wave-particle diagnostics for plasma simulation

WAVE-PARTICLE DIAGNOSTICS FOR PLASMA SIMULATION

Viktor K. Decyk University of California Dept. of Physics

405 Hilgard Avenue Los Angeles, CA 90024 USA

ABSTRACT. One important type of problem for particle simulation of plasmas, is one which is bounded and has external sources and sinks.

For example, there are problems with antennas for studying RF heating or current drive in fusion plasmas, or problems in space simulation where particles are injected at one boundary with some specified energy or momentum distribution. In understanding such simulation results, it

is useful to know how energy and momentum are flowing inside the plasma. This can be accomplished in electrostatic particle simulations on the basis of some theorems for energy and momentum flow. An important application of these theorems occurs when many waves are involved in producing some effect, e.g., generating a current by RF. One can then extract the contribution of each wave to the effect to

identify which are most important. Examples of such wave-particle diagnostics will be given.

Today I will talk about diagnostics for particle simulation. I will begin by giving a general overview of my philosophy of organizing

diagnostics and give some examples of generally useful[ diagnostics. Then I will specialize and give more details about diagnostics for bounded electrostatic plasmas which are driven by external sources of energy and momentum.

Diagnostics, I believe, are where the real physics get done in a simulation code, where you finally get to understand what is happening

and can explain it to someone else. Simulation codes generate much more information than one can possibly use--no one wants to know the trajectories of all the particles. Instead, the purpose of the

simulation, as of physics, is to explain the behavior of the physical system under study in terms of a small number of simple concepts.

Most of the time we want to represent the simulation data in terms

of concepts which are known to be useful, or which appear in some theory. For example, the idea of a particle distribution function or plasma potential function appear in most theoretical descriptions of plasmas and are necessary diagnostics to include in most any simulation. Occasionally, the concepts which best describe the phenomena under study are not even known, and their determination may,

Space Seience Reviews 42 (1985) 113-130. 0038-6308/85.15 �9 1985 by D. Reidel Publishing Company.

114 V . K . DECYK

in fact, be part of the research goal. For example, the best or most

simple way to describe strongly turbulent systems probably has not yet

been found, so one can be creative and try to contribute to the evolution of new ideas.

The dynamics part of a particle simulation code usually evolves into a fairly standard type that one does not change much. 1-z But the diagnostics one needs and uses are constantly changing as one tackles new physical problems. Therefore, I have found it useful to do most of my diagnostics in a separate post-processing program, perhaps even on a

different computer. The main simulation code generates a database, or a collection of

datafiles, which are very general, and stored on an external device, such as a disk, as the run proceeds. This includes things like the time history of the velocity distribution functions or the time history of a subset of the Fourier modes of the potential. In addition, the main program generates certain key diagnostics whose primary purpose is to point out those things in the database which need to be examined further. This would include things like the time-averaged fluctuation energy of the modes of the potential, from which one can decide which modes have been excited. Thus, the main program can be regarded as producing a database which is a first level reduction of data.

From the database, a further reduction of data occurs, as more detailed diagnostics specialized to the problem at hand are done. This could include things like frequency analysis, cross-correlations, etc.

This can be done interactively, which is especially useful in the preliminary runs on a new problem when one is still feeling ones way around. Once one has learned which particular diagnostics are most useful, then this post-processing can be done automatically as a batch job. As an example, when one first starts a problem, one may not know which frequencies are most significant. But eventually one decides what is worth looking at, so that future runs can produce output in the

frequency range of interest automatically. This style of doing diagnostics requires additional off-line

storage, but this is not usually a severe constraint, and it has the advantage that as one thinks of new particular diagnostics, they can often be calculated from the database, without having to do the run all over again. (It is rare that I think of the best diagnostics for a new

problem the first time around.) From that general outline, let me now illustrate by specific

examples. The model I will be talking about is electrostatic, with two spatial dimensions, and three velocity dimensions. One of the spatial co-ordinates is periodic, and the other is bounded. ~ Generally, there is an electrostatic antenna on one boundary driving the plasma, since this code was used for simulation of rf heating for fusion applications.

