The interaction of midlatitude eddies and the thermally driven Hadley circulation is studied using an idealized shallow-water model on the rotating sphere. The contributions of the annually averaged differential heating, vertical advection of momentum from a stationary boundary layer, and the gross effect of eddies, parameterized by Rayleigh damping, including a hemispherically asymmetric damping, are examined at steady state.
The study finds that the relative dominance of eddies, as quantified by the local Rossby number, is predicted by an effective macroturbulent Hadley circulation Prandtl number Pr. In addition, viscous solutions of the Hadley circulation width and strength, subtropical jet amplitude, and equator-to-pole temperature difference scale as deviations from the respective inviscid solutions.
Semianalytic solutions for the steady circulation are derived in the limit of weak eddy dominance (small Pr) as deviations from the respective inviscid solutions. These solutions follow a three-region paradigm: weak temperature gradient at the ascending branch of the Hadley circulation, monotonically decreasing angular momentum at the descending branch, and modified radiative–convective equilibrium at the extratropics. Using the three-region solutions, scaling relations found in the full solutions are reproduced analytically. The weak eddy-dominance solutions diverge from the full solutions as Pr increases and may become invalid for Pr > 1 due to the breakdown of the three-region global circulation structure.
The qualitative predictions of the response of the Hadley circulation to heating based on the weak eddy-dominance solutions and Pr are in agreement with the findings of more complex models and the observed atmosphere.
The Hadley circulation (HC) is a central element of Earth’s climate, transporting heat and momentum poleward and affecting the extent of the wet tropical and dry subtropical regions, and the properties of the midlatitude baroclinically unstable region. Understanding how the extent of the wet and dry regions, the HC strength (HCS), and the properties of the subtropical jets respond to variations of external parameters is therefore critical for understanding Earth’s climate.
Recent studies show fundamental discrepancies between the predictions of comprehensive global circulation models (GCMs) and observations of the HC width and strength (Mitas and Clement 2005; Johanson and Fu 2009). These discrepancies could result from the shortcomings of observational products (Santer et al. 2005), hydrological cycle parameterization (Held and Soden, 2006), or inherent sensitivity due to nonmonotonic dependence on heating (Levine and Schneider 2011). This highlights the need for improved understanding of the dynamics of the HC in the presence of eddies.
Outside the deep tropics, the mean meridional circulation momentum and heat fluxes are dominated by eddies. Eddies, however, are not necessary for the maintenance of the HC. Schneider (1977) has shown that an HC forms in the limit of vanishing viscosity (the so-called nearly inviscid limit) if advection of zonal momentum is not neglected. Following Schneider (1977), Held and Hou (1980, hereafter HH) argued that in the limit of vanishing viscosity, the steady global circulation divides into two regions: a tropical thermally driven HC, in which angular momentum is conserved; and an extratropical region in radiative–convective equilibrium, in which the meridional wind vanishes. By requiring that (i) the heating at the ascending branch is balanced by the cooling at the descending branch of the HC, (ii) the temperature field is continuous, and (iii) zonal winds at the surface are negligible relative to the winds aloft, the extent of the HC and the temperature and velocity fields can be calculated explicitly. For annually averaged heating, their resulting inviscid HC had a similar extent, weaker circulation strength, and stronger subtropical jets than the observed zonally and annually averaged atmosphere. These angular momentum–conserving (AMC) solutions of HH were extended to off-equatorial heating by Lindzen and Hou (1988) and time-dependent forcing by Fang and Tung (1999).
The weak horizontal temperature gradients characterizing the tropics can also be used as a simplifying assumption in dynamical models of the HC (e.g., Sobel et al. 2001). Satoh (1994) and Fang and Tung (1996) have shown that the HC width can be determined as the extent of the region in which temperature may be assumed uniform. The physical motivation for this assumption is that if convection is assumed to occur at a thin intertropical convergence zone, then the temperature at this zone constrains the temperature at the rest of the cell in the inviscid limit (cf. Emanuel et al. 1994).
Axisymmetric nearly inviscid theory poorly predicts the atmospheric circulation outside the deep tropics, both in observations and in eddy-permitting models (e.g., Schneider 2006). Existing eddy–mean flow interaction theory on the global scale is mostly restricted to sophisticated, highly parameterized models. A simple theory of the general circulation in which eddies and the thermal driving of the circulation play comparable roles in determining the properties of the HC does not exist (cf. Schneider 2006; Pfeffer 1981; Becker et al. 1997; Kim and Lee 2001).
