On application of asymmetric Kan-like exact equilibria to the Earth magnetotail modeling

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 1/x, contrary to the original Kan model (1/x3); magnetic configurations with a single X point are also available. The analytical solution is compared with the empirical T96 model in terms of the magnetic flux tube volume. It is found that parameters of the analytical model may be adjusted to fit a wide range of averaged magnetotail configurations. The best agreement between analytical and empirical models is obtained for the midtail at distances beyond 10–15RE at high levels of magnetospheric activity. The essential model parameters (current sheet scale, current density) are compared to Cluster data of magnetotail crossings. The best match of parameters is found for single-peaked current sheets with medium values of number density, proton temperature and drift velocity.

This result is in line with previous findings revealing that the configuration asymmetry can be an important factor of magnetosphere dynamics. Particularly, Kivelson and Hughes (1990) have first suggested that the CS bending may drop down the reconnection onset threshold. This idea was confirmed later, when Partamies et al. (2009) have noticed the seasonal variations in the number of substorm events with maximums in winter and summer periods, when dipole tilt angle is bigger (the known geomagnetic activity maximums, e.g., in Kp index, are registered contrary around the equinoxes). 5 Later, this effect was investigated in details in the paper of Kubyshkina et al. (2015), where it was shown that the substorm probability is higher for about 10 − 25% during the periods with tilt angle > 15 • , as compared to the periods with smaller tilt angles. The direction of the solar wind (SW) flow also affects the substorm probability, it grows for 10 − 20% when SW flow direction enforces the CS tilt to encrease. The statistical analysis has shown that the average substorm intensity (defined by AL value during the event) is lower for larger effective tilts (dipole tilt angle plus solar wind flow inclination). In other words, 10 a large number of weak substorms occur in those time intervals when effective tilt angles are high, and less number of more intense substorms is observed when tilt angles are small. This agrees also with the results of Nowada et al. (2009) study, where both AL and AU indices were analyzed for the intervals of negative interplanetary magnetic field B z .
In the same paper of Kubyshkina et al. (2015), the dependence of magnetotail lobe magnetic field (as a proxy of the magnetic flux) on the dipole tilt angle was studied by means of empirical modeling. The average lobe field was found to be smaller for 15 all radial distances in a case of non-zero tilt angles, the decrease reached 10 − 20% for maximum tilt angle. This result is reasonable under the assumption that substorm onsets require a lower energy input during the periods of increased dipole tilt.
Next, in the paper of Semenov et al. (2015) it was found that there is a clear dependence of the substorm probability on the jumps of the z component of the SW velocity (asymmetric factor), while the jumps of number density or plasma pressure (symmetric factor) turn out to be noneffective. At last, we should note that the Earth's dipole tilt angle undergoes daily and 20 seasonal variations in the interval of about ±35 • , so that it is equal to zero twice a day within about 4 months a year, and during other 8 months it is never zero. In addition, the solar wind flow direction varies for about ±6 • . These variations produce CS inclination, bending and shift from the ecliptic plane. Therefore, the simplest solar wind-magnetosphere configuration (vertical dipole, planar CS, radial solar wind) adopted by the majority of models, is rather untypical and the development of the relevant asymmetric CS models is highly-demanded. 25 The first exact solution for two-dimensional (2D) equilibrium bent CS with non-zero dipole tilt was presented in short notes of Semenov et al. (2015). This solution generalizes the well-known Harris-Kan-Fadeev-Manankova equilibria family (see Yoon and Lui (2005)). In the present paper we investigate the obtained solution to estimate its relevance for the magnetotail CS modeling and stability analysis. For this end, we compare the analytical asymmetric solution with the empirical Tsyganenko (1995) T96 model and define the analytical model parameters, providing the best agreement.

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The paper is organized as follows. In section 2 we describe the analytical solution for asymmetric CS. In section 3 we compare analytical and empirical T96 solutions. In section 4 we present the further generalization of the analytical model, providing more realistic profiles of B z in the equatorial plane. Discussion and conclusions finalize the paper in section 5.

