1 Introduction
Current sheets are important for the structure and dynamical behaviour of many plasma systems. In space and astrophysical plasmas current sheets play a crucial role in magnetic activity processes by, for example, aiding the release of magnetic energy by magnetic reconnection. Current sheet equilibria are often used as starting points for studying the dynamic behaviour of plasmas in, e.g. the solar atmosphere, the solar wind and planetary magnetospheres.
Many astrophysical plasmas can be described as collisionless and in this case the relevant equilibria are solutions of the steady-state Vlasov–Maxwell (VM) equations (e.g. Schindler Reference Schindler2007). Since current sheets are strongly localised in space, they can often be well approximated by one-dimensional (1-D) models (see, e.g. Roth, de Keyser & Kuznetsova Reference Roth, de Keyser and Kuznetsova1996; Zelenyi et al. Reference Zelenyi, Malova, Artemyev, Popov and Petrukovich2011; Kocharovsky et al. Reference Kocharovsky, Kocharovsky, Martyanov and Tarasov2016; Neukirch, Wilson & Allanson Reference Neukirch, Wilson and Allanson2018). An often used example of a 1-D current sheet model is the Harris sheet (Harris Reference Harris1962), which is a neutral sheet model that has been used extensively in studies of, e.g. magnetic reconnection (e.g. Kuznetsova, Hesse & Winske Reference Kuznetsova, Hesse and Winske1998; Shay et al. Reference Shay, Drake, Denton and Biskamp1998; Hesse et al. Reference Hesse, Schindler, Birn and Kuznetsova1999; Kuznetsova, Hesse & Winske Reference Kuznetsova, Hesse and Winske2000, Reference Kuznetsova, Hesse and Winske2001; Hesse, Birn & Kuznetsova Reference Hesse, Birn and Kuznetsova2001; Pritchett Reference Pritchett2001; Rogers, Denton & Drake Reference Rogers, Denton and Drake2003; Hesse, Kuznetsova & Birn Reference Hesse, Kuznetsova and Birn2004; Pritchett & Coroniti Reference Pritchett and Coroniti2004; Ricci et al. Reference Ricci, Brackbill, Daughton and Lapenta2004; Hesse et al. Reference Hesse, Kuznetsova, Schindler and Birn2005; Pritchett Reference Pritchett2005; Hesse Reference Hesse2006; Daughton & Karimabadi Reference Daughton and Karimabadi2007; Wan et al. Reference Wan, Lapenta, Delzanno and Egedal2008; Daughton et al. Reference Daughton, Roytershteyn, Karimabadi, Yin, Albright, Bergen and Bowers2011; Hesse et al. Reference Hesse, Neukirch, Schindler, Kuznetsova and Zenitani2011).
In some plasma systems, it can be more appropriate to use a current sheet model for which the pressure gradient is negligible. Such models are termed force free, and satisfy the condition 
$\boldsymbol{j}\times \boldsymbol{B}=0$, i.e. the current density 
$\boldsymbol{j}$ and magnetic field 
$\boldsymbol{B}$ are aligned with each other.
The Harris sheet magnetic field is kept in force balance by a pressure gradient, but one can also keep the system in a macroscopic force balance by adding a non-uniform guide field to the system while the plasma pressure is constant. The resulting configuration is often called the force-free Harris sheet. Equilibrium distribution functions (DFs) for this configuration have been found, for example, by Harrison & Neukirch (Reference Harrison and Neukirch2009a), Neukirch, Wilson & Harrison (Reference Neukirch, Wilson and Harrison2009), Wilson & Neukirch (Reference Wilson and Neukirch2011), Abraham-Shrauner (Reference Abraham-Shrauner2013), Kolotkov, Vasko & Nakariakov (Reference Kolotkov, Vasko and Nakariakov2015), Dorville et al. (Reference Dorville, Belmont, Aunai, Dargent and Rezeau2015), Allanson et al. (Reference Allanson, Neukirch, Wilson and Troscheit2015, Reference Allanson, Neukirch, Troscheit and Wilson2016), Wilson, Neukirch & Allanson (Reference Wilson, Neukirch and Allanson2017, Reference Wilson, Neukirch and Allanson2018) and Neukirch et al. (Reference Neukirch, Vasko, Artemyev and Allanson2020) (for further references on force-free Vlasov–Maxwell equilibria, see e.g. Moratz & Richter Reference Moratz and Richter1966; Sestero Reference Sestero1967; Channell Reference Channell1976; Correa-Restrepo & Pfirsch Reference Correa-Restrepo and Pfirsch1993; Attico & Pegoraro Reference Attico and Pegoraro1999; Bobrova et al. Reference Bobrova, Bulanov, Sakai and Sugiyama2001; Harrison & Neukirch Reference Harrison and Neukirch2009b; Vasko et al. Reference Vasko, Artemyev, Petrukovich and Malova2014).
