dc-Magnetic-Field Generation in Unmagnetized Shear Flows

The generation of dc magnetic fields in unmagnetized electron-ion shear flows is shown to be associated to either initial thermal effects or the onset of electron-scale shear instabilities, in particular the cold Kelvin-Helmholtz instability. This mechanism, intrinsic to shear gradients on the electron scale, is described through a kinetic model that predicts the growth and the saturation of the dc field in both scenarios. The theoretical results are confirmed by multidimensional particle-in-cell simulations, demonstrating the formation of long-lived magnetic fields ($t\sim 100’\mathrm{s}{\omega }_{pi}^{-1}$) along the full longitudinal extent of the shear layer, with a typical transverse width of $\sqrt{{\gamma }_0}c/{\omega }_{pe}$, reaching magnitudes $e{B}_{\mathrm{dc}}/m_{e}c{\omega }_{pe}\sim {\beta }_0\sqrt{{\gamma }_0}$ for an initial sharp shear. The case of an initial smooth shear is also discussed.

Figure 1

$B_z$ component of the magnetic-field structure generated by the cold KHI for $v_0=0.2c$ during (a) the linear regime, (b) near saturation, and (c) at $t=1000{\omega }_{pe}^{-1}=100{\omega }_{pi}^{-1}$. The insets on the right hand side represent the longitudinal average of the magnetic field, revealing the dc component. (${\mathrm{S}}_1$) and (${\mathrm{S}}_2$) show the magnetic field at saturation for an initial smooth shear $v_0(x)/c=0.2\tanh (x/L)$ with $L=10c/{\omega }_{pe}=c/{\omega }_{pi}$.

Figure 2

Evolution of the electron phase space [(a1)–(a3)] and (b1)–(b3)] and dc-magnetic-field peak [(a4) and (b4)]; log-log scale is used to display linear dc peak evolution in (a4), and log-linear scale is used to display exponential evolution in (b4). Left: 1D warm shear flow with $v_0=0.2c$ and $v_{\mathrm{th}}=0.016$. Right: 2D cold shear flow for the same $v_0$. The blue (red) color represents the electrons with a negative (positive) drift velocity $v_0$. The self-consistent dc magnetic field is represented by the solid curve, whereas the dashed curve represents the magnetic field given by the theoretical model. The flow velocity is positive (negative) on the right (left) side of the shear.

Figure 3

Magnitude of the dc-magnetic-field peak at saturation as a function of ${\beta }_0\sqrt{{\gamma }_0}$. The red, blue, and black markers represent the results of 1D, 2D, and 3D PIC simulations, respectively. The error bars are associated to the fluctuations of the peak value in the saturation stage. The lines represent best-fit curves to the simulation results, demonstrating the good agreement with the theoretical estimate.