Quasi-axisymmetric magnetic fields: weakly non-axisymmetric case in a vacuum
An asymptotic expansion is performed to obtain quasi-axisymmetric magnetic configurations that are weakly non-axisymmetric. A large space of solutions is identified, which satisfy the condition of quasi-axisymmetry on a single magnetic flux surface, while (non-axisymmetric) globally quasi-axisymmetric solutions are shown to not exist, agreeing with the conclusions of previous theoretical work. The solutions found are shown to be geometrically constrained at low aspect ratio or high toroidal period number. Solutions satisfying the more general condition of omnigeneity (generalized quasi-axisymmetry) are also shown to exist, and it is found that quasi-axisymmetric deformations can be superposed with an omnigenous solution, while preserving the property of omnigeneity, effectively extending the space of "good" configurations. A numerical solution of the first order quasi-axisymmetry problem is demonstrated and compared with solutions found with a widely used MHD equilibrium solver, independently verifying that quasi-axisymmetry is satisfied at the appropriate order. It is thereby demonstrated that approximately quasi-axisymmetric solutions can be directly constructed, i.e. without using numerical search algorithms.
Relationship between the shape of equilibrium magnetic surfaces and the magnetic field strength
A local analysis of the magnetic field near an equilibrium magnetic surface shows that there is generally no relationship between the magnetic field strength and the shape of the surface. However, the relationship exists under additional requirements such as the absence of the toroidal current, symmetry conservation, and the conservation of the magnetic field strength distribution on the nearest surface. An equilibrium magnetic surface can be calculated by specifying three functions of two angular variables—the magnetic field strength, the periodic component of the magnetic potential, and the mean curvature of the surface.
Velocity moments in alongshore bottom stress parameterizations
The time-averaged alongshore bottom stress is an important component of nearshore circulation models. In a widely accepted formulation the bottom stress is proportional to , the time average of the product of the instantaneous velocity magnitude |u→| and the instantaneous alongshore velocity component υ. Both mean and fluctuating (owing to random, directionally spread waves) velocities contribute to . Direct estimation of requires a more detailed specification of the velocity field than is usually available, so the term is parameterized. Here direct estimates of based on time series of near-bottom currents observed between the shoreline and 8-m water depth are used to test the accuracy of parameterizations. Common parameterizations that are linear in the mean alongshore current significantly underestimate for moderately strong alongshore currents, resulting in overestimation of a drag coefficient determined by fitting modeled (with a linearized bottom stress) to observed alongshore currents. A parameterization based on a joint-Gaussian velocity field with the observed velocity statistics gives excellent overall agreement with the directly estimated and allows analytic investigation of the statistical properties of the velocity field that govern . Except for the weakest flows, depends strongly on the mean alongshore current and the total velocity variance but depends only weakly on the mean wave angle, wave directional spread, and mean cross-shore current. Several other nonlinear parameterizations of are shown to be more accurate than the linear parameterizations and are adequate for many modeling purposes.