# Bibliography - Gareth P Williams

- Williams, Gareth P., 2006:
**Circulation Sensitivity to Tropopause Height**.*Journal of the Atmospheric Sciences*,**63(7)**, 1954-1961.The possibility that the tropopause could be lower during an ice-age cooling leads to an examination of the general sensitivity of global circulations to the tropopause height by altering a constant stratospheric temperature Ts in calculations with a dry, global, multilevel, spectral, primitive equation model subject to a simple Newtonian heating function. In general, lowering the tropopause by increasing the stratospheric temperature causes the jet stream to move to lower latitudes and the eddies to become smaller. Near the standard state with Ts = 200 K, the jets relocate themselves equatorward by 2° in latitude for every 5 K increase in the stratospheric temperature. A double-jet system, with centers at 30° and 60° latitude, occurs when the equatorial tropopause drops to 500 mb (for Ts = 250 K), with the high-latitude component extending throughout the stratosphere. The eddy momentum flux mainly traverses poleward across the standard jet at 40°, in keeping with the predominantly equatorward propagation of the planetary waves. But when the jet lies at 30° (for Ts = 225 K) the flux converges on the jet in keeping with planetary waves that propagate both equatorward and poleward. Two sets of such wave propagation occur in the double-jet system. As the troposphere becomes even shallower, the flux reverts to being primarily poleward across the jet (for Ts = 260 K) but then becomes uniquely primarily equatorward across the jet (for Ts = 275 K) before the circulation approaches extinction. Thus the existence of a predominantly poleward flux in the standard state appears to be parametrically fortuitous.

- Williams, Gareth P., 2006:
**Equatorial Superrotation and Barotropic Instability: Static Stability Variants**.*Journal of the Atmospheric Sciences*,**63(5)**, 1548-1557.Altering the tropospheric static stability changes the nature of the equatorial superrotation associated with unstable, low-latitude, westerly jets, according to calculations with a dry, global, multilevel, spectral, primitive equation model subject to a simple Newtonian heating function. For a low static stability, the superrotation fluxes with the simplest structure occur when the stratospheric extent and horizontal diffusion are minimal. Barotropic instability occurs on the jet's equatorward flank and baroclinic instability occurs on the jet's poleward flank. Systems with a high static stability inhibit the baroclinic instability and thereby reveal more clearly that the barotropic instability is the primary process driving the equatorial superrotation. Such systems produce a flatter equatorial jet and also take much longer to equilibrate than the standard atmospheric circulation.

- Williams, Gareth P., and Kirk Bryan, 2006:
**Ice Age Winds: An Aquaplanet Model**.*Journal of Climate*,**19(9)**, DOI:10.1175/JCLI3766.1 .Factors controlling the position and strength of the surface winds during the Last Glacial Maximum (LGM) are examined using a global, multilevel, moist, atmospheric model. The idealized aquaplanet model is bounded below by a prescribed axisymmetric temperature distribution that corresponds to an ocean-covered surface. Various forms of this distribution are used to examine the influence of changes in the surface cooling and baroclinicity rates. The model omits seasonal variations. Increasing the cooling lowers the tropopause and greatly reduces the moist convection in the Tropics, thereby causing a weakening and equatorward contraction of the Hadley cell. Such a cooling also weakens the surface westerlies and shifts the peak westerly stress equatorward. An extra surface baroclinicity in midlatitudes—implicitly associated with an increase in the polar sea ice—also shifts the peak westerly stress equatorward, but strengthens the surface westerlies. Thus, calculations with combined surface cooling and baroclinicity increases, representative of the Last Glacial Maximum, reveal an absence of change in the amplitude of the peak westerly stress but exhibit a substantial equatorward shift in its position, 7° for a 3-K cooling and 11° for a 6-K cooling. The easterlies, however, always increase in strength when the surface westerlies move equatorward. The application of these results to the LGM must take into account the model’s assumption of symmetry between the two hemispheres. Any changes in the climate’s hemispheric asymmetry could also cause comparable latitudinal shifts in the westerlies, probably of opposite sign in the two hemispheres. Published coupled-model simulations for the LGM give an equatorward shift for the peak westerlies in the Northern Hemisphere but give contradictory results for the Southern Hemisphere.

- Williams, Gareth P., 2003:
**Barotropic instability and equatorial superrotation**.*Journal of the Atmospheric Sciences*,**60(17)**, 2136-2152.Baroclinically unstable zones in midlatitudes normally produce medium-scale planetary waves that propagate toward the equator where they generate easterlies while transferring westerly momentum poleward, so that the jet lies in higher latitudes than in the corresponding axisymmetric (eddy-free) state. When the baroclinically unstable zone is moved into low latitudes, however, the equatorward side of the jet can also produce a barotropic instability whose large-scale eddies lead to a strong superrotating westerly current at the equator; the jet remains close to its axisymmetric location. For the earth, the transition between these two regimes occurs when the jet lies close to 30°, according to calculations with a global, multilevel, spectral, primitive equation model that examines superrotating flows for a wide range of rotation rates. The existence of a stable superrotating regime implies that an alternative climate could occur, but only under novel conditions.

