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Dispersion: A Full Two Fluid Solution

This notebook walks through the functionalty of the two_fluid_dispersion_solution() function. This function computes the wave frequencies for given wavenumbers and plasma parameters based on the analytical solution presented by Bellan 2012 to the Stringer 1963 two fluid dispersion relation. The two fluid dispersion equaiton assumes a uniform magnetic field, a zero D.C. electric field, and low-frequency waves \(\omega / k c \ll 1\) which equates to

\[\begin{split}\left( \cos^2 \theta - Q \frac{\omega^2}{k^2 {v_A}^2} \right) \left[ \left( \cos^2 \theta - \frac{\omega^2}{k^2 {c_s}^2} \right) - Q \frac{\omega^2}{k^2 {v_A}^2} \left( 1 - \frac{\omega^2}{k^2 {c_s}^2} \right) \right] \\ = \left(1 - \frac{\omega^2}{k^2 {c_s}^2} \right) \frac{\omega^2}{{\omega_{ci}}^2} \cos^2 \theta\end{split}\]

where

\[Q = 1 + k^2 c^2/{\omega_{pe}}^2\]

\[\cos \theta = \frac{k_z}{k}\]

\[\mathbf{B_o} = B_{o} \mathbf{\hat{z}}\]

\(\omega\) is the wave frequency, \(k\) is the wavenumber, \(v_A\) is the Alfvén velocity, \(c_s\) is the sound speed, \(\omega_{ci}\) is the ion gyrofrequency, and \(\omega_{pe}\) is the electron plasma frequency.

The approach outlined in Section 5 of Bellan 2012 produces exact roots to the above dispersion equation for all three modes (fast, acoustic, and Alfvén) without having to make additional approximations. The following dispersion relation is what the two_fluid_dispersion_solution() function computes.

\[\frac{\omega}{\omega_{ci}} = \sqrt{ 2 \Lambda \sqrt{-\frac{P}{3}} \cos\left( \frac{1}{3} \cos^{-1}\left( \frac{3q}{2p} \sqrt{-\frac{3}{p}} \right) - \frac{2 \pi}{3}j \right) + \frac{\Lambda A}{3} }\]

where \(j = 0\) represents the fast mode, \(j = 1\) represents the Alfvén mode, and \(j = 2\) represents the acoustic mode. Additionally,

\[p = \frac{3B-A^2}{3} \; , \; q = \frac{9AB-2A^3-27C}{27}\]

\[A = \frac{Q + Q^2 \beta + Q \alpha + \alpha \Lambda}{Q^2} \; , \; B = \alpha \frac{1 + 2 Q \beta + \Lambda \beta}{Q^2} \; , \; C = \frac{\alpha^2 \beta}{Q^2}\]

\[\alpha = \cos^2 \theta \; , \; \beta = \left( \frac{c_s}{v_A}\right)^2 \; , \; \Lambda = \left( \frac{k v_{A}}{\omega_{ci}}\right)^2\]

Contents:

1. Wave Propagating at 45 Degrees

2. Wave frequencies on the k-theta plane

3. Reproduce Figure 1 from Bellan 2012

%matplotlib inline

import astropy.units as u
import matplotlib.pyplot as plt
import numpy as np

from astropy.constants.si import c
from matplotlib import colors
from matplotlib.ticker import MultipleLocator
from mpl_toolkits.axes_grid1 import make_axes_locatable

from plasmapy.dispersion.two_fluid_dispersion import two_fluid_dispersion_solution
from plasmapy.formulary import parameters as pfp
from plasmapy.particles import Particle

plt.rcParams["figure.figsize"] = [10.5, 0.56 * 10.5]

Wave Propagating at 45 Degrees

Below we define the required parameters to compute the wave frequencies.

# define input parameters
inputs = {
    "k": np.linspace(10 ** -7, 10 ** -2, 10000) * u.rad / u.m,
    "theta": 45 * u.deg,
    "n_i": 5 * u.cm ** -3,
    "B": 8.3e-9 * u.T,
    "T_e": 1.6e6 * u.K,
    "T_i": 4.0e5 * u.K,
    "ion": Particle("p+"),
}

# a few useful plasma parameters
params = {
    "n_e": inputs["n_i"] * abs(inputs["ion"].integer_charge),
    "cs": pfp.ion_sound_speed(inputs["T_e"], inputs["T_i"], inputs["ion"],),
    "va": pfp.Alfven_speed(inputs["B"], inputs["n_i"], ion=inputs["ion"],),
    "wci": pfp.gyrofrequency(inputs["B"], inputs["ion"]),
}
params["lpe"] = pfp.inertial_length(params["n_e"], "e-")
params["wpe"] = pfp.plasma_frequency(params["n_e"], "e-")

The computed wave frequencies (\(rad/s\)) are returned in a dictionary with keys representing the wave modes and the values being an Astropy Quantity. Since our inputs had a scalar \(\theta\) and a 1D array of \(k\)’s, the computed wave frequencies will be a 1D array of size equal to the size of the \(k\) array.

