# On the Cascade-Dissipation Balance in Astrophysical Plasmas

D. Manzini<EMAIL_ADDRESS>Laboratoire de Physique des
Plasmas (LPP), CNRS, École Polytechnique, Sorbonne Université, Université
Paris-Saclay, Observatoire de Paris, 91120 Palaiseau, France  Dipartimento di
Fisica E.Fermi, University of Pisa, Italy  F. Sahraoui  Laboratoire de
Physique des Plasmas (LPP), CNRS, École Polytechnique, Sorbonne Université,
Université Paris-Saclay, Observatoire de Paris, 91120 Palaiseau, France  F.
Califano  Dipartimento di Fisica E.Fermi, University of Pisa, Italy

###### Abstract

The differential heating of electrons and ions by turbulence in weakly
collisional magnetized plasmas and the scales at which such energy dissipation
is most effective are still debated. Using a large data sample measured in the
Earth’s magnetosheath by the Magnetospheric Multiscale mission and the coarse-
grained energy equations derived from the Vlasov-Maxwell system we find
evidence of a balance over two decades in scales between the energy cascade
and dissipation rates. The decline of cascade rate at kinetic scales (in
contrasts with a constant one in the inertial range), is balanced by an
increasing ion and electron heating rates, estimated via the pressure-strain.
Ion scales are found to contribute most effectively to ion heating, while
electron heating originates equally from ion and electron scales. These
results can potentially impact current understanding of particle heating in
turbulent magnetized plasmas as well as their theoretical and numerical
modeling.

One of the central problems in turbulent media is to understand how energy is
transferred across scales and how it is eventually dissipated. For weakly
collisional plasma such as the solar wind or planetary magnetosheahts the
pioneering work of Politano and Pouquet (1998) enable us, in the framework of
incompressible Magnetohydrodynamics (MHD) and under the assumptions of fully
developed turbulence, to express the cascade rate as a function of third order
structure functions of the velocity and magnetic field (Sahraoui  _et al._ ,
2020). These results have been extended to account for compressibility
(Banerjee and Galtier, 2013; Andrés  _et al._ , 2018), the contribution of the
Hall current at subion scales (Galtier, 2008; Hellinger  _et al._ , 2018;
Ferrand  _et al._ , 2019), different fluid closure equations (Simon and
Sahraoui, 2021) and temperature anisotropy (Simon and Sahraoui, 2022) and have
been used extensively to measure the cascade rate in spacecraft data Sorriso-
Valvo  _et al._ (2007); MacBride  _et al._ (2008); Stawarz  _et al._ (2009,
2010); Coburn  _et al._ (2012); Hadid  _et al._ (2017, 2018); Bandyopadhyay
_et al._ (2018); Andrés  _et al._ (2019); Bandyopadhyay  _et al._ (2020);
Andrés  _et al._ (2021); Brodiano  _et al._ (2023); Pecora  _et al._ (2023).
In recent years the coarse-graining (CG) method, initially developed for
hydrodynamics (Germano, 1992; Eyink, 2005), gained popularity in the plasma
physics community (Aluie, 2017; Camporeale  _et al._ , 2018; Cerri and
Camporeale, 2020; Yang  _et al._ , 2017; Manzini  _et al._ , 2022). This
formulation provides an alternative way to measure the cascade rate and
enables us to overcome some limitations imposed by the stringent hypotheses of
fully developed turbulence (e.g., spatial homogeneity). Indeed, the CG
approach can be employed not only to measure the average cascade rate over
large plasma portions, but also to address localized cross-scale energy
transfer in reconnecting current sheets (Manzini  _et al._ , 2023; Adhikari
_et al._ , 2023). In this work we scale-filter the Vlasov-Maxwell system of
equations and measure the nonlinear energy cascade rate and the exchanges
between its various forms (kinetic, electromagnetic and thermal) as a function
of scale, with particular focus on the turbulent plasma heating given by the
pressure-strain interaction (Belmont  _et al._ , 2013; Fitzpatrick, 2022).
The Coarse Graining theory– To study cross-scale energy transfer we apply the
spatial CG approach to the moments of Vlasov equation, written for an
electron-ion plasma ($\alpha=e,i$) , and the Maxwell ones. All variables are
low-pass filtered at a scale $\ell$, _e.g_ $\overline{\bm{v}}_{\ell}=\int
d\bm{r}G_{\ell}(\bm{r})\bm{v}(\bm{x}+\bm{r})$, where $G_{\ell}$ is a centered,
normalized filtering kernel with variance of order $\ell^{2}$. To include the
effects of compressibility we introduce a density weighted filtering (_Favre
filtering_) defined for a given field $f$ as:
${\overline{\rho}_{\ell}}\widetilde{{f}}_{\ell}={\overline{\rho{f}}_{\ell}}$
(Aluie, 2011, 2013). For conciseness of the notations the filtering scale
$\ell$ is not written explicitly unless necessary.

