The construction of exact Taylor states. I: The full sphere

The dynamics of the Earth's fluid core are described by the so-called magnetostrophic balance between Coriolis, pressure, buoyancy and Lorentz forces. In this regime the geomagnetic field is subject to a continuum of theoretical conditions, which together comprise Taylor's constraint, placing restrictions on its internal structure. Examples of such fields, so-called Taylor states, have proven difficult to realize except in highly restricted cases. In previous theoretical developments, we showed that it was possible to reduce this infinite class of conditions to a finite number of coupled quadratic homogeneous equations when adopting a certain regular truncated spectral expansion for the magnetic field. In this paper, we illustrate the power of these results by explicitly constructing two families of exact Taylor states in a full sphere that match the same low-degree observationally derived model of the radial field at the core—mantle boundary. We do this by prescribing a smooth purely poloidal field that fits this observational model and adding to it an expediently chosen unconstrained set of interior toroidal harmonics of azimuthal wavenumbers 0 and 1. Formulated in terms of the toroidal coefficients, the resulting system is purely linear and can be readily solved to find Taylor states. By calculating the extremal members of the two families that minimize the Ohmic dissipation, we argue on energetic ground that the toroidal field in the Earth's core is likely to be dominated by low order azimuthal modes, similar to the observed poloidal field. Finally, we comment on the extension of finding Taylor states within a general truncated spectral expansion with arbitrary poloidal and toroidal coefficient

Eigenanalysis of the two-dimensional wind-driven ocean circulation problem

A barotropic model of the wind-driven circulation in the subtropical region of the ocean is considered. A no-slip condition is specified at the coasts and slip at the fluid boundaries. Solutions are governed by two parameters: inertial boundary-layer width; and viscous boundary-layer width. Numerical computations indicate the existence of a wedge-shaped region in this two-dimensional parameter space, where three steady solutions coexist. The structure of the steady solution can be of three types: boundary-layer, recirculation and basin-filling-gyre. Compared to the case with slip conditions (Ierley and Sheremet, 1995) in the no-slip case the wedge-shaped region is displaced to higher Reynolds numbers. Linear stability analysis of solutions reveals several classes of perturbations: basin modes of Rossby waves, modes associated with the recirculation gyre, wall-trapped modes and a “resonant” mode. For a standard subtropical gyre wind forcing, as the Reynolds number increases, the wall-trapped mode is the first one destabilized. The resonant mode associated with disturbances on the southern side of the recirculation gyre is amplified only at larger Reynolds number, nonetheless this mode ultimately provides a stronger coupling between the mean circulation and Rossby basin modes than do the wall-trapped modes

A Fully Pseudospectral Scheme for Solving Singular Hyperbolic Equations

With the example of the spherically symmetric scalar wave equation on Minkowski space-time we demonstrate that a fully pseudospectral scheme (i.e. spectral with respect to both spatial and time directions) can be applied for solving hyperbolic equations. The calculations are carried out within the framework of conformally compactified space-times. In our formulation, the equation becomes singular at null infinity and yields regular boundary conditions there. In this manner it becomes possible to avoid "artificial" conditions at some numerical outer boundary at a finite distance. We obtain highly accurate numerical solutions possessing exponential spectral convergence, a feature known from solving elliptic PDEs with spectral methods. Our investigations are meant as a first step towards the goal of treating time evolution problems in General Relativity with spectral methods in space and time.Comment: 24 pages, 12 figure

A comparison of no-slip, stress-free and inviscid models of rapidly rotating fluid in a spherical shell

We investigate how the choice of either no-slip or stress-free boundary conditions affects numerical models of rapidly rotating flow in Earth's core by computing solutions of the weakly-viscous magnetostrophic equations within a spherical shell, driven by a prescribed body force. For non-axisymmetric solutions, we show that models with either choice of boundary condition have thin boundary layers of depth E^(1/2), where E is the Ekman number, and a free-stream flow that converges to the formally inviscid solution. At Earth-like values of viscosity, the boundary layer thickness is approximately 1m, for either choice of condition. In contrast, the axisymmetric flows depend crucially on the choice of boundary condition, in both their structure and magnitude (either E^(-1/2) or E^(-1)). These very large zonal flows arise from requiring viscosity to balance residual axisymmetric torques. We demonstrate that switching the mechanical boundary conditions can cause a distinct change of structure of the flow, including a sign-change close to the equator, even at asymptotically low viscosity. Thus implementation of stress-free boundary conditions, compared with no-slip conditions, may yield qualitatively different dynamics in weakly-viscous magnetostrophic models of Earth's core. We further show that convergence of the free-stream flow to its asymptotic structure requires E ≤10^(-5)

Phase space analysis of the spurt phenomenon for the Giesekus viscoelastic fluid model

The spurt phenomenon is a flow instability which occurs in pressure-driven parallel shear flows of viscoelastic liquids. This phenomenon is characterized by an abrupt increase in the volumetric throughput at a critical value of the driving pressure gradient. Recently, non-monotone (steady shear response) constitutive equations have been proposed to model this phenomenon. We analyze the startup problem for the Giesekus model, both asymptotically and numerically, and compare the results to those obtained for other models. © 1989

Pulse dynamics in an unstable medium

A study is presented of a one-dimensional, nonlinear partial differential equation that describes evolution of dispersive, long-wave instability. The solutions, under certain specific conditions, take the form of trains of well-separated pulses. The dynamics of such patterns of pulses is investigated using singular perturbation theory and with numerical simulation. These tools permit the formulation of a theory of pulse interaction, and enable the mapping out of the range of behavior in parameter space. There are regimes in which steady trains form; such states can be studied with the asymptotic, pulse-interaction theory. In other regimes, pulse trains are unstable to global, wave-like modes or its radiation. This can precipitate more violent phenomena involving pulse creation, or generate periodic states which may follow Shil`nikov`s route to temporal chaos. The asymptotic theory is generalized lo take some account of radiative dynamics. In the limit of small dispersion, steady trains largely cease to exist; the system follows various pathways to temporal complexity and typical-bifurcation sequences are sketched out. The investigation guides us to a critical appraisal of the asymptotic theory and uncovers the wealth of different types of behavior present in the system

The construction of exact Taylor states: I: The full sphere

ISSN:0956-540XISSN:1365-246