Similarly to the Harris sheet, collisionless force-free configurations have been used as initial conditions for particle-in-cell simulations of collisionless reconnection, using for example the exact equilibrium by Harrison & Neukirch (Reference Harrison and Neukirch2009a) (e.g. Wilson et al. Reference Wilson, Neukirch, Hesse, Harrison and Stark2016), linear force-free equilibria (e.g. Bobrova et al. Reference Bobrova, Bulanov, Sakai and Sugiyama2001; Nishimura et al. Reference Nishimura, Gary, Li and Colgate2003; Bowers & Li Reference Bowers and Li2007) or approximate force-free equilibria (e.g. Hesse et al. Reference Hesse, Kuznetsova, Schindler and Birn2005; Liu et al. Reference Liu, Daughton, Karimabadi, Li and Roytershteyn2013; Guo et al. Reference Guo, Li, Daughton and Liu2014, Reference Guo, Liu, Daughton and Li2015; Zhou et al. Reference Zhou, Huang, Lu, Xie and Wang2015; Fan et al. Reference Fan, Huang, Lu, Xie and Wang2016; Guo et al. Reference Guo, Li, Daughton, Li and Liu2016a,Reference Guo, Li, Li, Daughton, Zhang, Lloyd-Ronning, Liu, Zhang and Dengb).
In their paper, Harrison & Neukirch (Reference Harrison and Neukirch2009a) also discussed the case of collisionless current sheets that are intermediate cases between the Harris sheet and the force-free Harris sheet, i.e. cases for which the macroscopic force balance is provided by a combination of the plasma pressure gradient and the gradient of the magnetic pressure component provided by the non-uniform guide field. These equilibria and their DFs self-consistently describe the transition from the Harris sheet to the force-free Harris sheet (or vice versa), but have so far not been studied in any detail. Hence, in this paper we present an investigation of these collisionless current sheet equilibria. As this investigation will show, there are actually some problems with the DFs presented in Harrison & Neukirch (Reference Harrison and Neukirch2009a), which limit their usefulness in practice. To circumvent these issues, we present a family of slightly modified magnetic field profiles and corresponding DFs which avoid these problems, but still describe a transition between the Harris sheet and force-free Harris sheet as limiting cases.
We remark that in this paper we focus on a case in which the plasma temperature is uniform across the current sheet (i.e. an isothermal case). Distribution functions for non-isothermal force-free current sheets have been found by, e.g. Kolotkov et al. (Reference Kolotkov, Vasko and Nakariakov2015), Wilson et al. (Reference Wilson, Neukirch and Allanson2017) and Neukirch et al. (Reference Neukirch, Vasko, Artemyev and Allanson2020). In principle the analysis carried out here could be generalised to these non-isothermal cases.
The paper is structured as follows; in § 2, we briefly discuss the macroscopic equilibria of the Harris sheet, the force-free Harris sheet and the intermediate cases. We then discuss the corresponding VM equilibrium distribution functions as given by Harrison & Neukirch (Reference Harrison and Neukirch2009a) in § 3 and illustrate the problem associated with the intermediate cases in the limit when the guide field amplitude tends to zero. In § 4, we present a modified magnetic field model, which allows us to avoid these problems with the distribution function. We close with our summary and conclusions in § 5.
