Weakly nonlinear shallow water magnetohydrodynamic waves

We consider an electrically conducting rotating fluid governed by the shallow water magnetohydrodynamic equations with no diffusion. We use an a priori asymptotic technique (the method of geometric optics or ray method) to study weakly nonlinear hydromagnetic waves. These waves are intermediate in length in the following sense: they are much longer than the fluid depth but much shorter than the radius of the earth. The time scale for the waves is much longer than that of the free surface oscillations and the approximation varies on an even longer timescale. The waves we are considering are studied in the beta plane approximation for an ambient magnetic field parallel to the equator which varies in the direction perpendicular to the equator. The leading order approximation gives a dispersion relation for the waves, which are generally found to be confined to bands about the equator as well as in bands at higher and lower latitudes. At the next order of approximation, a conservation law is found for the wave amplitude. We also obtain an equation governing the behavior of the leading order mean azimuthal velocity which is forced to grow linearly with time.     

Solubility of excess alumina in hydrous granitic melts in equilibrium with peraluminous minerals at 700–800 °C and 200 MPa,and applications of the aluminum saturation index

We have studied the controls on the Aluminum Saturation Index (ASI = molec. Al₂O₃/[(CaO)+(Na₂O)+(K₂O)]) and the concentration of normative corundum of granitic liquids saturated in alumina by equilibrating peraluminous minerals with initially metaluminous haplogranitic minimum composition liquids at 700–800 °C and 200 MPa, at, and below H₂O saturation. The ASI and normative corundum increase with increasing H₂O concentration in the melt (0.04 to 0.10 moles excess Al₂O₃ per mole of H₂O), temperature, and with addition of the non-haplogranitic components Fe, Mg, and B. The ASI parameter and concentration of normative corundum cannot be used to monitor aAl₂O₃ between different mineral assemblages and melt because other components that affect the solubility of alumina, including H₂O, Fe, Mg, and B, do not appear in their formulations. ASI and normative corundum, however, provide petrogenetic information about magmas generated by partial melting of strongly peraluminous protoliths by virtue of their regular and predictable variation with melt composition (e.g., H₂O concentration) and temperature. For the application of these data to natural rocks it is necessary to choose as an analogue system the ASI-solubility or normative corundum-solubility relations of the most chemically complex peraluminous mineral present in the rock. Comparison of ASI values of anatectic leucosomes and allochthonous leucogranites with experimentally predicted values suggests low H₂O concentrations in melt during crustal partial melting. Rapid melt segregation before equilibration with restitic peraluminous phases is also suggested in some cases.Editorial responsibility: I. Carmichael     

Nonlinear waves in rotating magnetohydrodynamics

We consider an electrically conducting fluid in rotating cylindrical coordinates in which the Elsasser and magnetic Reynolds numbers are assumed to be large while the Rossby number is assumed to vanish in an appropriate limit. This may be taken as a simple model for the Earth's outer core. Fully nonlinear waves dominated by the nonlinear Lorentz forces are studied using the method of geometric optics (essentially WKB). These waves are assumed to be of the form of an asymptotic series expanded about ambient magnetic and velocity fields which vanish on the equatorial plane. They take the form of short wave, slowly varying wave trains. The first-order approximation is sinusoidal and basically the same as in the linear problem, with a dispersion relation modified by the appearance of mean terms. These mean terms, as well the undetermined amplitude functions, are found by suppressing secular terms in a “fast” variable in the second-order approximation. The interaction of the mean terms with the dispersion relation is the primary cause of behaviors which differ from the linear case. In particular, new singularities appear in the wave amplitude functions and an initial value problem results in a singularity in one of the mean terms which propagates through the fluid. The singularities corresponding to the linear ones are shown to develop when the corresponding waves propagate toward the equatorial plane.     

Slow hydromagnetic oscillations in a rotating spheroidal shell

Abstract

The ray method is used to study slow hydromagnetic waves in an incompressible, inviscid, perfectly conducting fluid of constant density in the presence of a constant toroidal magnetic field. The fluid is bounded below by a rigid sphere and above by a rigid spheroidal surface, and the mean fluid layer thickness is assumed to be small. Both the general time-dependent and time-harmonic (free oscillation) problems are studied and dispersion relations and conservation laws are derived. These results are applied to free oscillations with constant azimuthal wave number in a spherical shell and then compared to those of previous authors. Such oscillations propagate to the east and are trapped between circles of constant latitude. Wave propagation in axisymmetric shells is then studied with emphasis on the relationship between shell shape and direction of propagation, and it is found that such shells can sustain westward propagating modes wherever the shell thickness decreases sufficiently rapidly from a maximum at the poles to zero at the equator; no shells exist which can sustain westward propagation at the equator.