Non-thermal particle acceleration and power-law tails via relaxation to universal Lynden-Bell equilibria

Collisionless and weakly collisional plasmas often exhibit non-thermal quasi-equilibria. Among these quasi-equilibria, distributions with power-law tails are ubiquitous. It is shown that the statistical-mechanical approach originally suggested by Lynden-Bell (Mon. Not. R. Astron. Soc., vol. 136, 1967, p. 101) can easily recover such power-law tails. Moreover, we show that, despite the apparent diversity of Lynden-Bell equilibria, a generic form of the equilibrium distribution at high energies is a ‘hard’ power-law tail ∝ε−2, where ε is the particle energy. The shape of the ‘core’ of the distribution, located at low energies, retains some dependence on the initial condition but it is the tail (or ‘halo’) that contains most of the energy. Thus, a degree of universality exists in collisionless plasmas

Non-thermal particle acceleration and power-law tails via relaxation to universal Lynden-Bell equilibria

Collisionless and weakly collisional plasmas often exhibit non-thermal quasi-equilibria. Among these quasi-equilibria, distributions with power-law tails are ubiquitous. It is shown that the statistical-mechanical approach originally suggested by Lynden-Bell (1967) can easily recover such power-law tails. Moreover, we show that, despite the apparent diversity of Lynden-Bell equilibria, a generic form of the equilibrium distribution at high energies is a `hard' power-law tail $\propto \varepsilon^{-2}$, where $\varepsilon$ is the particle energy. The shape of the `core' of the distribution, located at low energies, retains some dependence on the initial condition but it is the tail (or `halo') that contains most of the energy. Thus, a degree of universality exists in collisionless plasmas.Comment: 33 pages, 5 figure

Phase-space entropy cascade and irreversibility of stochastic heating in nearly collisionless plasma turbulence

We consider a nearly collisionless plasma consisting of a species of `test particles' in 1D-1V, stirred by an externally imposed stochastic electric field. The mean effect on the particle distribution function is stochastic heating. Accompanying this heating is the generation of fine-scale structure in the distribution function, which we characterize with the collisionless (Casimir) invariant $C_2 \propto \iint dx dv \, \langle f^2 \rangle$. We find that $C_2$ is transferred from large scales to small scales in both position and velocity space via a phase-space cascade enabled by both particle streaming and nonlinear interactions between particles and the stochastic electric field. We compute the steady-state fluxes and spectrum of $C_2$ in Fourier space, with $k$ and $s$ denoting spatial and velocity wavenumbers, respectively. Whereas even the linear phase mixing alone would lead to a constant flux of $C_2$ to high $s$ (towards the collisional dissipation range) at every $k$, the nonlinearity accelerates this cascade by intertwining velocity and position space so that the flux of $C_2$ is to both high $k$ and high $s$ simultaneously. Integrating over velocity (spatial) wavenumbers, the $k$-space ($s$-space) flux of $C_2$ is constant down to a dissipation length (velocity) scale that tends to zero as the collision frequency does, even though the rate of collisional dissipation remains finite. The resulting spectrum in the inertial range is a self-similar function in the $(k,s)$ plane, with power-law asymptotics at large $k$ and $s$. We argue that stochastic heating is made irreversible by this entropy cascade and that, while collisional dissipation accessed via phase mixing occurs only at small spatial scales rather than at every scale as it would in a linear system, the cascade makes phase mixing even more effective overall in the nonlinear regime than in the linear one.Comment: 26 pages, 6 figure

Modeling of electromagnetic, heat transfer, and fluid flow phenomena in an EM stirred melt during solidification

A methodology is presented to simulate the electromagnetic, heat transfer, and fluid flow phenomena for two dimensional electromagnetic solidification processes. For computation of the electromagnetic field, the model utilizes the mutual inductance technique to limit the solution domain to the molten metal and magnetic shields, commonly present in solidification systems. The temperature and velocity fields were solved using the control volume method in the metal domain. The developed model employs a two domain formulation for the mushy zone. Mathematical formulations are presented for turbulent flow in the bulk liquid and the suspended particle region, along with rheological behavior. An expression has been developed--for the first time--to describe damping of the flow in the suspended particle region as a result of the interactions between the particles and the turbulent eddies. The flow in the fixed particle region is described using Darcy's law. Calculations were carried out for globular and dendritic solidification morphologies of an electromagnetically-stirred melt in a bottom-chill mold. The coherency solid fraction for the globular solidification morphology was taken to be 0.5, while the coherency for dendritic morphology was 0.25. The results showed the flow intensity in the suspended particle region was reduced by an order of magnitude. The effect of the heat extraction rate on solidification time was investigated using three different heat transfer coefficients. The results showed that the decrease in solidification time is nonlinear with respect to increasing heat transfer coefficient. The influence of the final grain size on the damping of the flow in the suspended particle region was examined, and it was found that larger grain sizes reduce the extent of flow damping. (Published By University of Alabama Libraries

Mathematical modeling of solidification phenomena in electromagnetically stirred melts

A methodology is presented to simulate the electromagnetic, heat transfer, and fluid flow phenomena for two dimensional electromagnetic solidification processes. For computation of the electromagnetic field, the model utilizes the mutual inductance technique to limit the solution domain to the molten metal and magnetic shields, commonly present in solidification systems. The temperature and velocity fields were solved using the control volume method in the metal domain. The developed model employs a two domain formulation for the mushy zone. Mathematical formulations are presented for turbulent flow in the bulk liquid and the suspended particle region, along with rheological behavior. An expression has been developed--for the first time--to describe damping of the flow in the suspended particle region as a result of the interactions between the particles and the turbulent eddies. The flow in the fixed particle region is described using Darcy's law. Calculations were carried out for globular and dendritic solidification morphologies of an electromagnetically-stirred melt in a bottom-chill mold. The coherency solid fraction for the globular solidification morphology was taken to be 0.5, while the coherency for dendritic morphology was 0.25. The results showed the flow intensity in the suspended particle region was reduced by an order of magnitude. The effect of the heat extraction rate on solidification time was investigated using three different heat transfer coefficients. The results showed that the decrease in solidification time is nonlinear with respect to increasing heat transfer coefficient. The influence of the final grain size on the damping of the flow in the suspended particle region was examined, and it was found that larger grain sizes reduce the extent of flow damping. (Published By University of Alabama Libraries