2 The macroscopic picture: Harris sheet, force-free Harris sheet and intermediate cases
The macroscopic force balance for 1-D collisionless current sheet equilibria, with spatial variation only in the 
$z$-direction, is determined by (e.g. Mynick, Sharp & Kaufman Reference Mynick, Sharp and Kaufman1979; Neukirch et al. Reference Neukirch, Wilson and Allanson2018)
$$\begin{eqnarray}\frac{\text{d}}{\text{d}z}\left[\frac{B_{x}(z)^{2}+B_{y}(z)^{2}}{2\unicode[STIX]{x1D707}_{0}}+\unicode[STIX]{x1D617}_{zz}(z)\right]=0.\end{eqnarray}$$ For collisionless equilibria, we are usually dealing with a pressure tensor and 
$\unicode[STIX]{x1D617}_{zz}$ is the only component of the pressure tensor which contributes to the force balance equation.
In this paper we focus on the family of equilibria defined by
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{B}(z)=B_{0}\left(\tanh (z/L),\frac{B_{y0}}{B_{0}\cosh (z/L)},0\right), & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D617}_{zz}(z)=\frac{B_{0}^{2}-B_{y0}^{2}}{2\unicode[STIX]{x1D707}_{0}\cosh ^{2}(z/L)}+P_{b}, & \displaystyle\end{eqnarray}$$ where 
$L$ represents the half-thickness of the current sheet, and 
$P_{b}\geqslant 0$ is a constant background pressure. For completeness, we mention that the current density is given by
$$\begin{eqnarray}\boldsymbol{j}(z)=\frac{B_{0}}{\unicode[STIX]{x1D707}_{0}L}\left(\frac{B_{y0}\sinh (z/L)}{B_{0}\cosh ^{2}(z/L)},\frac{1}{\cosh ^{2}(z/L)},0\right).\end{eqnarray}$$ The case 
$B_{y0}=0$ gives the Harris sheet (Harris Reference Harris1962), which is a widely used 1-D VM equilibrium in, e.g. reconnection studies, Often a constant guide field component is added to the Harris sheet field, which is not included in our magnetic field model here. When 
$B_{y0}=B_{0}$, we obtain the force-free Harris sheet, for which both the pressure 
$\unicode[STIX]{x1D617}_{zz}=P_{b}$ and the magnetic pressure 
$(B_{x}^{2}+B_{y}^{2})/2\unicode[STIX]{x1D707}_{0}=B_{0}^{2}/2\unicode[STIX]{x1D707}_{0}$ are constant. For 
$0<B_{y0}<B_{0}$ we get intermediate cases between the Harris sheet and the force-free Harris sheet. Figure 1 shows the magnetic field, pressure, and current density profiles for the Harris sheet, two intermediate cases, and force-free Harris sheet. Note that in the figure we have set the background pressure 
$P_{b}$ (measured in units of 
$B_{0}^{2}/(2\unicode[STIX]{x1D707}_{0})$) to 
$B_{y0}^{2}/B_{0}^{2}+0.1$.
Figure 1. Magnetic field, pressure and current density profiles for (a) the Harris sheet, (b,c) intermediate cases with 
$B_{y0}/B_{0}\approx 0.25$ and 
${\approx}0.75$, respectively, and (d) the force-free Harris sheet. The background pressure 
$P_{b}$ (measured in units of 
$B_{0}^{2}/(2\unicode[STIX]{x1D707}_{0})$) has been set to 
$B_{y0}^{2}/B_{0}^{2}+0.1$ in each case.
At the macroscopic level discussed so far, there is no problem with varying 
$B_{y0}/B_{0}$ and in particular with letting this ratio go to 0. This changes, however, when we consider the microscopic picture.