Held (2000) suggested a simple principle that determines the HC width in the presence of eddies: the HC ends where the thermally driven jet either first becomes unstable or reaches the latitude predicted in the nearly inviscid limit. To obtain an explicit scaling relation for the HC boundary, which requires an explicit solution for the tropical region, Held assumed angular momentum conservation, even though this assumption does not hold in the presence of eddies. Despite this inconsistency, there is some evidence that the edge of the HC scales as Held suggested in GCMs of varying complexity (e.g., Lu et al. 2007; Walker and Schneider 2006; Kang and Lu 2012).
Shallow-water models (SWMs) provide a simple platform for modeling the upper branch of the HC (e.g., Polvani and Sobel 2002 and references herein). Indeed, the angular momentum–conserving solutions of HH and Lindzen and Hou (1988), derived for the upper branch of a Boussinesq fluid on the rotating sphere, are identical to solutions derived using a shallow-water model on the rotating sphere (Adam and Paldor 2009, 2010a). Similarly, Polvani and Sobel (2002) derived solutions of the upper branch of the HC based on a shallow-water model by assuming a weak temperature gradient throughout the HC. Unlike Satoh (1994) and Fang and Tung (1996), however, in which both angular momentum and temperature are assumed uniform, the weak temperature gradient solutions of Polvani and Sobel (2002) do not require the conservation of angular momentum and thus permit parameterized eddy damping. While neglecting the temperature gradient outside the ascending branch of the HC is not strictly justifiable, Polvani and Sobel (2002) showed that the weak temperature gradient solutions are in good agreement with the steady states of their model. More importantly, the assumption of a weak temperature gradient holds in the presence of eddies, making solutions based on this assumption unique, in that they represent the only known case in which semianalytic solutions of the HC width and strength can be obtained with parameterized eddy damping.
Inspection of the numerical solutions of the model employed by HH, Lindzen and Hou (1988), and Fang and Tung (1999) (Boussinesq fluid allowed to vary in latitude and height on a rotating sphere, bounded by a rigid lid) reveals that the ascending and descending branches of the HC differ considerably. As shown in Fig. 1, adapted from HH, even at the lowest viscosity for which stable solutions were obtained, angular momentum is conserved at the subtropics but is not conserved at the deep tropics. As suggested by HH and Lindzen and Hou (1988), strong vertical mixing prevents the conservation of angular momentum at the ascending branch of the HC. Schneider (1984, 1987) suggested that explicit inclusion of vertical momentum fluxes may be considered as a different nearly inviscid limit in better agreement with observations.
Adam and Paldor (2009) noted that while vertical advection of momentum (VAM) prevents angular momentum conservation in the ascending branch of the HC, a weak temperature gradient assumption is valid in this region. Vertical advection of momentum, however, does not affect the descending branch of the HC in the inviscid case, allowing for conservation of angular momentum there. Thus, they suggested a three-region paradigm of the mean meridional circulation for the steady, inviscid circulation: weak temperature gradient at the ascending branch of the HC, angular momentum conservation at the descending branch, and radiative–convective equilibrium at the extratropics. Adam and Paldor (2009) derived semianalytic solutions for the axisymmetric upper branch of the HC in the presence of vertical advection of momentum from a stationary boundary layer, which reproduce the angular momentum profile shown in Fig. 1.
Support for a three-region paradigm is found in the zonally and annually averaged atmosphere. The annually and zonally averaged upper troposphere can be described as being barotropic at the ascending branch of the HC and polar regions, and baroclinic at midlatitudes. Figure 2 shows the dependence on latitude of the vertical average between the 500- and 200-mb levels of the annually and zonally averaged temperature (solid), meridional wind (dashed), and vertical wind (dotted). The annually averaged ascending branch of the HC is thus very clearly observed as a positive vertical velocity (w) region, bounded by the maxima in the poleward meridional wind and characterized by a weak temperature gradient.
Thus, while inviscid models poorly predict the properties of simulated and observed HC, a three-region approach may still be justified in studying the macroturbulent HC. In the limit of weak macroturbulence, a weak-temperature-gradient approximation is still valid at the ascending branch of the HC, while angular momentum conservation at the descending branch of the HC and radiative–convective equilibrium at the extratropics are modified relative to their respective inviscid states.
