# Kinetic Dissipation Around a Dipolarization Front

## Abstract

Kinetic aspects of energy conversion and dissipation near a dipolarization front (DF) in the magnetotail are considered using fully kinetic 3-D particle-in-cell simulations. The energy conversion is described in terms of the pressure dilatation, as well as the double contraction of deviatoric pressure tensor and traceless strain rate tensor, also known as the *P**i*-*D* parameter in turbulence studies. It is shown that in contrast to the fluid dissipation measure, the Joule heating rate, which cannot distinguish between ion and electron dissipation and reveals deep negative dips at the DF, the *P**i*-*D* parameters, as kinetic analogs of the Joule heating rate, are largely positive and drastically different for ions and electrons. Further analysis of these parameters suggests that ions are heated at and ahead of the DF due to their reflection from the front, while electrons are heated at and behind the DF due to the long-wavelength lower-hybrid drift instability.

## Key Points

- Newly derived kinetic dissipation parameters are largely positive and different for ions and electrons
- Ion dissipation is dominated by ion reflection from fronts
- Electron dissipation is dominated by the lower-hybrid drift instability

## Plain Language Summary

We explore new measures of plasma dissipation in rapidly contracting tubes of magnetic flux and plasma on the nightside of the terrestrial magnetosphere. These contracting tubes make the stretched tail-like magnetic field more dipolar and have sharp profiles of plasma density and magnetic field at the leading edge. Relaxation of the stretched magnetic field releases the energy, which is spent for plasma acceleration and heating. Since collisions are extremely rare, the energy dissipation processes are different for electrons and ions and hence require special quantitative measures. Here we derive such measures from massively parallel three-dimensional particle-in-cell simulations of the tail plasmas and demonstrate that as expected for measures of dissipation, they are positive on average and different for ions and electrons. The new quantitative measures allow us to reveal specific physical processes responsible for energy dissipation.

## 1 Introduction

The Earth's magnetosphere is a unique natural laboratory of fully collisionless plasmas. Its magnetotail is particularly significant because it accumulates the energy of the solar wind/magnetosphere interaction and then releases it in the form of substorms, pseudobreakups, bursty bulk flows, and dipolarization fronts (DFs) (Angelopoulos et al., 2013). The recently launched Magnetospheric MultiScale (MMS) mission (Burch, Moore et al., 2016) provides for the first time a critical opportunity to investigate these dipolarization processes in collisionless plasmas. At the same time, understanding of magnetotail dipolarizations in observations, theory, and simulations is still strongly hindered by the fundamental problem of the missing kinetic description of dissipation processes in collisionless plasmas. There is a general consensus that dipolarization processes are caused by some kinetic effects, for example, magnetic reconnection (Baker et al., 1996) or current disruption (Lui, 1996). Independent of the specific scenario, they involve dissipation processes that make dipolarizations irreversible. Near Earth this may be provided by collisional dissipation in the ionospheric plasma (Panov et al., 2016). Farther in the tail, because of long Alfvén travel time to the ionosphere, the dipolarization instabilities require either electron (Coppi et al., 1966) or ion (Schindler, 1974) Landau dissipation (Landau, 1946), which provide entirely different reconnection regimes. In the first regime of the Electron Demagnetization-Mediated Reconnection the tail current sheet (CS) is thinned down to electron gyroradius scales, and one can expect electron dissipation in the corresponding diffusion region (Hesse & Schindler, 2001; Liu et al., 2014). In another regime of the Ion Demagnetization-Mediated Reconnection or IDMR the dipolarization starts from spontaneous generation of fast earthward flows followed by the formation of a new X line because of the flux starvation effect behind the DF (Bessho & Bhattacharjee, 2014; Pritchett, 2015; Sitnov et al., 2013). Dissipation in the IDMR regime is likely provided by ions (Schindler, 1974).

