H. Ralph Lewis

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.

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 use of Hamilton's principle to derive time-advance algorithms for ordinary differential equations

Hamiltons principle is applied to derive a class of numerical algorithms for systems of ordinary differential equations when the equations are derivable from a Lagrangian. This is an important extension into the time domain of an earlier use of Hamiltons principle to derive algorithms for the spatial operators in Maxwells equations. In that work, given a set of expansion functions for spatial dependences, the Vlasov-Maxwell equations were replaced by a system of ordinary differential equations in time, but the question of solving the ordinary differential equations was not addressed. Advantageous properties of the new time-advance algorithms have been identified analytically and by numerical comparison with other methods, such as Runge-Kutta and symplectic algorithms. This approach to time advance can be extended to include partial differential equations and the Vlasov-Maxwell equations. An interesting issue that could be studied is whether a collisionless plasma simulation completely based on Hamiltons principle can be used to obtain a convergent computation of average properties, such as the electric energy, even when the underlying particle motion is characterized by sensitive dependence on initial conditions.

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.

A second invariant for one‐degree‐of‐freedom, time‐dependent Hamiltonians given a first invariant

An explicit formula for a second invariant of a one‐degree‐of‐freedom time‐dependent Hamiltonian is derived in terms of the Hamiltonian and an assumed first invariant. If the first invariant is expressed as a function of two canonical functions, a transformation to an autonomous Hamiltonian system is possible.

A comparison of symplectic and Hamilton's principle algorithms for autonomous and non-autonomous systems of ordinary differential equations

After a thorough study of order and symmetry of variational methods based on Hamiltons principle, we study efficient implementations that make them competitive with symplectic methods for some of our test problems.

Nonlinear solutions of the Vlasov–Poisson equations

Based on the kinetic description, properties of collisionless unmagnetized plasma are studied. Time-dependent solutions of the nonlinear Vlasov–Poisson equations are found. Several dynamical structures of a Maxwellian type are presented.

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.

Periodic non-singular invariant series for periodic perturbations of autonomous one-degree-of-freedom Hamiltonians

Abstract For periodic perturbations of autonomous 1D Hamiltonians, to all orders in powers of the perturbation parameter, non-singular invariant series periodic in the independent variable are constructed. In applications to magnetohydrodynamic equilibria, the non-singularity has allowed magnetic island structure to be represented within the framework of a perturbation treatment.

Extension of Giacomini’s results concerning invariants for one‐dimensional time‐dependent potentials

One‐dimensional time‐dependent potentials are considered for which an invariant can be expressed in terms of the potential and the momentum according to the formulation of Giacomini. New solutions of Giacomini’s equations are derived. In addition, possibilities are discussed for extending Giacomini’s approach to more general systems.