For such problems, the main code could generate five different

data files. These were:

I. Trajectories of test particles, ~(t). II. Velocity distribution functions, either one dimensional,

such as

WAVE-PARTICLE DIAGNOSTICS FOR PLASMA SIMULATION 115

Vx+AV X ~

F(vx) ~ .[ dv x' [ dvy' [ dv z' f(vx' , Vy', Vz')

V X -~ --~

or two dimensional, such as

vl+Avl v TT+Av fr 2~ F(v i, wl t ) - ] vl'dv I' / dvlr' ] d~' f(vi' , vli' , 0')

v i v II o

These are calculated much like charge distribution is calculated by accumulating number density on a velocity-space grid. Occasionally, the distribution of some other quantity as a function of velocity can also be useful, such as particles' energy gain F'v, which can identify velocity space resonances.

Ill. Plasma profiles versus the bounded or inhomogeneous co-ordinate x. Usually, the particle density, drift velocity, and kinetic energy

versus x are calculated. Sometimes the energy absorption F.v or energy flux versus x are useful.

IV. Ion density for various Fourier modes in the periodic direction versus x,

Omi(x) 1 !LY pi (x, y) e-2~imy/Lydy

Ly o

This is expecially useful for looking at ion waves. Since the ions do not respond to high frequency waves, this data automatically has the high frequencies filtered out.

V. The potential for various Fourier modes versus x, ~m(X).

Most of these quantities are calculated and stored every 5 or i0 time steps.

In addition to generating these datafiles, the main program prints or displays the zeroth, first, and second moments of the velocity distribution functions, to tell me which components are evolving in time and need to be examined further. Occasionally, phase space diagrams of the particles are displayed. At the end of the run, the time histories of the field and kinetic energies and momentum are displayed. The work done and momentum imparted by the antenna is subtracted from the total energy and momentum of the plasma to determine how well the code is conserving these quantities. I will talk in more detail about this later. Finally, the integrated electric field fluctuation spectrum versus Fourier mode number is displayed to indicate which modes have high energy levels.

116 V.K. DECYK

From this kind of database, let me give some examples of what

kinds of things the post-processing programs calculate. The first few examples I will show come from a problem where we were driving lower hybrid waves by an external antenna for fusion applications. ~ The plasma was inhomogeneous with a density profile which increased away from the antenna in the bounded direction. The magnetic field was in

the periodic direction. The program which analyzes the potential datafile, ~m(X,t)

calculates quantities such as the integrated frequency spectrum per Fourier mode m:

L x ii T - 2 Pm(~)=/ dx ~ J V+ m (x,t)e-l~tdt

o o

The length of integration T depends on how much resolution one needs, but is also limited by how fast the waves are changing in the plasma. The best choice may not be known ahead of time. Figure I shows the spectrum for mode m = I, which corresponds to kll %DE = .034, and one sees different peaks which correspond to different eigenmodes in the x direction. Since the plasma is inhomogeneous in x, the eigenmodes are not sinusoidal functions of that coordinate. Picking one of the frequencies mo which correspond to the peaks, one can obtain the actual eigenfunction by going back to the original data and, using the

cross-correlation program, beat +m(x,t) with a reference signal at ~o to produce the interferogram defined by:

T-T 1

Cm(x'T) ---T-T I ~m(x,t+T)sin(~ot)dt- o

When the eigenfunction is excited only by the thermal fluctuations, as in this example, the length of integration time T must not be longer than the correlation time of tbe wave. The top graph of Fig. 2 shows the eigenfunction obtained from the simulation and the bottom one shows the WKB solution.

The spectrum in Fig. i showed the natural frequencies of oscillation of this inhomogeneous plasma. If the antenna frequency does not match one of these eigenfrequencies, then the plasma responds with constant amplitude driven oscillations, much like a harmonic oscillator which is driven off resonance. I will discuss later what happens if the frequencies do coincide.

In the next example, shown in Fig. 3a, one sees the interferogram of such a driven lower hybrid wave, obtained from

T-T 1 j - (x,t+T) sin( t~ot )dt E x,T t( ) -= T-T 3x

o


P (~1

'1 I I l 1 I 1

m = 1

0 ~/~.

i_.f

1.5

Fig. I One fourier component (m=l) of electric field spectrum for

plasma bounded in x and periodic in y.

1 18 V .K . DECYK

Fig. 2b.

m

t i I I I ; i I I J t s 4

r n = 1 ,I

i 1 1 I I l 1 ; l 1

0 x 64

Eigenmode of bounded, inbomogeneous plasma, extracted from thermal fluctuations in simulation.

