Finite-Larmor-radius equilibrium and currents of the Earth’s flank magnetopause

Abstract

We consider the one-dimensional equilibrium problem of a shear-flow boundary layer within an ‘extended-fluid model’ of a plasma that includes the Hall and the electron pressure terms in Ohm’s law, as well as dynamic equations for anisotropic pressure for each species and first-order finite-Larmor-radius (FLR) corrections to the ion dynamics. We provide a generalized version of the analytic expressions for the equilibrium configuration given in Cerri et al., (Phys. Plasmas, vol. 20 (11), 2013, 112112), highlighting their intrinsic asymmetry due to the relative orientation of the magnetic field $\boldsymbol{B}$, $\boldsymbol{b}=\boldsymbol{B}/|\boldsymbol{B}|$, and the fluid vorticity $\unicode[STIX]{x1D74E}=\unicode[STIX]{x1D735}\times \boldsymbol{u}$ (‘$\unicode[STIX]{x1D74E}\boldsymbol{b}$ asymmetry’). Finally, we show that FLR effects can modify the Chapman–Ferraro current layer at the flank magnetopause in a way that is consistent with the observed structure reported by Haaland et al., (J. Geophys. Res. (Space Phys.), vol. 119, 2014, pp. 9019–9037). In particular, we are able to qualitatively reproduce the following key features: (i) the dusk–dawn asymmetry of the current layer, (ii) a double-peak feature in the current profiles and (iii) adjacent current sheets having thicknesses of several ion Larmor radii and with different current directions.

1 Introduction

A comprehensive modelling of magnetized plasmas and of their multi-scale dynamics is an outstanding challenge in laboratory, astrophysical and space plasma research. In particular, given that direct numerical simulations are nowadays the main tool to address such complex dynamics, finding a compromise between an exhaustive theoretical model and its actual implementation represents a major goal for computational plasma physics.

A kinetic model based on the full Vlasov–Maxwell system of equations would need to be solved in a six-dimensional phase space (three real-space and three velocity-space dimensions), resolving length and time scales that typically span over several orders of magnitude. For this reason, fully kinetic simulations that adopt realistic parameters and/or complex geometries are still far from being realizable because of their colossal computational cost. Moreover, there is overwhelming difficulty in constructing an analytical description of Vlasov equilibria in realistic settings. In fact, the few existing examples typically consider very simplified cases (e.g. uniform and homogeneous magnetic field and/or only periodic functions) and still one cannot fully constrain the resulting velocity profiles beforehand and/or provide those equilibria without appealing to a numerical solution of the problem (see, e.g. Cai, Storey & Neubert 1990; Attico & Pegoraro 1999; Mahajan & Hazeltine 2000; Bobrova et al. 2001; Malara, Pezzi & Valentini 2018).

On the other hand, a model based on a fluid treatment such as the magnetohydrodynamic (MHD) equations neglects most of the characteristic length and time scales inherent to a kinetic description of the plasma dynamics and only needs to be solved in real space. The MHD description thus represents the simplest viable approach, which nevertheless has led to many fundamental theoretical results (e.g. Chapman & Ferraro 1930; Ferraro 1937; Alfvén 1942; Lüst & Schlüter 1954; Chandrasekhar 1956; Shafranov 1958; Grad 1960; Taylor 1974). Furthermore, in the last two decades, we have been able to afford well-resolved MHD global simulations providing useful insights (e.g. Groth et al. 2000; Siscoe et al. 2000; Jia et al. 2012, 2015; Merkin, Lyon & Claudepierre 2013; Liu et al. 2015; Dong et al. 2017; Sorathia et al. 2017). However, in a real system, the nonlinear plasma dynamics would naturally develop small scales and bring the effects associated with the neglected kinetic scales back to light, and so a MHD description eventually breaks down. Moreover, accounting for the leading kinetic effects may already be necessary to implement a correct initial plasma equilibrium, in order to avoid uncontrolled and spurious readjustments that can affect the subsequent dynamics or to explain certain features of the system under consideration (e.g. Cerri et al. 2013; Henri et al. 2013).

The fully kinetic and MHD descriptions actually represent the two extremes of a wide variety of plasma models. There are a large number of approaches that try to bridge the above antipodes in different ways: from the one side, by simplifying a fully kinetic description based on the dismissal of presumably unimportant effects; from the opposite side, by gradually including more and more kinetic effects within a fluid framework. The former class of models are usually referred to as ‘reduced-kinetic models’, such as the gyrokinetic (GK) (Brizard & Hahm 2007) and the hybrid Vlasov–Maxwell (HVM) (Valentini et al. 2007) approximations; the latter are known as ‘extended-fluid models’, in which kinetic effects are gradually included in a fluid description. This is the case, for instance, when retaining finite-Larmor-radius (FLR) corrections (Roberts & Taylor 1962; Macmahon 1965), or when including the effect of linear Landau damping (Landau 1946) by modelling it with a so-called Landau-fluid (LF) closure (e.g. Hammett & Perkins 1990). These two aspects can also be both included within a single framework, such as in the so-called finite-Larmor-radius Landau-fluid (FLRLF) model (Sulem & Passot 2015). However, within the range of validity defined by each model’s assumption (‘ordering’), reduced-kinetic models still unavoidably face the curse of high dimensionality, and so extended-fluid models still represent an attractive choice when seeking a compromise between kinetic and fluid descriptions.

The need to extend a standard fluid description of a collisionless plasma to include at least these effects related to a non-gyrotropic pressure tensor is particularly evident when a sheared flow is present: in the collisionless regime, due to FLR effects, the pressure tensor is indeed strongly coupled to the shear flow and they interact over very short time scales (Cerri 2012; Cerri et al. 2013, 2014; Del Sarto, Pegoraro & Califano 2016; Del Sarto, Pegoraro & Tenerani 2017; Del Sarto & Pegoraro 2018). This is exactly the case of the low-latitude boundary layer (LLBL) between the solar-wind flow and the Earth’s magnetosphere, where the velocity shear drives the Kelvin–Helmholtz instability (KHI) that generates the observed large-scale ‘MHD’ vortices (see, e.g. Faganello & Califano 2017 and references therein). In such a region, in addition to the vortex dynamics that naturally develops fluctuations on length scales comparable to (or even smaller than) the ion gyroradius $\unicode[STIX]{x1D71A}_{i}$ (or the ion inertial length $d_{i}$ ), the ‘large-scale’ equilibrium fields and the sheared flow itself vary over typical length scales $L_{0}$ that do not exceed the ion characteristic scales by a large amount, and so ‘ $\unicode[STIX]{x1D71A}_{i}/L_{0}$ corrections’ cannot be completely neglected. So far, such a system has been modelled by means of one-dimensional isotropic MHD equilibrium configurations that ensure the total pressure balance, i.e. a balance between the thermal and magnetic scalar pressures of the two plasmas without involving the properties of the background sheared flow. However, as soon as FLR effects and/or the full ion pressure tensor are taken into account, the shear-flow properties enter the pressure-balance conditions and the simple isotropic MHD configurations are generally no longer an equilibrium (Cerri 2012; Cerri et al. 2013, 2014). As a result, the system naturally develops shear-driven anisotropies (e.g. De Camillis et al. 2016; Del Sarto et al. 2016; Del Sarto & Pegoraro 2018). This is important for (at least) two practical reasons. First, a difficulty arises when comparing the linear evolution of the KHI using fluid and kinetic models. As discussed in Henri et al. (2013), in which the same isotropic MHD configuration was adopted as an initial condition for simulations using different plasma models (namely, MHD, two-fluid, hybrid and full Particle-In-Cell (PIC)), it was found that violent and uncontrolled readjustments were injecting large-amplitude fluctuations into the system (see also Del Sarto et al. 2017) and changing the configuration on top of which the instability develops (see also Nakamura, Hasegawa & Shinohara 2010). Therefore, these spurious effects would partially mask the actual kinetic effects on the KHI and make a genuine comparison difficult. Secondly, using ten years of observations made by the Cluster satellites, Haaland et al. (2014) have recently highlighted that the Earth’s magnetopause exhibits a current structure that is more complex than the simple MHD layer described by Chapman & Ferraro (1930), as well as a clear asymmetry between the dusk and the dawn sides. In addition to the implications for the current system of a planet magnetosphere, these ion-kinetic effects can indeed cause the asymmetric development of KHI at the dawn and the dusk sides of such magnetosphere, as well as other non-ideal effects (e.g. Nagano 1978; Huba 1996; Terada, Machida & Shinagawa 2002; Nakamura et al. 2010; Henri et al. 2012; Masters et al. 2012; Sundberg et al. 2012; Taylor et al. 2012; Delamere et al. 2013; Paral & Rankin 2013; Haaland et al. 2014; Johnson, Wing & Delamere 2014; Liljeblad et al. 2014; Walsh et al. 2014; Gershman et al. 2015; Gingell, Sundberg & Burgess 2015; De Camillis et al. 2016).