- Williams, Gareth P., 2003:
**Jet sets**.*Journal of the Meteorological Society of Japan*,**81(3)**, 439-476.To broaden the range of known circulations and to test existing theory, a variety of issues are examined concerning the dynamics of flows in thick, thin, and transitional atmospheric layers. The circulations are produced numerically using a primitive equation model subject to simple heating functions. To confine the motions to a thin upper layer, the heating is chosen to produce a flow with either an exponential (EXP) vertical structure, or one that is linear (LIN) aloft while vanishing below. Five sets of solutions are created to define the terrestrial and jovian axisymmetric states, some basic terrestrial states, and the transitional jovian states for the two structures. The axisymmetric cases examine how the surface drag, static stability, rotation rate, and layer thickness influence the flow character. The standard theory is extended to allow for a weaker drag and the solutions confirm that at lower rates the Hadley cells become wider, and the thermal fronts sharper and double. In the absence of any drag, the cells disappear and a thermal wind prevails globally. But in the absence of a background static stability, the cells become more intense and create their own stable temperature field. For normal parameter values, the Hadley cells adhere to the theoretical form as the rotation rate increases, except when their width falls below 3° of latitude. Furthermore, when the heated layer is thin and the jets are confined aloft, the cells develop vertically bimodal amplitudes, while remaining deep and exhibiting the usual widths. The basic 3-D terrestrial cases examine the role of the heating rate, static stability, surface drag, and rotation rate on the flow character. The mean jets exist within a limited latitudinal range, with their location being as much dependent on the heating amplitude as on the heating distribution. When the background static stability is absent, the standard circulation theory becomes less valid as the cells and baroclinic instability become more intense and act together to stabilize low and middle latitudes. However, when the drag is reduced, the baroclinic instability becomes much weaker and confined to lower levels because of suppression by the jet's stronger barotropic component. Other forms of baroclinic instability can be produced by creating double-jet flows, either by increasing the rotation rate or by adding an extra source of baroclinicity in low latitudes. The transitional jovian cases examine how the multiple jets behave as the active layer is varied between thick and thin for the LIN and EXP structures. In all cases, the jet widths remain constant with latitude, but their amplitudes vary, peaking either in low or middle latitudes depending on how the baroclinicity is distributed. An extra baroclinicity in low latitudes produces a jet whose barotropic instability can drive an equatorial superrotation, regardless of layer thickness. The eddy-driven jets have a similar dynamics for all layer thicknesses but, unlike the steady LIN jets, the EXP jets also migrate equatorward and, on rare occasions, poleward.

- Williams, Gareth P., 2003:
**Jovian Dynamics, Part III: Multiple, migrating, and equatorial jets**.*Journal of the Atmospheric Sciences*,**60(10)**, 1270-1296.Studies of the dynamical response of thin atmospheric layers overlying thick envelopes are extended to examine how multiple jets, such as those seen on Jupiter and Saturn, can be generated and maintained. The jets are produced by baroclinic instabilities and are examined numerically using a primitive equation model subject to simple heating functions. The motions are confined to a thin upper layer by a heating that produces a flow with either an exponential vertical structure or one that is linear aloft while vanishing below. The motions are driven by latitudinal heating distributions with a variety of global and local components. The calculations show that jets roughly resembling the main ovian ones in amplitude, scale, and form can be generated and maintained in a steady configuration when the flow has the confined linear structure. When the flow has the exponential structure, however, the jets migrate slowly but continuously equatorward while being regenerated in higher latitudes. For both structures, the flow is sensitive to the heating distribution in low latitudes where jets form only if a significant baroclinicity exists in that region; such jets can also be barotropically unstable and can generate a superrotating current at the equator. In midlatitudes, except for being confined to an upper layer, the baroclinic instabilities resemble the standard forms seen in terrestrial models with high rotation rates. Additional calculations show that superrotating equatorial currents can also be generated for deep layers or for Earth's atmosphere if the initial instabilities are developed in low latitudes. Broad easterly currents such as Neptune's can also be generated by elementary heating distributions, provided that the heated layer becomes progressively thicker with latitude. Finally, the hexagonal shape that high-latitude jets sometimes assume on Saturn when viewed in a polar projection can be attributed to nonlinear waves associated with baroclinic instabilities.

