3.1. Previous velocity shear-driven lower hybrid wave results

The first evidence of velocity shear-driven lower hybrid waves in a compressed gyro-scale current sheet was previously reported by DuBois et al. (Reference DuBois, Crabtree, Ganguli, Malaspina and Amatucci2022). The four MMS spacecraft traversed a gyro-scale current sheet of width ${L_{\textrm{cs}}}\sim 0.6{\rho _i}$ and crossed the magnetic reversal at approximately 5:27:07.02 UTC. DuBois et al. (Reference DuBois, Crabtree, Ganguli, Malaspina and Amatucci2022, figure 1) shows a full summary of the plasma parameters in LMN coordinates for this event.

At the B_L reversal (t = 7.02 s), there is a guide field (${B_M}\sim 10\ \textrm{nT}$), and the magnetic field normal to the current sheet (B_N) is flat and near-zero inside the current sheet. The electric field normal to the current sheet (E_N) is the perpendicular ambipolar electric field (figure 2(a) – solid, blue) and has a scale length ${\rho _e} < {L_E} < {\rho _i}$, calculated from the half-width at half-max of E_N. A spectrogram of the E_N burst data reveals lower hybrid fluctuations (figure 2, left vertical axis), which are intense around the peak of the ${V_{E \times B}}$ flow. The ion velocity (V_i) is significantly smaller than the electron velocity (V_e), which indicates that the ions experience negligible electric field due to gyro averaging (Ganguli et al. Reference Ganguli, Crabtree, Fletcher and Amatucci2020). The electron density (n_e) is nearly constant during the fluctuations (figure 2(a) – dashed, orange), which are localized around the field reversal region.

Non-gyrotropic distribution functions were observed in this event (Chen et al. Reference Chen, Wang, Le Contel, Rager, Hesse, Drake, Dorelli, Ng, Bessho and Graham2020). Figure 2(b–d) shows the electron distribution functions calculated for t = 6 s (figure 2(b), pink), t = 7.67 s (figure 2(c), green) and t = 9 s (figure 2(d), yellow). The vertical dotted lines in figure 2(a,e,f) note the times of distribution functions. Figure 2(c) shows that the agyrotropy is correlated to the sheared perpendicular ambipolar electric field, while in figure 2(b,d), the electron distribution function is observed to be significantly less agyrotropic where the shear of the ambipolar electric field is reduced. This indicates that the agyrotropy is proportional to the shear of the electric field. This observation is supported by a kinetic model (Ganguli et al. Reference Ganguli, Crabtree, Fletcher and Amatucci2020; DuBois et al. Reference DuBois, Crabtree, Ganguli, Malaspina and Amatucci2022), which establishes that the shear in the perpendicular ambipolar field is responsible for breaking the gyrotropy in the equilibrium distribution function, as opposed to the postulated wave heating (Chen et al. Reference Chen, Wang, Le Contel, Rager, Hesse, Drake, Dorelli, Ng, Bessho and Graham2020).