At each scale $\ell$ we can write the equations for the large-scale bulk flow
($\widetilde{\mathcal{E}}^{\mathrm{f}}_{\alpha}=\overline{\rho_{\alpha}}|\widetilde{\bm{v}_{\alpha}}|^{2}/2$),
electromagnetic (EM,
$\overline{\mathcal{E}}^{em}\\!=\\!({|\overline{\bm{E}}|^{2}+\\!|\overline{\bm{B}}|^{2}})/{8\pi}$)
and thermal energies
($\overline{\mathcal{E}}^{\mathrm{th}}_{\alpha}=\mathrm{Tr}(\overline{\bm{P}_{\alpha}})/2$):

$\displaystyle\frac{\partial}{\partial
t}\left(\widetilde{\mathcal{E}}^{\mathrm{f}}_{i}+\widetilde{\mathcal{E}}^{\mathrm{f}}_{e}\right)$
$\displaystyle=$
$\displaystyle-\nabla\cdot{\mathbfcal{F}}^{\mathrm{f}}_{\ell}+\overline{\bm{j}}\cdot\overline{\bm{E}}$
(1) $\displaystyle+$
$\displaystyle\overline{\bm{P}_{i}}:\nabla\overline{\bm{v}_{i}}+\overline{\bm{P}_{e}}:\nabla\overline{\bm{v}_{e}}-\pi(x,\ell)$
$\displaystyle\frac{\partial}{\partial t}\overline{\mathcal{E}}^{em}$
$\displaystyle=$
$\displaystyle-\nabla\cdot\mathbfcal{F}^{\mathrm{em}}_{\ell}-\overline{\bm{j}}\cdot\overline{\bm{E}}$
(2) $\displaystyle\frac{\partial}{\partial
t}\overline{\mathcal{E}}^{\mathrm{th}}_{\alpha}$ $\displaystyle=$
$\displaystyle-\nabla\cdot\mathbfcal{F}^{\mathrm{th}}_{\ell}-\nabla\cdot\overline{\mathbf{h}}_{\alpha}$
(3) $\displaystyle-$
$\displaystyle\overline{\bm{P}_{\alpha}}:\nabla\overline{\bm{v}_{\alpha}}-\phi_{\alpha}(x,\ell)$

where

$\begin{split}\pi(x,\ell)=&\sum_{\alpha=e,i}-\overline{\rho_{\alpha}}\left[\widetilde{\bm{v}_{\alpha}\bm{v}_{\alpha}}-\widetilde{\bm{v}_{\alpha}}\widetilde{\bm{v}}_{\alpha}\right]:\nabla\widetilde{\bm{v}_{\alpha}}\\\
&+(\nabla\overline{\bm{P}_{\alpha}})\cdot(\widetilde{\bm{v}_{\alpha}}-\overline{\bm{v}_{\alpha}})\\\
&-\overline{n_{\alpha}q_{\alpha}\bm{v}_{\alpha}}\cdot\left[(\widetilde{\bm{E}}-\overline{\bm{E}})+\frac{1}{c}\left(\widetilde{\bm{v}_{\alpha}\times\bm{B}}-\widetilde{\bm{v}_{\alpha}}\times\widetilde{\bm{B}}\right)\right]\end{split}$
(4)