- Williams, Gareth P., 2003:
**Super Circulations**.*Bulletin of the American Meteorological Society*,**84(9)**, 1190. - Williams, Gareth P., 2002:
**Jovian dynamics. Part II: The genesis and equilibration of vortex sets**.*Journal of the Atmospheric Sciences*,**59**, 1356-1370.To extend studies of the dynamics of thin atmospheric layers, the generation and equilibration of multiple anticyclonic vortex sets associated with long solitary baroclinic Rossby waves are examined numerically using a primitive equation model with jovian parameters subject to a simple heating function. We seek primarily to model the three main groups of anticyclones seen on Jupiter, namely, the Great Red Spot, the three White Ovals, and the dozen or so Small Ovals that occur at latitudes of -21°, -33°, and -41°, respectively. The motions are confined to thin upper layers by exponential vertical structures that favor absolute vortex stability. Calculations are also made to examine the regeneration, intrazonal and interscale interactions, and propagation rates of vortices. Vortex sets resembling the three main jovian groups in scale, form, and number can be simultaneously generated and maintained in a steady configuration by a heating that produces stable westerly and weakly unstable easterly jets. The steady configuration occurs when an optimal number of vortices exists in a balance between a weak heating and a weak dissipation. Vortex behavior can be more complex in the heated system because the generation of new storms offsets the tendency to merge into fewer vortices. The solutions also show that intrazonal vortex interactions can lead, in some situations, to the destruction of anticyclones modeling the Great Red Spot.

- Williams, Gareth P., 1997:
**Planetary vortices and Jupiter's vertical structure**.*Journal of Geophysical Research*,**102(E4)**, 9303-9308.Measurements of the vertical structure of Jupiter's circulation have recently been made near the equator by the Galileo spacecraft probe. In other regions, planetary vortices exist selectively for a limited range of generic (exponential) vertical forms and can be used to probe the atmospheric structure theoretically. A study of vortex genesis with a three-dimensional numerical model produces reasonably realistic simulations of the Great Red Spot for both the generic and galilean forms, provided that Jupiter's winds do not extend much beyond a 500 km depth. However, the actual depth of the winds remains uncertain.

- Williams, Gareth P., 1996:
**Jovian dynamics. Part I: Vortex stability, structure, and genesis**.*Journal of the Atmospheric Sciences*,**53(18)**, 2685-2734.The vertical structure of Jupiter's atmosphere is probed and isolated by evaluating the stability characteristics of planetary vortices over a wide parameter range. The resulting structures lead to simulating the genesis of single and multiple vortex states in Part I of this paper and the genesis of an equatorial superrotation and midlatitudinal multiple jets in Part II. The stability and genesis of baroclinic Rossby vortices, the vortices associated with long solitary Rossby waves in a stratified fluid, are studied numerically using a primitive equation model with Jovian and oceanic parameters and hypothetical structures. Vortex stability, that is, coherence and persistence, depends primarily upon latitude location and vertical structure and is used to deduce possible stratifications for Jupiter's atmosphere. The solutions suggest that Jupiter's large-scale motions are confined to a layer of depth h and are bounded by an abyss with an impermeable interface at a depth H, such that h/H < 1/20. Consequently, they also extend earlier results derived with the reduced-gravity, shallow-water model, particularly the explanation for the origin, uniqueness, and longevity of the Great Red Spot (GRS). Beginning at the equator, stable anticyclones are seen to exist only when they have the Hermitian latitudinal form, the Korteweg-deVries longitudinal form, the confined exponential vertical structure exp (Nz/H), and the amplitude range as prescribed by the analytical theory of Marshall and Boyd for N = 8. Soliton interactions occur between equatorial vortices of similar horizontal and vertical form. In middle and low latitudes, shallow anticyclones with an exponential structure of N = 20 exist quasi-stably for a variety of sizes. Such vortices remain coherent but tend to migrate equatorward (where they disperse) at rates that depend upon their size, location, and vertical structure:large and medium anticyclones propagate primarily westward while migrating slowly, whereas small storms just migrate rapidly and then collapse. The migration of these large, shallow vortices can be reduced, but not stopped, in low latitudes by an easterly jet with the same vertical structure. Anticyclones are stabler when they are thinner relative to the abyss. Thus, when N = 60, their migration is sufficiently slow that it can be stopped by a weak easterly jet. Furthermore, absolute stability sets in when N = 90 and migration ceases completely for the large, thin anticyclones that now just propagate westward. Such flows may also be usefully represented by a vertical structure that is linear in z for the velocity and static stability in the thin upper layer and vanishes in the abyss. Large, thin (N > 90) anticyclones can exist indefinitely either freely or when embedded within an anticyclonic zone of alternating jet streams of similar vertical structure. This holds true for the confined linear-z representation also. The permanence of GRS-like, low-latitude vortices in Jovian flow configurations occurs in a variety of lengthy calculations with thin structures. Ocean vortices are less persistent because the thermocline is relatively thick. The baroclinic instability of easterly jets is nonquasigeostrophic and takes on the form of solitary rather than periodic waves when the jets have a thin exponential (N > 90) or confined linear-z structure. Such nonlinear waves develop into vortices that exhibit a variety of configurations and evolutionary paths. In most cases multiple mergers tend toward an end state with a single large vortex. Two types of merging occur in which a stronger vortex either catches a weaker one ahead of it or reels in a weaker one from behind. This duality occurs because propagation rates depend as much on local as on global conditions. In a further complication, vortices generated by an unstable easterly tend to have an exponential structure for exponential jets but a first baroclinic eigenmodel structure for confined linear-z jets. Single vortex states resembling the GRS, with sizes ranging from 15 degrees to 50 degrees in longitude and with temperature gradients, velocities, and propagation rates near the observed range, can be generated either directly through the growth of a local front in a marginally unstable easterly jet or indirectly through a series of mergers of the multiple vortices generated by a more unstable easterly jet. Sets of vortices can be produced simultaneously in the anticyclonic zones centered about latitudes -21°, -33°, and -41°, and have the same relative scales as Jupiter's GRS, Large Ovals, and Small Ovals. Thin anticyclones can also be generated at the equator by the action of vortices lying in low latitudes. Equally realistic long-lived vortices can also be generated by jets with structures matching the recent Galileo spacecraft observations by using other hyperbolic forms and greater depth scales.

