Project 3a: Waves in Magnetized Plasmas

Imported and adapted from the UCLA Plasma Simulation Group's JupyterPIC Notebooks

Before anything else, we need to change to the stored notebook's directory, which contains input files for the PIC simulator.

cd /notebooks/x-and-o-mode-dispersion

In this project, we are going to look at the dispersion relation for electromagnetic waves in a uniform magnetic field.

In this project, you will study the accuracy of the dispersion relation we derived in class for waves that propagate across a constant magnetic field.

In class we began by stating that we are interested in waves with frequencies at or near the plasma frequency so that we can assume the ion motion is not important. We let the magnetic field point in the z^\hat z direction, B0=z^B0\vec B_0=\hat z B_0. We than assumed the wave moved in the x^\hat x direction, k=x^k\vec k=\hat x k. Under these conditions we found that there are two types of waves:

The O-wave (ordinary wave) in which the electric field of the wave points in the z^\hat z direction, E=z^E\vec E=\hat z E. For this wave the dispersion relation is:

c2k2ω2=1ωp2ω2\frac {c^2k^2}{\omega^2}=1-\frac{\omega_p^2}{\omega^2}

and the X-wave (extraordinary wave) in which the electric field of the wave points in both the y^\hat y and x^\hat x directions E=x^Ex+y^Ey\vec E=\hat x E_x + \hat y E_y. For this wave the dispersion relation is:

c2k2ω2=1ωp2ω2ω2ωp2ω2ωh2\frac {c^2k^2}{\omega^2}=1-\frac{\omega_p^2}{\omega^2} \frac{\omega^2-\omega_p^2}{\omega^2-\omega_h^2}

For this wave, the ratio of EyE_y to ExE_x is:

EyEx=iωωcω2ωh2ωp2\frac{E_y}{E_x}=i \frac{\omega}{\omega_c} \frac{\omega^2-\omega_h^2}{\omega_p^2}

In addition, for the X-wave there are several important frequencies. These are the cutoff frequencies where k vanishes and the wave is reflected:

ωR=12[ωc+(ωc2+4ωp2)1/2]\omega_R=\frac{1}{2}[\omega_c+(\omega_c^2+4\omega_p^2)^{1/2}]
ωL=12[ωc+(ωc2+4ωp2)1/2]\omega_L=\frac{1}{2}[-\omega_c+(\omega_c^2+4\omega_p^2)^{1/2}]

and the resonances where k goes to infinity and the wave is absorbed:

ωh=(ωc2+ωp2)1/2\omega_h=(\omega_c^2+\omega_p^2)^{1/2}

At a resonance the wave becomes longitudinal where E\vec E is parallel to k\vec k, the wave only has an ExE_x.

In this project you will study the spectrum of waves that exist in a magnetized plasma. The constant magnetic field will point in the x^3\hat x_3 direction z^\hat z. You will simulate a uniform plasma in which each plasma electron is initialized with positions (only in x or what we call x1). Each electron is also initialized with velocities (v1, v2, v3)=(.005c, .005c, .005c) or momentum (mv1, mv2, mv3) from a Maxwellian in each direction. The particles then begin to move in the self-consistent fields that their current and charge density produce:

  • The length of the simulation window is 50 c/ωp50 \ c/\omega_p.
  • The simulation will run for a time 400 [1/ωp]400 \ [1/\omega_p].
  • The simulation uses 50,000 particles.

We will have ωcωp\frac{\omega_c}{\omega_p}= 2.

1.
Dispersion Relation:

1.1.
X-wave dispersion

Here you can look at the dispersion relation of the X-wave and the frequencies described above. Just enter ωc\omega_c and ωp\omega_{p} and re-run the cell below:

from scipy.optimize import fsolve 
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

#first we define the range of k's of interest, "k" here means "ck"

karray=np.arange(0,2,0.05)
nk=karray.shape[0]

