Energy theorem

Energy theorem

The perpendicular, parallel, thermal and electromagnetic energies are defined respectively as $$\begin{align} E_\perp:=&\frac{nM_i(\mathbf{v}_E+\mathbf{v}_*^i)^2}{2}=\frac{nM_ic^2}{2B^2}\left|\nabla_\perp\phi+\frac{1}{en}\nabla_\perp\ p_i\right|^2, \\ E_\parallel:=&\frac{nM_iu_\parallel^2}{2}, \\ E_{\parallel e}:= & \frac{nm_e}{2}\left(\frac{j_\parallel}{en}\right)^2, \\ E_{the}:=&\frac{3}{2}nT_e, \\ E_{thi}:=&\frac{3}{2}nT_i, \\ E_{em}:=&\frac{\mathbf{B}_\perp^2}{8\pi}=\frac{1}{8\pi}\left|\nabla_\perp A_\parallel\right|^2. \end{align}$$ Via the electron/ion continuity equation, the following trick is obtained: $$\begin{align*} \frac{\partial}{\partial t}(nf)=n\frac{d_{i,e}}{dt}f-\nabla\cdot\left(nf\mathbf{v}_{i,e}\right), \end{align*}$$ which is important to derive the temporal changes of the energies.

The temporal evolution of the perpendicular kinetic energy is obtained by multiplying the vorticity equation by $\phi$, and noting that $\phi\nabla\cdot\mathbf{J}=\nabla\cdot\left(\phi\mathbf{J}\right)- \nabla\phi\cdot\mathbf{J}$, $$\begin{align} \frac{\partial}{\partial t}E_\perp+\nabla\cdot\left[E_\perp\mathbf{v}_i+\phi\mathbf{J}+\left(p_e+p_i\right)\mathbf{v}_E -\frac{G}{3}\left(\mathbf{v}_E+\mathbf{v}_*^i\right)\right] =J_\parallel\nabla_\parallel\phi+(p_e+p_i)\nabla\cdot\mathbf{v}_E-\mathbf{u}_{pol}\cdot \nabla p_i -\frac{G^2}{3\eta_i} - \frac{2}{3}\frac{G}{B^{3/2}}\nabla\cdot(\mathbf{b}u_\parallel B^{3/2}) + \Lambda_\perp. \end{align}$$

The temporal evolution of the parallel kinetic energy is obtained by multiplying the parallel momentum equation with $u_\parallel$, $$\begin{align} \frac{\partial}{\partial t}E_\parallel+\nabla\cdot\left[E_\parallel\left(\mathbf{v}_E+u_\parallel\mathbf{b}+\mathbf{u}_{pol}\right) + \frac{c}{eB^2}\mathbf{B}\times\nabla\left(E_\parallel T_i\right) +\frac{2}{3}Gu_\parallel\mathbf{b}\right]=-u_\parallel\nabla_\parallel (p_e + p_i) + \frac{2}{3}\frac{G}{B^{3/2}}\nabla\cdot(\mathbf{b}u_\parallel B^{3/2}) + \Lambda_{\parallel}. \end{align}$$

The temporal evolution of the electron thermal energy is obtained by multiplying the electron temperature equation by $3/2 n$, from which we obtain $$\begin{align} \frac{\partial}{\partial t}E_{the}+\nabla\cdot\left[\frac{3}{2}p_e\mathbf{v}_E+\frac{5}{2}p_ev_\parallel\mathbf{b}+\frac{5}{2}p_e\mathbf{v}_*^e+\mathbf{q}_e\right]=-p_e\nabla\cdot\mathbf{v}_E+v_\parallel\nabla_\parallel p_e+\eta_\parallel j_\parallel^2-0.71\frac{j_\parallel}{e}\nabla_\parallel T_e - Q_\Delta + \Lambda_{the}, \end{align}$$ where we note that $$\begin{align*} \frac{5}{2}p_e\mathbf{v}_*^e+\mathbf{q}_{\perp,e}=-\frac{5}{2}\frac{c}{eB^2}\mathbf{B}\times\nabla\left(p_eT_e\right). \end{align*}$$

