1D Hybrid Results

11/16/2005

Simulation results of the asymmetry problem are shown below.
Test case:

sulfur maxwellian core, isotropic, Tperp=Tpar= 100 eV
sulfur ring, vr = 57 km/s, thermal spread of ring with Tpar= 5 eV

run parameters:

ntimes=2000
dtwci=0.05D0
nx=64
xmax=20.0D0
wpiwci=3.074252D3
nsp=2
nspec(1)=10240
nspec(2)=10240
vbspec(1)=0.0D0
vbspec(2)=0.0D0
dnspec(1)=0.8D0
dnspec(2)=0.2D0
btspec(1)=0.0630D0
btspec(2)=0.00310D0
anspec(1)=1.0D0
anspec(2)=107.0D0
wspec(1)=1.0D0
wspec(2)=1.0D0
bete=1.0D-4
resis=0.0D0
theta=0.0D0
iemod=0

spatial step = 0.3125 c/wpi; particles per cell = 320; run time = 100 Omega_i*t

Initial velocity distribution
(red = background fixed, black = beam injected)
Fig 1

Velocity Space
Fig 2

Velocity Space
Fig 3

Velocity Space
Fig 4

Velocity Space (core is top, beam is bottom)
Fig 5

B and E field energy
Fig 6

Helicity components energy
Fig 7

Temps (normalized to simulation units)
Fig 8

Average velocities (normalized to simulation units)
Fig 9

Relative drift velocity (vd = avg(vx_beam) - avg(vx_core))
Fig 10

Numerical heating temps
Fig 12

Numerical heating average velocities
Fig 11

Numerical heating relative drift velocity
Fig 13

The asymmetry is produced because the L waves traveling along +x and -x have different growth rates and amplitudes, so the particles are accelerated differently.

Over time, the relative drift between the beam and the core (v_drift = avg(vx_beam) - avg(v_core)) grows larger (from ~0 to ~ 0.3 vA at saturation). Since there's a drift, then the velocity distribution function is no longer symmetric with respect to the magnetic field, and linear theory predicts that the two helicity modes will not have the same growth rate. But I don't know if the increasing relative drift is the cause of the different helicity energies or vice versa.

I was worried that the tiny nonzero current which exists in the simulation at t=0 was somehow getting magnified over time or that numerical heating was responsible. Figs 11-13 show the results when I set the ring density equal to 0 to see the numerical heating.

I've run many simulations and have found:
- vdrift increasing with time means positive helicity will be dominant and vice versa
- the asymmetry follows the magnetic field direction (if the field is switched to B=-B0x then the results also switch so in the frame of reference of the magnetic field everything looks same)
- increasing the #particles per cell increases the asymmetry and |v_drift| gets even larger (I think this is because it takes longer for the instability to start growing when there are more particles, by which time the relative drift is larger; this is despite the decrease in initial simulation noise by adding more particles)
- adding a small artificial drift (set vbspec > 0) to oppose the value of v_drift at the start of instability growth results in ~equal growth rate for both helicities
- things like switching the species order, changing the random number generator, changing the number of particles per cell can all influence the degree of asymmetry as well as its orientation
- the asymmetry is most apparent at and after saturation, and over time it does decrease, but is still apparent here after 400 Omega_i*t
- if I simulate a stronger instability by increase the ring density to 99%, the asymmetry is still there but to a lesser degree, and it decreases more quickly. The relative drift at saturation is also less, but still on the order of ~0.01 vA. However, if I instead increase the ring velocity by 50% to make a stronger instability and leave the beam density at 20%, the asymmetry becomes even more apparent (the v_drift at saturation is similar to its value before, however the drift velocity continues to increase after saturation so that at 100 Omega_i*t it is up to 0.05 vA).

Any thoughts about what's causing this and what I should do about it??

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