14. From Kinetic to MHD

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14. From Kinetic to MHD

Learning Objectives

  • Understand the systematic reduction from 6D kinetic theory to 3D single-fluid MHD
  • Derive single-fluid MHD equations from two-fluid theory by combining species
  • Identify the validity conditions and limitations of MHD approximations
  • Explain the CGL (Chew-Goldberger-Low) double adiabatic model for collisionless plasmas
  • Understand drift-kinetic and gyrokinetic theories as intermediate reductions
  • Compare different plasma models and know when to apply each

1. The Hierarchy of Plasma Models

1.1 Overview: From Full Kinetic to MHD

Plasma physics has a rich hierarchy of models, each with different levels of approximation and computational cost:

Full Kinetic (Vlasov-Maxwell)
    ↓  [average over gyration]
Drift-Kinetic (5D)
    ↓  [average over bounce motion / perturbative expansion]
Gyrokinetic (5D, with FLR)
    ↓  [take moments]
Two-Fluid (3D × 2 species)
    ↓  [combine species]
Extended MHD (Hall, FLR, etc.)
    ↓  [drop small terms]
Single-Fluid MHD (3D)
    ↓  [equilibrium, linearize]
MHD Waves, Instabilities

Each step down the hierarchy: - Reduces dimensionality or number of variables - Simplifies the equations - Loses some physics - Increases computational efficiency

The art of plasma physics is choosing the right model for the problem at hand.

1.2 What Does Each Model Capture?

Model Dimensions Captures Misses
Vlasov-Maxwell 6D (r,v,t) Everything: wave-particle, anisotropy, kinetic instabilities Computationally prohibitive
Drift-Kinetic 5D (R,v∥,μ,t) Parallel dynamics, trapped particles, collisionless damping Cyclotron resonance, gyrophase
Gyrokinetic 5D (R,v∥,μ,t) FLR, turbulence, microinstabilities Fast magnetosonic, compressibility
Two-Fluid 3D × 2 species Hall effect, electron pressure, separate species Kinetic effects (damping, instabilities)
Hall MHD 3D Whistler, fast reconnection, dispersive waves Kinetic damping, pressure anisotropy
Resistive MHD 3D Reconnection, resistive instabilities Fast processes at small scales
Ideal MHD 3D Alfvén/magnetosonic waves, gross equilibria Reconnection, kinetic physics, small scales

1.3 When to Use Which Model?

Use Ideal MHD when: - Large-scale equilibria and stability (tokamak, stellar atmosphere) - Low-frequency waves ($\omega \ll \omega_{ci}$) - Collisional plasmas with isotropic pressure - Magnetic Reynolds number $R_m \gg 1$

Use Resistive MHD when: - Magnetic reconnection (solar flares, substorms) - Resistive instabilities (tearing modes) - Current-driven dynamics

Use Hall MHD when: - Scales approaching $d_i$ (magnetopause, reconnection) - Fast reconnection with whistler outflow - Magnetic field generation (dynamo)

Use Two-Fluid when: - Separate electron and ion dynamics are important - Pressure anisotropy within each species - Kinetic effects are secondary

Use Gyrokinetic when: - Tokamak turbulence (ion-temperature-gradient modes, trapped-electron modes) - Microinstabilities with FLR effects - Collisionless plasmas with weak perturbations

Use Full Kinetic when: - Wave-particle resonances are crucial (Landau damping, cyclotron heating) - Strongly non-Maxwellian distributions (beam-plasma, runaway electrons) - Velocity-space instabilities (two-stream, bump-on-tail)

2. From Two-Fluid to Single-Fluid MHD

2.1 Defining Single-Fluid Variables

Recall the two-fluid equations for species $s$ (electrons $e$, ions $i$):

Continuity: $$\frac{\partial n_s}{\partial t} + \nabla \cdot (n_s \mathbf{u}_s) = 0$$

Momentum: $$m_s n_s \frac{d \mathbf{u}_s}{dt} = q_s n_s (\mathbf{E} + \mathbf{u}_s \times \mathbf{B}) - \nabla p_s + \mathbf{R}_s$$

Energy (adiabatic closure): $$\frac{d}{dt}\left( \frac{p_s}{n_s^\gamma} \right) = 0$$

To derive single-fluid MHD, we define center-of-mass (fluid) variables:

Mass density: $$\rho = m_i n_i + m_e n_e \approx m_i n$$

(using quasi-neutrality $n_i \approx n_e \equiv n$ and $m_i \gg m_e$)

Fluid velocity (center-of-mass velocity): $$\mathbf{v} = \frac{m_i n_i \mathbf{u}_i + m_e n_e \mathbf{u}_e}{\rho} \approx \mathbf{u}_i$$

Total pressure: $$p = p_i + p_e$$

Current density: $$\mathbf{J} = e(n_i \mathbf{u}_i - n_e \mathbf{u}_e) \approx en(\mathbf{u}_i - \mathbf{u}_e)$$

Charge density (quasi-neutrality): $$\rho_c = e(n_i - n_e) \approx 0$$

2.2 Combining Continuity Equations

Add the electron and ion continuity equations:

$$\frac{\partial n_e}{\partial t} + \nabla \cdot (n_e \mathbf{u}_e) = 0$$ $$\frac{\partial n_i}{\partial t} + \nabla \cdot (n_i \mathbf{u}_i) = 0$$

Multiply the electron equation by $m_e$ and the ion equation by $m_i$, then add:

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (m_e n_e \mathbf{u}_e + m_i n_i \mathbf{u}_i) = 0$$

Using $\rho \mathbf{v} = m_i n_i \mathbf{u}_i + m_e n_e \mathbf{u}_e \approx m_i n \mathbf{u}_i$:

$$\boxed{\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0}$$

This is the mass continuity equation for single-fluid MHD.

