Bennett Vorticity: A family of nonlinear Shear-Flow Stabilized Z-pinch equilibria

Abstract

The Bennett profile is a classic form for the plasma number density of an equilibrium Z-pinch that has been studied for almost a century by plasma physicists interested in nonlinear plasma pinch science, and fusion energy. By transferring the nonlinearity entirely from the number density to the plasma flow velocity the current density of the resulting flowing Z-pinch equilibrium remains unchanged whilst now being defined by a vortical flow which previously did not exist in the classic case. Due to the positive-definite structure of the nonlinearity's first derivative, in the ideal limit this equilibrium conforms globally to the validity criterion for a shear-flow stabilized Z-pinch when the form of the temperature profile satisfies certain constraints. An analytic equilibrium can be found for the case $T = \frac{T_{p}}{r_{p}^{3}}r^{3}$, and is investigated. The predictions are found to be in good agreement where they should with the observations from the Zap, and Zap-HD DD fusion devices, including a very accurate prediction of the shear, and an axial profile that can be seen developing at multiple instances. The minimum pinch length necessary for this cubic vortex equilibrium to form an SFS state can be arbitrarily small arbitrarily close to the pinch.

Figures (29)

• Figure 1: Normalized profile of the plasma current density for a Bennett Pinch, Equation (\ref{['eqn:bennett_current']}). $\tilde{J}_{z} = \frac{J_{z}}{en_{0}u_{z,0}}$, and $r^{\ast} = \frac{r}{L^{\ast}} = r\xi$. The key feature to notice is the evanescence of the profile over a handful of scale-lengths without the need for any piecewise constructions that introduce discontinuities in the solution.
• Figure 2: Normalized profile of the magnetic field for a Bennett Pinch, Equation (\ref{['eqn:bennett_field']}), with $\tilde{B}_{\theta} = \frac{\xi}{A}B_{\theta}$. Note the need to introduce an extra factor of $\xi$ into the form in order to properly normalize the length, completely. Outside of the region where the bulk plasma current goes to zero the magnetic field goes as $\sim \frac{1}{r}$ while inside this region the shape of the field is influenced by the nonlinear plasma current. Comparison with the other normalized profiles will convince that this $\frac{1}{r}$-dependence is a much slower falloff, comparatively. Also, observe that the theory gives a trivial solution for the value of the core magnetic field.
• Figure 3: Normalized profile of the magnetic tension for a Bennett Pinch, $M_{T} = \frac{B_{\theta}^{2}}{\mu r}$. The normalized profile displayed here is given by $\tilde{M}_{T} = M_{T}\frac{\mu\xi}{A^{2}} = \frac{\tilde{B}_{\theta}^{2}}{r^{\ast}}$. The lack of singularity in the profile is due to the nonlinear dependence of $B_{\theta}$, i.e., the Bennett profile.
• Figure 4: Normalized plasma pressure of the Bennett Pinch as a function of the dimensionless coordinate $r^{\ast} = r\xi$. $\tilde{p} = \frac{p}{p_{0}}$ is plotted here, and there is a specific requirement for $p_{0} = \frac{A^{2}}{8\xi^{2}\mu}$ to achieve the dimensionless form. This amounts to the trivial constraint that $T \neq 0$ which is an uninteresting limit in the theory because it represents a plasma with zero thermal energy, i.e., a trivial one as far as fusion plasma physics is concerned.
• Figure 5: The normalized flow profile of an $r^{3}$ Bennett Vortex. The asymptotic behavior of the normalized flow profile is to tend to unity, representing the edge flow state, as the dimensionless coordinate, $\phi$ goes to $\infty$. For values of $\phi < 1$ the flow profile has an approximately parabolic character that changes to more of a logarithmic appearance as the point of maximal shear is reached, which is intriguingly similar to the axial velocity profiles seen in the ZaPuri_2009 and FuZE machines.