The aim of the present work is to show how the non-ideal behaviour of the Chapman–Ferraro layer could be qualitatively understood in terms of a one-dimensional equilibrium of the shear-flow layer within an extended-fluid model that includes first-order ion-FLR corrections. The great simplicity of the treatment presented here allows us to derive analytical equilibrium profiles in which the ion-kinetic effects can be clearly identified. Therefore this study is meant to be a first step – a sort of ‘proof of concept’ – towards the identification of the effects possibly leading to the observed behaviour of the low-latitude magnetopause layer, rather than an exhaustive description of the actual system. In order to achieve a quantitative modelling of the global magnetopause current system within this (or a more comprehensive) extended-fluid model, a numerical approach to the solution of the full three-dimensional problem would likely be required.

The remainder of this paper is organized as follows. In § 2 we describe the extended two-fluid (eTF) model of Cerri et al. (2013) and we outline the procedure for the derivation of the equilibrium profiles (the actual derivation of a general family of solutions for the shear-flow boundary layer equilibrium is provided in appendix A); consequences for shear-flow instabilities, agyrotropy and links to turbulent environments are highlighted in §§ 2.3–2.5. In § 3 we show how these profiles can qualitatively explain the observed non-ideal behaviour of the LLBL between the solar wind and the Earth’s magnetosphere. Finally, in § 4 conclusions are drawn. Additionally, explicit considerations on the symmetries of the FLR expansion and on its convergence to a full pressure tensor case are reported in appendices B and C, respectively.

2 The extended two-fluid (eTF) model

Here, we consider a non-relativistic quasi-neutral proton–electron plasma ( $n_{\text{p}}\simeq n_{\text{e}}\equiv n$ ) in the limit of massless electrons, $m_{\text{e}}/m_{\text{p}}\rightarrow 0$ . The model includes the Hall and the electron pressure terms in the generalized Ohm’s law, as well as dynamic equations for the gyrotropic pressures of both species and first-order FLR corrections to the protons’ pressure tensor ¹ . The fluid hierarchy is closed with a double-adiabatic approximation, i.e. by neglecting the heat fluxes, $q_{\Vert }=0$ and $q_{\bot }=0$ . Such assumption is indeed justified within a finite-but-small Larmor-radius expansion and on time scales much longer than the ion cyclotron time scale, $\unicode[STIX]{x1D70C}/L\sim \unicode[STIX]{x1D714}/\unicode[STIX]{x1D6FA}\sim \unicode[STIX]{x1D700}\ll 1$ , where $\unicode[STIX]{x1D70C}$ is the thermal Larmor radius, $L$ is the typical length scale of variation for macroscopic quantities and $\unicode[STIX]{x1D6FA}$ is the cyclotron frequency (see Cerri et al. 2013, for explicit equations and further details about the eTF model ordering). In fact, by neglecting gradients in the direction of the magnetic field ( $\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}=\unicode[STIX]{x1D735}_{\Vert }=0$ ; see appendix A), the expressions for the perpendicular heat fluxes (see, e.g. Braginskii 1965; Ramos 2008) would give a second-order contribution which is ordered out in the eTF model ² . In this model, the thermal pressure tensor of the protons and of the electrons, $\unicode[STIX]{x1D72B}_{\text{p}}$ and $\unicode[STIX]{x1D72B}_{\text{e}}$ respectively, are written as

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D72B}_{\text{p}}=p_{\Vert \text{p}}\boldsymbol{b}\boldsymbol{b}+p_{\bot \text{p}}\unicode[STIX]{x1D749}+\unicode[STIX]{x03C0}_{\text{p}}^{(1)}, & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D72B}_{\text{e}}=p_{\Vert \text{e}}\boldsymbol{b}\boldsymbol{b}+p_{\bot \text{e}}\unicode[STIX]{x1D749}, & \displaystyle\end{eqnarray}$$

where $\boldsymbol{b}\equiv \boldsymbol{B}/|\boldsymbol{B}|$ is the magnetic-field unit vector, $\unicode[STIX]{x1D749}\equiv \unicode[STIX]{x1D644}-\boldsymbol{b}\boldsymbol{b}$ is the projector onto the plane perpendicular to $\boldsymbol{B}$ and $p_{\Vert \unicode[STIX]{x1D6FC}}$ and $p_{\bot \unicode[STIX]{x1D6FC}}$ are the gyrotropic thermal pressures of the $\unicode[STIX]{x1D6FC}$ species parallel and perpendicular to the magnetic field, respectively (Chew, Goldberger & Low 1956). In (2.1), $\unicode[STIX]{x03C0}_{\text{p}}^{(1)}$ is a traceless symmetric tensor taking into account first-order FLR corrections to the gyrotropic proton pressure (also known as gyroviscous tensor). Neglecting the heat fluxes, a general formulation for the gyroviscous tensor components can be written as (Macmahon 1965; Schekochihin et al. 2010; Sulem & Passot 2015)

(2.3) $$\begin{eqnarray}\unicode[STIX]{x03C0}_{\text{p},ij}^{(1)}=\frac{p_{\bot \text{p}}}{4\unicode[STIX]{x1D6FA}_{c\text{p}}}(\unicode[STIX]{x1D716}_{ilm}b_{l}S_{mk}H_{kj}-H_{ik}\unicode[STIX]{x1D716}_{jlm}S_{kl}b_{m})+\frac{2(p_{\bot \text{p}}-p_{\Vert \text{p}})}{\unicode[STIX]{x1D6FA}_{c\text{p}}}(b_{i}w_{j}+b_{j}w_{i}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FA}_{c\text{p}}=eB/m_{\text{p}}c$ is the proton gyro-frequency, $\unicode[STIX]{x1D716}_{ijk}$ is the completely antisymmetric Levi-Civita tensor, and we have introduced $S_{ij}\equiv \unicode[STIX]{x2202}_{i}u_{\text{p},j}+\unicode[STIX]{x2202}_{j}u_{\text{p},i}$ , $H_{ij}\equiv \unicode[STIX]{x1D6FF}_{ij}+3b_{i}b_{j}$ and $w_{i}\equiv \unicode[STIX]{x1D716}_{ijk}(\unicode[STIX]{x1D735}_{\Vert }u_{\text{p},j})b_{k}$ , with $\unicode[STIX]{x1D735}_{\Vert }\equiv \boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}$ . Note that the above formulation automatically takes into account for the asymmetry due to the magnetic field direction with respect to the vorticity (see also appendix B and Cerri et al. 2013, for explicit symmetry considerations).

