## 1. Introduction

We continue the presentation begun in Williams (1996, hereafter Part I) and extended in Williams (2002, hereafter Part II) of solutions to a primitive equation model that examine dynamical processes thought to be relevant to the global circulations of the Jovian atmospheres. Here, the main concern is with generating the multiple jet streams and an equatorial superrotation of the form and scale seen on Jupiter and Saturn. In addressing this problem, we again explore the hypothesis that the active atmosphere is driven by baroclinicity and is thin relative to a deep underlying envelope. The vertical structures confining the motions to the upper layer are assumed to be of the same form as those known to favor the existence of stable planetary vortices.

In Part I, thin jets were seen to be capable of generating and coexisting with thin planetary-scale vortices that are absolutely stable when all of the horizontal motions are limited to the upper layer by exponential stratifications with a high confinement rate. Such stratifications, denoted by EXP, have a related structure, denoted by LIN, whose horizontal flow and Brunt–Väisälä stability are linear in the thin upper layer but zero below; see Fig. 1. The LIN structure allows a similar vortex behavior but has different small-scale baroclinic instability properties, particularly as regards the latitudinal eddy heat and momentum fluxes. These similarities and differences are examined in this paper for issues relating to the genesis and equilibration of jets.

In Part II, the extension of vortex modeling to a wider range of latitudes indicates that the EXP and LIN structures could occur globally. The vortices are generated in anticyclonic zones by the long-wave baroclinic instability of baroclinic easterly jets and their relation to the long-wave solitary Rossby waves provides, via the propagation rate, a deformation radius of about 1000 km as an upper limit for the size of the eddies that sustain the jets. In addition, the creation of steady vortex configurations using a Newtonian heating function provides an estimate of about 300 days as a representative timescale for the thermal processes. These space scales and timescales are also used in this study.

The latitudinal distribution of Jupiter's zonal winds is well known at cloud level due to the high accuracy that the cloud tracking method allows; see Simon (1999) for a recent synthesis. Both hemispheres contain about six westerly jet streams, separated by weaker easterly flows, between the equator and ±60° latitude. Starting from a superrotation of about 100 m s^{−1} at the equator, the zonal flow increases into the Tropics to peak at ±8° latitude with a particularly intense maximum of 170 m s^{−1} occurring in the Southern Hemisphere. The winds then drop off rapidly with latitude and alternate from easterly to westerly with steadily decreasing amplitudes and inferior easterly components.

Little is known directly about the vertical wind structure on Jupiter apart from the measurement made by the *Galileo* entry probe at 6.5°N latitude, which revealed a zonal flow that remains uniform at 170 m s^{−1} over the 100-km layer between 4 and 21 bars, at least for the longitudinal range and time span sampled (Atkinson et al. 1998). This amplitude closely matches the value of the angular momentum conserving wind, *u*_{m} = *a*Ω sin*ϕ* tan*ϕ,* at this latitude (Williams 1997) and its compliance with a standard meteorological measure suggests that the motions could be shallow in the vertical. However, the nature of the winds measured by the *Galileo* probe has undergone reassessment (Young 1998; Allison 2000) following the realization that the spacecraft entered an anomalous warm vortex. As regards the related thermal structure, the *Galileo* probe revealed a statically stable troposphere (Seiff et al. 1998, Fig. 36) with complex variations similar to those predicted by vortex theory (Part I, Figs. 40 and 45).

The existence of multiple jets is usually explained theoretically in terms of the barotropic and baroclinic modes of energy and enstrophy cascade associated with the various forms of *β* turbulence, as described by Rhines (1975, 1977, 1994). Consequently, the generation and persistence of multiple jets have been explored numerically using a variety of atmospheric, oceanic, and planetary models. The models range from barotropic spheres (Williams 1978; Huang and Robinson 1998; Huang et al. 2001), to quasigeostrophic (QG) beta planes (Williams 1979; Panetta 1993; Vallis and Maltrud 1993; Treguier and Panetta 1994; Lee 1997), to shallow water spheres (Williams and Wilson 1988; Cho and Polvani 1996), to primitive equation general circulation models (GCMs) (Williams 1988).

The generation of multiple jets in thin baroclinic layers has not been evaluated until now, and it is not obvious that the baroclinic instability needed to energize the small eddies and the cascades needed to drive the jets can occur in such systems. In fact, linear QG theory for baroclinic ocean layers implies the opposite (Gill et al. 1974), that westerly currents are usually stable in exponential structures. Neither is the extent known to which baroclinic instability can be extended into low latitudes. Such processes could be responsible for, or at least influence, the onset of an equatorial superrotation. Other hypotheses, in various stages of development, have been suggested for the possible cause of the multiple jets; their processes range from deep convection (Busse 1994; Condie and Rhines 1994), to shallow convection (Williams and Robinson 1973; Williams 1978), to moist convection (Gierasch 1976; Ingersoll et al. 2000), and on to thin baroclinic layers overlying deep barotropic flows (Orsolini and Leovy 1993). Here, we chose to evaluate the simplest hypothesis first.

To examine the hypothesis that baroclinic instabilities can generate and maintain multiple jets in a thin atmosphere at all latitudes, calculations are made with a primitive equation model subject to a Newtonian heating function. The heating produces flows with either the LIN or EXP vertical structure and thence two forms of baroclinic instability and two classes of jet. The latitudinal heating distribution is set experimentally to develop a variety of global and local baroclinicities that examine the sensitivity of the circulation to the heating imbalance, especially in low latitudes.

The presentation begins in section 2 with a description of the basic model, the parameters, and the formulation of the heating distributions. This is followed by a brief review of those theories that help in designing the calculations and in understanding the solutions. Section 3 then gives an overview of those solutions that describe the range of jets found for the various heating arrangements, particularly those needed to isolate the conditions for the onset of equatorial westerlies in both structures. The LIN and EXP systems produce two distinct classes of circulation, the first giving steady flows and the latter producing jets that migrate equatorward and regenerate in high latitudes. Section 4 proceeds to examine in detail the two LIN solutions that exhibit the most basic and the most realistic circulations.

Likewise, section 5 concentrates on details of the most basic and the most realistic EXP solutions to examine the nature of baroclinic instabilities in a system that allows jets to migrate. To define processes more clearly, we then limit the domain to low latitudes in section 6 and examine the behavior of a solitary tropical jet. This also leads to a derivation of the conditions under which instabilities can produce equatorial westerlies in Earth's atmosphere. Then, to test the relevance of the thin-layer hypothesis even further, section 7 turns to some anomalous cases to show first how a major easterly current can be produced in mid- to low latitudes, as on Neptune, and then how hexagonal jets can occur in mid- to high latitudes, as on Saturn. The planetary implications of the main solutions are touched upon when concluding in section 8.

Finally, note that some of the new processes described by the solutions, particularly those in low latitudes, have also been reproduced in global atmospheric models, including a multilevel, spectral, pressure-coordinate GCM for Earth; the results for the latter are discussed in a separate paper (Williams 2003a). The variability of the jets as the extent of the baroclinic layer goes from shallow to deep is also discussed elsewhere (Williams 2003b).

## 2. Model and theory

### a. System of equations

The numerical studies use the primitive equations of motion with a Boussinesq equation of state, solved for a regional channel on a sphere. This model provides an adequate representation of the basic dynamical mechanisms under consideration and can be applied to either an ocean or an atmosphere provided that, for the latter, the variables are mapped from geopotential to pressure coordinates and reinterpreted appropriately, as described, for example, by Salmon (1998, p. 102).^{1} The primitive equation model for a thin hydrostatic fluid is specified by the zonal, meridional, and vertical velocity components *u,* *υ,* *w,* and by the pressure, density, and temperature fields *p,* *ρ,* *T.* The standard equations are written in spherical coordinates, as in (1)–(14) of Part I, and include simple biharmonic and Laplacian diffusion terms with coefficients *ν*_{4} and *ν*_{2} in the horizontal and vertical, respectively, as well as a heating function and a convective adjustment. The variables *λ,* *ϕ,* *z* represent the longitude, latitude, and height; *g* is gravity; Ω and *a* are the planetary angular velocity and radius; *f* = 2Ω sin*ϕ* and *β* = 2Ω*a*^{−1} cos*ϕ* are the Coriolis parameters; *T*_{s} and *α* are the background hydrostatic temperature and the Boussinesq coefficient; while *B* = *αg**T*^{*}_{z}*T** = *T* + *T*_{s} define the Brunt–Väisälä stability parameter and the total temperature.

