Magnetohydrodynamic Waves & Plasma Instabilities

1. The Ideal MHD System

Ideal magnetohydrodynamics treats the plasma as an electrically conducting, magnetized single fluid governed by:

$$\rho_m \left[ \frac{\partial \vec{v}}{\partial t} + (\vec{v}\cdot\nabla)\vec{v} \right] = \vec{J} \times \vec{B} - \nabla P$$ $$\frac{\partial \vec{B}}{\partial t} = \nabla \times (\vec{v} \times \vec{B})$$ $$\frac{\partial \rho_m}{\partial t} + \nabla \cdot (\rho_m \vec{v}) = 0$$ $$\nabla P = c_s^2 \nabla \rho_m, \quad \vec{J} = \frac{1}{\mu_0}\nabla \times \vec{B}, \quad \nabla\cdot\vec{B} = 0$$

where $c_s = \sqrt{\gamma P_0 / \rho_0}$ is the adiabatic sound speed.

2. First-Order Perturbation Expansion

Consider a static, homogeneous equilibrium embedded in a uniform magnetic field:

$$\vec{B} = B_0 \hat{z}, \quad \rho_m = \rho_0, \quad P = P_0, \quad \vec{v}_0 = 0$$

Applying small harmonic perturbations $\propto e^{i(\vec{k}\cdot\vec{r} - \omega t)}$:

$$-i\omega \rho_0 \vec{v}_1 = \frac{1}{\mu_0}(i\vec{k}\times\vec{B}_1)\times\vec{B}_0 - i\vec{k} P_1$$ $$-i\omega \vec{B}_1 = i\vec{k} \times (\vec{v}_1 \times \vec{B}_0)$$ $$-i\omega \rho_1 + i\rho_0 \vec{k}\cdot\vec{v}_1 = 0 \implies P_1 = c_s^2 \rho_1 = \rho_0 c_s^2 \frac{\vec{k}\cdot\vec{v}_1}{\omega}$$

1. The Pure Shear Alfvén Wave

Consider incompressible perturbations ($\nabla \cdot \vec{v}_1 = 0 \implies \rho_1 = 0, P_1 = 0$) propagating along the magnetic field with $\vec{k} = k_\parallel \hat{z}$.

Let the fluid velocity perturb along $\hat{x}$: $\vec{v}_1 = v_{1x} \hat{x}$. The induction equation gives:

$$-i\omega \vec{B}_1 = i k_\parallel \hat{z} \times (v_{1x}\hat{x} \times B_0\hat{z}) = i k_\parallel B_0 v_{1x} \hat{x} \implies \vec{B}_1 = -\frac{k_\parallel B_0}{\omega} v_{1x} \hat{x}$$

The linearized momentum equation becomes:

$$-i\omega \rho_0 v_{1x} \hat{x} = \frac{1}{\mu_0}(i k_\parallel \hat{z} \times B_{1x}\hat{x})\times(B_0\hat{z}) = \frac{i k_\parallel B_0}{\mu_0} B_{1x}\hat{x}$$

Substitute $B_{1x} = -\frac{k_\parallel B_0}{\omega} v_{1x}$:

$$-i\omega \rho_0 v_{1x} = \frac{i k_\parallel B_0}{\mu_0}\left( -\frac{k_\parallel B_0}{\omega} v_{1x} \right) = -i \frac{k_\parallel^2 B_0^2}{\mu_0 \omega} v_{1x}$$

Multiplying by $i\omega / (\rho_0 v_{1x})$:

$$\omega^2 = \frac{B_0^2}{\mu_0 \rho_0} k_\parallel^2 \equiv v_A^2 k_\parallel^2$$

where the characteristic propagation speed is the famous Alfvén velocity $v_A$:

$$v_A \equiv \frac{B_0}{\sqrt{\mu_0 \rho_0}}$$

2. The Plucked Magnetic String Analogy

A shear Alfvén wave is the magnetohydrodynamic analog of transverse vibrations on a plucked violin string of tension $T$ and mass per unit length $\mu$: $v = \sqrt{T / \mu}$. In a magnetized plasma:

$$T_{\text{mag}} = \frac{B_0^2}{\mu_0}, \quad \text{Inertia} = \rho_0 \implies v_A = \sqrt{\frac{T_{\text{mag}}}{\rho_0}} = \frac{B_0}{\sqrt{\mu_0 \rho_0}}$$

Magnetic tension provides the restoring force, while fluid mass density $\rho_0$ provides the inertia. Shear Alfvén waves carry magnetic and kinetic energy in exact equipartition: $\frac{1}{2}\rho_0 v_1^2 = \frac{B_1^2}{2\mu_0}$.

