A magnetic analog of pressure–strain interaction

Many modern studies of fundamental processes in magnetized plasmas rely on understanding the evolution of energy and its conversion between different forms.^1,2 While this is relatively well-understood for strongly collisional plasmas, it is challenging for collisionless or weakly collisional plasmas, such as many found in space, astrophysical, and fusion plasmas. In this brief communication, we present new insights about the time evolution of the local magnetic energy density in a magnetized plasma, specifically its evolution in the Lagrangian, i.e., comoving, reference frame that locally moves with the bulk velocity of the electrons.

Equations (1)–(3) are in the form of conservation laws, where the local energy density of each kind changes in time due to a locally diverging/converging energy flux (the divergence terms on the left-hand side) or through sources/sinks (the terms on the right-hand side). If the equations are integrated over a volume of interest to describe the change in energy over a whole domain, Gauss' divergence theorem implies the flux terms simply become the flux through the outer surface of the domain. If the domain being considered is closed or isolated, infinite in extent, or periodic (such as in a simulation), then the volume-integrated flux term vanishes and the total energy in each form only changes via the source/sink terms. However, we emphasize that this is only for “global” energy evolution; local energy density evolution is impacted by the divergence of the flux terms.

Equations (1)–(3) are written in the (Eulerian) stationary “laboratory” reference frame, which is at rest and the plasma and electromagnetic field energy move through it. However, at a point of interest in the Eulerian reference frame, the convection of a plasma or field from another point due to the flow of the plasma can change the energy density. Convection does not represent a genuine change in the properties of species $σ$⁠, so the study of energy evolution is preferably carried out in the reference frame comoving with species $σ$⁠, i.e., the Lagrangian reference frame. (See, e.g., Refs. 40 and 41.)

It is not uncommon to write Eqs. (1) and (2) in the Lagrangian frame of species $σ$⁠. It is less common to see Poynting's theorem [Eq. (3)] written in a Lagrangian frame (see Ref. 39 for one example). Notably, Zenitani et al.⁴² discussed the rate of energy density conversion due to the acceleration of charged particles in the reference frame comoving with the bulk velocity of the electrons. The so-called “Zenitani parameter” was introduced as a means to identify where non-ideal electron-scale physics takes place, such as the electron diffusion region of collisionless magnetic reconnection. It was originally thought to describe dissipation,⁴² but later it was recognized that it may include both reversible and irreversible contributions, particularly in weakly collisional plasmas.⁴³ 

In this study, we analyze the local magnetic energy density evolution in the Lagrangian reference frame comoving with the bulk velocity of electrons. By breaking the electric field into ideal and non-ideal contributions, we show the contribution due to the ideal electric field is described by terms analogous to those in ideal-magnetohydrodynamics (ideal-MHD) describing perpendicular compression and magnetic field line bending, but due to bulk electron motion. We recast these two terms to reveal a quantity analogous to the pressure–strain interaction term described in Eq. (4), but with the magnetic stress tensor $TM$ replacing the pressure tensor of the species, where the jkth element is given by $TM,jk=(1/4π)[BjBk−(1/2)δjkB2]$ and $δjk$ is the Kronecker delta. We call the magnetic analog to the pressure–strain interaction the “magnetic stress–strain interaction.” This term does not represent new physics concerning the evolution of magnetic energy density; rather it identifies a symmetry in the energy density evolution and conversion between the different forms of energy densities, which, given the broad interest in the pressure-strain interaction, may facilitate insight and understanding of the physical processes during energy conversion.

The form of Eq. (6) facilitates the identification of the physical processes at play. The first term on the left-hand side, $∂EM/∂t$⁠, quantifies the local change in magnetic energy density in the stationary (Eulerian) reference frame. The second term on the left-hand side, $(ue,⊥·∇)EM$⁠, describes the rate of change of magnetic energy density due to convection of magnetic energy density by electrons perpendicular to the magnetic field. These two terms together describe the rate of magnetic energy density conversion in the Lagrangian reference frame comoving with the perpendicular part of the bulk electron velocity $ue,⊥$⁠.

Consequently, the four terms on the right-hand side describe mechanisms by which the local magnetic energy density changes in the reference frame comoving with the perpendicular part of the bulk electron velocity. The first term on the right-hand side, $−(B2/4π)(∇·ue,⊥)$⁠, is the rate of change of magnetic energy density due to compression of electron bulk flow perpendicular to the magnetic field. The second term on the right-hand side, $(1/4π)B·[(B·∇)ue,⊥]$⁠, is the rate of change of magnetic energy density due to magnetic field bending by the electrons. Physically, the bending term is non-zero if the electron bulk flow perpendicular to the magnetic field changes along the direction of the magnetic field. Both of these terms are due to the ideal electric field and are analogous to the terms that arise in ideal-MHD, except the perpendicular electron bulk velocity $ue,⊥$ plays the role of the single-fluid bulk velocity in ideal-MHD. We note this analogy persists here despite the fact that we are not making any assumptions about the validity of ideal-MHD or Hall-MHD in our analysis.