The perpendicular ambipolar electric field breaks gyrotropy of the electron distribution function through the asymmetry in temperature in the gyro plane that arises due to the gradient in E_N included in the parameter $\eta (x) = 1 + (\textrm{d}{V_{E \times B}}/\textrm{d}x)/\varOmega$, where the gradient of the $\boldsymbol{E} \times \boldsymbol{B}$ velocity, $\textrm{d}{V_{E \times B}}/\textrm{d}x$, is the velocity shear, and $\varOmega$ is the cyclotron frequency (Ganguli et al. Reference Ganguli, Crabtree, Fletcher and Amatucci2020). Increasing compression intensifies shear making $\textrm{d}{V_{E \times B}}/\textrm{d}x \to - \varOmega$. Consequently, as $\eta \to 0$, the renormalized gyro frequency $\textrm{(}\varOmega \to \sqrt \eta \varOmega )$ becomes vanishingly small and the gyro-radius becomes large representing the larger temperature, but the particles still gyrate around the magnetic field, albeit in non-circular orbits. For $\eta < 0$, the orbits are no longer simple gyration (Ganguli et al. Reference Ganguli, Lee and Palmadesso1988b) and the particles are effectively unmagnetized and accelerated by the electric field to high energies. This explains the origin of agyrotropy in the equilibrium distribution of compressed plasma layers and is notable because Scudder & Mozer (Reference Scudder and Mozer2005) found from data analysis that non-gyrotropic distribution functions could be generated by a quasi-static electric field, but the source of such an electric field was unknown. Theoretical arguments by Mahajan & Hazeltine (Reference Mahajan and Hazeltine2000) and Scudder & Daughton (Reference Scudder and Daughton2008) suggest that the non-gyrotropic distribution function may be an indicator for imminent magnetic reconnection. Our data analysis confirms that non-gyrotropic distribution functions arise due to the localized perpendicular ambipolar electric field by their strong correlation. Physically, this is because the equilibrium electron $\boldsymbol{E} \times \boldsymbol{B}$ velocity is strongly sheared (DuBois et al. Reference DuBois, Crabtree, Ganguli, Malaspina and Amatucci2022), which makes the equilibrium electron ${\eta _e} \to 0$ and the ion ${\eta _i} < 0$. This configuration is unstable to EIH waves (discussed in § 1.1), which can heat the electrons and relax the velocity gradient resulting in a larger value of ${\eta _e}$ in the saturated state. This demonstrates the critical role of the perpendicular ambipolar electric field, which may be considered as a surrogate for compression, and its importance to current sheet evolution. It can ultimately lead to reconnection (Scudder & Mozer Reference Scudder and Mozer2005) if the electric field is sufficiently strong.

A large $\boldsymbol{E} \times \boldsymbol{B}$ velocity shear (figure 2(e) – solid, blue) in $\hat{L}({V_{E \times B - L}})$ peaks with the wave activity. Here, ${V_{E \times B - L}}$ is nearly equal to the total electron flow (figure 2(e) – dashed, grey) in $\hat{L}$, ${V_{eL}}$, which indicates that the electrons are $\boldsymbol{E} \times \boldsymbol{B}$ drifting. It has been previously shown that velocity shear becomes important when ${\omega _s}/{\omega _{LH}} > 1$ (Romero & Ganguli Reference Romero and Ganguli1993) and for this event, ${\omega _s}/{\omega _{LH}} \approx 8$. Additionally, the diamagnetic drift velocity $[{\boldsymbol{V}_{\textrm{drift}}} = \boldsymbol{\nabla }{P_e} \times \boldsymbol{B}/({n_e}{B^2})]$ in $\hat{L}$ is small in the region of the wave localization because ${V_{eL}} \approx {V_{E \times B - L}}$. The observed lower hybrid waves are characterized as shear-driven EIH waves because the diamagnetic drift frequency (${\omega ^\ast } ={-} {k_ \bot }{\boldsymbol{V}_{\textrm{drift}}}$ in figure 2(f) – dashed, orange, right vertical axis) is significantly smaller than the shear frequency (${\omega _s} \sim {\boldsymbol{V}_{E \times B}}/{L_E}$ in figure 2(f) – solid, blue, right vertical axis). The wavenumber $({k_ \bot })$ in the calculation of the diamagnetic drift frequency can be estimated using a time shift method (Norgren et al. Reference Norgren, Vaivads, Khotyaintsev and André2012). This rules out the density gradient as the driving mechanism (Romero & Ganguli Reference Romero and Ganguli1993; DuBois et al. Reference DuBois, Thomas, Amatucci and Ganguli2014, Reference DuBois, Crabtree, Ganguli, Malaspina and Amatucci2022; Ganguli et al. Reference Ganguli, Crabtree, Fletcher and Amatucci2020). The observed waves also have a long wavelength with ${k_ \bot }{\rho _e} \approx 0.18$ and ${k_ \bot }{L_E} \approx 1$ indicating they are shear-driven lower hybrid fluctuations.

