1. Introduction
Atmospheric superrotation refers to a local angular momentum maximum in the fluid interior. Because angular momentum must decrease toward the poles for the flow to be inertially stable, atmospheric superrotation usually means equatorial superrotation: that is, a local angular momentum maximum at the equator (Held 1999). Such atmospheric superrotation may be the norm rather than the exception. Venus and Titan have superrotating atmospheres (Schubert 1983; Gierasch et al. 1997; Kostiuk et al. 2001). Jupiter’s and Saturn’s atmospheres also superrotate (Porco et al. 2003; Sánchez-Lavega et al. 2007), but because they do not have a solid surface, they superrotate relative to the rotation of their cores and magnetic fields. For an atmosphere to superrotate, it needs to have angular momentum fluxes into the region of superrotation (Hide 1969). Inviscid axisymmetric circulations cannot accomplish this upgradient angular momentum transport; eddies must be involved (Held and Hou 1980; Schneider 2006). In general, in sufficiently rapidly rotating atmospheres, eddy angular momentum fluxes converge into the regions in which wave activity is generated, and they diverge where wave activity is dissipated (Held 1975; Andrews and Mcintyre 1976; Edmon et al. 1980). Thus, preferential wave activity generation near the equator is a prerequisite for superrotation. This must not be overcompensated by wave activity dissipation near the equator, for example, associated with baroclinic eddies that are generated in midlatitudes and dissipate preferentially in lower latitudes, as they do on Earth (Saravanan 1993).
Various mechanisms are available for preferential wave activity generation near the equator. A stationary heat source near the equator, for example, leads to the generation of stationary Rossby waves, which can dissipate away from the equator and so transport angular momentum toward the equator. This leads to superrotation when the heat source is strong enough (Suarez and Duffy 1992; Saravanan 1993; Kraucunas and Hartmann 2005; Arnold et al. 2012). The stationary wave mechanism is responsible for superrotation in simulations of tidally locked planets, in which stellar heating is radially symmetric around an equatorial focal point (Joshi et al. 1997; Merlis and Schneider 2010; Pierrehumbert 2011). For a planet without deviations from axisymmetry in boundary conditions, it is less obvious why waves should be preferentially generated near the equator. Wang and Mitchell (2014) and Pinto and Mitchell (2014) find that a Rossby–Kelvin instability produces angular momentum flux convergence at the equator that is responsible for the generation of superrotation in statically stable atmospheres. In convecting atmospheres, the variation of the Rossby number with latitude provides an alternative mechanism: Near the equator, where the Rossby number can be O(1), horizontal and temporal temperature variations are small when the Froude number is small. Therefore, fluctuations in convective heating must be balanced by vertical motion and hence by horizontal divergence at the level of the convective outflows in the upper troposphere (Charney 1963; Sobel et al. 2001). The horizontal divergence then can generate large-scale rotational flow and thereby Rossby waves, either by vortex stretching or vorticity advection (Sardeshmukh and Hoskins 1988). In contrast, in higher latitudes, where the Rossby number is small, convective heating fluctuations can, for example, be balanced by transient temperature fluctuations, which may relax radiatively without generating large-scale waves that dissipate in other latitude bands. The net result is preferential generation of Rossby waves near the equator by convective heating fluctuations. If some of these convectively generated Rossby waves dissipate at higher latitudes—for example, through interaction with the mean-flow shear—they will transport angular momentum toward the equator and thus can generate superrotation (Schneider and Liu 2009; Liu and Schneider 2010).
However, angular momentum flux convergence associated with preferential wave activity generation at the equator may be counterbalanced or overcompensated by angular momentum flux divergence associated with dissipation of wave activity that was generated at higher latitudes (e.g., by baroclinic instability) (Saravanan 1993). This is the case in Earth’s troposphere in the annual mean, and it may be the case on Uranus and Neptune, which are subrotating (Liu and Schneider 2010). Only when the baroclinically unstable region is moved into low latitudes by artificially increasing radiative heating gradients near the equator and reducing them in higher latitudes can baroclinic instability promote the onset of superrotation (Williams 2003).
