Gareth P. Williams

Planetary circulations: 1. Barotropic representation of Jovian and terrestrial turbulence

Abstract We seek the formative processes of the planetary circulations of Jupiter and Saturn. The study concentrates on examining whether processes known to control the terrestrial circulation, namely, two-dimensional turbulence and baroclinic instability, can produce Jovian circulations under Jovian conditions. The first numerical model involves a spherical barotropic vorticity equation subjected to a stochastic representation of baroclinic processes. The resulting solutions suggest that a strong affinity exists between the Jovian and terrestrial circulations. This leads to a reevaluation of terrestrial circulation theory from the broader perspective of parameter space. The solutions in the Jovian regime support the hypothesis that a variation of the Rhines effect—an interaction of the two-dimensional turbulence cascade and Rossby wave propagation—creates the pseudoaxisymmetry and scale Lβ=π(2U/β)½ of the bands (U is the rms zonal velocity, and β the northward gradient of the Coriolis force). The anisotr...

Geostrophic Regimes, Intermediate Solitary Vortices and Jovian Eddies

Abstract We examine the relevance to Jupiters atmosphere of the solitary vortices favored at scales intermediate to those of the quasi-geostrophic (QG) and planetary-geostrophic motions. Horizontal divergence plays a crucial role in the intermediate-geostrophic (IG) dynamics and leads to asymmetries in vortex behavior; in partcular, anticyclonic vortices are generally more stable than cyclonic vortices when the mean flow is weak or westerly. The IG vortices always propagate westward at close to the planetary long-wave speed, regardless of the mean zonal flow. Meridional shear influences only secondary aspects of vortex behavior. Although governed by a form of the Korteweg-deVries (KdV) equation, vortex encounters produce coalescence not soliton behavior. Jupiters Great Red Spot and Large Ovals appear to be in, or close to, an IG balance while the Small Ovals lie in a QG balance. The stability of anticyclonic IG vortices may explain why most of Jupiters super-eddies prefer anticyclonic spin. Solutions t...

Planetary Circulations: 2. The Jovian Quasi-Geostrophic Regime

Abstract The characteristics of the two-level quasi-geostrophic model are evaluated for a wide range of parameter values in the Jovian domain. The results support the hypothesis that baroclinic instability energizes the circulation of Jupiter and Saturn and that the blocking effect of planetary wave propagation on quasi-geostrophic turbulent cascades determines the width and zonality of the bands—the degree of zonality being higher in the absence of surface drag. The model circulations consist of multiple westerly jets, separated by strong easterly flows—the result of momentum partitioning by the Kuo vortex separation process. There are no large-scale vertical motions. A cyclic variation occurs (with a time scale of several years) during which phases with intense, large-scale baroclinic activity alternate with longer, more quiescent phases involving weak, small-scale baroclinic instability and neutral baroclinic waves. These neutral waves, generated by quasi-two-dimensional cascades and propagating at spe...

The stability and genesis of Rossby vortices

Abstract The stability and genesis of the vortices associated with long solitary divergent Rossby waves-the Rossby vortices–are studied numerically using the single-layer (SL) model with Jovian parameters. Vortex behavior depends on location and on balances among the translation, twisting, steepening, dispersion and advection processes. Advection is the main preserver of vortices. The solutions provide an explanation for the origin, uniqueness and longevity of the Great Red Spot (GRS). In midlatitudes, stable anticyclones exist in a variety of sizes and balances: from the large planetary-geostrophic (PG) and medium intermediate-geostrophic (IG) vortices that propagate westward, to the small quasi-geostrophic (QG) vortices that migrate equatorward. These vortices all merge during encounters. Geostrophic vortices in the f0-plane system adjust toward symmetry by rotating; those on the sphere adjust by rotating and propagating. Stable cyclones exist mainly at the QG scale or on the f0-plane. In low latitudes ...

Thermal Convection in a Rotating Fluid Annulus. Part I. The Basic Axisymmetric Flow

Abstract The thermally driven motion of a fluid contained in a rotating annulus is investigated by numerical integration of the Navier-Stokes equations as an initial value problem. Four distinct regimes of hydro-dynamical flow can exist in the annulus system. This paper will consider the nature and computational requirements of the axisymmetric state for its own sake and partly as a prelude to a quantitative study of the more complex irregular regime. Calculations were made for two flows whose parameters, with the exception of the rotation rates, are identical, and whose upper surfaces are free. How the axisymmetric state varies with the Rossby and Taylor parameters will be discussed in Part 2. The solutions show that the flow forms a direct circulation with countercurrents on both side walls, and with a strong flow from the base of the hot wall, across the interior, up toward the top of the cold wall. The thermal boundary layers form in small and isolated regions near the top of the cold wall and base of...