2.3 Combining Momentum Equations

Add the electron and ion momentum equations:

$$m_e n_e \frac{d \mathbf{u}_e}{dt} = -e n_e (\mathbf{E} + \mathbf{u}_e \times \mathbf{B}) - \nabla p_e + \mathbf{R}_e$$ $$m_i n_i \frac{d \mathbf{u}_i}{dt} = +e n_i (\mathbf{E} + \mathbf{u}_i \times \mathbf{B}) - \nabla p_i + \mathbf{R}_i$$

The collision terms cancel: $\mathbf{R}_e + \mathbf{R}_i = 0$ (momentum conservation).

The electric field terms cancel (using quasi-neutrality): $$-e n_e \mathbf{E} + e n_i \mathbf{E} = e(n_i - n_e) \mathbf{E} \approx 0$$

The Lorentz force terms give: $$-e n_e \mathbf{u}_e \times \mathbf{B} + e n_i \mathbf{u}_i \times \mathbf{B} = e n (\mathbf{u}_i - \mathbf{u}_e) \times \mathbf{B} = \mathbf{J} \times \mathbf{B}$$

The inertial terms: $$m_e n_e \frac{d \mathbf{u}_e}{dt} + m_i n_i \frac{d \mathbf{u}_i}{dt} \approx m_i n \frac{d \mathbf{u}_i}{dt} = \rho \frac{d \mathbf{v}}{dt}$$

(neglecting the electron inertia term $m_e n_e d\mathbf{u}_e/dt \ll m_i n_i d\mathbf{u}_i/dt$).

Putting it together:

$$\boxed{\rho \frac{d \mathbf{v}}{dt} = \mathbf{J} \times \mathbf{B} - \nabla p}$$

This is the momentum equation for single-fluid MHD.

2.4 Ideal Ohm's Law

The key step to ideal MHD is to derive Ohm's law. In Lesson 13, we derived the generalized Ohm's law:

$$\mathbf{E} + \mathbf{v} \times \mathbf{B} = \eta \mathbf{J} + \frac{1}{en} \mathbf{J} \times \mathbf{B} - \frac{1}{en} \nabla p_e + \frac{m_e}{e^2 n^2} \frac{d \mathbf{J}}{dt}$$

In ideal MHD, we make the following approximations:

  1. High conductivity ($\eta \to 0$): neglect resistive term
  2. Large scales ($L \gg d_i$): neglect Hall term
  3. Slow dynamics: neglect electron inertia
  4. Negligible electron pressure gradient (or isotropic electron pressure): neglect pressure term

This gives the ideal Ohm's law:

$$\boxed{\mathbf{E} + \mathbf{v} \times \mathbf{B} = 0}$$

This is the frozen-in condition: the magnetic field is frozen into the fluid and moves with it.

2.5 Faraday's Law and Induction Equation

From Maxwell's equations: $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$

Substituting the ideal Ohm's law $\mathbf{E} = -\mathbf{v} \times \mathbf{B}$:

$$\nabla \times (-\mathbf{v} \times \mathbf{B}) = -\frac{\partial \mathbf{B}}{\partial t}$$

Using the vector identity $\nabla \times (\mathbf{A} \times \mathbf{B}) = \mathbf{A}(\nabla \cdot \mathbf{B}) - \mathbf{B}(\nabla \cdot \mathbf{A}) + (\mathbf{B} \cdot \nabla)\mathbf{A} - (\mathbf{A} \cdot \nabla)\mathbf{B}$:

$$\nabla \times (\mathbf{v} \times \mathbf{B}) = \mathbf{v}(\nabla \cdot \mathbf{B}) - \mathbf{B}(\nabla \cdot \mathbf{v}) + (\mathbf{B} \cdot \nabla)\mathbf{v} - (\mathbf{v} \cdot \nabla)\mathbf{B}$$

Since $\nabla \cdot \mathbf{B} = 0$:

$$\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B}) = (\mathbf{B} \cdot \nabla)\mathbf{v} - \mathbf{B}(\nabla \cdot \mathbf{v}) - (\mathbf{v} \cdot \nabla)\mathbf{B}$$

Rearranging:

$$\boxed{\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B})}$$

Or equivalently:

$$\boxed{\frac{d \mathbf{B}}{dt} = (\mathbf{B} \cdot \nabla)\mathbf{v} - \mathbf{B}(\nabla \cdot \mathbf{v})}$$

where $d/dt = \partial/\partial t + \mathbf{v} \cdot \nabla$ is the convective derivative.

This is the induction equation (or magnetic evolution equation). It describes how the magnetic field evolves as the plasma flows.

2.6 Summary: Ideal MHD Equations

The ideal MHD equations are:

Mass continuity: $$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0$$

Momentum: $$\rho \frac{d \mathbf{v}}{dt} = \mathbf{J} \times \mathbf{B} - \nabla p$$

Induction: $$\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B})$$

Energy (adiabatic): $$\frac{d}{dt}\left( \frac{p}{\rho^\gamma} \right) = 0$$

Ampère's law (neglecting displacement current): $$\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$$

No magnetic monopoles: $$\nabla \cdot \mathbf{B} = 0$$

These are 8 equations for 8 unknowns: $\rho$, $\mathbf{v}$ (3 components), $p$, $\mathbf{B}$ (3 components), given the constraint $\nabla \cdot \mathbf{B} = 0$.

