Fluid Theory #
Recall that cold plasma theory assumed no finite temperature effects. We then added individual particle contributions to find quantities like currents.
In kinetic theory, we start from the full particle distribution function with a finite temperature, which considers a meaningful velocity distribution function. We considered waves with phase velocities such that we could not just “lump particles together” because \( |v_x - v_\phi| \) has a significant effect on wave propagation (e.g. Landau damping)
For a fluid picture, we’ll consider an “average” motion of particles of the same species, assuming that inter-species interactions come about only with collisions.
From Boltzmann equation,
\[\dv{f}{t} = \pdv{f}{t} + \vec v \cdot \grad f + \vec a \cdot \grad_{v} f = \left(\dv{f}{t} \right)_c\]At time scales long enough that particle correlations have negligible effects, the zeroth order equation would be the Vlasov equation:
\[\pdv{f}{t} + \vec v \cdot \grad f + \vec a \cdot \grad_v f = 0\]We can re-write the equation, making use of the fact that in phase space, \( \vec x \) and \( \vec v \) are independent variables:
\[f \div \vec v = 0\] \[f \grad_v \cdot \vec a = 0\] \[\pdv{f}{t} + \div (f \vec v) + \grad _v \cdot (f \vec a) = 0\]We can average a scalar \( Q(\vec v) \) over the velocity distribution:
\[\langle Q \rangle = \frac{\int Q f \dd \vec v}{\int f \dd v}\]If we multiply the Boltzmann equation by \( Q(\vec v) \) and integrate over \( \vec v \):
\[\int Q \pdv{f}{t} \dd \vec v + \int Q \vec v \cdot \grad f \dd \vec v + \int Q \vec a \cdot \grad_v f \dd \vec v = 0\] \[\pdv{}{t} \int Q f \dd \vec v + \div \int Q \vec v f \dd \vec v + \oint _v (Q \vec a f) \cdot \dd \vec S_v - \int f \vec a \cdot \grad _v Q \dd \vec v = 0\] \[\rightarrow \pdv{}{t}(n \langle Q \rangle) + \div (n \langle Q \vec v \rangle) - n \langle \vec a \cdot \grad_v Q \rangle = 0\]If we take \( Q = 1 \) we get the zeroth moment: the continuity equation
\[\pdv{n}{t} + \div (n \vec u) = 0\]where \( \vec u \equiv \langle \vec v \rangle \) is the average (drift) velocity.
The momentum equations come from taking the first moment
\[\pdv{}{t} (n m u_x) + \div (n m \langle v_x \vec v \rangle) - n m \langle a_x \rangle = 0\]We call the stress tensor \( \vec \Psi = n m \langle (\vec v - \vec u) (\vec v - \vec u) \rangle \)
\[n m \left[ \pdv{\vec u}{t} + (\vec u \cdot \grad) \vec u \right] + \div \vec \Psi - n q (\vec E + \vec u \cross \vec B) = 0\]The off-diagonal elements of \( \vec \Psi \) are viscosity effects.
For isotropic pressure, we can write the stress tensor as a scalar pressure \( p \) times the identity tensor \( \vec 1 \). Then \( \div \vec \Psi = \grad p \).
To close out the fluid equations, we need to define a closure relation to relate higher-moment terms. This is our equation of state:
\[p = C \rho^\gamma\]where \( C \) is a constant and \( \gamma \) is the ratio of specific heats (adiabatic constant)
\[\gamma = \frac{c_p}{c_t} = \frac{2 + N}{N}\]where \( N \) is degrees of freedom. For 1-D system \( \gamma = 3 \).
Fluid Equations #
\[\sigma = n_i q_i + n_e q_e\] \[\vec J = n_i q_i \vec u_i + n_e q_e \vec u_e\] \[\epsilon_0 \div \vec E = n_i q_i + n_e q_e\]\[\curl \vec E = - \pdv{\vec B}{t}\] \[\div \vec B = 0\] \[\frac{1}{\mu_0} \curl \vec B = n_i q_i \vec u_i + n_e q_e \vec u_e + \epsilon_0 \pdv{\vec E}{t}\]
With the ion and electron continuity conservation equations (2 equations), the ion and electron momentum conservation equations (2 equations), and the ion and electron equations of state (2 equations), we have 16 equations in total and 16 unknowns: \( n_i \), \( n_e \), \( p_i \), \( p_e \), \( \vec v_i \), \( \vec v_e \), \( \vec E \), \( \vec B \).
