Nature of Kinetic Scale Turbulence in the Earth's Magnetosheath

The observational analysis is based on data from the four MMS spacecraft (Burch et al. 2016) during a period in the Earth's magnetosheath (2015 October 16, 09:24:11–09:25:24) for which burst mode data are available. During this time, the spacecraft were located 11.9 R_E from Earth, close to the dusk side magnetopause, with an inter-spacecraft separation of ∼14 km. This period does not contain large-amplitude variations in the magnetic field magnitude (associated with mirror modes) or high-frequency wave packets at electron scales (thought to be whistler waves), allowing the pure turbulent cascade to be studied.

For the magnetic field ${\boldsymbol{B}}$, data from the FGM (Fluxgate Magnetometer, Russell et al. 2016) and SCM (Search-Coil Magnetometer, Le Contel et al. 2016) instruments were combined using a wavelet technique (Chen et al. 2010a) to produce 8192 samples s^–1 data containing the full range of frequencies (with the crossover between instruments at ∼8 Hz). The electric field ${\boldsymbol{E}}$ was measured by the SDP (Spin-Plane Double Probe, Lindqvist et al. 2016) and ADP (Axial Double Probe, Ergun et al. 2016) instruments at the same resolution. FPI (Fast Plasma Investigation, Pollock et al. 2016) was used for the ion and electron densities n_i and n_e, velocities ${{\boldsymbol{v}}}_{{\rm{i}}}$ and ${{\boldsymbol{v}}}_{{\rm{e}}}$, and temperatures T_i and T_e; the resolution of these moments is 150 ms for the ions and 30 ms for the electrons. A time series of the data from MMS3 for this interval is shown in Figure 1. The average plasma conditions were: $B\approx 39$ nT, ${n}_{{\rm{i}}}\approx {n}_{{\rm{e}}}\,\approx$ $14\,{\mathrm{cm}}^{-3},{v}_{{\rm{i}}}\,\approx {v}_{{\rm{e}}}\approx 180\,\mathrm{km}\,{{\rm{s}}}^{-1},{T}_{{\rm{i}}}\approx 210\,\mathrm{eV}$, ${T}_{{\rm{e}}}\approx 23\,\mathrm{eV}$, with temperature anisotropies ${({T}_{\perp }/{T}_{\parallel })}_{{\rm{i}}}\approx 1.6$ and ${({T}_{\perp }/{T}_{\parallel })}_{{\rm{e}}}\approx 1.0$. These parameters result in average ion and electron plasma betas ${\beta }_{{\rm{i}}}\approx 0.79$ and ${\beta }_{{\rm{e}}}\approx 0.087$ (where ${\beta }_{s}=2{\mu }_{0}{n}_{s}{k}_{{\rm{B}}}{T}_{s}/{B}^{2}$).

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From the magnitude of the ion and electron betas, ${\beta }_{{\rm{e}}}\ll {\beta }_{{\rm{i}}}\sim 1$, it can be seen that for the ions, the gyroradius ${\rho }_{{\rm{i}}}={v}_{\mathrm{th}\perp {\rm{i}}}/{{\rm{\Omega }}}_{{\rm{i}}}$ (where ${v}_{\mathrm{th}\perp {\rm{i}}}=\sqrt{2{k}_{{\rm{B}}}{T}_{\perp {\rm{i}}}/{m}_{{\rm{i}}}}$ is the thermal speed, ${{\rm{\Omega }}}_{{\rm{i}}}={q}_{{\rm{i}}}B/{m}_{{\rm{i}}}$ is the gyrofrequency, m_i is the mass, and q_i is the charge) and inertial length ${d}_{{\rm{i}}}={v}_{{\rm{A}}}/{{\rm{\Omega }}}_{{\rm{i}}}$ (where ${v}_{{\rm{A}}}=B/\sqrt{{\mu }_{0}\rho }$ is the Alfvén speed and ρ is the mass density) are similar ${\rho }_{{\rm{i}}}\sim {d}_{{\rm{i}}}$, whereas for the electrons, the gyroradius is much smaller, ${\rho }_{{\rm{e}}}\ll {d}_{{\rm{e}}}$. This results in two sub-ranges between the ion and electron gyroscales: one above the electron inertial scale, $1/{\rho }_{{\rm{i}}}\lt k\lt 1/{d}_{{\rm{e}}}$, and one below, $1/{d}_{{\rm{e}}}\lt k\lt 1/{\rho }_{{\rm{e}}}$. The following sections describe the nature of the fluctuations in each of these ranges.