(The electric field $\mathbf{E}$ is determined by Ohm's law: $\mathbf{E} = -\mathbf{v} \times \mathbf{B}$.)

3. Validity Conditions for MHD

3.1 Low Frequency: $\omega \ll \omega_{ci}$

MHD is a low-frequency approximation. The time scale of phenomena must be much longer than the ion cyclotron period:

$$\omega \ll \omega_{ci} = \frac{eB}{m_i}$$

This ensures that ions have time to gyrate and respond to the fields in a fluid-like manner, rather than exhibiting individual particle behavior.

Example: For $B = 1$ T, $\omega_{ci} \approx 10^8$ rad/s ($f \approx 16$ MHz). MHD is valid for phenomena slower than ~10 MHz.

3.2 Large Scale: $L \gg \rho_i$

The spatial scale must be much larger than the ion gyroradius:

$$L \gg \rho_i = \frac{v_{th,i}}{\omega_{ci}}$$

At scales $\lesssim \rho_i$, finite Larmor radius (FLR) effects become important, and MHD breaks down.

Example: For $T_i = 10$ keV and $B = 1$ T, $\rho_i \approx 0.5$ cm. MHD is valid for scales $\gg 1$ cm.

3.3 Collisional: $\lambda_{mfp} \ll L$

For isotropic pressure (as assumed in ideal MHD), collisions must be frequent enough to isotropize the distribution function:

$$\lambda_{mfp} = v_{th} \tau \ll L$$

where $\tau$ is the collision time.

In collisionless plasmas, the pressure tensor is anisotropic ($p_\parallel \neq p_\perp$), requiring a more general closure (e.g., CGL, discussed below).

Example: In the solar wind, $\lambda_{mfp} \sim 1$ AU $\gg L$ for any reasonable structure. Standard MHD is not valid—CGL or kinetic models are needed.

3.4 Non-Relativistic: $v \ll c$

Plasma flows and thermal velocities must be non-relativistic:

$$v, v_{th} \ll c$$

This allows us to neglect displacement current in Ampère's law and use non-relativistic momentum equations.

Example: For $T = 10$ keV, $v_{th,e} \approx 0.04c$ (relativistic corrections ~few percent). For higher temperatures, relativistic MHD is needed.

3.5 Quasi-Neutrality: $n_e \approx n_i$

The plasma must be quasi-neutral on the scales of interest:

$$L \gg \lambda_D = \sqrt{\frac{\epsilon_0 k_B T}{n e^2}}$$

This allows us to neglect charge separation and drop the displacement current.

Example: For $n = 10^{19}$ m$^{-3}$ and $T = 10$ eV, $\lambda_D \approx 10$ μm. MHD is valid for $L \gg 10$ μm.

3.6 High Magnetic Reynolds Number: $R_m \gg 1$

For ideal MHD (frozen-in), the magnetic Reynolds number must be large:

$$R_m = \frac{\mu_0 V L}{\eta} \gg 1$$

where $V$ is a characteristic flow velocity, $L$ is a length scale, and $\eta$ is the resistivity.

When $R_m \sim 1$, resistivity becomes important → resistive MHD.

Example: In a tokamak, $V \sim 100$ m/s, $L \sim 1$ m, $\eta \sim 10^{-8}$ Ω·m → $R_m \sim 10^{10}$. Ideal MHD is excellent.

3.7 Validity Regime Summary

Ideal MHD is valid when ALL of the following hold:

1. ω << ω_ci           (low frequency)
2. L >> ρ_i            (large scale)
3. λ_mfp << L          (collisional, for isotropic p)
4. v << c              (non-relativistic)
5. L >> λ_D            (quasi-neutral)
6. R_m >> 1            (frozen-in)

Violations → need extended MHD or kinetic models.

4. CGL (Double Adiabatic) Model

4.1 Motivation: Collisionless Magnetized Plasmas

In many astrophysical plasmas (solar wind, magnetosphere, galaxy clusters), the collision mean free path is enormous:

$$\lambda_{mfp} \gg L$$

In such plasmas, particles can travel long distances without colliding. The pressure tensor becomes anisotropic:

$$\overleftrightarrow{P} = p_\perp \overleftrightarrow{I} + (p_\parallel - p_\perp) \hat{\mathbf{b}} \hat{\mathbf{b}}$$

where $\hat{\mathbf{b}} = \mathbf{B}/B$ and: - $p_\parallel$: pressure parallel to $\mathbf{B}$ - $p_\perp$: pressure perpendicular to $\mathbf{B}$

Standard MHD assumes $p_\parallel = p_\perp = p$ (isotropic), which is invalid in collisionless plasmas.

4.2 Chew-Goldberger-Low (1956) Model

Chew, Goldberger, and Low (CGL) derived a closure for collisionless, strongly magnetized plasmas by assuming conservation of adiabatic invariants:

First adiabatic invariant (magnetic moment): $$\mu = \frac{m v_\perp^2}{2B} = \text{const}$$

This implies: $$\frac{d}{dt}\left( \frac{p_\perp}{n B} \right) = 0$$

Second adiabatic invariant (longitudinal action): $$J = \oint v_\parallel ds = \text{const}$$

For a local fluid element (not bouncing between mirrors), this becomes: $$\frac{d}{dt}\left( \frac{p_\parallel B^2}{n^3} \right) = 0$$

These are the CGL equations (also called double adiabatic equations).

