Quantified Plasma Heating and Energy Dissipation in the Earth’s Quasi-perpendicular Bow Shock

The high-speed solar wind decelerates as it travels through the Earth’s bow shock, during which a portion of the incident flow energy is converted into the plasma thermal energy. However, the energy partition of plasma heating among different species, as well as the energy dissipation mechanism, remains elusive. In this paper, we quantify the plasma heating and the energy dissipation by calculating the pressure–strain interactions in 33 quasi-perpendicular bow shocks observed by the Magnetospheric Multiscale spacecraft. Our results show that the Joule dissipation measured by J·E′ and the quasi-viscous dissipation quantified by PiD are distinct in the examined shocks. We also reveal that ions gain more energy than electrons, and the compressive effect is more important than the incompressible channel in plasma heating. PiD contributed by the gyrotropic pressure tensor is consistently positive for electrons; however, there is no discernible distinction between the gyrotropic and nongyrotropic contribution to PiD for ions.


Introduction
Collisionless shocks represent fundamental plasma phenomena in the Universe, arising from the interaction between a supersonic flow and an obstacle.They serve as pivotal locations where a fraction of the upstream bulk flow energy undergoes irreversible conversion into thermal energy (Petschek 1958;Fishman et al. 1960;Shu 1992).Despite the significance of collisionless shocks, the mechanism underlying energy dissipation and plasma heating across them remains inadequately comprehended.
Since the proposal of shocks in collisionless plasma (Kellogg 1962), numerous potential dissipation mechanisms have been postulated in recent decades.These include dispersive radiation (e.g., Sundkvist et al. 2012), wave-particle interactions (e.g., Wilson et al. 2014aWilson et al. , 2014b)), particle reflection (e.g., Su et al. 2012), and macroscopic quasi-static field effects (e.g., Scudder et al. 1986aScudder et al. , 1986bScudder et al. , 1986c)).It has been suggested that the dominant mechanism of energy dissipation is significantly influenced by upstream macroscopic parameters, such as the angle between the upstream magnetic field and the shock normal, Mach number, and plasma β (e.g., Sagdeev 1966;Kennel et al. 1985).Theoretically, dissipation within the shock layer is attributed to anomalous resistivity at low Mach numbers.When the Mach number exceeds the critical Mach number, anomalous resistivity can no longer dissipate all the incoming bulk flow energy and particle reflection becomes important to maintain a stationary shock.(e.g., Edmiston & Kennel 1984;Kennel et al. 1985;Kennel 1987;Balogh & Treumann 2013).Wilson et al. (2014aWilson et al. ( , 2014b))) quantified the contribution of wave-particle interactions, facilitated by high-frequency waves, to the total energy dissipation budget.Their findings suggest that highfrequency waves contribute more significantly to energy dissipation than quasi-static fields.Schwartz et al. (2022) study quasi-perpendicular shocks observed by the Magnetospheric Multiscale (MMS) mission, suggesting that the primary ram energy of incident protons is predominantly converted into downstream proton enthalpy flux, facilitated mainly by a small fraction of suprathermal protons.Recently, magnetic reconnection has been observed in the transition region of the Earthʼs bow shock (e.g., Gingell et al. 2019;Wang et al. 2019;Gingell et al. 2020).Given that reconnection is inherently a dissipative process, it may play a crucial role in energy dissipation for shocks.
Establishing a methodology for quantifying the energy dissipation is crucial to understand the dissipation mechanism within the shock layer.In closed thermodynamic systems, irreversibly dissipating energy is associated with an increase in system entropy.While the bow shock cannot be considered a closed system, an increase in kinetic entropy density across Earthʼs bow shock from upstream to downstream has been measured (Parks et al. 2012;Lindberg et al. 2022).Recently, the parameter + ´= + has been deemed as a reliable measure of Joule dissipation (Zenitani et al. 2011).J E • ¢ finds extensive application in evaluating energy dissipation rates within various collisionless plasma scenarios (e.g., Wan et al. 2015;Burch et al. 2016).The pressure-work term , also dubbed as pressurestrain interaction, delineates the energy conversion between flow energy and internal energy for each species.It can be decomposed into three terms, denoted as pθ α , PiD g,α , and PiD ng,α , which are listed on the right-hand side of Equation (3), respectively.The superscript α = e, i represents the species and g, ng represents the gyrotropic component and the nongyrotropic component, defined in Equations (1)-( 4).The pθ α term accounts for the increase in internal energy due to compressive effects, while the PiD α (PiD g,α +PiD ng,α ) term represents the heating contributed by incompressive processes, also referred to as "collisionless viscosity" (Yang et al. 2017;Zhou et al. 2021).If dissipation is defined as the energy conversion to internal energy, then PiD is also considered as a reliable proxy for energy dissipation, i.e., quasi-viscous dissipation (Yang et al. 2022).

