A specific class of solutions of the Vlasov–Maxwell equations, developed by means of generalization of the well-known
Harris–Fadeev–Kan–Manankova family of exact two-dimensional equilibria, is studied. The examined model reproduces the current
sheet bending and shifting in the vertical plane, arising from the Earth dipole tilting and the solar wind nonradial
propagation. The generalized model allows magnetic configurations with equatorial magnetic fields decreasing in a tailward direction as
slow as

Studies of magnetosphere dynamics, including substorm events, require a relevant current sheet (CS) stability analysis. This in turn
requires a proper choice of the background magnetoplasma configuration. In applications to collisionless plasma, the background
equilibrium is to be derived from a solution of the kinetic Vlasov–Maxwell equations. A number of such solutions are derived both
numerically

This result is in line with previous findings revealing that the configuration asymmetry can be an important factor of magnetosphere
dynamics. Particularly,

Later, this effect was investigated in detail in the paper of

In

The first exact solution for two-dimensional (2-D) equilibrium bent CS with nonzero dipole tilt was presented in short notes of

The paper is organized as follows. In Sect.

For two-component (proton

A series of analytical solutions of Eq. (

The particular choice of the generating
function

The solution for a bent CS is developed in the paper of

Magnetic potential

X-point location vs. the effective tilt angle

Flux tube volume

Flux tube volumes: analytical solution (red curves) and T96 (black curves) for quiet

Comparison of analytical and empirical models for symmetric (dipole tilt

The values of magnetic potential

Profiles

Current density

Magnetic field component

Analytical model units: current density

Typical current density

The set of magnetic configurations for dipole tilt angle PHI

Topologically, magnetic configurations plotted in Fig.

To explore the appropriateness of the here presented analytical solution for bent CS, we compare the predicted magnetic flux tube volume
(a proxy for the entropy) with that calculated from the empirical model of

The FTV is determined in the same way for both analytical and empirical models: we integrate

Then, FTVs, calculated by means of analytical and empirical models, are compared at different levels of magnetospheric activity,
characterized by input parameters of the T96 model (Dst index, the SW dynamical pressure,

One can see that the agreement between two models is quite good, with the maximal SDs varying within 2–11 %. The values of

Figure

The results of the previous section show that parameters of the asymmetric Kan-like model may be adapted to provide rather good
agreement with the magnetotail CS, especially in a distant tail beyond 15–20

Parameter

In two dimensions, contributions of parameters

Figure

Figure

The solution (Eqs.

Figure

In empirical models (T89, T96, T01, TS05, etc.) magnetic field configurations with any plasma populations are not force-balanced since

To validate the obtained analytic solution for bent CS we performed a comparison with the T96 model, used as a proxy of realistic
averaged magnetospheric configuration. It is shown that the proposed model provides a reasonable approximation for the magnetotail CS
in a wide range of dipole tilt angles and geomagnetic activity levels. Particularly, the parameters of the analytical model can always be
adjusted to fit the behavior of the magnetic FTV with an accuracy of about

Notably, such a good agreement is obtained for the simplest three-parametric Kan-like model
(Eqs.

Of course, the suggested analytical model is still far from universality. One significant limitation of this model is related to the
isothermal constraint. This constraint may be released for four-component (two positive

The constancy of the proton temperature is not reflected in observations

Other model limitations are the two-dimensionality and isotropy of the plasma pressure. Even with these limitations, the model stays
appropriate for a wide class of problems, mentioned in the beginning of the current section. Particularly, we lay hopes that
application of the presented model can stimulate investigations on the magnetotail CS stability to resolve the questions suggested by

Our findings are summarized as follows:

An exact 2-D bent CS equilibrium, built by means of generalization of the Harris–Fadeev–Kan–Manankova family of symmetric solutions of the Vlasov–Maxwell equations, is considered. The examined model reproduces the effects, related to the Earth dipole tilt and CS bending. The further generalization releases degeneracy of the original model, which caused of the normal magnetic component to decrease too rapidly.

Parameters of the asymmetric model may be adjusted to reproduce the realistic distribution of the magnetic flux tube volume at
any level of geomagnetic activity; with enhancing activity the model relevance improves. The model-typical scales for CS width and
current density match the corresponding parameters of the in situ registered single-peaked current sheets with medium values
of number density, proton temperature and drift velocity; disagreement does not exceed a factor of

The asymmetric solution does not contain any limitation for the tilt angle values, and hence the model is appropriate for any Earth-like magnetosphere with arbitrary dipole inclination.

The obtained bent CS solution contains the X point, moving from infinity toward the dipole with the dipole tilt increase, staying
still far beyond the lunar orbit for the Earth magnetotail realistic tilt angles. The location of the X point
is much more effectively controlled by the new parameter

No data sets were used in this article.

The authors declare that they have no conflict of interest.

This study has been supported by the Austrian Science Fund (FWF), P 27012-N27 and I 3506-N27, and by Russian Science Foundation (RSF) grant no. 18-47-05001. The authors thank Anna V. Egorova for her help with preparation of the images, and reviewers for their help in improving the paper. The topical editor, Elias Roussos, thanks two anonymous referees for help in evaluating this paper.