The description of dissipation in collisionless plasmas is very challenging. In particular, it cannot be limited by the standard resistive magnetohydrodynamic (MHD) parameters (Birn & Hesse, 2005; Zenitani et al., 2011), i.e., the energy conversion rates in the frame moving with ions or electrons
, where **j** = **j**_{i}+**j**_{e},
, **j**_{e,i} are the electron/ion currents in the laboratory frame of reference and **v**_{e} and **v**_{i} are the electron and ion bulk flow velocities. The problem is that assuming quasi-neutrality (*n*_{e}≈*n*_{i}), such parameters for ion and electron reference frames are the same
. Thus, they are reduced to a single MHD parameter
, also known as the Joule heating rate, which cannot be used to distinguish between electron and ion dissipation processes (see, e.g., similar profiles of
and
in Figure 6 in Yao et al., 2017). In addition, recent MMS observations and analysis (Cassak et al., 2017; Genestreti et al., 2017; Yao et al., 2017) revealed regions of large negative values of the
parameter, comparable in magnitude to its positive peaks in the electron diffusion region (EDR). They are hard to interpret since in MHD theory
determines the increase of the entropy (e.g., Birn & Hesse, 2005) and hence is expected to be positive.

In this paper we consider new kinetic measures of dissipation, which have recently been proposed in studies of collisionless plasma turbulence (Yang, Matthaeus, Parashar, Haggerty, et al., 2017; Yang, Matthaeus, Parashar, Wu, et al., 2017). The new measures are applied for the analysis of magnetotail dipolarization regimes associated with the IDMR, which have recently been reproduced in 3-D particle-in-cell (PIC) simulations (Sitnov et al., 2017). It is shown that in contrast to the fluid parameter , the new dissipation measures do not reveal any strong negative dips and they are drastically different for ions and electrons. Moreover, the analysis of their components provides interesting implications for the underlying instabilities and particle acceleration mechanisms. New kinetic dissipation parameters also suggest that substantial plasma dissipation in magnetotail reconnection may occur at the DF, away from the X line and the EDR, and that it may be dominated by ions.

## 2 Kinetic Dissipation Parameters

*u*in the form

*p*= 2

*u*/3 is the scalar pressure,

**v**is the plasma bulk flow velocity vector, and

*∂*

_{t}denotes the time derivative. Using the relation between

*p*and

*u*as well as the continuity equation, the left-hand side of 1 can be reduced to the expression , where

*ρ*is the plasma number density. Thus, in 1 determines the evolution of the entropy , which may be directly related to the statistical measure of disorder for an ideal gas (e.g., Wolf et al., 2009).

*u*

^{(α)}for each species

*α*=

*e*,

*i*, derived directly from the Vlasov equation, which have the form

**v**

^{(α)},

**h**

^{(α)}, and are the bulk flow velocity and heat flux vectors as well as the pressure tensor for the species

*α*=

*e*,

*i*; and

*∂*

_{i}denotes the derivative over the coordinate

*x*

_{i}. To make the left-hand side similar to its MHD analog 1, they extracted in its last term the “isotropic part”

*ρ*

^{(α)}are the isotropic component of the plasma pressure and the density of the species

*α*. Since the parameter resembles the entropy of the species

*α*, it is reasonable to suppose that the difference between the parameters

*p*

^{(α)}

*θ*

^{(α)}and is a kinetic analog of the MHD dissipation rate

*p*

^{(α)}

*θ*

^{(α)}defined in 3 is usually termed in the turbulence theory as the pressure dilatation (Yang, Matthaeus, Parashar, Haggerty, et al., 2017; Yang, Matthaeus, Parashar, Wu, et al., 2017, and references therein).

The fundamental role of tensors
and
had been recognized already in early works on plasma transport processes (Braginskii, 1965). In particular, in weakly collisional and weakly magnetized plasmas
, where
is the viscosity coefficient, and hence
. In their recent works Yang, Matthaeus, Parashar, Wu, et al. (2017) and Yang, Matthaeus, Parashar, Haggerty, et al. (2017) applied the pressure dilatation and *P**i*-*D*^{(α)} parameters to the analysis of collisionless plasma turbulence in 2-D PIC simulations. They have found that these parameters correlate with velocity gradients and are highly localized in similar spatial regions, with their quantities integrated over the simulation box being persistently positive. Here we investigate these parameters in 3-D PIC simulations of magnetotail dipolarizations.

