The Euler potentials for two current layers aligned to an ambient homogeneous magnetic field are found. Previous treatment of such a system assumed constant current density in the layers. However, the magnetic field becomes infinite at the edges. The new approach eliminates this inconsistency by introducing an inhomogeneous current density. Euler potentials are constructed semi-analytically for such a system. Charged-particle motion and trapping in it are examined by this representation. Using Euler potentials, the influence of current sheets of zero and non-zero thicknesses on energetic-particle fluxes is investigated, and characteristic flux variations near the sheets are presented. The results can be applied to Birkeland currents.

Euler potentials represent an alternative way of describing a magnetic field.
They are two scalar functions, usually denoted as

Such a relationship relies on the fact that magnetic fields are always
solenoidal; that is, they fulfill the condition

Figure

Upper panels:

First we consider one sheet only and centered at the origin. The magnetic
field is determined from the Biot–Savart law:

Profile of the surface current density

One can verify that the boundary conditions for the magnetic field at the
current sheet are satisfied. They generally read

Magnetic field at the edge is

Magnetic field of our system is given by

When

First we search Euler potentials

We construct a set of curvilinear orthogonal coordinates

The curvilinear coordinates of the system of two current sheets (plus

Figure

Lead

To construct the pair

We shall consider anti-parallel case, which has relevance to Birkeland
currents, and drop “

Figure

Examples of particle mirroring at magnetic field lines for

Figure

Magnetic field magnitude at the sheet edges as a function of

Figure

The current density along the sheet (Fig.

In order to compare effects on charged particles of the current systems with
zero and non-zero sheet thicknesses, we investigate particle flux variations
which would observe a hypothetic spacecraft moving perpendicularly through
the sheets. Besides the comparison, the simulated observations may serve as
examples of what a spacecraft could measure during a superstorm. At present,
such observations are not available because no superstorm with a strength
comparable to the 1859 Carrington event

Consider charged particles moving adiabatically in a magnetic field
system. The number density of the particles is given by

Figure

Profiles of the surface current density

The hemispheric flux along the field line is calculated using the Liouville
theorem. The flux

For the initially isotropic distribution (

Sketch of a magnetic field line with values of magnetic field
magnitudes

Figure

Magnetic field and particle flux variations near the sheets.

Changing

Figure

Figure

The

In a previous work

In the present paper, we consider a current distribution in which the current
falls to zero at the edges (in a trapezoidal profile). This removes the field
divergence while retaining the field configuration very close to the former
one. Therefore former results and conclusions of

Using Euler potentials we investigated the influence of current sheets of zero
and non-zero thicknesses on energetic-particle fluxes and analyzed their
characteristic variations near the sheets. The variations might serve as a
tool to reveal geometric parameters of the sheets. Figure

This work is supported by project 14-19376S from GA ČR. We also acknowledge support from the AV ČR grant RVO:67985815. The topical editor, E. Roussos, thanks two anonymous referees for their help in evaluating this paper.