4.3 CGL Closure Relations

The CGL equations are:

$$\boxed{\frac{d}{dt}\left( \frac{p_\perp}{nB} \right) = 0}$$

$$\boxed{\frac{d}{dt}\left( \frac{p_\parallel B^2}{n^3} \right) = 0}$$

These can be rewritten as:

$$\frac{1}{p_\perp} \frac{dp_\perp}{dt} = \frac{1}{n} \frac{dn}{dt} + \frac{1}{B} \frac{dB}{dt}$$

$$\frac{1}{p_\parallel} \frac{dp_\parallel}{dt} = 3 \frac{1}{n} \frac{dn}{dt} - 2 \frac{1}{B} \frac{dB}{dt}$$

Physical interpretation:

  • When the field increases ($dB/dt > 0$), $p_\perp$ increases (betatron heating), but $p_\parallel$ decreases (magnetic mirror effect).
  • Compression ($dn/dt > 0$) increases both $p_\perp$ and $p_\parallel$.

4.4 CGL Pressure Tensor

The momentum equation with the CGL pressure tensor becomes:

$$\rho \frac{d \mathbf{v}}{dt} = \mathbf{J} \times \mathbf{B} - \nabla \cdot \overleftrightarrow{P}$$

where: $$\nabla \cdot \overleftrightarrow{P} = \nabla p_\perp + (p_\parallel - p_\perp) \left[ \frac{\nabla \cdot \mathbf{B}}{B} \hat{\mathbf{b}} + \frac{(\mathbf{B} \cdot \nabla) \mathbf{B}}{B^2} \right]$$

Using $\nabla \cdot \mathbf{B} = 0$ and $(\mathbf{B} \cdot \nabla)\mathbf{B} = B^2 \boldsymbol{\kappa}$ (where $\boldsymbol{\kappa}$ is the curvature):

$$\nabla \cdot \overleftrightarrow{P} = \nabla p_\perp + (p_\parallel - p_\perp) \boldsymbol{\kappa}$$

So the momentum equation is:

$$\rho \frac{d \mathbf{v}}{dt} = \mathbf{J} \times \mathbf{B} - \nabla p_\perp - (p_\parallel - p_\perp) \boldsymbol{\kappa}$$

The anisotropy creates an extra force $-(p_\parallel - p_\perp) \boldsymbol{\kappa}$ along the field curvature.

4.5 CGL Instabilities

The CGL model predicts pressure-anisotropy-driven instabilities when:

  1. Mirror instability: If $p_\perp / p_\parallel$ is too large $$\frac{p_\perp}{p_\parallel} > 1 + \frac{1}{\beta_\perp}$$ where $\beta_\perp = 2\mu_0 p_\perp / B^2$.

The plasma creates local magnetic mirrors (enhanced $B$ regions) to reduce $p_\perp$.

  1. Firehose instability: If $p_\parallel / p_\perp$ is too large $$\frac{p_\parallel}{p_\perp} > 1 + \frac{2}{\beta_\parallel}$$ where $\beta_\parallel = 2\mu_0 p_\parallel / B^2$.

The magnetic field line "kinks" like a firehose under pressure.

These instabilities are observed in the solar wind and Earth's magnetosheath.

4.6 Limitations of CGL

  1. No heat flux: CGL assumes no parallel heat conduction. In reality, heat flux is important on long parallel scales.

  2. No collisions: CGL is for collisionless plasmas. Adding even weak collisions modifies the evolution.

  3. Local approximation: CGL assumes the second adiabatic invariant holds locally, which breaks down for trapped particles bouncing on long scales.

  4. Slow dynamics: CGL assumes slow evolution compared to the gyro-period and bounce period.

Despite these limitations, CGL captures essential physics of anisotropic pressure in collisionless plasmas and is widely used in space physics.

5. Beyond MHD: Drift-Kinetic and Gyrokinetic Theory

5.1 Drift-Kinetic Theory

Drift-kinetic theory reduces the dimensionality from 6D to 5D by averaging over the gyrophase.

The idea: in a magnetized plasma, particles rapidly gyrate around field lines. If we only care about slow dynamics ($\omega \ll \omega_c$), we can average over the fast gyration.

Variables: - $\mathbf{R}$: guiding center position (3D) - $v_\parallel$: parallel velocity (1D) - $\mu$: magnetic moment (adiabatic invariant, parameter) - Time $t$

Distribution function: $F(\mathbf{R}, v_\parallel, \mu, t)$ (5D instead of 6D)

Drift-kinetic equation (simplified): $$\frac{\partial F}{\partial t} + \mathbf{v}_d \cdot \nabla_\mathbf{R} F + \frac{d v_\parallel}{dt} \frac{\partial F}{\partial v_\parallel} = C[F]$$

where $\mathbf{v}_d$ includes the parallel motion and perpendicular drifts: $$\mathbf{v}_d = v_\parallel \hat{\mathbf{b}} + \mathbf{v}_E + \mathbf{v}_{\nabla B} + \mathbf{v}_\kappa + \cdots$$

What it captures: - Parallel motion and bounce dynamics (trapped particles) - All guiding-center drifts - Collisionless (Landau) damping

What it misses: - Cyclotron resonance (averaged out with gyrophase) - Finite Larmor radius (FLR) effects

Applications: - Neoclassical transport (tokamak collisional diffusion) - Bounce-averaged kinetic theory (trapped-particle instabilities) - Radiation belt dynamics (drift-loss-cone)

5.2 Gyrokinetic Theory

Gyrokinetic theory is the most sophisticated reduced model, capturing finite Larmor radius (FLR) effects while still averaging over gyrophase.

Key innovation: Expand in small parameters: $$\delta = \frac{\rho_i}{L} \sim \frac{\omega}{\omega_{ci}} \sim \frac{\delta f}{f_0} \ll 1$$

This is the gyrokinetic ordering: slow, small-amplitude, long-wavelength fluctuations.