- Williams, Gareth P., 1988:
**The dynamical range of global circulations - I**.*Climate Dynamics*,**2**, 205-260.The dynamical range of atmospheric circulations is examined by integrating a global circulation model (GCM) over a wide range of parameter values. We study the influence of rotation rate on moist and dry atmospheres with regular, drag-free, and interior-heated surfaces in Part I, and on axisymmetric, oblique, and diurnally heated moist atmospheres in Part II. Despite their variety, the circulations are composed of only a few elementary forms whose existence, scale, and mix alter as the parameters vary. These elements can be interpreted in terms of standard symmetric-Hadley (SH) and quasi-geostrophic (QG) theories. The natural-Hadley (NH) circulation consists of a polar jet and a hemispheric direct cell, such as occur in slowly rotating SH flows, together with Rossby waves generated by moist convection and barotropic cascades. The quasi-Hadley (QH) circulation consists of a tropical westerly jet and a narrow direct cell, such as occur in the low-latitude part of rapidly rotating SH flows, together with Rossby waves generated by baroclinic instabilities in the neighboring midlatitude part of the SH flows; it occurs only in moist atmospheres. The two QG circulations represent the two extremes of eddy momentum flux produced during eddy cycles - the special form of enstrophy cascade describing nonlinear baroclinically unstable wave growth and barotropic wave dispersion. The QGγ element has a latitudinally asymmetric wave dispersion that gives a poleward, jet-traversing momentum transport, while QGβ has a symmetric wave dispersion that gives a jet-converging momentum transport. Both elements have a westerly jet and three cells. (In Part II, we describe the solstitial symmetric-Hadley, the QG-Hadley, the diurnally modified NH, and the Hadley circulations.) In moist atmospheres, NH circulations exist in the rotational low range (Ω* = 0 - 1/4); overlapping QG, and QH elements in the midrange (Ω* = 1/2 - 1); and QGγ, QGβ, and QH elements in the high range (omega* = 2-8); here Ω* = Ω/ΩE is the rotation rate normalized by the terrestrial value. In dry atmospheres, circulations follow a similar progression but have a simpler blend because they lack a QH element. Kinetic energy peaks at Ω* = 1/8 in the moist, Hadley-dominated atmospheres but at Ω* = 1/2 in the dry, QG-dominated atmospheres. Instability-generated Rossby waves propagate equatorward more easily in the westerlies of the diabatically driven (moist) Hadley cell than in the easterlies of the eddy-induced (dry) direct cell. Temperatures vary from almost barotropic at Ω* = 0 to almost radiative-convective at Ω* = 8, while maintaining almost constant global means. In modified-surface systems, freeslip conditions eliminate the QH element from a moist atmosphere and allow strong deep easterlies to arise in low latitudes to balance the strongly barotropic westerly jets that occur in midlatitudes. In a regular dry atmosphere, enhanced surface heating in low latitudes imitates latent heating and produces a tropical circulation resembling that of the moist QH element. Overall, circulation theory works well in explaining the GCM states but does not, as yet, describe the interactions among elements or reveal how jet scales are determined, nor explain phenomena at the extremes of the parameter range.