To determine the nature of the fluctuations, first the anisotropy was measured, using the multi-spacecraft technique described by Chen et al. (2010a). Two-point structure functions $\delta {{\boldsymbol{B}}}^{2}({\boldsymbol{l}})={\langle | {\boldsymbol{B}}({\boldsymbol{x}}+{\boldsymbol{l}})-{\boldsymbol{B}}({\boldsymbol{x}}){| }^{2}\rangle }_{{\boldsymbol{x}}}$ were calculated from the time-lagged magnetic field measurements between pairs of spacecraft. The technique mixes spatial and temporal measurements, and assumes the Taylor hypothesis (Taylor 1938) to be satisfied (i.e., that the measured temporal variations correspond to spatial variations in the plasma frame), which appears to be the case, despite the low flow speed and dispersive regime (see Section 2.4). Figure 2 shows $\delta {{\boldsymbol{B}}}^{2}({\boldsymbol{l}})$, binned and averaged as a function of length scale parallel and perpendicular to the local mean field. The range of scales covered is $11\lt k{\rho }_{{\rm{i}}}\lt 57$, or equivalently $0.29\lt {{kd}}_{{\rm{e}}}\lt 1.5$. It can be seen that the contours of $\delta {{\boldsymbol{B}}}^{2}$ are elongated in the parallel direction, and that the value of $\delta {{\boldsymbol{B}}}^{2}$ at a scale of 15 km is ∼10 times larger in the perpendicular direction than in the parallel direction. This indicates strongly anisotropic fluctuations ${k}_{\perp }\gg {k}_{\parallel }$, consistent with previous findings for magnetosheath (Mangeney et al. 2006; Alexandrova et al. 2008) and solar wind (Chen et al. 2010a) turbulence in the kinetic range.

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The two possible modes in this regime for an isotropic Maxwellian plasma are the kinetic Alfvén wave,

$$\begin{eqnarray}&&{\omega }^{2}=\displaystyle \frac{{k}_{\parallel }^{2}{v}_{{\rm{A}}}^{2}{k}_{\perp }^{2}{\rho }_{{\rm{i}}}^{2}}{{\beta }_{{\rm{i}}}+2/(1+{T}_{{\rm{e}}}/{T}_{{\rm{i}}})},\end{eqnarray} \tag{ 1 }$$

and the oblique whistler wave,

$$\begin{eqnarray}&&{\omega }^{2}={k}_{\parallel }^{2}{v}_{{\rm{A}}}^{2}{k}_{\perp }^{2}{d}_{{\rm{i}}}^{2}.\end{eqnarray} \tag{ 2 }$$

To distinguish these, the correlation between $\delta n$ and $\delta {B}_{\parallel }$ can be used, which is negative for the kinetic Alfvén wave and positive for the whistler wave. Figure 3 shows the magnitude-squared wavelet coherence, γ, between $\delta {n}_{{\rm{e}}}$ and $\delta {B}_{\parallel }$, and the phase lag ϕ (black arrows) for $\gamma \gt 0.5$, measured by MMS3. To avoid complications with the definition of ${{\boldsymbol{B}}}_{0},\delta | {\boldsymbol{B}}|$ was used as a proxy for $\delta {B}_{\parallel }$, which requires $\delta {{\boldsymbol{B}}}^{2}/{B}_{0}^{2}\ll 2\delta {B}_{\parallel }/{B}_{0}$, a condition well-satisfied here. For spacecraft-frame frequencies $0.5\,\mathrm{Hz}\lesssim {f}_{\mathrm{sc}}\lesssim 5\,\mathrm{Hz}$, corresponding to $1\lesssim {k}_{\perp }{\rho }_{{\rm{i}}}\lesssim 10,\gamma \sim 1$ and there is a strong anti-correlation. The average phase lag in this range is $\langle \phi \rangle =(172\pm 7)^\circ$ (where the uncertainty is the standard deviation). For ${f}_{\mathrm{sc}}\gtrsim 10\,\mathrm{Hz}$, the anti-correlation is lost due to noise in the density measurement. This strong anti-correlation, along with the ${k}_{\perp }\gg {k}_{\parallel }$ anisotropy, indicates the predominantly kinetic Alfvén nature of the turbulence in the first decade of the kinetic range.

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Figure 4 shows the spectra of various quantities measured by MMS3, calculated using the multitaper method (Percival & Walden 1993). In the range $1/{\rho }_{{\rm{i}}}\lt k\lt 1/{d}_{{\rm{e}}}$, the trace magnetic fluctuation spectrum has a power-law index of −2.8 before steepening at electron scales, similar to solar wind observations (e.g., Alexandrova et al. 2009; Kiyani et al. 2009; Chen et al. 2010a, 2012; Sahraoui et al. 2013) and kinetic Alfvén turbulence simulations (e.g., Howes et al. 2011; Boldyrev & Perez 2012), and not far from the ${k}_{\perp }^{-8/3}$ prediction for intermittent kinetic Alfvén turbulence (Boldyrev & Perez 2012). The trace electric field spectrum, Lorentz transformed into the zero mean velocity frame (Chen et al. 2011), has a spectral index of −0.8, a factor of k² shallower than the magnetic spectrum, before also steepening at electron scales. The ratio $(| \delta {\boldsymbol{B}}| /| \delta {\boldsymbol{E}}| )/{v}_{{\rm{A}}}$ is around unity for $k{\rho }_{{\rm{i}}}\lt 1$ then displays linear scaling for $k{\rho }_{{\rm{i}}}\gt 1$, as also seen by Matteini et al. (2017). This is because kinetic Alfvén fluctuations, although electromagnetic in nature, have a significant potential component of the electric field; the ion motion satisfies $\omega \ll {k}_{\perp }{v}_{\mathrm{th},{\rm{i}}}$, so the density adjusts to the electric potential as $\delta n\propto \phi$, and since $\delta n\propto \delta B$, the electric field is given by $E(k)\approx -{ik}\phi (k)\propto {kB}(k)$. The electron velocity spectral index is −0.9, close to k² shallower than the magnetic field and similar to the electric field, since the electron velocity is dominated by the ${\boldsymbol{E}}\times {{\boldsymbol{B}}}_{0}$ drift. The ion velocity spectrum, however, is much steeper, similar to in the upstream solar wind (Šafránková et al. 2013, 2016), reaching the instrumental noise around $k{\rho }_{{\rm{i}}}\approx 3$. This is because the ions no longer participate in the same ${\boldsymbol{E}}\times {{\boldsymbol{B}}}_{0}$ drift as the electrons, since their gyroradius is much larger than the scales of the electric field fluctuations in this range. Figure 4(d) shows the normalized ratio of density and magnetic fluctuations,

