Higher-order nonequilibrium term: Effective power density quantifying evolution towards or away from local thermodynamic equilibrium

A common approach to assess the nature of energy conversion in a classical fluid or plasma is to compare power densities of the various possible energy conversion mechanisms. A leading research area is quantifying energy conversion for systems that are not in local thermodynamic equilibrium (LTE), as is common in a number of fluid and plasma systems. Here we introduce the “higher-order nonequilibrium term” (HORNET) effective power density, which quantifies the rate of change of departure of a phase space density from LTE. It has dimensions of power density, which allows for quantitative comparisons with standard power densities. We employ particle-in-cell simulations to calculate HORNET during two processes, magnetic reconnection and decaying kinetic turbulence in collisionless magnetized plasmas, that inherently produce non-LTE effects. We investigate the spatial variation of HORNET and the time evolution of its spatial average. By comparing HORNET with power densities describing changes to the internal energy (pressure dilatation, $\mathrm{Pi}-\mathrm{D}$, and divergence of the vector heat flux density), we find that HORNET can be a significant fraction of these other measures (8% and 35% for electrons and ions, respectively, for reconnection; up to 67% for both electrons and ions for turbulence), meaning evolution of the system towards or away from LTE can be dynamically important. Applications to numerous plasma phenomena are discussed.

Figure 1

Particle-in-cell simulation results of power densities for electrons and ions during the nonlinear growth phase of reconnection ($t=13$). (a, e) Higher-order nonequilibrium terms effective power density $P_{e,\mathrm{HORNET}}$ and $P_{i,\mathrm{HORNET}}$, (b, f) pressure dilatation $-{\mathcal{P}}_e(\mathbf{\nabla }·{\mathbf{u}}_e)$ and $-{\mathcal{P}}_i(\mathbf{\nabla }·{\mathbf{u}}_i)$, (c, g) ${\mathrm{Pi}-\mathrm{D}}_e$ and ${\mathrm{Pi}-\mathrm{D}}_i$, and (d, h) vector heat flux density divergence $-\mathbf{\nabla }·{\mathbf{q}}_e$ and $-\mathbf{\nabla }·{\mathbf{q}}_i$.

Figure 2

Instantaneous spatially averaged power densities in the reconnection electron diffusion region as a function of time for (a) electrons and (b) ions. $\langle P_{\sigma ,\mathrm{HORNET}}\rangle$ are the solid green lines, $\langle -{\mathcal{P}}_{\sigma }(\mathbf{\nabla }·{\mathbf{u}}_{\sigma })\rangle$ are the dotted orange lines, $\langle {\mathrm{Pi}-\mathrm{D}}_{\sigma }\rangle$ are the dashed purple lines, and $\langle -\mathbf{\nabla }·{\mathbf{q}}_{\sigma }\rangle$ are dash-dotted magenta lines.

Figure 3

Particle-in-cell simulation results comparing power densities for electrons and ions during decaying turbulence at $t=30$. (a, e) Higher-order nonequilibrium terms effective power density $P_{e,\mathrm{HORNET}}$ and $P_{i,\mathrm{HORNET}}$, (b, f) pressure dilatation $-{\mathcal{P}}_e(\mathbf{\nabla }·{\mathbf{u}}_e)$ and $-{\mathcal{P}}_i(\mathbf{\nabla }·{\mathbf{u}}_i)$, (c, g) ${\mathrm{Pi}-\mathrm{D}}_e$ and ${\mathrm{Pi}-\mathrm{D}}_i$, and (d, h) vector heat flux density divergence $-\mathbf{\nabla }·{\mathbf{q}}_e$ and $-\mathbf{\nabla }·{\mathbf{q}}_i$.

Figure 4

Instantaneous spatially averaged power densities in the turbulence simulation domain as a function of time for (a) electrons and (b) ions. The inset in (b) is a plot from time $t=10-50$. The same color scheme as Fig. 2 is used. In this simulation, ${\tau }_{nl}\simeq 17$.

Figure 5

Reduced electron phase space densities $f_e(v_x,v_y)$ from a particle-in-cell magnetic reconnection simulation (using the simulation with PPG = 25 600 from Ref. [14]). (a) Inside the EDR, close to its upstream edge, (b) farther inside the EDR, and (c) near the X-line.

Figure 6

Particle-in-cell magnetic reconnection simulation results (using the simulation with PPG = 25 600 from Ref. [14]) to explain the vector heat flux density profiles observed. Reduced electron phase space densities $f_e(v_x,v_z)$ at the (a) green, (b) magenta, and (c) cyan “x”'s overlayed on the $x$ component of electron heat flux density ${\mathbf{q}}_e$ shown in panel (d). Panels (e) and (f) show the $y$ and $z$ components of ${\mathbf{q}}_e$. The magenta dashed lines in (a)–(c) denote the bulk flow velocity components $u_{e,x}$ and $u_{e,z}$.

Figure 7

Plot of $P_{e,\mathrm{HORNET}}$ identical to Fig. 3, except with colorbar saturated at values identical to that of $P_{i,\mathrm{HORNET}}$ in Fig. 3.