Here, we focus on equatorial superrotation on terrestrial planets: that is, planets with solid surfaces with a distribution of radiative heating rates resembling Earth’s. We explore a wide parameter regime that encompasses subrotating (Earth like) and superrotating atmospheres. Our goal is to elucidate the mechanisms that generate and maintain tropospheric superrotation in convecting atmospheres and quantify the conditions under which superrotation generally arises.1 We quantify the relative importance of the angular momentum fluxes associated with equatorial convectively generated waves and off-equatorial baroclinic eddies. We use simulations with an idealized GCM to demonstrate that whether superrotation occurs in terrestrial atmospheres depends on the competition between the two, and we use scaling arguments to estimate their relative importance in terms of mean-flow quantities and external parameters.
2. Idealized GCM and simulations
The idealized GCM used for the simulations is based on the dynamical core of the Geophysical Fluid Dynamics Laboratory’s Flexible Modeling System. It performs a time integration of the primitive equations of motion on a sphere with Earth’s radius, using the spectral transform method in the horizontal and using 30 σ levels in the vertical. Here, 
Neither seasonal nor diurnal cycles of insolation are included in this model, and there is no topography. The GCM treats the atmosphere as an ideal gas without a hydrologic cycle. The effects of moisture are generally ignored, but they are implicit in a convection parameterization, which relaxes atmospheric temperatures to a profile with lapse rate equal to a fraction 
We performed 60 simulations by varying three model parameters: the pole–equator temperature contrast in radiative equilibrium 
Parameters varied in the 60 simulations: planetary rotation rate 
All simulations were integrated for at least 1500 days, and the model output is averaged over the last 400 days of each simulation.
3. Results
a. Circulation variations
Our goal is to determine why some simulations superrotate and some do not. Figure 1 displays two superrotating and two subrotating simulations. The left column shows the eddy angular momentum flux divergence (colors) and zonal wind (black contours), and the right column shows the mass flux streamfunction, with solid lines for counterclockwise rotation and dotted lines for clockwise rotation. The top row shows an Earth-like reference simulation, with Earth’s rotation rate 
(left) Zonal-mean zonal wind (black contours; contour interval: 5 m s−1) and eddy angular momentum flux divergence (colors; 10−6 m s−2) in the latitude–sigma plane. The thick black line is the zero zonal-wind contour. (right) Eulerian-mean mass flux streamfunction, with contour intervals given in each panel. Four simulations are shown, with their parameter values indicated in the left column.
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0030.1
The simulation in the second row has the same Earth-like parameter values for convective lapse rate and pole–equator temperature contrast but a planetary rotation rate 
The simulation in the third row is one with Earth’s parameter values, except for a smaller, more stable convective lapse rate 
The simulation in the bottom row differs from the simulation above it only in the reduced pole–equator radiative-equilibrium temperature contrast of 
All other parameters held constant, a decrease in the planetary rotation rate generally leads to an increase in the average equatorial wind speed (Fig. 2). There are deviations from this behavior that occur at the lowest rotation rates, when the midlatitude westerly jets migrate toward the poles, as was already seen in simulations by Del Genio and Zhou (1996).
Upper-tropospheric zonal wind at the equator vs 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0030.1
For faster planetary rotation rates, the equatorial winds become more westerly with decreasing 
b. Relation to wave activity sources
Figure 3 displays all 59 simulations as a function of 
All simulations shown as a function of 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0030.1
The simulations that lie to the right of the one-to-one line (i.e., with 
There are two strongly superrotating simulations that lie on the left side of the line and for which the scalings do not work well in the following figures. These simulations both have 
4. Scaling theory
To understand more completely under which conditions superrotation arises, we develop a scaling theory for the wave activity generation 
a. Equatorial wave activity generation
To relate 
Convective heating fluctuations vs mean convective heating.