In this work we seek to study general aspects of the effect of eddies on the HC by (i) assuming a three-region structure of the mean meridional circulation; (ii) using simplified eddy parameterization; (iii) comparing and relating the inviscid and viscous (i.e., eddy permitting) steady states of the simplified model; and (iv) establishing an analytic framework for deriving semianalytic solutions of the steady states of the viscous system, which is applicable to general forms of eddy parameterization. We therefore employ a shallow-water model in which vertical advection of momentum and eddy momentum flux divergence are given simplified parameterizations.
In section 2 we present the model. In section 3 we review the known inviscid solutions of the model. In section 4 we report the results of a wide parameter sweep of the steady states at long times of the time-dependent model and compare these steady states with the respective inviscid solutions. In section 5 we derive semianalytic solutions for the steady viscous states as small deviations from the inviscid solutions, followed by a summary and discussion in section 6.
2. Model equations
To model the upper branch of the HC, we employ a differentially heated SWM on the rotating sphere, with vertical advection of momentum from a stationary boundary layer and Rayleigh damping. This model differs from the one described in Adam and Paldor (2010b) and Shell and Held (2004) only in its representation, which better serves the semianalytic solutions presented in section 5. A similar model is discussed in Held and Hoskins (1985). For clarity, the model representation is derived below.
The shallow-water equations (e.g., Gill 1982) on the spherical Earth [with radius a, and latitude (φ) and longitude (λ) coordinates] rotating about its polar axis (frequency Ω but no centrifugal acceleration) with a source of mass (height) Q, vertical advection of momentum from a stationary bottom layer and Rayleigh damping are given by
where denotes the material (total, Lagrangian) derivative in spherical coordinates; g is the gravitational constant; r is the Rayleigh coefficient of damping; u and υ denote the longitudinal and meridional velocity components, respectively; and h denotes the thickness of the free troposphere. The source is given as Newtonian relaxation to a prescribed radiative–convective equilibrium height hf as follows:
Here, τ is a constant atmospheric thermal relaxation time (typically 2–3 weeks), H0 is the tropospheric height scale (~10 km), ΔH, the equator-to-pole temperature (height) difference (EPTD ~ 3 km) at radiative–convective equilibrium, and φ0 is the latitude of maximal heating. This prescribed form of hf follows the upper branch of the radiative forcing used by HH for heating centered at the equator (φ0 = 0, equinoctial case) and Lindzen and Hou (1988) for off-equatorial heating (φ0 ≠ 0). Following Schneider (1987), the term hf/τ is ascribed to the contributions of solar and latent convective heating, and −h/τ is ascribed to longwave radiative cooling.
The heaviside function I is given by
so that momentum is conserved locally when mass from the stationary bottom layer enters the upper layer.
Equations (1a)–(1f) are nondimensionalized using the following scales: u and υ on Ωa, t, τ; 1/r on 1/Ω; and h and hf on H0. Absolute angular momentum, , is scaled by . To simplify the subsequent analysis, we make the following substitutions in system (1): M for u, V = υ cosφ for υ, and μ = sinφ for φ. Axial symmetry is introduced by setting the longitudinal derivative equal to zero. Using (·)′ to denote , these substitutions yield the following nondimensional system:
is the planetary Burger’s number with a typical value of 0.45. Since g, Ω and a may be assumed fixed in Earth’s atmosphere, variations in α are attributable to H0, which we also interpret as the mean global temperature. Similarly, we attribute Δh to the differential heating of the sun.
We interpret r as modeling the gross effect of eddies on the HC. Figure 3 shows a long-term annual mean of the zonally and vertically averaged U = u cos(φ) (solid) and sum of transient and stationary eddy momentum flux convergence (EMFC, dashed; cf. Fig. 1 in Lee 1999). Eddy parameterization using Rayleigh damping is therefore in agreement with the observed poleward momentum flux at midlatitudes but in disagreement with the observed equatorward momentum fluxes in the deep tropics and polar regions. From Fig. 3, we find that the typical value of U/EMFC is on the order of 1000 days, which is much longer than the typical eddy–mean flow response time of 1 week at midlatitudes (Lorenz and Hartmann 2001). This suggests that r represents the damping rate of the entire tropical circulation, which this model implies is much longer.
