Fusion energy may hold the key to a sustainable future of electricity production. However some technical stumbling blocks remain to be overcome. One central challenge of the fusion enterprise is how to effectively withstand the high heat load emanating from the core plasma. Even the sturdiest solid solutions suffer damage over time, which could be avoided by adding a thin liquid coating. This thin liquid-metal layer must be maintained in the presence of the strong magnetic field needed to confine the core plasma and under the influence of the hot particles that escape the confinement and strike the liquid. To reduce the effective heat load it has been proposed to sweep these incoming particles up and down the liquid layer. However it is unclear how this sweeping motion will affect the liquid.

In this work Oxford Mathematicians Davin Lunz and Peter Howell in collaboration with Tokamak Energy examine how the sweeping motion affects the deflections of the liquid layer. They find that there are effectively three regimes: one in which the sweeping motion's influence is neutral as it has little effect on the deflections, one in which the deflections are minimised (that is a positive outcome in the fusion context as the liquid provides a more uniform covering), and one in which the deflections are dangerously amplified (that is a negative outcome in this context, as large deflections can leave the underlying solid exposed and result in liquid particles detaching from the layer and impurities reaching the core plasma). To uncover this, they focus on the (appropriately normalised) governing equation, namely, \begin{align} \frac{\partial h}{\partial t} + \frac \partial{\partial x} \left[ Q(h) \left( 1 - \frac{\partial p}{\partial x} + \frac{\partial ^3 h}{\partial x^3} \right) \right] = 0, \end{align} where $x$ is a horizontal coordinate, $t$ denotes time, $h(x,t)$ is the liquid deflection, $p(x,t)$ is the oscillating pressure exerted on the layer from the impinging plasma, and $Q$ is a flux function. This is a thin film equation where the fluid is driven by gravity and the applied pressure while being modulated by surface tension.

One key observation is that there are two independent time scales in the problem: the first is the time scale at which the surface equilibrates, and the second is the time scale at which the pressure completes one oscillation. Their work shows that if the pressure time scale is much longer than the deflection time scale (that is, if the sweeping is sufficiently slow) then the deflections are largely unaffected by the motion. In the opposite case - the moving pressure time scale is much shorter than the deflection time scale, that is, the sweeping motion is very fast - the load of the impacting particles is effectively averaged out and this more even distribution minimises deflections. When the two time scales are of similar order, the pressure can oscillate at a speed similar to the natural speed that deflections propagate along the free surface. This has the effect of dangerously amplifying the free-surface deflections and should be avoided in the context of confined fusion.

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