I I 1 I I i 1 l I * ij _ ~,.'~,., *. 2 5 7 J

- 'n= ! i

- 1 I l l ; l i l I I

0 x 6 4

Fig. 2b Eigenmode of bounded, inhomogeneous plasma, from W.K.B. solution of fluid equations.


where ~0 o is the driving frequency and ~m/3X was obtained from the database by finite differencing. By displaying the waveform at different values of r, one can determine the direction of propagation of the wave fronts. In this example, the antenna is on the left, so that one can see that we have a backward wave, as one expected for a

lower hybrid wave. Once can also see that the wave amplitude becomes very small, indicating that the wave is nearly completely absorbed.

Another quantity one can calculate from this datafile is the local wavenumber. If we have a traveling wave of the form

Ei(x,z) = A(x) sin[ J ki(x')dx' - mot + 4]

Then the amplitude is given by

A(x) = L.M. [EI(x,T)]

where L.M. refers to the local maximum as a function of time for each x. The derivative

d__ [Ei(x,T) 1 = kL(x ) cos[ f kl(x')dx' - ~o T + 4] dx L A(x) J

so that

El(x, T) kl(x ) = L.M. [d { }] dx A(x)

The dots shown in Fig. 3b represent these measured values from the simulation and the solid line comes from a local (at each x) solution

of the dispersion relation. These are just a sample of the kinds of diagnostics which are

useful from this data file. The datafile of ion density is treated by the same post-processing programs to analyze ion waves.

Figure 3c shows an example of a plot from the datafile which

contains the plasma profiles. It shows the energy absorption F.v versus x, averaged over time to remove the oscillating part. What it illustrates is where in space ion heating occurs. One sees that heating occurs in a rather localized region of space in the interior, as well as in a region just in front of the antenna. This was not seen in a plot of particle kinetic energy versus x, because the amount of heating was small, and because the particles could move away from the heating region, giving a much more uniform distribution of kinetic energy.

Having such data in a file also means that one could superimpose graphs illustrating the change in time of some quantity, which makes for clearer presentations.

The trajectories of test particles in phase space can sometimes be the most revealing diagnostic of all. However, it can be the most difficult to use in practice, since one is usually looking for those few particles which clearly illustrate the effect one is looking for out of a large number.

Fig. 3a.

X

eE.:'slj Mivi / /

g

E~ f o r p r o p a g a t i n 9 LH wave

~ o ' r . 2.16

Illl\\\\ ~

Fig. 3b.

Perpendicular electric field of driven lower hybrid wave at different phases.

p e r p e n d i c u l a r pha~e v e l o c i t L j

t I I I" I I i ~ n "',,1~;= 8 Te/Ti = 1

kji ~ theorLj

5 . . . . . . . . . . . . ".. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . �9 l p , , �9 ,

0

3# �84

Perpendicular phase velocity from fluid theory (solid line) and simulation (dots)�9

X

O

power absorption

120 V.K. DECYK

Fig. 3c. Power absorption by ions as a function of space.


One trick is to take advantage of the fact that simulation runs

are absolutely reproducible, at least on most computers. In the case of the lower hybrid heating simulations, at the end of the run, we identified which ions had heated the most. The current co-ordinates had been saved in a restart file and were therefore available, and the initial co-ordinates could be reconstructed with the initial random number generator. Once they had been identified, we ran the same run over again, this time saving the trajectories of those test ions. Then we could examine the time histories of these particles to determine how they had gained their energy.

The next graph (Fig. 4) shows one such orbit in ]perpendicular velocity space. One sees that the ion is mostly going around in its cyclotron orbit, then occasionally makes a quantum jump to another

orbit. The reason for this is that if v I > mo/kl, then the particle can be traveling along with the wave during part of its gyroperiod and can interact strongly with the wave. If the perturbation electric field is strong enough, the subsequent interaction in the next gyroperiod is decoupled from the previous one, and the particle performs a random walk in velocity space. This is called stochastic heating, b

I think that this has given you a flavor of the variety of diagnostics which are useful and possible in a particle simulation. The only advise I would give to someone just starting out is to be

flexible and creative--all the information you could possibly want is out there in the data somewhere.

