a. System of equations

The numerical model is based on the dynamical core of the Geophysical Fluid Dynamics Laboratory's spectral GCM and is driven by a simple heating function, along the lines devised byHeld and Suarez (1994). The primitive equations have the standard hydrostatic, vorticity–divergence form that is preferred for the semi-implicit, spectral transform scheme as summarized byGordon and Stern (1982). The model predicts the zonal, meridional, and vertical velocity components (u,υ,ω), plus the temperature and surface pressure fields (T,p_*), as a function of the latitude, longitude, and sigma coordinates (ϕ, Λ,σ), whereσ =p/p_* is the normalized pressure. The variableψ(ϕ,σ) = −∫υ cosϕ dσ defines a quasi-streamfunction for the zonally averaged meridional motion.

As well as a heating function, the equations include ∇⁸ diffusion terms³ in the horizontal and, in the vertical, a linear boundary layer drag of the form

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wherek⁻¹_V andσ_b define the time scale and the extent of the mixing. Topography, moisture, vertical diffusion, and convective adjustment are all omitted. The numerical procedure uses a triangular truncation at wavenumber 42 in the horizontal with 30 equally spacedσ levels in the vertical.

b. Heating function

All flows are developed from an isothermal state of rest and are maintained by a Newtonian heating function of the form

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where the heating rate is proportional to the difference between the atmospheric temperature and a specified radiative relaxation temperatureT_r with a relaxation damping ratek_T(ϕ,σ). The following distribution

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provides the heating, whereT₀ andT_s are the tropospheric and stratospheric reference temperatures, respectively. The constants Δ_HT and Δ_VΘ define the amplitude of the horizontal temperature and vertical potential temperature gradients. Note that, unlikeHeld and Suarez (1994), Δ_VΘ is not modified by a function of latitude. The resulting constant static stability eliminates small-scale convection at all latitudes, leading to cleaner solutions. In addition, the baroclinic zone is located in low latitudes by using a high value for then power in(4).

c. Parameter values

The fixed physical parameters needed in the calculations use the following standard values:a = 6370 × 10³ m and Ω = 7.292 × 10⁻⁵ s⁻¹ for the planetary radius and rotation rate;g = 9.8 m s⁻² for the acceleration of gravity;c_p = 1004 J kg⁻¹ K⁻¹ for the specific heat of air;κ ≡R/c_p = 2/7, whereR is the gas constant;p₀ = 1000 mb for a mean surface pressure based on the total massp₀/g; andT₀ = 315 K andT_s = 150 K for the reference temperatures. The other parameters areσ_b = 0.7, (k_f ,k_T) = (1, 1/40) day⁻¹, Δ_HT = 40 K, and Δ_VΘ = (10, 20, 40, 60, 80, 100) K.

Only two cases, A and B, are presented for systems with low and high static stabilities, as given by the two values of Δ_VΘ = 10 and 80 K, respectively. Other cases with Δ_VΘ values lying between these two extremes exhibit intermediate characteristics, so are not presented. Equilibration takes longer for low-latitude jets and for higher static stabilities, so these two cases have to be extended to 3000 and 9000 days, respectively.⁴ SettingT_s = 150 K gives a minimal stratosphere, which helps to give simpler circulations—particularly eddy momentum fluxes. To ensure that all flows lie well within the superrotation regime, then power is set at 32; seeWilliams (2003c) for a discussion of the regime dependence on this parameter.

In presenting the solutions, the fields shown are time-averaged quantities, based on zonal means sampled once a day over the last 200 days of the calculations.⁵ For brevity and clarity, the main jet and equatorial westerly are sometimes referred to using theW₁ andW₀ symbols, respectively.

d. Analysis functions

The solutions are described using standard analysis procedures and notation, with the overbar and prime denoting the zonal mean and eddies. In particular, the Eliassen–Palm (EP) flux vectorF = {F^(ϕ),F^(p)} and its scaling, together with the flux divergenceE, are defined followingEdmon et al. (1980) as

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for the dominant geostrophic components, wheref = 2Ω sinϕ. Additional analysis with the ageostrophic terms included in(5) and(6) revealed little difference, so the terms are omitted for simplicity.