The equations are solved using the finite difference methods documented in Part I; these involve a leapfrog time differencing and a centered spatial differencing on the so-called B grid. The computational domain consists of a Southern Hemisphere channel with periodic boundary conditions in longitude, symmetry conditions at the northern boundary on the equator, together with a no-slip, no-flux condition on the southern wall. In the vertical, both surfaces are taken to be horizontal rigid lids with free-slip, no-flux conditions at *z* = 0 and −*H,* where *H* is the fluid thickness. Near the lower surface, a weak linear drag with a timescale *τ*_{D} helps equilibrate some flows.

### b. Parameter values

The calculations use parameter values that are thought to be appropriate for Jupiter's atmosphere, values that produce zonal jets with amplitudes, scales, and form comparable to those observed at cloud level. The planetary parameters, *a* = 71 300 km, Ω = 1.76 × 10^{−4} s^{−1}, and *g* = 26 m s^{−2}, remain fixed while the following represent the basic values about which variations are made in Tables 1–3. For evaluation purposes, *H* is nominally set at 15 000 km but, as discussed in Part II, it can be reduced by a factor of 10 or more and all solutions can be rescaled in the vertical without altering their form, provided that the temperature gradients are increased (within the observational limits) by a corresponding factor to maintain identical winds. The Boussinesq coefficient is kept fixed at *α* = 0.005°C^{−1}, an arbitrary value, and the diffusion coefficients are set close to *ν*_{4} = −10^{17} m^{4} s^{−1} and *ν*_{2} = 0.

The domains generally range over 60° longitude and 70° latitude, with Δ*λ* = 1° and Δ*ϕ* = 1° forming the standard grid spacing, and with Δ*t* = (1/100) day being a typical time step.^{2} For the EXP cases whose structure goes as exp(*Nz*′), the vertical grid Δ*z* also varies exponentially in its spacing, usually as exp(7*z*′) when the confinement rate *N* equals 200, to put more than half the grid points in the active layer, where *z*′ = *z*/*H.* In the LIN cases, a simple split grid with Δ*z* = [Δ*z*_{1}, Δ*z*_{2}] = [0.05, 0.95]2*H*/*KZ,* puts one-half of the grid points, the thinly spaced ones, in an upper layer of thickness *h*_{1} = *H*/20 that more than contains the main motion, and the other half, the thickly spaced ones, in the abyss; the number of grid points KZ usually equals 20.

For the two Earth cases discussed in section 6b, the basic parameters are *a* = 6400 km, Ω = 7.3 × 10^{−5} s^{−1}, *g* = 9.8 m s^{−2}, *H* = 8 km, *α* = 0.003°C^{−1}, *ν*_{4} = −10^{15} m^{4} s^{−1}, *ν*_{2} = 0; for a domain resolved by grids with Δ*λ* = 3°, Δ*ϕ* = 1°, Δ*z* = *H*/20, and Δ*t* = (1/100) day.

In presenting the solutions, the figures use solid contour lines to plot values greater than or equal to zero, while dashed lines denote negative values. For the time-mean eddy transports, evaluated using fields sampled at 1-day intervals, the zero-value contours are omitted from the plots for greater clarity. The temperature plots exclude the background component except when referred to as the total temperature. Altitude is measured in kilometers in the vertical cross sections. Indices are used to refer to phenomena by zone, starting at the equator, so that W_{0} refers to the superrotating westerly, followed by W_{1} and E_{1} for the low-latitude westerly and easterly, and thereafter by W_{i} and E_{i} (*i* = 2, 3, …) for the numerous midlatitude currents.

### c. Heating functions

*T*

_{r}(

*ϕ,*

*z*). The radiative–convective damping time

*τ*(

*ϕ,*

*z*) is set to constant values but, for computational expediency, uses the split form with

*τ*

*τ*′ to maintain the zonal mean (bar) fields without significantly dampening the eddy (prime) fields. This distinction has little effect. Following Part II, the axisymmetric state is usually spun up with

*τ*

*τ*

*τ*

_{D}= 300 days and

*τ*′ = 1000 days.

*T*

_{r}

*T P*

*ϕ*

*S*

*z*

*T*

_{s}

*z*

*S*= (

*d*/

*dz*)[sech(

*Nz*′)] to produce

*u*(

*z*) profiles that are exponential at depth while having a vanishing shear over the upper atmosphere (see Fig. 1) in line with the

*Galileo*data (Williams 1997). The hydrostatic temperature,

*T*

_{s}(

*z*) =

*δT*exp(

*Nz*′), is independent of latitude. The amplitudes Δ

*T*and

*δT*set the baroclinicity and hydrostatic stability rates. For the LIN system, the structures are defined, as in Part I, by the split functions

*S*=

*C*[1, 0] and

*T*

_{s}=

*δT C*[

*z*

^{2}

_{c}

*C*symbolizes confinement and where the first factor defines the distribution in an upper region of depth

*h*≈ 0.7

*h*

_{1}, with

*z*

_{c}= 1 − |

*z*/

*h*| over |

*z*| ≤

*h.*The second factor defines the abyssal distribution. The upper region extends over about seven grid points when

*KZ*= 20, so the main motions always lie well within the highly resolved layer and experience no computational problems at the Δ

*z*

_{1}to Δ

*z*

_{2}interface.

^{3}The ratio

*δ*=

*h*/

*H*defines the confinement parameter for the LIN system.

To provide a global baroclinicity, the latitudinal heating distribution *P*(*ϕ*) is first based on the second Legendre polynomial and set equal to cos^{2}*ϕ*—denoted as *P*_{2}(*ϕ*) in form 1 of Fig. 2—where only positive values are used so as to avoid cooling and inducing large-scale convection in the confined layer. As the *P*_{2} distribution is not universal enough to yield the full range of circulations, the form of *P*(*ϕ*) is varied and evolved toward greater complexity by introducing separate barotropic and baroclinic zones. We begin by examining the influence of a barotropic (BT) zone at the equator by varying its extent into the Tropics, in effect replacing *P*_{2} by *P*_{BT+BC}, a combination of a tropical BT zone and a midlatitude baroclinic (BC) zone^{4}—form 2 in Fig. 2.

We are particularly interested in determining what factors control the location, especially the lowest latitude, at which jets can be generated and what heating forms are needed to develop an equatorial superrotation. To begin addressing such issues, a simple *P*_{BC} heating—form 3 in Fig. 2—is used to extend the baroclinicity to the equator to gauge its effect upon the tropical flow. Then, to encourage jets to form more readily in low latitudes, a stronger baroclinicity is introduced there, together with the normal barotropic and baroclinic zones elsewhere, to give a *P*_{BT+2BC} distribution—form 4 in Fig. 2. Form 5 does the same but excludes the barotropic zone, as does form 6, which is reserved for developing solitary low-latitude jets.

The evolution of *P*(*ϕ*) ends with a *P*_{DIP} distribution—form 7 in Fig. 2—which is equivalent to form 2 with a Gaussian cooling component added to boost the baroclinicity in low latitudes, making it similar in effect to form 4 or 5 but allowing a different way of creating and interpreting the distribution. Heating profiles resembling *P*_{DIP} have been used in dry Earth models to allow implicitly for the intense heating gradient produced by condensation at the equator (Smagorinsky 1963, Fig. A5). All of the *P*(*ϕ*) profiles in Fig. 2 are smoothed and normalized to vary between 0 and 1; the tables document the variety of amplitudes *a*_{i} and latitudes *ϕ*_{i} associated with each zone, starting at the equator. In retrospect, a more systematic approach would involve setting *P*(*ϕ*) = *c*_{2} cos^{2}(*ϕ*) + *c*_{n} cos^{n}(*ϕ*), where *c*_{2} and *c*_{n} are constants for *n* = 4, 8, or 16, as in Williams (2003a).