$$\begin{eqnarray}&&\displaystyle \frac{\delta {\tilde{n}}^{2}}{\delta {\tilde{{\boldsymbol{b}}}}^{2}}=\displaystyle \frac{{\beta }_{{\rm{i}}}}{2}\left(1+\displaystyle \frac{{T}_{{\rm{e}}}}{{T}_{{\rm{i}}}}\right)\left[1+\displaystyle \frac{{\beta }_{{\rm{i}}}}{2}\left(1+\displaystyle \frac{{T}_{{\rm{e}}}}{{T}_{{\rm{i}}}}\right)\right]\displaystyle \frac{{(\delta n/{n}_{0})}^{2}}{{(\delta {\boldsymbol{B}}/{B}_{0})}^{2}},\end{eqnarray} \tag{ 3 }$$

which is $\delta {\tilde{n}}^{2}/\delta {\tilde{{\boldsymbol{b}}}}^{2}\sim 1$, confirming that the turbulence is predominantly low frequency, $\omega \ll {k}_{\perp }{v}_{\mathrm{th},{\rm{i}}}$, and kinetic Alfvén rather than whistler (Boldyrev et al. 2013; Chen et al. 2013b). The increase in $\delta {\tilde{n}}^{2}/\delta {\tilde{{\boldsymbol{b}}}}^{2}$ for ${f}_{\mathrm{sc}}\gtrsim 10\,\mathrm{Hz}$ is not physical, but due to the density spectrum reaching the noise level.

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While the first decade of the magnetosheath kinetic range, $1\lesssim k{\rho }_{{\rm{i}}}\lesssim 10$, described in Section 2.2, is similar to that in the upstream solar wind, the nature of the turbulence changes in the second decade. The kinetic Alfvén wave (Equation (1)) is derived assuming $\omega \ll {k}_{\parallel }{v}_{\mathrm{th},{\rm{e}}}$; however, it can be seen that (for ${T}_{{\rm{i}}}\gg {T}_{{\rm{e}}}$) this breaks down at the scale

$$\begin{eqnarray}&&{k}_{\perp }^{2}{\rho }_{{\rm{e}}}^{2}\sim {\beta }_{{\rm{i}}}(2+{\beta }_{{\rm{i}}}){({T}_{{\rm{e}}}/{T}_{{\rm{i}}})}^{2},\end{eqnarray} \tag{ 4 }$$

about halfway between ${k}_{\perp }{\rho }_{{\rm{i}}}\sim 1$ and ${k}_{\perp }{\rho }_{{\rm{e}}}\sim 1$ for the measured parameters. In the range ${k}_{\parallel }{v}_{\mathrm{th},{\rm{e}}}\ll \omega \ll {k}_{\perp }{v}_{\mathrm{th},{\rm{i}}}$, the kinetic Alfvén wave transforms into a mode with dispersion relation

$$\begin{eqnarray}&&{\omega }^{2}=\displaystyle \frac{{k}_{\parallel }^{2}{v}_{{\rm{A}}}^{2}{k}_{\perp }^{2}{\rho }_{{\rm{i}}}^{2}}{{\beta }_{{\rm{i}}}(1+{k}_{\perp }^{2}{d}_{{\rm{e}}}^{2})(1+2/{\beta }_{{\rm{i}}}+{k}_{\perp }^{2}{d}_{{\rm{e}}}^{2})},\end{eqnarray} \tag{ 5 }$$

which we call the inertial kinetic Alfvén wave (see Appendix A for the derivation). This should be distinguished from the standard inertial Alfvén wave derived under the conditions ${\beta }_{{\rm{e}}}\ll {m}_{{\rm{e}}}/{m}_{{\rm{i}}}$ and ${\beta }_{{\rm{i}}}\ll 1$ (e.g., Lysak & Lotko 1996), which are not satisfied here. Note also that the term "inertial kinetic Alfvén wave" has occasionally been applied to the standard inertial Alfvén wave (Shukla et al. 2009; Agarwal et al. 2011), although the conditions assumed in these works are essentially the same as in Lysak & Lotko (1996) and different from those leading to Equation (5).