b P
The advantage of PiD lies in its capability to estimate dissipation across different species and its independence from the choice of inertial frame.While both J E • ¢ and PiD have independently been demonstrated as capable of quantifying the energy dissipation, it remains uncertain whether they are reliable measurements of energy dissipation within the bow shock.In this paper, we ascertain plasma heating across Earthʼs bow shock by calculating the pressure-strain interaction based on MMS observations.Although pθ and PiD have been utilized to study energy conversion in plasma turbulence and reconnection, this marks the first application of these measures to shocks.Here, we present the common characteristics of plasma heating and energy dissipation at Earthʼs quasiperpendicular bow shock.The remainder of the paper is organized as follows: Section 2 details a case study; Section 3 provides statistical results; and Section 4 discusses and summarizes the main findings from our analysis.

Overview
The data utilized in this study were acquired through three instruments on board the MMS spacecraft (Burch et al. 2016).The Fluxgate Magnetometer supplied the magnetic field data (Russell et al. 2016), the Electric Double Probe provided the electric field data (Ergun et al. 2016;Lindqvist et al. 2016), and the Fast Plasma Instrument (FPI) recorded the plasma data (Pollock et al. 2016).
This section presents a case study illustrating the heating and dissipation processes across the bow shock.MMS crossed the Earthʼs bow shock at approximately 12:07 UT on 2015 October 7, at the location of (7.9, 9.0, −0.5)R E in the Geocentric Solar Ecliptic coordinate system.Figure 1 provides an overview of MMS1 observations from 11:56:10 to 12:18:10 UT, when the spacecraft moved from downstream to upstream through the bow shock.The angle between the averaged upstream magnetic field, B up = (−10.06,11.10, −2.61) nT, and the shock normal, n sh = (0.90, 0.43, −0.03), (obtained from the empirical model proposed by Farris et al. 1994), was approximately 74°.This angle indicates that the spacecraft crossed a quasi-perpendicular bow shock.Relying on variations in magnetic field magnitude and electron density, we can discern the foot, ramp, and overshoot of this bow shock.These features are hallmark structures of quasi-perpendicular shocks.
As indicated in Figures 1(a) and (b), the magnetic field exhibits minor variations in the foot region, followed by a sharp increase in the ramp, and eventually stabilizes downstream with notable oscillations.Figures 1(c) and (d) illustrate that both electron temperature and density experience rapid increments across the transition layer toward the downstream.The electron temperature displays a slight anisotropy in the transition layer, while it is almost isotropic downstream.On the contrary, ions exhibit anisotropy in the transition region and downstream, manifested as a considerably larger perpendicular temperature than the parallel temperature.Fluxes of high-energy electrons and ions also increase substantially from the upstream to the downstream, often accompanied by plasma heating and energy dissipation that we aim to investigate.