The third term on the right-hand side, $−∇·S′$⁠, is the rate of change of magnetic energy density due to the divergence of the non-ideal Poynting flux.³⁹ This term describes the transport of magnetic energy density rather than a pure source/sink of it, as integrating it over the entire volume of a system in question results solely in the Poynting flux across the boundary of the system.

Finally, the fourth term on the right-hand side, $−J·E′$⁠, is the local rate of change of magnetic energy density due to the acceleration of charged particles by the non-ideal electric field.³⁹ The Zenitani parameter⁴² is defined as $De=γe[J·(E+ue×B/c)−ρc(ue·E)],$ where $γe=(1−ue2/c2)−1/2$ is the relativistic gamma factor for electrons and $ρc=∑σqσnσ$ is the net charge density. In the non-relativistic quasineutral limit, $γe≃1$ and $ρc≃0$⁠, so the Zenitani parameter is approximately given by $J·E′$⁠. The appearance of the Zenitani parameter as a source term in Eq. (6) is consistent with the expectations in Ref. 42.

We now present the terms in the magnetic energy density evolution equation [Eq. (6)] using particle-in-cell simulations of magnetic reconnection. The simulations are identical to those in Ref. 27 and are thoroughly described therein. The simulation data are presented only for the lower current sheet at time $t=13 Ωci0−1$ with black solid lines denoting in-plane projections of magnetic field lines, where $Ωci0$ is the ion cyclotron frequency based on the initial asymptotic magnetic field $B0$⁠. This time is when the reconnection rate is rising most rapidly during the nonlinear growth phase, so the system is not in steady-state. The plots are centered at the X-line $(x0,y0)$⁠, and all distances are given in units of the ion inertial scale $di0$ based on the difference between the maximum initial number density and the asymptotic value. To minimize particle noise in the simulation, we smooth the raw simulation data over a width of five cells, followed by calculating the desired spatial or temporal derivatives, and finally, smooth the results over five cells. This smoothing approach is chosen after testing various options for the number of cells to smooth over and the number of recursions, ensuring that the signal structure remains largely unaffected by the smoothing. All power densities are given in units of $(B02/4π)Ωci0$⁠.

Panel (a) of Fig. 1 displays the left-hand side of Eq. (6) that denotes the time rate of change of $EM$ in the Lagrangian frame comoving with $ue,⊥$⁠. We see that the magnetic energy density decreases (purple) with time in this Lagrangian frame upstream of the X-line and there is a weak increase (green) along the horizontal neutral line (the dashed line) downstream of the X-line. Since the reconnection process converts magnetic energy to energy in the plasma, a decrease in magnetic energy in this Lagrangian frame is expected. The positive signal along the outflow region of the neutral line is because the reconnected magnetic field $By$ increases with distance in the outflow direction from the X-line while the reconnecting magnetic field $Bx=0$⁠, so $EM$ increases in the reference frame moving with the electrons.

We now examine the terms on the right-hand side of Eq. (6). Figures 1(b)–1(e) contain the perpendicular compression term $−EM(∇·ue,⊥)$⁠, the magnetic stress–strain interaction $(TM·∇)·ue,⊥$⁠, the negative divergence of the non-ideal Poynting flux $−∇·S′$⁠, and the power density by the non-ideal electric field $−J·E′$⁠, respectively. We consider the action of these terms in a few key regions for the reconnection process.

Near the X-line, within $|x−x0|<1$ and $|y−y0|<0.3$⁠, perpendicular compression [in (b)] and magnetic stress–strain interaction [in (c)] are small because they are proportional to the magnetic energy and magnetic stress, respectively, which are small because the magnetic field is weak near the X-line. As known from previous work,⁴² there is a strong $−J·E′$ [in (e)] in the inner electron diffusion region where the (non-ideal) reconnection electric field $E′$ in the $−z$ direction accelerates electrons in the z direction. Since the magnetic energy density near the X-line does not change much in time, the non-ideal Poynting flux [in (d)] essentially balances $−J·E′$⁠. Thus, $S′$ converges near the X-line to provide the energy necessary for the reconnection electric field to accelerate the electrons. Physically, $E′$ is in the $−z$ direction while $Bx>0$ for $y−y0>0$ and $Bx<0$ for $y−y0<0$⁠, so $E′×B$ is toward the neutral line both above and below the X-line.

