Frozen-in condition for ions and electrons: implication on magnetic flux transport by dipolarizing flux bundles

The ability of dipolarizing flux bundles (DFBs) in transporting magnetic flux from the mid-tail reconnection site for near-Earth dipolarization is evaluated by two methods: the generalized Ohm’s law and the concept of flux preserving and line preserving. From the generalized Ohm’s law, the breakdown of the frozen-in condition (FIC) for ions is shown to be intimately related to that for electrons. When FIC is not satisfied for the ion fluid associated with energy conversion, it also implies the same for the electron fluid. When FIC holds, the plasma has the flux preserving property. It further guarantees that charged particles on a given magnetic field line will stay together on a magnetic field line at later times, i.e., line preserving. Conversely, when line preserving does not hold, flux preserving does not hold also. Previous detailed examination on the FIC for DFBs revealed that the majority of DFBs associated with energy conversion violate the FIC for the ion fluid. This implies that FIC does not hold for the electron fluid also. Furthermore, plasmas in substorm injections come from vastly different locations, violating the line preserving property and implying that FIC is broken for the magnetic flux tubes associated with substorm injection and dipolarization. These observations indicate that DFBs are not an effective agent to transport magnetic flux within the magnetosphere and further imply that midtail magnetic reconnection is rather ineffective in transporting magnetic flux for near-Earth dipolarization. © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Background Magnetic field reconfiguration, commonly referred to as dipolarization, is frequently observed with plasma injection in the near-Earth region (XGSM > − 15 RE) for typical magnetospheric substorms (e.g., Akasofu 1968; DeForest and McIlwain 1971; McPherron et al. 1973). One common idea of linking mid-tail magnetic reconnection at XGSM ≈ − 20 RE to near-Earth dipolarization is through the consideration of magnetic flux transport by plasma flows from magnetic reconnection. Magnetic structures associated with Earthward reconnection flows have been observed to exhibit transient large northward swings in the magnetic field called dipolarization fronts (DFs) (Nakamura et al. 2002; Runov et al. 2009, 2011; Schmid et al. 2011). Liu et al. (2013, 2014) studied the magnetic flux transport associated with DFs, referring them as dipolarizing flux bundles (DFBs) and proposing them as elementary elements for the substorm current wedge (SCW). In this viewpoint, DFBs are considered to transport magnetic flux from the mid-tail to the near-Earth region for SCW development. For this scenario to be realizable, DFBs must satisfy the criterion for visualizing magnetic flux within them to be carried by the plasma bulk flow, which is the criterion for the frozen-in condition (FIC). In contrast to the above viewpoint, there are several proposed processes in which substorm dipolarization in the near-Earth region is produced close to the Earth (e.g., Lui et al. 1988; Lopez et al. 1988, 1989; Roux et al. 1991; Lui 1991, 2011, 2013; Cheng and Lui 1998; Henderson 1994, 2009; Liu 1997; Cheng 2004; Liu et al. 2012; Haerendel et al. 2012; Haerendel and Frey 2014; Akasofu 2013, 2017). For comparison between these two viewpoints, it is important to address how effective are DFBs in transporting magnetic flux from the mid-tail region to the near-Earth region. Open Access *Correspondence: [email protected] JHU/APL, Laurel, MD 20723-6099, USA Page 2 of 7 Lui Geosci. Lett. (2018) 5:5 The frozen‐in condition The FIC was introduced by Alfvén (1942) to visualize the properties of the low-frequency electromagnetic Alfvén waves resulting from the combination of the hydrodynamic equations with Maxwell equations. It is expressed by the criterion E + V × B = 0, where E is the electric field, V is the plasma bulk velocity, and B is the magnetic field. The validity of this condition allows one to visualize transport of magnetic flux through the plasma fluid motion. If this condition does not hold, then magnetic field line motion is not applicable, since it is ill-defined as it does not tie to the plasma fluid motion. As a result, magnetic flux transport cannot be visualized to be carried by the plasma fluid motion. Therefore, the ability of DFBs to transport magnetic flux from the mid-tail to the near-Earth region requires the validity of the FIC along their entire paths. Whether or not the FIC is satisfied can be examined with the generalized Ohm’s law, which is essentially the electron momentum equation. Adopting the good approximation that the ion fluid velocity Vi represents well the plasma bulk velocity, this law can be written in International system (SI) units as (e.g., Parks, 2004, p. 296): where J is the current density, n is the number density, e is the elementary electric charge, and D denotes the sum of terms associated with non-ideal magnetohydrodynamics (MHD) effects from inertial, electron viscosity, and anomalous resistivity. If the FIC is broken only by the first term on the RHS (the Hall term), then there is no energy conversion (dissipation or dynamo), because the triple product J·J × B is precisely zero. Without energy conversion, DFBs lack the ability to have dynamo effect to drive field-aligned currents and to be a part of an elementary SCW. Furthermore, they cannot energize particles, because no dissipation is involved in this situation. Therefore, for DFBs to possess these properties, the breakdown of FIC for DFBs must involve D ≠ 0. By moving the Hall term on the RHS to the LHS and noting that J/ne = Vi − Ve, Eq. (1) becomes where Ve is the electron bulk velocity. Therefore, for situation when DFBs exhibit dynamo effects and/or particle energization associated with breakdown of the FIC for the ion fluid, the FIC does not hold for the electron fluid also because D ≠ 0 in this situation. In other words, when energy conversion exists and the magnetic flux cannot be viewed as carried by the ion flow, then magnetic flux cannot be viewed as carried by the electron flow also. (1) E + V i × B = J × B