Variables (same as drift-kinetic): - $\mathbf{R}$: gyrocenter position - $v_\parallel$: parallel velocity - $\mu$: magnetic moment

Gyrokinetic distribution: $g(\mathbf{R}, v_\parallel, \mu, t)$ (perturbed part)

Gyrokinetic equation (schematic): $$\frac{\partial g}{\partial t} + \mathbf{v}_d \cdot \nabla g + \frac{dv_\parallel}{dt} \frac{\partial g}{\partial v_\parallel} = \text{(source terms with FLR)}$$

The key difference from drift-kinetic: FLR effects are retained through: - Gyroaveraged electric field: $\langle \phi \rangle_\alpha$ (averaged over gyro-orbit) - Gyroaveraged magnetic perturbation

What it captures: - FLR effects (ion Landau damping, wave-particle resonances with FLR) - Microinstabilities: ITG (ion temperature gradient), TEM (trapped electron mode), ETG (electron temperature gradient) - Turbulence cascades with FLR

What it misses: - Compressible Alfvén waves (fast magnetosonic) - Low-frequency approximation: $\omega \ll \omega_{ci}$

Applications: - Tokamak turbulence: gyrokinetic simulations (GENE, GS2, GYRO) predict turbulent transport, which limits confinement - Microinstability analysis: determine growth rates of ITG, TEM, ETG modes - Zonal flows: self-generated sheared flows that regulate turbulence

Gyrokinetic simulations are the state-of-the-art for tokamak physics and run on the world's largest supercomputers.

5.3 Comparison: Drift-Kinetic vs. Gyrokinetic

Feature Drift-Kinetic Gyrokinetic
Dimensions 5D 5D
FLR effects No Yes
Gyrophase-averaged Yes Yes
Ordering None (exact gyroaverage) $\delta \ll 1$ (perturbative)
What it solves Bounce motion, drifts Turbulence, microinstabilities
Typical application Neoclassical, radiation belts Tokamak turbulence, ITG/TEM
Computational cost Moderate Very high

6. Extended MHD Models

6.1 Hall MHD

Hall MHD includes the Hall term in Ohm's law:

$$\mathbf{E} + \mathbf{v} \times \mathbf{B} = \frac{1}{en} \mathbf{J} \times \mathbf{B}$$

This allows ions and electrons to decouple at scales $\sim d_i$ (ion skin depth).

Key features: - Whistler waves at high $k$ - Fast magnetic reconnection (Petschek rate) - Dispersive Alfvén waves

Applications: - Magnetic reconnection (magnetopause, magnetotail, solar corona) - Dynamo theory (magnetic field generation) - Space plasma turbulence

6.2 Two-Temperature MHD

Separate electron and ion temperatures:

$$\frac{d p_e}{dt} + \gamma p_e \nabla \cdot \mathbf{v} = Q_{ei} + Q_e$$ $$\frac{d p_i}{dt} + \gamma p_i \nabla \cdot \mathbf{v} = -Q_{ei} + Q_i$$

where $Q_{ei}$ is electron-ion energy exchange, and $Q_{e,i}$ are external heating.

Applications: - Heating and energy partition (e.g., shocks heat ions more than electrons initially) - Radiative cooling (electrons radiate more efficiently)

6.3 FLR-MHD

Include finite Larmor radius corrections to the pressure tensor:

$$\overleftrightarrow{P} = p \overleftrightarrow{I} + \overleftrightarrow{\Pi}^{\text{FLR}}$$

where $\overleftrightarrow{\Pi}^{\text{FLR}}$ includes gyroviscosity and other FLR effects.

Applications: - Kinetic Alfvén waves - FLR stabilization of MHD instabilities

6.4 Inertial MHD (Electron MHD)

At very small scales ($d_e$), electron inertia becomes important:

$$\mathbf{E} + \mathbf{v}_e \times \mathbf{B} = \frac{m_e}{e^2 n} \frac{d \mathbf{J}}{dt}$$

This is electron MHD (EMHD), where ions are stationary and only electrons move.

Dispersion relation (whistler in EMHD): $$\omega = k^2 V_A d_e$$

Applications: - Magnetic reconnection diffusion region - Electron-scale turbulence

7. Python Code Examples

7.1 Validity Regime Diagram

import numpy as np
import matplotlib.pyplot as plt

# Parameter space: length scale vs. frequency
L = np.logspace(-4, 6, 200)  # 0.1 mm to 1000 km
omega = np.logspace(2, 10, 200)  # 100 rad/s to 10 GHz

L_grid, omega_grid = np.meshgrid(L, omega)

# Plasma parameters (typical tokamak)
n = 1e20  # m^-3
B = 2.0   # T
T = 5e3   # eV (5 keV)

e = 1.6e-19
m_i = 1.67e-27
m_e = 9.11e-31
k_B = 1.38e-23

# Characteristic scales and frequencies
omega_ci = e * B / m_i
omega_ce = e * B / m_e
omega_pi = np.sqrt(n * e**2 / (m_i * 8.85e-12))
omega_pe = np.sqrt(n * e**2 / (m_e * 8.85e-12))

v_th_i = np.sqrt(2 * k_B * T * e / m_i)
v_th_e = np.sqrt(2 * k_B * T * e / m_e)

rho_i = v_th_i / omega_ci
rho_e = v_th_e / omega_ce
d_i = 3e8 / omega_pi
d_e = 3e8 / omega_pe
lambda_D = np.sqrt(8.85e-12 * k_B * T * e / (n * e**2))

print("Characteristic scales and frequencies:")
print(f"  Ion gyrofrequency ω_ci = {omega_ci:.2e} rad/s ({omega_ci/(2*np.pi):.2e} Hz)")
print(f"  Ion gyroradius ρ_i = {rho_i*100:.2f} cm")
print(f"  Ion skin depth d_i = {d_i:.2f} m")
print(f"  Electron skin depth d_e = {d_e*100:.2f} cm")
print(f"  Debye length λ_D = {lambda_D*1e6:.2f} μm")
print()

