Quantifying the Effect of Non-Larmor Motion of Electrons on the Pressure Tensor.

In space plasma, various effects of magnetic reconnection and turbulence cause the electron motion to significantly deviate from their Larmor orbits. Collectively these orbits affect the electron velocity distribution function and lead to the appearance of the "non-gyrotropic" elements in the pressure tensor. Quantification of this effect has important applications in space and laboratory plasma, one of which is tracing the electron diffusion region (EDR) of magnetic reconnection in space observations. Three different measures of agyrotropy of pressure tensor have previously been proposed, namely, $A\varnothing_e$, $D_{ng}$ and $Q$. The multitude of contradictory measures has caused confusion within the community. We revisit the problem by considering the basic properties an agyrotropy measure should have. We show that $A\varnothing_e$, $D_{ng}$ and $Q$ are all defined based on the sum of the principle minors (i.e. the rotation invariant $I_2$) of the pressure tensor. We discuss in detail the problems of $I_2$-based measures and explain why they may produce ambiguous and biased results. We introduce a new measure $AG$ constructed based on the determinant of the pressure tensor (i.e. the rotation invariant $I_3$) which does not suffer from the problems of $I_2$-based measures. We compare $AG$ with other measures in 2 and 3-dimension particle-in-cell magnetic reconnection simulations, and show that $AG$ can effectively trace the EDR of reconnection in both Harris and force-free current sheets. On the other hand, $A\varnothing_e$ does not show prominent peaks in the EDR and part of the separatrix in the force-free reconnection simulations, demonstrating that $A\varnothing_e$ does not measure all the non-gyrotropic effects in this case, and is not suitable for studying magnetic reconnection in more general situations other than Harris sheet reconnection.