The key observational feature of the transition to inertial kinetic Alfvén turbulence is the magnetic compressibility,

$$\begin{eqnarray}&&\displaystyle \frac{\delta {B}_{\parallel }^{2}}{\delta {B}_{\perp }^{2}}=\displaystyle \frac{1+{k}_{\perp }^{2}{d}_{{\rm{e}}}^{2}}{1+2/{\beta }_{{\rm{i}}}+{k}_{\perp }^{2}{d}_{{\rm{e}}}^{2}}.\end{eqnarray} \tag{ 6 }$$

For ${k}_{\perp }^{2}{d}_{{\rm{e}}}^{2}\ll 1,\delta {B}_{\parallel }^{2}/\delta {B}_{\perp }^{2}=1/(1+2/{\beta }_{{\rm{i}}})$, and for ${k}_{\perp }^{2}{d}_{{\rm{e}}}^{2}\,\gg 1+2/{\beta }_{{\rm{i}}},\delta {B}_{\parallel }^{2}/\delta {B}_{\perp }^{2}=1$, i.e., in general, the magnetic compressibility increases as energy cascades through ${k}_{\perp }{d}_{{\rm{e}}}\sim 1$. This is because for ${k}_{\perp }{d}_{{\rm{e}}}\lt 1$, the ion pressure reduces plasma compressibility, which due to pressure balance causes a ${\beta }_{{\rm{i}}}$-dependent reduction in $\delta {B}_{\parallel }$. For ${k}_{\perp }{d}_{{\rm{e}}}\gt 1$, however, the compressibility caused by the electron polarization drift becomes stronger than the compressibility due to the ${\boldsymbol{E}}\times {\boldsymbol{B}}$ drift, leading to $\delta {B}_{\parallel }^{2}/\delta {B}_{\perp }^{2}=1$.

Figure 5 shows the spectra of ${B}_{\perp }$ and ${B}_{\parallel }$, and the ratio of these, as measured by MMS3. As for Section 2.2, $\delta | {\boldsymbol{B}}|$ was used as a proxy for $\delta {B}_{\parallel }$, and $\delta {B}_{\perp }^{2}=\delta {{\boldsymbol{B}}}^{2}-\delta {B}_{\parallel }^{2}$. It can be seen that at $k{\rho }_{{\rm{i}}}\sim 1$, the magnetic compressibility is $\approx 1/(1+2/{\beta }_{{\rm{i}}})$, then this increases through ${{kd}}_{{\rm{e}}}\sim 1$, becoming $\approx 1$ by the time $k{\rho }_{{\rm{e}}}\sim 1$ is reached. In fact, the red line shows the measured compressibility to be following Equation (6) over the whole range between the ion and electron gyroscales. Note that the increase in compressibility is not due to noise; random fluctuations would produce equal power in all three components, resulting in $\delta {B}_{\parallel }^{2}/\delta {B}_{\perp }^{2}=0.5$ (which occurs only for ${f}_{\mathrm{sc}}\gt 200\,\mathrm{Hz}$, where the noise level is reached). It is also not due to parallel propagating whistler wave packets, which would appear as enhancements in the spectrum with a strong circular polarization (Matteini et al. 2017), but are not present in this interval. The observed magnetic compressibility, therefore, is consistent with a transition to inertial kinetic Alfvén turbulence for ${{kd}}_{{\rm{e}}}\gtrsim 1$.

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The use of the Taylor hypothesis to interpret Figures 2–5 in the spatial domain requires careful consideration. This assumes that the plasma-frame frequencies are small compared to the frequency of the convected spatial variations $| \omega | \ll | {\boldsymbol{k}}\cdot {\boldsymbol{v}}|$. If the fluctuations follow the kinetic Alfvén wave (Equation (1)) and inertial kinetic Alfvén wave (Equation (5)) dispersion relations, which the measurements of Sections 2.2 and 2.3 are consistent with, this condition (for ${T}_{{\rm{i}}}\gg {T}_{{\rm{e}}}$) becomes ${k}_{\parallel }{\rho }_{{\rm{i}}}\ll (v/{v}_{{\rm{A}}})\sqrt{{\beta }_{{\rm{i}}}(1+{k}_{\perp }^{2}{d}_{{\rm{e}}}^{2})(1+2/{\beta }_{{\rm{i}}}+{k}_{\perp }^{2}{d}_{{\rm{e}}}^{2})}$ (see also Howes et al. 2014). For ${k}_{\perp }^{2}{d}_{{\rm{e}}}^{2}\ll 1$, this reduces to ${k}_{\parallel }{\rho }_{{\rm{i}}}\ll (v/{v}_{{\rm{A}}})\sqrt{2+{\beta }_{{\rm{i}}}}\approx 1.3$. Due to the anisotropy ${k}_{\perp }\gg {k}_{\parallel }$, this can be well-satisfied, even for ${k}_{\perp }{\rho }_{{\rm{i}}}\gg 1$. For ${k}_{\perp }^{2}{d}_{{\rm{e}}}^{2}\gg 1+2/{\beta }_{{\rm{i}}}$ this reduces to ${k}_{\parallel }{\rho }_{{\rm{i}}}\ll (v/{v}_{{\rm{A}}})\sqrt{{\beta }_{{\rm{i}}}}{k}_{\perp }^{2}{d}_{{\rm{e}}}^{2}$, which can be written ${k}_{\parallel }/{k}_{\perp }\ll {k}_{\perp }v/{{\rm{\Omega }}}_{{\rm{e}}}$. Therefore, as long as ${k}_{\parallel }/{k}_{\perp }$ does not grow faster than ${k}_{\perp }$ (theoretical considerations suggest it actually grows more slowly; see Section 3.2), the Taylor condition would remain valid down to ${k}_{\perp }{\rho }_{{\rm{e}}}\sim 1$.

