Multiple solutions and advection-dominated flows in the wind-driven circulation. Part I: Slip

We consider steady solutions of the barotropic quasigeostrophic vorticity equation for a single subtropical gyre with dissipation in the form of lateral friction. Solutions are governed by two parameters: inertial boundary-layer width; and viscous boundary-layer width. Numerical computations for slip conditions indicate a wedge-shaped region in this two-dimensional parameter space, where three solutions coexist. One of these is a viscous solution with weak recirculation; one a solution of intermediate recirculation; and one a strongly nonlinear recirculation gyre. Parametric scalings based on elementary solutions are numerically corroborated as the first and third of these solutions are continued away from the vicinity of the wedge. The multiplicity of solutions is anticipated by a severely truncated Fourier modal representation paralleling Veronis (1963). The Veronis work was originally applied to predict the possibility of multiple solutions in Stommel\u27s (1948) bottom friction model of the circulation. Paradoxically, it appears the solutions are, in that case, unique

Macrodynamics of ± \u3c sup\u3e 2 \u3c/sup\u3e dynamos

Two distributions of the ±-effect in a sphere are considered. The inviscid limit is approached both by direct numerical solution and by solution of a simpler nonlinear eigenvalue problem deriving from asymptotic boundary layer analysis for the case of stress-free boundaries. The inviscid limit in both cases is dominated by the need to satisfy the Taylor constraint which states that the integral of the Lorentz force over cylindrical (geostrophic) contours in a homogeneous fluid must tend to zero. For a small supercritical range in ±, this condition can only be met by magnetic fields which vanish as the viscosity goes to zero. In this range, the agreement of the two approaches is excellent. In a portion of this range, the method of finite amplitude perturbation expansion is useful, and serves as a guide for understanding the numerical results. For larger a, evidence from the nonlinear eigenvalue problem suggests both that the Taylor state exists, and that the transition from small to large amplitude can require a finite amplitude (oscillatory) instability in accord with the findings of Soward and Jones (1983). However, solutions of the full equations have not been found which are independent of viscosity at larger values of ±. © 1985, Taylor & Francis Group, LLC. All rights reserved

Spectral methods for the solution of nonlinear boundary value problems, a case study

Spectral methods enjoy a variety of well known virtues for the solution of ordinary and partial differential equations. The coefficient spectrum falls off exponentially (as for the use of any basis set which is the solution of a singular Sturm-Liouville problem). Nonlinear terms can be effectively accomodated by a true spectral calculation in one dimensional problems, or two dimensional problems with modest resolution, and pseudospectrally by use of the FFT algorithm for larger two dimensional or three dimensional problems. Boundary conditions are frequently easily imposed by natural means of the tau method as an obvious (linear) algebraic constraint equation on the coefficients. Since spectral methods are global in character, and infinite in order, no special techniques are required to calculate derivatives at boundaries (unlike finite difference algorithms). Finally, a variety of diagnostic integral tests of the accuracy of the solution are usually easily implemented. For the model problem considered in this talk, a nonlinear boundary value problem, conventional shooting methods may encounter several difficulties including the occurrence of spontaneous singularities, and intrinsic instability of the desired solution. Numerical solution of this problem using the Gear package, an adaptive step Romberg integration routine of the author, and a spectral program were implemented. (Details will be discussed in a forthcoming article by Ruehr and Ierley, 1985.) For accuracy and efficiency spectral methods proved obviously superior (unless quite high accuracy is required of high derivative terms near the origin). Greater programming time is, for this problem, the only significant drawback. © 1986

Theoretical estimates of the westward drift

Virtually all dynamo models may be expected to give rise to a permanent differential rotation between mantle and core. Weak conductivity in the mantle permits small leakage currents which couple to the radial component of the magnetic field, producing a Lorentz torque. Mechanical equilibrium is achieved when a zero net torque is established at a critical rotation rate. An estimate of the drift is determined easily given the magnetic field structure predicted by any dynamo model. The result for the drift rate at the core-mantle interface along the equator is given by the product of three factors Uφ*=UφR λ* L*. The first of these is a geometrical factor which depends only on the structural character of the field. For a variety of model fields, this factor ranges from 16 to 35. The second factor is the ratio of r.m.s. toroidal to poloidal field. This ratio is an (implicitly) adjustable parameter of both α2 and α-ω dynamos, and is a measure of the relative efficiency of the generation process for each component. The third (dimensional) term is the ratio of core magnetic diffusivity to core radius, 10-4 cm s-1. The result is essentially independent of the value of mantle diffusivity and its effective depth. The sign of the result may be positive or negative. For α2 dynamos a westward drift is produced by choosing α \u3e 0 in the Northern Hemisphere, which constitutes a dynamical assertion about the dynamo process. For an r.m.s. toroidal field of the order of 15 Gs, based on fairly general considerations, a drift rate comparable to observation is expected. © 1984

A modal α2-dynamo in the limit of asymptotically small viscosity

A spherical α2-dynamo is presented as an expansion in the free decay modes of the magnetic field. In the limit of vanishing viscosity the momentum equation yields various asymptotic expansions for the flow, depending on the precise form of the dissipation and boundary conditions applied. A new form for the dissipation is introduced that greatly simplifies this asymptotic expansion. When these expansions are substituted back into the induction equation, a set of modal amplitude equations is derived, and solved for various distributions of the α-effect. For all choices of α the solutions approach the Taylor state, but the manner in which this occurs can vary, as previously found by Soward and Jones (1983). Furthermore, as hypothesized by Malkus and Proctor (1975), but not previously demonstrated, the post-Taylor equilibration is indeed independent of the viscosity in the asymptotic limit, and depending on the choice of a may be either steady-state or oscillatory. © 1991 Gordon and Breach Science Publishers S.A

Quasi-L p norm orthogonal Galerkin expansions in sums of Jacobi polynomials: Orthogonal expansions

In the study of differential equations on [ − 1,1] subject to linear homogeneous boundary conditions of finite order, it is often expedient to represent the solution in a Galerkin expansion, that is, as a sum of basis functions, each of which satisfies the given boundary conditions. In order that the functions be maximally distinct, one can use the Gram-Schmidt method to generate a set orthogonal with respect to a particular weight function. Here we consider all such sets associated with the Jacobi weight function, w(x) = (1 − x) α (1 + x) β . However, this procedure is not only cumbersome for sets of large degree, but does not provide any intrinsic means to characterize the functions that result. We show here that each basis function can be written as the sum of a small number of Jacobi polynomials, whose coefficients are found by imposing the boundary conditions and orthogonality to the first few basis functions only. That orthogonality of the entire set follows—a property we term “auto-orthogonality”—is remarkable. Additionally, these basis functions are shown to behave asymptotically like individual Jacobi polynomials and share many of the latter’s useful properties. Of particular note is that these basis sets retain the exponential convergence characteristic of Jacobi expansions for expansion of an arbitrary function satisfying the boundary conditions imposed. Further, the associated error is asymptotically minimized in an L p(α) norm given the appropriate choice of α = β. The rich algebraic structure underlying these properties remains partially obscured by the rather difficult form of the non-standard weighted integrals of Jacobi polynomials upon which our analysis rests. Nevertheless, we are able to prove most of these results in specific cases and certain of the results in the general case. However a proof that such expansions can satisfy linear boundary conditions of arbitrary order and form appears extremely difficult