2.1 Shear-flow layer equilibrium with FLR

Within this model, we now outline the derivation of equilibrium profiles for a one-dimensional velocity-shear layer separating, for instance, two different plasmas. The explicit derivation of this class of analytical solutions that generalize the results provided in Cerri et al. (2013) and that include a much wider range of configurations of interest for what concerns magnetospheric observations will be provided in appendix A. The goal is to provide an equilibrium configuration with FLR corrections for the flank magnetopause, and to discuss the implications on the low-latitude boundary layer (LLBL) profiles. For the sake of simplicity, here we consider the one-dimensional equilibrium problem, which can be seen as a local approximation of the LLBL. A global treatment of the magnetospheric structure should take into account curvature terms, as well as possible gradients parallel to the magnetic field and compressible flows. This may need to include additional equilibrium conditions that involve all the gyroviscous components and eventually to go beyond the simple adiabatic FLR treatment presented here by, for instance, including heat fluxes (see, e.g. Sulem & Passot 2015; Del Sarto & Pegoraro 2018).

We consider a given $x$ -dependent incompressible MHD flow in the $y$ – $z$ plane,

(2.4a,b ) $$\begin{eqnarray}\boldsymbol{u}=u_{y}(x)\boldsymbol{e}_{y}+u_{z}(x)\boldsymbol{e}_{z},\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0,\end{eqnarray}$$

such that it becomes constant at the boundaries (i.e. we consider a localized velocity-shear layer). The magnetic field also lies on the $y$ – $z$ plane,

(2.5) $$\begin{eqnarray}\boldsymbol{B}(x)=B_{y}(x)\boldsymbol{e}_{y}+B_{z}(x)\boldsymbol{e}_{z}.\end{eqnarray}$$

We further simplify the problem by assuming a polytropic relation for the thermal pressures ³ . This assumption is not strictly necessary in order to derive the equilibrium, but it is useful for providing density and temperature profiles from the obtained pressure profiles. In general, the equilibrium for this configuration is found by imposing total pressure balance:

(2.6) $$\begin{eqnarray}\frac{\text{d}}{\text{d}x}[\unicode[STIX]{x1D72B}_{\text{p}}(x)+\unicode[STIX]{x1D72B}_{\text{e}}(x)+\unicode[STIX]{x1D72B}_{\text{B}}(x)]=0,\end{eqnarray}$$

where $\unicode[STIX]{x1D72B}_{\text{B}}\equiv (B^{2}/8\unicode[STIX]{x03C0})\unicode[STIX]{x1D644}-\boldsymbol{B}\boldsymbol{B}$ is the magnetic pressure tensor ( $\unicode[STIX]{x1D644}$ being the identity tensor). Within an (anisotropic) MHD model of plasma, the shear flow does not play a role in the equilibrium profile. In fact, when $\unicode[STIX]{x03C0}_{\text{p}}^{(1)}$ is neglected, the equilibrium condition for the above configuration simply consists of a balance between the magnetic pressure, $P_{B}(x)=B^{2}(x)/8\unicode[STIX]{x03C0}$ , and the perpendicular thermal pressures, $P_{\bot }(x)=p_{\bot \text{p}}(x)+p_{\bot \text{e}}(x)$ . In particular, that includes the widely adopted uniform and homogeneous plasma configuration, namely $p_{\bot \unicode[STIX]{x1D6FC}}=p_{\Vert \unicode[STIX]{x1D6FC},0}$ , $p_{\bot \unicode[STIX]{x1D6FC}}=p_{\bot \unicode[STIX]{x1D6FC},0}$ , $B_{y}=B_{0y}$ and $B_{z}=B_{0z}$ , that is not allowed anymore when FLR corrections (or the full pressure-tensor equations) are included in the fluid description (Cerri 2012; Cerri et al. 2013, 2014). In general, the solution of the MHD equilibrium condition is completely described by the chosen magnetic profile in (2.5), which determines all the profiles of the other relevant quantities. Let us now consider the changes of a given MHD equilibrium profile that are induced by a velocity shear of the type described above when first-order FLR corrections are taken into account. In this case, the only component of $\unicode[STIX]{x03C0}_{\text{p}}^{(1)}$ that is relevant to the equilibrium condition is

(2.7) $$\begin{eqnarray}\unicode[STIX]{x03C0}_{\text{p},xx}^{(1)}=-\frac{1}{2}\frac{p_{\bot \text{p}}}{\unicode[STIX]{x1D6FA}_{c\text{p}}}\left(b_{z}\frac{\text{d}u_{y}}{\text{d}x}-b_{y}\frac{\text{d}u_{z}}{\text{d}x}\right).\end{eqnarray}$$

From (2.7), one directly identifies the connection between the fluid vorticity, $\unicode[STIX]{x1D74E}\equiv \unicode[STIX]{x1D735}\boldsymbol{\times }\boldsymbol{u}$ , and the magnetic field direction $\boldsymbol{b}$ , arising as a consequence of the FLR effects:

(2.8) $$\begin{eqnarray}\unicode[STIX]{x03C0}_{\text{p},xx}^{(1)}=-\frac{1}{2}\frac{p_{\bot \text{p}}}{\unicode[STIX]{x1D6FA}_{c\text{p}}}(\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D74E})\quad \longrightarrow \quad \frac{\text{d}}{\text{d}x}\left[\left(1-\frac{m_{\text{p}}c}{eB}\frac{\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D74E}}{2}\right)p_{\bot \text{p}}+p_{\bot \text{e}}+\frac{B^{2}}{8\unicode[STIX]{x03C0}}\right]=0,\end{eqnarray}$$

where $\unicode[STIX]{x1D714}_{y}=-u_{z}^{\prime }$ and $\unicode[STIX]{x1D714}_{z}=u_{y}^{\prime }$ are the components of the fluid vorticity in our configuration. Therefore, the FLR corrections give rise to an intrinsic asymmetry in the system’s configurations, pressure anisotropy (and most likely also the subsequent dynamics), which depends on the degree of alignment (or anti-alignment) between the flow vorticity and the magnetic field, namely on the sign of $\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D74E}$ . Such asymmetry has been highlighted in previous numerical simulations and analytical studies (see, e.g. Nagano 1978; Hazeltine, Hsu & Morrison 1987; Cai et al. 1990; Huba 1996; Ramos 2005b ; Nakamura et al. 2010; Cerri et al. 2013; Henri et al. 2013; Del Sarto et al. 2016, 2017; Franci et al. 2016; Parashar & Matthaeus 2016; Yang et al. 2017; Del Sarto & Pegoraro 2018). We stress, however, that the simple dependence on $\unicode[STIX]{x1D74E}$ and $\boldsymbol{b}$ in (2.7) is related to the simplified character of the configuration considered here.