3 The microscopic picture
3.1 One-dimensional Vlasov–Maxwell equilibria
We assume a 1-D Cartesian set-up, in which all quantities depend only on the 
$z$-coordinate, and consider magnetic field profiles of the form 
$\boldsymbol{B}=(B_{x},B_{y},0)$, for which 
$\boldsymbol{B}=\unicode[STIX]{x1D735}\times \boldsymbol{A}$ (for vector potential 
$\boldsymbol{A}=(A_{x},A_{y},0)$). In this paper we will always impose conditions on the microscopic parameters of the DFs such that the electric potential 
$\unicode[STIX]{x1D719}$ (and hence the electric field) vanishes (this can always be achieved for the cases we discuss here, see e.g. Neukirch et al. Reference Neukirch, Wilson and Allanson2018). On the microscopic level, we assume that the distributions functions, 
$f_{s}$, are functions of the particle energy, 
$H_{s}=m_{s}(v_{x}^{2}+v_{y}^{2}+v_{z}^{2})/2$, and the 
$x$- and 
$y$-components of the canonical momentum, 
$\boldsymbol{p}_{s}=m_{s}\boldsymbol{v}+q_{s}\boldsymbol{A}$ (for 
$m_{s}$ the mass and 
$q_{s}$ the charge of species 
$s$, respectively), since these are known constants of motion for a time-independent system with spatial invariance in the 
$x$- and 
$y$-directions.
Under the assumptions described above, the VM equations reduce to Ampère’s law in the form
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}^{2}A_{x}}{\text{d}z^{2}}=-\unicode[STIX]{x1D707}_{0}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D617}_{zz}}{\unicode[STIX]{x2202}A_{x}}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}^{2}A_{y}}{\text{d}z^{2}}=-\unicode[STIX]{x1D707}_{0}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D617}_{zz}}{\unicode[STIX]{x2202}A_{y}}, & \displaystyle\end{eqnarray}$$ where 
$\unicode[STIX]{x1D617}_{zz}$ is the only component of the pressure tensor that plays a role in the force balance of the 1-D equilibrium, defined by
$$\begin{eqnarray}\unicode[STIX]{x1D617}_{zz}(A_{x},A_{y})=\mathop{\sum }_{s}m_{s}\int v_{z}^{2}f_{s}(H_{s},p_{xs},p_{ys})\,\text{d}^{3}v.\end{eqnarray}$$ For a specified magnetic field profile, therefore, one needs to determine 
$\unicode[STIX]{x1D617}_{zz}(A_{x},A_{y})$ such that the vector potential associated with the given magnetic field is a solution of Ampère’s law. Regarding (3.3) as an integral equation for 
$f_{s}$ and solving it, will give DFs that self-consistently reproduce this macroscopic field profile (e.g. Alpers Reference Alpers1969; Channell Reference Channell1976; Mottez Reference Mottez2003). For some examples of the application of this approach, see Harrison & Neukirch (Reference Harrison and Neukirch2009a,Reference Harrison and Neukirchb), Neukirch et al. (Reference Neukirch, Wilson and Harrison2009), Wilson & Neukirch (Reference Wilson and Neukirch2011), Abraham-Shrauner (Reference Abraham-Shrauner2013), Kolotkov et al. (Reference Kolotkov, Vasko and Nakariakov2015), Allanson et al. (Reference Allanson, Neukirch, Wilson and Troscheit2015, Reference Allanson, Neukirch, Troscheit and Wilson2016), Wilson et al. (Reference Wilson, Neukirch and Allanson2017) and Wilson et al. (Reference Wilson, Neukirch and Allanson2018).
