Equations (cgs)

Drift-reduced equations in cgs units

Electron continuity equation

The electron continuity equation becomes under the drift approximation $$\begin{align*} \frac{\partial}{\partial t}n+\nabla\cdot n\left(\mathbf{v}_E+\mathbf{v}_*^e+v_\parallel\mathbf{b}\right)=0, \end{align*}$$ which can be written as: $$\begin{align} \frac{\partial}{\partial t}n+\mathbf{v}_E\cdot\nabla n +\nabla\cdot\left(nu_\parallel\mathbf{b}\right)=n\mathcal{C}(\phi)-\frac{n}{e}\mathcal{C}(T_e)-\frac{T_e}{e}\mathcal{C}(n)+\frac{1}{e}\nabla\cdot\left(j_\parallel\mathbf{b}\right). \label{drift_continuity} \end{align}$$

Vorticity equation

The quasi-neutrality condition $\nabla\cdot\mathbf{j}=0$ is $$\begin{align*} \nabla\cdot\left(en\mathbf{u}_{pol}-en\mathbf{v}_*^e+en\mathbf{v}_*^i+j_\parallel\mathbf{b}\right)=0, \end{align*}$$ which can be written as $$\begin{align} \nabla\cdot\left[\frac{M_ic^2}{B^2}n\frac{d_i}{dt}(\nabla_\perp\phi+\frac{\nabla_{\perp}p_i}{en})\right] = -T_e\mathcal{C}(n)-n\mathcal{C}(T_e)-T_i\mathcal{C}(n)-n\mathcal{C}(T_i)+\nabla\cdot\left(j_\parallel\mathbf{b}\right)-\frac{1}{6}\mathcal{C}(G). \label{drift_vorticity} \end{align}$$

Parallel momentum equation

For the (total) parallel momentum of the plasma, we add the parallel components of the electron and ion momentum equations together. The Lorentz force and friction terms add up to zero. Due to the small electron mass, $m_e\ll M_i$, we can neglect their inertia. Therefore, the total plasma momentum is approximately carried exclusively by the ions, $M_i u_\parallel + m_e v_\parallel \approx M_i u_\parallel$, yielding

$$\begin{align} %M_in\frac{d_i}{dt}u_\parallel = -\nabla_\parallel(p_e + p_i) + \frac{M_ip_i}{e}\mathcal{C}(u_\parallel)+\frac{\nabla_\parallel G}{3}-\nabla\cdot\left(\mathbf{b}G\right) M_in\frac{d_i}{dt}u_\parallel = -\nabla_\parallel(p_e + p_i) + \frac{M_ip_i}{e}\mathcal{C}(u_\parallel) - \frac{2}{3} B^{3/2} \nabla_\parallel \frac{G}{B^{3/2}}. \label{drift_parallel_momentum} \end{align}$$

Ohm's law

The parallel current is defined as $j_\parallel = en(u_\parallel - v_\parallel)$, i.e. as the difference between ion and electron currents. However, due to $m_e/M_i \ll 1$, electrons are much faster and carry the majority of the current. To obtain Ohm's law, the parallel current conservation equation, we scale the ion parallel momentum equation by $m_e/M_i$ and subtract it from the electron parallel momentum equation, obtaining

$$\begin{align} -\frac{m_e}{e}\frac{d_e}{dt}\frac{j_\parallel}{en} = \eta_\parallel j_\parallel + \frac{1}{c}\frac{\partial}{\partial t}A_\parallel + \nabla_\parallel\phi - \frac{1}{en}\nabla_\parallel p_e - 0.71\frac{1}{e}\nabla_\parallel T_e. \label{drift_ohms_law} \end{align}$$ On the right-hand side, all terms of order $m_e/M_i$ have been neglected, i.e. the parallel ion momentum equation contributes only to the inertia. The inertia term is important to physically limit the wave propagation speed below light speed Dudson 2021.

The parallel resistivity is defined as $$\begin{align} \eta_\parallel := \frac{0.51m_e}{e^2\tau_en}, \end{align}$$ where we note that $\eta_\parallel\propto T_e^{-3/2}$, but independent of $n$.

Ampere's law

Ohm's law is solved for $A_\parallel$, connected to $j_\parallel$ through Ampere's law $$\begin{align} \nabla_\perp^2A_\parallel=-\frac{4\pi}{c}j_\parallel. \label{drift_ampere} \end{align}$$