# here we specify the plasma conditions 
#
wp=1
wc=0.7
#
#

def xwave_disp(w,omegap,omegac,ck):
    ratio=omegac/omegap
    y=(ck*ck)/(omegap*omegap)-(w*w)/(omegap*omegap)+(w*w/(omegap*omegap)-1)/(w*w/(omegap*omegap)-(1+ratio*ratio))
    return y

wR=0.5*(wc+np.sqrt(4*wp*wp+wc*wc))
wL=0.5*(np.sqrt(4*wp*wp+wc*wc)-wc)

warrayL=np.zeros(karray.shape[0])
warrayR=np.zeros(karray.shape[0])
wLarray=np.zeros(karray.shape[0])
wRarray=np.zeros(karray.shape[0])
wHarray=np.zeros(karray.shape[0])

wLarray[:]=wL
wRarray[:]=wR
wHarray[:]=np.sqrt(wp*wp+wc*wc)

warrayL[0]=wL
warrayR[0]=wR
for ik in range(1,nk):
    warrayL[ik]=fsolve(xwave_disp,warrayL[ik-1],args=(wp,wc,karray[ik]))
    warrayR[ik]=fsolve(xwave_disp,warrayR[ik-1],args=(wp,wc,karray[ik]))

plt.plot(karray,warrayR,'b',label='$\omega$ > $\omega_R$')
plt.plot(karray,warrayL,'r',label=' $\omega_L$ < $\omega$ < $\omega_H$')
plt.plot(karray,wLarray,'--',label='$\omega_L$')
plt.plot(karray,wRarray,'--',label='$\omega_R$')
plt.plot(karray,wHarray,'--',label='$\omega_H$')
plt.plot(karray, karray,'--', color='fuchsia',label='$\omega/k=c$')
plt.xlabel('wave number $[ck/\omega_{pe}]$')
plt.ylabel('frequency $[\omega_{pe}]$')
plt.title('X wave dispersion relation,')
plt.legend()
plt.xlim([0,karray[nk-1]])
plt.ylim([0,karray[nk-1]+1.0])
#plt.grid(b=True, which='major', axis='both')
plt.legend(loc=0, fontsize=8)
plt.show()

2.
O-wave dispersion

# Plotting w(k)
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
N = 5
k = np.linspace(-N,N,N*20)
w_p = 1
c = 1
w = np.sqrt(w_p**2 + c**2 * k**2)
cline = k
plt.plot(k,w, label='$\omega$(k)')
plt.plot(k,cline, label='slope = c')
plt.xlabel('k [$c/\omega_p$]')
plt.ylabel('$\omega$ (in units of $\omega_p$)')
plt.xlim(-N,N)
plt.ylim(0,N+1)
plt.grid(b=True, which='major', axis='both')
plt.legend(loc=0)
plt.show()

3.
Simulations with a Particle-in-Cell Code

In this project you simulate plasmas with similar conditions as in Project 1a and 2a.

Each plasma electron is initialized with positions (only in xx or what we call x1x_1) such that the density is uniform. The ions are initialized at the same positions but they have an infinite mass. Each electron is also initialized with velocities (v1v_1, v2v_2, v3v_3) or momentum (mv1mv_1, mv2mv_2, mv3mv_3) from a Maxwellian in each direction. The particles then begin to move in the self-consistent fields that their current and charge density produce.

  • The length of the plasmas is 50 c/ωpc/\omega_p
  • The simulation will run for a time 400 1/ωp1/\omega_p.
  • The simulation uses 50,000 particles.

You will be looking at plots of the electric field in the x3x_3 direction, E3E_3, which corresponds to O-waves, and at electric fields in the x1x_1 and x2x_2 directions, E1E_1 and E2E_2, which corresponds to X-waves.

3.1.
The following lines must always be executed before running anything else.

Reminder: Hit Shift+Enter to run a cell, or select the cell and click on the "Run" button in the top menu bar

import osiris
%matplotlib inline

4.
Run cases in which Ωce=0.5ωpe\Omega_{ce} = 0.5 \omega_{pe} and 2.0ωpe2.0 \omega_{pe}.

dirname = 'therm-b05'
osiris.runosiris(rundir=dirname,inputfile='therm-b05.txt')
dirname = 'therm-b20'
osiris.runosiris(rundir=dirname,inputfile='therm-b20.txt')

5.
Case for which Ωce=0.5ωpe\Omega_{ce} = 0.5 \omega_{pe}.

After the simulation is finished, plot E3(x1)E_3(x_1) at t100t \approx 100 (run the next cell).

  • Do you see any evidence of a coherent wave?
  • Does the plot make sense?
dirname = 'therm-b05'
osiris.field(rundir=dirname, dataset='e2', time=100)

Next, plot E3(t)E_3(t) at x1=5c/ωpx_1=5 c/\omega_p (i.e., at cell=100).