Evaluating the Plasma Heating and Energy Dissipation
Following the method outlined above, we calculate the joule dissipation J E • ¢, pressure-strain interaction P u ( • ) •  and its various components to quantify the plasma heating and energy dissipation in this event.Beyond calculating the instantaneous value at each time point, we determine their cumulative values to assess the accumulated effect across the shock.We set a starting point in the immediate upstream of the shock and calculate the accumulation of each parameter as the sum of the instantaneous values from the starting point.
Figure 2 illustrates the rates of plasma heating and energy dissipation around the transition region of the bow shock.We see from Figure 2 • u e exhibit predominantly positive values with significant fluctuations across the shock.Nevertheless, the peak of −(P i • ∇) • u i is situated in the foot, with a value of 28.3 nW m -3 , while the peak of −(P e • ∇) • u e is located in the ramp and overshoot, measuring 23.2 nW m -3 .This is consistent with the observation that the accumulation of −(P i • ∇) • u i primarily increases in the foot, while the accumulation of −(P e • ∇) • u e rises predominantly in the ramp and overshoot.In this event, electrons primarily experience heating in the ramp and overshoot, while ions undergo predominant heating in the foot.In the transition region, the accumulation of −(P e • ∇) • u e is 202 nW m -3 , whereas the accumulation of −(P i • ∇) • u i is 646 nW m -3 , indicating that ions gain much more energy than electrons across the shock.The components PiD and pθ, which constitute P u ( • ) •  -, are depicted in Figures 2(c) and (d), with their ion component being larger than their electron counterpart.Figures 2(b)-(d) reveal that pθ e closely matches −(P e • ∇) • u e due to the negligible contribution of PiD e .Furthermore, pθ i , with a cumulative value of 468.1 nW m −3 , accounts for about 72% of −(P i • ∇) • u i .This suggests that compressive processes constitute the primary channel of plasma heating in this event, especially for electrons.Figure 2(d) also illustrates that PiD i displays a substantial positive peak in the foot, with the accumulation demonstrating a positive increasing trend.
Figure 2(e) demonstrates that the amplitude of J E • ¢ is predominantly high in the ramp and overshoot of the shock, exhibiting positive accumulation.The accumulation of PiD is 180.5 nW m −3 , surpassing that of J E • ¢ with a value of 134.1 nW m −3 .In Figure 2(f), PiD g,i and PiD ng,i are primarily positive in the foot, with their accumulations increasing at a comparable rate.As the ramp is approached, PiD ng,i reverses from positive to negative, leading to a subsequent decrease in accumulation.The amplitude of PiD g,i and PiD ng,i closely matches in the ramp, balancing each other and resulting in no significant change in PiD i .Figure 2(g) illustrates that PiD ng,e is almost negligible across the shock and PiD g,e undergoes a positive-to-negative reversal in the overshoot, similar to PiD ng,i .

Statistical Results
In addition to the previously mentioned event (event 1), we have analyzed 32 other quasi-perpendicular shocks.These events were selected from 2015 October, when the MMS first observed the bow shock, to 2018 June, when MMS4 experienced issues with its electron measurements.These events span a broad range of positions along the Y-axis in space, extending from the subsolar region to the flank of the bow shock.For all 33 events, we ensured the availability of burst mode data in the transition region and the formation of a well-defined tetrahedron in space by requiring a tetrahedron quality factor greater than 0.7 (Fuselier et al. 2016).In each event, we intend to identify a transition region akin to event 1.However, establishing a universal standard for determining the transition region is challenging due to the unique characteristics of each event.The upstream edge is positioned approximately at the boundary between the solar wind and the foot, while the downstream edge is situated roughly at the boundary between the overshoot and the magnetosheath.These edges are discerned based on the amplitude of the magnetic field and electron density.Notably, making slight adjustments to the transition regionʼs edges does not qualitatively impact the statistical results.We calculate the average values of the parameters, denoted as 〈L〉, within the shock transition region and compare 12 pairs of parameters, as illustrated in Figure 3.Each pair of parameters is divided by the sum of the absolute values of the two parameters in order to emphasize their signs and relative magnitudes across all events.
Figure 3(a) reveals that both J E • á ¢ñ and 〈PiD〉 are negative in five events, positive in 13 events, and show different signs in the remaining 15 events.The result indicates that the average values of Joule dissipation and quasi-viscous dissipation are not consistently positive within the shocks.Additionally, it is noteworthy that the magnitude of 〈PiD〉 exceeded that of J E • á ¢ñ in 22 events.In Figure 3 In Figures 3(c) and (d), 〈pθ i 〉 + 〈pθ e 〉 is predominantly positive in most events, implying that compression mainly heats the plasma.Conversely, 〈PiD〉 is negative in 14 events.
Although 〈PiD e 〉 is positive in most of the 14 events, 〈PiD i 〉 is negative and its absolute value exceeds that of 〈PiD e 〉.This implies that the incompressive processes not only heat plasma but also cool them.Figures 3(e) and (f) illustrate that 〈pθ〉 is typically larger than 〈PiD〉 for both ions and electrons in the majority of events, which suggests that the compressive channel plays a more significant role than the incompressive channel in the plasma heating within the shock transition region.This also accounts for the similarity in statistics between 〈pθ α 〉 and 〈−(P α • ∇) • u α 〉 as depicted in Figures 3(b) and (d).
Figures 3(g) and (h) compare the contribution to PiD from the gyrotropic and nongyrotropic pressure tensor for ions and electrons, respectively.It is shown that, for both ions and electrons, the gyrotropic contribution PiD g is almost always positive, while the nongyrotropic component PiD ng can be either positive or negative.Moreover, PiD g,i is larger than PiD ng,i in about half of the events (15 events), while PiD g,e is larger than PiD ng,e in 30 events.Considering all the events, the contribution from the electron gyrotropic pressure tensor to PiD is significantly larger than that from the nongyrotropic counterpart, while the ion gyrotropic pressure tensor contributes equally to PiD as the nongyrotropic component.