5
The quantity Ψ is normalized for (−B 0 L), where L = 2cT i /(eB 0 V i ) is the typical length and B 0 = 8πn 0 (T e + T i ) is the lobe magnetic field, n 0 = n 0e = n 0i is the typical number density, T e,i are the electron and ion temperatures, respectively, and V e,i are the corresponding drift velocities, fulfilling the condition Eq.
(2) expresses the condition of the zero electrostatic potential. The model of an ion-dominated CS, where |V i /V e | > T i /T e is 10 considered in the paper of Yoon and Lui (2004). In the case of Maxwellian distribution functions condition (2) can be satisfied by means of the proper choice of the reference system, in general it cannot be fulfilled (Schindler and Birn (2002)).
A series of analytical solutions of Eq. (1) was found by Walker (1915), shown that the solution may be expressed via an arbitrary generating function g of the complex variable ζ = x + iz, With the solution (3), the equilibrium magnetoplasma configuration takes the form where p is the plasma pressure. By definition, the dimensionless magnetic field components are B x = −∂Ψ/∂z and B z = +∂Ψ/∂x. 20 The particular choice of the generating function g specifies the particular CS model. In the current paper we consider the family of Harris-like models, including the classical Harris (1962) current sheet, the Fadeev et al. (1965) solution (Harris sheet complemented by an infinite chain of magnetic islands along the neutral plane), the Kan (1973) solution (Harris sheet with quasi-dipole), and the Manankova et al. (2000) solution, representing the combination of all previous models. The last one is specified by the generating function Solution (6) where R 2 = x 2 + (z − a) 2 . Asymmetric configurations of this type possess a dipole singularity at (x, z) = (0, a) and two Noteworthy, in the symmetric Kan model the X-point (B x = B z = 0) is located at infinity, but in asymmetric configurations the X-point starts to approach the dipole with increasing tilt angle. The X-point position as a function of ϕ is shown in Fig. 3 for three values of the parameter b 0 . For realistic values of ϕ/2 ≤ 30 • the X-point is still located in the distant tail (far beyond 60 R E ), the approach to 14 R E is achieved for ϕ/2 = 60 • . This X-point is not produced by magnetic reconnection, it is just an 20 attribute of the bent CS steady state equilibrium. However, the appearance of the X-point can be considered as a manifestation of potentially unstable configuration. In such a case, the X-point motion towards dipole with increasing tilt angle could mean that CS evolves toward an unstable state.

Comparison with the T96 model
Topologically, magnetic configurations plotted in Fig. 2 are very similar to that of the Earth's magnetosphere. However, to 25 estimate the relevance of the analytical solution one should compare some important numerical characteristics of the CS model with the corresponding values registered in real observations. This can be done utilizing empirical magnetic field models, providing realistic averaged magnetospheric configurations at various levels of magnetospheric activity. Of course, we should keep in mind that the real magnetosphere is an essentially tree-dimensional structure. Following the dipole tilt (and solar wind flow direction) variations, the magnetotail CS bends and shifts from the equatorial plane in z direction (at most ∼ 3 R E for 30 maximum tilt) and also warps in the y direction. These effects are well-pronounced in empirical magnetospheric models, but the 2D analytical model is evidently unable to reproduce all these complex deformations. Therefore, we restrict our study to the noon-midnight plane y = 0, and the two main effects manifested in that plane: CS bending and shifting in z direction.
To explore the appropriacy of the here presented analytical solution for asymmetric CS, we compare the predicted magnetic flux tube volume (a proxy for the entropy) with that calculated from the empirical model of Tsyganenko (1995) T96. We consider the flux tube volume (FTV) instead of the entropy, since the analytical solution is isothermal. This quantity is chosen 5 due to its importance for the magnetotail dynamics. As it was claimed by Birn et al. (2009) and verified by in-situ data analysis (Sergeev et al. (2014)), any bursty bulk flow (BBF), produced by reconnection in the magnetotail and moving toward the Earth, stops at that particular point, where the entropy of the ambient plasma is equal to that inside the BBF. The distribution of entropy along the magnetotail is also an important factor for the stability analysis (Birn et al. (2009)) and for the study of wave (oscillations) generation and dissipation (Panov et al. (2016)). 10 The FTV is determined in the same way for both analytical and empirical models: we integrate dS/B along magnetic field lines, where dS is the field line length element and B = B 2 x + B 2 z . In the T96 model the location of the flux tube is computed by means of field line tracing, in the analytical model this is a curve of constant Ψ. As a first step, the values of FTV of a single flux tube are compared. The model parameters correspond to the quiet magnetospheric conditions with tilt angle of 30% (see the legend of Fig. 5b). FTVs are calculated along the magnetic field line with a node at (x, z) = (30, −2.4). To eliminate 15 singularities, we excluded the near-Earth region x < 5 R E , so that the total FTVs are calculated as  One can see that the agreement between two models is quite good with the maximal standard deviations varying within 2 − 11%. The values of d max = σ max / F T V are given in legends of Fig. 5, where σ is the standard deviation and F T V is 30 the average FTV. The better agreement is achieved for disturbed magnetospheric conditions, i.e. the analytical model describes the stretched CSs even better than the thicker ones. It is found that minimal difference between two models is obtained when parameter a is very close to the medium neutral sheet position determined from the empirical model. The best-fit value of the parameter b 0 , controlling the field lines stretching and the CS thinning, depends on the level of activity and the dipole tilt angle. It grows from 8.8 for the quiet magnetosphere to 51 for storm conditions. At any fixed distance, the stretching of field lines makes the FTV to decrease with growing magnetospheric activity. E.g., at the distance of x = 30 R E it changes from ≈ 5 R E /nT for "quiet" conditions to ≈ 3 R E /nT for "substorm" conditions and to the ≈ 1.6 R E /nT for "storm-time" conditions. Contrary, the asymmetric deformation of CS (dipole tilt angle) enforces the FTV to increase.