Eddy parameterizations more complex than Rayleigh damping have been shown to capture some of the basic features of seasonal climatology (e.g., Schneider 1984, Sobel and Schneider 2009). In this work we neglect the momentum fluxes associate with equatorial waves and consider only the effect of midlatitude momentum fluxes on the HC. However, since EMFC in the deep tropics and polar regions is small relative to EMFC at midlatitudes, and since this work is of a qualitative nature, we justify the parameterization of EMFC using simple Rayleigh damping for its mathematical and conceptual simplicity (e.g., Shell and Held 2004; Polvani and Sobel 2002).
3. Inviscid steady states
In this section we briefly describe the known semianalytic solutions of the inviscid SWM for the case of annually averaged (equinoctial) heating (μ0 = 0) with vertical advection of momentum (Adam and Paldor 2009) and without it (HH). We will refer to these known solutions in the following sections, and in particular to their predictions of the HC strength and width.
a. AMC solutions
For the case of the inviscid SWM (r = 0) at steady state, without vertical advection of momentum (I = 0), Eq. (2b) becomes
so that the SWM has only two steady states: uniform-M steady state, given by
where and h0 are constants; and a radiative–convective equilibrium (V, Q = 0) steady state, given by
Since neither radiative–convective equilibrium nor uniform M are physically realistic as global steady states, the circulation divides into a tropical uniform-M region and an extratropical radiative–convective equilibrium region (cf. HH; Adam and Paldor 2009, 2010a).
Requiring conservation of mass and continuity of h, and assuming that the zonal velocity vanishes at the equator, the latitude separating the two regions and marking the polar boundary of the HC (μH) is obtained by the roots of
A small angle approximation then yields
where RT = αΔh is the thermal Rossby number. These are the well-known angular momentum–conserving solutions of HH, in which due to Hide’s theorem, (i.e., u vanishes at the equator; cf. HH; Adam and Paldor 2009).
Since Q = (hf − h)/τ changes sign at the polar boundary of the heated region (ascending branch), from Eq. (4) we find that at this latitude, denoted as μA, Vh reaches its maximum. Using Eq. (4) and the continuity of h, we get
We refer to the value of the mass flux at this latitude as the HCS, defined as
While some qualitative differences exist between alternative definitions of the HC strength (e.g., Levine and Schneider 2011), we find the above-mentioned form suitable for this study.
b. Inviscid model with VAM
The steady states of the SWM for the case of the inviscid (r = 0) model with vertical advection of momentum follow a three-region paradigm where each region is unique to sign(Q): a radiative–convective equilibrium (Q = 0) extratropical region given by Eq. (5), a subtropical uniform-M cooling (Q < 0, descending HC branch) region given by Eq. (4), and a tropical heating (Q > 0, ascending HC branch) region characterized by a uniform height (i.e., weak temperature gradient), given by
Requiring conservation of mass and continuity of h, μA and μH are found by the roots of
From Eq. (10a), a small angle approximation yields
Note that the vertical advection of momentum solutions of Adam and Paldor (2009) assume a weak temperature gradient only at the ascending branch of the HC, and not throughout the entire HC as was done by Polvani and Sobel (2002). In addition, the main difference of the vertical advection of momentum solutions from those of Polvani and Sobel (2002) and HH is that they offer a dynamic definition of the tropics, in agreement with the observed zonally averaged atmosphere, that is, the equatorial ascent region, characterized by a weak temperature gradient and bounded by the latitude of maximal poleward mass flux.
Since Vh is maximal at μA, and since we also have , from Eq. (9b) the HC strength is given by
4. Steady states of the viscous (r > 0) SWM
To study the effect of eddies, as parameterized by Rayleigh damping r, we begin by examining the steady states obtained by long time integrations of Eq. (2). Time integrations were performed using a staggered-grid (where h is staggered from V and M) leapfrog scheme with a Robert–Asselin time filter (temporal damping coefficient = 0.001), as described in Adam and Paldor (2009). A spatial resolution of 1000 meridional grid points, evenly spaced in μ, was used in the results shown below. This translates into a resolution of 0.01, 0.015, and 0.05° of latitude in the tropics, midlatitudes, and polar regions, respectively.