For the last part of my talk, I will discuss energy and momentum

flow in electrostatic systems. I was first faced with these issues when someone always asked me after I presented by heating results, how did I know that this heating was not numerical. When a simulation is closed, one can always point out that the code conserved energy and momentum to some acceptable level. But when antennae or other sources are present, the plasma energy and momentum are not constant and I never had a good answer.

When I first started simulation, I was taught that the Poynting vector described how electromagnetic energy was flowing into or out of

a system, but that for electrostatic models such as I was using, the Poynting vector did not exist. Eventually I came to the realization that this was not correct, b

An electrostatic Poynting vector can be derived very easily. Let

Jt = J - ~ a--t

This can be regarded as the total plasma current, conduction plus displacement. These are well-defined objects in electrostatic systems. From the equation of charge continuity and Poisson's equation, one readily can show that

+ + 1 ~ = g . ~ + - - = 0 V.jt = V.j V2~ ~0 47 ~t ~t

+

Thus, Jt is also the transverse plasma current. + divergence of the vector R defined by

Since V'j t = O, the

122 V . K . D E C Y K

t ra jectory of single ion

Vx t- Bo V:~

s-" j.. --.

\

\

i

i : /

�9 y

�9 7 �9 . j , .

~ ' ~ ~.t j

Fig. 4. Ion gyro-orbit in velocity space shows stochastic heating.


= jt s

can be shown to be:

7"~ = ~t'VS = ~'VO - 1 7~ vs. ~ w

+

since E = - VS. energy

*g_ 3 E 2

Here we have an equation of continuity for electric

E 2 = -

with the right hand side representing an energy sink or source.

Thus, R is an energy flow vector for electrostatics, an electrostatic Poynting vector, if you will.

One thing most of us have read somewhere but tend to lose sight of is that the Poynting vector is not unique. 0nly its divergence has

physical significance, and other vectors with equal divergence could do + +

just as well, although we've all rather gotten used to E x B for electromagnetics.

To illustrate this, let's derive another energy flow vector. + +

Since 7"jt = 0, then i$ means that Jt can be expressed[ as the curl of some vector potential V,

+ +

Jt = 7xV + +

Another vector R' with the same divergence as R can be constructed merely by adding the curl of some vector to it, since the divergence of a curl always vanishes. Let us then construct the vector

-> + +

R' = R - 7x(SV)

Multiplying this out

since the first two terms on the right hand side cancel by construction. This alternative form of the energy flow vector is in

+

fact exactly the Poynting vector, since the defining equation for V

=it =j +----

->

1 3E 4~ ~t

is just Ampere's Law without the constants 4~/c! But magnetic fields

124 V.K. DECYK

do not exist in an electrostatic universe, so @ is merely the vector potential for the transverse current.

Thus if one had calculated the magnetic field from the currents in the electrostatic simulation, ignoring the constants 4~/c, and constructed the Poynting vector one would have gotten the correct energy flow in a purely electrostatic system, even though magnetic fields nowhwere enter into the dynamics. This equally well applies to gravitational systems, even though no one believes that a gravitational analogue to the magnetic field exists.

I have already pointed out that the energy flow vector is not unique. It is equally true that the energy density itself is also not uniquely defined. One can always add the divergence of any vector to the definition of energy and subtract the time derivative of the same amount from the energy flow vector. As an illustration, the set

E z [~_w) w ~ 8-7- V " F

= +-ff , F ,

also satisfy the equation of continuity for energy

(w) = - v.~" + ~

If we multiply these out, we find that W is just

E 2 VO-V~ ~VzO 1 W = 8w 8w 8w - 2 o0,

a definition of potential energy we have been using all along. The corresponding energy flow vector can be simplified to

~" j~ +~- -~] = [wvr -

With all these choices, I will chose the following form because it is convenient to calculate in my particle simulation code:

3 E z v.~ + ~t (~) = - }'~

-> ->

The energy source, represented by the term - j'E is energy that came from the particles. So to complete the picture, we need to examine the flow of kinetic energy. In a simulation with the finite-size

particles, with shape function S(~), one can define the kinetic energy density and flux as follows:


+ l + U(r,t) = ~ !mivi2(t) S[~ - ri(t)]