Normal magnetic component
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 R E , and especially for bent current sheets.
However, until now the practical usage of this model encountered the substantial obstacle, related to the behavior of the normal 15 magnetic field component. It can be easily checked that in the distant tail the Kan model yields B z ∼ 1/x 3 , while in reality B z decreases as 1/x or even slower (e.g., Behannon and Ness (1966); Mihalov et al. (1968); Behannon (1970); Wang and Lyons (2004); Yue et al. (2013)). This problem may be solved by introducing one more parameter in the generating function g(ζ).
With the additional parameter n, general asymmetric model takes the form (compare to Eq. 16 of Yoon and Lui (2005))

20
Assuming {f, n, k} to be real values, a = a 1 + ia 2 , and b = b 0 exp(iϕ), we derive For symmetric Kan-like CS without plasmoids (a = 0, f = 0, ϕ = 0), the quantity B z at the x axis takes the simple form

Discussion and conclusions
In empirical models (T89, T96, T01, TS05, etc.) magnetic field configurations with any plasma populations are not force-10 balanced since ∇ × [j × B] = 0, or there is no ∇P to balance Ampere's force (Zaharia et al. (2003)). That is why we crucially need kinetic force-balanced CS models for many magnetospheric studies, such as wave generation in plasma, CS stability analysis, and numerical simulations of magnetotail dynamics. So far these studies were restricted by purely symmetric background equilibria. In this paper we present the extension of the well-known family of exact kinetic Harris-Kan-Faddev-Manankova solutions to the asymmetric 2D CS. This extension is really important, since the Earth dipole is tilted most of the time.

15
To validate the obtained analytic solution for asymmetric CS we performed a comparison with 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 ever be adjusted to fit the behavior of the magnetic FTV with an accuracy of about 10% for all distances from 5 to 30 R E tailward. For short segments (5 R E ) of the CS, located beyond 15 R E , the agreement may be improved up 20 to 5% (except the case of the bent CS at quiet magnetospheric conditions). The agreement between analytical and empirical models is found to be better for the stretched magnetic configuration, i.e., for the pre-substorm conditions.
Notably, such a good agreement is obtained for the simplest three-parametric Kan-like model (7-9), where parameter a controls the CS displacement from the equatorial plane, parameter b 0 controls magnetic field lines stretching, and parameter ϕ specifies the CS bending. For further studies the more general model (11-16) can be considered, where additional parameters n 25 and k provide the more accurate adjustment of the magnetoplasma quantities. More over, for sub-Alfvénic plasma, i.e., for the low-activity periods, all model parameters may be treated as time-dependent quantities (Wolf (1983);Semenov et al. (2015)).
The time-dependent approach in such a modeling is not appropriate for the periods of explosive activity, such as storms and substorms, when BBFs with Alfvénic speed are produced.
Of course, the suggested analytical model is still far from universality. One significant limitation of this model is related to 30 the isothermal constraint. This constraint may be released for four-component (two positive + two negative) plasma with bi-Maxwellian distribution functions for each particle specie (Kan (1973); Voronina and Kan (1993)). In such a case the condition velocity, V i2 −V e2 = 0, the Eq. (1) stays valid for nonuniform plasma temperature (Voronina and Kan (1993)). In application to the Earth magnetotail studies this means that Kan-like models are to be best appropriate for high activity levels. Indeed, in the quiet magnetotail the population of ions {O + , O ++ , He ++ }, penetrating from ionosphere, is less than 1% (Lennartsson et al. (1986)), hence the approximation of "proton+electron" plasma is relevant. In such a case the Kan-like model assumes isothermality, which is not reflected in observations (e.g., Kissinger et al., 2012;Wang et al., 2012). With the growth of geomagnetic 5 activity, the O + contribution becomes essential during intensive storm events in a main and recovery phases, hence the isothermal constraint may be released. This conclusion is in line with the results of our study.
Other model limitations are the two-dimensionality, and isotropy of the plasma pressure. Even with these limitations, the model stays appropriate for the 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 10 question suggested by Kivelson and Hughes (1990): why symmetric CS can accumulate magnetic flux energy more effective, and does the threshold of substorm-initiating instability depend on degree of the CS bending.