The current density, calculated from the moments data $[\boldsymbol{J} = e({n_i}{\boldsymbol{V}_i} - {n_e}{\boldsymbol{V}_e})]$, shows the electron current density $({J_e})$ is dominant (figure 3(a) – solid, orange) because ${V_i} \ll {V_e}$ and it is comparable to the cross-field current, ${J_{\textrm{cf}}} = en(\boldsymbol{E} \times \boldsymbol{B})$ (figure 3 – dot, purple). The large peak in J_eL (figure 3(a) – solid, orange) is localized to the region of E_N and the shear-driven lower hybrid waves, and the scale size is less than ${\rho _i}$. This indicates the formation of a gyro-scale current sheet. This is notable because previous studies (Hwang et al. Reference Hwang, Goldstein, Moore, Walsh, Baishev, Moiseyev, Shevtsov and Yumoto2014; Xu et al. Reference Xu, Fu, Norgren, Hwang and Liu2018) have shown a connection between current sheets with substructures in J and the onset of magnetic reconnection.

Figure 3. Current densities (a) J_L, (b) J_M and (c) J_N. The electron (orange, solid) and ion (blue, dashed) current densities are calculated from moments data. The total current density (red, dash-dot) is the sum of the electron and ion components. The cross-field current (purple, dot) is shown for comparison.

These previous results demonstrated that a localized perpendicular ambipolar electric field develops in a gyro-scale current sheet with scale size ${L_{\textrm{CS}}} < {\rho _i}$ and results in a sheared $\boldsymbol{E} \times \boldsymbol{B}$ velocity that drives lower hybrid fluctuations. Not only does this indicate a new mechanism to generate intense electrostatic turbulence near the magnetic field reversal, but these shear-driven lower hybrid waves can be used as a diagnostic to indicate if the system is in a path to magnetic reconnection since they imply the existence of a sheared electron flow and a non-gyrotropic electron distribution function, both of which may be prerequisites for magnetic reconnection (Mahajan & Hazeltine Reference Mahajan and Hazeltine2000; Scudder & Mozer Reference Scudder and Mozer2005; Scudder & Daughton Reference Scudder and Daughton2008). The shear-driven lower hybrid waves are also ideally localized in the current sheet centre and can contribute to anomalous dissipation, indicating they may play a significant role in magnetic reconnection (Romero & Ganguli Reference Romero and Ganguli1993; Ganguli et al. Reference Ganguli, Keskinen, Romero, Heelis, Moore and Pollock1994; Walker et al. Reference Walker, Amatucci, Ganguli, Antoniades, Bowles and Koepke1997; Amatucci et al. Reference Amatucci, Walker, Ganguli, Duncan, Antoniades, Bowles, Gavrishchaka and Koepke1998; Matsubara & Tanikawa Reference Matsubara and Tanikawa2000; Kumar, Mattoo & Jha Reference Kumar, Mattoo and Jha2002).

3.2. Impact on magnetic reconnection

One of the essential questions in reconnection physics is what effect breaks the frozen-in condition (where magnetic field lines move with the plasma)? One way for the frozen-in condition to be broken is through anomalous processes caused by the interaction of wave turbulence with electrons. Anomalous terms can be added to the generalized Ohm's Law,

(3.1)\begin{equation}\boldsymbol{E} + {\boldsymbol{V}_{\boldsymbol{e}}} \times \boldsymbol{B} = {\boldsymbol{\eta }_{\boldsymbol{R}}}\boldsymbol{\cdot }\boldsymbol{J} + \boldsymbol{T} - \frac{1}{{e{n_e}}}\boldsymbol{\nabla }\boldsymbol{\cdot }{\bar{\boldsymbol{P}}_e} + \frac{{{m_e}}}{e}{\boldsymbol{V}_{\boldsymbol{e}}}\boldsymbol{\cdot }\boldsymbol{\nabla }{\boldsymbol{V}_{\boldsymbol{e}}},\end{equation}