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0030.1
Root-mean-square equatorial divergence fluctuations 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0030.1
To obtain a scaling for the wave activity generation, it remains to find a scaling for the enstrophy source 
Variance of convective heating fluctuations 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0030.1
Equatorial wave activity source 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0030.1
b. Baroclinic angular momentum flux divergence
Eddy angular momentum flux divergence 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0030.1
5. Discussion
a. Mechanisms and origin of parameter dependences
Although idealized, the simulations performed here have the basic ingredients to produce superrotation in terrestrial atmospheres. The most important quantities controlling whether superrotation occurs or not in our simulations are the planetary rotation rate and the meridional temperature gradient, with, for example, the static stability in the tropics and extratropics playing secondary roles.
The dependence of 
Thus, for faster rotation rates, the baroclinic eddy angular momentum flux divergence increases with rotation rate primarily because eddies get smaller and the angular momentum fluxes become more concentrated in narrower baroclinic zones. Increases in meridional temperature contrasts because of reduced efficiency of poleward energy transport also modify the baroclinic eddy angular momentum flux divergence (Schneider and Walker 2008). By contrast, the equatorial wave activity generation decreases with rotation rate both because the equatorial Rossby radius decreases and the convective heating fluctuations and divergence fluctuations weaken. For our simulations, the equatorial wave activity generation effect dominates and leads to 
The dependence of 
To look at the equatorial wave structures responsible for the generation of superrotation more closely, Fig. 9 shows the correlation coefficient between equatorial divergence fluctuations at a reference point and horizontal streamfunction and wind fluctuations at 300 hPa for two superrotating simulations, one with half Earth’s rotation rate (
Correlation coefficients between divergence fluctuations at a reference point on the equator (black dot) and horizontal streamfunction fluctuations (colors) and wind (arrows) at 300 hPa. (a) Superrotating simulation with 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0030.1
The correlations between wind fluctuations and divergence fluctuations at the equator (tilt of arrows) indicate angular momentum transport toward the equator. They also show that no more than the usual tilt of phase lines with latitude is needed to generate the angular momentum transport. It is not necessary to see outright meridional propagation of wave packets for angular momentum transport to occur. It suffices to have a meridional group velocity away from the equator, as indicated by the tilt of phase lines, accompanied by dissipation of waves preferentially in the off-equatorial wings: for example, by critical-layer rollup or shearing by the mean flow (e.g., Farrell 1987; Lindzen 1988; Huang and Robinson 1998; O’Gorman and Schneider 2007; Ait-Chaalal and Schneider 2015). This overall picture is consistent with our theoretical considerations, which assign primary importance to equatorial Rossby waves and their equatorward angular momentum transport in the generation of superrotation.
Figure 10 shows the eddy angular momentum flux cospectra (black contours for positive and blue contours for negative fluxes) versus latitude at the level 
Eddy angular momentum flux cospectra vs latitude at 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0030.1
b. Relation to prior work
Our scaling theory and simulation results are consistent with previous simulations of superrotating terrestrial atmospheres. The scaling theory provides a unifying framework within which the generation of superrotation, for example, in slowly rotating (e.g., Del Genio et al. 1993; Del Genio and Zhou 1996; Walker and Schneider 2006) or strongly convective (e.g., Schneider and Liu 2009; Caballero and Huber 2010; Liu and Schneider 2011) atmospheres can be interpreted as arising from a common set of principles. It provides a quantitative criterion to determine when superrotation occurs, which is more generally applicable than previous criteria.