A recent numerical study (Klein et al. 2014) concluded that the Taylor hypothesis is indeed satisfied for kinetic Alfvén turbulence at $v/{v}_{{\rm{A}}}\sim 1$ and if it was violated significantly shallower spectra would result. The fact that the spectra in Figure 4 match those in the solar wind, as well as expectations from simulations and theory, is consistent with the interpretation that the measured fluctuations are spatial. As a more direct test, Figure 6 shows the normalized magnetic fluctuation amplitudes, as calculated from the second-order structure function, from both the single-spacecraft method (converting from the temporal to spatial domain assuming the Taylor hypothesis) and as direct spatial measurements from the six pairs of MMS spacecraft. There is some scatter in the multi-spacecraft measurements, due to the spacecraft separation vectors being at different angles to the mean field direction, but the average value of $| \delta {\boldsymbol{B}}| /{B}_{0}=0.018$ is similar to the single-spacecraft value of $| \delta {\boldsymbol{B}}| /{B}_{0}=0.019$, consistent with the Taylor hypothesis being valid down to ${{kd}}_{{\rm{e}}}\sim 1$.

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To understand the nonlinear properties of the cascade, we first derive the dynamical equations for inertial kinetic Alfvén turbulence. We consider strongly anisotropic (${k}_{\perp }\gg {k}_{\parallel }$) fluctuations, and use the ordering ${k}_{\parallel }/{k}_{\perp }\sim \delta B/{B}_{0}\sim \delta n/{n}_{0}\,\sim \omega /{{\rm{\Omega }}}_{{\rm{e}}}\ll 1$.

The starting equations are the electron continuity and momentum equations. In the electron continuity equation,

$$\begin{eqnarray}&&\displaystyle \frac{\partial n}{\partial t}+{{\rm{\nabla }}}_{\perp }\cdot (n{{\boldsymbol{v}}}_{\perp })+{{\rm{\nabla }}}_{\parallel }({{nv}}_{\parallel })=0,\end{eqnarray} \tag{ 7 }$$

the parallel electron velocity can be expressed through the parallel electron current, ${v}_{\parallel }=-{J}_{\parallel }/({ne})$, which can, in turn, be expressed through the z component of the magnetic vector potential, $\psi \equiv -{A}_{z}\approx -{A}_{\parallel }$, as ${J}_{\parallel }=(1/{\mu }_{0}){{\rm{\nabla }}}_{\perp }^{2}\psi$. Here, e is the magnitude of the electron charge. Due to the small amplitude of the magnetic fluctuations, $\delta B/{B}_{0}\ll 1$, the deviation of the magnetic field lines from the z direction (large-scale mean field) is small, so the parallel components of the vector fields can be approximated by their z components. This is, however, not true for the parallel gradients, since ${k}_{\parallel }\ll {k}_{\perp }$, which are given by

$$\begin{eqnarray}&&{{\rm{\nabla }}}_{\parallel }=\partial /\partial z+{B}_{0}^{-1}(\hat{z}\times {\rm{\nabla }}\psi )\cdot {\rm{\nabla }}.\end{eqnarray} \tag{ 8 }$$

The perpendicular electron velocity is more complicated. It has two parts, the ${\boldsymbol{E}}\times {\boldsymbol{B}}$ drift and the polarization drift,

$$\begin{eqnarray}&&{{\boldsymbol{v}}}_{\perp }={{\boldsymbol{v}}}_{E}-\displaystyle \frac{{m}_{{\rm{e}}}}{{{eB}}^{2}}\,{\boldsymbol{B}}\times \displaystyle \frac{{d}_{E}}{{dt}}{{\boldsymbol{v}}}_{E}.\end{eqnarray} \tag{ 9 }$$

In this expression, ${{\boldsymbol{v}}}_{E}=({\boldsymbol{E}}\times {\boldsymbol{B}})/{B}^{2}$ is the ${\boldsymbol{E}}\times {\boldsymbol{B}}$ drift velocity, the convective derivative is ${d}_{E}/{dt}\equiv \partial /\partial t+{{\boldsymbol{v}}}_{E}\cdot {\rm{\nabla }}$, and we note that ${B}^{2}\approx {B}_{0}^{2}+2{B}_{0}\delta {B}_{z}$. The second term on the right-hand side of Equation (9) is smaller than the first by $\omega /{{\rm{\Omega }}}_{{\rm{e}}}$; however, it needs to be retained since in the continuity equation the leading contribution from the first term cancels out. The electric field has both potential and non-potential components, ${\boldsymbol{E}}=-{\rm{\nabla }}\phi +{{\boldsymbol{E}}}_{\mathrm{np}}$. The non-potential component is small compared to the potential one as it includes the (small) time derivative of the (small) magnetic fluctuations,

$$\begin{eqnarray}&&{\rm{\nabla }}\times {{\boldsymbol{E}}}_{\mathrm{np}}=-\displaystyle \frac{\partial {\boldsymbol{\delta }}B}{\partial t},\end{eqnarray} \tag{ 10 }$$

although it also needs to be retained for the same reason. The fluctuations of the perpendicular magnetic field component are given by the z component of the magnetic vector potential, $\delta {{\boldsymbol{B}}}_{\perp }=\hat{z}\times {\rm{\nabla }}\psi$.