### d. Theoretical background

^{5}linear baroclinic instability theory can be used to explain eddy origin and scale, and nonlinear theory can be used to explain eddy evolution and fluxes. In the classic ocean study of Gill et al. (1974) for exponentially structured flows, the latitudinally independent, quasigeostrophic flows on a beta plane can be unstable if the potential vorticity gradientchanges sign internally or opposes the sign of

*u*

_{z}at the upper boundary, where

*s*=

*B*/

*f*

^{2}and

*y*is the meridional coordinate. Instability is primarily determined by the slope of the isopycnals. When the slope is uniform, only easterly flows are unstable. To destabilize the westerly flow produced by the standard atmospheric heating requires that the symmetry between the

*u*(

*z*) and

*B*(

*z*) distributions in exponential systems be broken.

*d*<

*b,*where

*d*and

*b*are depth parameters. The first combination, as Gill et al. (1974) show, can have a maximum zonal flow below the upper surface where it produces a weak instability. Both forms of asymmetry occur in the numerical solutions where the static stabilities have a strong jet-related component

*B*

_{j}that tends to dominate the background component

*B*

_{s}aloft.

*u*(

*z*) profiles by Spar (1957) and Wiin-Nielsen (1967), apply to both the LIN and EXP structures. The vertical heat transport term

*Bw*is assumed to be negligible in this model, which makes it useful for understanding flows that may have a non-QG balance or may have a complex or negligible

*B*(

*z*). The analysis of Wiin-Nielsen (1967) for arbitrary

*u*(

*z*) profiles shows that disturbances of the form exp[

*ik*(

*x*−

*ct*)] have a phase speed given bywhere

*c*

_{r}=

*β*/

*k*

^{2},

*I*

_{1}=

^{1}

_{0}

*u*

*dz*′,

*I*

_{2}=

^{1}

_{0}

*u*

^{2}

*dz*′. Instability occurs according to Schwartz's inequality when

*I*

_{2}−

*I*

^{2}

_{1}

*c*

^{2}

_{r}

*L*<

*L*

_{β}(2/

*N*)

^{1/4}and

*L*<

*L*

_{β}(2

*h*/

*H*)

^{1/2}in the EXP and LIN systems, respectively, where

*L*

_{β}= (

*U*/

*β*)

^{1/2}for the velocity scale

*U.*For the numerical cases with

*N*∼ 200 or

*δ*∼ 1/20, the cutoff occurs at

*L*∼

*L*

_{β}/4. This upper limit to the scale of the baroclinic instability implies that a horizontal grid of approximately 1000 km is needed in the calculations. Similar criteria for the instability of thin-layer westerlies are given by Killworth (1980, section 9b) and Benilov (1995, section 5).

The nonlinear baroclinic instability theories that follow eddy development through cycles of growth and decay, and help explain eddy evolution and fluxes in Earth's atmosphere, appear to be relevant to thin layers also. Detailed analyses of such “eddy cycles” indicate that, for a specified zonal flow, linear theory applies initially, with the eddy energy growing at all heights but transporting heat mainly at lower levels (Simmons and Hoskins 1978; Edmon et al. 1980). The upper-level eddies, fed by upward wave radiation, generate planetary waves that propagate to other latitudes. These waves produce a large momentum flux that either traverses poleward, as in the standard terrestrial case, or converges on the jet cores, as in terrestrial models with higher rotation rates (Williams 1988).

Quasigeostrophic turbulence theory analyzes more general forms of energy and enstrophy cascade and gives insight into processes that determine the jet scale. According to this theory (Rhines 1975, 1977), the Coriolis gradient *β* stops the nonlinear barotropic cascade at a wavenumber *k̃*_{β} = (*β*/*Ũ*)^{1/2} that separates the wave and turbulence regimes and thereby defines the Rhines jet scale as *L̃*_{R} = *π*(*Ũ*/*β*)^{1/2}, where *Ũ* is the root-mean-square barotropic eddy velocity. For baroclinically unstable flows, this barotropic velocity can be related to the baroclinic eddy velocity *Û* by using scaling arguments to connect the barotropic energy level to the rate of eddy energy production, Held and Larichev (1996). This leads to a “baroclinic” version of the Rhines wavenumber, *k̂*_{β} = *β*Λ/*Û,* where Λ = *hB*^{1/2}/*f* is the deformation radius. This analysis can be further extended to thin layers (Smith and Vallis 2002) to give an alternative scale for the jet widths, *L̂*_{R} = *π*(2*δ*)^{1/2}(*Û*/*β*Λ). Although the processes involved in the solutions differ from those invoked in the theoretical derivation, this length scale seems to apply to the numerical and Jovian jets and seems to be preferable for flows with a small or uncertain barotropic component, with *Ũ* ≪ *Û.* In particular, the typical modeling scales *Û* = 50 m s^{−1}, *δ* = 1/30, *β* = 0.4 × 10^{−8} km^{−1} s^{−1}, and Λ = 10^{3} km yield a reasonable value, *L̂*_{R} = 10 000 km, comparable to Jupiter's jet widths. The *L̂*_{R} scale also resembles the optimal eddy size, *L*_{C} = 2*πhU*_{z}/*β*Λ, predicted by Charney's linear instability analysis (Gill 1982, section 13.4). In practice, the various scales are too close to each other in value to determine which processes are involved in defining the jet scales.

## 3. The range of multiple jets

We begin our discussion of the numerical solutions by giving an overview of the circulation range as defined by the mean zonal flows in Figs. 3 and 4. The flows are generated by the various heating distributions described in section 2c using the parameters listed in Tables 1 and 2. The heating profile evolves from the elementary *P*_{2} form to the complex *P*_{DIP} profile so that the baroclinicity can be extended into lower latitudes to produce a tropical W_{1} jet whose barotropic instability generates a W_{0} superrotating current at the equator. Two classes of circulation are obtained as the LIN and EXP structures support significantly different flows, with the latter exhibiting novel equatorward-migrating jets. The dynamics of the main cases are discussed in detail later in sections 4 and 5.

### a. The LIN jet range

The representative set of solutions for the LIN structure, L1–L6 in Fig. 3, shows how the circulation progresses as the level of complexity in the *P*(*ϕ*) heating profile increases. The simplest circulation occurs for the elementary *P*_{2}(*ϕ*) heating form and consists of four westerly jets, W_{1–4}, over the 70° domain (Fig. 3a). Weak easterly currents form near the equator and between the westerly jets. The lowest latitude at which a jet core lies is 17°S, where the stable W_{1} current is essentially a residual of the axisymmetric thermal wind that is truncated on its poleward side by the action of the eddies associated with the midlatitude jets.

To examine the sensitivity of the W_{1} jet to the way the heating distribution allocates the barotropic and baroclinic zones, the *P*_{2} variation is first replaced by a two-component function that has a purely barotropic part from the equator to 10°S and a linear baroclinic part over the rest of the domain, as in form 2 of Fig. 2. The main effect of this *P*_{BT+BC} heating for the L2 case in Fig. 3b is to produce a wider W_{1} jet with a more poleward core at 20°S than in the L1 case. The L1–L2 similarity raises questions about the dependence of the W_{1} jet and the E_{0} current that lies equatorward of it, on the width of the tropical barotropic zone and on the existence of baroclinicity in low latitudes. To address these issues in the L3 case, the baroclinicity is extended to within 2° of the equator, as in the *P*_{BC} profile of Fig. 2. The resulting flow in Fig. 3c resembles the L1 and L2 forms, though the jets are fewer because they are stronger. The tropical baroclinicity does not shift the W_{1} jet equatorward because wave propagation produces a strong E_{0} easterly current near the equator.

