Euler Potentials for the Earth Magnetic Field With Field‐Aligned Currents

Abstract

Euler potentials of the Earth's dipole magnetic field with field‐aligned currents are found. The results generalize ones obtained earlier for systems of plane currents parallel to an ambient uniform magnetic field. A real current system corresponding to an observed increase of the magnetic field measured by a low‐altitude spacecraft is used in this modeling. Parameters of charged particle trajectories, such as length and period, are determined for typical and strongly disturbed conditions. A particle distribution function is calculated.

Euler Potentials for Dungey Magnetosphere With Axisymmetric Ring and Field‐Aligned Currents

“…Romashets and Vandas (2020) treated as an example a superstorm case of a Carrington‐Event caliber (Gonzalez et al., 2011; Tsurutani et al., 2003) with an estimated total current I 0 = 30 MA. Here are some other estimated values for the Carrington Event: the average speed of the ejectum about 2,400 km s −1 , E y ≈ 200 mV m −1 , D st ≈ −1,100 nT, and Δ H ≈ 1,600 nT.…”

“…The parameters of the RC yield the value of Δ H ≈ 20 nT as a contribution to the equatorial horizontal surface field. Our g function corresponds to slightly disturbed conditions with the interplanetary electric field E y = 4 mV m −1 (Korth et al., 2010; Romashets & Vandas, 2020) and the total current I 0 ≈ 4.5 MA (as the total current we take an average of unsigned upward and downward currents). For our ordinary case, we consider E y = 2 mV m −1 which is characterized by I 0 = 2 MA (Korth et al., 2010), so our g function is scaled by 0.45 to yield approximately this value.…”

“…Obviously, the magnetic field in the volume is not current‐free. Romashets and Vandas (2020) suggested the following model for it, $$${B}_{tr}=0,\hspace*{.5em}{B}_{t\theta }=0,\hspace*{.5em}{B}_{t\varphi }=\frac{{B}_{0}g\left({\alpha }_{d}\right){r}_{0}}{r\mathrm{sin}\theta },$$$ and showed how the function g can be determined from observations. Its derivative is proportional to FAC.…”

“…We consider a situation with two current layers of zero thickness with FACs (i.e., they are surface currents flowing on two magnetic surfaces of the field B d ), inducing a potential toroidal magnetic field between the layers and no field outside the layers. This magnetic field is (Romashets & Vandas, 2020):…”

“…We consider a situation with two current layers of zero thickness with FACs (i.e., they are surface currents flowing on two magnetic surfaces of the field B d ), inducing a potential toroidal magnetic field between the layers and no field outside the layers. This magnetic field is (Romashets & Vandas, 2020): $$${B}_{tr}=0,\hspace*{.5em}{B}_{t\theta }=0,\hspace*{.5em}{B}_{t\varphi }=\frac{{B}_{0}{g}_{0}{r}_{0}}{r\mathrm{sin}\theta },$$$ where g 0 is a constant determining the strength of the field. The goal is to find a β t potential of this field with α t equal to α d .…”