# Define validity regions
# 1. MHD: ω << ω_ci, L >> ρ_i
MHD = (omega_grid < 0.1 * omega_ci) & (L_grid > 10 * rho_i)

# 2. Hall MHD: ω << ω_ci, L ~ d_i
Hall_MHD = (omega_grid < 0.1 * omega_ci) & (L_grid > 10 * rho_i) & (L_grid < 100 * d_i)

# 3. Two-fluid: ω << ω_ce, L > d_e
Two_Fluid = (omega_grid < 0.1 * omega_ce) & (L_grid > 10 * d_e)

# 4. Gyrokinetic: ω ~ ω_ci, L ~ ρ_i
Gyrokinetic = (omega_grid > 0.01 * omega_ci) & (omega_grid < omega_ci) & \
              (L_grid > rho_i) & (L_grid < 100 * rho_i)

# 5. Full kinetic: always valid (but expensive)
Full_Kinetic = np.ones_like(L_grid, dtype=bool)

# Plot
fig, ax = plt.subplots(figsize=(11, 8))

# Color regions
ax.contourf(L_grid, omega_grid, MHD.astype(int), levels=[0.5, 1.5],
            colors=['lightblue'], alpha=0.6)
ax.contourf(L_grid, omega_grid, Hall_MHD.astype(int), levels=[0.5, 1.5],
            colors=['lightcoral'], alpha=0.6)
ax.contourf(L_grid, omega_grid, Gyrokinetic.astype(int), levels=[0.5, 1.5],
            colors=['lightgreen'], alpha=0.6)

# Boundary lines
ax.axhline(omega_ci, color='r', linestyle='--', linewidth=2, label=f'$\omega_{{ci}}$ = {omega_ci:.2e} rad/s')
ax.axhline(omega_ce, color='m', linestyle='--', linewidth=1.5, label=f'$\omega_{{ce}}$ = {omega_ce:.2e} rad/s')

ax.axvline(rho_i, color='b', linestyle='--', linewidth=2, label=f'$\\rho_i$ = {rho_i*100:.1f} cm')
ax.axvline(d_i, color='g', linestyle='--', linewidth=2, label=f'$d_i$ = {d_i:.1f} m')
ax.axvline(d_e, color='orange', linestyle='--', linewidth=1.5, label=f'$d_e$ = {d_e*100:.1f} cm')

# Labels for regions
ax.text(1e0, 1e3, 'MHD', fontsize=16, weight='bold', color='blue')
ax.text(1e-1, 1e4, 'Hall MHD', fontsize=14, weight='bold', color='red')
ax.text(1e-2, 1e7, 'Gyrokinetic', fontsize=14, weight='bold', color='green')
ax.text(1e-3, 1e9, 'Full Kinetic', fontsize=14, weight='bold', color='black')

ax.set_xscale('log')
ax.set_yscale('log')
ax.set_xlabel('Length scale L (m)', fontsize=13)
ax.set_ylabel('Frequency ω (rad/s)', fontsize=13)
ax.set_title('Plasma Model Validity Regimes (n=$10^{20}$ m$^{-3}$, B=2 T, T=5 keV)', fontsize=14)
ax.legend(fontsize=10, loc='upper left')
ax.grid(True, which='both', alpha=0.3)
ax.set_xlim(1e-4, 1e6)
ax.set_ylim(1e2, 1e10)

plt.tight_layout()
plt.savefig('validity_regimes.png', dpi=150)
plt.show()

7.2 CGL vs. Isotropic MHD: Mirror Instability

import numpy as np
import matplotlib.pyplot as plt

def mirror_instability_threshold(beta_perp):
    """
    Mirror instability threshold: p_perp/p_parallel > 1 + 1/beta_perp
    """
    return 1 + 1/beta_perp

# Beta range
beta_perp = np.logspace(-2, 2, 200)

# Threshold
threshold = mirror_instability_threshold(beta_perp)

# Plot
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(13, 5))

# Threshold curve
ax1.plot(beta_perp, threshold, 'r-', linewidth=3, label='Mirror instability threshold')
ax1.fill_between(beta_perp, 1, threshold, alpha=0.3, color='red', label='Unstable')
ax1.fill_between(beta_perp, threshold, 10, alpha=0.3, color='green', label='Stable')

ax1.set_xscale('log')
ax1.set_xlabel(r'$\beta_\perp = 2\mu_0 p_\perp / B^2$', fontsize=12)
ax1.set_ylabel(r'$p_\perp / p_\parallel$', fontsize=12)
ax1.set_title('Mirror Instability Threshold', fontsize=13)
ax1.set_ylim(1, 10)
ax1.legend(fontsize=11)
ax1.grid(alpha=0.3)