To really move the W_{1} jet toward the equator, without altering the midlatitude jets, requires the introduction of a stronger and separate baroclinicity in lower latitudes, as in form 4 of Fig. 2. For such a *P*_{BT+2BC} heating in the L4 case, the barotropic zone goes from the equator to 6°S while the stronger baroclinic zone extends from 6° to 12°S (Table 1). The latter produces a strong W_{1} jet centered at 10°S, clearly a significant equatorward shift and one that leads to a five-jet circulation in the domain as a whole (Fig. 3d). Although there is much eddy activity in low latitudes, it does not lead to the onset of a W_{0} current at the equator. This result, however, is conditional and depends on the width of the barotropic zone. When the barotropic zone extends from the equator to a critical latitude that lies somewhere between 3° and 6°S, other calculations reveal that the W_{1} jet then lies sufficiently near the equator for its instability to generate a W_{0} current. For the parameter range under consideration, the W_{1} jet must be centered at |*ϕ*| ≤ 8° and must be sufficiently strong for this to happen.

In the limiting L5 case, a stronger baroclinicity extends all the way to the equator from 10°S in the *P*_{2BC} heating and produces a W_{1} jet at 8°S whose strong instability generates eddies capable of driving a strong W_{0} superrotation at the equator (Fig. 3e). Another way of increasing the tropical baroclinicity is by introducing a local Gaussian cooling dip at the intersection between the barotropic and baroclinic zones, at 10°S in form 7 of Fig. 2. In the L6 case, such a *P*_{DIP} heating produces an unstable W_{1} jet at 8°S and a W_{0} superrotation at the equator (Fig. 3f). As noted in section 2c, the stronger baroclinicity needed to produce an equatorial westerly can be represented by either a *P*_{2BC} or a *P*_{DIP} heating distribution, which in turn can be ambiguously interpreted as being due either to a powerful latent heating at the equator or to albedo variations in a cloudy atmosphere.

All of the LIN jets are steady once the flow has fully evolved and are fully represented by the typical *u**ϕ,* *t*) diagram for the L6 case in Fig. 5a. Most of the jets equilibrate completely within the first 1000 days but the equatorial W_{0} current, when it occurs, grows more slowly and takes about 5000 days to equilibrate, which it does by modifying the W_{1} instability. Figure 5 also contains a representative *u**ϕ,* *t*) diagram for the jets in the EXP system, showing their equatorward migration and regeneration in high latitudes; we turn to them next.^{6}

### b. The EXP jet range

The cases chosen to illustrate the progression of the flow with heating complexity for the EXP structure, E1–E6 in Fig. 4 and Table 2, differ from the LIN set because the jet migration alters the dependence on local heating components. There is no need, for example, for the *P*(*ϕ*) forms involving two separate baroclinic zones. The *u**ϕ,* *t*) diagram for the E6 case in Fig. 5b is typical and shows the timescales to be about 5000 days between mergers involving the W_{2} jet and about 15 000 days for the complete migration of a W_{5} jet from high latitudes to its merger with a W_{2} jet at 23°S; the latter interval corresponds to a migration speed of 3 cm s^{−1}. The migration, merger, and renewal of jets also complicates the use of a positional terminology, which is now refined so that W_{i} refers to the *i*th westerly jet from the equator *at a particular time.*

The circulation given by the elementary *P*_{2}(*ϕ*) heating distribution consists of five westerly jets plus an E_{0} easterly current at the equator, for the E1 case in Fig. 4a. The jets have similar widths, with amplitudes that decrease poleward, and are separated by zones having almost no zonal flow rather than by easterlies. But the jets now reach lower latitudes than their LIN counterparts, with the W_{1} jet core settling at 12°S, due in part to the continuous equatorward migration. The migration does not extend to the equator, however, as it is blocked by the E_{0} current.

Turning next to the two-zone *P*_{BT+BC} heating distribution, we find that for the EXP system the width of the barotropic zone is crucial as it determines whether or not a sufficiently strong E_{0} current can develop to stop the migration reaching the equator and forming a W_{0} flow. The critical width for the barotropic zone to exclude an equatorial westerly lies somewhere between 6° and 9° of latitude, compared to the 3° to 6° needed by the LIN system with two baroclinic zones. Thus, in the E2 case in Fig. 4b, the westerlies are well blocked by the 12°-wide barotropic zone and the strong E_{0} current that it allows. Although the W_{1} jet core ends up at 15°S, it initially forms at 10°S, but mergers with migrating W_{2} jets at 2000 days and 5500 days, together with the gradually strengthening E_{0} current, move the jet poleward.

To produce a westerly current at the equator, the E_{0} flow must not be allowed to become too strong. This is so in the E3 case in Fig. 4c where the barotropic zone is only 6° wide, not enough to stop the jets from migrating all the way to the equator even though an E_{0} easterly persists there for the first 3000 days. Mergers between the W_{1} and W_{2} jets at 1500 days and 4500 days produce a westerly current strong enough to displace the E_{0} flow. The resulting conditional W_{0} westerly at the equator is, however, just a part of the W_{1} jet whose core lies at 4°S and, unlike in the LIN cases, does not form a distinctly separate current.

Given that jet migration can overcome the easterlies of a narrower barotropic zone, a one-component *P*_{BC} heating distribution should produce the same circulation as the E3 case, which is what the E4 solution in Fig. 4d confirms. Again this differs from behavior in the LIN system and again an E_{0} current prevails initially until mergers between the W_{1} and W_{2} jets at 1100 days and 3600 days produce a stronger westerly that spreads to the equator. Details in the *P*(*ϕ*) heating distribution between the equator and 9°S appear to be of little consequence given the inexorable migration so we can proceed directly to the most complex distribution.

Thus, consider the two cases E5 and E6 in Figs. 4e,f that describe the circulations produced by the complex *P*_{DIP} distribution. The two cases differ in that E5 has the simpler eddy fluxes while E6 has the stronger heating rate and a deeper equatorial flow. The lengthy E6 calculation extends to 21 000 days to establish the steadiness of the flow configuration when individual jets migrate and regenerate (Fig. 5b). Most jets form quickly but the W_{0} current has a longer timescale and takes about 10 000 days to reach its full amplitude; see Fig. 11 later. The midlatitude jets migrate steadily and continuously equatorward but do not penetrate beyond 20°S after the first merger between the W_{2} and W_{3} jets at 5000 days. An unusual equatorial westerly undercurrent forms in the E6 case at 4000 days and merges vertically with the upper W_{0} current at 6000 days to give the deeper flow seen in Fig. 4f.

In general, the jets in the EXP set resemble those in the LIN set when the heating has either the elementary *P*_{2} or the complex *P*_{DIP} form, despite the extra migration, merger, and regenesis processes. The cool Gaussian dip, in particular, helps create and anchor the more robust tropical currents that limit migration to latitudes poleward of 20°S. Jet migration can be thought of as a form of wave propagation that is blocked at a critical latitude when the tropical currents modify the eigenfunctions appropriately. Such a transfer of energy toward the equator may also be considered as the ultimate turbulent cascade toward a larger scale (Salmon 1998, p. 284). Note, however, that in the EXP solutions of Part II the jets do not migrate, presumably because they are imposed and maintained by a sinusoidal heating that also creates strong easterly currents.^{7}

The EXP jets in midlatitudes are almost fully baroclinic as the barotropic component never exceeds 1.5 m s^{−1} anywhere. Although the LIN flows have a more significant (5 m s^{−1}) barotropic contribution, in neither case is it clear what role this plays in determining the jet scales.

## 4. Steady multiple jets

To define the processes involved in the formation of the steady multiple jets, the time-averaged eddy transports are examined for the two main states realized in the LIN set. The first case, L6 in Figs. 5–9, is considered *realistic* as it displays a Jupiter-like set of jets. Then follows the L1 case, considered *basic* as it involves the elementary *P*_{2}(*ϕ*) heating function. These two states mainly differ in low latitudes and their differences help isolate the phenomena that can occur near the equator.

