Magnetic flux conservation in an imploding plasma

The theory of magnetic flux conservation is developed for a subsonic plasma implosion and used to describe the magnetic flux degradation in the MagLIF concept [S. A. Slutz et al., Phys. Plasmas 17, 056303 (2010)]. Depending on the initial magnetic Lewis and Péclet numbers and the electron Hall parameter, the implosion falls into either a superdiffusive regime in which the magnetization decreases or a magnetized regime in which the magnetization increases. Scaling laws for magnetic field, temperature, and magnetic flux losses in the hot spot of radius $R$ are obtained for both regimes. The Nernst velocity convects the magnetic field outwards, pushing it against the liner and enhancing the magnetic field diffusion, thereby reducing the magnetic field compression and degrading the implosion performance. However, in the magnetized regime, the core of the hot spot becomes magnetically insulated and undergoes an ideal adiabatic compression ($T\sim R^{-4/3}$ compared to $T\sim R^{-2/3}$ without magnetic field), while the detrimental Nernst term is confined to the outer part of the hot spot. Its effect is drastically reduced, improving the magnetic flux conservation.

Figure 1

Dependence of transport coefficients on the magnetization (electron Hall) parameter $x_e$.

Figure 2

Unmagnetized plasma implosion. (a) Temperature and velocity self-similar profiles. (b) Stationary values for ${\lambda }_B$ and $B_c$. (c, d) Magnetic field self-similar profiles.

Figure 3

(a), (b) Magnetized implosion profiles. Initial parameters: ${\text{Pe}}_0=50, {\text{Le}}_0=100, x_{ec0}=1.5, {\theta }_0\left(\eta \right)=cos\left(\pi {\eta }^6/2\right), {\varphi }_0\left(\eta \right)=1$. From light to darker gray: $\tau =0$, 1, 2, ${\tau }_f=log\left(30\right)\approx 3.4$, 5. (c) Normalized hot spot mass in implosions with the same initial parameters but different initial magnetization. (d) Normalized final magnetic flux remaining in the hot spot in implosions with different initial magnetization.

Figure 4

Parameter frontier for superdiffusive and magnetized implosion regimes. The dashed line corresponds to ${\text{Le}}_0{\text{Pe}}_0^{10/7}=75$. The crosses indicate the position in the chart of the three schemes in Table 1.