where the first term on the right side of the equation $({\boldsymbol{\eta }_{\boldsymbol{R}}}\boldsymbol{\cdot }\boldsymbol{J})$ is the anomalous resistivity (drag) due to the interaction of wave turbulence and electrons, the second term $(\boldsymbol{T})$ is the anomalous viscosity (momentum transport), the third term $(\boldsymbol{\nabla }\boldsymbol{\cdot }{\bar{\boldsymbol{P}}_e})$ is the electron pressure tensor, and the last term $({\boldsymbol{V}_{\boldsymbol{e}}}\boldsymbol{\cdot }\boldsymbol{\nabla }{\boldsymbol{V}_{\boldsymbol{e}}})$ is the electron inertia. The anomalous terms can compete with other effects during the reconnection process. Due to measurement uncertainties, it is difficult to determine the dominance of any term unambiguously. For example, the magnitude of the anomalous terms depends on the wave amplitudes and is likely to change throughout the current sheet evolution due to the magnitude of compression. The time evolution of the pressure tensor term may also vary with current sheet evolution. Therefore, it is reasonable to account for all effects in determining the generalized Ohm's Law balance unless an unambiguous determination can be made for the dominance of a single process. In this section, we focus on estimating the anomalous, off-diagonal pressure tensor and electron inertia terms directly from MMS data in a gyro-scale current.

It was previously thought that density gradient driven lower hybrid drift waves could provide the anomalous resistivity necessary to trigger magnetic reconnection (Davidson & Gladd Reference Davidson and Gladd1975; Silin et al. Reference Silin, Büchner and Vaivads2005). However, magnetotail observations and theoretical calculations now suggest that lower hybrid drift waves are unlikely to be the source because the waves are localized far from the magnetic field reversal region (Huba & Ganguli Reference Huba and Ganguli1983) where magnetic reconnection is likely to be initiated, and the centre of current sheets are characterized by the lack of a density gradient (Runov et al. Reference Runov, Sergeev, Nakamura, Baumjohann, Apatenkov, Asano, Takada, Volwerk, Vörös and Zhang2006), which is necessary to drive lower hybrid drift waves. Drake, Gladd & Huba (Reference Drake, Gladd and Huba1981) argued that the lower hybrid drift instability may be supported at the centre due to a strongly varying magnetic field; but the model does not consider a guide field and the strong ambipolar electric field, which will also exist and intensify due to compression and substantially affect particle dynamics. One-dimensional PIC simulations by Birn, Schindler & Hesse (Reference Birn, Schindler and Hesse2004) that were initialized with a Harris equilibrium but were compressed by inflowing boundary conditions show the self-consistent formation of a transverse electric field, consistent with our kinetic model. In figure 2(e), we show that the net cross-field electron drift is close to the electron $\boldsymbol{E} \times \boldsymbol{B}$ drift, implying that the diamagnetic drift that is necessary to drive the lower hybrid drift instability is almost non-existent in the data. Two-dimensional simulations (Daughton Reference Daughton2003; Silin et al. Reference Silin, Büchner and Vaivads2005) showed that it may be possible for an electromagnetic lower hybrid drift wave to radiate inward to the centre if the current sheet is sufficiently thin. However, these simulations do not consider a realistic 3-D geometry in which the wave propagation characteristics may not convect the energy towards the centre. We find (DuBois et al. Reference DuBois, Crabtree, Ganguli, Malaspina and Amatucci2022) that the shear-driven EIH waves, which are ideally generated in the centre of the current sheet where magnetic reconnection is likely to be initiated, need no supplementary arguments to justify their presence in the centre of the current sheet. In addition, their predominant electrostatic nature ensures that they remain localized to the generation region to contribute robustly to anomalous dissipation.

Laboratory experiments (Matsubara & Tanikawa Reference Matsubara and Tanikawa2000; Kumar et al. Reference Kumar, Mattoo and Jha2002) have shown that the EIH instability can contribute to anomalous dissipation. The EIH anomalous resistivity estimated from PIC simulations by Romero & Ganguli (Reference Romero and Ganguli1993) showed that the sheared cross-field electron current is greatly reduced due to the growth of the EIH instability, indicating the presence of both anomalous resistivity and viscosity. It was also shown that the resistivity was proportional to the shear frequency, which is proportional to the global compression, thus implying that increasing compression may result in a larger anomalous resistivity and a faster reconnection rate. Additionally, theoretical arguments (Mahajan & Hazeltine Reference Mahajan and Hazeltine2000; Scudder & Daughton Reference Scudder and Daughton2008) suggest that non-gyrotropic distribution functions are an indicator of the possibility for magnetic reconnection to occur, and the source of non-gyrotropic distribution functions is the localized ambipolar electric field (§ 3.1).