### a. The realistic LIN circulation

The axisymmetric spinup of the L6 system produces a broad westerly flow in midlatitudes and, thanks to the Gaussian cooling dip, a sharp westerly jet of 180 m s^{−1} at 8°S, with a weak easterly flow in between; see the first curve in Fig. 5a. Once perturbed, the broad westerly soon develops baroclinic instabilities that change it completely into three midlatitude westerly jets. The sharp axisymmetric component, however, survives in a modified form as a W_{1} jet while its instability helps generate the westerly at the equator. The original easterly also survives but is broadened by eddy action into an E_{1} current.

The equilibrated jets extend 500 km into the fluid and end just below the thermal interface (Figs. 3f and 6a). The fluid remains statically stable everywhere. Although the Brunt–Väisälä stability is strongest at the interface due to the latitudinally varying heating component, the active layer also has a significant value due to the background component.^{8} The zonality of the jets is evident in the horizontal sections, except in high latitudes where *β* is weak (Fig. 7). The axisymmetric residual of the W_{1} jet is most apparent in the enstrophy field, which elsewhere contains eddies lying on both sides of each jet core. The enstrophy eddies are less than 3° wide and become smaller with latitude; in the spectrum for the W_{2} jet, they exhibit a peak at wavenumbers *k* = 11 and 12 over the 180° sector. Although the observed eddies are half this scale, their action is reasonably well represented in the L6 solution.

The midlatitude eddy transports in Figs. 8b,c,f reflect the action of three sets of nonlinear baroclinic instabilities, all centered on the cores of the W_{2–4} jets that they sustain at *ϕ* = (26, 40, 55)°S. All transports have the same form within each jet: (a) a strong upward heat flux, *w*′*T*′*υ*′*T*′*u*′*υ*′

**F**= {

*F*

^{(ϕ)},

*F*

^{(z)}} and flux divergence

*E*for the Boussinesq model can be defined following Andrews and McIntyre (1978) aswhere

*ζ*

*a*cos

*ϕ*)

^{−1}(

*u*

*ϕ*)

_{ϕ}. The

*E*field is plotted in Fig. 9 for the L6B case (a recreation of the L6 case on a different computer) using contour values that are powers of 2 to reveal the weaker contributions. The flux divergence is positive within the jet cores due to the

*F*

^{(ϕ)}term though elsewhere the

*F*

^{(z)}term is dominant and the wave propagation mainly upward.

Turning to low latitudes, the tropical eddy transports occur primarily during the first 1000 days and must be plotted separately, on the right side of Fig. 8, to reveal their form. The processes producing these relatively weak transports are multiple and difficult to isolate. In particular, the eddy momentum transport *u*′*υ*′_{0} current. The baroclinic instability of the W_{1} jet, as defined by *υ*′*T*′_{1} jet in section 6 gives a clearer view of low-latitude dynamics while the following null case provides a reference state and further perspective.

### b. The basic LIN circulation

For the *basic* L1 case with the elementary *P*_{2}(*ϕ*) heating in Fig. 10, the two main midlatitude jets W_{2} and W_{3} have the following standard features: converging *u*′*υ*′*w*′*T*′*υ*′*T*′_{4} jet, however, has a poleward traversing *u*′*υ*′_{1} jet centered at 17°S is baroclinically stable, though weak eddies do exist to produce a weak poleward momentum flux across it and support the E_{1} current near the equator (Fig. 10d). This W_{1} jet is the residual of a broad axisymmetric westerly that has been truncated at 22°S by the currents produced by the midlatitude instabilities.

The L1 case shows that without a significant baroclinicity in low latitudes, the W_{1} jet is both too weak to be unstable and too far from the equator to activate a W_{0} current. Furthermore, the transitional L4 case shows that even a jet centered at as low a latitude as 10°S is still too far away, even if it is unstable. For the LIN system and the present parameter range, the critical latitude for the W_{1} jet to generate a W_{0} current lies close to the 8°S value realized by the *realistic* L6 case.

In all cases, the midlatitude jets all have the classic eddy transports normally associated with the nonlinear baroclinic instabilities of thick layers at high rotation rates but that now appear to occur in thin layers as well.

### c. Minor LIN variants

To try to isolate the process that determines the scale of the jets, the L1 case can be recalculated using lower and higher baroclinicities, such as Δ*T* = 4° and 12°C. These lead, respectively, to six narrower jets with amplitudes of 20 ± 5 m s^{−1}, and to two wider jets of 80 and 100 m s^{−1}, compared to the original four jets of 50 ± 10 m s^{−1}. Such values are in keeping with variations in the Λ, *L̂*_{R}, and *L*_{C} scales discussed in section 2d, mainly because Δ*T* also influences the static stability, even though *δT* is unaltered. Consequently, the variations do not reveal which process actually determines the jet width and amplitude.

## 5. Migrating multiple jets

To examine the dynamics of the migrating jets in the EXP solutions of Fig. 4, we now consider the *realistic* case E6 in Figs. 11–13 with its Jupiter-like jets. This is followed by the *basic* E1 case with the elementary *P*_{2}(*ϕ*) heating to illustrate the main alternative state in low latitudes. The dynamics of the EXP system differs from the LIN due to the jet migration and due to differences in the baroclinic instability character, especially in low latitudes. Because of the jet migration, relatively short (500 day) averaging periods must be used when defining the eddy fields in Figs. 12–14. Longer periods lead to a latitudinal blurring or smoothing of fluxes such as *u*′*υ*′

### a. The realistic EXP circulation

Driven by the complex *P*_{DIP} heating, the axisymmetric spinup of the E6 system produces a broad westerly flow in midlatitudes plus a sharp westerly of 110 m s^{−1} at 9°S, with a weak westerly flow in between; see the first curve in Fig. 5b. Baroclinic instabilities convert the broad current into four westerly jets and modify the sharp current into a W_{1} jet, all within the first 1000 days. The jets immediately begin to migrate equatorward but the W_{1} jet stops near 5°S at 5000 days when the first merger of the W_{2} and W_{3} jets occurs. Further W_{2,3} mergers then follow at 5000-day intervals, each time resulting in a new W_{2} jet whose position moves poleward from 17° to 22°S. The blocking of the migration by the W_{2} jet allows the W_{1} and W_{0} currents to develop in relative isolation.

Consequently, the W_{1} jet reaches its peak at 6000 days, just after the first W_{2,3} merger, and then remains steady in form while undergoing rapid fluctuations of about ±10 m s^{−1} in strength (Fig. 11a). Throughout, the W_{1} jet has baroclinic and barotropic instabilities whose eddies also produce an equatorial W_{0} current that grows fastest over the first 3000 days, then more slowly until 7000 days, after which it undergoes a sharp drop before reaching an oscillatory equilibrium (Fig. 11a). An unusual westerly undercurrent forms at the equator at 4000 days and merges vertically with the developing W_{0} current to deepen it at 6000 days; compare Figs. 12a and 12e. This merger accounts for the sharp rise in the barotropic kinetic energy around 7000 days (Fig. 11b).

Given the continuous migration, the meridional structure of the circulation shown in Figs. 4f and 12e represents a nonmerging phase only, one in which the three main midlatitude jets extend downward about 200–250 km into the fluid to where the Brunt–Väisälä stability peaks (Fig. 6b). Only the equatorial current reaches the base of the heated layer at the 400-km depth, and then only after merging with the undercurrent. The near-neutral stability layer occupying the top 100 km occurs because the background static stability cannot compensate for the cooling created by the (*d*/*dz*) sech(*Nz*′) distribution used to produce a vanishing shear at the top surface. Despite this near-neutral layer, baroclinic instabilities occur at all latitudes, as might be expected from Fjortoft's (1951) theory.

Turning to the E6 eddies, Figure 12 displays the various transports for both the early formative phase (on the left side) and the steady configuration phase (on the right side). At all times, the eddy transports lie primarily within the near-neutral 100-km-deep layer, for which the deeper nonzero *B* field may be acting as a broad thermal interface. Extending only 50 km down, the eddy momentum transport *u*′*υ*′*u*′*υ*′_{1} jet to produce the W_{0} superrotation, but it eventually alters in Fig. 12f to converge on the W_{1} jet core as the W_{0} flow equilibrates.