# compute
omegas = two_fluid_dispersion_solution(**inputs)

(list(omegas.keys()), omegas["fast_mode"], omegas["fast_mode"].shape)

(['fast_mode', 'alfven_mode', 'acoustic_mode'],
 <Quantity [1.63839709e-02, 1.80262570e-01, 3.44262572e-01, ...,
            1.52032171e+03, 1.52047365e+03, 1.52062560e+03] rad / s>,
 (10000,))

Let’s plot the results of each wave mode.

fs = 14  # default font size
figwidth, figheight = plt.rcParams["figure.figsize"]
figheight = 1.6 * figheight
fig = plt.figure(figsize=[figwidth, figheight])

# normalize data
k_prime = inputs["k"] * params["lpe"]

# plot
plt.plot(
    k_prime, np.real(omegas["fast_mode"] / params["wpe"]), "r.", ms=1, label="Fast",
)
ax = plt.gca()
ax.plot(
    k_prime, np.real(omegas["alfven_mode"] / params["wpe"]), "b.", ms=1, label="Alfvèn",
)
ax.plot(
    k_prime,
    np.real(omegas["acoustic_mode"] / params["wpe"]),
    "g.",
    ms=1,
    label="Acoustic",
)

# adjust axes
ax.set_xlabel(r"$kc / \omega_{pe}$", fontsize=fs)
ax.set_ylabel(r"$Re(\omega / \omega_{pe})$", fontsize=fs)
ax.set_yscale("log")
ax.set_xscale("log")
ax.set_ylim(1e-6, 2e-2)
ax.tick_params(
    which="both", direction="in", width=1, labelsize=fs, right=True, length=5,
)

# annotate
text = (
    f"$v_A/c_s = {params['va'] / params['cs']:.1f} \qquad "
    f"c/v_A = 10^{np.log10(c / params['va']):.0f} \qquad "
    f"\\theta = {inputs['theta'].value:.0f}"
    "^{\\circ}$"
)
ax.text(0.25, 0.95, text, transform=ax.transAxes, fontsize=18)
ax.legend(loc="upper left", markerscale=5, fontsize=fs)

<matplotlib.legend.Legend at 0x7fc4fe64cf50>

Wave frequencies on the k-theta plane

Let us now look at the distribution of \(\omega\) on a \(k\)-\(\theta\) plane.

# define input parameters
inputs = {
    "k": np.linspace(10 ** -7, 10 ** -2, 10000) * u.rad / u.m,
    "theta": np.linspace(5, 85, 100) * u.deg,
    "n_i": 5 * u.cm ** -3,
    "B": 8.3e-9 * u.T,
    "T_e": 1.6e6 * u.K,
    "T_i": 4.0e5 * u.K,
    "ion": Particle("p+"),
}

# a few useful plasma parameters
params = {
    "n_e": inputs["n_i"] * abs(inputs["ion"].integer_charge),
    "cs": pfp.ion_sound_speed(inputs["T_e"], inputs["T_i"], inputs["ion"],),
    "va": pfp.Alfven_speed(inputs["B"], inputs["n_i"], ion=inputs["ion"],),
    "wci": pfp.gyrofrequency(inputs["B"], inputs["ion"]),
}
params["lpe"] = pfp.inertial_length(params["n_e"], "e-")
params["wpe"] = pfp.plasma_frequency(params["n_e"], "e-")

Since the \(\theta\) and \(k\) values are now 1-D arrays, the returned wave frequencies will be 2-D arrays with the first dimension matching the size of \(k\) and the second dimension matching the size of \(\theta\).

# compute
omegas = two_fluid_dispersion_solution(**inputs)

(
    omegas["fast_mode"].shape,
    omegas["fast_mode"].shape[0] == inputs["k"].size,
    omegas["fast_mode"].shape[1] == inputs["theta"].size,
)

((10000, 100), True, True)

Let’s plot (the fast mode)!

fs = 14  # default font size
figwidth, figheight = plt.rcParams["figure.figsize"]
figheight = 1.6 * figheight
fig = plt.figure(figsize=[figwidth, figheight])

# normalize data
k_prime = inputs["k"] * params["lpe"]
zdata = np.transpose(np.real(omegas["fast_mode"].value)) / params["wpe"].value

# plot
im = plt.imshow(
    zdata,
    aspect="auto",
    origin="lower",
    extent=[
        np.min(k_prime.value),
        np.max(k_prime.value),
        np.min(inputs["theta"].value),
        np.max(inputs["theta"].value),
    ],
    interpolation=None,
    cmap=plt.cm.Spectral,
)
ax = plt.gca()