Thermal Convection in a Rotating Fluid Annulus: Part 2. Classes of Axisymmetric Flow

Abstract This paper presents the solutions obtained for various axisymmetric thermal convection flows in a rotating annulus. Initially, a solution is obtained for a flow whose interior structure has been observed in detail. A comparison reveals the similarity of the experimental and computed temperature fields and shows the discrepancy to be independent of the computational resolution. On increasing the resolution, the Nusselt number decreases and converges to a value close to that observed. For this particular flow the rotation rate is zero and the flow consists of a direct meridional cell with a large stagnant interior. The associated isotherms lie horizontally in the interior such that the vertical temperature gradient is constant. Secondly, we present solutions of five flows with a rigid surface. These flows cover a wide range of values of the external driving parameters so that physical processes vary from predominately viscous and conduction diffusion to free convection transports. Despite them diff...

Friction Term Formulation and Convective Instability in a Shallow Atmosphere

Abstract The form of the friction terms for a shallow layer of fluid on a sphere is discussed for isotropic and transversely-isotropic fluids. We then examine the nature of convection in a transversely-isotropic fluid and find that long flat convection cells with a width to height ratio of 2(νH/νV)½ are produced, where νH, νV are the horizontal and vertical diffusion coefficients. The critical Rayleigh number is given by Rc=4π4νH/νV, but another Rayleigh number P=βgΔTd5/(νVκVL), with a constant critical value Rc=4π4 is shown to be a more relevant parameter. Results for convection in a rotating system are also given.

Geostrophic Regimes on a Sphere and a Beta Plane

Abstract A general geostrophic equation is derived for a shallow layer of fluid on a sphere. This equation encompasses the planetary, intermediate, and quasi-forms of geostrophy and produces their equations directly when the appropriate parametric ordering relationships are chosen. The three regimes have proven useful for defining and describing oceanic and Jovian eddies and currents on the planetary, intermediate and synoptic scales respectively. The general geostrophic equation may be most useful in describing the interactions among these three different regimes of motion and between motions in high and low latitudes. The accuracy of the β-plane version of these equations is also examined in detail.

Dynamics of a Convectively Unstable Atmosphere.: Jupiter?

Abstract We test the hypothesis that the atmospheric circulations of Jupiter are a manifestation of large-scale convective instability brought about primarily by the presence of an internal heat source. This is done by examining the nature of convection in an unstable rotating atmosphere through numerical integration of the Boussinesq equations. The general properties of convection are obtained from solutions with laboratory-scale parameters while particular Jovian characteristics are studied through calculations with planetary-scale parameters. In the Jupiter calculations, physical and theoretical constraints on parametric freedom produce a desirably under-determined system in which there remain more observational criteria to be explained than free parameters to manipulate. The solutions indicate that a tropical westerly jet can be produced by an axisymmetric flow provided that the atmosphere is relatively shallow (d<500 km). A strong equatorial westerly flow can occur provided that there is a strong dif...

Thermal Convection in a Rotating Fluid Annulus: Part 3. Suppression of the Frictional Constraint on Lateral Boundaries

Abstract In certain rotating fluid systems such as the atmosphere, the flow must maintain a zero net torque on the horizontal surface. The character of such flows is sought through numerical integration of the Navier-Stokes equations. The fluid occupies a torus shaped region whose vertical boundaries are assumed to be frictionless. The solutions relate to either a laboratory annulus with hypothetical free-slip sidewalls or to a zonal strip of the atmosphere or ocean. All the solutions are qualitatively similar despite parametric differences; their flows have a westerly-easterly zonal wind distribution near the horizontal boundary together with direct and indirect cells in a manner reminiscent of that proposed by classical theory for the general circulation of the atmosphere. Under a strong external temperature differential the isotherms concentrate into a front. The meridional circulation assumes the form of gliding motion parallel to the front together with frictionally driven secondary circulations. Cer...