The anomalous resistivity $({\boldsymbol{\eta }_R})$ due to the shear-driven EIH waves can be estimated directly from MMS data,

(3.2)\begin{equation}{\boldsymbol{\eta }_R} = 4\mathrm{\pi }{\boldsymbol{\nu }_\alpha }/\omega _{pe}^2,\end{equation}

where, ${\boldsymbol{\nu }_\alpha }$ is the anomalous collision frequency,

(3.3)\begin{equation}{\boldsymbol{\nu }_\alpha } = e\langle \delta {n_e}\delta \boldsymbol{E}\rangle /\langle {m_e}{n_e}{\boldsymbol{V}_e}\rangle .\end{equation}

The electron density fluctuation quantity $(\delta {n_e})$ is calculated by taking the raw FPI qmoments burst data and subtracting the quasi-static profile (indicated by $\langle {n_e}\rangle$). The qmoments data products (Rager et al. Reference Rager, Dorelli, Gershman, Uritsky, Avanov, Torbert, Burch, Ergun, Egedal and Schiff2018) are computed at one quarter of the 30 ms resolution time of the FPI moments burst data and have sufficient resolution for $\delta {n_e}$ determination. The electric field fluctuation quantity $(\delta \boldsymbol{E})$ can be similarly calculated using the electric field burst data. The anomalous resistivity in $\hat{M}({\eta _{RM}})$ is the resistivity in the reconnection direction of the current sheet. Figure 4 shows ${\eta _{RM}}$ calculated from $\delta {E_M}$ and $\delta {n_e}$ using the methodology described in § 2, overlaid on the spectrogram of shear-driven lower hybrid waves. The anomalous resistivity is relatively constant in the time prior to the fluctuations, and then begins to quickly increase during the time of the shear-driven EIH waves. The anomalous resistivity due to the interaction between shear-driven lower hybrid wave turbulence and electrons is estimated to have a peak value of ${\eta _{RM}} = 4 \pm 2.9\ \mathrm{k\Omega }\ \textrm{m}$. This is similar to what Silin et al. (Reference Silin, Büchner and Vaivads2005) estimated using spacecraft potential to infer the electron density for lower hybrid fluctuations observed in Cluster data. The anomalous resistivity term in the generalized Ohm's Law can be calculated using ${\eta _{RM}}$ and the current density, J_M (figure 3(b) – dash-dot, red). Figure 5(a) shows that the anomalous resistivity term (solid, white) is nearly 0 prior to the shear-driven lower hybrid waves and then, during the time of the waves, begins to decrease and has a minimum value of ${({\eta _R}J)_M} ={-} 0.18 \pm 0.12\ \textrm{mV}\ {\textrm{m}^{ - 1}}$.

Figure 4. Anomalous resistivity in $\hat{M}$ (white) overlaid on spectrogram of lower hybrid fluctuations.

Figure 5. (a) Anomalous resistivity term ${({\eta _R}J)_M}$ (solid, white) and anomalous viscosity term T (yellow, dashed) estimated from data. (b) Cross-field diffusion coefficient (solid, white). (c) Electron inertial term (solid, white). Plots are overlaid on the spectrogram of lower hybrid fluctuations (left vertical axis).

It has also been shown that the anomalous viscosity $(\boldsymbol{T})$ associated with lower hybrid waves can be calculated directly from MMS data (Graham et al. Reference Graham, Khotyaintsev, André, Vaivads, Divin, Drake, Norgren, Le Contel, Lindqvist and Rager2022). To estimate the anomalous viscosity in $\hat{M}$, we use