# # adjust axes
ax.set_xscale("linear")
ax.set_xlabel(r"$kc/\omega_{pe}$", fontsize=fs)
ax.set_ylabel(r"$\theta$ [$deg.$]", fontsize=fs)
ax.tick_params(
    which="both",
    direction="in",
    width=2,
    labelsize=fs,
    right=True,
    top=True,
    length=10,
)

# Add colorbar
divider = make_axes_locatable(ax)
cax = divider.append_axes("top", size="5%", pad=0.07)
cbar = plt.colorbar(
    im, cax=cax, orientation="horizontal", ticks=None, fraction=0.05, pad=0.0,
)
cbar.ax.tick_params(
    axis="x",
    direction="in",
    width=2,
    length=10,
    top=True,
    bottom=False,
    labelsize=fs,
    pad=0.0,
    labeltop=True,
    labelbottom=False,
)
cbar.ax.xaxis.set_label_position("top")
cbar.set_label(r"$\omega/\omega_{pe}$", fontsize=fs, labelpad=8)

Reproduce Figure 1 from Bellan 2012

Figure 1 of Bellan 2012 chooses parameters such that \(\beta = 0.4\) and \(\Lambda=0.4\). Below we define parameters to approximate Bellan’s assumptions.

# define input parameters
inputs = {
    "B": 400e-4 * u.T,
    "ion": Particle("He+"),
    "n_i": 6.358e19 * u.m ** -3,
    "T_e": 20 * u.eV,
    "T_i": 10 * u.eV,
    "theta": np.linspace(0, 90) * u.deg,
    "k": (2 * np.pi * u.rad) / (0.56547 * u.m),
}

# a few useful plasma parameters
params = {
    "n_e": inputs["n_i"] * abs(inputs["ion"].integer_charge),
    "cs": pfp.cs_(inputs["T_e"], inputs["T_i"], inputs["ion"]),
    "wci": pfp.wc_(inputs["B"], inputs["ion"]),
    "va": pfp.va_(inputs["B"], inputs["n_i"], ion=inputs["ion"]),
}
params["beta"] = (params["cs"] / params["va"]).value ** 2
params["wpe"] = pfp.wp_(params["n_e"], "e-")
params["Lambda"] = (inputs["k"] * params["va"] / params["wci"]).value ** 2

(params["beta"], params["Lambda"])

(0.4000832135717194, 0.4000017351804854)

# compute
omegas = two_fluid_dispersion_solution(**inputs)

# generate data for plots
plt_vals = {}
for mode, arr in omegas.items():
    norm = (np.absolute(arr) / (inputs["k"] * params["va"])).value ** 2
    plt_vals[mode] = {
        "x": norm * np.sin(inputs["theta"].to(u.rad).value),
        "y": norm * np.cos(inputs["theta"].to(u.rad).value),
    }

fs = 14  # default font size
figwidth, figheight = plt.rcParams["figure.figsize"]
figheight = 1.6 * figheight
fig = plt.figure(figsize=[figwidth, figheight])

# Fast mode
plt.plot(
    plt_vals["fast_mode"]["x"], plt_vals["fast_mode"]["y"], linewidth=2, label="Fast",
)
ax = plt.gca()

# adjust axes
ax.set_xlabel(r"$(\omega / k v_A)^2 \, \sin \theta$", fontsize=fs)
ax.set_ylabel(r"$(\omega / k v_A)^2 \, \cos \theta$", fontsize=fs)
ax.set_xlim(0.0, 1.5)
ax.set_ylim(0.0, 2.0)
for spine in ax.spines.values():
    spine.set_linewidth(2)
ax.minorticks_on()
ax.tick_params(which="both", labelsize=fs, width=2)
ax.tick_params(which="major", length=10)
ax.tick_params(which="minor", length=5)
ax.xaxis.set_major_locator(MultipleLocator(0.5))
ax.xaxis.set_minor_locator(MultipleLocator(0.1))
ax.yaxis.set_major_locator(MultipleLocator(0.5))
ax.yaxis.set_minor_locator(MultipleLocator(0.1))


# Alfven mode
plt.plot(
    plt_vals["alfven_mode"]["x"],
    plt_vals["alfven_mode"]["y"],
    linewidth=2,
    label="Alfv$\`{e}$n",
)

# Acoustic mode
plt.plot(
    plt_vals["acoustic_mode"]["x"],
    plt_vals["acoustic_mode"]["y"],
    linewidth=2,
    label="Acoustic",
)

# annotations
plt.legend(fontsize=fs, loc="upper right")

<matplotlib.legend.Legend at 0x7fc4fbe61bd0>