Cheng-Chin Wu

On a dynamo driven by topographic precession

More than half a century ago, Bullard suggested that the Earths dynamo might be driven by the motions created in the Earths core by the luni-solar precession. The precessionally forced motion of the mantle drives core flow through viscous forces and also, because of the electrical conductivity of the deep mantle, through magnetic forces. Both these couplings are thought to be insignificant in comparison with the topographical coupling created by the oblateness of the core-mantle interface. Because of technical difficulties in studying dynamo action in non-spherical bodies of electrically conducting fluid, this is the first serious attempt to study dynamo action by topographically forced flows. It describes the novel numerical methods that were employed, the tests that were devised to validate these methods, and the successful outcome of those tests. Some preliminary results for these dynamos are presented. It is shown that, in some parameter ranges, the magnetic field produced by the dynamos enhances the vigor of the precessional motions.

Structure and stability of a spherical implosion

Abstract Similarity solutions and the stability for strong spherical implosions are studied for both ideal and van der Waals gases. When the van der Waals excluded volume is sufficiently large, a new type of solution is found and the shock may be linearly stable. Implications for inertial confinement fusion and sonoluminescence are discussed.

Bubble shape instability and sonoluminescence

Abstract Light from a sonoluminescing bubble is extinguished if the amplitude of the acoustic field that drives the bubble oscillation exceeds a certain threshold. It has been suggested that shape instability of the bubble surface is responsible. The effect of viscosity on such an instability is examined here.

On a dynamo driven topographically by longitudinal libration

Variation in the angular velocity Ω of a planetary body is called libration or longitudinal libration when the Ω-axis is fixed in direction. This motion of the bodys solid mantle drives motions in its fluid core, either by viscous coupling across the core-mantle interface S, or topographically when S is asymmetric with respect to the Ω-axis, the only case considered in this article. A significant topographically-driven flow is identified having uniform vorticity within S and no component parallel to the Ω-axis. Its dynamic stability depends on the amplitude, Ω 1, of the sinusoidally varying part of Ω and on the ratio, b/a, of the lengths of the principal axes of S, assumed spheroidal. In (Ω 1/Ω 0, b/a) parameter space where Ω 0 is the average Ω, islands are shown to exist where the constant vorticity states are dynamically unstable. These are surrounded by a sea in which they are stable. When the fluid is slightly viscous, a state in the stable sea retains its uniform vorticity structure except in a viscous boundary layer on S in which the flow acquires a component parallel to the Ω-axis. For (Ω 1/Ω 0, b/a) on an island where the uniform vorticity state is unstable, an “alternative flow” exists, which is three-dimensional and is examined here. Assuming that the core is electrically conducting, kinematic dynamos are sought. Uniform vorticity flow appears to be non-regenerative but, when it is stable and viscosity acts to create a sufficiently strong boundary layer flow, dynamo action may occur. It is shown that the alternative flow that exists on an instability island in (Ω 1/Ω 0,u2009b/a) space can be vigorously regenerative.

Evolution of small-amplitude intermediate shocks in a dissipative and dispersive system

Our study of the relationship between shock structure and evolutionarity is extended to include the effects of dispersion as well as dissipation. We use the derivative nonlinear Schrodinger-Burgers equation (DNLSB), which reduces to the Cohen-Kulsrud-Burgers equation (CKB) when finite ion inertia dispersion can be neglected. As in our previous CKB analysis, the fast shock solution is again unique, and the intermediate shock structure solutions are non-unique. With dispersion, the steady intermediate shock structure solutions continue to be labelled by the integral through the shock of the non-co-planar component of the magnetic field, whose value now depends upon the ratio of the dispersion and dissipation lengths. This integral helps to determine the solution of the Riemann problem. With dispersion, this integral is also non-zero for fast shocks. Thus, even for fast shocks, the solution of the Riemann problem depends upon shock structure.

On magnetostrophic mean-field solutions of the geodynamo equations

A dynamo driven by motions unaffected by viscous forces is termed magnetostrophic. Although such a model might describe well magnetic field generation in Earth’s core, its existence is in doubt as numerical simulators have to impose substantial viscosity to stabilize solutions of the full MHD dynamo equations. An attempt is made here to revive interest in a procedure proposed by Taylor [Proc. R. Soc. Lond. A, 1963, 274, 274] for finding inertialess magnetostrophic dynamos. The evolution of the magnetic field from the fluid flow follows the usual kinematic path, but the creation of the zero viscosity flow from the magnetic field was reduced by Taylor to the solution of a second-order ordinary differential equation. Roberts and Wu [Geophys. Astrophys. Fluid Dyn., 2014, 108] derived an exact solution of this equation for axisymmetric mean-field dynamos. Numerical solutions of this equation are presented here, leading to the first truly magnetostrophic dynamos ever found. The magnetic field and fluid flow are derived and discussed for - and -dynamos.

High order instabilities of the Poincaré solution for precessionally driven flow

Sloudsky in 1895 and Poincaré in 1910 were the first to derive solutions for the flow driven in the Earths fluid core by the luni-solar precession. In 1993, Kerswell investigated the stability of this so-called “Poincaré flow” by applying a method devised in 1992 by Ponomarev and Gledzer to study the instability of flows with elliptical streamlines. They represented the components of the perturbed flow by sums of polynomials. Kerswell restricted attention to the linear and quadratic cases. Here cubic, quartic, quintic, and sextic generalizations are developed. Instabilities are located in new areas of parameter space, including some that verge on the small oblateness of the Earths core

On the modified Taylor constraint

A dynamo driven by motions unaffected by viscous forces is termed magnetostrophic. Although such a model might describe magnetic field generation in Earth’s core well, a magnetostrophic dynamo has not yet been found even though Taylor [Proc. R. Soc. Lond. A 1963, 274, 274–283] devised an apparently viable method of finding one. His method for determining the fluid velocity from the magnetic field and the energy source involved only the evaluation of integrals along lines parallel to the Earth’s axis of rotation and the solution of a second-order ordinary differential equation. It is demonstrated below that an approximate solution of this equation for a broad family of magnetic fields is immediate. Furthermore inertia, which was neglected in Taylor’s theory, is restored here, so that the modified theory includes torsional waves, whose existence in the Earth’s core has been inferred from observations of the length of day. Their theory is reconsidered.

A high order WENO finite difference scheme for incompressible fluids and magnetohydrodynamics

We present a high order accurate weighted essentially non-oscillatory (WENO) finite difference scheme for solving the equations of incompressible fluid dynamics and magnetohydrodynamics (MHD). This scheme is a direct extension of a WENO scheme that has been successfully applied to compressible fluids, with or without magnetic fields. A fractional time-step method is used to enforce the incompressibility condition. Two basic elements of the WENO scheme, upwinding and wave decomposition, are shown to be important in solving the incompressible systems. Numerical results demonstrate that the scheme performs well for one-dimensional Riemann problems, a two-dimensional double-shear flow problem, and the two-dimensional Orszag–Tang MHD vortex system. They establish that the WENO code is numerical stable even when there are no explicit dissipation terms. It can handle discontinuous data and attain converged results with a high order of accuracy.