1. Compressible Oblique Perturbations

When the fluid is compressible ($ abla \cdot \vec{v}_1 \ne 0$) and the wave propagates at an arbitrary angle $\theta$ relative to $\vec{B}_0$ ($\vec{k} = k_\perp \hat{x} + k_\parallel \hat{z}$, with $\cos\theta = k_\parallel / k$):

Substituting $\vec{B}_1$ and $P_1$ into the momentum equation yields the master MHD wave dispersion relation:

$$\left( \frac{\omega^2}{k^2} - v_A^2 \cos^2\theta \right) \left[ \frac{\omega^4}{k^4} - (v_A^2 + c_s^2)\frac{\omega^2}{k^2} + v_A^2 c_s^2 \cos^2\theta \right] = 0$$

2. The Three Fundamental MHD Modes

1. Shear Alfvén Wave (Intermediate Mode): $$\frac{\omega^2}{k^2} = v_A^2 \cos^2\theta \implies \omega = k v_A \cos\theta = k_\parallel v_A$$
2. Fast Magnetosonic Wave: $$\left(\frac{\omega}{k}\right)^2_{\text{fast}} = \frac{1}{2}\left( v_A^2 + c_s^2 + \sqrt{(v_A^2 + c_s^2)^2 - 4 v_A^2 c_s^2 \cos^2\theta} \right)$$
3. Slow Magnetosonic Wave: $$\left(\frac{\omega}{k}\right)^2_{\text{slow}} = \frac{1}{2}\left( v_A^2 + c_s^2 - \sqrt{(v_A^2 + c_s^2)^2 - 4 v_A^2 c_s^2 \cos^2\theta} \right)$$

The phase velocities satisfy the strict hierarchy:

$$v_{ph,\text{slow}} \le v_{ph,\text{Alfvén}} \le v_{ph,\text{fast}}$$

1. Friedrichs Polar Diagrams

A polar plot of phase velocity $v_{ph}(\theta) = \omega / k$ as a function of propagation angle $\theta$ relative to the magnetic field $\vec{B}_0$ is known as a Friedrichs diagram:

• Shear Alfvén Wave: Traces two tangent circles touching at the origin along the magnetic field axis ($v_{ph} = v_A |\cos\theta|$). Phase velocity is strictly zero perpendicular to $\vec{B}_0$ ($\theta = 90^\circ$).
• Fast Magnetosonic Wave: Traces an oval (nearly spherical) outer shell. At $\theta = 90^\circ$, it propagates at the combined magnetosonic speed: $$v_{ph,\text{fast}}(\theta=90^\circ) = \sqrt{v_A^2 + c_s^2}$$ Here, thermal acoustic pressure and magnetic pressure compress in phase, reinforcing each other.
• Slow Magnetosonic Wave: Traces two cusped lobes inside the Alfvén circles. At $\theta = 90^\circ$, $v_{ph,\text{slow}} = 0$. Thermal and magnetic pressure compress out of phase, partially canceling.

1. Free Energy and Instability Mechanism

A plasma equilibrium is characterized by time-independent macroscopic state variables. If a small initial perturbation $\psi_1(t) \propto e^{-i\omega t}$ has a complex frequency $\omega = \omega_r + i\gamma$:

$$\psi_1(t) = \psi_1(0) e^{-i(\omega_r + i\gamma)t} = \psi_1(0) e^{\gamma t} e^{-i\omega_r t}$$

• If $\gamma < 0$: The perturbation is damped; the plasma is stable.
• If $\gamma > 0$: The perturbation grows exponentially with growth rate $\gamma$; the plasma is unstable.

Instabilities are powered by tapping reservoirs of free energy stored in the non-equilibrium plasma:

• Spatial gradients: Density $\nabla n$, temperature $\nabla T$, or pressure gradients $\nabla P$.
• Currents and magnetic shear: Non-zero $\nabla \times \vec{B} = \mu_0 \vec{J}$.
• Velocity space anisotropy: Non-Maxwellian beams ($v_d > v_{\text{th}}$) or pitch angle anisotropy ($T_\perp \ne T_\parallel$).