3.2 The distribution functions
Harrison & Neukirch (Reference Harrison and Neukirch2009a) used Channell’s method (Channell Reference Channell1976) to find the following DF for the force-free Harris sheet
$$\begin{eqnarray}\displaystyle f_{s}(H_{s},p_{xs},p_{ys})=\frac{n_{0s}}{(\sqrt{2\unicode[STIX]{x03C0}}v_{th,s})^{3}}\text{e}^{-\unicode[STIX]{x1D6FD}_{s}H_{s}}[\text{e}^{\unicode[STIX]{x1D6FD}_{s}u_{ys}p_{ys}}+a_{s}\cos (\unicode[STIX]{x1D6FD}_{s}u_{xs}p_{xs})+b_{s}], & & \displaystyle\end{eqnarray}$$ where 
$n_{0s}$ is a typical particle density for species 
$s$, 
$\unicode[STIX]{x1D6FD}_{s}=(k_{\text{b}}T_{s})^{-1}$ is the usual inverse temperature parameter and 
$v_{th,s}^{2}=k_{\text{b}}T_{s}/m_{s}=(m_{s}\unicode[STIX]{x1D6FD}_{s})^{-1}$ is the square of the thermal velocity of species 
$s$. As discussed in detail in, for example, Neukirch et al. (Reference Neukirch, Wilson and Harrison2009), the additional parameters 
$a_{s}$, 
$b_{s}$, 
$u_{xs}$ and 
$u_{ys}$ have to satisfy further constraints to (i) have a positive DF (
$b_{s}>a_{s}\geqslant 0$), (ii) guarantee that the electric potential vanishes and (iii) ensure that the magnetic vector potential associated with the given macroscopic magnetic field is a solution of Ampère’s law.
The DF (3.4) is the sum of the Harris sheet DF (Harris Reference Harris1962) and an additional part, depending on 
$H_{s}$ and 
$p_{x,s}$
$$\begin{eqnarray}f_{s}(H_{s},p_{xs},p_{ys})=f_{s,\text{Harris}}(H_{s},p_{ys})+\frac{n_{0s}}{(\sqrt{2\unicode[STIX]{x03C0}}v_{th,s})^{3}}\text{e}^{-\unicode[STIX]{x1D6FD}_{s}H_{s}}(a_{s}\cos (\unicode[STIX]{x1D6FD}_{s}u_{xs}p_{xs})+b_{s}),\end{eqnarray}$$where the Harris sheet DF is given by
$$\begin{eqnarray}f_{s,\text{Harris}}(H_{s},p_{ys})=\frac{n_{0s}}{(\sqrt{2\unicode[STIX]{x03C0}}v_{th,s})^{3}}\text{e}^{-\unicode[STIX]{x1D6FD}_{s}(H_{s}-u_{ys}p_{ys})}.\end{eqnarray}$$ Harrison & Neukirch (Reference Harrison and Neukirch2009a) pointed out that by varying 
$a_{s}$ the DF (3.4) can in principle describe all the intermediate cases between the force-free and Harris cases and it looks as if in the limit 
$a_{s}\rightarrow 0$ one should recover the Harris sheet DF.
However, if one looks more carefully one finds that the parameter 
$a_{s}$ has to satisfy the relation (similarly to e.g. Neukirch et al. Reference Neukirch, Wilson and Harrison2009)
$$\begin{eqnarray}a_{s}=\frac{B_{y0}^{2}}{2B_{0}^{2}}\exp \left(\frac{u_{xs}^{2}}{2v_{th,s}^{2}}\right)\exp \left(\frac{u_{ys}^{2}}{2v_{th,s}^{2}}\right),\end{eqnarray}$$where
$$\begin{eqnarray}u_{xs}^{2}=\frac{4}{B_{y0}^{2}\unicode[STIX]{x1D6FD}_{s}^{2}q_{s}^{2}L^{2}}.\end{eqnarray}$$ This particular form for 
$a_{s}$ results from the consistency relations that the distribution functions have to satisfy for the electric potential calculated from the quasi-neutrality condition to vanish identically (this is a pre-requisite for being able to apply the method by Channell (Reference Channell1976)). Hence, in the limit 
$B_{y0}\rightarrow 0~a_{s}$ does not go to zero, but to 
$\infty$, which is unacceptable for the distribution function. For finite 
$B_{y0}$, 
$a_{s}$ is finite but it increases rapidly as a function of 
$B_{y0}$.
