We measure the thermal electron energization in 1D and 2D particleincell simulations of quasiperpendicular, lowbeta (
We present an approach to understand geometricincompatibility–induced rigidity in underconstrained materials, including subisostatic 2D spring networks and 2D and 3D vertex models for dense biological tissues. We show that in all these models a geometric criterion, represented by a minimal length
 NSFPAR ID:
 10088612
 Publisher / Repository:
 Proceedings of the National Academy of Sciences
 Date Published:
 Journal Name:
 Proceedings of the National Academy of Sciences
 Volume:
 116
 Issue:
 14
 ISSN:
 00278424
 Page Range / eLocation ID:
 p. 65606568
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract β _{p}= 0.25) collisionless ion–electron shocks with mass ratiom _{i}/m _{e}= 200, fast Mach number –4, and upstream magnetic field angle ${\mathcal{M}}_{\mathrm{ms}}=1$θ _{Bn}= 55°–85° from the shock normal . It is known that shock electron heating is described by an ambipolar, $\stackrel{\u02c6}{\mathit{n}}$ parallel electric potential jump, ΔB ϕ _{∥}, that scales roughly linearly with the electron temperature jump. Our simulations have –0.2 in units of the preshock ions’ bulk kinetic energy, in agreement with prior measurements and simulations. Different ways to measure $\mathrm{\Delta}{\varphi}_{\parallel}/(0.5{m}_{\mathrm{i}}{{u}_{\mathrm{sh}}}^{2})\sim 0.1$ϕ _{∥}, including the use of de Hoffmann–Teller frame fields, agree to tensofpercent accuracy. Neglecting offdiagonal electron pressure tensor terms can lead to a systematic underestimate ofϕ _{∥}in our lowβ _{p}shocks. We further focus on twoθ _{Bn}= 65° shocks: a ( ${\mathcal{M}}_{\mathrm{s}}\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}4$ ) case with a long, 30 ${\mathcal{M}}_{\mathrm{A}}\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}1.8$d _{i}precursor of whistler waves along , and a $\stackrel{\u02c6}{\mathit{n}}$ ( ${\mathcal{M}}_{\mathrm{s}}\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}7$ ) case with a shorter, 5 ${\mathcal{M}}_{\mathrm{A}}\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}3.2$d _{i}precursor of whistlers oblique to both and $\stackrel{\u02c6}{\mathit{n}}$ ;B d _{i}is the ion skin depth. Within the precursors,ϕ _{∥}has a secular rise toward the shock along multiple whistler wavelengths and also has localized spikes within magnetic troughs. In a 1D simulation of the , ${\mathcal{M}}_{\mathrm{s}}\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}4$θ _{Bn}= 65° case,ϕ _{∥}shows a weak dependence on the electron plasmatocyclotron frequency ratioω _{pe}/Ω_{ce}, andϕ _{∥}decreases by a factor of 2 asm _{i}/m _{e}is raised to the true proton–electron value of 1836. 
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