Taking advantage of the simplicity of the SWM, we performed a numerical parameter sweep (~1000 model runs) of the steady solutions at long times of the time-dependent equations of the SWM for a wide range of parameter values. It is readily shown that if t is scaled by the thermal relaxation time τ, Eq. (2) can be represented by an effective Rayleigh coefficient of damping rτ [cf. Adam and Paldor (2010b) and Fang and Tung (1997) for the scaling of τ in the inviscid case]. It is therefore the product of r and τ (i.e., ratio of thermal and effective eddy-damping time scales) that affects solutions at steady state. This will also become evident in the subsequent analysis. We therefore performed a parameter sweep for the parameter ranges: α = 0.15–1.7, Δh = 2/15–2/3 (RT = 0.1–0.225), and rτ = 0.001–1. These parameter ranges represent the physically realistic extremities of the model parameters α and Δh for Earth, and a wider range of rτ values.
Figure 4 presents the steady states for the zonal wind (Fig. 4a), meridional wind (Fig. 4b), and height (Fig. 4c) at long times for α = 0.45, Δh = ⅓ (RT = 0.15), and rτ = 0 (VAM solutions), 0.005, 0.05, and 0.5 (cf. Figs. 5 and 7 in HH).
Several robust features of the steady states of the SWM emerge from the wide parameter sweep, which we also derive analytically in the following section.
The local Rossby number, defined as
(where ζ is relative vorticity and f is the Coriolis parameter), can be perceived as a measure of the degree to which angular momentum is conserved meridionally (e.g., Schneider 2006). Since in the inviscid case angular momentum is conserved in the descending branch (where Q < 0), the value of Ro may be used to quantify the dominance of eddy momentum fluxes there. As shown in Fig. 5, Ro approximately depends on a single parameter, which represents the competition between the differential heating of the sun (which drives the HC and subtropical jets) and eddy momentum fluxes (which feed on the energy of the subtropical jets). This ratio, denoted by Pr, can be thought of as an effective HC Prandtl number, which relates eddy and thermal driving of the HC on the rotating sphere, is defined as
Shown as dots are Ro values calculated at μA for the entire parameter sweep (other representations of Ro, e.g., the mean of Ro over the entire descending branch region, yield similar dependence on Pr). A clear separation of the steady states into three regimes is observed:
Weak eddy dominance: Ro ~ 1, Pr → 0
Intermediate eddy dominance: 0 < Ro < 1, Pr ~ 1
Strong eddy dominance: Ro ~ 0, Pr ≫ 1
The steady states shown in Fig. 4 for rτ = 0, 0.005, 0.05, and 0.5 correspond to Pr = 0, 0.02, 0.1, 1, and 10, respectively.
The dependence of Ro on Pr shown in Fig. 5 predicts that Ro increases (i.e., eddy dominance decreases) in response to mean (α) and differential (Δh) heating, as well as in response to a decreased rotation rate Ω. We further elaborate on the agreement of this dependence with existing theory and observations in the discussion (section 6).
We find that steady-state variables for r > 0 generally do not scale with Pr, as shown in Fig. 6 for the HC strength, for example. The different branches seen in Fig. 6 correspond to different values of r, with the HC strength increasing with r. The HC strength increases with within branches, as in the inviscid case. Accordingly, we find that when the HC strength and width are scaled by their respective value in the inviscid case (vertical advection of momentum solutions for r = 0 and same α, Δh, and τ), a dependence on Pr very similar to that observed for Ro (Fig. 5) emerges.
The scaling of the HC strength in the inviscid case, for VAM (solid) and AMC (dashed) solutions, is shown in the left panel of Fig. 7 for the parameter ranges used in the parameter sweep. The dependence on Pr of the inverse of the HC strength, scaled by its respective inverse inviscid value (hereafter referring to VAM solutions) is shown in the right panel of Fig. 7. The similarity of the dependence of the scaled HC strength and Ro on Pr implies that it is the deviation from inviscid solution that scales with Pr, and that this deviation from inviscid solutions is closely associated with the level of eddy dominance.