Now assume that ${\mathcal{F}}_{\bot }(x)$ , ${\mathcal{G}}_{\bot }(x)$ and ${\mathcal{H}}(x)$ are the solutions for the anisotropic MHD equilibrium describing the profiles of the proton perpendicular pressure, $p_{\bot \text{p}}=p_{\bot \text{p},0}{\mathcal{F}}_{\bot }(x)$ , of the electron perpendicular pressure, $p_{\bot \text{e}}=p_{\bot \text{e},0}{\mathcal{G}}_{\bot }(x)$ , and of the magnetic pressure, $P_{B}(x)=(B_{0}^{2}/8\unicode[STIX]{x03C0}){\mathcal{H}}(x)$ (here $p_{\bot \text{p},0}$ , $p_{\bot \text{e},0}$ and $B_{0}$ are the asymptotic constant values of the pressures and of the magnetic field away from the shear layer, on one of the two sides – here we do not assume a symmetric shear layer; see appendix A for details). We now seek FLR-corrected equilibrium profiles in the form $\widetilde{{\mathcal{F}}}_{\bot }(x)={\mathcal{F}}_{\bot }(x)f_{\bot }(x)$ , $\widetilde{{\mathcal{G}}}_{\bot }(x)={\mathcal{G}}_{\bot }(x)g_{\bot }(x)$ and $\widetilde{{\mathcal{H}}}(x)={\mathcal{H}}(x)h(x)$ , where $f_{\bot }$ , $g_{\bot }$ and $h$ are the ‘correction functions’. By requiring quasi-neutrality and that the MHD profile $\unicode[STIX]{x1D6FD}_{\bot \text{p}}(x)$ does not change when passing to the corresponding FLR-corrected profile, the solution can be given in term of one function only, i.e.  $f_{\bot }(x)=g_{\bot }(x)=h(x)$ (see appendix A):

(2.9) $$\begin{eqnarray}f_{\bot }(x)=\left\{\frac{\widetilde{U}^{\prime }(x)}{2}+\sqrt{1+\left(\frac{\widetilde{U}^{\prime }(x)}{2}\right)^{2}}\right\}^{2},\end{eqnarray}$$

where we have defined

(2.10) $$\begin{eqnarray}\widetilde{U}^{\prime }(x)\equiv \frac{\widetilde{\unicode[STIX]{x1D6FD}}_{\bot \text{p},0}}{2}\frac{m_{\text{p}}c}{eB_{0}}\frac{{\mathcal{F}}_{\bot }(x)}{{\mathcal{H}}(x)}\left(\frac{B_{0z}}{B_{0}}H_{z}(x)u_{y}^{\prime }(x)-\frac{B_{0y}}{B_{0}}H_{y}(x)u_{z}^{\prime }(x)\right),\end{eqnarray}$$

with $\widetilde{\unicode[STIX]{x1D6FD}}_{\bot \text{p},0}\equiv \unicode[STIX]{x1D6FD}_{\bot \text{p},0}/(1+\unicode[STIX]{x1D6FD}_{\bot ,0})$ for brevity. Note that the solution in (2.9) has been obtained taking into account the FLR corrections computed with the self-consistent (i.e. FLR-corrected) equilibrium magnetic field profile, $B(x)=B_{0}\sqrt{{\mathcal{H}}(x)f_{\bot }(x)}$ . The equilibrium profiles resulting from (2.9) are then naturally asymmetric with respect to the sign of $\unicode[STIX]{x1D74E}\boldsymbol{\cdot }\boldsymbol{b}$ .

2.2 FLR profiles and approximate kinetic equilibria

The profiles derived above can be used to initialize the ion distribution function in order to set-up an approximate kinetic equilibrium (see Cerri et al. 2013). For instance, assuming the inhomogeneity direction to be along $x$ , the magnetic field to be in the $z$ -direction, $\boldsymbol{B}=B_{z}(x)\boldsymbol{e}_{z}$ and the flow to be along the $y$ -axis, $\boldsymbol{u}=u_{y}(x)\boldsymbol{e}_{y}$ , one obtains the following temperatures:

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle T_{x}(x)=\frac{p_{\bot \text{p},0}}{n_{0}}(1-\unicode[STIX]{x1D712}(x))({\mathcal{F}}_{\bot }(x)\,f_{\bot }(x))^{(\unicode[STIX]{x1D6FE}_{\bot }-1)/\unicode[STIX]{x1D6FE}_{\bot }}, & \displaystyle\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle T_{y}(x)=\frac{p_{\bot \text{p},0}}{n_{0}}(1+\unicode[STIX]{x1D712}(x))({\mathcal{F}}_{\bot }(x)\,f_{\bot }(x))^{(\unicode[STIX]{x1D6FE}_{\bot }-1)/\unicode[STIX]{x1D6FE}_{\bot }}, & \displaystyle\end{eqnarray}$$

(2.13) $$\begin{eqnarray}\displaystyle & \displaystyle T_{z}=T_{\Vert \text{p}}=\frac{p_{\Vert \text{p},0}}{n_{0}}({\mathcal{F}}_{\bot }(x)\,f_{\bot }(x))^{(\unicode[STIX]{x1D6FE}_{\Vert }-1)/\unicode[STIX]{x1D6FE}_{\bot }}, & \displaystyle\end{eqnarray}$$

from which the three thermal velocities, $v_{\text{th},x}(x)$ , $v_{\text{th},y}(x)$ and $v_{\text{th},z}(x)$ can be defined. The parameter $\unicode[STIX]{x1D712}$ is defined by the first-order FLR correction to the pressure tensor in (2.8), and provides the agyrotropy of the distribution as a function of the alignment between the flow vorticity, $\unicode[STIX]{x1D74E}$ , and the self-consistent FLR-corrected magnetic field. In our transverse case with $\boldsymbol{u}=u_{y}(x)\boldsymbol{e}_{z}$ and $\boldsymbol{B}=B_{z}(x)\boldsymbol{e}_{z}$ , it reads

(2.14) $$\begin{eqnarray}\unicode[STIX]{x1D712}(x)\equiv \frac{1}{2}\frac{m_{\text{p}}c}{e|\boldsymbol{B}|}(\unicode[STIX]{x1D74E}\boldsymbol{\cdot }\boldsymbol{b})=\frac{1}{2}\frac{m_{\text{p}}c}{eB_{0}}\frac{u_{y}^{\prime }(x)}{H_{z}(x)\sqrt{f_{\bot }(x)}},\end{eqnarray}$$

where $u_{y}^{\prime }(x)=\text{d}u_{y}/\text{d}x$ . The ‘Maxwellian-like’ particle distribution function corresponding to the above profiles reads

(2.15) $$\begin{eqnarray}F_{\text{M}}^{\text{(FLR)}}(x,v_{x},v_{y},v_{z})=\frac{(2\unicode[STIX]{x03C0})^{-3/2}n(x)}{\sqrt{T_{x}(x)T_{y}(x)T_{z}(x)}}\exp \left\{-\frac{v_{x}^{2}}{2T_{x}(x)}-\frac{(v_{y}-u_{y}(x))^{2}}{2T_{y}(x)}-\frac{v_{z}^{2}}{2T_{z}(x)}\right\}.\end{eqnarray}$$

