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Gough, D. O.

Publications and source records attributed to Gough, D. O..

Sources of uncertainty in direct seismological measurements of the solar helium abundance

The methods by which Dappen et al. (1988, 1990, 1991) and Dziembowski et al. (1990, 1991, 1992) recently obtained discrepant estimates of the helium abundance in the solar convection zone are compared. The aim of the investigation reported in this paper is to identify the main source of the discrepancy. Using as proxy data eigenfrequencies of a set of modes of a theoretical solar model, computed with the same physics as were the frequencies of a reference model with which these data are compared, the two methods yield similar results. Thus we ascertain that the principal source of the discrepancy is not in the inversions themselves, which yield essentially a measure of the variation of the adiabatic exponent gamma of the material in the He II ionization zone. Instead it is in the approximations adopted in the treatment of heavy elements in the equation of state used to relate the variation of gamma to chemical composition. We obtain acceptably consistent results when inverting solar data by two methods using the same equation of state. We attempt to identify the likely residual sources of uncertainty.

Kosovichev, A. G.

On the rate of destruction of lithium in late-type main-sequence stars

It it shown that the rate of destruction of lithium by nuclear reactions in the outer mixed layers of late-type main-sequence stars depends on both the depth of the region where mixing occurs and the stratification within it. The mixed region includes and probably extends beneath the base of the convection zone. As the star evolves on the main sequence, the properties of the convection zone vary in a simple manner. If the mixed layer behaves similarly, then the mean destruction rate can easily be related to the present local value evaluated at the base of the mixed layer. In the case of the sun, it is found that if mixing is rapid compared with the nuclear destruction rate, then the mean destruction rate is approximately equal to one-half the present local value.

Christensen-Dalsgaard, J.

The depth of the solar convection zone

The transition of the temperature gradient between being subadiabatic and adiabatic at the base of the solar convection zone gives rise to a clear signature in the sound speed. Helioseismic measurements of the sound speed therefore permit a determination of the location of the base of the convection zone. Two techniques were tested by applying them to artifical data, obtained by adding simulated noise to frequencies computed from two different solar models. The determinations appear to be relatively insensitive to uncertainties of the physics of the solar interior. From an analysis of observed frequencies of solar oscillation it is concluded that the depth of the solar convection zone is (0.287 + or - 0.003) solar radii.

Christensen-Dalsgaard, J.

Speed of sound in the solar interior

The sound speed of the solar interior is directly determinable on the basis of the frequencies of solar 5-min oscillations, irrespective of solar model, and relying only on a simple asymptotic description of the oscillations in terms of trapped acoustic waves. It is plausible that, by using this asymptotic determination as an initial trial in a more accurate inversion, and imposing constraints of smoothness on the solution resulting from the iteration, a good model representing the large scale structure of the sun which satisfies the observed frequencies may be determined.

Christensen-Dalsgaard, J.

Internal rotation of the sun

The frequency difference between prograde and retrograde sectoral solar oscillations is analyzed to determine the rotation rate of the solar interior, assuming no latitudinal dependence. Much of the solar interior rotates slightly less rapidly than the surface, while the innermost part apparently rotates more rapidly. The resulting solar gravitational quadrupole moment is J2 = (1.7 + or - 0.4) x 10 to the -7th and provides a negligible contribution to current planetary tests of Einstein's theory of general relativity.

Duvall, T. L., Jr.

On the identification of normal modes of oscillation from observations of the solar periphery

The decomposition of solar oscillations into their constituent normal modes requires a knowledge of both the spatial and temporal variation of the perturbation to the sun's surface. The task can be especially difficult when only limited spatial information is available. Observations of the limb-darkening function, for example, are probably sensitive to too large a number of modes to permit most of the modes to be identified in a power spectrum of measurements at only a few points on the limb, unless the results are combined with other data. In this paper a procedure is considered by which the contributions from quite small groups of modes to spatially well resolved data obtained at any instant can be extracted from the remaining modes. Combining these results with frequency information then permits the modes to be identified, at least if their frequencies are low enough to ensure that modes of high degree do not contribute substantially to the signal.

Gough, D. O.

Immediate and long-term prospects for helioseismology

Recent extensive measurements of frequencies of free oscillation of the sun have permitted a first direct estimate of the variation of sound speed and angular velocity throughout the sun. The results hint that the answers to some tantalizing questions concerning the sun's interior structure and its history are almost within grasp. Optimists like myself believe that in a few years a worldwide network of ground-based observing stations will give us imortant clues. However, it may be necessary to make observations from space before we can be sure of the answers.