is the cross-scale energy transfer (or _turbulent cascade_) rate across the
scale $\ell$. The quantities $\overline{\bm{j}}\cdot\overline{\bm{E}}$ and
$\mathrm{PS}_{\alpha}\equiv\overline{\bm{P}_{\alpha}}:\nabla\overline{\bm{v}_{\alpha}}$
are the only sink terms of the large-scale EM and Thermal energy,
respectively, and appear as a source in the large scale bulk flow energy. This
implies that any process that changes the large scale thermal energy must go
through the $\mathrm{PS}_{\alpha}$ channel. At the same time equation (1)
shows that (a fraction of) $\bm{j}.\bm{E}$ can lead to plasma heating (via the
$\mathrm{PS}_{\alpha}$) without having to modify the large scale bulk flow
energy. This can be seen even more clearly by summing equations (1) and (3),
which indicates that $\bm{j}\cdot\bm{E}$ acts as a source for the total
kinetic energy of the plasma particles (bulk flow and thermal energy) (Howes
_et al._ , 2017; Afshari  _et al._ , 2021). The quantity
$\phi_{\alpha}=\overline{\bm{P}_{\alpha}:\nabla\bm{v}_{\alpha}}-\overline{\bm{P}_{\alpha}}:\nabla\overline{\bm{v}_{\alpha}}$
stands for a nonlinear cascade of thermal energy. This term transfers thermal
energy from large to small spatial scales and will not be discussed in this
work since we are interested only in the transfer to the thermal energy. While
$\overline{\bm{j}}\cdot\overline{\bm{E}}$ and
$\overline{\bm{P}_{\alpha}}:\nabla\overline{\bm{v}_{\alpha}}$ are cumulative
quantities, encompassing the energy exchanges from all scales larger than
$\ell$, the cross-scale terms $\pi,\phi$ measure the transfer _across_ scale
$\ell$. The spatial fluxes in the form $\nabla\cdot\mathbfcal{F}$, including
the divergence of the filtered heat flux
$\nabla\cdot\overline{\mathbf{h}_{\alpha}}$, move the large scale energies in
space and disappear after integration over a suitable domain.
A similar set of equations was derived in (Yang  _et al._ , 2017). In this
work the cascade rate includes the term
$\nabla\overline{\bm{P}}_{\alpha}\cdot(\widetilde{\bm{v}_{\alpha}}-\overline{\bm{v}_{\alpha}})$,
also present in compressible hydrodynamics (Aluie, 2011).
Summing equations (1)-(2) and averaging over a portion of plasma yields :

$\displaystyle\frac{\partial}{\partial
t}\left\langle\widetilde{\mathcal{E}}^{\mathrm{f}}_{i}+\widetilde{\mathcal{E}}^{\mathrm{f}}_{e}+\overline{\mathcal{E}}^{em}\right\rangle+\nabla\cdot\left\langle{\mathbfcal{F}}^{\mathrm{f}}_{\ell}+{\mathbfcal{F}}^{\mathrm{em}}_{\ell}\right\rangle$
$\displaystyle=$ $\displaystyle\mathrm{PS}_{i}(\ell)+\mathrm{PS}_{e}(\ell)$
$\displaystyle-$ $\displaystyle\Pi(\ell)$ (5)

where in the right hand side we find the average $\mathrm{PS}$ interaction,
$\mathrm{PS}_{\alpha}(\ell)=\langle\overline{\bm{P}_{\alpha}}:\nabla\overline{\bm{v}_{\alpha}}\rangle$,
filtered at scale $\ell$, and the net energy cascade
$\Pi(\ell)=\langle\pi_{\ell}\rangle$.
Equation (5) states that, under the assumption of suitable boundary
conditions, what is lost by the large scale energies either cascades to
smaller scales or is transferred to thermal energy. In this view the pressure-
strain interaction plays the role of an energy sink, reason for which we will
refer to it, even if somehow inappropriately, as _dissipation_.