- Williams, Gareth P., 1988:
**The dynamical range of global circulations - II**.*Climate Dynamics*,**3**, 45-84.The dynamical range of global atmospheric circulations is extended to specialized parameter regions by evaluating the influence of the rotation rate (Ù) on axisymmetric, oblique, and diurnally heated moist models. In Part I, we derived the basic range of circulations by altering omega for moist and dry atmospheres with regular and modified surfaces. Again we find the circulations to be composed of only a few elementary forms. In axisymmetric atmospheres, the circulations consist of a single jet in the rotational midrange (Ù* = 1/2-1) and of double jets in the high range (Ù* = 2-4), together with one or two pairs of Hadley and Ferrel cells; where Ù* = Ù/ÙE is the rotation rate normalized by the terrestrial value. These circulations differ from those predicted by first-order symmetric-Hadley (SH1) theory because the moist inviscid atmosphere allows a greater nonlinearity and prefers a higher-order meridional mode. The axisymmetric circulations do, however, resemble the mean flows of the natural system - but only in low latitudes, where they underlie the quasi-Hadley (QH) element of the MOIST flows. In midlatitudes, the axisymmetric jets are stronger than the natural jets but can be reduced to them by barotropic and baroclinic instabilities. Oblique atmospheres with moderate to high tilts (è = 25° - 90°) have the equator-straddling Hadley cell and the four basic zonal winds predicted by the geometric theory for the solstitial-symmetric-Hadley (SSH) state: an easterly jet and a westerly tradewind in the summer hemisphere, and a westerly jet and an easterly tradewind in the winter hemisphere. The nonlinear baroclinic instability of the winter westerly produces a Ferrel cell and the same eddy fluxes as the quasi-geostrophic QGã element, while the instability of the summer easterly jet produces a QG-Hadley (QGH) element with a unique, vertically bimodal eddy momentum flux. At high è and low Ù*, the oblique atmospheres reach a limiting state having global easterlies, a pole-to-pole Hadley cell, and a warm winter pole. At low tilts (è < 10°), the oblique circulations have a mix of solstitial and equinoctial features. Diurnal heating variations exert a fundamental influence on the natural-Hadley (NH) circulations of slowly rotating systems, especially in the singular range where the zonal winds approach extinction. The diurnality just modifies the NH element in the upper singular range (1/45 < Ù * < 1/16 ), but completely transforms it into a subsolar-antisolar Halley circulation in the lower singular range (0 < omega* < 1/45). In the modified NH flows, the diurnality acts through the convection to enhance the generation of the momentum-transferring planetary waves and, thereby, changes the narrow polar jets of the non-diurnal states into broad, super-rotating currents. Circulation theory for these specialized flows remains rudimentary. It does not explain fully how the double jets and the multiple cells arise in the axisymmetric atmospheres, how the QGH element forms in the oblique atmospheres, or how waves propagate in the slowly rotating diurnal atmospheres. But eventually all theories could, in principle, be compared against planetary observation: with Mars testing the QGH elements; Jupiter, the high-range elements; Titan, the equinoctial and solstitial axisymmetric states; and Venus, the diurnally modified NH flows.

- Williams, Gareth P., and R John Wilson, 1988:
**The stability and genesis of Rossby vortices**.*Journal of the Atmospheric Sciences*,**45(2)**, 207-241.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 fo-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 fo-plane. In low latitudes stable anticyclones exist only when a strong equatorial westerly jet and a significant easterly current are present to elininate the highly dispersive equatorial modes. The permanence of a GRS-like, low-latitude vortex in a Jovian flow configuration is established by a 100-year simulation. At the equator, stable anticyclones exist only when they have the Hermite latitudinal form and the Korteweg-DeVries longitudinal form and amplitude range as prescribed by Boyd (1980). Soliton interactions occur between equatorial vortices of similar order. Vortices can be generated at the equator by the collapse of low-latitude anticyclones. In mid or low latitudes, unstable easterly jets generate vortices whose final number depends mainly on the interaction history. Stochastically forced eddies cascade by wave interactions into zonal currents and by eddy mergers into a single Rossby vortex that thrives on the turbulence. Directly forced ageostrophic jets can make vortex drift more westerly and can change it from free state values of -10 ms

^{-1}to forced state values of -5 ms^{-1}(as the GRS) or of +5 ms^{-1}(as the Large Ovals). - Williams, Gareth P., 1985:
**Geostrophic regimes on a sphere and a beta plane**.*Journal of the Atmospheric Sciences*,**42(12)**, 1237-1243.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 beta-plane version of these equations is also examined in detail.

- Williams, Gareth P., 1985:
**Jovian and comparative atmospheric modeling**.*Advances in Geophysics*,**28A**, 381-429. - Williams, Gareth P., and T Yamagata, 1984:
**Geostrophic regimes, intermediate solitary vortices and Jovian eddies**.*Journal of the Atmospheric Sciences*,**41(4)**, 453-478.We examine the relevance to Jupiter's 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 particular, 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. Jupiter's 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 Jupiter's super- eddies prefer anticyclonic spin. Solutions to the shallow water (SW) equations, using Jovian parameters, show that an IG vortex with the scale and environment of the Great Red Spot has great longevity and that such a vortex may originate in a weak barotropic instability of the zonal currents. Strong barotropic instability on the IG scale differs from its counterpart on the QG scale and produces multiple, steep, isolated vortices resembling the Large Ovals. Equations are derived for all forms of geostrophic balance (three basic classes, ten subsets) to investigate the uniqueness of the IG system. Numerical studies use the IG beta-plane equation to examine basic modal properties and the full SW equations to examine the Jovian eddies.