## 3 Simulation Setup and Results

We use for our analysis the results of 3-D PIC simulations of magnetotail dipolarizations in the IDMR regime (Sitnov et al., 2017). They were performed using an open boundary modification (Divin et al., 2007; Sitnov & Swisdak, 2011) of the explicit massively parallel code P3D (Zeiler et al., 2002) in a 3-D box with dimensions *L*_{x}×*L*_{y}×*L*_{z}=60*d*_{i}×10*d*_{i}×20*d*_{i}, where *d*_{i}=*c*/*ω*_{pi} is the ion inertial scale and *ω*_{pi}=(4*π**e*^{2}*ρ*_{0}/*m*_{i})^{1/2} is the plasma frequency; *ρ*_{0} is the plasma number density at the earthward side of the simulation box near the neutral plane (*z* = 0) of the tail CS given by the vector potential **A**^{(0)}=(0,−*ψ*(*x*,*z*),0), where
, *L* is the characteristic CS thickness parameter, and the *x* axis points from Earth to Sun. Its global shape is determined by the function
, with *ξ* = *x*/*L*, *ϵ*_{1}≪1 and
, which provides a region of accumulated magnetic flux near *ξ* = *ξ*_{0}. This is seen from the magnetic field profile
having a characteristic hump.

An interval of the tailward gradient earthward of the hump peak is critically important for spontaneous dipolarization because it relaxes the sufficient condition for the tearing stability (Sitnov & Schindler, 2010), which otherwise coincides with the WKB (Wentzel-Kramers-Brillouin) approximation (*k**L**β* > *π**B*_{z}/*B*_{0}, where *k* is the wave number) making the tearing mode universally stable (Lembege & Pellat, 1982). Simulations with the mass ratio *m*_{i}/*m*_{e}=128, ion-to-electron temperature ratio *T*_{i}/*T*_{e}=3, CS thickness parameter *L* = 0.75*d*_{i}, and the equilibrium magnetic field parameters *ϵ*_{1}=0.03, *ϵ*_{2}=0.2, *α* = 3, and *ξ*_{0}=−24 (further details are provided in the supporting information file) showed that the “humped” CS is indeed unstable. Its evolution starts from the earthward acceleration of the hump and its transformation into a DF. Tailward of the DF a new X line forms because of the flux starvation effect (e.g., Bessho & Bhattacharjee, 2014; Pritchett, 2015; Sitnov et al., 2013 as is seen from Figure 1a showing the distribution of the equatorial magnetic field *B*_{z}(*x*,*z* = 0) at *ω*_{0i}*t* = 32. Figure 1b shows the corresponding equatorial distribution of the conventional dissipation parameter
. It reveals in particular the formation of the EDR as identified by the enhancement of
near the X line (Birn & Hesse, 2014; Burch, Torbert, et al., 2016). Figure 1b also reveals strong negative dips of the
parameter earthward of the X line in the DF region. They were identified earlier (Sitnov et al., 2014) as signatures of the long-wavelength (
) modification (Daughton, 2003) of the lower-hybrid drift instability (LHDI) of thin CSs with zero *B*_{z} component (Huba et al., 1977), which is also similar in case of nonzero *B*_{z} to the kinetic ballooning/interchange instability (Pritchett & Coroniti, 2010). Similar variations of
near DFs, including its negative dips, were reported in simulations of other reconnection regimes (Khotyaintsev et al., 2017; Lapenta et al., 2014). Negative
dips were also reported in Cluster and MMS observations of DFs (Khotyaintsev et al., 2017; Yao et al., 2017), as well as MMS observations of the EDR vicinity at the magnetopause (Genestreti et al., 2017). Figure S1 shows linear profiles of the energy conversion and dissipation across the DF demonstrating both positive and negative excursions of
with comparable amplitudes. It also shows another problem, a very small difference between the parameters
and
, which challenges the use of those parameters in the quantitative description of electron or ion dissipation.