The mean quasigeostrophic potential vorticity gradient is defined, again followingEdmon et al. [1980, Eq. (3.8)], on normalizing by 2Ω, as

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whereζ = −(a cosϕ)⁻¹(u cosϕ)_ϕ is the mean relative vorticity. The condition for inertial instability

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is also calculated, but such an instability is never realized. Although the quasigeostrophic approximation is less accurate in low latitudes, the diagnostics remain useful as the barotropic component and planetary wave fluxes dominate near the equator.

a. Superrotation at low static stability

Case A, for Δ_VΘ = 10 K inFig. 1, takes 2000 days to equilibrate to a jet withW₁ = 100 m s⁻¹ and a strong equatorial westerly withW₀ = 69 m s⁻¹. The solution differs from its predecessor (case B inWilliams 2003c) due to differences in the values of the static stability and baroclinicity, the use of a ∇⁸ horizontal diffusion, and the presence of a shallower stratosphere. The Hadley cell is particularly novel in that it splits into two vertical components, thereby altering the influence of the surface drag on the zonal flow. Otherwise, the Ferrel cell and a second direct cell at 40° latitude coincide with the baroclinic instability as usual.

The poleward eddy heat transport, produced by the baroclinic instability, extends from the equator to 40° latitude while being confined to the lower atmosphere and peaking atϕ = 20° andσ = 0.8 (Fig. 1d). The eddy momentum transport,, is again made up of three primary components per hemisphere, consisting of a deep equatorward flux on the jet's poleward flank, a midlevel equatorward flux on the jet's equatorward flank, and a weak split poleward flux in between (Fig. 1e). According to the cospectra (inFig. 5), these components are associated with both baroclinic and barotropic instabilities, but at different wavenumbers—seesection 3d. The midlevel equatorward flux, extending overϕ = 0°–20° andσ = 0.2–0.7, acts to produce and maintain the equatorial superrotation. The poleward flux, however, intervenes less than in the predecessor case, and this results in the simpler form being sought. In addition, according to the energy field, the eddies are strongest on the jet's poleward flank at the tropopause.

b. Superrotation at high static stability

The character of the case B flow for a high static stability (Δ_VΘ = 80 K) changes with time, so we document both the early growth phase at 1000 days and the equilibrated phase at 9000 days (Figs. 2 and3). This system takes much longer (8000 days) to equilibrate, presumably because of the absence of any vigorous mixing by the baroclinic instability. Between 1000 and 9000 days theW₁ andW₀ westerlies go from 108 and 51 m s⁻¹ to 241 and 232 m s⁻¹, respectively, so the latitudinal jet profile becomes much flatter with time.

At 1000 days, the barotropic instability is clearly responsible for the equatorial westerly as the baroclinic instability, measured by the poleward flux, is very weak,Fig. 2d. The equatorward, countergradient flux in the middle and upper troposphere is significant and forced by the barotropic instability. The meridional circulation is relatively simple and consists of just a Hadley cell, though one that has two vertical components. The flux is also simple and equatorward with two strong low-latitude components at the upper and middle levels (σ = 0.15 and 0.45), both associated with the barotropic instability, together with a weaker midlatitude component at low level (σ = 0.7), associated with the weak baroclinic instability (Fig. 2e). The first two, large-scale components, however, occur at zonal wavenumberk = 3 (Fig. 5b), while the third, medium-scale component occurs atk = 9 (not shown). According to the field the strongest eddies now occur right at the equator.

By 9000 days (seeFig. 3), the system has equilibrated with the Hadley cell being further confined to the lower troposphere. The flux component associated with the barotropic instability has weakened and reversed sign, whereas the component associated with the weak baroclinic instability has strengthened. The two upper-level flux components (due to barotropic instability) merge and weaken, while the lower level component (due to baroclinic instability) increases. According to the field, strong large-scale (k = 2, not shown) eddies occur at the equator, while two sets of medium-scale (k = 7, not shown) eddies occur in low latitudes, atϕ = 15° and 30°.