The E6 poleward eddy heat transport −*υ*′*T*′^{9} while being more uniform with height due to the neutral stability layer. This transport gradually weakens in low latitudes as the flow approaches equilibrium. The *υ*′*T*′*w*′*T*′_{1} jet remains baroclinically unstable throughout, though this instability becomes more confined to the poleward flank of the jet during the later phase.

In the E5 *realistic* case, with its weaker *P*_{DIP} heating, the resulting jets and eddy fields are found to be similar to those of the early phase of E6 but to lack the complications caused by the westerly equatorial undercurrent. This means that *simple* W_{0} currents can form as readily in the EXP system as in the LIN setup. Generally, the baroclinic instabilities that occur in the EXP system seem to be close to the classic form, but in low latitudes the eddy transports and their variation between the early and late phases are novel, as is the jet migration in midlatitudes.

### b. The basic EXP circulation

The *basic* state given by the elementary *P*_{2}(*ϕ*) heating function for the EXP system, E1 in Fig. 14, consists of five jets of comparable width at 2000 days. No mergers have occurred as yet but a slow migration is underway, with the W_{2} jet, for example, having moved to 24°S from its original position at 30°S. The W_{1} jet is an exception, however, and remains at 12°S, which, when compared to the 18°S position of its L1 counterpart, shows that the EXP structure allows baroclinic instability to occur in significantly lower latitudes than does the LIN system. But at the equator, an easterly exists to prevent the W_{1} jet, unstable though it may be, from ever generating a W_{0} current. To achieve a superrotation, even the EXP system requires more baroclinicity in low latitudes than *P*_{2}(*ϕ*) provides.

The variation of the EXP circulations with latitude is most clearly seen in the E1 solution, and shows that while the jets extend 200–300 km into the fluid and become *shallower* with increasing latitude, the corresponding baroclinic instabilities (as defined by the eddy heat fluxes) extend 70–100 km in depth and become *deeper* with latitude (Fig. 14). All of the jets have converging eddy momentum transports while the eddy heat transports differ significantly from their LIN counterparts; the eddies transport heat poleward between the jets but not across them (Fig. 14e). The eddies are strongest and deepest in the W_{3} and W_{4} jets but the jets themselves decrease poleward in amplitude while keeping their widths constant. All of these features of the multiple jets and their instabilities may contain clues as to how they actually arise, set their scale, sustain themselves, and migrate.

Concerning the cause of the migration, closer inspection of the eddy momentum transports reveals that although the *u*′*υ*′*u*′*w*′*u*′*w*′*F*^{(z)} Eliassen–Palm flux to reduce the jet at a given location and through *u*′*υ*′*F*^{(ϕ)} to push it equatorward. Presumably, the real origin of the migration lies in the basic character of the baroclinic instabilities and wave dispersion favored by the EXP system, though the weakness of the process makes it difficult to detect.

To summarize, we see that the EXP system favors slowly migrating jets driven by baroclinic instabilities that occur as readily in low latitudes as in midlatitudes. When the baroclinicity extends into latitudes lower than 9°S the barotropic instability of the resulting W_{1} jet can lead to an equatorial superrotation. In such cases, the jet migration only reaches 20°S and thus does not directly influence the W_{0} onset or maintenance.

## 6. Equatorial jets

The generation of the W_{0} equatorial superrotation in the L6 case discussed above involves low-latitude baroclinic and barotropic instabilities whose characteristics are complicated by planetary waves emanating from the midlatitude instabilities. To isolate the low-latitude instability more clearly, consider now a more limited LIN system, Q1 in Table 3, for which the heating creates only a W_{1} jet in a narrower channel. Creating a single jet at 8°S then leads to a well-defined instability whose features have much in common with those of the L6 case and also with the EXP system in low latitudes.

Given that the low-latitude instabilities in the two structures have similar momentum transports, the question arises as to whether the W_{0} onset depends crucially on the active layer being thin and confined, or whether such superrotations can also occur in thick or terrestrial atmospheres. On switching from Jovian to terrestrial parameters (Table 3), we find that the classic baroclinic instability seen in a reference case, Q2, can be replaced by a quite different mode in which the barotropic and baroclinic instabilities generate a W_{0} current in the Q3 solution. This new form of instability defines the essence of the equatorial process.

### a. Jovian superrotation

To create a solitary low-latitude jet centered at 8°S, the fluid is subjected to a simple linear heating with a constant baroclinicity extending from 3° to 15°S, as in form 6 of Fig. 2. The 2.2°C heating amplitude used for Q1 leads to W_{1} and W_{0} flows with peaks of 214 and 187 m s^{−1}, respectively, in Fig. 15, whereas a 2.0°C heating produces winds of 168 and 69 m s^{−1}, in which W_{0} is much reduced. This indicates that for the Q1 configuration to occur, the W_{1} jet must reach a critical value close to 150 m s^{−1} before its instability can generate a significant W_{0} flow.

In the Q1 solution, the W_{0} current grows quickly to 50 m s^{−1} from the W_{1} instability but then increases in steps (Fig. 15a) before leveling off at 10 000 days; later, the two westerlies merge into a single current (Fig. 15c). During the W_{0} growth phase, the eddy amplitudes lie in the 30–40 m s^{−1} range for about 7000 days but then rapidly decay, allowing the zonal current to equilibrate (Fig. 15b). Apparently, the eddies drive the W_{0} current until it approaches W_{1} in strength, at which point the currents equilibrate by eliminating the eddy source.

When averaged over the growth phase, the mean zonal flow in the Tropics has the same form as in the L5 and L6 multijet cases (Figs. 16a and 3e,f), but toward the end of the calculation the contours become flatter and the W_{0} and W_{1} components are indistinguishable (Fig. 15c).^{10} The eddy heat transports produced by the instability peak just above the interface in the W_{1} core at 7°S and extend to within 3° of the equator (Fig. 16e,f). The eddy kinetic energy and eddy momentum transport, however, are strongest farther away from the equator in the jet flank at |*ϕ*| > 10° and are produced by waves generated by the instability (Fig. 16b,d).

Over the complete growth phase, the eddies mainly transport momentum equatorward across the cyclonic part of the W_{1} jet between 8° and 18°S (Fig. 16b). But when averaged over the 300–800-day period of rapid W_{0} growth, the *u*′*υ*′_{0} current and is produced by the barotropic instability of the equatorward flank of the W_{1} jet. Looking back to the *u*′*υ*′_{1} instability if we assume that the weak poleward contribution that disrupts the predominantly equatorward transport near the upper surface is accredited to externally generated planetary waves.

### b. Terrestrial superrotation

To examine the onset of W_{0} currents in a terrestrial context, we now impose heating distributions with cos^{n}*ϕ* profiles to create jets that lie in mid- or low latitudes, depending on the value of *n.* This section considers the two solutions produced by *n* = 2 for the Q2 case and by *n* = 8 for the Q3 case. The terrestrial parameters are listed in section 2b and Table 3 for the solutions plotted in Figs. 17–19. The dynamical processes involved are discussed in greater detail for a more realistic model in Williams (2003a).

The axisymmetric states created during the spinup of the Q2 and Q3 solutions have similar forms but differ from those of the standard theory (Held and Hou 1980) in that the jet core and the Hadley cell limit lie near 20°S for the 70°S domain rather than at the 30°S location produced by the standard *P*_{2}(*ϕ*) heating. When the two axisymmetric states are perturbed, Q2 develops a standard circulation in Fig. 17 while Q3 has a novel circulation with an equatorial superrotation in Fig. 18. This simple switch in regimes as the baroclinic zone moves to lower latitudes occurs most readily when the flows start from a fully formed axisymmetric state whose jet lies in low latitudes. The novel regime may be less pervasive than the classic state but, nevertheless, exhibits a permanent superrotation, a mode related to those described by Suarez and Duffy (1992) and Saravanan (1993), despite differences in the forcing.