In the outflow region around $1<|x−x0|<2$⁠, there is a small region of magnetic energy density increase near the neutral line, and otherwise the magnetic energy density is mostly decreasing [panel (a)]. The increase in magnetic energy density is caused by perpendicular compression [panel (b), and it also contributes to the magnetic stress–strain interaction in panel (c) from Eq. (8)]. Near the neutral line, magnetic field lines are strongly kinked, and the y component of $ue,⊥$ points toward the neutral line, thus giving strong compression $−EM(∇·ue,⊥)$⁠. Away from the neutral line, the compression and magnetic stress–strain interaction are negative and significant, consistent with the strongly kinked magnetic field lines losing energy [see Fig. 1(a)] as they straighten; we return to this shortly to identify the physical cause. We note there is also a positive $−J·E′$ in the outflow region because of a non-zero divergence of the electron pressure tensor.^46,47 Since the sign of $Ez′$ flips to positive in this region, $∇·S′$ is positive which nearly balances $−J·E′$⁠, so the non-ideal electric field is not the cause of the change in $EM$ at this region.

We next treat the region around the upstream edge of the electron diffusion region, $|x−x0|<1$ and $0.3<|y−y0|<0.5$⁠. The dominant term is $∇·S′$⁠, which arises because the non-ideal reconnection electric field $Ez′$ is reasonably uniform, but the strength of the reconnecting magnetic field $Bx$ decreases approaching the X-line. This leads to a converging $S′$ associated with the decrease in magnetic energy density in the Lagrangian frame of the electrons. Further upstream, outside the electron diffusion region, the magnetic energy density decreases in the Lagrangian frame of the electrons [panel (a)]; this is dominated by a decrease in $EM$ in the Eulerian frame, i.e., $∂EM/∂t<0$ (not shown). At the time slice plotted here, the system is not in the steady-state, so the magnetic field energy density decreases locally due to the magnetic field getting pulled in toward the X-line as the reconnection process speeds up.

To assess the physical cause of the magnetic stress–strain interaction in Fig. 1(c), we show in Fig. 2 its decomposition, with the compressible $TM(∇·ue,⊥)$⁠, incompressible normal $TDnormal$⁠, and incompressible shear $TDnormal$ contributions in panels (a)–(c), respectively. The compressible part in Fig. 2(a) is simply $1/3$ of the compression term $−EM(∇·ue,⊥)$ in Fig. 1(b). To see how these plots can help elucidate the physics taking place, consider the outflow region near the neutral line. The increase in $EM$ near the neutral line is caused by the positive compression terms as stated previously, but we also see in Fig. 2(b) that the increase in magnetic energy density is reduced because there is incompressible normal deformation also occurring. Figures 2(a) and 2(c) show that the decrease in $EM$ away from the neutral line in the exhaust is due to negative compression (expansion) and incompressible shear as a result of magnetic field line bending as the strongly curved magnetic field line straightens. Importantly, both Figs. 2(b) and 2(c) reveal changes to the magnetic energy via incompressible effects. This is a reasonable result because the Alfvénic and whistler wave-like effect driving the outflow^48–50 are incompressible.

Finally, we make two observations. First, the non-ideal terms [in (d) and (e)] are predominantly localized to the electron diffusion region and within an electron inertial length size surrounding the separatrix. This is consistent with the electron frozen-in condition breaking down at sub-electron inertial scales in anti-parallel magnetic reconnection so that $E′≠0$⁠. Second, although the data presented here are from a time slice when the time evolution is the most rapid, we do not expect significant differences in the structure of these plots near the electron diffusion region in the steady-state because these plots are in the Lagrangian reference frame of the electrons. In the steady-state, $∂/∂t=0$⁠, but the convection term $ue,⊥·∇$ remains significant.

The reconnection process is used as an example, but we expect the analysis performed here and its interpretation could be useful in other fundamental processes in magnetized plasmas, including turbulence, collisionless shocks, and wave–particle interactions.

This work was partially motivated by P.A.C.'s participation in the Magnetospheric Multiscale (MMS) 10th Community Meeting. We gratefully acknowledge support from NASA Grant Nos. 80NSSC24K0172 and 80NSSC23K0409, NSF Grant No. PHY-2308669, and DOE grant DE-SC0020294. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No. DE-AC02-05CH11231 using NERSC award FES-ERCAP0027083.