(3.4)\begin{align} {T_M} & = \langle \delta {V_L}\delta {B_N}\rangle - \langle \delta {V_N}\delta {B_L}\rangle\nonumber\\ & \quad + \left[ {\frac{{\langle \delta {n_e}\delta {V_L}\rangle \langle {B_N}\rangle + \langle \delta {n_e}\delta {B_N}\rangle \langle {V_L}\rangle - \langle \delta {n_e}\delta {V_N}\rangle \langle {B_L}\rangle - \langle \delta {n_e}\delta {B_L}\rangle \langle {V_N}\rangle }}{{\langle {n_e}\rangle }}} \right],\end{align}

where $\delta V$ is the fluctuation quantity of the electron flow (subscript indicates direction in LMN), $\delta B$ is the fluctuation quantity of the magnetic field and $\langle \delta V\delta B\rangle$ is calculated by multiplying the fluctuation quantities together, followed by calculating the boxcar average. Figure 5(a) shows the anomalous viscosity in $\hat{M}$ estimated directly from MMS data (dashed, yellow). The anomalous viscosity begins to increase during the time of the shear-driven lower hybrid waves, remains localized to the region of the waves and has a peak value of ${T_M} = 0.43 \pm 0.28\ \textrm{mV}\ {\textrm{m}^{ - 1}}$, and is in the opposite direction of the anomalous resistivity.

The cross-field diffusion coefficient $({D_ \bot })$, which is related to anomalous viscosity, can be estimated directly from MMS data using

(3.5)\begin{equation}{D_ \bot } ={-} \frac{{\langle \delta {n_e}\delta {V_N}\rangle }}{{\boldsymbol{\nabla }{{\langle {n_e}\rangle }_N}}},\end{equation}

where the density gradient is taken in $\hat{N}$ or the direction of motion of the spacecraft across the current sheet. The cross-field diffusion coefficient is shown in figure 5(b), where there is a large increase as soon as shear-driven lower hybrid waves are observed and then quickly decreases back to near-zero as the shear-driven waves disappear. This sudden increase in the cross-field diffusion indicates anomalous electron drift and diffusion across the current sheet, which can modify the reconnection process. This was similarly observed in laboratory experiments (Matsubara & Tanikawa Reference Matsubara and Tanikawa2000).

The electron inertia term in $\hat{M}({I_{eM}})$ can also be calculated directly from the MMS data (figure 5c) using

(3.6)\begin{equation}{I_{eM}} = \frac{{{m_e}}}{e}({\boldsymbol{V}_{\boldsymbol{e}}}\boldsymbol{\cdot }\boldsymbol{\nabla }){\boldsymbol{V}_{\boldsymbol{e}}} = \frac{{{m_e}{V_{eN}}}}{e}\frac{{\textrm{d}{V_{eM}}}}{{\textrm{d}N}},\end{equation}

where, again, the gradient is taken in $\hat{N}$ because this is the direction of motion of the spacecraft across the current sheet, and V_eN and V_eM are boxcar averaged (quasi-static) electron velocities. The electron inertial term peaks at $- 0.7 \pm 0.15\ \textrm{mV}\ {\textrm{m}^{ - 1}}$ during the time of the intense ambipolar electric field and is comparable to the other terms in the generalized Ohm's Law. As we have already shown, the majority of the electron flow consists of the $\boldsymbol{E} \times \boldsymbol{B}$ velocity. Since the $\boldsymbol{E} \times \boldsymbol{B}$ velocity also has a component in $\hat{M}$, the quasi-static electron velocities are related to the quasi-static ambipolar electric field. This shows how the ambipolar effect can enhance the inertial term, which is typically considered to be less important and usually ignored in estimates of the generalized Ohm's Law balance.

A previous study (Egedal et al. Reference Egedal, Ng, Le, Daughton, Wetherton, Dorelli, Gershman and Rager2019) of MMS data has shown that the off-diagonal pressure tensor can be responsible for breaking the frozen-in condition, but significant lower hybrid fluctuations were not detected in the reconnecting current sheet. Another study (Torbert et al. Reference Torbert, Burch, Giles, Gershman, Pollock, Dorelli, Avanov, Argall, Shuster and Strangeway2016a) has shown that the pressure and inertial terms may not actually be sufficient to balance the generalized Ohm's Law. We calculate the off-diagonal pressure tensor terms in $\hat{M}$ from MMS moments data using