Fluid Equations - Wave Propagation #
Starting with our two moment equations above, we perform the regular linearization process with perturbations
\[n = n_0 + n_1\] \[E = 0 + E_1\] \[u = u_0 + u_1\]
\[m_e n_0 \pdv{u_1}{t} = - e n_0 E_1 - \gamma k_b T_e \pdv{n_1}{x}\] \[\pdv{n_1}{t} + n_0 \pdv{u_1}{x} = 0\] \[\pdv{E_1}{x} = - \frac{e}{\epsilon_0} n_1\]Applying Fourier transform: \[m n_0 (-i \omega \tilde{u}_1) = - e n_0 \tilde{E}_1 - \gamma k_b T_e (i k \tilde{n}_1)\] \[- i \omega \tilde{n}_1 + i k n_0 \tilde{u}_1 = 0\] \[i k \tilde{E}_1 = - \frac{\rho}{\epsilon_0} \tilde{n}_1\] \[\rightarrow \tilde{n}_1 = \frac{k n_0 \tilde{u}_1}{\omega}\] \[\rightarrow \tilde{E}_1 = \frac{i \rho}{\epsilon_0} \frac{\tilde{n}_1}{k} = \frac{i \rho}{\epsilon_0} \frac{n_0 \tilde{u}_1}{\omega}\] \[- i \omega m n_0 \tilde{u}_1 = - e n_0 \frac{i \rho}{\epsilon_0} \frac{n_0 \tilde{u}_1}{\omega} - \gamma k_b T_e i k \left( \frac{k n_0 \tilde{u}_1}{\omega}\right)\] \[\omega ^2 m n_0 \tilde{u}_1 = \frac{e^2 n_0 ^2}{\epsilon_0} \tilde{u}_1 + \gamma k_b T_e k^2 n_0 \tilde{u}_1\] \[\omega ^2 = \omega_p ^2 + \frac{\gamma k_b T_e}{m} k^2\]
\[\omega^2 = \omega_p ^2 + \frac{3 k_b T_e}{m} k^2 = \omega_p ^2 + 3 k^2 v_t ^2\]Recall from the kinetic theory that this expression is only valid when the phase velocity is much greater than the thermal velocity (\( v_x / v_\phi \ll 1 \))
Low-Frequency Waves #
Waves in the fluid equations conveniently break up into two regions when \( m_e \ll m_i \). For the high-frequency waves, ion motions are completely neglected, while in the low frequency region we neglect terms of order \( m_e / m_i \).
If one combines the fluid continuity equation and momentum equation for an electron-ion plasma, we eventually find that the low-frequency dispersion relation (LFDR) is:
\[\left( 1 - \frac{\omega ^2}{k^2 v_A ^2} - \frac{\omega ^2}{\omega_{ce} \omega_{ci}} + \frac{k^2 c_s ^2 \sin ^2 \theta}{\omega ^2 - k^2 c_s ^2} \right) \left( \cos ^2 \theta - \frac{\omega ^2}{k^2 v_A ^2} - \frac{\omega ^2}{\omega_{ce} \omega_{ci}} \right) = \frac{\omega ^2 \cos ^2 \theta}{\omega_{ci}^2}\]where \( \theta \) is the angle between the direction of propagation and the static magnetic field, \( c_s \) is the ion acoustic frequency:
\[c_s ^2 \equiv \frac{\gamma_e k_b T_e + \gamma_i k_b T_i}{m_e + m_i}\]\( \gamma_\alpha \) is the ratio of specific heats for species \( \alpha \), and \( v_A \) is the Alfven speed
\[v_A ^2 = \frac{B_0 ^2}{\mu_0 \rho_0} \qquad \rho_0 = \sum n_{0\alpha} m_\alpha\]LFDR Waves, Dispersion Relation, and Transitions #
In the MHD region, when \( \omega / \omega_{ci} \ll 1 \), then the second, fifth, and sixth terms are very small compared to the others. The LFDR becomes
\[\left( 1 - \frac{\omega ^2}{k^2 v_A ^2} + \frac{k^2 c_s ^2 \sin ^2 \theta}{\omega ^2 - k^2 c_s ^2} \right) \left( \cos ^2 \theta - \frac{\omega^2}{k^2 v_A ^2} \right) = 0\]The factor on the right is the shear Alfven wave:
\[\omega ^2 = k^2 v_A ^2 \cos ^2 \theta\]The remaining factor gives two roots:
\[\omega ^4 - k^2 (v_A ^2 + c_s ^2) \omega ^2 + k^4 v_A ^2 c_s ^2 \cos ^2 \theta = 0\]For \( c_s / v_A \ll 1 \), the approximate roots are:
Magnetized ion sound wave #
\[\omega ^2 = k^2 c_s ^2 \cos ^2 \theta\]####Magnetosonic / magnetoacoustic wave
\[\omega ^2 = k^2 (v_A ^2 + c_s ^2 \sin ^2 \theta)\]We can think of this as a cold, compressional Alfven wave coupled with an acoustic wave at low frequency. If \( v_A / c_s \gg 1 \) we just get the compressional Alfven wave \( \omega ^2 = k^2 v_A ^2 \). The compressional Alfven wave has linear dispersion, since the group and phase velocity are both \( v_A \).
