Formulary

Kinetic Description

$\dv{\vec v}{t} = \frac{q_i}{m_i} (\vec E + \vec v_i \cross \vec B) + \sum_{j \neq i} \left[ \left. \dv{\vec v_{ij}}{t} \right|_{coll} \delta(\vec r_i - \vec r_j) \right]$

$\pdv{\vec B}{t} = - \curl \vec E$

$\frac{1}{c^2} \pdv{\vec E}{t} = \curl \vec B - \mu_0 \sum_i q_i \vec v_i \delta (\vec r - \vec r_i)$

$\div \vec B = 0$

$\div \vec E = \frac{1}{\epsilon_0} \sum_i q_i \delta (\vec r - \vec r_i)$

Klimontovich equation:

$\dv{N}{t} = 0 = \pdv{N}{t} + \pdv{}{q_i} \cdot (\dot{q_i} N) \\ N \equiv \sum_i \delta (p - p_i) \delta(q - q_i)$

Plasma Fluid Description

Boltzmann Equation

$\pdv{f_\alpha}{t} + \vec v \cdot \pdv{f_\alpha}{t} + \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} = \left. \pdv{f_\alpha}{t} \right|_{coll} = \sum_{\beta \neq \alpha} C_{\alpha \beta}$

Maxwellian distribution:

$f_\alpha (\vec v) = n_\alpha \left( \frac{m_\alpha}{2 \pi T} \right)^{3/2} e^{- \frac{m_\alpha(\vec v - \vec v_\alpha)^2}{2T}}$

Moments of fluid model (moments of distribution $\rightarrow$ moments of Boltzmann equation:

$\text{Continuity:} \qquad n_\alpha = \int f_\alpha \dd \vec v \\ \quad \rightarrow \pdv{n_\alpha}{t} + \div (n_\alpha \vec v_\alpha) = 0$

$\text{Momentum:} \qquad n_\alpha \vec v_\alpha = \int \vec v f_\alpha \dd v \\ \quad \rightarrow \quad \pdv{}{t} (n_\alpha \vec v_\alpha ) + \div (n_\alpha \vec v_\alpha \vec v_\alpha) + \frac{1}{m_\alpha} \div \vec P_\alpha - \frac{q_\alpha}{m_\alpha} n_\alpha ( \vec E + \vec v _\alpha \cross \vec B) = \sum_{\beta \neq \alpha} \int \vec w C_{\alpha \beta} \dd \vec v$

$\rightarrow \rho_\alpha \left(\pdv{\vec v_\alpha}{t} + \vec v_\alpha \cdot \grad \vec v_\alpha \right) + \grad \vec P_\alpha + \div \vec \Pi_\alpha - q_\alpha n_\alpha (\vec E + \vec v_\alpha \cross \vec B) = \sum_{\beta \neq \alpha} \vec R_{\alpha \beta}$

$\text{Energy:} \qquad \int \vec v \vec v \pdv{f_\alpha}{t} \dd \vec v = \pdv{}{t} \int \vec v \vec v f_\alpha \dd \vec v = \pdv{}{t} \vec E_\alpha / m_\alpha \rightarrow \pdv{}{t} \vec P_\alpha$

$\rightarrow \quad \frac{3}{2} n_\alpha \left( \pdv{T_\alpha}{t} + \vec v_\alpha \cdot \grad T_\alpha \right) + P_\alpha \div \vec v_\alpha + \vec \Pi_\alpha \cdot \cdot \grad \vec v_\alpha + \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta}$

Closure relations

$\vec h_\alpha = - \kappa \grad T_\alpha$

$\overline \Pi_ \alpha = \nu \grad \vec v_\alpha$

Ideal MHD

Conservation Law Form of Ideal MHD

$\pdv{\rho}{t} + \div (\rho \vec v) = 0$

$\pdv{(\rho \vec v)}{t} + \div \left[ \rho \vec v \vec v - \frac{\vec B \vec B}{\mu_0} + \left( p + \frac{B^2}{2 \mu_0} \right) \overline{I} \right] = 0$

$\pdv{\epsilon}{t} + \div \left[ \left( \epsilon + p + \frac{B^2}{2 \mu_0} \right) \vec v - (\vec B \cdot \vec v) \frac{\vec B}{\mu_0} \right] = 0$