Substituting these expressions into Equation (7), and keeping the leading order terms, we obtain

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}\left(\displaystyle \frac{\delta n}{{n}_{0}}-\displaystyle \frac{\delta {B}_{z}}{{B}_{0}}+\displaystyle \frac{{m}_{{\rm{e}}}}{{{eB}}_{0}^{2}}{{\rm{\nabla }}}^{2}\phi \right)\\ &&\quad +\displaystyle \frac{1}{{B}_{0}}(\hat{z}\times {\rm{\nabla }}\phi )\cdot {\rm{\nabla }}\left(\displaystyle \frac{\delta n}{{n}_{0}}-\displaystyle \frac{\delta {B}_{z}}{{B}_{0}}+\displaystyle \frac{{m}_{{\rm{e}}}}{{{eB}}_{0}^{2}}{{\rm{\nabla }}}^{2}\phi \right)\\ &&\quad -\displaystyle \frac{1}{{\mu }_{0}{n}_{0}e}{{\rm{\nabla }}}_{\parallel }{{\rm{\nabla }}}_{\perp }^{2}\psi =0.\end{eqnarray} \tag{ 11 }$$

We now turn to the electron momentum equation. The parallel component of the equation is

$$\begin{eqnarray}&&\displaystyle \frac{\partial {v}_{\parallel }}{\partial t}+({{\boldsymbol{v}}}_{E}\cdot {\rm{\nabla }}){v}_{\parallel }=-\displaystyle \frac{e}{{m}_{{\rm{e}}}}{E}_{\parallel }-\displaystyle \frac{1}{{m}_{{\rm{e}}}n}{{\rm{\nabla }}}_{\parallel }{p}_{{\rm{e}}}.\end{eqnarray} \tag{ 12 }$$

Here, we have made use of ${k}_{\perp }\gg {k}_{\parallel }$ and ${v}_{\parallel }\sim {v}_{\perp }$. The latter condition can be checked a posteriori from the expressions for the parallel and perpendicular velocity components. The electric field is expressed through the scalar and vector potentials,

$$\begin{eqnarray}&&{E}_{\parallel }=-{{\rm{\nabla }}}_{\parallel }\phi -\partial {A}_{\parallel }/\partial t.\end{eqnarray} \tag{ 13 }$$

As can be checked, the electron pressure fluctuations can be neglected compared to the electric potential fluctuations, ϕ, for ${T}_{{\rm{e}}}\ll {T}_{{\rm{i}}}$. We then get the equation for the z component of the magnetic potential ψ,

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}(\psi -{d}_{{\rm{e}}}^{2}{{\rm{\nabla }}}_{\perp }^{2}\psi )-{d}_{{\rm{e}}}^{2}({{\boldsymbol{v}}}_{E}\cdot {\rm{\nabla }}){{\rm{\nabla }}}_{\perp }^{2}\psi ={{\rm{\nabla }}}_{\parallel }\phi .\end{eqnarray} \tag{ 14 }$$

To close Equations (11) and (14), we need to find two additional relations among the fluctuating fields. One of these can be found from the perpendicular component of the electron momentum equation. Neglecting the small electron pressure, the force balance in the perpendicular direction is

$$\begin{eqnarray}&&{{\boldsymbol{E}}}_{\perp }+{{\boldsymbol{v}}}_{\perp }\times {\boldsymbol{B}}=0.\end{eqnarray} \tag{ 15 }$$

Writing the perpendicular velocity as ${{\boldsymbol{v}}}_{\perp }=-{{\boldsymbol{J}}}_{\perp }/({ne})\,=-{({\rm{\nabla }}\times {\boldsymbol{B}})}_{\perp }/({\mu }_{0}{ne})$, the electric field is then given by

$$\begin{eqnarray}&&{{\boldsymbol{E}}}_{\perp }\approx [({\boldsymbol{B}}\cdot {\rm{\nabla }})\delta {{\boldsymbol{B}}}_{\perp }-{B}_{0}{{\rm{\nabla }}}_{\perp }\delta {B}_{z}]/({\mu }_{0}{ne}).\end{eqnarray} \tag{ 16 }$$

Since, as we will see later, $\delta {B}_{z}\sim \delta {B}_{\perp }$, the first term on the right-hand side of Equation (16) is small. Using ${k}_{\perp }\gg {k}_{\parallel }$, the relation between the electric potential and the z component of the fluctuating magnetic field is obtained,

$$\begin{eqnarray}&&\phi =\displaystyle \frac{{B}_{0}}{{\mu }_{0}{ne}}\delta {B}_{z}.\end{eqnarray} \tag{ 17 }$$

For the second relation, we use the fact that kinetic Alfvén fluctuations exist for $\omega \ll {{kv}}_{\mathrm{th},{\rm{i}}}$. In this case, the ion (and, by quasineutrality, electron) density fluctuations adjust to the electric potential according to the Boltzmann formula,

$$\begin{eqnarray}&&\phi =-\displaystyle \frac{{T}_{{\rm{i}}}}{e}\displaystyle \frac{\delta n}{{n}_{0}}.\end{eqnarray} \tag{ 18 }$$

