Plasma as a Fluid: The Dielectric Multi-Fluid Equations & MHD

1. From Microscopic Distribution to Fluid Fields

Tracking $10^{20}$ individual charged particles through single-particle equations of motion is computationally intractable. In the fluid description, we average over velocity space by taking moments of the phase space distribution function $f_\alpha(\vec{r}, \vec{v}, t)$:

$$\langle \chi \rangle_\alpha = \frac{1}{n_\alpha} \int \chi(\vec{v}) f_\alpha(\vec{r}, \vec{v}, t) d^3v$$

The fundamental macroscopic fluid variables are:

• Number Density (Zeroth Moment): $$n_\alpha(\vec{r}, t) = \int f_\alpha(\vec{r}, \vec{v}, t) d^3v$$
• Mean Fluid Velocity (First Moment): $$\vec{u}_\alpha(\vec{r}, t) = \langle \vec{v} \rangle = \frac{1}{n_\alpha} \int \vec{v} f_\alpha(\vec{r}, \vec{v}, t) d^3v$$
• Stress Tensor (Second Moment): $$\mathbf{P}_\alpha(\vec{r}, t) = m_\alpha \int (\vec{v} - \vec{u}_\alpha)(\vec{v} - \vec{u}_\alpha) f_\alpha(\vec{r}, \vec{v}, t) d^3v$$

1. The Continuity Equation (Zeroth Moment)

Integrating the Boltzmann equation over all velocity space $d^3v$:

$$\int \left[ \frac{\partial f_\alpha}{\partial t} + \nabla \cdot (\vec{v} f_\alpha) + \frac{q_\alpha}{m_\alpha}\nabla_v \cdot \left( (\vec{E} + \vec{v}\times\vec{B})f_\alpha \right) \right] d^3v = \int \left( \frac{\partial f_\alpha}{\partial t} \right)_{\text{coll}} d^3v$$

Assuming elastic collisions that conserve particle number:

$$\frac{\partial n_\alpha}{\partial t} + \nabla \cdot (n_\alpha \vec{u}_\alpha) = 0$$

2. The Fluid Momentum Equation (First Moment)

Multiplying the Boltzmann equation by $m_\alpha \vec{v}$ and integrating over velocity space:

$$m_\alpha n_\alpha \left[ \frac{\partial \vec{u}_\alpha}{\partial t} + (\vec{u}_\alpha \cdot \nabla)\vec{u}_\alpha \right] = q_\alpha n_\alpha (\vec{E} + \vec{u}_\alpha \times \vec{B}) - \nabla \cdot \mathbf{P}_\alpha + \vec{R}_{\alpha\beta}$$

where $\vec{R}_{\alpha\beta} = -m_\alpha n_\alpha \nu_{\alpha\beta}(\vec{u}_\alpha - \vec{u}_\beta)$ represents the inter-species collisional frictional drag.

1. Scalar Pressure & The Closure Problem

Each moment equation contains the next higher moment: continuity contains fluid velocity $\vec{u}$, momentum contains the pressure tensor $\mathbf{P}$, and energy contains the heat flux tensor $\vec{Q}$. This is the famous closure problem of kinetic theory.

To close the fluid equations, we assume an isotropic velocity distribution, reducing the stress tensor to a scalar pressure:

$$\mathbf{P}_\alpha = P_\alpha \mathbf{I} \implies \nabla \cdot \mathbf{P}_\alpha = \nabla P_\alpha$$

where $P_\alpha = n_\alpha k_B T_\alpha$ is the ideal gas equation of state.

2. Polytropic Equation of State

The thermodynamic response is modeled by the polytropic relation:

$$P_\alpha n_\alpha^{-\gamma} = \text{const} \iff \nabla P_\alpha = \gamma k_B T_\alpha \nabla n_\alpha$$

where:

• $\gamma = 1$: Isothermal compression (rapid thermal conduction along field lines).
• $\gamma = 5/3$: Adiabatic compression in 3 dimensions ($C_p / C_v = (d+2)/d = 5/3$).
• $\gamma = 3$: Adiabatic compression in 1 dimension along strong magnetic field lines ($d=1$).