Gough, D. O.

On the detection of subphotospheric convective velocities and temperature fluctuations

A procedure is outlined for estimating the influence of large-scale convective eddies on the wave patterns of five-minute oscillations of high degree. The method is applied to adiabatic oscillations, with frequency omega and wave number k, of a plane-parallel polytropic layer upon which is imposed a low-amplitude convective flow. The distortion to the k-omega relation has two constituents: one depends on the horizontal component of the convective velocity and has a sign which depends on the sign of omega/k; the other depends on temperature fluctuations and is independent of the sign of omega/k. The magnitude of the distortion is just at the limit of present observational sensitivity. Thus there is reasonable hope that it will be possible to reveal some aspects of the large-scale flow in the solar convection zone.

Gough, D. O.

Time-dependent solutions of multimode convection equations

Truncated modal equations are used to study the time evolution of thermal convection. In the Boussinesq approximation these nonlinear equations are obtained by expanding the fluctuating velocity and temperature fields in a finite set of planforms of the horizontal coordinates. Numerical studies dealing with two or three modes with triad interactions are discussed. Rich time dependence was found in these cases: periodic and aperiodic solutions can be obtained, along with various steady solutions. Three-mode solutions reproduce the qualitative appearance of spoke-pattern convection as observed in experiments at high Prandtl numbers. Though the values of the periods of the time-dependent solutions do not agree with those of the experiments, their variation with Rayleigh number compares favorably. Except at the highest Rayleigh number considered (10,000,000), the theoretical Nusselt numbers agree well with experiment.

Toomre, J.

Single-mode theory of diffusive layers in thermohaline convection

A two-layer configuration of thermohaline convection is studied, with the principal aim of explaining the observed independence of the buoyancy-flux ratio on the stability parameter when the latter is large. Temperature is destabilizing and salinity is stabilizing, so diffusive interfaces separate the convecting layers. The convection is treated in the single-mode approximation, with a prescribed horizontal planform and wavenumber. Surveys of numerical solutions are presented for a selection of Rayleigh numbers R, stability parameters lambda and horizontal wavenumbers. The solutions yield a buoyancy flux ratio chi that is insensitive to lambda, in accord with laboratory experiments. However chi increases with increasing R, in contradiction to laboratory observations.

Gough, D. O.

Numerical solutions of single-mode convection equations

In the Boussinesq approximation, single-mode equations describing thermal convection are constructed by expanding the fluctuating velocity and temperature fields in a complete set of functions (or planforms) of the horizontal coordinates and retaining just one term. Numerical solutions of the single-mode equations are investigated, chief consideration being given to hexagonal planforms. Extensive surveys of steady solutions are presented for various Rayleigh numbers, Prandtl numbers, and horizontal wavenumbers. The dependences on Rayleigh number and Prandtl number at very large Rayleigh number are in satisfactory agreement with the results of asymptotic expansions.

Toomre, J.

Modal equations for cellular convection

We expand the fluctuating flow variables of Boussinesq convection in the planform functions of linear theory. Our proposal is to consider a drastic truncation of this expansion as a possible useful approximation scheme for studying cellular convection. With just one term included, we obtain a fairly simple set of equations which reproduces some of the qualitative properties of cellular convection and whose steady-state form has already been derived by Roberts (1966). This set of 'modal equations' is analyzed at slightly supercritical and at very high Rayleigh numbers. In the latter regime the Nusselt number varies with Rayleigh number just as in the mean-field approximation with one horizontal scale when the boundaries are rigid. However, the Nusselt number now depends also on the Prandtl number in a way that seems compatible with experiment. The chief difficulty with the approach is the absence of a deductive scheme for deciding which planforms should be retained in the truncated expansion.

Gough, D. O.

Nonlinear cellular motions in Poiseuille channel flow

In a number of nonlinear solutions to the equations of channel flow the velocity is decomposed into a mean part plus a nonlinear disturbance. The idea that nonlinear effects place a limitation on the amplitudes of the disturbance flow is considered. In the reported investigation the disturbance flow is represented by drastically limited Fourier expansions in the downstream coordinate. The resulting equations are solved numerically with high accuracy to obtain a good representation of the cross-stream structure of the solution. The results of the investigation show that indeed the nonlinear terms always limit the amplitude of the disturbance flow in this approximation.

Zahn, J.-P.

Rotating stars

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Gough, D. O.