Note that, in the general case, a distribution function reproducing the FLR-corrected profiles would be more complicated, since it may have to give non-diagonal pressure terms. Nevertheless, the equilibrium profiles derived from the FLR correction function $f_{\bot }(x)$ in (2.9) still holds for a generic flow and magnetic-field profile (given that they lie in the plane perpendicular to the inhomogeneity direction; see § A.1) and can be used to set up such ‘Maxwellian-like’ distributions. We stress anyway that a distribution function built in this way is only an approximate kinetic equilibrium, which nevertheless can strongly reduce the spurious fluctuations arising from a readjustment induced by adopting MHD-like equilibrium profiles within a kinetic (or a hybrid kinetic) framework. Unfortunately, exact solutions of the kinetic (or of the hybrid kinetic) problem usually need to consider simplified configurations, e.g. of the magnetic field, and/or cannot exactly constrain the resulting velocity profiles beforehand (see, e.g. Cai et al. 1990; Attico & Pegoraro 1999; Mahajan & Hazeltine 2000; Bobrova et al. 2001; Malara et al. 2018). Which solution is better to use clearly depends on the problem under consideration. For instance, in the context of the Earth’s flank magnetopause we are dealing with inhomogeneous magnetic field and density profiles (and directions), so the approach presented here is more appropriate for that case.

2.3 Readjustment time scale of unbalanced equilibria

As mentioned in the Introduction, taking into account the leading kinetic effects (such as the above first-order ion-FLR correction) may be necessary already at the level of the initial plasma configuration. In fact, adopting an ideal MHD initial equilibrium in a kinetic framework will result in a quick readjustment and in the development of spurious large-amplitude fluctuations (see Cerri et al. 2013; Henri et al. 2013).

When MHD equilibria are employed in kinetic simulations where a sheared flow is present, the unbalanced leading ion-FLR corrections will induce a readjustment on time scales $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x03C0}}$ of the order ⁴

(2.16) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x03C0}}^{-1}\sim \unicode[STIX]{x1D6FD}_{i,\bot }^{-1/2}M_{A}\left(\frac{\unicode[STIX]{x1D70C}_{i}}{L_{u}}\right)^{2}\unicode[STIX]{x1D6FA}_{c,i}\sim \unicode[STIX]{x1D6FD}_{i,\bot }^{1/2}M_{A}\left(\frac{d_{i}}{L_{u}}\right)^{2}\unicode[STIX]{x1D6FA}_{c,i},\end{eqnarray}$$

where $M_{A}\equiv u_{0}/v_{A}$ and $L_{u}$ are the Alfvénic Mach number and length scale of the background shear flow. It may be useful to compare this readjustment time scale with the growth rate of the fastest-growing mode (FGM) for the Kelvin–Helmholtz instability,

(2.17) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{\text{FGM}}^{\text{(KHI)}}\sim \frac{1}{4}k_{\text{FGM}}u_{0}\sim 0.1\unicode[STIX]{x1D6FD}_{i,\bot }^{-1/2}M_{A}\left(\frac{\unicode[STIX]{x1D70C}_{i}}{L_{u}}\right)\unicode[STIX]{x1D6FA}_{c,i}\sim 0.1M_{A}\left(\frac{d_{i}}{L_{u}}\right)\unicode[STIX]{x1D6FA}_{c,i},\end{eqnarray}$$

where we have used the relation $k_{\text{FGM}}L_{u}\sim 0.4$ derived in the compressible MHD limit (see Faganello & Califano 2017, and references therein). Therefore, the effects of such readjustment on the KHI growth are of order

(2.18) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{\text{FGM}}^{\text{(KHI)}}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x03C0}}\sim 0.1\left(\frac{L_{u}}{\unicode[STIX]{x1D70C}_{i}}\right)\sim 0.1\unicode[STIX]{x1D6FD}_{i,\bot }^{-1/2}\left(\frac{L_{u}}{d_{i}}\right),\end{eqnarray}$$

which is typically smaller than (or of the order of) unity for the magnetopause case, meaning that any readjustment happens faster than the instability itself and therefore will strongly change the equilibrium on top of which the KHI develops.

Figure 1. (a) Iso-surfaces of $\log (\unicode[STIX]{x1D6FA}_{c\text{i}}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x03C0}})$ in the $L_{u}/d_{i}$ versus $\unicode[STIX]{x1D6FD}^{1/2}M_{A}$ plane. The same iso-surfaces apply to the $L_{u}/\unicode[STIX]{x1D70C}_{i}$ versus $M_{s}$ plane ( $M_{s}\equiv u_{0}/c_{s}$ is the Mach number, $c_{s}$ being the sound speed). (b) Iso-surfaces of $\log (\unicode[STIX]{x1D6FE}_{\text{FGM}}^{\text{(KHI)}}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x03C0}})$ in the $L_{u}/d_{i}$ versus $\unicode[STIX]{x1D6FD}$ plane.

A sketch of the behaviour of time scales in (2.16) and (2.18) with respect to the relevant parameters is provided in figure 1. MHD-like behaviour is recovered in the parameter space denoted by yellow/white colours.

2.4 Sustainability of pressure agyrotropy

An interesting feature of the interaction between the pressure tensor and a sheared flow is the sustainability and/or the generation of pressure ‘agyrotropy’ (Cerri et al. 2013, 2014; Del Sarto et al. 2016, 2017). This means that, in addition to the typical pressure anisotropy with respect to the magnetic-field direction that is typical of collisionless plasmas ( $p_{\bot }\neq p_{\Vert }$ ), now additional pressure anisotropy can be present in the plane perpendicular to $\boldsymbol{B}$ , e.g.  $p_{\bot ,1}\neq p_{\bot ,2}\neq p_{\Vert }$ , where ( $\boldsymbol{e}_{\bot ,1}$ , $\boldsymbol{e}_{\bot ,2}$ , $\boldsymbol{e}_{\Vert }$ ) is any orthogonal basis within which the pressure tensor is diagonal and where $\boldsymbol{e}_{\bot ,1}$ and $\boldsymbol{e}_{\bot ,2}$ define the plane perpendicular to the magnetic field. In this section we analyse this aspect in terms of equilibrium configurations and their corresponding agyrotropy. However, we stress that this feature has consequences for the dynamics of a collisionless plasma as well, e.g. modifying linear properties of perturbations (e.g. Del Sarto et al. 2016, 2017), enhancing the kinetic activity related to vorticity, current sheets, reconnection and energy transfer in turbulence (e.g. Greco et al. 2012; Servidio et al. 2012, 2014; Yang et al. 2017) and possibly affecting the regulation of anisotropies in accretion disks (e.g. Kunz, Stone & Quataert 2016).

Figure 2. Pressure anisotropy in the plane perpendicular to the magnetic field direction, $\unicode[STIX]{x1D6E5}_{\bot }$ , versus $\unicode[STIX]{x1D712}\equiv \unicode[STIX]{x1D74E}\boldsymbol{\cdot }\boldsymbol{b}/2\unicode[STIX]{x1D6FA}_{c\text{i}}$ obtained from first-order FLR corrections (dashed line, equation (2.19)) and from the full pressure-tensor equation (continuous line, equation (2.20)). Positivity of pressure from the full- $\unicode[STIX]{x1D6F1}$ treatment requires $\unicode[STIX]{x1D712}\geqslant -1/2$ (see Cerri et al. 2014), while the FLR treatment holds for $|\unicode[STIX]{x1D712}|\ll 1$ .

