Extended magnetohydrodynamic effects in Z-pinch plasmas
File(s)
Author(s)
Farrow, Griffin
Type
Thesis
Abstract
Magnetic fields are central to magneto-inertial fusion (MIF) schemes that aim to achieve controlled nuclear fusion as well as in laboratory astrophysics experiments designed to explore fundamental physics. Accurate numerical modelling of the coupled behaviour of magnetic fields and dense plasmas is crucial for understanding results, interpreting diagnostics and designing future experiments. However, conventional resistive magnetohydrodynamic (MHD) models are inaccurate in low density plasmas such as at a vacuum-plasma interface. These regions are frequently encountered in pulsed power experiments. In this work, we establish which are the dominant physics effects in this regime in order to develop a more physical model of this region.
We show that the equations of Braginskii MHD in one dimensional planar geometry are amenable to the method of self-similar solutions and use this to develop a semi-analytic model of pressure equilibria in magnetised plasmas. This tool is useful for validating numerical MHD codes and it is used to provide a test problem for the Ohmic heating and Ettingshausen effects. The self-similar code is used to perform a parameter scan over plasma $\beta$. This shows that in the high $\beta$ plasma characteristic of the stagnation region of MIF experiments such as MagLIF, the most relevant transport effects are magnetised thermal conduction and the Nernst effect. However, we find that as the $\beta$ is reduced, the Ohmic heating and resistive diffusion terms become increasingly important. At a vacuum-plasma interface, the Ohmic heating becomes the dominant term in the energy balance. However, we show the Ettingshausen effect plays a key role in reducing the unphysically high electron temperatures in this region. This may be of use in resistive MHD codes which traditionally lack the Ettingshausen term, but also tend to overestimate the electron temperature at vacuum-plasma interfaces. We then use the self-similar code to design a potential experiment to measure the Nernst effect in a pulsed power plasma. We show that it is the dimensionless magnetic Lewis number which dictates the impact of the Nernst effect. The lower temperatures generated in pulsed power experiments when compared to laser platforms attempting to measure the same effect means that a measurement of the Nernst effect would be difficult in this regime.
To support theoretical analysis of the different physics effects at a vacuum-plasma interface, fully kinetic simulations of the edge of a z-pinch using the EPOCH particle-in-cell (PIC) code are then carried out. These simulations show how an initial MHD equilibrium undergoes a transient phase which leads to the formation of a long-lasting charge separation layer at the edge of the pinch. This layer has width comparable to the ion Larmor radius and the steady state electric field in this region is dominated by the Hall term, indicating it is the dominant beyond-MHD effect at a vacuum-plasma interface. Through studying the evolution of the edge of the z-pinch in 2D, we show that macroscopically the Hall effect leads to shearing of plasma flows in the direction of current flow.
Based on the analysis in previous sections, we then implement the Hall effect within the Gorgon extended MHD code. The Hall term is notoriously difficult to handle numerically, so we develop a new semi-implicit method for it involving electron inertia. This is tested thoroughly by comparing to MHD test problems and theoretical dispersion relations for wave modes driven by the Hall effect. This new solver is then used to study the evolution of an $m=0$ sausage instability in a z-pinch. We find that similarly to the PIC results, the Hall effect causes the lobes of the instability to shear in the direction of current flow by diverting magnetic field with the electron drift velocity. This has the effect of increasing the growth rate of the instability slightly compared to the resistive MHD case, though this does depend on the plasma parameters used.
Further work is needed to explore other beyond-MHD terms, such as anomalous resistivity and finite Larmor radius effects, and implement them within a fluid code to provide a more physical model of vacuum-plasma interfaces. The Hall solver developed as part of this work can be used in the future to study beyond-MHD effects in magneto-inertial fusion and laboratory astrophysics experiments.
We show that the equations of Braginskii MHD in one dimensional planar geometry are amenable to the method of self-similar solutions and use this to develop a semi-analytic model of pressure equilibria in magnetised plasmas. This tool is useful for validating numerical MHD codes and it is used to provide a test problem for the Ohmic heating and Ettingshausen effects. The self-similar code is used to perform a parameter scan over plasma $\beta$. This shows that in the high $\beta$ plasma characteristic of the stagnation region of MIF experiments such as MagLIF, the most relevant transport effects are magnetised thermal conduction and the Nernst effect. However, we find that as the $\beta$ is reduced, the Ohmic heating and resistive diffusion terms become increasingly important. At a vacuum-plasma interface, the Ohmic heating becomes the dominant term in the energy balance. However, we show the Ettingshausen effect plays a key role in reducing the unphysically high electron temperatures in this region. This may be of use in resistive MHD codes which traditionally lack the Ettingshausen term, but also tend to overestimate the electron temperature at vacuum-plasma interfaces. We then use the self-similar code to design a potential experiment to measure the Nernst effect in a pulsed power plasma. We show that it is the dimensionless magnetic Lewis number which dictates the impact of the Nernst effect. The lower temperatures generated in pulsed power experiments when compared to laser platforms attempting to measure the same effect means that a measurement of the Nernst effect would be difficult in this regime.
To support theoretical analysis of the different physics effects at a vacuum-plasma interface, fully kinetic simulations of the edge of a z-pinch using the EPOCH particle-in-cell (PIC) code are then carried out. These simulations show how an initial MHD equilibrium undergoes a transient phase which leads to the formation of a long-lasting charge separation layer at the edge of the pinch. This layer has width comparable to the ion Larmor radius and the steady state electric field in this region is dominated by the Hall term, indicating it is the dominant beyond-MHD effect at a vacuum-plasma interface. Through studying the evolution of the edge of the z-pinch in 2D, we show that macroscopically the Hall effect leads to shearing of plasma flows in the direction of current flow.
Based on the analysis in previous sections, we then implement the Hall effect within the Gorgon extended MHD code. The Hall term is notoriously difficult to handle numerically, so we develop a new semi-implicit method for it involving electron inertia. This is tested thoroughly by comparing to MHD test problems and theoretical dispersion relations for wave modes driven by the Hall effect. This new solver is then used to study the evolution of an $m=0$ sausage instability in a z-pinch. We find that similarly to the PIC results, the Hall effect causes the lobes of the instability to shear in the direction of current flow by diverting magnetic field with the electron drift velocity. This has the effect of increasing the growth rate of the instability slightly compared to the resistive MHD case, though this does depend on the plasma parameters used.
Further work is needed to explore other beyond-MHD terms, such as anomalous resistivity and finite Larmor radius effects, and implement them within a fluid code to provide a more physical model of vacuum-plasma interfaces. The Hall solver developed as part of this work can be used in the future to study beyond-MHD effects in magneto-inertial fusion and laboratory astrophysics experiments.
Version
Open Access
Date Issued
2023-08
Date Awarded
2023-10
Copyright Statement
Creative Commons Attribution NonCommercial Licence
License URL
Advisor
Chittenden, Jeremy
Kagan, Grigory
Sponsor
Engineering and Physical Sciences Research Council
Grant Number
EP/R513052/1
Publisher Department
Physics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)