The standard Q2 circulation that forms the reference state in Fig. 17 has all the classic features seen in dry GCM circulations (cf. Williams 1988, Figs. 16c–26c). According to the eddy heat transports, the baroclinic instability occurs within the jet core, between 30° and 40°S, with *υ*′*T*′*w*′*T*′*u*′*w*′*u*′*υ*′

On the other hand, the novel Q3 circulation in Fig. 18 differs in almost all aspects from the standard form. The most original feature, the equatorial superrotation, grows immediately as the axisymmetric state is perturbed, but does so quite slowly over 500 days. Elsewhere, the jet and the thermal front now form in low rather than middle latitudes, near 20°S, and weak easterlies fill the region poleward of 35°S. The meridional circulation is strongly influenced by the instability, with the Hadley cell becoming smaller, much weaker, and significantly constrained by a Ferrel cell of comparable strength.

According to the Q3 eddy heat transports, the novel baroclinic instability is centered in low latitudes and confined to lower levels (Figs. 18f,g). The eddy momentum transport *u*′*υ*′_{0} current. The vertical eddy momentum flux, upward at the equator, also supports the W_{0} current but is mostly downward in the jet flank. The Eliassen–Palm flux divergence is positive aloft, in keeping with the production of eddy-driven zonal flows by upward-propagating waves (Fig. 19b). The weaker *E* flux divergence and F flux vectors in low latitudes (Fig. 19c) reveal the existence of an equatorward wave propagation at low levels together with a poleward propagation aloft, if the eddies are wavelike. The role of such processes in the equatorial superrotation is discussed in Williams (2003a).

The scale of the eddies, as defined by a spectral analysis of the enstrophy at the top surface, is not too different between the two states, with a maximum wavenumber of *k* = 4 over a 180° sector in both cases.

## 7. Easterly and hexagonal jets

To complete the set of circulations, consider now a couple of simple variations on the above theme that may be of singular planetary relevance. The first variant shows how *easterly* currents can be generated by elementary heating arrangements. The solutions discussed above are notable for their lack of significant easterly currents and although this may be reasonable for Jupiter and Saturn, it is not valid for Neptune where a strong easterly current occupies the equatorward half of each hemisphere. But by removing the a priori assumption that the active layer have a constant depth and by allowing the layer to deepen toward the pole, simple easterly currents can be readily realized. The P1 and P2 cases examine these novel easterlies in the EXP system for the axisymmetric and three-dimensional states, respectively. The second variant, the P3 case, involves minor parameter changes in the LIN system but yields a circulation with hexagonal jets of possible relevance to Saturn.

### a. Easterly currents

*N*of the exponential system is made to decrease with latitude in the baroclinic part of the heating function (2) but not in the background static stability, so that

*T*

_{r}

*T P*

_{2}

*ϕ*

*N*

*ϕ*

*z*

*δT*

*N*

_{0}

*z*

*N*(

*ϕ*) =

*N*

_{0}/[1 +

*d*(0.5 −

*P*

_{2})] varies from 400 to 100 between 0° and 45°S when

*N*

_{0}= 100 and

*d*= 1.5, where

*d*is the depth variation parameter (Table 3). The

*N*(

*ϕ*) variation can be interpreted as implying either that (a) the penetration depth of the external heat source increases with latitude, or (b) the depth at which the internal heat source becomes effective increases with latitude.

Although a monotonic heating is imposed on the fluid by *P*_{2}(*ϕ*) in (10), it extends over different depths at different latitudes and can, when *d* is large enough, produce a negative baroclinicity in lower latitudes and a positive baroclinicity in higher latitudes; these in turn can produce, through the thermal wind balance, an easterly and westerly current in the two regions, such as those seen in the axisymmetric P1 solution in Fig. 20. A reverse (Ferrel) cell forms between the equator and 25°S and advects heat downward to deepen the thermal layer, which elsewhere corresponds closely to the heating function. Consequently, the active layer has only a modest increase in depth between the equator and 45°S, according to the temperature field.

*T*

_{r}, is given by the latitudinal heating gradient:where

*z*′ in the last term provides the only negative factor for reversing the baroclinicity in lower latitudes. The line along which the baroclinicity given by (11) vanishes

^{11}matches the zero zonal flow contour in Fig. 20a to within 3° of latitude, except near the upper surface where the Ferrel cell is influential.

Clearly, variations in the depth to which the heating penetrates (or originates at) can influence a latitudinally monotonic distribution to the extent that they reverse the baroclinicity and zonal wind in lower latitudes. The depth variations needed to do this are not extreme and so perhaps could be realized on a planet. The zonal flow found on Neptune (Limaye and Sromovsky 1991) has some features in common with the P1 solution, although the planet also has strong easterlies at the equator.

To examine the stability of the axisymmetric P1 circulation, a three-dimensional calculation is made for the related P2 system. The resulting solution in Fig. 21 shows that small-scale baroclinic instabilities arise at 10° and 50°S that are weak in the easterly current but strong in the westerly. Because the eddies are small and weak, the original broad easterly and westerly currents remain intact and provide the dominant mode.

The eddies in the westerly produce two small-scale minijets within the main current. The singular peak in the eddy kinetic energy and the unusually narrow eddy heat transport *υ*′*T*′*u*′*υ*′*u*′*υ*′*u*′*υ*′*w*′*T*′

As regards any planetary application, the minijets and other three-dimensional features introduced by the baroclinic instabilities could be eliminated or enhanced by altering the background static stability.

### b. Hexagonal currents

Finally, consider a solution P3 in Fig. 22 that displays a feature of singular planetary relevance. For this case, we return to the LIN system and make a slight variation in the L6 parameters (Table 3). The usual five-jet circulation develops, but when this is viewed in a polar projection, some jets take on a hexagonal form that lasts for about 100 days, first in the W_{3} jet at 55°S and 500 days, and then in the W_{2} jet at 30°S and 600 days. The calculation uses a 60° sector and this no doubt influences the existence and order of the polyhedron that occurs. The hexagonal appearance is actually due to a localized nonlinear wave that occurs within the jets near the middle of each sector.

The nonlinear wave is associated with concentrated heat fluxes: the *υ*′*T*′*w*′*T*′*Voyager* spacecraft of a similar hexagonal pattern on Saturn at 70°N, Allison et al. (1990) interpret it as being due to a stationary Rossby wave. Here it is a feature of a nonlinear baroclinic instability that also involves planetary waves. The existence of the hexagonal jets provides a further test on the relevance of the thin layer hypothesis.

## 8. Conclusions

Multiple jets can be generated and maintained in thin layers when their primitive equation representation is subject to a simple but appropriate Newtonian heating function. Heating functions designed to produce a globally uniform baroclinicity lead to a broad axisymmetric westerly flow that can be baroclinically unstable when confined to a thin upper layer by either an exponential (EXP) or locally linear (LIN) vertical heating structure. Broad unstable westerlies evolve into multiple westerly jets that achieve a steady state in the LIN system but only a steady configuration in the EXP system where they migrate continuously equatorward and regenerate in high latitudes. All jets are zonally aligned due to the smallness of the eddies but can occasionally, due to nonlinear waves, appear to be hexagonal (or polyhedral) when viewed in a polar projection. Boosting the baroclinicity in low latitudes helps generate tropical jets whose barotropic instability can, in turn, lead to equatorial westerlies. Such superrotations do not depend on the layer being thin and may, for example, also be generated for Earth's atmosphere provided that the circulation is developed by perturbing a strong axisymmetric state whose jet lies in the Tropics. Most solutions are marked by a lack of significant easterly currents but these too can be generated if the heating is allowed to penetrate to, or originate at, depths that increase with latitude.

Flows in the confined linear (LIN) system behave very much like those in a deep linear system as the eddy transports in both closely resemble those produced by classic nonlinear baroclinic instabilities. In particular, the eddy momentum transport *u*′*υ*′*υ*′*T*′_{1} jet can develop a barotropic instability that generates waves that slowly drive a westerly current at the equator, a current that equilibrates by gradually eliminating the eddy source. The W_{1} jet and its instabilities do not, however, produce an equatorial westerly if they are centered much farther from the equator than the critical latitude at 8°S for Jovian parameters and at 25°S for terrestrial parameters.