+

+ + i ~ 2(t)$i(t) S[~ ri(t)] K(r,t) = ~ miv i

where the sum is over particles, and I have neglected, as before, the effects of grids and finite time steps. By straightforward differentiation, one can show that these qualities satisfy

+

3U = ~miv i+ (t)'dvi(t)dt S[~ + v-~ +-fs 1 - r i ( t ) ]

The acceleration of a particle in the simulation is given by the Lorentz force

+

mi dvi(t) + + § + + dt qi { <E[ri(t)]> + Eext + vi(t) x Bext:/c }

where .-> ..~ + -)-

<E[ri(t)]> = -/S[r' - ri(t)]V~(r')d~r '

is the averaged self-consistent field accelerating the particle. If we use the definition

j(r,t) = )~qivi(t ) S[r - ri(t)] 1

Then one can add the equations of continuity for field and energy to give:

3 E 2 t + 7"(~ + ~) +-~ (U +~-~ ) = j.Eex t +

-> -> + -> -> +

~qivi(t)" )]> [r {<E[ri(t - E(r)} S - ri(t)] 1

The second term on the right hand side represents an internal tension force which is constraining the finite size particle to move together even when different parts of the particle feel different forces. When we integrate over the plasma volume that term vanishes, and using the divergence theorem one can write:

• S(~ + 3 E z 1 vJ'Eext dV + ~)'da + -~ [JV 8-~ dV + ~ ~imiviZ(t)] = ~ + +

+

This is the result we have been seeking. If Eex t = 0, meaning that

126 V.K. DECYK

only self-consistent electric forces are included in the equation of motion, then by integrating in time, one obtains an expression for global energy conversion:

W E + W K + It SS (~ + R)'d~ = constant o

E 2 1 !i mivi 2(t) where W E = J 8~ dV and W K =

Let me show you how I applied this to the case of RF heating simulations 4. The antenna could be represented by specifying on one boundary ~(x = 0,y,t) and setting r = 0 at x = L x. In the y direction the system was periodic. Since we used specular reflection for

^ +

particles in the x direction, the surface integral involving K vanishes, and only the surface at x = 0 contributes to the surface

+

integral over R. We then have:

3 ~ x = i jt dt +I Ly r (~x)dY+ = 0 constant WE + WK +4-~ o o

The last term on the left hand side represents the negative of the work done on the plasma by the antenna. Since the system is periodic in y, this expression can be written in terms of the Fourier harmonics:

W E + W K - ~ W m = constant m=-~

where Wm =- ~ +I~dt~m* ~ (~m]l -~ ~x ! x = 0 is the work done by each i

Fourier mode. The following example was a case where we set the frequency of the

antenna equal to one of the eigenfrequencies of the plasma which I showed earlier. In such a case, the plasma responds with secular growth of the mode in time, much like a harmonic oscillator, until some nonlinearity stops the growth. This particular example is a very

severe test. The wave phase velocity along Bex t was fast, so that the wave amplitude grow large until particle trapping occurred. At peak the wave electric field was i/3 of the initial plasma kinetic energy. By the end of the run, the plasma kinetic energy increased by more than a factor of five. The calculated work done, E W m was able to account for all but 0.2% of that growth, m

Figure 5 shows the antenna spectrum versus parallel mode number m, where kll = 2m~/Ly. It also shows a plot of W m versus m, which shows the amount of work done by each component of kll. One can see that mode m = 1 did almost all of the work, which was to be expected since it was one of the eigenmodes with that mode number which was being excited. This is a very important diagnostic in RF heating because it tells us which modes in an antenna are most effective and one which cannot be readily obtained by any other means.


4 r r ~

i .65 / "1 . s ~ , ]

.,, J ]

. r i;

'~ i . 1 5 i

M

Fig. 5a. Antenna input spectrum for each mode.

SSO00

r

5oooo

+ ~ o o o

r

3 5 0 0 O

~ o o o o

~5000

20000

15005

1OO0O

50OO

L

o [ A ' 1 1 , , , l i ~

M

Fig. 5b. Work done by antenna for each mode.