In our configuration it is easy to show that the FLR effects introduce an agyrotropy, $\unicode[STIX]{x1D6E5}_{\bot }$ , i.e. an anisotropy in the plane perpendicular to the magnetic field (see, e.g. Scudder & Daughton 2008, for a general formulation), given by

(2.19) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{\bot }=\frac{|\unicode[STIX]{x1D74E}\boldsymbol{\cdot }\boldsymbol{b}|}{\unicode[STIX]{x1D6FA}_{ci}}\equiv |2\unicode[STIX]{x1D712}|.\end{eqnarray}$$

Since only the first-order FLR corrections have been retained in the present description, only small deviations from gyrotropy are correctly described in this case, i.e. the condition $|\unicode[STIX]{x1D712}|=|\unicode[STIX]{x1D74E}\boldsymbol{\cdot }\boldsymbol{b}/2\unicode[STIX]{x1D6FA}_{c\text{i}}|\ll 1$ should hold. Also, in this approximation the equilibrium exhibits an asymmetry with respect to the sign of $\unicode[STIX]{x1D74E}\boldsymbol{\cdot }\boldsymbol{b}$ , but $\unicode[STIX]{x1D6E5}_{\bot }$ does not. In order to have such asymmetry in the agyrotropy, next-order corrections or the full pressure tensor must be retained. In the latter case, the agyrotropy in the plane perpendicular to $\boldsymbol{B}$ would be (Cerri et al. 2014)

(2.20) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{\bot }=\left|\frac{2\unicode[STIX]{x1D712}}{1+\unicode[STIX]{x1D712}}\right|,\end{eqnarray}$$

where the condition $\unicode[STIX]{x1D712}\geqslant -1/2$ must hold because of the positivity constraint on pressure.

In figure 2 we report a comparison between the pressure anisotropy in the plane perpendicular to the magnetic field, $\unicode[STIX]{x1D6E5}_{\bot }$ , as a function of the parameter $\unicode[STIX]{x1D712}$ , obtained via the full pressure-tensor equation (Cerri et al. 2014) and via first-order FLR corrections.

2.5 A broader view: relevance to other instabilities and turbulent environments

As we will show in § 3, the main consequences related to the ion-FLR effects reported in this paper have a direct effect in the current system of a planetary magnetopause. Moreover, these ion-kinetic effects can cause the asymmetric development of KHI at the dawn and the dusk sides of such magnetosphere, as well as other non-ideal features (see, e.g. Nagano 1978; Huba 1996; Terada et al. 2002; Nakamura et al. 2010; Henri et al. 2012; Masters et al. 2012; Sundberg et al. 2012; Taylor et al. 2012; Delamere et al. 2013; Paral & Rankin 2013; Haaland et al. 2014; Johnson et al. 2014; Liljeblad et al. 2014; Walsh et al. 2014; Gershman et al. 2015; Gingell et al. 2015; De Camillis et al. 2016). However, ion-FLR effects and their relations with anisotropy, vorticity and current sheets can have implications on a wide variety of astrophysical and space scenarios.

In fact there are further shear-driven instabilities that may also get relevant feedback from anisotropy (and agyrotropy) developed (or sustained) by the underlying shear flow within a kinetic description such as, for instance, for the case of magneto-rotational instability (MRI) in accretion disks (e.g. Ferraro 2007; Riquelme et al. 2012; Kunz et al. 2016; Squire, Quataert & Kunz 2017b ). Furthermore, ion-kinetic effects such as FLR and pressure-tensor dynamics can affect anisotropy-driven instabilities themselves (e.g. Schekochihin et al. 2010; Rosin et al. 2011; Sarrat, Del Sarto & Ghizzo 2016; Squire et al. 2017a ), which are relevant, e.g. in the evolution of the solar wind (e.g. Hellinger et al. 2006; Tenerani, Velli & Hellinger 2017; Yoon 2017) and in magnetic reconnection (e.g. Schoeffler, Drake & Swisdak 2011; Cassak et al. 2015).

Finally, current sheets and the associated reconnection processes are fundamental ingredients of turbulent plasmas (e.g. Matthaeus & Lamkin 1986; Biskamp 2008; Servidio et al. 2010, 2011; Lazarian, Eyink & Vishniac 2012; Karimabadi et al. 2013a ; Servidio et al. 2015; Franci et al. 2016; Cerri et al. 2017). In this context, currents and coherent structures are typically related to simultaneous enhancement of vorticity, kinetic activity, turbulent transfer and dissipation (e.g. Servidio et al. 2012, 2014; Karimabadi et al. 2013b ; Valentini et al. 2014, 2016; Wan et al. 2015; Franci et al. 2016; Parashar & Matthaeus 2016; Yang et al. 2017; Grošelj et al. 2017; Camporeale et al. 2018; Sorriso-Valvo et al. 2018). Furthermore, reconnection/structures have been recently proved to enhance/trigger the kinetic turbulent cascades in real space (Cerri & Califano 2017; Franci et al. 2017; Camporeale et al. 2018) and also to be related to simultaneous velocity-space cascades (Servidio et al. 2017; Cerri, Kunz & Califano 2018; Pezzi et al. 2018). These reconnecting current sheets and the resulting magnetic structures are quasi-equilibrium pressure-balanced structures with embedded sheared flows even within a turbulent environment (see e.g. Cerri & Califano 2017). Therefore ion-kinetic effects such as FLR contributions (or the full pressure tensor; see Cerri et al. (2014), Yang et al. (2017) and Del Sarto & Pegoraro (2018)) may play a relevant role in the complex interplay between currents, vorticity, reconnection, non-Maxwellian features, velocity-space cascades and dissipation in turbulent plasmas.

3 Application to the LLBL of the Earth’s magnetopause

Let us now consider an explicit application to the LLBL of the Earth’s magnetopause, the goal being to show that the observed deviations from the ideal Chapman–Ferraro current system highlighted in Haaland et al. (2014) can be qualitatively explained with the ion-FLR corrections. We want to stress that this is not meant to be a quantitative explanation of the observed profiles, since also the three-dimensional geometry and other effects may contribute to the actual profiles. In what follows, equations are normalized to the proton mass, inertial length and cyclotron frequency ( $m_{\text{p}}$ , $d_{\text{p}}$ and $\unicode[STIX]{x1D6FA}_{c\text{p}}$ , respectively), and the Alfvén speed ( $v_{A}$ ).