We evaluate equation (5) between scales $\ell_{0}$ and $\ell<\ell_{0}$. Under
the hypotheses of negligible spatial fluxes and energy stationarity at scales
smaller than $\ell_{0}$ we find:

$\Pi(\ell_{0})-\Pi(\ell)=-\Delta\mathrm{PS}_{i}(\ell)-\Delta\mathrm{PS}_{e}(\ell)$
(6)

where
$-\Delta\mathrm{PS}_{\alpha}(\ell)=-\mathrm{PS}_{\alpha}(\ell)+\mathrm{PS}_{\alpha}(\ell_{0})$
is the cumulative contribution to thermal heating rate of species $\alpha$ in
the range $[\ell,\ell_{0}]$. (6) shows that any difference between
$\Pi(\ell_{0})$ and $\Pi(\ell)$ indicates the amount of energy that is lost to
thermal energy between those two scales. In this perspective a constant
cascade rate indicates an inertial range where dissipation is negligible,
while a scale-dependent cascade rate is signature of active dissipation. It
must be stressed that relation (5) comes directly from Vlasov-Maxwell
equations, and as such is not limited by any fluid approximation since no
closure equation is imposed on the pressure. This implies that if kinetic
effects play a role in heating the plasma, this will be captured by the
$\mathrm{PS}$ interaction, which explains why the energy cascade rate
(inherently a fluid quantity) could capture dissipation via Landau damping in
turbulence simulations (Ferrand  _et al._ , 2021). However, the interpretation
of $\mathrm{PS}$ as a measure of change in the thermal energy (i.e., heating)
is grounded on the assumption that the spatial fluxes contribution (including
the heat flux) in equation (3) are negligible.

Data selection and methods– We use data from the Magnetospheric
Multispacecraft (MMS) mission (Burch  _et al._ , 2016), which enables us to
compute the spatial derivatives in $\mathrm{PS}_{\alpha}$ and $\Pi$ using the
gradiometer technique (Chanteur, 1998). We use FluxGate Magnetometer data for
the magnetic field, the Spin-Plane Double Probe (Lindqvist  _et al._ , 2016)
and the Axial Double Probe (Ergun  _et al._ , 2016) for the electric field and
the Fast Plasma Investigation (Pollock  _et al._ , 2016) for the plasma data.
Spin-tone removal is applied to the electron velocity data. The CG operation
in spacecraft data at a given time-scale $\tau$ is computed as
$\overline{f}_{\tau}(t)=\int dt^{\prime}G_{\tau}(t^{\prime})f(t+t^{\prime})$
where $G_{\tau}$ is a 1-dimensional filtering kernel with characteristic width
$\sim\tau$. To minimize the finite sample size effect, the maximum scale
$\tau_{0}$ should be significantly smaller than the duration of the interval
under considerations. Here we set $\tau_{0}\approx 20-30\,s$, which limits the
largest scales of the study to $k\rho_{i}\gtrsim 0.05-0.1$ depending on the
value of the Taylor shifted Larmor radius. The smallest accessible scale
$\tau_{\mathrm{min}}$ is given by the instrument resolution: when combining
MMS products of different time resolution, we re-sample the data at the
frequency of the least resolved quantities (typically the ions at 150 m$s$),
and set $\tau_{\mathrm{min}}$ at twice the resolution of the instrument. It is
worth noting that the electron contributions to the cascade rate and
$\mathrm{PS}_{e}$ do not involve ion data, and as such they are computed down
to (twice) the electron data time resolution of $30\,\mathrm{m}s$.
To ensure the robustness of the results with respect to the choice of the
start an final time of each interval we follow this pipeline: we compute
$\pi,\overline{P_{i}}:\nabla\overline{v_{i}},\overline{P_{e}}:\nabla\overline{v_{e}}$
for all the data points in the interval
$[t_{\mathrm{start}},t_{\mathrm{end}}]$. We then average the above quantities
in an interval $[t_{\mathrm{start}}+\Delta t_{1},t_{\mathrm{end}}-\Delta
t_{2}]$. By varying independently $\Delta t_{1},\Delta t_{2}$ between 0 and
10% of the interval duration we obtain different estimates of the cascade rate
$\Pi$ and $\mathrm{PS}_{\alpha}$. At each scale we take the mean value as our
best estimate and use the standard deviation as error bar estimate.