In contrast, Figures 1c and 1d suggest that distributions of the kinetic parameters *P**i*-*D* are drastically different for ions and electrons. Moreover, they do not reveal any strong negative dips seen in Figure 1b and are largely positive on average (the distributions are averaged over the scale 0.39*d*_{i}; the necessity of averaging in the description of Landau dissipation is elaborated, for example, by Mouhot & Villani, 2011). The latter feature is seen even in comparison with the pressure dilatation distributions shown in Figures 1e and 1f, which are also different for ions and electrons, but do not reveal any characteristic patterns other than stronger variations ahead of the DF compared to its trailing region. It becomes particularly clear from Table 1, which shows means of the kinetic parameters over the region −20 < *x*/*d*_{i}<0 normalized by their standard deviations in the same region. In particular, 〈*P**i*-*D*^{(i)}〉_{σ} is greater than −〈*p*^{(i)}*θ*^{(i)}〉_{σ} by a factor of 2, whereas for electrons the ratio between similar parameters exceeds 7.

ω_{0i}t |
−〈p^{(i)}θ^{(i)}〉_{σ}a |
〈Pi-D^{(i)}〉_{σ} |
−〈p^{(e)}θ^{(e)}〉_{σ} |
〈Pi-D^{(e)}〉_{σ} |
---|---|---|---|---|

26 | 0.0033 | 0.0098 | 0.0054 | 0.091 |

32 | 0.0068 | 0.013 | 0.0079 | 0.06 |

34 | 0.0047 | 0.022 | 0.0035 | 0.052 |

^{a}〈...〉_{σ}denotes the mean over the region −20 <*x*/*d*_{i}<0 normalized by the standard deviation.

Further analysis of the kinetic dissipation parameters *P**i*-*D*^{(α)} is presented in Figure 2. This figure shows contributions of the specific components
to the total sum *P**i*-*D*^{(α)} along the lines *y* = *c**o**n**s**t* chosen to represent the largest values of those parameters for ions and electrons. It shows in particular that for ions the *P**i*-*D*^{(i)} peak is provided by the first two components of the double contraction sum:
and
(they are marked by the yellow frame in Figure 2). This peak is located at the forefront part of the DF, and it is apparently related to the reflection of ions from the DF, which was demonstrated earlier both in test particle (Ukhorskiy et al., 2013; Zhou et al., 2010) and in PIC simulations (Wu & Shay, 2012). This is also consistent with the ion Landau dissipation mechanism for the tearing mode perturbation in the *x* direction (Pritchett et al., 1991; Schindler, 1974), although the particle interaction with soliton-like DF is quite different from their interaction with monochromatic waves that are usually considered in the Landau dissipation theory (e.g., Chust et al., 2009, and references therein).

In contrast, the main contribution to the electron dissipation parameter comes from the off-diagonal components of the pressure tensor and traceless strain rate tensor
(blue frames in Figure 2) suggesting the energy transfer between the perpendicular and parallel to the magnetic field components of the electron velocity distribution. At the same time, the spatial distribution of the electron dissipation *P**i*-*D*^{(e)} shown in Figure 1d suggests that it is provided by the LHDI, which is indeed known to heat electrons (Daughton et al., 2004; Huba et al., 1977; Le et al., 2018). Strong LHDI activity was found in 3-D PIC simulations of reconnection (Roytershteyn et al., 2012) and DFs (Divin et al., 2015; Lapenta et al., 2014; Sitnov et al., 2014) as well as in their space (Khotyaintsev et al., 2011; Zhou et al., 2009) and laboratory (Carter et al., 2002; Fox et al., 2010) observations. Figure 1d also shows that in contrast to the ion dissipation peaking at and ahead of the DF, the electron dissipation occupies the trailing DF region between *B*_{z} ridge and the X line. In addition to the main *P**i*-*D*^{(e)} peak, Figure 2 reveals signatures of somewhat less pronounced tearing-type electron (i. e., similar to the ion mechanism discussed above) dissipation provided by the diagonal components
and
marked by the gray frame and similar to the ion dissipation terms.