We consider a local one-dimensional model the LLBL region in which the inhomogeneity direction ( $x$ ) is perpendicular to the plane ( $yz$ ) where both the flow and the magnetic field lie. Typically, hyperbolic tangent give a reasonably realistic modelling of the flow,

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle u_{y}(x)=u_{0}\sin \unicode[STIX]{x1D719}\tanh \left(\frac{x-x_{u,0}}{L_{u}}\right), & \displaystyle\end{eqnarray}$$

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle u_{z}(x)=u_{0}\cos \unicode[STIX]{x1D719}\tanh \left(\frac{x-x_{u,0}}{L_{u}}\right), & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D719}$ is the angle between the $z$ -axis and the plane where the sheared flow velocity lies, and of the magnetic field,

(3.3) $$\begin{eqnarray}\displaystyle B_{y}(x) & = & \displaystyle B_{0}\left\{\frac{B_{G}}{B_{0}}\sin \unicode[STIX]{x1D717}\left[1+\frac{\unicode[STIX]{x0394}B_{\Vert }}{2B_{G}}\left(1-\tanh \left(\frac{x-x_{B,0}}{L_{B}}\right)\right)\right]\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\frac{\unicode[STIX]{x0394}B_{\bot }}{2B_{0}}\cos \unicode[STIX]{x1D717}\left[1+\tanh \left(\frac{x-x_{B,0}}{L_{B}}\right)\right]\!\right\},\end{eqnarray}$$

(3.4) $$\begin{eqnarray}\displaystyle B_{z}(x) & = & \displaystyle B_{0}\left\{\frac{B_{G}}{B_{0}}\cos \unicode[STIX]{x1D717}\left[1+\frac{\unicode[STIX]{x0394}B_{\Vert }}{2B_{G}}\left(1-\tanh \left(\frac{x-x_{B,0}}{L_{B}}\right)\right)\right]\right.\nonumber\\ \displaystyle & & \displaystyle -\left.\frac{\unicode[STIX]{x0394}B_{\bot }}{2B_{0}}\sin \unicode[STIX]{x1D717}\left[1+\tanh \left(\frac{x-x_{B,0}}{L_{B}}\right)\right]\!\right\},\end{eqnarray}$$

where $B_{0}=\sqrt{B_{G}^{2}+\unicode[STIX]{x0394}B_{\bot }^{2}}$ and $\unicode[STIX]{x1D717}$ is the angle between the $z$ -axis and the magnetic field at $x\rightarrow -\infty$ ⁵ . The above magnetic profile accounts both for variations that are purely in magnitude, through $\unicode[STIX]{x0394}B_{\Vert }$ , and for rotations (magnetic shear) of the magnetic-field direction, through $\unicode[STIX]{x0394}B_{\bot }$ (see, e.g. Fadanelli et al. 2018, for the effects of $\unicode[STIX]{x0394}B_{\bot }$ on KHI at the Earth’s magnetospheric flanks). Note that usually $x_{u,0}=x_{B,0}$ and $L_{u}=L_{B}$ are assumed in numerical simulations (see, e.g. Miura 1987; Fujimoto & Terasawa 1995; Otto & Fairfield 2000; Nykyri & Otto 2004; Nakamura & Fujimoto 2005; Faganello, Califano & Pegoraro 2008; Palermo et al. 2011; Tenerani et al. 2011; Faganello et al. 2012). However, recent satellite measurements have shown that the magnetic (and density) profiles can be slightly shifted with respect to the velocity shear and/or that the shear length scales of these quantities may differ, i.e.  $x_{u,0}\neq x_{n,0}$ and/or $L_{u}\neq L_{n}$ (Foullon et al. 2008; Haaland et al. 2014; Rossi 2015). This idea has been also recently implemented in numerical simulations in order to explain some observational features (Rossi 2015; Leroy & Keppens 2017). Therefore, here we also take into account these features. For a magnetic profile as in (3.3)–(3.4) the MHD magnetic pressure function, ${\mathcal{H}}$ , is given by

(3.5) $$\begin{eqnarray}\displaystyle {\mathcal{H}}(x)=\frac{B_{G}^{2}}{B_{0}^{2}}\left\{\!\left[1+\frac{\unicode[STIX]{x0394}B_{\Vert }}{2B_{G}}\left(1-\tanh \left(\frac{x-x_{B,0}}{L_{B}}\right)\right)\right]^{2}+\frac{\unicode[STIX]{x0394}B_{\bot }^{2}}{4B_{G}^{2}}\left[1+\tanh \left(\frac{x-x_{B,0}}{L_{B}}\right)\right]^{2}\right\},\hspace{-9.60004pt} & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

and the corresponding MHD thermal profiles are obtained in terms of

(3.6) $$\begin{eqnarray}\displaystyle {\mathcal{F}}_{\bot }(x)={\mathcal{G}}_{\bot }(x) & = & \displaystyle 1+\frac{\unicode[STIX]{x0394}B_{\bot }^{2}}{\unicode[STIX]{x1D6FD}_{\bot ,0}B_{0}^{2}}-\frac{B_{G}\unicode[STIX]{x0394}B_{\Vert }}{\unicode[STIX]{x1D6FD}_{\bot ,0}B_{0}^{2}}\left[1-\tanh \left(\frac{x-x_{B,0}}{L_{B}}\right)\right]\nonumber\\ \displaystyle & & \displaystyle -\,\frac{\unicode[STIX]{x0394}B_{\Vert }^{2}}{4\unicode[STIX]{x1D6FD}_{\bot ,0}B_{0}^{2}}\left[1-\tanh \left(\frac{x-x_{B,0}}{L_{B}}\right)\right]^{2}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{\unicode[STIX]{x0394}B_{\bot }^{2}}{4\unicode[STIX]{x1D6FD}_{\bot ,0}B_{0}^{2}}\left[1+\tanh \left(\frac{x-x_{B,0}}{L_{B}}\right)\right]^{2},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FD}_{\bot ,0}B_{0}^{2}=2P_{\bot ,0}\equiv 2(p_{\bot \text{p},0}+p_{\bot \text{e},0})$ and the positivity condition on pressure (see (A 10) in § A.3) here reads as

(3.7) $$\begin{eqnarray}B_{G}\unicode[STIX]{x0394}B_{\Vert }+\frac{\unicode[STIX]{x0394}B_{\Vert }^{2}}{2}\leqslant P_{\bot ,0}+\frac{\unicode[STIX]{x0394}B_{\bot }^{2}}{2}.\end{eqnarray}$$

The FLR corrections to the above MHD profiles are then given in terms of

(3.8) $$\begin{eqnarray}\displaystyle \widetilde{U}^{\prime }(x) & = & \displaystyle \frac{\widetilde{\unicode[STIX]{x1D6FD}}_{\bot \text{p},0}}{2}\frac{u_{0}}{B_{0}L_{u}}\frac{{\mathcal{F}}_{\bot }(x)}{{\mathcal{H}}(x)}\cosh ^{-2}\left(\frac{x-x_{u,0}}{L_{u}}\right)\nonumber\\ \displaystyle & & \displaystyle \times \,\left\{\frac{B_{G}}{B_{0}}\left[1+\frac{\unicode[STIX]{x0394}B_{\Vert }}{2B_{G}}\left(1-\tanh \left(\frac{x-x_{B,0}}{L_{B}}\right)\right)\right]\sin (\unicode[STIX]{x1D719}-\unicode[STIX]{x1D717})\right.\nonumber\\ \displaystyle & & \displaystyle -\left.\frac{\unicode[STIX]{x0394}B_{\bot }}{2B_{0}}\left[1+\tanh \left(\frac{x-x_{B,0}}{L_{B}}\right)\right]\cos (\unicode[STIX]{x1D719}-\unicode[STIX]{x1D717})\right\},\end{eqnarray}$$

which is again related to the sign of the scalar product between the fluid vorticity and the magnetic field through the $\sin (\unicode[STIX]{x1D719}-\unicode[STIX]{x1D717})$ and $\cos (\unicode[STIX]{x1D719}-\unicode[STIX]{x1D717})$ coefficients.

