Jason W. Bates

Some numerical studies of exotic shock wave behavior

For shock waves propagating in materials with nonideal equations of state, a variety of nonstandard phenomena can occur. Here, we present numerical studies of two such exotic shock effects: (i) “anomalous” behavior, in the terminology of Zel’dovich and Raizer; and (ii) a search for “acoustic emission instabilities.” The motivation is in part the possibility of such phenomena in the implosion of inertial confinement fusion (ICF) pellet materials, whose equations of state are currently far from well known. In shock wave theory, anomalous materials are those whose isentropes have regions of negative curvature (in the plane of pressure versus specific volume) through which the shock adiabatic passes. The existence of such regions is significant because they can interfere with the steepening of compressive pulses into shocks, lead to the formation of rarefactive shock waves, and even cause shocks to “split.” A van der Waals fluid with a large heat capacity is one example of a material possessing such anomalous...

Resistive magnetohydrodynamic equilibria in a torus

It was recently demonstrated that static, resistive, magnetohydrodynamic equilibria, in the presence of spatially uniform electrical conductivity, do not exist in a torus under a standard set of assumed symmetries and boundary conditions. The difficulty, which goes away in the “periodic straight cylinder approximation,” is associated with the necessarily non-vanishing character of the curl of the Lorentz force, j×B. Here, we ask if there exists a spatial profile of electrical conductivity that permits the existence of zero-flow, axisymmetric resistive equilibria in a torus, and answer the question in the affirmative. However, the physical properties of the conductivity profile are unusual (the conductivity cannot be constant on a magnetic surface, for example) and whether such equilibria are to be considered physically possible remains an open question.

Toroidal visco-resistive magnetohydrodynamic steady states contain vortices

Poloidal velocity fields seem to be a fundamental feature of resistive toroidal magnetohydrodynamic (MHD) steady states. They are a consequence of force balance in toroidal geometry, do not require any kind of instability, and disappear in the “straight cylinder” (infinite aspect ratio) limit. If a current density j results from an axisymmetric toroidal electric field that is irrotational inside a torus, it leads to a magnetic field B such that ∇×(j×B) is nonvanishing, so that the Lorentz force cannot be balanced by the gradient of any scalar pressure in the equation of motion. In a steady state, finite poloidal velocity fields and toroidal vorticity must exist. Their calculation is difficult, but explicit solutions can be found in the limit of low Reynolds number. Here, existing calculations are generalized to the more realistic case of no-slip boundary conditions on the velocity field and a circular toroidal cross section. The results of this paper strongly suggest that discussions of confined steady st...

A toroidal boundary‐value problem in resistive magnetohydrodynamics

Under a standard set of assumptions, the magnetic flux function for a static, resistive magnetofluid is determined in axisymmetric toroidal geometry for two different scalar‐conductivity profiles; in one case the conductivity is spatially uniform, and in the other it is proportional to the square of the distance from the toroidal axis. Exact analytic expressions are found in both cases for a magnetofluid that is surrounded by a perfectly conducting toroidal shell with a circular cross section.

The geometry and symmetries of magnetohydrodynamic turbulence: Anomalies of spatial periodicity

It has become common to formulate theories and computations of magnetohydrodynamic turbulent effects in rectangular periodic boundary conditions, proceeding by analogy with what is seen as a useful framework for Navier–Stokes fluid turbulence. It is shown here that because of certain features of Maxwell’s equations for electrodynamics, it is inconsistent to invoke three-dimensional, rectangular, periodic boundary conditions and symmetry at the same time that the displacement current is neglected. The difficulty does not arise in the two-dimensional case. In three dimensions, the difficulty can be remedied by a reformulation in cylindrical geometry, imposing symmetry in the azimuthal and axial directions, but not in the radial one; a geometry that is closer to laboratory possibilities than the wholly three-dimensional periodic assumption. The reformulation seems particularly necessary in cases with a net flux of magnetic field and/or electric currents through the system. These cases no longer seem disconti...

Toroidal vortices in resistive magnetohydrodynamic equilibria

When a time-independent electric current flows toroidally in a uniform ring of electrically conducting fluid, a Lorentz force results, j×B, where j is the local electric current density, and B is the magnetic field it generates. Because of purely geometric effects, the curl of j×B is nonvanishing, and so j×B cannot be balanced by the gradient of any scalar pressure. Taking the curl of the fluid’s equation of motion shows that the net effect of the j×B force is to generate toroidal vorticity. Allowed steady states necessarily contain toroidal vortices, with flows in the poloidal directions. The flow pattern is a characteristic “double smoke ring” configuration. The effect seems quite general, although it is analytically simple only in special limits. One limit described here is that of high viscosity (low Reynolds number), with stress-free wall boundary conditions on the velocity field, although it is apparent that similar mechanical motions will result for no-slip boundaries and higher Reynolds numbers. A...

Time-dependent perturbation theory for the construction of invariants of Hamiltonian systems

Abstract A time-dependent perturbation theory is presented for iteratively constructing invariants for a Hamiltonian consisting of a time-independent zeroth-order term plus a time-dependent perturbation. The procedure involves only a single canonical transformation and small divisors can be avoided. The Mathieu equation is treated as an example.

Toroidal flows in resistive magnetohydrodynamic steady states

We consider the resistive steady states of a uniformly conducting magnetofluid inside a toroidal boundary. The problem becomes tractable in the limit of slow flow: i.e., low Reynolds number, which may be in turn justified when the viscous Lundquist number is small. Previous calculations are extended to apprehend the toroidal component of the necessary flow. The emerging pattern is one of helical vortices which seem likely to be ubiquitous in toroidal geometry, and which disappear in the “straight-cylinder approximation.”

On toroidal Green’s functions

Green’s functions are valuable analytical tools for solving a myriad of boundary-value problems in mathematical physics. Here, Green’s functions of the Laplacian and biharmonic operators are derived for a three-dimensional toroidal domain. In some sense, the former result may be regarded as “standard,” but the latter is most certainly not. It is shown that both functions can be constructed to have zero value on a specified toroidal surface with a circular cross section. Additionally, the Green’s function of the biharmonic operator may be chosen to have the property that its normal derivative also vanishes there. A “torsional” Green’s function is derived for each operator which is useful in solving some boundary-value problems involving axisymmetric vector equations. Using this approach, the magnetic vector potential of a wire loop is computed as a simple example.

Hamiltonian description of toroidal magnetic fields in vacuum

A Hamiltonian description of vacuum magnetic fields in toroidal geometry is presented. Using a magnetic scalar potential in conjunction with Boozer’s canonical representation of a magnetic field [A. Boozer, Phys. Fluids 26, 1288 (1983)], a field-line Hamiltonian for general nonaxisymmetric vacuum fields has been derived.