- Williams, Gareth P., 1983:
**Review of "Weather and Climate on Planets," by Kondratyev, K. Y. and G. E. Hunt**.*Bulletin of the American Meteorological Society*,**64(6)**, 647-649. - Williams, Gareth P., and J L Holloway, Jr, 1982:
**The range and unity of planetary circulations**.*Nature*,**297(5864)**, 295-299.Altering the rotation rate, obliquity and diurnal period of an Earth-like model atmosphere produces a wide range of circulation forms, some of which resemble those observed on Venus, Mars, Jupiter, Saturn and (perhaps) on Uranus and Neptune. These unified solutions suggest: that Jupiter and Saturn resemble a larger, faster-spinning Earth and possess a stress-bearing or momentum-exchanging sublayer; that easterly winds prevail in Uranus' summer hemisphere; and that Venus resembles a slowly rotating Earth if diurnal heating variations are included.

- Rossow, W, and Gareth P Williams, 1979:
**Large-scale motion in the Venus stratosphere**.*Journal of the Atmospheric Sciences*,**36(3)**, 377-389.To examine a postulate that the horizontal momentum exchanges in Venus' stratosphere are quasi-nondivergent, we investigate the properties of two- dimensional turbulence on a slowly rotating sphere in a high-resolution, one- level numerical model. We conclude that the forcing which maintains the stratospheric flow is weak and influences the dynamics far less than the inertial effects. Consequently, the behavior resembles that of vorticity- conserving, two-dimensional flows. On a slowly rotating planet like Venus, such flows are dominated by solid-body rotation and by a planetary wave of unit zonal wavenumber - this wave corresponds to the observed Y-shaped UV feature. Although these largest scales of motion stand out, the dynamic balances of the flow are fundamentally nonlinear, in contrast to the quasi- linear Rossby wave regime on rapidly rotating planets.

- Williams, Gareth P., 1979:
**Ultra-long baroclinic waves and Jupiter's Great Red Spot**.*Journal of the Meteorological Society of Japan*,**57(2)**, 196-198. - Williams, Gareth P., 1979:
**Planetary circulations: 2. The Jovian quasi-geostrophic regime**.*Journal of the Atmospheric Sciences*,**36(5)**, 932-968.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. - Williams, Gareth P., 1979:
**Planetary circulations: 3. The terrestrial quasi-geostrophic regime**.*Journal of the American Water Resources Association*,**36(8)**, 1409-1435.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 the Coriolis gradient and by the time scale of the surface drag, acting through the influence of Rhines' transitional wavenumber, where U2 is the barotropic energy level. Two extreme types of circulation occur: jets when the wavenumber 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.

- Williams, Gareth P., 1978:
**Planetary Circulations: 1. Barotropic representation of Jovian and terrestrial turbulence**.*Journal of the Atmospheric Sciences*,**35**, 1399-1426.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 barotropical 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 pseudo-axisymmetry and scale of the bands. 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 forces midlatitude zones, and Rossby-wave propagation from those zones causes the broad easterly trade winds. Enstrophy cascade and B effects control the formation of momentum converging eddy patterns. L

_{B}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. - Williams, Gareth P., 1975:
**Jupiter's atmospheric circulation**.*Nature*,**257(5529)**, 778. - Williams, Gareth P., 1975:
**Some ocean-Jupiter connections**.*MODE (Mid-Ocean Dynamics Experiment) Hot Line News*,**78**, 1-5. - Williams, Gareth P., 1974:
**Generalized Eady waves**.*Journal of Fluid Mechanics*,**62(4)**, 643-655.Solutions are obtained for the baroclinic instability problem for situations in which the static stability and mean shear vary geminately with height. The simple solution given by Eady is shown to be a special limiting case of a class of exact solutions for flows whose basic states have a vanishing interior potential vorticity gradient. The generalized solutions show that the temperature amplitude distribution is particularly sensitive to vertical variations in static stability but that phases and other amplitudes are only slightly influenced by such variations. When the static stability and shear increase (decrease) with height an enhanced temperature maximum occurs at the upper (lower) surface in comparison with the standard Eady solution. The generalized solutions also help to explain the character of annulus waves and predict a short-wave cut-off that is the same as that given by Eady's theory provided that it is based on the vertically averaged gravitational frequency.

- Williams, Gareth P., and J B Robinson, 1974:
**Generalized Eady waves with Ekman pumping**.*Journal of the Atmospheric Sciences*,**31(7)**, 1768-1776.The effects of Ekman layers on generalized Eady waves (i.e., height-varying static stability and shear) are examined. The non-constancy of N(z) and u z modify the classical Eady results but do not introduce any new effects. Thus, a short-wave cutoff is always found for flows with double Ekman layers but never for flows with a single Ekman layer. By comparing the analytical solutions with a numerically simulated annulus wave, we are able to categorize the latter quite accurately.