Although they migrate, jets in the EXP system also exhibit the classic eddy characteristics associated with nonlinear baroclinic instabilities but with all fluxes confined to an upper layer that is close to having a neutral static stability. The eddy momentum transport *u*′*υ*′*υ*′*T*′*u*′*υ*′*υ*′*T*′

Again, four or five jets form in the EXP system under the elementary *P*_{2}(*ϕ*) heating distribution but the W_{1} jet now lies in lower latitudes, at 12°S, than its LIN counterpart. However, the system still requires that the baroclinicity be extended into lower latitudes to give a W_{1} jet near enough to the equator for its barotropic instability to generate a W_{0} current; but, unlike the LIN case, the baroclinicity does not need to be stronger in low latitudes than in midlatitudes to do this. Nevertheless, the resulting W_{1} instability is significantly stronger in the EXP system than the LIN, and this results in a blending of the W_{0} and W_{1} currents into a broad flow that blocks other jets migrating from higher latitudes at 20°S, a latitude where the W_{2} and W_{3} jets proceed to merge every 5000 days. To prevent westerlies from occurring at the equator, the heating has to produce a barotropic zone between the equator and 9°S that allows an easterly to form therein that is strong enough to stop the jets from displacing it.

The nature of the low-latitude instability is more clearly apparent when the W_{1} jet exists alone in either the thin Jovian or thick terrestrial system. For Earth, the type of instability appears to depend primarily on the location of the baroclinic zone, for the novel form mainly occurs when the flow is driven by a baroclinicity that lies in lower latitudes than usual; otherwise, a classic jet and instability occur; see Williams (2003a) for further details. For midlatitude jets, the eddy momentum flux *u*′*υ*′

Although both structural systems fail to produce any real easterly jets (except at the equator), such flows can be generated quite easily, even for the elementary *P*_{2}(*ϕ*) source of heat, by allowing the heat to penetrate to (or originate at) depths that increase monotonically with latitude. A factor of 2 in depth between the equator and 45°S suffices to reverse the baroclinicity in lower latitudes and split the hemispheric flow into broad easterly and westerly currents; these currents can also be modified by small-scale baroclinic instabilities that generate minijets within them.

Theoretically, the dynamics of the multiple jets in thin systems is not well understood, particularly as regards the nonlinear processes that determine the jet scales. Given the weakness of the barotropic component, the Rhines *β* turbulence mechanism remains relevant only if viewed from the Held–Larichev perspective. Neither are the standard midlatitude and novel low-latitude forms of baroclinic instability understood for multiple jets in thin or thick layers, especially for the nonlinear phases that affect the jet formation and maintenance.

Turning to the planets, the solutions have some bearing on attempts to define the three-dimensional character of the flows on Jupiter and Saturn, within the limits set by the Boussinesq assumption. In particular, they indicate that multiple jets with realistic features can be generated from a *global*-scale heating that is confined to a thin vertical layer, a structure that also favors vortex formation (Part I). An extra baroclinicity component in low latitudes, however, helps make the flows more realistic in the Tropics and at the equator; either latent heating or albedo variations could provide such a component. However, the absence of the baroclinic easterly jets needed to generate large vortices such as the Great Red Spot suggests that once the *P*_{2}(*ϕ*) heating has created the westerly jets, the flow needs to produce stronger easterlies by activating, possibly through the albedo or latent heating, a sinusoidal *P*_{10}(*ϕ*) ∼ cos(5*ϕ*) heating mode commensurate with the scale of the five jets.

To do this we envision a heating profile of the form *P*(*ϕ*) = {*P*_{2}[1 − *t*)] + *P*_{10}*t*)}, where *t*) = [1 − exp(−(*t*/*t*_{0})^{4})], and *t*_{0} is a timescale long enough to let the original jets form and set the latitudinal scale before the jet-dependent albedo or latent heat variations become influential. The possibility of albedo-controlled circulations is noted by Green (1999, section 12.7). However, the scarcity of vortices in Jupiter's Northern Hemisphere undermines the need for such a complete *P*_{2} → *P*_{10} transition in the heating. Elsewhere, the hexagonal jet seen on Saturn could indicate the presence of a nonlinear baroclinic instability in a thin dynamical layer; while the easterly current on Neptune could be due to the heated layer becoming thicker with increasing latitude.

So is the baroclinic thin-layer hypothesis adequate for theorizing about the Jovian atmospheres or does it need modification? So far, the hypothesis has mainly been studied using a variety of standard dynamical models so the conditions under which it gives relevant solutions are well defined. But the need for baroclinic easterlies implies that physical processes must be included and these may introduce ambiguities. However, other representations of the thin layer, other parameter ranges, or other heat sources may be less constrained. On the other hand, the limitations may be real and indicative of a need to extend or even replace the basic hypothesis. There may be many ways of producing multiple jets. Clearly, we have barely begun to approach the subtleties of the Jovian system.

## Acknowledgments

Special thanks go to Catherine Raphael for organizing the graphics. I am also greatly indebted to Robert Hallberg, Isaac Held, Conway Leovy, and the reviewers for perceptive comments that helped improve this presentation, while discussions with Shafer Smith improved its content.

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Basic heating parameters for the functions defined in section 2c and Fig. 2, for the jet-genesis cases with the LIN vertical structure. The L6 case has a longer 180° domain, plus a Gaussian cooling dip, written as *g*(−10°, 4°, −0.25) to denote, its latitudinal center, half-width, and amplitude relative to unity

Basic heating parameters for the functions defined in section 2c and Fig. 2, for the jet-genesis cases with the EXP vertical structure. The vertical confinement rate *N* equals 200 in all cases. The E5 and E6 cases have a Gaussian cooling dip, written as *g*(−12°, 4°, −0.1) to denote its latitudinal center, half-width, and amplitude relative to unity

Basic heating parameters for the functions defined in section 2c and Fig. 2, for the cases relating to (i) the equatorial superrotation on Jupiter in Q1 and on Earth in Q2 and Q3; (ii) planets with atmospheres of variable depth in P1 and P2; and (iii) planets with hexagonal flows in P3. The P3 case has a Gaussian cooling dip, written as *g*(−9°, 5°, −0.25) to denote its latitudinal center, half-width, and amplitude relative to unity. Earth cases have τ = 20 days, τ_{D} = 1 day, a thick LIN2 structure with *S* (*z*) = 1 and *T _{s}*(

*z*) = δ

*T z*′, and diffusion coefficients ν

_{4}= −10

^{15}m

^{4}s

^{−1}, with ν

_{2}= 0 for Q2 and ν

_{2}= 0.2 m

^{2}s

^{−1}for Q3. (a) For Q2 the heating profile is normalized over 70° of latitude. Cases P1 and P2 have the special

*N*(ϕ) structure defined in (10)

^{1}

The model fails, however, to represent the dynamics of the upper atmosphere or the deepest layers but these lie beyond the intended scope of this study.

^{2}

A day equals 86 400 s in this paper.

^{3}

This was verified by comparison with calculations using uniform or exponential grids.

^{4}

By *barotropic* or *baroclinic* zones, we mean regions with latitudinally uniform or latitudinally linear heating distributions, respectively.

^{5}

By *thin* or *thick* layers, we mean layers for which the motion is either confined aloft or extends over the entire fluid.

^{7}

The E1–E6 potential vorticity distributions are similar to the EXP form seen in Fig. 6 of Part II, but with the stationary dips now replaced by equatorward-moving (escalator) steps.

^{10}

Note that, after 10 000 days, W_{1} > W_{0} in Fig. 15a when the velocities are sampled at all heights, while W_{1} ≃ W_{0} in Fig. 15c when sampled at a fixed height.

^{11}

The *P*_{2} profile is normalized over 80°.