It causes dispersion across \( \vec B \) if \( v_A \sim O(c_s) \). At intermediate frequencies, if the wave travels along the magnetic field then it is a Whistler wave. When \( \theta = 0 \) it’s just the R-wave, and when \( \theta = 90 \) it is the X-wave.
Recall from the cold plasma dispersion relation discussion, we defined \( \alpha \) to be the angle between the phase velocity and the group velocity:
\[\tan \alpha = - \frac{1}{k} \left. \pdv{k}{\theta} \right|_{\omega}\] \[k^2 = \omega ^2 (v_A ^2 + c_s ^2 \sin ^2 \theta)^{-1}\]\[2 k \left. \pdv{k}{\theta} \right|_{\omega} = - \omega ^2 (v_A ^2 c_s ^2 \sin ^2 \theta)^{-2} c_s ^2 2 \sin \theta \cos \theta\] \[= - k ^2 \frac{(v_A ^2 + c_s ^2 \sin ^2 \theta) c_s ^2 \cos \theta \sin \theta}{(v_A ^2 c_s ^2 \sin ^2 \theta )^2}\] \[-\frac{1}{k} \left. \pdv{k}{\theta} \right|_\omega = \frac{\cos \theta \sin \theta}{v_A ^2 / c_s ^2 + \sin ^2 \theta} = \tan \alpha\]
simplifies to (for \( \beta \equiv \theta + \alpha \))
\[\tan(\theta + \alpha) = \tan \beta = (1 + \frac{c_s ^2}{v_A ^2}) \tan \theta\]As the acoustic component increases, wave energy is routed away from the magnetic field.
Ion Acoustic Waves #
From the electrostatic dispersion relation:
\[1 + \frac{1}{k^2 \lambda_{Di}^2} + \frac{1}{k^2 \lambda_{De}^2} = \qquad \qquad \qquad \\ \frac{1}{k^2 \lambda_{Di}^2 \left[ 1 - k^2 c_i ^2 \left( \frac{\sin ^2 \theta}{\omega ^2 - \omega_{ci}^2} + \frac{\cos ^2 \theta}{\omega ^2} \right) \right]} \\ + \cancel{\frac{1}{k^2 \lambda_{De}^2 \left[1 - k^2 c_e ^2 \left( \frac{\sin ^2 \theta}{\omega ^2 - \omega_{ce}^2} + \frac{\cos ^2 \theta}{\omega ^2} \right) \right]}}{\omega / k \ll c_e}\]At \( \theta = 0 \):
\[1 + \frac{1}{k^2} \left( \frac{\lambda_{De}^2 + \lambda_{Di}^2}{\lambda_{De}^2 \lambda_{Di}^2} \right) = \frac{\omega ^2}{k^2 \lambda_{Di}^2 (\omega ^2 - k^2 c_i ^2)}\]With some more simplification
\[\omega ^2 = k^2 c_i ^2 \left[ 1 + \frac{T_e}{T_i (1 + k^2 \lambda_{De}^2)} \right]\]If \( k \lambda_{De} \ll 1 \), then \( \omega \approx k c_s \).
If \( k \lambda_{De} \gg 1 \), then \( \omega \approx k c_i \)
Stringer Diagrams #
Stringer diagrams are useful for visualizing and identifying waves in different regions of \( \omega - k \) space. They have abscissa \( k c_s / \omega_{ci} \) and ordinate \( \omega/\omega_{ci} \) with a log scale.
In this format, waves traveling at a constant phase velocity appear as straight lines in the Stringer diagram.
The key speeds are:
- \( c \): the highest phase velocity we’ll see
- \( c_e \)