A similar dependence on Pr is found for μA and μH. While in the inviscid case, μH denotes the latitude of both HC width and subtropical jet, this is no longer the case in the viscous case. If we assume a three-region structure at steady state (ascending branch, descending branch, and modified radiative–convective equilibrium regions), then we expect a transition region from the descending branch to a modified radiative–convective equilibrium poleward of the latitude of the subtropical jet μSTJ [latitude of max(U)]. We therefore define μH as the latitude where the second derivative of the mass flux is maximal, poleward of the latitude of the subtropical jet: . This definition is also valid in the inviscid case, where μH = μSTJ. The left panels of Fig. 8 show the dependence of μH (top) and μA (bottom) on the RT, in the inviscid case for VAM (solid) and AMC (dashed) solutions. The right panels of Fig. 8 show the inverse of μH (top) and μA (bottom), scaled by their respective inverse values in the inviscid case. Steady states for which μH = 1, indicating that the assumption of a three-region structure is not valid, are removed from the top panel. Similar to the HC strength, the resemblance of the dependence on Pr of the scaled inverse of μH and μA and Ro (Fig. 5) allows for associating the deviation from the inviscid solution with the level of eddy dominance. A similar dependence to that of μH is found for μSTJ (not shown).
Thus, we observe that the deviation from values in the inviscid case of the HC strength, and the latitudes of the μH, the μA, and the μSTJ increase with Pr at a rate that depends on the associated Ro regime. We also find in our runs that for Pr > 7.4 (Ro ~ 0.1; e.g., Fig. 4, thick solid line), the viscous HC encompasses the entire globe, so that an extratropical modified radiative–convective equilibrium region ceases to exist. Such an extended HC is unlikely to occur on Earth but may be relevant for more slowly rotating planets.
From the left panels of Figs. 7 and 8, it is evident that in the inviscid case, both the HC width and strength monotonically increase in response to α and Δh (cf. Hou 1984; Hou and Lindzen 1992). We wish now to examine whether the addition of eddy damping (r) affects this monotonic behavior. The right panels of Figs. 7 and 8 suggest that a decrease in the HC strength or width in response to heating is allowed only at the intermediate or eddy-dominated Ro regime, and only if r is affected by the heating [i.e., r = r(α, Δh)] such that it decreases with mean or differential heating. In other words, while the inviscid response to heating is an increase in the HC strength, a sufficient decrease in the eddy-driven component of the circulation will lead to an overall decrease in the HC strength. Such a decrease in r can only occur for high enough Pr (low enough Ro), suggesting that for high enough Pr values, a nonmonotonic response of the HC strength to heating is allowed, while for low Pr values we expect a monotonic increase in the HC strength. A decrease of r with heating was shown, for example, by Levine and Schneider (2011, Figs. 4–6), who found a monotonic increase of Ro in response to heating, but a nonmonotonic response of the HC strength to heating, coupled to a nonmonotonic response of the eddy component of the HC. In their study, both the HC strength and eddy component of the HC strength increased with mean temperature for temperatures below today’s climate conditions and then decreased as the mean temperature increased (and similarly, eddy kinetic energy, mean available potential energy, and eddy momentum flux convergence). Thus, we conclude that a nonmonotonic response of the HC strength to heating is allowed if the effect of eddies depends on the heating.
We turn now to examine the amplitude of the subtropical jet. The subtropical jet amplitude in the inviscid case can be obtained from Eq. (5),
As shown on the left panel of Fig. 9, in the inviscid case the subtropical jet amplitude scales with RT. We find that for r > 0, monotonically (and close to linearly) decreases with ln(rτα). By analogy to the thermal Rossby number RT = Δhα, the parameter RE ≡ rτα can be thought of as a macroturbulent Rossby number, in which the ratio of the thermal and effective eddy-damping time scales replaces differential thermal driving. Since RT and the inviscid solutions are known, this implies that RE can be used to predict the value of USTJ for r > 0.
A similar dependence on RE is found for EPTD. The left panel of Fig. 10 shows that in the inviscid case, EPTD/Δh scales with RT. For r > 0, the EPTD, scaled by its respective inviscid EPTD, monotonically decreases with RE, also reflecting a decrease in available potential energy (not shown).