$\pdv{\vec B}{t} + \div ( \vec v \vec B - \vec B \vec v) = 0$

where

$\epsilon = \frac{1}{\gamma - 1} p + \frac{1}{2} \rho v^2 + \frac{B^2}{2\mu_0}$

Static Equilibrium:

$\vec j \cross \vec B = \grad p$

$\frac{B^2}{\mu_0} \vec K = \grad_\perp (p + \frac{B^2}{2 \mu_0})$

$\vec K \equiv \frac{\vec B}{|B|} \cdot \grad \frac{ \vec B}{|B|}$

Conservation of flux:

$\vec E + \vec v \cross \vec B = 0$

$\pdv{\vec B}{t} = - \curl \vec E$

$\rightarrow \dv{}{t} \left( \frac{\vec B}{\rho} \right) = \frac{\vec B}{\rho} \cdot \grad \vec v$

1D Equilibria

$\theta$-pinch

$B_\theta = 0$

$j_\theta B_z = \dv{p}{r}$

$j_\theta = - \frac{1}{\mu_0} \dv{B_z}{r}$

$\rightarrow p + \frac{B_z ^2}{2 \mu_0} = \frac{B_0 ^2}{2 \mu_0}$

$\langle \beta \rangle = \frac{2}{a^2} \int_0 ^a \frac{r p}{B_0 ^2 / 2 \mu_0} \dd r$

$q = \infty$

$W = \frac{\mu_0 r}{B_z ^2} \dv{}{r} \left( p + \frac{B_z ^2}{2 \mu_0} \right) = 0$

Z-pinch

$B_z =0$

$\grad p = \dv{p}{r} = - j_z B_\theta$

$- \dv{}{r} \left( p + \frac{B_\theta ^2}{2 \mu_0} \right) = \frac{B_\theta ^2}{\mu_0 r}$

$\langle \beta \rangle = \frac{2 \mu_0}{B_0 ^2 \pi a^2} \int _0 ^a 2 \pi r p \dd r = 1 \quad \text{ if } \quad p(a) = 0$

$q = S = 0$

$W = 1$

Screw pinch

$\dv{}{r} \left( p + \frac{B^2}{2 \mu_0} \right) = - \frac{B_\theta ^2}{\mu_0 r}$

$\beta_t = \frac{2 \mu_0}{B_z (a) ^2} \left( \frac{1}{\pi a^2} \int_0 ^a 2 \pi r p \dd r \right)$

$\beta_p = \left( 1 - \frac{\alpha_t}{\beta _t} \right)^{-1} \qquad \alpha_t \equiv \frac{2}{a^2} \int_0 ^a \left(1 - \frac{B_z ^2}{B_0 ^2} \right) r \dd r$

$q = \frac{2 \pi r B_z}{L B_\theta}$

$q_a = \frac{4 \pi ^2 a^2 B_0}{\mu_0 I_a}$

$S = \frac{r}{q} \dv{q}{r}$

$W = - \frac{B_\theta ^2}{B_\theta ^2 + B_z ^2}$

Stability

Shear:

$S = 2 \frac{ dq / q}{dV / V} = 2 \frac{d \ln q}{d \ln V}$

$q = \frac{\text{\# long windings}}{\text{\# short windings}} = \dv{\psi_t}{\psi_p}$

Shear for toroid

$q = \frac{r B_\phi}{R B_\theta}$

Shear for cylinder

$q = 2 \pi \frac{r B_z}{L B_\theta}$

Well

$W = \frac{ d \langle p + B^2 / 2 \mu_0 \rangle / \langle B^2 / 2 \mu_0 \rangle}{dV / V}$

For stabilization, $B^2 / 2 \mu_0$ should increase faster than $p$ decreases

2D Equilibria

Grad-Shafranov Equation: Static toroidal equilibrium

$\grad p = \vec j_\theta \cross \vec B_\phi + \vec j_\phi \cross \vec B_\theta$

$A_\phi = \frac{\phi}{R} \vu \phi$

$\phi = \frac{\phi_p}{2 \pi}$

$\vec B_\theta = - \frac{\vu R}{R} \pdv{\psi}{z} + \frac{ \vu z}{R} \pdv{\psi}{R} = \frac{ \grad \psi}{R} \cross \vu \phi$