Results–

Figure 1: Panel (a) shows the time series of the magnetic field measured by
MMS1 in Geocentric Solar Ecliptic (GSE) coordinates. Panel (b) displays the
power spectrum of the magnetic field data computed using the Welch method
(Welch, 1967). Power-law fit and compensated spectra are shown.

We show in Fig. 1 the data from MMS1 taken in the Earth’s magnetosheath
(2016/02/23, 20:02:35 - 20:04:44). During this time the average plasma
conditions were: $B\approx 35\,$n$T$, $n_{e}\approx 19\,cm^{-3}$,
$T_{i}\approx 175\,eV$ $T_{e}\approx 27\,eV$. The ratio of thermal to magnetic
pressure is $\beta_{i}\approx 1.07,\beta_{e}\approx 0.16$. The mean flow speed
$V_{f}\approx 300\,km/s$ and the angle between the flow and the magnetic field
$\theta_{\mathbf{v}\mathbf{B}}\approx 50^{o}$. The inter-spacecraft separation
is of $\sim 11\,$k$m$. The magnetic field power spectrum (Fig. 1) displays a
$f^{-5/3}$ scaling in the MHD range and steepens to $f^{-2.74}$ at higher
frequencies.
Fig. 2(a) shows the balance between the energy cascade rate $\Pi$ and the
dissipation rate as a function of scale $\tau$. At large scales, within the
inertial range, the dominant process is the cascade, while at smaller scales,
approaching $k\rho_{i}\sim 0.2$, dissipation grows and the turbulent cascade
is progressively weakened. In accordance with equation (6) the decline in the
cascade rate is counterbalanced by a rise in ion and electron’s $\mathrm{PS}$,
maintaining the sum of the three quantities constant over two decades of
scales. Furthermore, Fig. 2(a) shows that the small scale edge of the MHD
range is highly dissipative as the cascade rate weakens by a factor $\sim 5$,
consistent with the idea of increased dissipation around the spectral break
(Hellinger  _et al._ , 2022). Throughout the weakly dissipative subion range
the cascade rate keeps weakening progressively.

Figure 2: Panel (a) shows the different terms of equation (6) as a function of
scale for the interval shown in Fig. 1. Shaded region denote the error bars.
At time lags smaller than the time resolution of the ion data we assume that
there is no additional contribution to ion heating (hence a constant
$\Delta\mathrm{PS}_{i}(\tau)$, dotted), thus only the electron contribution to
the cascade is computed. The (blue) dotted line is a cubic spline
interpolation to aid the visualization. Panel (b) shows the scale-by-scale ion
and electron heating rates (see text).

Looking at the behaviour of quantities $\Delta\mathrm{PS}_{\alpha}$ it is not
immediate to infer at which scales ions and electrons are heated most.
$\mathrm{PS}_{\alpha}$ being a cumulative quantity, the contribution to the
heating rate of a given scale range ($\tau$,$\tau+\Delta\tau$) is simply
$q_{\alpha}(\tau)\Delta\tau\equiv-\mathrm{PS}_{\alpha}(\tau)+\mathrm{PS}_{\alpha}(\tau+\Delta\tau)\sim-[\partial\mathrm{PS}_{\alpha}/{\partial\tau}]\Delta\tau$,
which is the quantity plotted in Fig. 2(b) after binning logarithmically the
range of scales.
The total heating rate for each species is defined as
$Q_{\alpha}=\int_{\tau_{\mathrm{min}}}^{\tau_{\mathrm{max}}}q_{\alpha}(\tau)d\tau=\mathrm{PS}_{\alpha}(\tau_{\mathrm{max}})-\mathrm{PS}_{\alpha}(\tau_{\mathrm{min}})$,
$\tau_{\mathrm{min}}=0.3\,s\;(0.06\,s)$ for ions (electrons) is twice the
plasma data resolution and $\tau_{\mathrm{max}}=30\,s$ for this interval. For
ions we obtain a total heating rate $Q_{i}=(1.9\pm 0.3)\times
10^{-2}\,nW/m^{3}$ (assuming no additional contribution to ion heating
originates from scales smaller than $0.3\,s$). For electrons, we estimate the
heating rate $Q_{e}$ similarly to ions. However, that quantity is complemented
by an extra term given by the energy cascade rate at the smallest available
time lag, namely $\Pi(\tau=\tau_{min})$ (highlighted in Fig. 2(a) for
$\tau_{min}\sim 0.06$s), i.e. $Q_{e}^{\star}=Q_{e}+\Pi(\tau_{\mathrm{min}})$.
This is based on the assumption that the residual cascade rate
$\Pi(\tau_{min})$ will be entirely converted into electron heating at the
smallest scales. Thus, we obtain the value $Q_{e}^{\star}=(1.2\pm 0.2)\times
10^{-2}\,nW/m^{3}$ for the total electron heating rate. The cascade rate at
MHD scales has a value of $\Pi(\tau_{\mathrm{max}})=(3.04\pm 0.05)\times
10^{-2}\,nW/m^{3}$ placing the ratio
$\Pi(\tau_{\mathrm{max}})/(Q_{i}+Q_{e}^{\star})\simeq 1$. We thus verify (6) :
what is lost by the energy cascade has been transferred to thermal energy.
This observation confirms previous numerical results where the balance between
the cascade rate and dissipation via $\mathrm{PS}$ was first reported (Yang
_et al._ , 2017; Hellinger  _et al._ , 2022; Yang  _et al._ , 2022) and
improves over the comparison between $\mathrm{PS}$ and the cascade rate
estimate via incompressible third order laws presented in (Roy  _et al._ ,
2022).