- Williams, Gareth P., and J B Robinson, 1973:
**Dynamics of a convectively unstable atmosphere: Jupiter?***Journal of the Atmospheric Sciences*,**30(4)**, 684-717.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 diffusion of the tropical jet. The strength of such a diffusion is of a magnitude that suggests that it can only realistically be brought about by large-scale non-axisymmetric disturbances. The axisymmetry of the convective rolls, i.e., their longitudinal stability, is controlled by the latitudinal variation of omega costheta. This differential rotation suppresses the organization of large-scale convective motion poleward of 45 degrees while toward the equator such motions can set in strongly. The banded structure and zonal velocity field of the most realistic theoretical solution resemble the observed, having five zones (w >0) and four belts (w<0) each with its characteristic differential zonal motion. The square-shaped form of the mean vertical velocity variation with latitude produces sharply bounded zones of uniform intensity. Calculations to test the stability of the axisymmetric flow to longitudinal perturbations indicate that ovals and streaks are the natural form of the disturbance elements.

- Williams, Gareth P., 1972:
**The field distributions and balances in a baroclinic annulus wave**.*Monthly Weather Review*,**100(1)**, 29-41.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.

- Williams, Gareth P., 1972:
**Friction term formulation and convective instability in a shallow atmosphere**.*Journal of the Atmospheric Sciences*,**29(5)**, 870-876.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 (v

_{H}/v_{V})1/2 are produced, where v_{H}/v_{V}are the horizontal and vertical diffusion coefficients. The critical Rayleigh number is given by R^{c}= 4 pi^{4}v_{H}/v_{V}, but another Rayleigh number with a constant critical value is shown to be a more relevant parameter. - Williams, Gareth P., 1971:
**Baroclinic annulus waves**.*Journal of Fluid Mechanics*,**49(3)**, 417-449.The thermally driven motion of water contained in a rotating annulus of square cross-section and having a free surface is investigated by numerical integration of the three-dimensional non-linear Navier-Stokes equations. The nature of steady wave flow is examined in detail and a comparison made with the corresponding axisymmetric solution in parameter space. # The steady wave solution proves to be consistent kinematically, dynamically and energetically with Lorenz's hypothesis that the wave can be attributed to the baroclinic instability mechanism. The deviatoric* wave possesses some of the characteristics of the theoretical Eady wave and it is possible to define the complete deviatoric wave structure by means of two-dimensional quasi-phase, amplitude diagrams. These diagrams may also typify the nature of certain solutions to the non-separable baroclinic instability problem. The wave motion is almost completely independent of the side boundary layers which make little contribution to the characteristics and energetics of the deviatoric flow. These side layers are approximately axisymmetric and appear qualitatively indistinguishable from their counterparts in the axisymmetric solution. However, significant Ekman layer features appear in the deviatoric wave structure. # Away from the boundaries the dynamical balance of terms is hydrostatic and quasi-geostrophic with changes of vertical vorticity influenced by stretching and viscous diffusion. Heat conduction is completely unimportant except in the side boundary layers. The angular momentum transport by the deviatoric motion is largest at the free surface and is mainly against the angular momentum gradient. A strong outward deviatoric flux of momentum is found in the Ekman layer. The dissipation of deviatoric kinetic energy occurs in the Ekman layer and jet whilst most of the dissipation of the mean kinetic energy occurs in the boundary layer of the inner wall. The large differences between the axisymmetric and zonal mean states is not strictly relevant to an understanding of the wave formation. The character of the wave suggests that the mean environment with which the deviatoric wave interacts is the wave-present zonal mean state. Only a non-linear finite amplitude baroclinic instability analysis (as yet undeveloped) could possibly explain the wave formation.

- Piacsek, S A., and Gareth P Williams, 1970:
**Conservation properties convection difference schemes**.*Journal of Computational Physics*,**6**, 392-405.The so-called conservative or flux form of the finite difference formulation of convection terms is shown to be inadequate for preventing nonlinear instability in some cases. A preferred scheme for the convection terms which has the property of absolute spatial conservation is obtained. Illustrative examples are given for (i) the Navier-Stokes equations; (ii) a forced convection equation; and (iii) a Burger's type equation.

- Williams, Gareth P., 1970:
**Axisymmetric annulus convection at unit Prandtl number**.*Geophysical Fluid Dynamics*,**1**, 357-369.We examine the role played in annulus flows by mechanisms dependent upon the Prandtl number, sigma. Solutions are obtained at sigma = 1 for both the real annulus system and for the hypothetical "free annulus" system (free slip lateral boundaries). These solutions are compared with previously obtained solutions at sigma = 7.7. In the free annulus, the solution at sigma = 1 differs radically from that at sigma = 7. The sigma = 1 solution appears to be essentially a finite amplitude mode due to Solberg instability whereas the solution at sigma = 7 manifests a flow caused by the diffusive overturning mechanism. The variation with sigma of the real annulus flow is not so fundamental but some differences in the dynamical structures are noted.

