New Theory for Zonal and Meridional Flows in the Solar Convection Zone; and a Simple Model for Secular Orbit Oscillations
This theoretical work first studies the fundamental force balances associated with the rotation and internal motion of the Sun. The relative motions take place in the convection zone. They include the dominant global mean zonal circulation, and the order of magnitude smaller mean meridional circulation. The observed zonal circulation is shown to be consistent with a dynamic equilibrium between global pressure gradients, centrifugal forces and Coriolis forces. This force balance is revealed only by using the most precise observations of global shape and by avoiding the effective gravity approximation. The latter is the more challenging conceptual task, as it has become an almost universally applied assumption in geophysical and astrophysical fluid dynamics. Explanation of the meridional circulation requires a similar theoretical advance. A generalized astrophysical Ekman theory is developed in spherical polar coordinates, and in the absence of the customary approximations for geostrophic balance, in order to remain consistent with the dynamic equilibrium of the zonal flow. Observed zonal velocity anomalies, departures from the dynamic equilibrium, are interpreted as the effects of friction forces retarding the bulk flow at the top and bottom boundaries. Astrophysical Ekman theory shows how friction then forces a meridional overturning consistent with observations. For both zonal and meridional circulations the forces governing internal fluid motions are shown to be strongly influenced by the inertial forces of rotation - Coriolis and centrifugal forces. Next the precise global shape is explained by returning to the friction free dynamic equilibrium model. A new equipotential surface is derived from first principles. The radial force, and the associated potential function, now include contributions from the Coriolis forces of relative motion - as well as the typical gravity and centrifugal terms. Using an empirical description of the observed relative motions leads to surface shape function as a sum of Legendre polynomials and their amplitudes. Shape anomalies predicted by this method are near to their observed values. Additional analysis details the specific angular momentum content of the zonal flows on the Sun and outer planets. Their totals consistently fit within a hierarchy between the values of the host body and its natural satellites. Within each body the specific angular momentum broken down by latitude appears to have a distribution consistent with a narrow variance around a mean value. Examination of kinetic and potential energy indicates the non-spherical component of the potential may be identified with the kinetic energy. These studies suggest the global shape and internal motion of the Sun and outer planets may be traced back to an effectively constant angular momentum of the fluid envelop. The inherited angular momentum content, simply described by its mean and variance, may determine the prograde or retrograde character and overall pattern of the body's relative zonal motions. The initial condition and slow evolution of this angular momentum may explain the variety of surface flows observed on the Sun and outer planets. The last part of the dissertation makes an initial foray into the angular momentum dynamics of the solar system as a whole. Motivated by the central role of rotation and angular momentum for the Sun’s internal fluid dynamics, it constructs a highly simplified theoretical model for interacting planetary orbits. Its design recognizes the significant history of secular perturbation theory, and the relatively recent vector theory of angular momentum. The latter is used in a relatively simple analytical model to obtain an approximate subset of the known secular frequencies of orbit variations. Obtaining these oscillations, with periods of order 10 Kyr - 10 Myr, previously required long time scale integrations or complex perturbation techniques. The model provides a highly accessible introduction to the secular dynamics of planetary orbits. More than that, it provides insight into the physical mechanisms, which can be difficult to extract from the classical approach. Nodal precession dynamics are shown to be of central importance in the exchange of angular momentum between planetary orbits, and among the angular momentum components of a single orbit. These insights suggest new possibilities for interpreting Earth's geological record, and suggest model advancements that could provide a new basis for investigating and modeling secular dynamics.
differential rotation, equilibrium shape, meridional circulation, orbit oscillations, secular frequencies, zonal circulation
Networked Digital Library of Theses and Dissertations