  • Do you see any evidence of a coherent wave?
  • Does the plot make sense?
dirname = 'therm-b05'
osiris.field(rundir=dirname, dataset='e2', space=100)

Next, in the following two cells, we are going to plot ω(k)\omega(k). This is generated by taking E3(x1,t)E_3(x_1,t) and Fourier analyzing in both position and time.

  • ω(k)\omega(k) with wavenumber in units of [k] = ωpe/c\omega_{pe}/c:
# omode
dirname = 'therm-b05'
osiris.plot_wk(rundir=dirname, wlim=[0,5], klim=[0,15], vth = 0.1, b0_mag=0.5, vmin=-5, vmax=2, plot_or=3) 
osiris.plot_wk(rundir=dirname, wlim=[0,5], klim=[0,15], vth = 0.1, b0_mag=0.5, vmin=-5, vmax=2, plot_or=3, 
               show_theory=True) 
  • Do these plots make sense?
  • ω(k)\omega(k) with wavenumber in units of [k] = λDe\lambda_{De}:
# xmode
dirname = 'therm-b05'
osiris.plot_wk(rundir=dirname, wlim=[0,5], klim=[0,25], vmin=-1, vmax=3, vth = 0.02, b0_mag=0.5, plot_or=2) 
osiris.plot_wk(rundir=dirname, wlim=[0,5], klim=[0,25], vmin=-1, vmax=3, vth = 0.02, b0_mag=0.5, plot_or=2, 
               show_theory=True) 
  • What are the range of frequencies where the x-wave exists?
  • Why do you think you do not see any signal in E2E_2 for kcωp\frac{kc}{\omega_p} > 10 for the waves near ωH\omega_H?
  • What is the order of ωR\omega_R, ωL\omega_L, ωH\omega_H, and ωp\omega_p? (highest to lowest)
# xmode
dirname = 'therm-b05'
osiris.plot_wk(rundir=dirname, wlim=[0,5], klim=[0,25], vmin=-1, vmax=3, vth=0.02, b0_mag=0.5, plot_or=1) 
osiris.plot_wk(rundir=dirname, wlim=[0,5], klim=[0,25], vmin=-1, vmax=3, vth=0.02, b0_mag=0.5, plot_or=1, 
               show_theory=True) 
  • Why don't you see much signal in E1E_1 for the frequencies above ωR\omega_R
  • Do you have any guesses as to what the extra curve is due to?

6.
Case for which Ωce=2.0ωpe\Omega_{ce} = 2.0 \omega_{pe}.

dirname = 'therm-b20'
osiris.plot_wk(rundir=dirname, wlim=[0,5], klim=[0,15], vmin=-5, vmax=2, vth = 0.1, b0_mag=2.0, plot_or=3) 
osiris.plot_wk(rundir=dirname, wlim=[0,5], klim=[0,15], vmin=-5, vmax=2, vth = 0.1, b0_mag=2.0, plot_or=3, 
               show_theory=True) 
  • Why didn't the ω(k)\omega (k) plots for E3E_3 change when B3B_3 was increased?
dirname = 'therm-b20'
osiris.plot_wk(rundir=dirname, wlim=[0,5], klim=[0,15], vmin=-5, vmax=2, vth=0.02, b0_mag=2.0, plot_or=2) 
osiris.plot_wk(rundir=dirname, wlim=[0,5], klim=[0,15], vmin=-5, vmax=2, vth=0.02, b0_mag=2.0, plot_or=2, 
               show_theory=True) 
  • When B3B_3 was increased did the order of ωR\omega_R, ωL\omega_L, ωH\omega_H, and ωp\omega_p change?
dirname = 'therm-b20'
osiris.plot_wk(rundir=dirname, wlim=[0,5], klim=[0,15], vmin=-5, vmax=2, vth = 0.1, b0_mag=2.0, plot_or=1) 
osiris.plot_wk(rundir=dirname, wlim=[0,5], klim=[0,15], vmin=-5, vmax=2, vth = 0.1, b0_mag=2.0, plot_or=1, 
               show_theory=True) 
  • Why is the signal stronger (weaker) for the modes with ω\omega near ωH\omega_H for kcωp\frac{kc}{\omega_p}>4?
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