- Williams, Gareth P., 1969:
**Numerical integration of the three-dimensional Navier-Stokes equations for incompressible flow**.*Journal of Fluid Mechanics*,**37(4)**, 727-750.A method of numerically integrating the Navier-Stokes equations for certain three-dimensional incompressible flows is described. The technique is presented through application to the particular problem of describing thermal convection in a rotating annulus. The equations, in cylindrical polar co-ordinate form, are integrated with respect to time by a marching process, together with the solving of a Poisson equation for the pressure. A suitable form of the finite difference equations gives a computationally-stable long-term integration with reasonably faithful representation of the spatial and temporal characteristics of the flow. Trigonometric interpolation techniques provide accurate (discretely exact) solutions to the Poisson equation. By using an auxiliary algorithm for rapid evaluation of trigonometric transforms, the proportion of computation needed to solve the Poisson equation can be reduced to less than 25% of the total time needed to advance one time step. Computing on a UNIVAC 1108 machine, the flow can be advanced one time-step in 2 sec for a 14 x 14 x 14 grid upward to 96 sec for a 60 x 34 x 34 grid. As an example of the method, some features of a solution for steady wave flow in annulus convection are presented. The resemblance of this flow to the classical Eady wave is noted.

- Williams, Gareth P., 1968:
**Thermal convection in a rotating fluid annulus: Part 3. Suppression of the frictional constraint on lateral boundaries**.*Journal of the Atmospheric Sciences*,**25(6)**, 1034-1045.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 fronttogether 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. 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.

- Williams, Gareth P., 1967:
**Thermal convection in a rotating fluid annulus: Part 1. The basic axisymmetric flow**.*Journal of the Atmospheric Sciences*,**24(2)**, 144-161.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 hydrodynamical 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 more 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 baroclinic 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.

- Williams, Gareth P., 1967:
**Thermal convection in a rotating fluid annulus: Part 2. Classes of axisymmetric flow**.*Journal of the Atmospheric Sciences*,**24(2)**, 162-174.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 predominantely viscous and conduction diffusion to free convection transports. Despite these 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 (delta tau/omega)0.5 dependency. Compared to the omega 1 dependency of free surface flows, the rigid surface system forms the better transporting mechanism and is less inhibited by rotation.

- Williams, Gareth P., and D R Davies, 1967:
**A time average model of the general circulation**In*Proceedings of International Symposium on Dynamics of Large-scale Atmospheric Processes*, Moscow, 148-151. - Williams, Gareth P., and D R Davies, 1965:
**A mean motion model of the general circulation**.*Quarterly Journal of the Royal Meteorological Society*,**91(390)**, 471-489.Equations are constructed to represent quasi-stationary mean flow of momentum and heat on a spherical earth, averaged over a long period of time such as a year and over latitude circles. The crucial shearing Reynolds stress associated with meridional transfer of zonal velocity is assumed to depend linearly on a product of the earth's angular velocity, Omega, and the meridional gradient of mean temperature; the shearing stresses associated with vertical transfer of zonal velocity and of meridional velocity are assumed to depend linearly on the vertical gradients of zonal and of meridional mean velocities respectively, and the mean eddy transfer of heat along a meridian is assumed to depend linearly on the mean meridional temperature gradient. All proportionality coefficients are taken to be independent of latitude. Two forms are assumed for the non-adiabatic atmospheric heat source function, Q, used in the thermodynamic equation. In the first case, Q is assumed known (from analyses of observations) as a function of height and latitude. In the second case, Q incorporates a heating term which is partly controlled by the model itself and represents some of the characteristics of sensible and latent heat transfer. A solution of the basic equations is obtained in both cases in the form of double expansions in powers of two parameters, one depending on omega and the other on delta tau, the mean annual temperature difference between equator and pole. The solution is evaluated using Fourier techniques. The series expansions are found to be reasonably convergent for realistic values of the various parameters involved, three terms only being required in the delta tau expansion and five terms at most in the omega expansion, but extensive numerical evaluation by digital computer is involved: the region considered is bounded by the tropopause and lies between the equator and 70 degrees latitude. The computed zonal velocity has the characteristic east-west variation with latitude and a broad band maximum of expansion, but extensive numerical evaluation by digital computer is involved: the region considered is bounded by the tropopause and lies between the equator and 70 degrees latitude. The computed zonal velocity has the characteristic east-west variation with latitude and a broad band maximum of 19 m sec

^{-1}and the meridional velocity the characteristic tricellular structure. A poleward eddy angular momentum flux and polar inversion are predicted. The results, through verification of the postulates, add support to the Rossby view of the general circulation in which the cyclonic-scale eddies act to release potential energy of the atmosphere to supply their own kinetic energy and form the mean zonal kinetic energy. They further indicate the value of the reconstructed 'austausch' approach for this problem.

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