# Growth rate (simplified)
# γ/Ω_i ~ sqrt(β_perp) * (p_perp/p_parallel - 1 - 1/β_perp) for unstable
beta_example = 1.0
anisotropy = np.linspace(1, 5, 100)
threshold_value = mirror_instability_threshold(beta_example)

gamma_normalized = np.where(anisotropy > threshold_value,
                             np.sqrt(beta_example) * (anisotropy - threshold_value),
                             0)

ax2.plot(anisotropy, gamma_normalized, 'b-', linewidth=3)
ax2.axvline(threshold_value, color='r', linestyle='--', linewidth=2,
            label=f'Threshold at $\\beta_\\perp$ = {beta_example}')
ax2.fill_between(anisotropy, 0, gamma_normalized, alpha=0.3, color='blue')

ax2.set_xlabel(r'$p_\perp / p_\parallel$', fontsize=12)
ax2.set_ylabel(r'Growth rate $\gamma / \Omega_i$', fontsize=12)
ax2.set_title(f'Mirror Instability Growth Rate ($\\beta_\\perp$ = {beta_example})', fontsize=13)
ax2.legend(fontsize=11)
ax2.grid(alpha=0.3)

plt.tight_layout()
plt.savefig('mirror_instability.png', dpi=150)
plt.show()

print(f"Mirror instability:")
print(f"  At β_perp = 0.1: threshold p_perp/p_parallel > {mirror_instability_threshold(0.1):.2f}")
print(f"  At β_perp = 1.0: threshold p_perp/p_parallel > {mirror_instability_threshold(1.0):.2f}")
print(f"  At β_perp = 10:  threshold p_perp/p_parallel > {mirror_instability_threshold(10):.2f}")
print()
print("Physical interpretation:")
print("  High β_perp (strong pressure): easier to go unstable (lower threshold)")
print("  Low β_perp (weak pressure): harder to go unstable (higher threshold)")

7.3 Dispersion Comparison: MHD vs. Hall MHD vs. Kinetic

import numpy as np
import matplotlib.pyplot as plt

# Plasma parameters
n = 1e19
B = 0.1
T_e = 10  # eV
T_i = 10

e = 1.6e-19
m_i = 1.67e-27
m_e = 9.11e-31
mu_0 = 4e-7 * np.pi
k_B = 1.38e-23

# Derived quantities
omega_ci = e * B / m_i
omega_ce = e * B / m_e
omega_pi = np.sqrt(n * e**2 / (m_i * 8.85e-12))

v_A = B / np.sqrt(mu_0 * n * m_i)
c_s = np.sqrt(k_B * (T_e + T_i) * e / m_i)
d_i = 3e8 / omega_pi

v_th_e = np.sqrt(2 * k_B * T_e * e / m_e)
v_th_i = np.sqrt(2 * k_B * T_i * e / m_i)

print("Plasma parameters:")
print(f"  V_A = {v_A:.2e} m/s")
print(f"  c_s = {c_s:.2e} m/s")
print(f"  d_i = {d_i:.2e} m")
print(f"  ω_ci = {omega_ci:.2e} rad/s")
print()

# Wavenumber range
k = np.logspace(-3, 3, 500) / d_i  # normalized to d_i

# 1. MHD Alfvén wave
omega_MHD = k * v_A / omega_ci * (k * d_i)  # normalized to omega_ci

# 2. Hall MHD (whistler)
omega_Hall = k * v_A / omega_ci * (k * d_i) * np.sqrt(1 + (k * d_i)**2)

# 3. Kinetic Alfvén wave (warm plasma, with electron Landau damping)
# Approximate dispersion (electrostatic limit)
k_perp = k / 2  # assume oblique
rho_s = c_s / omega_ci
omega_KAW = k * v_A / omega_ci * (k * d_i) * np.sqrt(1 + (k_perp * d_i * rho_s / d_i)**2)

# 4. Ion acoustic wave
omega_ion_acoustic = k * c_s / omega_ci * (k * d_i)

# Plot
fig, ax = plt.subplots(figsize=(11, 7))

ax.loglog(k * d_i, omega_MHD, 'b-', linewidth=3, label='MHD Alfvén: $\omega = k_\parallel V_A$')
ax.loglog(k * d_i, omega_Hall, 'r--', linewidth=3, label='Hall MHD (whistler): $\omega = k_\parallel V_A \sqrt{1+(kd_i)^2}$')
ax.loglog(k * d_i, omega_KAW, 'g-.', linewidth=3, label='Kinetic Alfvén (warm)')
ax.loglog(k * d_i, omega_ion_acoustic, 'm:', linewidth=3, label='Ion acoustic: $\omega = k c_s$')

# Reference lines
ax.axvline(1, color='k', linestyle=':', alpha=0.5, linewidth=2, label='$k d_i = 1$')
ax.axhline(1, color='gray', linestyle=':', alpha=0.5, linewidth=2, label='$\omega = \omega_{ci}$')

# Asymptotic slopes
k_ref = np.logspace(-2, 0, 50)
ax.loglog(k_ref * d_i, (k_ref * d_i)**1 * 0.01, 'k--', alpha=0.4, label='slope = 1')
ax.loglog(k_ref * d_i * 10, (k_ref * d_i * 10)**2 * 0.001, 'k-.', alpha=0.4, label='slope = 2')

ax.set_xlabel(r'$k d_i$ (normalized wavenumber)', fontsize=13)
ax.set_ylabel(r'$\omega / \omega_{ci}$ (normalized frequency)', fontsize=13)
ax.set_title('Dispersion Relations: MHD vs. Hall MHD vs. Kinetic', fontsize=14)
ax.legend(fontsize=10, loc='upper left')
ax.grid(True, which='both', alpha=0.3)
ax.set_xlim(1e-3, 1e3)
ax.set_ylim(1e-4, 1e2)

plt.tight_layout()
plt.savefig('dispersion_comparison.png', dpi=150)
plt.show()

print("Dispersion relations:")
print("  MHD: ω ∝ k (linear, non-dispersive)")
print("  Hall MHD: ω ∝ k² at k d_i >> 1 (whistler, dispersive)")
print("  Kinetic: includes Landau damping (not shown, requires complex ω)")

Summary

In this lesson, we traced the systematic reduction from kinetic theory to MHD:

  1. Two-fluid to single-fluid: By combining electron and ion equations, we obtain the MHD momentum and continuity equations. The key step is deriving the ideal Ohm's law $\mathbf{E} + \mathbf{v} \times \mathbf{B} = 0$ from the generalized Ohm's law by dropping resistive, Hall, pressure, and inertia terms.