(3.7)\begin{equation}\frac{1}{{e{n_e}}}{(\boldsymbol{\nabla }\boldsymbol{\cdot }{\bar{\boldsymbol{P}}_{\boldsymbol{e}}})_M} = \frac{1}{{e{n_e}}}\left( {\frac{{\textrm{d}{P_{LM}}}}{{\textrm{d}L}} + \frac{{\textrm{d}{P_{MN}}}}{{\textrm{d}N}}} \right).\end{equation}

The pressure tensor terms depend on gradients in $\hat{L}$ and $\hat{N}$, assuming $\partial /\partial M\sim 0$ (Egedal et al. Reference Egedal, Ng, Le, Daughton, Wetherton, Dorelli, Gershman and Rager2019). Figures 6(a) and 6(b) show the pressure terms, P_LM and P_MN, respectively, for each spacecraft. The signals have been shifted in time using the method described in § 2. There are several methods that can be used to calculate the gradients of the two terms in (3.7). First, if the spacecraft are moving across the current sheet in the same direction of the gradient, then one can take the direct gradient of the pressure by converting time to distance using the current sheet velocity. If there is a steady increase or decrease in the signals with no variations between the spacecraft signals, and the spacecraft motion is in the same direction as the gradient, then the gradient can also be calculated using $(\max - \min )/({x_{\max }} - {x_{\min }})$, where ${x_{\max }}$ is the position of the spacecraft in the gradient direction when the signal is the maximum point and ${x_{\min }}$ is the position of the spacecraft in the gradient direction when the signal is at the minimum point. A more straightforward method depends on if there are variations in the signals for each spacecraft and can be used regardless of whether the spacecraft direction and the gradient direction are the same.

Figure 6. Off-diagonal pressure tensor terms (a) P_LM and (b) P_MN measured by MMS1 (solid, black), MMS2 (dashed, yellow) and MMS3 (dash-dot, green). (c) Resulting pressure tensor gradient estimated from P_MN data (solid, white) overlaid on the spectrogram of shear-driven lower hybrid fluctuations.

For our case of interest, the spacecraft are moving across the current sheet in $\hat{N}$, making the second term in (3.7) straightforward to calculate using the third method. Figure 6(b) illustrates that during the time of the shear-driven lower hybrid waves, the MMS1 (solid, black) and MMS2 (dashed, yellow) measure relatively the same signal, while MMS3 (dash dot, green) is noticeably different. Therefore, the gradient can be calculated using $({P_{MN1}} - {P_{MN3}})/({N_1} - {N_3})$. This calculation can likewise be completed using MMS2 and MMS3 data, and then the average can be taken. The first term in (3.7) requires taking the gradient with respect to $\hat{L}$, which means only the third method is applicable to this calculation since spacecraft motion across the current sheet is in $\hat{N}$. Looking at figure 6(a), the P_LM data have no variations between spacecraft during the region of interest (between 7 and 8 seconds). Therefore, the gradient of the P_LM term in (3.7) cannot be estimated directly from the data. The pressure tensor gradient is therefore estimated using only the P_MN term (figure 6c). There is a peak of approximately $- 2.8 \pm 4.5\ \textrm{mV}\ {\textrm{m}^{ - 1}}$ in the pressure tensor gradient during the time of the shear-driven lower hybrid fluctuations.

Interestingly, the related off-diagonal terms in the pressure tensor can be generated due to compression in a 2-D magnetic field, as was shown in studies of dipolarization fronts by Ganguli et al. (Reference Ganguli, Crabtree, Fletcher, Tejero, Malaspina and Cohen2018, Reference Ganguli, Crabtree, Fletcher and Amatucci2020). In a 2-D magnetic field, compression creates the 2-D ambipolar potential, $\varPhi \textrm{(}x,z\textrm{)}$, which generates ${E_\parallel } \propto \partial \varPhi /\partial z$ along the magnetic field line. Pressure anisotropy must arise to balance the ${E_\parallel }$ in a collisionless plasma. This indicates that the off-diagonal pressure tensor terms may be related to the ambipolar electric field through compression.