This leads to further unwanted properties of the DF, which we illustrate in figures 2 and 3. Figure 2 shows the DF as a function of 
$v_{x}/v_{th,s}$, for 
$z/L=0.5$ and 
$v_{y}=v_{z}=0$, and how it changes as 
$B_{y0}/B_{0}$ decreases from 1.0 to 0.1. The maximum values of the DFs in figure 2 are normalised to unity (i.e. we have divided the DFs by their maximum value for a given ratio 
$B_{y0}/B_{0}$). As one can clearly see in figure 2 the DF develops more and more maxima and minima in the 
$v_{x}$-direction, due to the dominance of the cosine term in the distribution function caused by the increase in 
$a_{s}$. It must be suspected that this filamentation in velocity space might lead to instabilities. We also point out that the parameter 
$b_{s}$ has to increase as well so that 
$b_{s}>a_{s}$ to keep the DF positive (in the plots we have used 
$b_{s}=1.5\,a_{s}$). Figure 3 shows on a logarithmic scale how the maximum value of the DF (for the fixed values of 
$z$, 
$v_{y}$ and 
$v_{z}$) increases dramatically as 
$B_{y0}/B_{0}$ decreases.
Figure 2. Variation of the DF with 
$v_{x}/v_{th,s}$ for four different values of 
$B_{y0}/B_{0}$: panel (a) 
$B_{y0}/B_{0}=1.0$, panel (b) 
$B_{y0}/B_{0}=0.7$, panel (c) 
$B_{y0}/B_{0}=0.4$, panel (d) 
$B_{y0}/B_{0}=0.1$. Here, each DF has been normalised by its maximum value and we have chosen 
$z/L=0.5$ and 
$v_{y}=v_{z}=0$.
Figure 3. Variation of the DF maximum with 
$B_{y0}/B_{0}$.
Taken together his clearly shows not only that the limit 
$B_{y0}/B_{0}\rightarrow 0$ does not exist and that hence there is no smooth transition to the Harris sheet DF, but that the family of DFs will not be very useful even for finite, but small values of the ratio 
$B_{y0}/B_{0}$. The question that arises is: can one find a family of VM equilibrium DFs which provides a smooth transition from the force-free Harris sheet to the Harris sheet?
4 Alternative intermediate cases
In this section, we will consider an alternative magnetic field profile to that in (2.2), of the form
$$\begin{eqnarray}\boldsymbol{B}(z)=B_{0}\left(\tanh (z/L),\frac{\unicode[STIX]{x1D706}}{\cosh (\unicode[STIX]{x1D706}z/L)},0\right),\end{eqnarray}$$where we have defined the abbreviation
$$\begin{eqnarray}\unicode[STIX]{x1D706}=\frac{B_{y0}}{B_{0}}.\end{eqnarray}$$A similar, albeit not totally identical magnetic field profile has previously been used by Huang et al. (Reference Huang, Xu, Yan, Zhang and Yu2017) to study instabilities using this type of collisionless current sheet. We will show that this magnetic field profile can be used to consistently describe a transition from the Harris sheet to the force-free Harris sheet.
For completeness we here also state the current density and 
$zz$-component if the pressure tensor associated with this field, which are given by
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{j}(z)=\frac{B_{0}}{\unicode[STIX]{x1D707}_{0}L}\left(\frac{\unicode[STIX]{x1D706}^{2}\sinh (\unicode[STIX]{x1D706}z/L)}{\cosh ^{2}(\unicode[STIX]{x1D706}z/L)},\frac{1}{\cosh ^{2}(z/L)},0\right), & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D617}_{zz}(z)=\frac{B_{0}^{2}}{2\unicode[STIX]{x1D707}_{0}}\left[\frac{1}{\cosh ^{2}(z/L)}+\unicode[STIX]{x1D706}^{2}\left(1-\frac{1}{\cosh ^{2}(\unicode[STIX]{x1D706}z/L)}\right)\right]+P_{b2}, & \displaystyle\end{eqnarray}$$ respectively, where 
$P_{b2}\geqslant 0$ is a constant background pressure. We remark that we have written the non-background part of the pressure in such a way that it is always positive, regardless of the value of the positive constant 
$P_{b2}$.