  2. Validity conditions: MHD is valid for low-frequency ($\omega \ll \omega_{ci}$), large-scale ($L \gg \rho_i$), collisional ($\lambda_{mfp} \ll L$), non-relativistic ($v \ll c$), quasi-neutral ($L \gg \lambda_D$), high-$R_m$ plasmas. Violations require extended MHD or kinetic models.

  3. CGL model: For collisionless plasmas, the pressure is anisotropic ($p_\parallel \neq p_\perp$). The CGL (double adiabatic) closure uses conservation of adiabatic invariants: $p_\perp/(nB) = \text{const}$ and $p_\parallel B^2 / n^3 = \text{const}$. This predicts mirror and firehose instabilities.

  4. Drift-kinetic and gyrokinetic: These 5D models average over gyrophase while retaining kinetic effects. Drift-kinetic captures bounce dynamics; gyrokinetic includes FLR effects and is used for tokamak turbulence simulations.

  5. Extended MHD: Hall MHD, two-temperature MHD, FLR-MHD, and electron MHD extend standard MHD to capture additional physics at the cost of increased complexity.

  6. Model comparison: Each model has a regime of validity. The choice depends on the scales, frequencies, and physics of interest. MHD is simple and captures large-scale dynamics; kinetic theory is comprehensive but computationally expensive.

Understanding the hierarchy of plasma models is essential for choosing the appropriate level of description for a given problem.

Practice Problems

Problem 1: Ideal MHD from Generalized Ohm's Law

Starting from the generalized Ohm's law: $$\mathbf{E} + \mathbf{v} \times \mathbf{B} = \eta \mathbf{J} + \frac{1}{en} \mathbf{J} \times \mathbf{B} - \frac{1}{en} \nabla p_e + \frac{m_e}{e^2 n^2} \frac{d \mathbf{J}}{dt}$$ For a tokamak plasma with $n = 10^{20}$ m$^{-3}$, $T_e = 10$ keV, $B = 5$ T, $L = 1$ m, $V = 100$ m/s: (a) Calculate the characteristic time scale $\tau = L/V$. (b) Estimate the magnitude of each term on the RHS relative to the LHS. (c) Which terms can be neglected to obtain ideal MHD? Justify your answer.

Problem 2: CGL Pressure Evolution

A collisionless plasma is compressed adiabatically by increasing the magnetic field from $B_0$ to $2B_0$ while keeping the density constant ($n = n_0$). (a) Using the CGL equations, find the final values of $p_\perp$ and $p_\parallel$ in terms of the initial values. (b) If initially $p_{\perp 0} = p_{\parallel 0} = p_0$, what is the anisotropy ratio $p_\perp / p_\parallel$ after compression? (c) For $\beta_{\perp 0} = 0.5$, does the compressed plasma exceed the mirror instability threshold?

Problem 3: Frozen-In Flux

In ideal MHD, the magnetic flux through any closed loop moving with the fluid is conserved: $$\frac{d\Phi}{dt} = 0, \quad \text{where } \Phi = \int_S \mathbf{B} \cdot d\mathbf{A}$$ (a) Prove this frozen-in theorem using the ideal Ohm's law and the induction equation. (b) A circular flux tube of initial radius $r_0 = 10$ cm has magnetic field $B_0 = 0.1$ T. The plasma is compressed radially to $r = 5$ cm. What is the final magnetic field (assuming incompressible flow)? (c) What is the physical meaning of "frozen-in"? Can field lines reconnect in ideal MHD?

Problem 4: Gyrokinetic Ordering

In gyrokinetic theory, the ordering is: $$\frac{\rho_i}{L} \sim \frac{\omega}{\omega_{ci}} \sim \frac{\delta f}{f_0} \sim \delta \ll 1$$ (a) For a tokamak with $L = 1$ m, $\rho_i = 5$ mm, what is $\delta$? (b) If $\omega_{ci} = 10^8$ rad/s, what is the maximum frequency resolved by gyrokinetics? (c) The fast magnetosonic wave has $\omega \sim k V_A$ with no upper frequency limit. Why can't gyrokinetics capture this wave?

Problem 5: Hall MHD Reconnection

In resistive MHD, the Sweet-Parker reconnection rate is: $$V_{in} \sim \frac{\eta}{L} \sim \frac{V_A}{S^{1/2}}$$ where $S = L V_A / \eta$ is the Lundquist number.

In Hall MHD, the reconnection rate becomes (Petschek): $$V_{in} \sim 0.1 V_A$$ independent of resistivity!

(a) For a solar flare with $B = 0.01$ T, $n = 10^{16}$ m$^{-3}$, $L = 10^4$ km, $T_e = 10^6$ K, calculate the Alfvén speed and the ion skin depth. (b) Estimate the Sweet-Parker reconnection time $\tau_{SP} \sim L / V_{in}$ (use Spitzer resistivity). (c) Estimate the Hall MHD reconnection time $\tau_{Hall}$. (d) Solar flares release energy on time scales of minutes. Which model is consistent with observations?

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