From the growth and equilibration states of case B inFigs. 2 and3, we conclude that the barotropic instability acts to generate and maintain the superrotation. The instability gradually weakens and the zonal flow's latitudinal profile becomes flatter near the equator as equilibrium is approached.

c. Superrotation variability

Calculating additional cases with Δ_VΘ = 20, 40, 60, and 100 K shows that only the first of these cases resembles the low static stability case A in its eddy fluxes; the three other cases more closely resemble the high static stability case B, with the influence of the barotropic instability over the baroclinic instability progressively increasing. As a result, the case with Δ_VΘ = 40 K takes 7000 days to equilibrate and produces zonal flows withW₁ = 196 m s⁻¹ andW₀ = 187 m s⁻¹. Thus the superrotation form seen in case B is quite general and does not need an extreme value of the static stability to be realized, though higher values do help reveal the cause of the equatorial westerly.

d. Superrotation dynamics

All of the above eddy fluxes appear to have forms consistent with the theories describing nonlinear baroclinic instability, barotropic instability, and Rossby wave propagation. Consequently, the EP cross sections inFig. 4 give further insight into how the eddies originate and function in the superrotating flows. According to theory, theE flux divergence provides a measure of the source and magnitude of the transient and irreversible eddy processes, as well as the eddy forcing of the zonal mean circulation. TheF flux vectors give a measure of the wave propagation from one location to another.

To identify the sources of the eddies, we also examine the mean quasigeostrophic potential vorticity gradientq_ϕ as defined in(8). We recall that the equation for the eddy potential enstrophy on a quasigeostrophic beta plane with lateral coordinatey may be written as

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whereE =∇ ·F =, andS represents the nonconservative processes. Wave sources (instabilities) are expected to occur whereq_y has the opposite sign toE.

For case A during the equilibrated phase, the mainE flux divergence and convergence regions (associated with the baroclinic instability and the vertical wave propagation) lie mainly in the lower half of the troposphere (Fig. 4a). TheirF flux vectors are upward with a slight poleward tilt, then turn equatorward at midheight, and gradually descend into the subtropics. In upper levels and at low latitudes, however, the EP fluxes are quite different, with theF flux vectors mainly pointing poleward. A weak but effective divergentE flux forms over the equator in the upper troposphere in keeping with the eddy forcing of theW₀ current. TheF flux vectors are consistent with the action of a tropical instability that generates the large-scale waves that extend aloft from the equator to about 20° latitude. A secondary divergentE flux also forms at the tropopause at 30° and is also associated with poleward pointingF flux vectors.

The EP fields for case B differ in detail but not in effect between the growth and equilibrated phases (Figs. 4b, c); they primarily reflect the changes seen in the fields ofFigs. 2e,3e. At 1000 days, the divergentE flux and the weak polewardF flux vectors associated with the weak baroclinic instability indicate that this process has no direct influence on the dynamics equatorward of 15°. The latter is determined by the two strong divergentE fluxes lying aloft over the equator that produce two distinct groups of polewardF flux vectors. In other words, two barotropic instability sources at the equator produce two sets of poleward propagating Rossby waves. By 9000 days, the two equatorialE fluxes have merged but are still associated with strong polewardF flux vectors, especially at middle levels. In addition, the EP fields affiliated with the baroclinic instability strengthen and indicate that the wave propagation remains poleward.

The quasigeostrophic potential vorticity gradients, defined in(8), have negative values in low latitudes during the growth stages of cases A and B—a necessary but not sufficient condition for barotropic instability—only to be replaced by extensive regions with small or zero values as the flows equilibrate (Figs. 4d–f).

The zonal spectra for the main eddy fluxes inFig. 5 are for the equilibrated phase of case A and the growth phase of case B. For case A, they show that the baroclinic instability occurs at a medium scale (k = 6,Fig. 5a) and the barotropic instability at a large scale (k = 2,Fig. 5b), a clear separation of scales. For case B, the weak baroclinic instability occurs at a smaller scale (k = 9) and a lower level (σ = 0.7, not shown), while the barotropic instability peaks at a smaller scale (k = 3) and an upper level (σ = 0.2),Figs. 5d–f.