Electron temperature equation

The following identities are useful for the derivation of the drift reduced electron temperature equation, as they are responsible for the diamagnetic cancellation, $$\begin{align} &p_e\nabla\cdot\left(\mathbf{v}_E+\mathbf{v}_*+v_\parallel\mathbf{b}\right) = -T_e\frac{d_e}{dt}n+n\mathbf{v}_*\cdot\nabla T_e, \\ &\nabla\cdot\mathbf{q}_{e,\perp}=\nabla\cdot\left[-\frac{5}{2}\frac{cp_e}{eB^2}\mathbf{B}\times\nabla T_e\right] = -\frac{5}{2}n\mathbf{v}_*\cdot\nabla T_e +\frac{5}{2}\frac{p_e}{e}\mathcal{C}(T_e). \end{align}$$ This yields $$\begin{align} \frac{3}{2}n\frac{d_e}{dt}T_e=T_e\frac{d_e}{dt}n-\frac{5}{2}\frac{p_e}{e}\mathcal{C}(T_e)+0.71\frac{T_e}{e}\nabla\cdot\left(j_\parallel\mathbf{b}\right)+\nabla\cdot \left(\chi_\parallel^e\mathbf{b}\nabla\parallel T_e\right)+\eta_\parallel j_\parallel^2 - Q_\Delta. \end{align}$$ Using the continuity equation we obtain $$\begin{align} \frac{d_e}{dt}T_e=&-\frac{2}{3}\left[\frac{7}{2}\frac{T_e}{e}\mathcal{C}(T_e)+\frac{T_e^2}{en}\mathcal{C}(n)-T_e\mathcal{C}(\phi)\right]-\frac{2}{3}T_e\nabla\cdot \left(v_\parallel\mathbf{b}\right)+\frac{2}{3}0.71\frac{T_e}{en}\nabla\cdot\left(j_\parallel\mathbf{b}\right)+\frac{2}{3}\frac{\eta_\parallel j_\parallel^2}{n} \notag\\ &+\frac{2}{3}\frac{1}{n}\nabla\cdot\left(\chi_\parallel^e\nabla_\parallel T_e\right) - \frac{2m_e}{M_i \tau_e}(T_e-T_i), \label{drift_electron_temperature} \end{align}$$ where the parallel electron heat conductivity is defined as: $$\begin{align*} \chi_\parallel^e=3.16\frac{nT_e\tau_e}{m_e}\propto T_e^{5/2}, \end{align*}$$ noting that $\chi_\parallel\propto T_e^{5/2}$, but again independent of $n$.

Ion temperature equation

The derivation follows in the same spirit as the electron temperature equation. Again, the useful identities responsible for the diamagnetic cancellation are $$\begin{align} &p_i\nabla\cdot\left(\mathbf{v}_E+\mathbf{v}_*^i+\mathbf{u}_{pol}+ u_\parallel\mathbf{b}\right)=-T_i\frac{d_i}{dt}n+n\mathbf{v}_*^i\cdot\nabla T_i, \\ &\nabla\cdot\mathbf{q}_{i,\perp} = -\frac{5}{2}n\mathbf{v}_*^i\cdot\nabla T_i -\frac{5}{2}\frac{p_i}{e}\mathcal{C}(T_i), \end{align}$$ yielding $$\begin{align} \frac{3}{2}n\frac{d_i}{dt}T_i=T_i\frac{d_i}{dt}n+\frac{5}{2}\frac{p_i}{e}\mathcal{C}(T_i)+\nabla\cdot\left(\chi_\parallel^i\mathbf{b}\nabla\parallel T_i\right) - P_i:\nabla\mathbf{v}_i + Q_\Delta. \end{align}$$ The gyroviscous term yields $P_i:\nabla\mathbf{v}_i=-\frac{1}{3\eta_i}G^2$ according to Zeiler's habilitation. We use the electron continuity equation to express the ion advective derivative of the density, neglecting advection with the polarisation velocity, i.e. $$\begin{align} \frac{d_i}{dt}n=\frac{\partial}{\partial t}n+\left(\mathbf{v}_E+u_\parallel\mathbf{b}\right)\cdot\nabla n = -\nabla\cdot\left(n\mathbf{v}_*^e\right)-n\nabla\cdot\mathbf{v}_E-n\nabla\cdot\left(v_\parallel\mathbf{b}\right)+\frac{j_\parallel}{n}\nabla_\parallel n. \end{align}$$ Putting everything together we obtain $$\begin{align} \frac{d_i}{dt}T_i=&-\frac{2}{3}\left[\frac{T_i}{e}\mathcal{C}(T_e)+\frac{T_e T_i}{en}\mathcal{C}(n)-T_i\mathcal{C}(\phi)-\frac{5}{2}\frac{T_i}{e}\mathcal{C}(T_i)\right]-\frac{2}{3}T_i\nabla\cdot \left(v_\parallel\mathbf{b}\right)+\frac{2}{3}T_i\frac{j_{\parallel}}{en^2}\nabla_\parallel n \notag\\ &+\frac{2}{3}\frac{1}{n}\nabla\cdot\left(\chi_\parallel^i\nabla_\parallel T_i\right)+\frac{2}{9n\eta_i}G^2 + \frac{2m_e}{M_i \tau_e}(T_e-T_i), \label{drift_ion_temperature} \end{align}$$ where the parallel ion heat conductivity is defined as: $$\begin{align*} \chi_\parallel^i=3.9\frac{nT_i\tau_i}{m_i}\propto T_i^{5/2}. \end{align*}$$