For 
$\unicode[STIX]{x1D706}=0$, the magnetic field (4.1) becomes the Harris sheet field, and for 
$\unicode[STIX]{x1D706}=1$ it becomes the force-free Harris sheet field. Setting 
$P_{b}=P_{b2}+B_{0}^{2}/2\unicode[STIX]{x1D707}_{0}$ in (2.3) for 
$\unicode[STIX]{x1D706}=1$ (
$B_{y0}=B_{0}$) will make the two pressure functions equal in that case. The range 
$0<\unicode[STIX]{x1D706}<1$ can be thought of as describing intermediate fields between the Harris and force-free Harris sheets, although the guide field and pressure profile deviates from the previous intermediate cases. Figure 4 shows magnetic field, pressure and current density profiles for the Harris sheet (
$\unicode[STIX]{x1D706}=0.0$), two intermediate cases with 
$\unicode[STIX]{x1D706}=0.253$ and 
$\unicode[STIX]{x1D706}=0.747$ and the force-free Harris sheet (
$\unicode[STIX]{x1D706}=1.0$). The background pressure has been chosen in the same way as for figure 1. The only obvious difference to the plots shown in figure 1 are the slight dips in the pressure profile (local minima) at the edges of the current sheet. On comparison with figure 1, we also see that decreasing 
$\unicode[STIX]{x1D706}$ results in a widening of the 
$B_{y}$ profile, due to the 
$\unicode[STIX]{x1D706}$ factor inside the 
$[\cosh (\unicode[STIX]{x1D706}z/L)]^{-1}$ in the 
$y$-component of (4.1). The amplitude of 
$B_{y}$ decreases in the same way as in the other case as 
$\unicode[STIX]{x1D706}$ decreases, and eventually heads to zero as 
$\unicode[STIX]{x1D706}\rightarrow 0$.
Figure 4. Magnetic field, pressure and current density for the alternative intermediate cases with (a) 
$\unicode[STIX]{x1D706}=0.0$, (b) 
$\unicode[STIX]{x1D706}=0.253$, (c) 
$\unicode[STIX]{x1D706}=0.747$ and (d) 
$\unicode[STIX]{x1D706}=1.0$. The only noticeable change compared to the profiles shown in figure 1 are the slight dips (minima) in the pressure profile at the edge of the current sheet. The changes in the 
$B_{y}$ and 
$j_{x}$ profiles are not immediately obvious without direct comparison.
It is straightforward to show that this macroscopic magnetic field profile is consistent with the DF in (3.4). One could suspect that this leads to the same problem with the limit 
$B_{y0}\rightarrow 0$ as before, but when checking the constraints on the parameters of the DF one finds that while one still has
$$\begin{eqnarray}a_{s}=\unicode[STIX]{x1D706}^{2}\exp \left(\frac{u_{xs}^{2}}{2v_{th,s}^{2}}\right)\exp \left(\frac{u_{ys}^{2}}{2v_{th,s}^{2}}\right),\end{eqnarray}$$ as before, the condition for 
$u_{xs}$ has changed to
$$\begin{eqnarray}u_{xs}^{2}=\frac{4\unicode[STIX]{x1D706}^{2}}{B_{y0}^{2}\unicode[STIX]{x1D6FD}_{s}^{2}q_{s}^{2}L^{2}}=\frac{4}{B_{0}^{2}\unicode[STIX]{x1D6FD}_{s}^{2}q_{s}^{2}L^{2}},\end{eqnarray}$$ which no longer varies with 
$B_{y0}$. Therefore, in this case we indeed find that 
$\unicode[STIX]{x1D706}\rightarrow 0$ implies 
$a_{s}\rightarrow 0$, as desired.
