I. Thermal Evolution of Ganymede and Implications for Surface Features. II. Magnetohydrodynamic Constraints on Deep Zonal Flow in the Giant Planets. III. A Fast Finite-Element Algorithm for Two-Dimensional Photoclinometry

Citation

Kirk, Randolph Livingstone (1987) I. Thermal Evolution of Ganymede and Implications for Surface Features. II. Magnetohydrodynamic Constraints on Deep Zonal Flow in the Giant Planets. III. A Fast Finite-Element Algorithm for Two-Dimensional Photoclinometry. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/T5PT-S948. https://resolver.caltech.edu/CaltechETD:etd-11142006-131417

Abstract

The work is divided into three independent papers:

PAPER I:

Thermal evolution models are presented for Ganymede, assuming a mostly differentiated initial state of a water ocean overlying a rock layer. The only heat sources are assumed to be primordial heat (provided by accretion) and the long-lived radiogenic heat sources in the rock component. As Ganymede cools, the ocean thins, and two ice layers develop, one above composed of ice I, and the other below composed of high-pressure polymorphs of ice. Subsolidus convection proceeds separately in each ice layer, its transport of heat calculated using a simple parameterized convection scheme and the most recent data on ice rheology. The model requires that the average entropy of the deep ice layer exceed that of the ice I layer. If the residual ocean separating these layers becomes thin enough, then a Rayleigh-Taylor-like ("diapiric") instability may ensue, driven by the greater entropy of the deeper ice and merging the two ice mantles into a single convective layer. This instability is not predicted by linear analysis but occurs for plausible finite amplitude perturbations associated with large Rayleigh number convection. The resulting warm ice diapirs may lead to a dramatic "heat pulse" at the surface and to fracturing of the lithosphere, and may be directly or indirectly responsible for resurfacing and grooved terrain formation on Ganymede. The timing of this event depends rather sensitively on poorly known rheological parameters but could be consistent with chronologies deduced from estimated cratering rates. Irrespective of the occurrence or importance of the heat pulse, we find that lithospheric fracturing requires rapid stress loading (on a timescale ≾ 10⁴) years). Such a timescale can be realized by warm ice diapirism, but not directly by gradual global expansion. In the absence of any quantitative and self-consistent model for the resurfacing of Ganymede by liquid water, we favor resurfacing by warm ice flows,which we demonstrate to be physically possible, a plausible consequence of our models, compatible with existing observations, and a hypothesis testable by Galileo. We discuss core formation as an alternative driver for resurfacing, and conclude that it is less attractive. We also consider anew the puzzle of why Callisto differs so greatly from Ganymede, offering several possible explanations. The models presented do not provide a compelling explanation for all aspects of Ganymedean geological evolution, since we have identified several potential problems, most notably the apparently extended period of grooved terrain formation (several hundred million years), which is difficult to reconcile with the heat pulse phenomenon.

PAPER II:

The observed zonal flows of the giant planets will, if they penetrate below the visible atmosphere, interact significantly with the planetary magnetic field outside the metalized core. The appropriate measure of this interaction is the Chandrasekhar number Q = (H²)/(4πρνα²λ) (where H = radial component of the magnetic field, ν = eddy viscosity, λ = magnetic diffusivity, α^-1 = lengthscale on which λ varies); at depths where Q ≳ 1 the velocity will be forced to oscillate on a small lengthscale or decay to zero. We estimate the conductivity due to semiconduction in H² (Jupiter, Saturn) and ionization in H²O (Uranus, Neptune) as a function of depth; the value λ ≃ 10¹⁰ cm²s^-1 needed for Q = 1 is readily obtained well outside the metallic core (where λ ≃ 10² cm²s^-1).

These assertions are quantified by a simple model of the equatorial zonal jet in which the flow is assumed uniform on cylinders concentric with the spin axis, and the viscous and magnetic torques on each cylinder are balanced. We solve this "Taylor constraint" simultaneously with the dynamo equation to obtain the velocity and magnetic field in the equatorial plane. With this model we reproduce the widely differing jet widths of Jupiter and Saturn (though not the flow at very high or low latitudes) using ν = 2500 cm²s^-1, consistent with the requirement that viscous dissipation not exceed the specific luminosity. A model Uranian jet consistent with the limited Voyager data can also be constructed, with appropriately smaller ν, but only if one assumes a two-layer interior. We tentatively predict a wide Neptunian jet.

For Saturn (but not Jupiter or Uranus) the model has a large magnetic Reynolds number where Q = 1 and hence exhibits substantial axisymmetrization of the field in the equatorial plane. This effect may or may not persist at higher latitudes. The one-dimensional model presented is only a first step. Variation of the velocity and magnetic field parallel to the spin axis must be modeled in order to answer several important questions, including: 1) What is the behavior of flows at high latitudes, whose Taylor cylinders are interrupted by the region with Q ≳ 1? 2) To what extent is differential rotation in the envelope responsible for the spin-axisymmetry of Saturn's magnetic field?

PAPER III:

It is shown that the problem of two-dimensional photoclinometry (PC) -- the reconstruction of a surface z(x,y) from a brightness image B(x,y) -- may be formulated in a natural way in terms of finite elements. The resulting system of equations is underdetermined as a consequence of the lack of boundary conditions for z, but a unique solution may be chosen by minimizing a function S expressing the "roughness" of the surface. An efficient PC algorithm based on this formulation is presented, requiring ~ 10.66 (four-byte) memory locations and ~10⁴ floating multiplications/additions per pixel, and incorporating: 1) Minimization of the roughness by the penalty method, which yields the smallest set of equations. 2) Iterative solution of the nonlinear equations by Newton's method. 3) Solution of the linearized equations by an inner iterative cycle of successive over-relaxation, which takes advantage of the extreme sparseness of the system. 4) Multigridding, in which the solutions to the smaller problems obtained by reducing the resolution are used recursively to greatly speed convergence at the higher resolutions, and 5) A rapid noniterative initial estimate of z obtained by exploiting the special symmetry of the equations obtained in the first linearization.

The algorithm is extensively demonstrated on 200 by 200 pixel synthetic "images" generated from digital topographic data for northern Utah over a range of phase angles. Rms error in the solution is ~ 22 m, out of ~ 660 m total relief. The error is dominated by "stripes" with the same azimuth as the light source, resulting from use of the roughness criterion in lieu of boundary conditions; the rms error along profiles parallel to the stripes is only ~ 2-8 m, depending on the phase angle. Satisfactory solutions are obtained even in the presence of quantization error, noise, and moderate blur in the image.

Applications of the PC algorithm to both remote sensing and photomicrography are sketched; a photoclinometric map of a low-relief Precambrian era fossil is presented as an example of the latter. Prospects for dealing with photometrically inhomogeneous surfaces, and an extension of the method to the analysis of side-looking radar data ("radarclinometry") are also discussed.

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