The EMFC distribution shown in Fig. 3 has a clear north–south asymmetric component. To examine the effect of asymmetric EMFC, we performed an additional parameter sweep similar to the one described above, in which the value of r was allowed to change per hemisphere, such that
where rNH and rSH are positive constants. To avoid instabilities arising from the sharp transition between rNH and rSH, r was smoothed in the vicinity of the equator. The ratio rSH/rNH was varied between 1 and 5. Figure 11 shows the dependence on latitude of the meridional (υ, dashed), vertical [w = (Vh)′, dotted] and zonal (u, gray) winds, and h (solid) at steady state for the symmetric (top panel) and asymmetric case (bottom panel). In both cases τrNH = 0.006, α = 0.45, and Δh = ⅓. The ratio rSH/rNH was set to 1 and 3 for the symmetric and asymmetric cases, respectively. The nondimentional vertical wind is given here as w = (Vh)′ (cf. Gill 1980), with a scaling of ΩH0. For presentational clarity, winds are brought to scale by using units of 100, 0.01, and 10−6 m s−1, for the zonal, meridional, and vertical winds, respectively. The ascending branch (w > 0) is shaded as in Fig. 2.
Since only r is allowed to vary between hemispheres, the ratio rSH/rNH also equals the Pr ratio of the two hemispheres. Accordingly, we find that Ro is smaller in the SH, the SH HC strength increases partly at the expense of the NH HC strength, the ITCZ (the latitude where υ changes sign) is shifted toward the NH (3° in Fig. 11b), and the SH subtropical jet is weakened and shifted poleward. We found all of these tendencies to monotonically increase with the rSH/rNH ratio. Hemispheric asymmetry in the HC strength and location of the ITCZ comparable to the observed annually averaged atmosphere (Fig. 2) are obtained for rSH/rNH > 3. Such strong asymmetry in EMFC is not evident in Fig. 3, suggesting that asymmetry in the annually averaged midlatitude total eddy momentum fluxes plays a minor role in determining the asymmetric features of the annually averaged HC.
5. Semianalytic steady solutions for r > 0
The findings of the previous section suggest that steady solutions for r > 0 possess a strong dependence on the respective inviscid (r = 0) solutions. In addition, the deviation from inviscid solutions in general is found to be proportional to the ratio of the thermal and effective eddy forcing time scales rτ and related to the associated Ro regime. Moreover, from Fig. 5 we find that the “weak eddy-dominance limit” can be defined as
We now proceed to derive the steady solutions for the case of r > 0 as deviations from the respective inviscid solutions in the weak eddy-dominance limit. We assume that in this limit, the solutions are small deviations from the inviscid three-region vertical advection of momentum solutions given by Eqs. (4), (5), and (9). For clarity, we shall refer to these regions as the tropical (uniform h, Q > 0, in the inviscid case), subtropical (uniform M, Q < 0, in the inviscid case), and extratropical (radiative–convective equilibrium in the inviscid case) regions. We use , , and to denote the steady inviscid vertical advection of momentum solutions, and V, M, and h to denote steady-state solutions for r > 0. In addition, unless stated otherwise, a scaled variable shall refer to a variable scaled by its respective inviscid value, as in the previous section.
a. Extratropical region
We begin by solving for the functional forms of h, V, and M in the weak eddy-dominance limit at the extratropical region. We note that for r > 0, a radiative–convective equilibrium (V = 0) steady state does not exist because from Eq. (2b), eddy vorticity fluxes must be balanced by the mean circulation (M′ = absolute vorticity). Thus, the SWM is relaxed to a state that is not a steady state of the time-dependent equations. We therefore refer to the extratropical steady state for r > 0 as “modified radiative–convective equilibrium.”
Assuming small deviations from inviscid solutions to leading order, the steady form of Eq. (2) becomes
where from Eq. (5)
We therefore find from Eq. (18b) that
Using Eq. (18c) we obtain for h, defined as
From Eqs. (5) and (9), the EPTD in the inviscid case is given by . Thus, since in the inviscid case scales with RT, EPTD/Δh also scales with RT, as shown in the left panel of Fig. 10. Assuming that h remains uniform at the tropics for r > 0, from Eq. (20b), for small values of RT, the EPTD for r > 0 is given by
We therefore find that the scaled EPTD monotonically decreases with RE, as shown in the right panel of Fig. 10.
b. Tropical region
We now proceed to approximate the functional form of M and h in the tropical region, in which h is assumed constant (weak temperature gradient approximation). Thus, combining this assumption with steady Eqs. (2b) and (2c) and integrating, we obtain
We note that in the inviscid case, Eq. (22) is the equivalent of Hide’s theorem (cf. HH), that is, everywhere (Adam and Paldor 2010a). Similarly for r > 0, from Eq. (2b) U must vanish where V = 0 (cf. Held 2000). This, however, is an intrinsic property of the eddy parameterization of the model rather than a general result applicable to the physical atmosphere (cf. Shell and Held 2004; Sobel and Schneider 2009).