128 V.K. DECYK

I have shown you how to account for energy flow in electrostatic

simulations. The same kind of development can be carried out for momentum. Since the derivation is almost the same, I will not go through the details except to mention the things that are different. I

The equation analogous to the equation of continuity for energy 1 1

V.~ = oE where ~ =~ [EE - ~ (E.E)~] is the Maxwell stress tensor with

the magnetic field missing. This can be shown easily by carrying through the indicated differentiation:

+ +

I 1 ~ (E'E)I = 0~ Ex(VxE) _ 0~ V-% = ~ [(E.V) E + E (V.E) - ~ ~ ~ 4~

+

since V x E = 0. Note that there is no term involving the time derivative of field momentum, as there was for energy. Evidently, electrostatic fields do not carry momentum. One can derive an equation of continuity for particle momentum,

+ +

V.~ +-~ ~P = 1~ mi dvi(t)dt S[~ - ~i(t)l

where

21 mivi(t)+ S[~ + = - r •

1 + +

= ~ m i x i ( t ) X i ( t ) S [ r - r i ( t ) ]

are the particle momentum density and momentum flux tensor, respecti- vely. Adding the field and particle eqations then gives:

+ +

V.M= + ~oL = ~P V.T= + 0~ext + ~ x Bext/C

-> -> -> + ->

+ ~ qi{<E[ri(t)]> - E(r)} S[r - ri(t) ] i

Integrating over the plasma volume yields:

+

S (~ - ~)'d~ +~-~ = Jv[pEext + j • Bext/c]dV

-> ->

where ..I[ = ~ mivi(t) is the total momentum content inside the 1

s i m u l a t i o n vo lum e .


In fusion applications one is interested in the momentum along the magnetic field. For that case, the expression for global momentum conservation is expressed in terms of a vector:

t

~,. + fodt /s(M,, - ~,)'~a = constant

Further simplifying to the same geometry as before, one has:

lIll + -~-~- f d t ] Y E x E y d y x ; 0 = c o n s t a n t

I n t e r m s o f t h e F o u r i e r h a r m o n i c s , o n e c a n w r i t e :

IIT[- >~ Pm = c o n s t a n t m

w h e r e

= Pm ~ Itdt ~m 4~ 3x x = 0

is the momentum input for each Fourier mode. For the same example I used to illustrate the energy conservation,

we did not expect to generate any momentum in the parallel direction, because the antenna elements were phased symmetrically and waves were being launched in both the parallel and antiparallel directions equally. However, the plasma was not symmetric and the plasma momentum in fact did not remain constant. But the term Z Pm, representing the

m net momentum input from the antenna, was able to account for the momentum fluctuation within about a percent. In this case, the

momentum input per mode had a large variation, with some modes contributing positive momentum and some negative.

When we changed the wave phasing to launch waves primarily in one direction, then the electrons acquired a net drift of 2.5Vth e. The term ~ Pm was able to account for all but 0.3% of this momentum

increase, which was all coming from mode m = i. The last illustration shows an example from some work carried out

by Hiratada Abe and myself at Kyoto University. ~ We applied a wave spectrum similar to that in lower hybrid current drive experiments, including all the very low amplitude secondary peaks. We discovered tht the momentum input from the secondary peaks was very significant and in fact crucial to the success of the current drive. When the low amplitude modes were suppressed, the current was drastically reduced.

Acknowledgments

This work has been supported by NSF and USDOE.

130 V.K. DECYK

References

i. R.W. Hockney and J.W. Eastwood, Computer Simulation Using Particles, (McGraw-Hill, New York, 1981).

2. Charles K. Birdsall and A. Bruce Langdon, Plasma Physics via Computer Simulation, (McGraw-Hill, New York, 1985).

3. V.K. Decyk and J.M. Dawson, J. Computational Phys., 30, 407 (1979).

4. V.K. Decyk, G.J. Morales, and J.M. Dawson, Second Joint Grenoble- Varenna International Symposium on Heating in Toroidal Plasmas, Como, Italy, Sept., 1980 [Commission of the European Communities, Brussels, 1980], Vol. I, p. 365.

5. C.F.F. Karney, Phys. Fluids, 21, 1584 (1979).

6. V.K. Decyk, Phys. Fluids, 25, 1205 (1982).

7. V.K. Decyk, J. Computational Phys., 56, 461 (1984).

8. V.K. Decyk and H. Abe, Bull. Am. Phys. Soc., 29, 1307 (1984).

wave-particle diagnostics for plasma simulation

Documents