Summary and Discussion
This paper quantifies plasma heating and energy dissipation using the Joule dissipation J E • ¢ and pressure-strain interaction −(P α • ∇) • u α , along with its components, in the transition region of the Earthʼs quasi-perpendicular bow shock.The primary results of our investigation are summarized as follows: 1.There is no significant correlation between the Joule dissipation measurement J E • ¢ and the quasi-viscous dissipation measurement PiD.Furthermore, both J E • á ¢ñ and 〈PiD〉 are not consistently positive across the shock in all examined events.However, it is noteworthy that 〈PiD〉 tends to have a larger magnitude in the majority of events.2. Plasmas are indeed heated, with the heating process primarily occurring through the compressive channel.However, in 14 observed events, ions exhibit a distinct behavior from heated electrons, undergoing cooling in the incompressive process.This distinction serves as the primary explanation for the negative 〈PiD〉.The significant compressive heating is easily comprehensible, given that the bow shock decelerates the fast solar wind, leading to substantial plasma compression.This deceleration induces a convergent flow within the shock layer, i.e., ∇ • u < 0, resulting in positive values for pθ. 3.In the majority of examined events, ions exhibit a propensity to gain or lose more energy than electrons in both compressive and incompressive processes.On average, ions gain or lose approximately twice as much energy to that compared to electrons.4. The gyrotropic pressure tensor contribution to PiD, i.e., PiD g , consistently registers as positive.Notably, for electrons, the contribution of PiD g to PiD is larger than PiD ng , setting them apart from ions.Cassak et al. (2022) propose that combining the normal deformation of flow with the pressure dilatation allows for the characterization of power density arising from converging (diverging) flows, with the remaining portion describing the work solely due to shear deformation.They used PiD shear to quantify the increase in internal energy resulting from shear deformation and PDU to represent the power density arising from converging and diverging flows, encompassing both dilatation and normal deformation.The definitions of PiD shear and PDU are The sum of PiD shear and PDU is equivalent to P This similarity is due to the dominance of the ion term over the electron term and the prevalence of the compressive channel over the incompressive channel.These findings also suggest that the heating due to shear deformation does not serve as the primary cause of plasma heating.
The MMS plasma instruments were specifically designed to measure hot, slow magnetospheric and magnetosheath plasmas.Consequently, they have a tendency to overestimate the temperature and underestimate the number density of particles in the solar wind (e.g., Schwartz et al. 2022;Wilson et al. 2022).The low density and cold nature of the solar wind lead to some uncertainties and errors in determining upstream parameters and calculating PiD.However, once the plasma starts being heated from the shock foot the plasma measurements by FPI become reliable (e.g., Yang et al. 2020).Since the primary focus of this study is the transition region, even when considering a small portion of the solar wind, the impact of the very small PiD in the solar wind on the overall results is negligible.
The intricate relationship between the shock parameters and the processes of energy conversion has been a long-standing problem.A number of previous studies have investigated the dependence of plasma heating on upstream parameters such as the shock normal angle θ Bn , the Alfvén Mach number M A and plasma beta β, revealing no discernible relationships with heating, etc. (e.g., Schwartz et al. 1988;Wilson et al. 2007Wilson et al. , 2014bWilson et al. , 2020)).Here we also performed a preliminary analysis of the relationship between the upstream parameters (θ Bn , β, M A ) and the average values of J E • ¢, P u ( • ) •  -, PiD and pθ.Upstream parameters were derived from timeshifted data obtained from spacecraft located upstream of the MMS, as provided by the OMNI database (King & Papitashvili 2005).Our results indicate that there is no apparent linear correlation (all the Pearson and Spearman coefficients are less than 0.25), which is consistent with the conclusions that weak correlations are found between some velocity distribution function parameters and macroscopic shock parameter across interplanetary shocks (Wilson et al. 2020).
This paper predominantly focuses on average values of the pressure-strain interaction in the transition region, omitting an examination of their values across different frequency bands.As depicted in Figure 2, PiD and pθ exhibit rapid variations, likely corresponding to fluctuations with varying frequencies.These fluctuations may involve a variety of waves or smallscale structures.Various plasma waves have been observed in the Earthʼs bow shock, including fast-mode whistler waves, ion-acoustic waves, etc. (e.g., Vasko et al. 2018;Hull et al. 2020;Vasko et al. 2022;Hao et al. 2023).It has been proposed that wave-particle interactions play essential roles in energy dissipation and plasma heating for shocks (e.g., Wilson et al. 2014b).One should be aware that the pressure-strain interaction P u ( • ) •  is directly derived from the Vlasov equation.Thus, P u ( • ) •  contains all the pathways for energy conversion to thermal energy, including wave-particle interactions.The only constraint to quantifying energy dissipation and plasma heating using and its various components is the data cadence, with a time resolution of 150 ms for ions and 30 ms for electrons.This constraint means that higher-frequency processes are neglected in this analysis.Investigating the decomposition of PiD and pθ in different frequency bands and their respective contributions holds promise for shedding new light on the role of waves in plasma heating and energy dissipation within shock layers.
It is well known that, for the quasi-parallel shocks, reflected ions travel upstream and interact with incident particles, inducing waves and turbulence and resulting in the formation of the foreshock.The quasi-parallel shock transition is thereby lengthened and more dynamic (Schwartz & Burgess 1991).Consequently, assessing plasma heating and energy dissipation rates across quasi-parallel bow shocks becomes more challenging.Further research on quasi-parallel shocks is imperative for a comprehensive understanding of energy dissipation and partition in heliospheric shocks.