Figure 5. Variation of the DF with 
$v_{x}/v_{th,s}$ for different values of 
$\unicode[STIX]{x1D706}=B_{y0}/B_{0}$. Here, the DF has been normalised by its maximum value and we have chosen 
$z/L=0.5$ and 
$v_{y}=v_{z}=0$. As one can see there is very little change as 
$\unicode[STIX]{x1D706}$ decreases.
It is, however, prudent to also have a look at the DFs and their maximum value as 
$\unicode[STIX]{x1D706}\rightarrow 0$. We show plots of the variation of the DF with 
$v_{x}$ (for 
$z/L=0.5$ and 
$v_{y}=v_{z}=0$) in figure 5, for decreasing values of 
$\unicode[STIX]{x1D706}$. For this case we can actually take the limit 
$\unicode[STIX]{x1D706}\rightarrow 0$ without any problem (see panel d). As in figure 2, we have normalised each DF to its maximum value. The variation of this maximum value with decreasing 
$\unicode[STIX]{x1D706}$ is shown in figure 6. For this case the maximum of the DF actually decreases as 
$\unicode[STIX]{x1D706}$ and hence 
$B_{y0}$ decreases. With a relatively simple modification of the magnetic field profile we have managed to eliminate the singular limit.
Figure 6. Variation of the maximum of the DF with 
$B_{y0}/B_{0}$ for the alternative magnetic field profile. As one can see the maximum decreases with decreasing 
$\unicode[STIX]{x1D706}$ and it does not diverge in the limit 
$\unicode[STIX]{x1D706}\rightarrow 0$. In contrast to figure 3 here a linear scale can be used for the plot.
5 Summary and conclusions
In this paper, we have discussed collisionless current sheet equilibria that are intermediate cases between the Harris sheet (current density perpendicular to magnetic field direction) and the force-free Harris sheet (current density exactly parallel to the magnetic field direction). Such a family of Vlasov–Maxwell equilibrium DF had already been briefly mentioned in Harrison & Neukirch (Reference Harrison and Neukirch2009a). However, as the more detailed investigation presented in this paper shows, this family of DFs is of limited usefulness due to the fact that first of all the limit of the guide field amplitude 
$B_{y0}\rightarrow 0$ is singular in the sense that the maximum of the DF tends to 
$\infty$ and that with decreasing 
$B_{y0}$ the velocity space structure of the DF in the 
$v_{x}$ direction becomes more and more filamentary. We proposed an alternative family of intermediate collisionless current sheet equilibria with a magnetic guide field that has a slightly modified spatial structure. Formally, the DFs associated with this magnetic field remain the same, but the constraints imposed on the DF parameters by the self-consistency condition now allow the maximum value of the DFs to remain not only finite, but at a reasonable level as 
$B_{y0}\rightarrow 0$.
We consider it important both from a theoretical and from a modelling and observational point of view that reasonable self-consistent equilibria of collisionless current sheets are available not only for the two limiting cases of force-free Harris sheet and normal Harris sheet. While some observations can be explained by, for example, force-free current sheet models (e.g. Panov et al. Reference Panov, Artemyev, Nakamura and Baumjohann2011; Artemyev, Angelopoulos & Vasko Reference Artemyev, Angelopoulos and Vasko2019a; Artemyev et al. Reference Artemyev, Angelopoulos, Vasko, Runov, Avanov, Giles, Russell and Strangeway2019b; Neukirch et al. Reference Neukirch, Vasko, Artemyev and Allanson2020), it is to be expected that versions of the intermediate current sheet models are encountered with a greater likelihood than the limiting cases.
Acknowledgements
The authors acknowledge the support of the Science and Technology Facilities Council (STFC) via the consolidated grants ST/K000950/1, ST/N000609/1 and ST/S000402/1 (T.N. and F.W.) and the Natural Environment Research Council (NERC) Highlight Topic grant no. NE/P017274/1 (Rad-Sat) (O.A.). T.N. and F.W. would also like to thank the University of St Andrews for general financial support.
Editor Francesco Califano thanks the referees for their advice in evaluating this article.
