Asymptotic Estimation for Minimal Plasma Radius in a Spherically Symmetric Magnetized Target Fusion Reactor Model

Nonlinear conservation laws with moving boundaries often arise in applications such as gas and plasma dynamics and more recently in the context of nuclear fusion reactors. Gaining insight into the nature of solutions to such models and their dependence upon design parameters is often done numerically. Instead of relying upon numerics, in this paper we use formal asymptotics to study a model of a magnetized target fusion reactor for producing energy. We will asymptotically solve the compressible euler equations with two free moving boundaries, with an algebraic coupling of fluid pressure at one of the free boundaries. The model consists of an intense pressure being imparted on the boundary of a large sphere holding molten lead-lithium over a very short time scale, in the middle of which is a smaller concentric sphere containing plasma. A localized disturbance quickly propagates towards the plasma, whereby part of the energy is reflected and part compresses the plasma. The plasma pressure is modelled by a magnetic pressure, dependent upon its radius. Both the outer boundary of the lead-lithium sphere and the plasma boundary are free. Using matched asymptotics and various techniques for solving linear hyperbolic systems including Riemann Invariants and using the velocity potential formulation of the linear acoustic equations, we estimate the maximum compression of the plasma. This maximum compression is obtained in two ways: with an energy argument originally attributed to Lord Rayleigh in studying bubble dynamics and a new asymptotic argument for the maximum compression of a bubble subjected to the Rayleigh-Plesset equation. This estimation gives insights into the factors relevant in designing an energy-producing reactor.