## 1. Introduction

[2] The Generalized Ohm's Law is

where **E** and **B** are the electric and magnetic fields, **U**_{I} is the velocity of an element of ion fluid, **j** and n are the current and plasma densities, respectively, η is the resistivity associated with ion-electron interactions, and ∇ · **P**_{e} is the divergence of the electron pressure tensor. The first term on the right side of this equation is called the Hall MHD term. *Vasyliunas* [1975] showed that this term becomes important on scale sizes the order of the ion skin depth, c/*ω*_{pI}, where *ω*_{pI} is the ion plasma frequency, while the remaining terms on the right side of equation (1) manifest themselves on the 40 times shorter scale, c/*ω*_{pe}, where *ω*_{pe} is the electron plasma frequency. Thus, sub-solar magnetopause crossings that do not pass through an electron diffusion region are controlled by the physics of the first term on the right side of equation (1). Because this paper deals with the physics on scale sizes of many ion skin depths, the remaining terms on the right side are neglected in the following discussions.

[3] Two dimensional simulations of Hall MHD physics for reconnecting systems having equal plasma densities and magnetic field strengths on the two sides of the reconnection region showed an out-of-plane quadrupolar magnetic field and in-plane bipolar electric field [*Birn et al.*, 2001]. The out-of-plane quadrupolar magnetic field has positive then negative components across the current sheet on one side of the X-line and negative then positive components on the other side. The in-plane bipolar electric field is normal to the current sheet and points toward the center of the region from either side. These features have been observed in sub-solar magnetopause [*Mozer et al.*, 2002], and tail reconnection events [*Wygant et al.*, 2005] and in the laboratory [*Ren et al.*, 2005]. It is sometimes assumed that the presence of these fields provides necessary and sufficient evidence for reconnection. However, it has been emphasized [*Mozer and Retinò*, 2007] that these field geometries are observed in ≤1% of sub-solar reconnection events and it was conjectured that the absence of such fields was a consequence of asymmetric boundary conditions on the two sides of the magnetopause. Theories and simulations of asymmetric reconnection have been published [*Nakamura and Scholer*, 2000; *Borovsky and Hesse*, 2007; *Cassak and Shay*, 2007]. *Pritchett* [2007] has studied the impact of density and magnetic field asymmetries on the quadrupolar B and bipolar E structures with results similar to those reported in this paper.

[4] The absence of the bipolar electric field can be understood through comparing the two measured estimates of the Hall electric field, **E** + **U**_{I} × **B** and **j** × **B**/en. Pioneering comparisons of this type have been reported [*Khotyaintsev et al.*, 2006, and references therein]. In this paper, the first quantitative comparisons are made, based on two sub-solar magnetopause crossings on July 20, 2007 by three of the five THEMIS spacecraft, C, D, and E (the electric field antennas were not yet deployed on the other spacecraft).