2. Classification of Instabilities

1. Macro-Instabilities (MHD / Hydrodynamic): Driven by macroscopic spatial gradients. Wavelengths are macroscopic ($\lambda \gg r_L$). They rapidly destroy global plasma confinement on microsecond timescales (e.g., kink, sausage, Rayleigh-Taylor).
2. Micro-Instabilities (Kinetic): Driven by non-thermal features in velocity space. Wavelengths are microscopic ($\lambda \sim r_L, \lambda_D$). They cause anomalous cross-field transport and turbulent diffusion (e.g., two-stream, drift waves, loss cone modes).

1. The Plasma Rayleigh-Taylor Instability

Consider a dense plasma of mass density $\rho_0$ supported against gravity $\vec{g} = -g\hat{y}$ by a magnetic field $\vec{B} = B_0\hat{z}$ or by a lighter fluid below (an "inverted density gradient").

If a small rippled perturbation $y_1 \propto e^{i(k_x x - \omega t)}$ forms at the interface:

• Ions and electrons experience opposite gravitational drifts: $$\vec{v}_g = \frac{m}{q}\frac{\vec{g}\times\vec{B}}{B^2} = -\frac{m g}{q B}\hat{x}$$
• Positive ions drift along $-\hat{x}$; electrons drift along $+\hat{x}$.
• At ripple crests and troughs, this differential drift accumulates positive charge on one side of a crest and negative charge on the other side.
• This charge separation produces a perturbed electric field $\vec{E}_1$ along $\hat{x}$.
• The resulting $\vec{E}_1 \times \vec{B}_0$ drift velocity $\vec{v}_E = \frac{\vec{E}_1\times\vec{B}_0}{B_0^2}$ is directed upward at crests and downward at troughs!

The $\vec{E}\times\vec{B}$ drift amplifies the initial perturbation, driving runaway exponential growth with classical growth rate:

$$\omega^2 = -g k \implies \gamma = \sqrt{g k}$$

2. Magnetic Field Line Bending Stabilization

If the magnetic field possesses a component along the perturbation wavevector $\vec{k}$ (so that $\vec{k}\cdot\vec{B}_0 \ne 0$), rippling the interface bends magnetic field lines. Magnetic tension opposes the deformation, modifying the dispersion relation to:

$$\omega^2 = -g k + \frac{(\vec{k}\cdot\vec{B}_0)^2}{\mu_0 \rho_0}$$

The interface is completely stabilized ($\omega^2 \ge 0$) against Rayleigh-Taylor modes whenever:

$$\frac{(\vec{k}\cdot\vec{B}_0)^2}{\mu_0 \rho_0} \ge g k$$

This magnetic line-tying and shear stabilization is essential for stabilizing solar prominences and inertial confinement fusion (ICF) implosions.

1. Current-Driven MHD Instabilities

In a cylindrical current-carrying plasma column (Z-pinch or tokamak), perturbations are decomposed into azimuthal Fourier modes $e^{i(m\theta - k_z z)}$:

• $m = 0$ Mode (Sausage Instability): Axisymmetric constrictions along the column. Where the radius narrows ($r < a$), the azimuthal magnetic field $B_\theta \propto I/r$ intensifies, increasing magnetic pressure $B_\theta^2 / (2\mu_0)$ and pinching the neck even tighter until the column snaps.
• $m = 1$ Mode (Kink Instability): The entire column bends into a helix. Field lines on the inside of the bend crowd together, raising magnetic pressure, while field lines on the outside spread out, lowering pressure. The net pressure imbalance kicks the bend outward, causing violent helical disruption.

2. The Kruskal-Shafranov Stability Criterion

To stabilize the dangerous $m=1$ kink instability in tokamaks, a strong toroidal magnetic field $B_z$ (or $B_\phi$) is applied along the column. As the column kinks, it is forced to stretch the strong axial field lines, which resists the deformation via magnetic tension.

Defining the safety factor $q(r)$:

$$q(r) \equiv \frac{r B_z(r)}{R B_\theta(r)}$$

where $R$ is the major radius and $r$ is the minor radius. The column is completely stable against the fundamental external kink mode if and only if the safety factor at the plasma edge exceeds unity:

$$q(a) > 1$$

This is the celebrated Kruskal-Shafranov stability limit. It sets an absolute upper bound on the maximum toroidal current $I_p$ that can be stably driven in any tokamak:

$$I_p < \frac{2\pi a^2 B_z}{\mu_0 R}$$

Exceeding this current triggers immediate catastrophic kink disruption.