pitch angle scattering off the strongly fluctuating fields within the cavity. In this way a trapped population of particles develops.
The trapped population can now migrate back and forth between the equatorial regions and the mirror regions. Barring other considerations, in the equatorial regions the trapped population has a nearly isotropic temperature distri- bution, while in the mirror regions it has a distribution that is dominated by Tñ. In the mirror regions a particle sorter works to remove those particles with large axial velocities, while returning those particles with low axial velocities to the equatorial regions. Hence, after the first initial increase in entropy in the mirror region the entropy is allowed to decrease locally by virtue of this particle sorting. However, the population of particles that is returned to the equatorial regions is characterized by high entropy, because of the free expansion that occurred early on.
As this reflected population, which has high entropy, high Tñ, and low Tll, migrates back to the equatorial regions, it should evolve into a population with approximate tempera- ture isotropy and high entropy. However, the configuration is now much bigger than it was initially, and its expansion speed is now larger than the current ion thermal speed (since the population has cooled). This means that only a small part of the reflected population can ever get back to the central region, namely that part with axial speeds considerably
higher than the average. Thus Tll is observed to rise well above Tñ on account of this reflected population, and this rise occurs last in the central boxes.
The reheating in Tll is seen in the outer radial boxes before it is seen in the inner ones. This is because the ions having large pitch angles will mirror earliest, and these will tend to go out to large radial distances when they return to the equatorial regions.
Confirmation of this basic picture is given by the axial phase space plots shown in Figure 8, where it is seen that the distribution is perceptibly thinner at the center than farther out and that the thickening has the appearance of a front that is propagating inward from the axial edges.
In this way the plasma within the cavity reaches temper- atures greater than expected for adiabatic expansions. De- pending upon the rise in entropy at the axial ends that is subsequently shared with the plasma in the cavity interior, the temperatures achieved may be comparable to and indeed greater than the initial temperature. There is, of course, nothing particularly special about the initial temperature: once the plasma has been shifted to an adiabat of higher entropy, its subsequent adiabatic evolution could bring it to the initial temperature or greater at densities significantly lower than the initial density.
GENERALIZING FROM ONE WELL-DIAGNOSED RUN
Over the past few years we have run a large variety of simulations of diamagnetic expanding plasmas, all of them prior to, and not as thoroughly diagnosed as, the one presented here. Physical quantities we have independently varied include the initial thermal energy, the external mag- netic field strength, the size and shape of the initial plug, the ion-to-electron temperature ratio, and the ion-to-electron mass ratio. We have also varied quantities of mainly numer- ical interest: the cell size, the time step, the number of
GISLER AND ONSAGER: THERMODYNAMICS OF PLASMA EXPANSIONS 4273
Fig. 8. Ion axial phase space plots at two times during the expansion. The distribution in axial momentum is thinner at the center than toward the ends of the expansion. The region of thicker momentum-space distribution propagates back toward the center as particles with progressively lower pitch angles are reflected back from the mirrors at the expansion necks.
particles, and the size of the computational box compared to the maximum extent of the expanded plasma.
Examination of the limited diagnostics installed in the earlier runs leads to the conclusion that nonadiabatic behav-
ior similar to that described here probably occurred in all or most of them as well. Final temperatures an order of magnitude greater than the initial temperature are not un- common in the earlier runs. Unfortunately, without carrying out the expensive and time-consuming postprocessing that we did for the run reported here, we are unable to quantify the entropy increase in a way that has predictive value.
We have also performed a simulation of a freely expanding thermal plasma, with no external magnetic field, and sub- jected it to the same sort of analysis described here. In that case we found the expansion to be largely isothermal, as predicted by elementary thermodynamics. More specifi- cally, the outer boxes were entirely isothermal while the inner, initially confined boxes began in adiabatic expansion and then suffered a pare increase in entropy when the rarefaction wave reached them. All boxes end up expanding isothermally, with very nearly the same temperature.
CONCLUSIONS
The sudden heating of a small region of plasma in a weak uniform magnetic field produces a barrel-shaped cavity that expands and eventually recontracts. Plasma is continually lost from the cavity as it expands almost freely along the magnetic field. As the cavity develops, mirrors form, and a fraction of the plasma becomes trapped within the cavity although low-pitch angle particles continue to leak out along the axis.
Because of the loss of plasma out the ends of the barrel, the work done on the field during the expansion is returned to fewer particles during the subsequent contraction, with the result that the internal temperature rises during the cycle. This may account for the observed high temperatures seen in "hot flow anomalies" upstream of the Earth's bow shock (see, for example, Onsager et al. [1990]). The mech- anism by which the ions are heated seems to be a sharing of the entropy generated during the free expansion at the ends of the barrel with the population of particles that becomes trapped in the cavity when the mirrors form. We are led to conclude that diamagnetic plasma expansions are not entirely adiabatic but have phases that are adiabatic and other phases in which entropy is generated and transported, so as to produce internal temperatures that are higher than expected.
There is one further proviso that limits the applicability of the results reported here. The simulations performed in the course of this work have all been in an external vacuum.
This is, of course, not the case with real-life expansions in nature. Nevertheless, the essential feature giving rise to the generation and subsequent trapping of entropy is the asym- metry between expansion along the field and expansion across it. This asymmetry does not go away if the external medium is not a vacuum: expansion along the field is still much easier than across it.
Acknowledgments. We would like to acknowledge helpful con- versations with Don Lemons, Peter Gary, and Bob Holzworth.
The Editor thanks T.-Z. Ma and another referee for their assis-
tance in evaluating this paper.
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(Received April 29, 1991' revised October 23, 1991'