It can be seen that Equations (17) and (18) agree with the force balance in the single-fluid momentum equation, $\delta {B}_{z}/{B}_{0}\,=-({\beta }_{{\rm{i}}}/2)(\delta n/{n}_{0})$. We can now eliminate the $\delta {B}_{z}$ and ϕ fluctuations from Equations (11) and (14) in favor of the density fluctuations. Using the dimensionless variables $t^{\prime} =t| {{\rm{\Omega }}}_{{\rm{e}}}|$, ${\boldsymbol{x}}^{\prime} ={\boldsymbol{x}}/{d}_{{\rm{e}}}$, $\psi ^{\prime} =\psi /({d}_{{\rm{e}}}{B}_{0})$, $\delta n^{\prime} =({\beta }_{{\rm{i}}}/2)(\delta n/{n}_{0})$, and omitting the prime signs, we obtain the nonlinear system of equations for inertial kinetic Alfvén turbulence:

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}(1-{{\rm{\nabla }}}_{\perp }^{2})\psi +[(\hat{z}\times {\rm{\nabla }}\delta n)\cdot {\rm{\nabla }}]{{\rm{\nabla }}}_{\perp }^{2}\psi =-{{\rm{\nabla }}}_{\parallel }\delta n,\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}\left(1+\displaystyle \frac{2}{{\beta }_{{\rm{i}}}}-{{\rm{\nabla }}}_{\perp }^{2}\right)\delta n+[(\hat{z}\times {\rm{\nabla }}\delta n)\cdot {\rm{\nabla }}]{{\rm{\nabla }}}_{\perp }^{2}\delta n={{\rm{\nabla }}}_{\parallel }{{\rm{\nabla }}}_{\perp }^{2}\psi .\end{eqnarray} \tag{ 20 }$$

The nonlinearities in these equations appear in the second terms on the left-hand sides, and in the parallel gradients on the right-hand sides,

$$\begin{eqnarray}&&{{\rm{\nabla }}}_{\parallel }=\partial /\partial z+(\hat{z}\times {\rm{\nabla }}\psi )\cdot {\rm{\nabla }}.\end{eqnarray} \tag{ 21 }$$

Without the nonlinear terms, these equations reproduce the inertial kinetic Alfvén wave dispersion relation (Equation (36)), and the relation between $\delta n$ and ψ for these modes,

$$\begin{eqnarray}&&\delta n=\sqrt{\displaystyle \frac{1+{k}_{\perp }^{2}}{1+2/{\beta }_{{\rm{i}}}+{k}_{\perp }^{2}}}{k}_{\perp }\psi .\end{eqnarray} \tag{ 22 }$$

The last equation agrees with Equation (6) since ${k}_{\perp }\psi$ represents the perpendicular magnetic fluctuations, and $\delta n=-\delta {B}_{z}$ in the above normalization.

From Equations (19) and (20), we can derive the spectrum of inertial kinetic Alfvén turbulence. In the absence of energy supply and dissipation, the equations conserve the energy

$$\begin{eqnarray}&&E=\int \left[\delta n\left(1+\displaystyle \frac{2}{{\beta }_{{\rm{i}}}}-{{\rm{\nabla }}}_{\perp }^{2}\right)\delta n-{{\rm{\nabla }}}_{\perp }^{2}\psi (1-{{\rm{\nabla }}}_{\perp }^{2})\psi \right]{d}^{3}{\boldsymbol{x}}.\end{eqnarray} \tag{ 23 }$$

In a turbulent state, both terms of E are of the same order. For scales ${k}_{\perp }^{2}\gg 1+2/{\beta }_{{\rm{i}}}$, this means that $\delta {n}_{\lambda }\sim {\psi }_{\lambda }/\lambda$, where $\delta {n}_{\lambda }$ and ${\psi }_{\lambda }$ denote the typical fluctuations of the fields at the scale λ across the background magnetic field. In the same limit, the nonlinearity is dominated by the terms on the left-hand sides of Equations (19) and (20), and the nonlinear time can be estimated as $\tau \sim {\lambda }^{2}/\delta {n}_{\lambda }$. Assuming a constant energy flux through scales, $\varepsilon \sim (\delta {n}_{\lambda }^{2}/{\lambda }^{2})/\tau \sim \delta {n}_{\lambda }^{3}/{\lambda }^{4}$, leads to the scaling of the density and magnetic fluctuations $\delta {n}_{\lambda }\sim {\psi }_{\lambda }/\lambda \sim {\varepsilon }^{1/3}{\lambda }^{4/3}$. Applying the Fourier transform in the plane perpendicular to the background magnetic field, the spectrum of density and perpendicular fluctuations is obtained, ${E}_{n,B}({k}_{\perp })\propto {k}_{\perp }^{-11/3}$. In the data interval discussed in Section 2, the scaling range between ${{kd}}_{{\rm{e}}}\sim 1$ and $k{\rho }_{{\rm{i}}}\sim 1$ is not large (a factor of 3.4), so a well-developed power-law spectrum is not present, but the measured spectral index can still be compared to the prediction in the limited range. Figure 5 shows the spectral index to be −3.6 for the $\delta {B}_{\perp }$ fluctuations between ${{kd}}_{{\rm{e}}}=1$ and $k{\rho }_{{\rm{e}}}=1$, which is not far from the predicted value of $-11/3$.