3.1 Current profiles at the Earth’s flank magnetopause: an example

Let us now consider a few explicit examples relevant for the magnetopause layer and see how the first-order FLR corrections qualitatively modify its current profile. For the sake of simplicity, we consider the case of $\unicode[STIX]{x1D717}=0$ and $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/2$ and two slightly different regimes are taken into account. A summary of the parameters adopted for the example profiles is given in table 1. These parameters are chosen so that they are as realistic as possible for the low-latitude flanks of the magnetopause (Haaland et al. 2014), and they are also able to somewhat emphasize some of the resulting features  ⁶ .

Table 1. Summary of the parameters used for profiles in figure 3. All the parameters are normalized with respect to quantities characteristic of the SW region: flow speed is in $v_{\text{A}}^{\text{(SW)}}$ units, lengths are in $d_{\text{i}}^{\text{(SW)}}$ units and magnetic-field variations are in $B_{0}^{\text{(SW)}}=1$ units (from which $B_{G}=\sqrt{1-\unicode[STIX]{x0394}B_{\bot }^{2}}$ follows). The plus and minus sign in $u_{0}$ and in $x_{u,0}$ are for the dusk and for the dawn side, respectively. We also remind the reader that $\unicode[STIX]{x1D717}=0$ and $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/2$ in both cases.

Figure 3. Current profiles for cases reported in table 1. (a,b) Case A, dawn (a) and dusk (b) sides. (c,d) Case B, dawn (c) and dusk (d) sides. The MHD current profiles, $J_{y}^{\text{(MHD)}}$ and $J_{z}^{\text{(MHD)}}$ , are reported with dashed light blue and dot-dashed orange lines, respectively, whereas the corresponding FLR-corrected profiles, $J_{y}^{\text{(FLR)}}$ and $J_{z}^{\text{(FLR)}}$ , are drawn in blue and red solid lines, respectively.

In figure 3 we report the current profile arising from a simple MHD configuration, $J_{y}^{\text{(MHD)}}$ and $J_{z}^{\text{(MHD)}}$ (light blue dashed line and orange dot-dashed line, respectively), as well as the profile accounting for the first-order FLR corrections in (3.8), $J_{y}^{\text{(FLR)}}$ and $J_{z}^{\text{(FLR)}}$ (blue and red solid lines, respectively). The MHD profiles of the dusk and of the dawn sides, apart from the sign, have the same shape, i.e. it is the classic Chapman–Ferraro current layer (Chapman & Ferraro 1930). On the other hand, the corresponding FLR-corrected profiles of the dawn and of the dusk sides are qualitatively different. This is the effect of the ‘ $\unicode[STIX]{x1D74E}\boldsymbol{b}$ asymmetry’ intrinsically encoded in the FLR contributions. Furthermore, the current structure of the shear layer in this latter case is much more complex than the Chapman–Ferraro MHD layer. In fact, a double-peak feature asymmetrically arises in $J^{\text{(FLR)}}$ on the two sides of the flank magnetopause and the different modification of the two components of the current results in adjacent current sheets with different current directions (see figure 4, where we report the $x$ -dependence of the angle between $\boldsymbol{J}$ and the $z$ -axis, $\unicode[STIX]{x1D6FC}=\arctan (J_{y}/J_{z})$ , for the cases shown in figure 3). These three peculiar features, namely (i) the dusk–dawn asymmetry of the current layer, (ii) the double-peak feature in the current profiles and (iii) two (or more) adjacent current sheets having thickness of several ion Larmor radii and with different current directions, are qualitatively consistent with the Cluster observations reported in Haaland et al. (2014). Taking into account these FLR effects can also be a relevant starting point for explaining certain anomalies occurring during magnetopause distortions related to large-scale magnetosheath plasma jets (see, e.g. Dmitriev & Suvorova 2012).

Figure 4. Profiles of the angle between the current $\boldsymbol{J}$ and the $z$ -axis,  $\unicode[STIX]{x1D6FC}$ , versus $x$ for cases reported in figure 3. (a) Case A, dawn (a,b) and dusk (b) sides. (c,d) Case B, dawn (c) and dusk (d) sides.

Finally, we want to stress that here we focused on the FLR corrections to the magnetic and current structures, as most of the analysis done on satellite data for the purpose of reconstructing the characteristics of the Earth’s flank magnetopause has been carried out in this direction. However, there are other relevant features and signatures of non-ideal effects that one could seek for in the available satellite data, as, for instance, the equilibrium profiles presented here would be supported by agyrotropic particle distribution functions localized in the large-scale shear-flow layer at the Earth’s magnetopause ⁷ .

4 Conclusions

We have derived the one-dimensional equilibrium solutions for a shear-flow boundary layer within a so-called ‘extended two-fluid’ (eTF) model accounting for first-order ion finite-Larmor-radius (FLR) corrections in the double-adiabatic limit. These analytical solutions represent a generalization of the solutions given in Cerri et al. 2013.

We have explicitly shown that first-order FLR corrections exhibit what we have called ‘ $\unicode[STIX]{x1D74E}\boldsymbol{b}$ asymmetry’, i.e. an asymmetry that depends on the relative orientation of the fluid vorticity, $\unicode[STIX]{x1D74E}$ , and of the magnetic-field direction, $\boldsymbol{b}$ , through the scalar product $\unicode[STIX]{x1D74E}\boldsymbol{\cdot }\boldsymbol{b}$ . Moreover, depending again on the parameter $\unicode[STIX]{x1D74E}\boldsymbol{\cdot }\boldsymbol{b}$ , it has been demonstrated that the free energy available in the shear flow is able to develop and sustain a non-negligible level of agyrotropy, i.e. a pressure (and temperature) anisotropy that is not limited to the directions parallel and perpendicular to the magnetic field (the so-called gyrotropy), but that manifests also within the plane perpendicular to $\boldsymbol{b}$ as $p_{\Vert }\neq p_{\bot ,1}\neq p_{\bot ,2}$ .

Finally, we have applied these FLR-corrected equilibrium profiles to few cases with parameters typical of the low-latitude flanks of the Earth’s magnetopause. The resulting current structure has been shown to be more complex than the MHD layer by Chapman & Ferraro (1930), in qualitative agreement with the Cluster observations recently reported in Haaland et al. (2014). In particular, by accounting for ion-FLR effects, we have been able to qualitatively reproduce the following key observational features: (i) an asymmetry of the current layer with respect to the dusk and the dawn sides of the magnetopause, (ii) a double-peak feature arising in the current profiles and (iii) the presence of adjacent current sheets having thickness of several ion Larmor radii and with different current directions. We want to stress that other effects that may contribute to further corrections have been neglected, e.g. the full ion pressure-tensor dynamics and the electron kinetic effects, so a quantitative comparison between the Cluster data and our profiles would be beyond the scope of the present work. Nevertheless, the good qualitative agreement between our one-dimensional analytical profiles and the Cluster observations reported in Haaland et al. (2014) shows that ion-FLR corrections are a relevant ingredient to correctly describe the Earth’s flank magnetopause layer. Further effects, including a three-dimensional treatment of the magnetosphere–wind interface, as well as the full ion pressure tensor and self-consistent electron kinetic effects, will clearly have to be considered for a more quantitative comparison. In this regard, new and future space missions will also provide better measurements of the Earth’s magnetopause structure and allow for a deeper understanding of the relevant plasma physics at play.

Finally, we underline that the main consequences of the ion-FLR effects reported in this work, and their relation to anisotropy, agyrotropy, vorticity and current sheets, may have implications for a wide variety of astrophysical and space collisionless plasmas, from the turbulent solar wind to low-luminosity accretion flows around compact objects.