The study of the scale dependent heating rate $q_{\alpha}(\tau)\Delta\tau$
informs us about the scales most effective in heating the two species. Figure
2(b) shows that the largest contribution to ion heating comes from
$\tau\approx 10\,s$ (or $k\rho_{i}\approx 0.1$ using the Taylor hypothesis
$\ell\sim\tau V_{f}\sim 1/k$), in the same range of scales electrons are
substantially heated. The residual cascade rate at $\tau_{\mathrm{min}}=0.06$,
assumed to sustain the turbulence and eventually heat electrons at scales
$k\rho_{i}\gtrsim 13$ (Sahraoui  _et al._ , 2009), accounts for
$\Pi(\tau_{\mathrm{min}})/Q_{e}^{\star}\sim 30\%$ of the total electron
heating rate.

Statistics – To confirm the statistical robustness of the previous results we
perform the same analysis for a large sets of MMS intervals taken in the
magnetosheath between October 2015 and May 2018 (we paid attention to exclude
intervals that showed sharp jumps in the background plasma parameters). We
further narrowed down the selection to keep only data intervals that satisfy
the energy balance defined (within the error bars) by
$\Pi(\tau_{\mathrm{max}})/(Q_{i}+Q_{e}^{\star})\in[0.4,1.6]$ to ensures that
spatial fluxes and time derivatives in equation (5) are negligible in the
range of scales we are considering. This guarantees a reliable estimation of
the cascade rate, the ion and electron heating rates and the related effective
dissipation scales. To increase the size of the statistical sample, for a
given interval we consider each spacecraft as an independent realization,
although the spatial derivatives involved in equation (6) are computed from
the four spacecraft and thus are identical for a given event.

Figure 3: Cascade-Dissipation balance for the selected intervals that satisfy
the balance condition (see text) binned according to the value of the cascade
rate $\Pi(\tau_{\mathrm{max}})$. The star denotes the case study presented in
the text. The inset shows the histogram of the cascade-dissipation ratio for
all the inervals.

