Fig. 1. XMHD parameter space for various values of |B| and Z, with ion mass number A = 2Z. The red region is magnetic-pressure-dominated. The blue region indicates intermediate Hall parameter 0.01 < χ < 100. The region above the green line is where the Nernst advection is larger than the resistive diffusion. The black line shows the Coulomb logarithm and the gray line is the Fermi temperature. Regions of validity for different MHD models and parameters for several experiments are indicated.

Fig. 2. Mock-up schematic of the Biermann magnetic field (red/blue) at a laser ablation front, showing the effects of each term in the induction equation [Eqs. (5)–(7)]. We have assumed an ion charge Z = 5 and mass number A = 10. The target is at x = 0, with a laser centered on y = 0, and there is a positive temperature gradient (from 1 to 2 keV) and uniform fluid velocity (40 km s⁻¹) along x. Streamlines show the magnetic field velocity due to (a) ideal advection, (b) Hall advection, (c) Nernst advection, (d) Ohmic resistance, and (e) total of (a)–(d). The magnetic field produces maximal χ ≃ 1, deflecting the Nernst streamlines in (c). There is a similar deflection of the heat flux. Within each panel, the streamline thicknesses are proportional to the advection speed. For these parameters, both ideal and Nernst advection are ≃40 km s⁻¹, whereas the Hall and resistive terms are ≃1 km s⁻¹.

Fig. 3. Line-outs of the XMHD parameter space terms in Eqs. (5)–(7), assuming sin θ = 0.1. Conditions are chosen for (a) and (d) solid targets in HED experiments with Z = 5, (b) and (e) laser ablation fronts with Z = 5, and (c) and (f) ICF hot-spots with Z = 1. We have assumed an ion mass number A = 2Z. The upper panels show the variation of each term with temperature at a fixed magnetic field. The lower panels show the variation of each term with magnetic field strength at a fixed temperature. A fixed scale-length of L = 20 µm has been assumed for the gradients of all quantities. However, with the exception of the ideal advection, changing this value does not change the relative strength of each term. The ideal term assumes |u| to be equal to the sound speed. The maximum value of the temperature axis in (a)–(c) is where the electron mean free path exceeds L/10 and so the local transport theory breaks down.

Fig. 4. Estimated minimum electron temperature required to achieve χ > 0.1 self-magnetization of the heat flux, shown for fixed ion charge and mass numbers Z = 1, A = 2 (solid) and Z = 5, A = 10 (dashed). These are given by the numerical solution of Eq. (12) with transport coefficient α_∥(Z) calculated using the fit function from Ref. 16. We have assumed highly misaligned density and temperature gradients with sin θ ≃ 0.1, typical of a shear layer.