A new shock fitting procedure for the MHD Rankine‐Hugoniot relations for the case of small He²⁺ slippage

Abstract

[1] To study MHD shocks in space, it is important to find the shock frame of reference from the observed plasma and magnetic field parameters. These shock parameters have to satisfy the Rankine-Hugoniot relations. In this study we present a novel procedure for shock fitting of the one-fluid anisotropic Rankine-Hugoniot relations and of the time difference between two spacecraft observations in the case of small He 2+ slippage. Here, a Monte-Carlo calculation and a minimization technique are used. The observed variables including the upstream and downstream magnetic fields, plasma densities, plasma betas, plasma anisotropies, W (the difference between the downstream and upstream velocities, W V 2 À V 1 ), and Dt (the time difference between two spacecraft observations) are used in our procedure where V is defined as the center of mass velocity of plasmas. A loss function based on a difference between the calculated and the observed values is defined, and the best fit solution is found by searching for the minimum loss function value. For shocks that cannot be fitted well, we introduce two new parameters in the modified RH relations, one in the normal momentum flux and the other in the energy flux equations. These two parameters are interpreted as the equivalent ''normal momentum'' and ''heat'' fluxes needed in the RH relations. They provide two degrees of freedom in the system, and their amounts can be estimated from our procedure. Several synthetic shocks are given to verify our procedure. We also apply this procedure to two interplanetary shocks observed by both the WIND and Geotail spacecraft. The results demonstrate that our method works for both the synthetic and the real shocks. We have shown that our method can provide accurate shock normal estimations for perpendicular and parallel shocks as well. Given that our model is based on the RH relations that do not include the effect of alpha particle (He 2+ ) slippage, it can only be applied to the cases with an ignorable slippage pressure tensor. We have investigated the pressure tensor due to alpha particle slippage using the WIND spacecraft data. It is found that in general the slippage pressure is small in comparison with the thermal pressure of the system and can be ignored. Thus our model can be applied to most interplanetary shocks observed near the ecliptic plane. However, when the slippage pressure is large, the magnetic coplanarity theorem is not valid any more. A more general model that involves slippage pressure tensor is a major and important development that is beyond the scope of the present study.

Observations of an Interplanetary Intermediate Shock Associated With a Magnetic Reconnection Exhaust

“…Then a minimization technique is used; with this, a best fit solution that satisfies the R-H relations and that is closest to the data mean is obtained. More detailed descriptions about the procedure were given by Lin et al (2006).…”

“…In searching for a shock frame, several methods have been developed to determine the shock normal vector n s , such as the coplanarity method mentioned above, the minimum variance analysis method, and the triangulation method (Knetter et al 2004). Here, we apply a shock fitting procedure based on the Monte Carlo calculation proposed by Lin et al (2006). Lin et al (2006) use the upstream and downstream observed mean variables and associated errors to randomly generate arrays of the variables needed for the fitting procedure, and every generated array for these variables satisfy the R-H relations.…”

“…Here, we apply a shock fitting procedure based on the Monte Carlo calculation proposed by Lin et al (2006). Lin et al (2006) use the upstream and downstream observed mean variables and associated errors to randomly generate arrays of the variables needed for the fitting procedure, and every generated array for these variables satisfy the R-H relations. Each array is generated by a random number generator function, called Rnd (σ).…”

“…In order to determine whether or not the leading boundary is an IS, we fitted its upstream and downstream data with the R-H relations using the fitting procedure of Lin et al (2006). The selected up-and downstream intervals of the DD are indicated by the vertical lines in Figure 2.…”

A theoretical and simulation study of the contact discontinuities based on a Vlasov simulation code

“…For example, the ion-electron temperature ratio, the pressure anisotropy, and the gradient of the heat flux are all free parameters in the MHD R-H jump conditions. It has been shown that both the pressure anisotropy and the gradient of the heat flux can modify the R-H jump conditions of the shock waves [e.g., Chao, 1970;Chao and Goldstein, 1972;Lyu and Kan, 1986;Lin et al, 2006] and the rotational discontinuities [Hudson, 1971]. It should also be true for the CDs.…”