To compare how well 
Equatorial zonal wind in the upper troposphere vs (a) 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0030.1
Simulations in three series are connected with thick green lines in Fig. 11a. These simulations have the same values of γ and 
The nondimensional number 
c. Limitations and possible extensions
Our simulations and scaling theory ignored several factors that are known to affect whether superrotation occurs. For example, we ignored the seasonal cycle. Yet the seasonal cycle decelerates the zonal wind in the equatorial upper troposphere in the annual mean, because a Hadley circulation whose ascending branch is displaced off the equator is associated with equatorial easterlies (e.g., Lindzen and Hou 1988; Lee 1999; Kraucunas and Hartmann 2005; Mitchell et al. 2014). This disfavors superrotation for planets with nonzero obliquities. Additionally, the convective wave activity generation G in the presence of a seasonal cycle would be maximal off the equator for part of the year, decreasing the numerator of 
We also focused on transient sources of equatorial Rossby waves, rather than stationary sources, which in several previous studies have been shown to be able to generate superrotation if they are strong enough (e.g., Suarez and Duffy 1992; Saravanan 1993; Joshi et al. 1997; Kraucunas and Hartmann 2005; Merlis and Schneider 2010; Arnold et al. 2012; Pierrehumbert 2011). Our scaling theory can be extended to take stationary equatorial wave activity generation into account by considering how the stationary divergence perturbation and equatorial wave activity generation scale with parameters controlling the strength of the stationary wave source (e.g., Merlis and Schneider 2011). In particular, it is to be expected that the importance of equatorial wave activity generation owing to stationary heat sources increases in importance relative to baroclinic eddy angular momentum flux divergence as the planetary rotation rate decreases for the same reasons the importance of transient equatorial wave activity generation increases. Thus, we expect that stationary heat sources (e.g., stellar heating focused on a substellar point on tidally locked planets) more easily lead to equatorial superrotation on slowly rotating planets.
6. Conclusions
We have presented simulations and a scaling theory that establish conditions under which superrotation occurs in terrestrial atmospheres. By varying the planetary rotation rate, the pole–equator temperature contrast in radiative equilibrium, and a scaling parameter for the convective lapse rate, we generated a wide range of atmospheric flows, some superrotating and some subrotating.
The theory presented here is based on a simple idea, going back to Saravanan (1993), about two competing sources for eddy angular momentum flux convergence at the equator. The first is a source at the equator: Rossby waves generated by convective heating fluctuations. As these waves dissipate preferentially away from the equator—whether after propagation of wave packets or by preferential dissipation in the off-equatorial wings of the waves (e.g., by shearing through the mean flow)—they converge angular momentum into the equatorial region, increasing the propensity for superrotation. The other source is baroclinic instability in midlatitudes, which generates Rossby waves in midlatitudes that dissipate farther equatorward, thus extracting momentum from lower latitudes. This mechanism decreases the propensity for superrotation. Quantifying the magnitude of the two mechanisms, introducing their nondimensional ratio 
- Superrotation occurs when the eddy angular momentum flux convergence associated with equatorial wave activity generation exceeds eddy angular momentum flux divergence near the equator produced by midlatitude baroclinic eddies (i.e.,
- Superrotation is favored for low planetary rotation rates and/or strong diabatic heating.
- Superrotation is favored when midlatitude baroclinicity is weak.
Our simulations confirm that superrotation is preferred for slowly rotating planets like Venus and Titan, a result that was already obtained by Del Genio et al. (1993) and Del Genio and Zhou (1996) in a similar set of simulations. The scaling arguments presented here help us understand why slowly rotating planets exhibit superrotation. In our simulations, equatorial convective heating fluctuations strengthen with decreasing rotation rate, generating waves that transport momentum upgradient toward the equator, leading to superrotation. Such convective heating fluctuations may play a role on Venus, in the shallow convective layers observed in the upper troposphere (Markiewicz et al. 2007). The equatorial wave activity generation strengthens as the planetary rotation rate decreases primarily because the diabatic heating rate strengthens. Thus, similar arguments may also apply for other mechanisms that increase the diabatic heating or more generally, divergence fluctuations at the equator. For example, stationary or nearly stationary heat sources that generate equatorial Rossby waves have been suggested to play a role on Venus near the subsolar point (Gierasch et al. 1997).
Strengthening equatorial convective heating fluctuations may also explain why some Earth climate models exhibit a transition to superrotation under extreme global warming (e.g., Caballero and Huber 2010). In this scenario, convective heating at the equator strengthens because of greenhouse gas forcing, and meridional temperature gradients decrease because of polar amplification of the warming. Both factors favor superrotation.
Some of the scaling results in this paper were presented at the 18th Conference on Atmospheric and Oceanic Fluid Dynamics in 2011. The research was supported by the U.S. National Science Foundation through Grant AGS-1049201 and a Graduate Research Fellowship.
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Superrotation can also occur in the stratosphere, for example, during the westerly phase of the quasi-biennial oscillation. The mechanisms responsible for that are different from those in the troposphere, which is our focus here.