It is interesting to compare the new kinetic dissipation picture provided by the parameters *P**i*-*D*^{(i,e)} with distributions of the resulting ion and electron thermal energy density *u*^{(i,e)} shown in Figures 1g and 1h. Apart from a more shallow energy density distribution for electrons caused by the fact that electrons are heated mainly behind the DF, it reveals two sharp peaks of *u*^{(e)} near the corresponding *B*_{z} peaks ((*x*,*y*)≈(−5,6.5) and (−6,4.5)). These peaks form in spite of the corresponding plasma density drops, and they are likely caused by the betatron acceleration of electrons. Since these peaks of *u*^{(e)} do not correlate with peaks of *P**i*-*D*^{(e)}, one can conclude that their mechanism is adiabatic and reversible as is indeed the case for betatron acceleration.

The dissipation picture near the DF before the X line formation (*ω*_{0i}*t* = 26) is presented in Figure 3. It shows that dissipation is weaker compared to *ω*_{0i}*t* = 32, and it is mainly provided by electrons via the LHDI. This picture is consistent with recent 2-D MHD simulations (Birn et al., 2018; Merkin et al., 2015), which suggest that dipolarization and reconnection in the IDMR regime start from the ideal MHD-like instability, which follows (Merkin et al., 2015) from the corresponding MHD energy principle (Schindler, 2006). Interestingly, in spite of the fact that the dissipation at this moment is small, it is still more persistently positive compared to the pressure dilatation with 〈*P**i*-*D*^{(i)}〉_{σ}/〈−*p*^{(i)}*θ*^{(i)}〉_{σ}≈3 and 〈*P**i*-*D*^{(e)}〉_{σ}/〈−*p*^{(e)}*θ*^{(i)}〉_{σ}≈17, according to Table 1. The analysis of electron dissipation presented in Figure S2 suggests that the energy conversion takes place between the perpendicular and parallel to the magnetic field components of the electron velocity shown by blue rectangles in that figure and corresponding to the dissipation components
and
with the additional contribution from the diagonal component
and the off-diagonal term
shown by the gray rectangles and localized near the *B*_{z} peak (Figure S2k).

According to Figure 4, which shows parameters similar to those in Figure 1 at the later moment *ω*_{0i}*t* = 34, the kinetic dissipation effects found at the moment *ω*_{0i}*t* = 32 persist and are even strengthened. As follows from the comparison of Figures 4b, 4e, and 4f with Figures 4c and 4d, the kinetic dissipation parameters are still more positive on average and they have no strong negative dips, in contrast to the Joule heating rate and the pressure dilatation. Moreover, according to Table 1, the normalized mean of the kinetic dissipation for ions becomes even stronger compared to the pressure dilatation: 〈*P**i*-*D*^{(i)}〉_{σ}/〈−*p*^{(i)}*θ*^{(i)}〉_{σ}≈5, while the corresponding ratio for electrons remains large: 〈*P**i*-*D*^{(e)}〉_{σ}/〈−*p*^{(e)}*θ*^{(i)}〉_{σ}≈15. At the same time, the evolution of the original *P**i*-*D* parameters, from 0.0098 to 0.022 for ions and from 0.091 to 0.052 for electrons, suggests that in the considered time period the plasma modes associated with the ion dissipation grow in time, whereas the electron dissipation-related processes decay. Figures 4c and 4g reveal correlation between the ion thermal energy increase ahead of two *B*_{z} peaks (the corresponding enhancement of the ion temperature is shown in Figure S3a) and the enhancement of the ion kinetic dissipation in that region. For electrons, the thermal energy increase is provided adiabatically near *B*_{z} peaks, where the corresponding kinetic dissipation is relatively small, and in the halo region behind the DF (−12 < *x*/*d*_{i}<−5), where the parameter *P**i*-*D*^{(e)} is relatively large. The latter effect is rather hidden in the distribution of the thermal energy density *u*^{(e)}, and it is much more prominent in the corresponding electron temperature distribution (Figure S3b; see also Figure 4d in Sitnov et al., 2017).