The anisotropy implied by the critical balance condition can also be determined. Balancing the linear and nonlinear terms in Equations (19) and (20), ${\psi }_{\lambda }/({l}_{\parallel }{\lambda }^{2})\sim \delta {n}_{\lambda }^{2}/{\lambda }^{4}$, we obtain the relation between the parallel and perpendicular scales ${l}_{\parallel }\sim {\lambda }^{5/3}$. In Fourier space this means that the turbulent energy is concentrated in the region ${k}_{\parallel }\lesssim {k}_{\perp }^{5/3}$, which becomes progressively broader in ${k}_{\parallel }$ and less anisotropic as ${k}_{\perp }$ increases. This suggests, therefore, that in contrast to standard Alfvén and kinetic Alfvén turbulence, the energy cascade in inertial kinetic Alfvén turbulence supplies energy more efficiently to ${k}_{\parallel }$ rather than ${k}_{\perp }$ modes. This anisotropy also implies that the Taylor condition becomes better satisfied as ${k}_{\perp }$ increases (see Section 2.4). The current data interval does not allow the scale-dependence of the anisotropy to be tested, but this could be done in future studies with larger data sets, and also tested in numerical simulations.

The condition derived in Section 3.2 that critically balanced inertial kinetic Alfvén turbulence becomes more isotropic toward smaller scales leads to the interesting possibility that the cascade may transition to inertial whistler turbulence if the anisotropy reduces sufficiently. This is because there is a maximum value of ${k}_{\parallel }/{k}_{\perp }$ at which inertial kinetic Alfvén waves can exist (see Appendix A); the broadening of the spectrum in the ${k}_{\parallel }$ direction would lead to the turbulence reaching $\omega \gt {{kv}}_{\mathrm{th},{\rm{i}}}$, which is the whistler frequency range.

For completeness, we give here the dynamical equations for inertial whistler turbulence. The basic Equations (11) and (14) hold for whistler turbulence as well; however, the additional condition (18) needs to be modified. Due to the high frequency of the whistlers, the density fluctuations do not follow the electric potential and remain small,

$$\begin{eqnarray}&&\displaystyle \frac{\delta n}{{n}_{0}}\ll -\displaystyle \frac{e\phi }{{T}_{{\rm{i}}}}.\end{eqnarray} \tag{ 24 }$$

We can thus neglect the density fluctuations in Equations (11) and (14), and use Equation (17) to remove the electric potential fluctuations. Using again the normalized variables, and also $\delta {B}_{z}^{\prime }=\delta {B}_{z}/{B}_{0}$, with the primes omitted, we obtain the nonlinear equations for inertial whistler turbulence:

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}(1-{{\rm{\nabla }}}_{\perp }^{2})\psi -[(\hat{z}\times {\rm{\nabla }}\delta {B}_{z})\cdot {\rm{\nabla }}]{{\rm{\nabla }}}_{\perp }^{2}\psi ={{\rm{\nabla }}}_{\parallel }\delta {B}_{z},\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}(1-{{\rm{\nabla }}}_{\perp }^{2})\delta {B}_{z}-[(\hat{z}\times {\rm{\nabla }}\delta {B}_{z})\cdot {\rm{\nabla }}]{{\rm{\nabla }}}_{\perp }^{2}\delta {B}_{z}=-{{\rm{\nabla }}}_{\parallel }{{\rm{\nabla }}}_{\perp }^{2}\psi .\end{eqnarray} \tag{ 26 }$$

Linearization of these equations leads to the inertial whistler wave dispersion relation, which (in unnormalized variables) is given by

$$\begin{eqnarray}&&{\omega }^{2}=\displaystyle \frac{{k}_{\parallel }^{2}{v}_{{\rm{A}}}^{2}{k}_{\perp }^{2}{\rho }_{{\rm{i}}}^{2}}{{\beta }_{{\rm{i}}}{(1+{k}_{\perp }^{2}{d}_{{\rm{e}}}^{2})}^{2}},\end{eqnarray} \tag{ 27 }$$

and is also derived in Appendix B.

Due to the structural similarity of the inertial kinetic Alfvén and inertial whistler equations, (19), (20) and (25), (26), the spectrum of magnetic fluctuations in inertial whistler turbulence is the same as that derived in Section 3.2, with the difference that the transition to the inertial regime occurs at ${k}_{\perp }^{2}\gt 1$ rather than ${k}_{\perp }^{2}\gt 1+2/{\beta }_{{\rm{i}}}$. This spectrum for inertial whistler turbulence has also been discussed previously (Biskamp et al. 1996, 1999; Meyrand & Galtier 2010; Andrés et al. 2014). The structural similarity between the equations means that the anisotropy implied by the critical balance condition (discussed in Section 3.2) is also the same, meaning that the increasing isotropization would continue if the inertial kinetic Alfvén cascade transitions to inertial whistler turbulence.

The main physical difference between inertial kinetic Alfvén and inertial whistler turbulence is the ion dynamics, as discussed above, which leads to negligibly small density fluctuations in inertial whistler turbulence (see Appendix B). The magnetic compressibility, however, is the same, with $\delta {B}_{\perp }^{2}/\delta {B}_{\parallel }^{2}=1$ for both types of turbulence at sub-electron-inertial scales. The measurements in Figure 5, therefore, allow for the possibility of a transition to inertial whistler turbulence between ${{kd}}_{{\rm{e}}}\sim 1$ and $k{\rho }_{{\rm{e}}}\sim 1$. Further measurements will be required to determine whether, and under what conditions, such a transition occurs.