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Gareth P. Williams

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 speeds of O(1 m s−1), provide the major mode of adjustment in the quasi-steady phase and form the gyres endemic to multiple jet circulations.

Similar large-scale motions occur for all the parameter values considered: for weak and strong static-stabilities, for eddy sizes ranging from 2000–9500 km and for pole-to-equator temperature differences varying from 5–90 K. The weak thermal gradients maintain strong dynamical activity by their association, in geostrophic motion, with the large value of the specific-heat constant for hydrogen.

For Jupiter, a correspondence between the theoretical perturbation pressure and the observed planetary-scale features suggests that condensation processes related to the geostrophically balanced pressure variations produce the main cloud bands and Great Red Spot, while local temperature changes due to baroclinic instability and frontogenesis create the eddy cloud systems embedded within the main bands. An analogy between the Great Red Spot and the warm high-pressure region of a neutral baroclinic wave leads us to suggest that the scale selectivity and energy source of ultralong, baroclinically unstable waves could explain the size and persistence of the Jovian feature.

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Gareth P. Williams

Abstract

The characteristics of the two-level quasi-geostrophic model are evaluated for a wide range of parameters in the terrestrial domain. Flow form is determined primarily by β (the Coriolis gradient) and by τD, (the time scale of the surface drag), acting through the influence of Rhines' transitional wave-number k β = (β/2U)1/2 where U 2 is the barotropic energy level. Two extreme types of circulation occur: jets when k β is large, and gyres when wave propagation and drag are negligible.

The present terrestrial circulation, in its quasi-geostrophic representation, is extremely efficient: the system can cope with increased heating rates without a significant rise in the pole-to-equator temperature differential. Although each hemisphere is, on occasion, near to transforming into a double-jet state, multi-jet circulations–corresponding to those in the Jovian regime–occur more readily at higher rotation rates. For the existing circulation to switch to a gyre form requires a large, unrealizable drop in surface drag.

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Gareth P. Williams

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. Certain mesoscale geophysical phenomena also possess these characteristics.

The solutions provide good examples of Eliassen's theory of vortex circulations.

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GARETH P. WILLIAMS

Abstract

The detailed structure of a steady wave occurring in a rotating annulus of square cross-section and having a free surface is presented. The field distributions are obtained by numerical integration of the three-dimensional nonlinear Navier-Stokes equations.

The distributions of pressure, temperature, and the three velocity components are displayed for the total fields and for the fields of deviation from the zonal means. Their dynamical balances are also discussed. The deviation wave is a type of Eady wave and the solution is used to discuss the structure of such waves in finite amplitude steady-state form under the influence of variations in baroclinicity, shear, and boundary layers.

The side layers make little contribution to the characteristics of the wave in the deviation field although significant Ekman layer features do appear. The flow is essentially in hydrostatic and geostrophic balance except in the boundary layers. Heat conduction is important only in the side layers.

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Gareth P. Williams

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 L β=π(2U/β)½ of the bands (U is the rms zonal velocity, and β the northward gradient of the Coriolis force). The anisotropy of the interaction produces zonally oriented flows, composed of a series of alternating easterly and westerly jets, between which lie characteristic ovals. Equatorial jets occur readily when vorticity sources that lie symmetrically about the equator act on the atmosphere. Frictionally induced Ekman circulations provide a possible mechanism for cloud formation.

Integrations with terrestrial parameters support Kuo’s (1951) forced vorticity-transfer theory for the Earth’s circulation: westerly jets form in the forced midlatitude zones, and Rossby-wave propagation from those zones causes the broad easterly trade winds. Enstrophy cascade and β effects control the formation of momentum converging eddy patterns. L β also provides a measure of the width of the terrestrial jet. Cascade blocking by a stronger surface drag prevents terrestrial flows from approaching the same degree of zonality as Jovian ones.

Jupiter also appears to be dynamically equivalent to a hypothetical (or primeval) global ocean that has neither continental boundaries nor surface winds.

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Gareth P. Williams

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.

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Gareth P. Williams

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(νHV)½ are produced, where νH, νV are the horizontal and vertical diffusion coefficients. The critical Rayleigh number is given by Rc=4π4νHV, but another Rayleigh number P=βgΔTd 5/(νVκV L), 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.

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Gareth P. Williams

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 the hot wall. The fluid and container effect most of their heat exchange through these discrete regions. The isotherms slope up toward the cold wall and a large region of constant temperature exists near the fluid surface. The higher rotation rate makes the isotherms ware vertical and, as a consequence, the Nusselt number is inversely proportional to the first power of the rotation rate. The upper three-fourths of the fluid flows in the same zonal direction as the rotation, while the remainder flows in the opposite direction. Although the fluid interior is essentially geostrophic, the nonlinear terms do make a significant contribution to the vorticity balance. The angular momentum has a single sink region; this occurs at the top of the cold inner cylinder, and the fluid ignores the potential maximum source at the (hot) outer cylinder. The contributions of the sidewall boundary layers to the energy transformation oppose each other; this leaves the interior region of the fluid as a significant source of energy. Application of Eady's criterion for baxoclinic instability when applied to the solutions, shows one flow to be stable, the other unstable. This conclusion agrees with observation.

Contours of the transient fields show the predominately isothermal evolution of the flow towards a steady state. The close association of the sidewall countercurrents to the sidewall boundary layers appears at all stages of development.

Changing the number of grid points used and repeating the calculations demonstrated the accuracy of the solutions.

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Gareth P. Williams

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 differences, all five flows exhibit a similar structure, i.e., the interior flows form direct (Hadley) cells with sidewall countercurrents and the zonal flow reverses sign near the center of the fluid. Interpolation of the Nusselt number values yields a (ΔT/Ω)0.5 dependency. Compared to the Ω−1 dependency of free surface flows, the rigid surface system forms the better transporting mechanism and is less inhibited by rotation.

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Gareth P. Williams and Kirk Bryan
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