Finally, in Figures S4 and S5 we provide velocity distributions obtained near the peaks of ion and electron dissipation (*x*,*y*,*z*) = (−3.91,6.1,0) and (−9.46,8.9,0). Figures S4a and S4b confirm the original interpretation of the *P**i*-*D*^{(i)} peak by ion reflection from fronts. Figure S5 suggests that, in addition to the LHDI, a contribution to the electron heating behind the DF may also be due to acceleration of electrons trapped near the X line, resulting in flat-top distributions with counterstreaming electrons in reconnection exhausts (Asano et al., 2008; Egedal et al., 2010). At the same time, as follows from 5, even the comprehensive description of particle velocity distributions at a single point remains insufficient for the assessment of the kinetic dissipation measures, which requires also the knowledge of spatial gradients of velocities.

## 4 Conclusion

In this paper new measures of kinetic dissipation, the pressure dilatation, and the *P**i*-*D*^{(i,e)} parameters, recently proposed by Yang, Matthaeus, Parashar, Wu, et al. (2017) and Yang, Matthaeus, Parashar, Haggerty, et al. (2017) for studies of collisionless plasma turbulence, have been applied for the analysis of spontaneous dipolarizations reproduced in 3-D PIC simulations of the magnetotail CS (Sitnov et al., 2017). The analysis shows that in contrast to the fluid dissipation measure, the Joule heating rate
, which cannot distinguish between ion and electron dissipation and reveals deep negative spikes at the DF, the *P**i*-*D*^{(i,e)} parameters are largely positive and drastically different for ions and electrons. This is consistent with the heuristic arguments showing that *P**i*-*D*^{(i,e)} appear as kinetic analogs of the Joule heating rate. The *P**i*-*D*^{(i,e)} parameters are also found to be more persistently positive, compared to pressure dilatation parameters, in terms of their normalized means (Table 1).

Further analysis of these parameters suggests that ions are heated at and ahead of the DF due to their reflection from the front, while electrons are heated at and behind the DF due to the long-wavelength LHDI. These findings resurrect early ideas of the ion tearing instability as a mechanism of substorm dipolarizations (Schindler, 1974) and the LHDI as a source of anomalous resistivity in the magnetotail (Huba et al., 1977). The important clarifications resulting from this and other recent works are as follows. The ion tearing can only be unstable for CSs with regions of accumulated magnetic flux and the corresponding regions of the tailward *B*_{z} gradient. The LHDI is found to be a source of electron viscosity, while its impact on the resistive models of collisionless plasmas is still to be investigated.

Thus, the new kinetic dissipation parameters *P**i*-*D*^{(i,e)} proved to be important quantitative measures of the kinetic energy transfer in collisionless plasmas. Their assessment requires information about 3-D vectors of electric and magnetic fields, ion and electron bulk flow velocities, pressure tensor components, and, crucially, their spatial derivatives. It is amazing that all these parameters can now be provided by the MMS mission (Burch, Moore et al., 2016), which should therefore further clarify the collisionless dissipation picture at the magnetopause and in the magnetotail.

## Acknowledgments

The authors thank J. Birn, P. Cassak, D. Gershman, and T. Motoba for useful discussions. The work on this paper also benefited greatly from the discussions at the ISSI workshops on “Explosive Processes in the Magnetotail: Reconnection Onset and Associated Plasma Instabilities” held in Bern, Switzerland, in October 2016 and 2017. This research was supported by the NASA HSR and LWS programs and by the NSF GEM program. Simulations were made possible by the NASA High-End Computing Program through the NASA Advanced Supercomputing Division at Ames Research Center as well as NCAR's Computational and Information Systems Laboratory (https://doi.org/10.5065/D6RX99HX), supported by the NSF. The data used to produce figures in the paper are preserved at the NCAR High Performance Storage System and are available upon request.