$F \equiv R B_\phi$

$\Delta ^\star \equiv R \pdv{}{R} \frac{1}{R} \pdv{}{R} + \pdv{^2}{z^2}$

$\Delta ^\star \psi = \pdv{^2 \psi}{z^2} + \pdv{^2 \psi}{R^2} - \frac{1}{R} \pdv{\psi}{R}$

$\vec j_\phi = - \frac{1}{\mu_0 R} \Delta ^\star \psi \vu \phi$

$\vec j_\theta = \frac{1}{\mu_0 R} \grad (F) \cross \vu \phi$

$R^2 \mu_0 \pdv{p}{\psi} = - \Delta ^\star \psi - F \pdv{F}{\psi}$

$q(\psi) = \frac{F(\psi)}{2 \pi} \oint_{p} \frac{r \dd \theta}{R^2 B_\theta}$

Limits:

$\text{Force-free:} \qquad \vec j \parallel \vec B$

$\rightarrow \Delta ^\star \psi + F F' = 0$

$\text{Connected $\theta$ pinch:} \qquad FF' \gg \Delta ^\star \psi$

$\rightarrow \grad p \approx \vec j_\theta \cross \vec B_\phi$

$\text{Connected Z-pinch:} \qquad FF' \ll \Delta ^\star \phi$

$\rightarrow \grad p \approx j_\phi \cross B_\theta$

MHD Stability

Linear stability

$\pdv{\rho_1}{t} = - \vec v_1 \grad \rho_0 - \rho_0 \div \vec v_1$

$\pdv{\vec B_1}{t} = \curl ( \vec v_1 \cross \vec B_0)$

$\rho_0 \pdv{\vec v_1}{t} = - \grad p_1 + \vec j_0 \cross \vec B_1 - \vec j_1 \cross \vec B_0$

$\pdv{p_1}{t} = - \vec v_1 \cdot \grad p_0 - \gamma p_0 \div \vec v_1$

For linear perturbation $\vec \xi = \int_0 ^t \vec v_1 \dd t$ the momentum equation becomes

$\rho_0 \pdv{^2 \xi}{t^2} = \vec F(\xi)$

where

$F(\xi) = \grad (\xi \cdot \grad p_0 + \gamma p_0 \div \xi) + \frac{1}{\mu_0} \left[ ( \curl \vec B_0) \cross \curl (\xi \cross \vec B_0) + \curl \curl (\xi \cross \vec B_0) \cross \vec B_0 \right]$

Eigenvalues of $\frac{1}{\rho_0} \vec F (\xi) = \omega^2 \xi$ are real and ordered. Only need to check $n=0$ to determine stability/instability of configuration.

$\delta W$ Approach

$\delta W =$ change in potential energy due to a displacement $\xi$

$\delta W < 0 \rightarrow \text{instability}$

$\delta W = - \frac{1}{2} \int \xi \cdot F(\xi) \dd V = \delta W_F + \delta W_S + \delta W_V$

Surface term:

$\delta W_s = \frac{1}{2} \oint \dd S (\vu n \cdot \xi) ^2 \left( \vu n \cdot \grad p_0 + \left[ \vu n \cdot \grad \frac{B_0 ^2}{2 \mu_0} \right]_{jump} \right)$

Vacuum term:

$\delta W_V = \int_{vac} \dd V \frac{B_1 ^2}{\mu_0}$

Plasma (free) term:

$\delta W_F = \frac{1}{2} \int \dd V \frac{ |B_{1, \perp}|^2}{\mu_0} \quad \leftarrow \text{Shear Alfven} \\ + \mu_0 \left| \frac{B_{1, \parallel}}{\mu_0} - \frac{B_0 \xi \cdot \grad p_0}{B_0} ^2 \right|^2 \quad \leftarrow \text{Fast magnetosonic} \\ + \Gamma p_0 |\div \xi|^2 \quad \leftarrow \text{Acoustic}\\ + \frac{\vec j_0 \cdot \vec B_0}{B_0 ^2} (\vec B_0 \cross \vec \xi) \cdot \vec B_1 \quad \leftarrow \text{Current-driven (kink)} \\ - 2 ( \vec \xi \cdot \grad p_0)(\vec \xi \cdot \vec \kappa) \quad \leftarrow \text{pressure-driven (interchange/balooning)}$

Shear Alfven, fast magnetosonic, and acoustic modes are stabilizing. Current-driven and pressure-driven modes can lead to instability.