1. The Apparent Paradox of Quasi-Neutrality

In fluid plasma physics, we universally employ the plasma approximation:

$$n_e \approx Z n_i \equiv n$$

However, Poisson's equation states $\nabla \cdot \vec{E} = \frac{e(n_i - n_e)}{\varepsilon_0}$. If $n_e = n_i$, one might naively conclude that $\nabla \cdot \vec{E} = 0$, implying that electrostatic fields cannot exist inside a plasma!

2. Resolution of the Paradox

The paradox is resolved by recognizing that the fractional charge imbalance $\delta n / n = |n_i - n_e| / n_0$ required to create enormous macroscopic electric fields is vanishingly small:

$$\frac{\delta n}{n_0} = \frac{\varepsilon_0 \nabla \cdot \vec{E}}{e n_0} \sim \frac{\varepsilon_0 (E/L)}{e n_0} \sim \left( \frac{\lambda_D}{L} \right)^2 \ll 1$$

Because $\lambda_D / L \sim 10^{-4}$ in typical laboratory plasmas, a fractional charge imbalance of only 1 part in $10^8$ creates millions of volts per meter! Therefore, in the fluid momentum equations we safely set $n_e = n_i = n$ everywhere except inside Poisson's equation itself. In practice, we determine $\vec{E}$ directly from the fluid equations of motion rather than from Poisson's equation.

1. The Steady-State Perpendicular Fluid Equation

Consider a steady-state ($\partial / \partial t = 0$), low-velocity ($(\vec{u}\cdot\nabla)\vec{u} \approx 0$), collisionless fluid equilibrium in a uniform magnetic field $\vec{B}$:

$$0 = q n (\vec{E} + \vec{u}_\perp \times \vec{B}) - \nabla P$$

Taking the vector cross product with $\vec{B}$:

$$q n \vec{E} \times \vec{B} + q n (\vec{u}_\perp \times \vec{B}) \times \vec{B} - \nabla P \times \vec{B} = 0$$

Using $(\vec{u}_\perp \times \vec{B}) \times \vec{B} = -B^2 \vec{u}_\perp$ and solving for $\vec{u}_\perp$:

$$\vec{u}_\perp = \frac{\vec{E} \times \vec{B}}{B^2} - \frac{\nabla P \times \vec{B}}{q n B^2} = \vec{v}_E + \vec{v}_D$$

2. The Diamagnetic Drift Velocity

The second term is the diamagnetic drift velocity $\vec{v}_D$:

$$\vec{v}_D \equiv -\frac{\nabla P \times \vec{B}}{q n B^2}$$

Fundamental properties of diamagnetic drift:

• Not a Single-Particle Guiding Center Drift: Individual guiding centers do not drift with velocity $\vec{v}_D$! $\vec{v}_D$ is a purely macroscopic fluid effect: in the presence of a density or pressure gradient, more particles gyrate across an area element in one direction than in the reverse, resulting in a net fluid flux.
• Charge Dependence: Ions and electrons drift in opposite directions, driving a net macroscopic diamagnetic current $\vec{J}_D$: $$\vec{J}_D = n e (\vec{v}_{Di} - \vec{v}_{De}) = -\frac{(\nabla P_i + \nabla P_e) \times \vec{B}}{B^2} = -\frac{\nabla P \times \vec{B}}{B^2}$$

1. Single-Fluid Center-of-Mass Variables

Combining electron and ion fluid equations yields the single-fluid Magnetohydrodynamic (MHD) description:

$$\rho_m \equiv n_i M_i + n_e m_e \approx n M_i, \quad \vec{v} \equiv \frac{M_i \vec{u}_i + m_e \vec{u}_e}{M_i + m_e} \approx \vec{u}_i$$ $$\vec{J} \equiv n e (\vec{u}_i - \vec{u}_e), \quad P \equiv P_i + P_e$$