The temporal evolution of the ion thermal energy is obtained by multiplying the ion temperature equation by $3/2 n$, from which we obtain, by adding the ion continiuty equation, $$\begin{align} \frac{\partial}{\partial t}E_{thi} + &\nabla\cdot\left[\frac{3}{2}p_i\mathbf{v}_E+\frac{5}{2}p_iu_\parallel\mathbf{b}+\frac{5}{2}p_i\mathbf{u}_{pol}+q_{i,\parallel}\mathbf{b}+\frac{5}{2}\frac{c}{eB^2}\mathbf{B}\times\nabla(p_iT_i)\right] \\ &= -p_i\nabla\cdot\mathbf{v}_E+u_\parallel\nabla_\parallel p_i \notag + \mathbf{u}_{pol}\cdot\nabla p_i + Q_\Delta + \frac{G^2}{3\eta_i} + \Lambda_{thi}. \end{align}$$

The temporal evolution of the electromagnetic kinetic energy is obtained by multiplying Ohm's law by $-j_\parallel$. Here, we neglect electron inertia $m_e \rightarrow 0$. $$\begin{align} \frac{\partial}{\partial t}E_{em}+\nabla\cdot\left[-\frac{\nabla_\perp A_\parallel}{4\pi}\frac{\partial}{\partial t}A_\parallel\right]=-\eta_\parallel j_\parallel^2-j_\parallel\nabla_\parallel\phi+u_\parallel\nabla_\parallel p_e-v_\parallel\nabla_\parallel p_e+0.71\frac{j_\parallel}{e}\nabla_\parallel T_e + \Lambda_{em} . \end{align}$$

Each term on the right hand side has a counter term in another equation, so summing up the contributions yields as time derivative for the total energy $$\begin{align} \frac{\partial}{\partial t}\left(E_\perp+E_\parallel+E_{the} +E_{thi}+E_{em}+E_{\parallel e}\right)+\nabla\cdot\left[\dots\right]=\Lambda_\perp + \Lambda_\parallel + \Lambda_{the} + \Lambda_{thi} + \Lambda_{em}, \end{align}$$ i.e. the total energy of the system is conserved up to sources/sinks and dissipation.

Sources and Sinks

The $\Lambda$ terms in the balance equations arise due to extra terms added to the dynamical equations of the pure Braginskii system, i.e. source and dissipation terms (see Implemented equations).

The source/sink terms for perpendicular kinetic energy are: $$\begin{align} \Lambda_\perp = S_\perp + D_\perp, \quad & \text{where} \\ S_\perp = & \frac{E_\perp}{n}S_n + \phi S_{\Omega}, \\ D_\perp = & \frac{E_\perp}{n}\mathcal{D}_n - \phi \mathcal{D}_{\Omega}. \end{align}$$

For parallel kinetic energy: $$\begin{align} \Lambda_\parallel = S_\parallel + D_\parallel, \quad & \text{where} \\ S_\parallel = & \frac{Mu_\parallel^2}{2}S_n + nu_\parallel S_{u_\parallel}, \\ D_\parallel = & \frac{Mu_\parallel^2}{2}\mathcal{D}_n + nu_\parallel\mathcal{D}_{u_\parallel}. \end{align}$$

The electron thermal energy sources/sinks are: $$\begin{align} \Lambda_{the} = S_{the} + D_{the}, \quad & \text{where} \\ S_{the} = & \frac{3}{2}T_eS_n + \frac{3}{2}nS_{T_e}, \\ D_{the} = & \frac{3}{2}T_e\mathcal{D}_n + \frac{3}{2}n\mathcal{D}_{T_e}, \end{align}$$ and the ion thermal energy sources/sinks: $$\begin{align} \Lambda_{thi} = S_{thi} + D_{thi}, \quad & \text{where} \\ S_{thi} = & \frac{3}{2}T_iS_n + \frac{3}{2}nS_{T_i}, \\ D_{thi} = & \frac{3}{2}T_i\mathcal{D}_n + \frac{3}{2}n\mathcal{D}_{T_i}. \end{align}$$ We note that if dissipation is applied on pressure instead of temperature (see Implemented equations), $D_{thi}$ is a total divergence of a dissipative flux. So the dissipation is not a sink/source then but absorbed into the background.

There are no sources/sinks for electromagnetic energy, but only dissipation: $$\begin{align} \Lambda_{em} = D_{em}, \quad & \text{where} \\ D_{em} = & j_\parallel\mathcal{D}_{A_\parallel}. \end{align}$$