Figure 1 .
Figure 1.Overview of MMS1 observations during 11:56:10-12:18:10 UT on 2015 October 7. From the top to the bottom are (a) three components of the magnetic field in the Geocentric Solar Ecliptic coordinate system; (b) the magnetic field magnitude; (c) electron densities; (d) electron parallel and perpendicular temperatures; (e) ion parallel and perpendicular temperatures; (f) the electron and (g) ion differential energy fluxes; blue, orange, and green bars at the top of this figure correspond to the downstream, transition, and upstream regions, respectively.The overshoot, ramp, and foot of the bow shock are indicated by arrows with different colors.
(b), it is evident that 〈−(P i • ∇) • u i 〉 and 〈−(P e • ∇) • u e 〉 are predominantly positive in most events.Only two events exhibit both 〈−(P i • ∇) • u i 〉 and 〈−(P e • ∇) • u e 〉 as negative.Although 〈−(P e • ∇) • u e 〉 is negative in another five events, the sum of 〈−(P i • ∇) • u i 〉 and 〈−(P e • ∇)• u e 〉 is positive in these events.These observations strongly indicate that plasmas experience heating across the bow shock.Moreover, it is evident that 〈−(P i • ∇) • u i 〉 is consistently greater than 〈−(P e • ∇) • u e 〉 in more than 25 events.In other words, ions gain or lose more energy than electrons during the heating or cooling processes.Averaged over 33 events, 〈−(P i • ∇) • u i 〉 is approximately 2 times as large as 〈−(P e • ∇) • u e 〉.

Figure 2 .
Figure 2. Measurements of the plasma heating and the energy dissipation in the bow shock transition region.(a) magnetic field magnitude; (b)-(d) P u • • ( )  -, pθ and PiD and their accumulations (cyan and green) for ions (red) and electrons (blue); (e) J E• ¢ (blue) and its accumulation (red); (f)-(g) The gyrotropic (red) and nongyrotropic (blue) components of PiD i and PiD e and their accumulations (cyan and green).The right axis and the thick lines correspond to the cumulative values, while the left axis and the thin lines correspond to the instantaneous values.The transition region is highlighted by the yellow shadow and the vertical dashed line marks the boundary between the ramp and foot.

Figure 3 .
Figure 3. Histograms of the average values of 12 pairs of quantities for all events.The values are normalized by the sum of their absolute values.(a) J E • ¢ and PiD; (b) −(P i • ∇) • u i and −(P e • ∇) • u e ; (c) PiD i and PiD e ; (d) pθ i and pθ e ; (e) PiD i and pθ i ; (f) PiD e and pθ e ; (g) PiD g,i and PiD ng,i ; (h) PiD g,e and PiD ng,e .
) and (j) show the average PiD shear,i versus PiD shear,e (PDU i versus PDU e ), and Figures3(k) and (l) shows the average PiD shear,i versus PDU i (PiD shear,e versus PDU e ) for each event.Notably, the results presented in Figures 3(i)-(l) are similar to those in Figures 3