2. The Ideal MHD Equations

Adding the electron and ion momentum equations yields the MHD equation of motion:

$$\rho_m \left[ \frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \nabla)\vec{v} \right] = \vec{J} \times \vec{B} - \nabla P$$

Subtracting the electron equation from the ion equation (and neglecting small electron inertia terms) yields the generalized Ohm's law:

$$\vec{E} + \vec{v} \times \vec{B} = \eta \vec{J} + \frac{1}{n e}\vec{J}\times\vec{B} - \frac{1}{n e}\nabla P_e$$

In ideal MHD, resistivity $\eta \to 0$, Hall effect, and electron pressure gradient terms are neglected, yielding the simple ideal Ohm's law:

$$\vec{E} + \vec{v} \times \vec{B} = 0$$

1. The Magnetic Induction Equation

Combining Faraday's law $\frac{\partial \vec{B}}{\partial t} = -\nabla \times \vec{E}$ with Ohm's law $\vec{E} = -\vec{v}\times\vec{B} + \eta \vec{J}$ and Ampère's law $\mu_0 \vec{J} = \nabla \times \vec{B}$:

$$\frac{\partial \vec{B}}{\partial t} = \nabla \times (\vec{v} \times \vec{B}) + \frac{\eta}{\mu_0} \nabla^2 \vec{B}$$

The ratio of the convective (advective) term to the resistive diffusion term defines the magnetic Reynolds number $R_m$:

$$R_m \equiv \frac{|\nabla \times (\vec{v}\times\vec{B})|}{|(\eta/\mu_0)\nabla^2\vec{B}|} = \frac{v L}{\eta_m} = \frac{\mu_0 v L}{\eta}$$

In high-temperature fusion plasmas and astrophysical systems ($L \sim 10^6\text{ m}$, $T \sim 10^7\text{ K}$), $R_m \gg 10^6$, rendering resistivity entirely negligible.

2. Alfvén's Flux-Freezing Theorem

When $R_m \to \infty$, the magnetic flux $\Phi = \int_S \vec{B}\cdot d\vec{S}$ through any closed fluid contour moving with the plasma velocity $\vec{v}$ is strictly conserved in time:

$$\frac{d\Phi}{dt} = 0$$

This is Hannes Alfvén's Flux-Freezing Theorem: the magnetic field lines are "frozen into" the fluid and move synchronously with the plasma.

3. Magnetic Pressure, Magnetic Tension & The Plasma Beta

Using the vector identity $(\nabla \times \vec{B}) \times \vec{B} = (\vec{B}\cdot\nabla)\vec{B} - \nabla(B^2/2)$:

$$\vec{J} \times \vec{B} = \frac{1}{\mu_0}(\nabla \times \vec{B}) \times \vec{B} = -\nabla\left( \frac{B^2}{2\mu_0} \right) + \frac{(\vec{B}\cdot\nabla)\vec{B}}{\mu_0}$$

The Lorentz force decomposes into:

• Magnetic Pressure: $P_{\text{mag}} = \frac{B^2}{2\mu_0}$ (isotropic perpendicular expansion force).
• Magnetic Tension: $\frac{(\vec{B}\cdot\nabla)\vec{B}}{\mu_0} = \frac{B^2}{\mu_0}\frac{\hat{R}_c}{R_c}$ (restoring force along curved field lines).

The ratio of thermal kinetic pressure to magnetic pressure is the fundamental plasma beta parameter $\beta$:

$$\beta \equiv \frac{P_{\text{thermal}}}{P_{\text{magnetic}}} = \frac{n k_B(T_e + T_i)}{B^2 / (2\mu_0)} = \frac{2\mu_0 n k_B T}{B^2}$$

In tokamaks, $\beta \sim 0.05$ (low-beta magnetically dominated); in the solar photosphere, $\beta \sim 1$; and in the solar wind, $\beta \sim 1\text{ to }10$.