Fig. 3 summarizes the cascade-dissipation balance for the selected intervals,
which we use to estimate the value of the energy cascade rate in the Earth
magnetosheath at different scales: the small scale edge of the MHD range
$k\rho_{i}=0.2$, around the ion scale $k\rho_{i}=2$ and at the subion scale
$k\rho_{i}=10$. Histograms of the cascade rate at different scales are
displayed in Fig. 4. To provide a statistically significant measure we
identify the minimum number of consecutive logarithmically spaced bins
containing over 60% of the data set. Figure 4(a) shows that MHD scales
generally exhibit a cascade rate in the interval $[0.3-1.4]\times
10^{-2}\,\mathrm{n}W/m^{3}$, comparable with the values reported in (Hadid
_et al._ , 2018; Andrés  _et al._ , 2019) using third order laws in the
Earth’s magnetosheath. Notably, this rate diminishes by a factor of two upon
reaching $k\rho_{i}=2$ [Fig. 4(b)] and further weakens by an additional factor
of two at $k\rho_{i}=10$ [Fig. 4(c)] reaching a rate in the range
$[0.1-0.4]\times 10^{-2}\mathrm{n}W/m^{3}$. This shows that the subion range
is weakly dissipative and that at $k\rho_{i}\sim 10$ a significant ratio
($\sim 30\%$) of the cascade rate at MHD scales remains available to sustain
the turbulence cascade all the way down to electron scales (Sahraoui  _et al._
, 2009).

Figure 4: Histogram of the cascade rate at different scales,
$k\rho_{i}=0.2,2,10$ respectively. The shaded blue region denotes the minimum
number of contiguous bins that contains 60% of the data set.

We now wish to delineate the scales at which the $\mathrm{PS}$ interaction is
effective in heating the plasma. For each species we calculate the fraction of
heating coming from the MHD range ($k\rho_{i}<0.5$), around the ion Larmor
scale ($0.5<k\rho_{i}<2$) and the subion range ($k\rho_{i}>2$). The results in
Fig. 5(a-c) show that the largest contribution to the ion heating rate comes
from MHD scales (the median contribution being 60%). The relative importance
decreases to 30% at the ion Larmor scale and then 10% at sub-Larmor scales.
This result corroborates the assumption made above that the residual cascade
at $k\rho_{i}\gtrsim 10$ translates predominantly into electron heating.
The picture that emerges for electrons is more complex: Fig. 5(a-c) show a
nearly equal contribution from all scale ranges: 30%, 20% and 40% median
contribution from the MHD, ion Larmor and subion scales, respectively. This
result demonstrates that electron heating can be significant at scales
comparable with the ion Larmor radius (including the edge of the MHD range) in
line with some numerical results (Howes  _et al._ , 2011).

Figure 5:  Histogram of the relative contribution to the heating rate from the
three range of scales $k\rho_{i}<0.5$ (a), $0.5<k\rho_{i}<2$ (b) $k\rho_{i}>2$
Ions (red,filled) and electrons (green, empty). In each panel a vertical sign
denotes the median for each populatuion.

Conclusions – In this work we measure for the first time using in-situ data
the scale dependence of the cascade rate and the dissipation rate and show
that there exist a balance that holds for over two decades of scales: the
depletion of the energy cascade as turbulence proceeds from MHD to kinetic
scales is compensated by a net positive transfer to the thermal energy. On a
statistical data set we show that electrons can get substantial heating at
scales comparable with the ion Larmor radius, against the conventional wisdom
that electron heating occurs solely at electron scales. This questions the
validity of kinetic-hybrid models for plasmas where electrons are treated as a
politropic fluid whose dissipation is confined to the small (electron)
resistive scales.
Despite the net decline of the cascade rate in the subion range, the magnetic
energy spectra still show clear power-laws all the way down to the electron
scales. In the absence of a rigorous explanation to this observation, we
speculate that any residual turbulent energy is bound to cascade to small
scales following the scaling law of one of the existing modes (e.g., Kinetic
Alfvén Modes (Sahraoui  _et al._ , 2010; Howes  _et al._ , 2011).
The technique used in this work enables us to quantify turbulent heating rate,
but it does not inform us about the processes responsible for it. In this view
methods based on the analysis of the velocity distribution functions (Howes
_et al._ , 2017; Schekochihin  _et al._ , 2016; Servidio  _et al._ , 2017;
Cassak  _et al._ , 2023) should be seen as complementary to this study as they
can provide the missing part of the information about the nature of the
processes at play.

###### Acknowledgements.

DM acknowledges useful discussions with A. Chasapis. MMS data come from
CDPP/AMDA and NASA GSFC’s Space Physics Data Facility’s CDAWeb. The python
client SPEASY was used for data retrieval.

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