Exact solutions of Eq. (22) for hf given by Eq. (2d) involve known but unwieldy hypergeometric functions. To avoid unnecessary cumbrous and unrevealing algebra, we assume that in the second term on the rhs of Eq. (22) and obtain
so that at the polar boundary of this region
[with ]. The functional form of V in this region is found from Eq. (9) by assuming as in the inviscid case.
c. Subtropical region
Therefore, for small RT, we expect the relative change in ΔM with respect to the inviscid case, , to scale with Pr = rτ/RTΔh. The right panel of Fig. 12 shows the log–log dependence of the scaled ΔM on Pr. Unlike the HC strength and region boundaries that possess a dependence on the associated Ro regime, the scaled ΔM exhibits a power-law dependence, .
While for r > 0 M is continuous everywhere, in the inviscid case M is discontinuous at μH (the latitude separating the subtropical angular momentumx–conserving region and extratropical radiative convective equilibrium). Therefore, even in the weak eddy-dominance limit, in the vicinity of μH deviations from the inviscid solutions of M cannot be regarded as small compared with the inviscid solutions. We therefore make the following heuristic assumption of the functional form of M at the subtropical region:
where [given by Eq. (23b)], and M1 and M2 are constants set to satisfy the boundary conditions and , defined as
This functional form of M was chosen because it possesses the following properties: (i) it allows a smooth transition from continuous viscous solutions of M to discontinuous inviscid solutions; (ii) in the weak eddy-dominance limit, Pr ≪ 1, M given by Eq. (25) behaves like a small deviation from the value of M in the inviscid case everywhere except near μH; and (iii) it has a finite known integral, allowing us to integrate Eq. (2a) and obtain an expression for h.
where h0 is a constant of integration. Finally, V is found from
We have now obtained expressions for M, V, and h at the tropics, subtropics, and the extratropics. These expressions depend on the constants . To solve for , we impose the continuity of M, V, and h at the regions’ boundaries and conservation of mass, yielding a set of five equations, which we solve semianalytically. Explicit derivation of the solutions of the five equations is found in the appendix.
Figure 13 shows the functional forms of υ (left) and u (right) for steady states at long time integrations of Eq. (2) (thick solid), the weak eddy-dominance semianalytic solutions (thick dashed), vertical advection of momentum (thin solid), and angular momentum–conserving (thin dotted) solutions for Pr = 0.133, which we find relevant for present climate conditions. Dashed vertical lines mark the region boundaries. We find that the weak eddy-dominance solutions for u deviate from the solutions at long times in the vicinity of the subtropical jet maxima, but they approximate u well elsewhere. The weak eddy-dominance solutions also overestimate the meridional wind, reflecting an underestimation of the height at the tropics.
Figure 14 shows the convergence of the weak eddy-dominance solutions (dashed) to the steady solutions at long times (solid). Weak eddy-dominance solutions, derived from the roots of Eqs. (A7) and (A8) in the appendix, for RT = 0.113, Δh = ⅓ (as in Fig. 13), and Pr < 1 are shown for μA and μSTJ (top left), subtropical jet amplitude (top right), circulation strength (bottom left), and EPTD (bottom right).
A note on how the semianalytic solutions of the weak eddy-dominance solutions are obtained is in order. The weak eddy-dominance solutions involve nonlinear and transcendental terms whose roots may not be unique. We find that the roots obtained for Pr ≤ 1 shown in Fig. 14 are unique in the sense that they provide the only solutions with smooth dependence on Pr. We do not present solutions for Pr > 1, as the assumption that the solutions are small deviations from the inviscid three-region structure may not be valid in this case.
6. Summary and discussion
Our aim was to gain qualitative understanding of the interaction of midlatitude eddies with the mean Hadley circulation (HC). In particular, we examined the response of the HC strength and width, the latitude (μSTJ) and amplitude (USTJ) of the subtropical jets, the boundary of the ascending branch of the HC (μA), the equator-to-pole temperature difference (EPTD), and the local Rossby number (Ro) to variations in heating and eddy strength.
For this purpose we employed a shallow-water model (SWM) on the rotating Earth to model the upper branch of the HC, forced by annually averaged heating, in which eddies are parameterized as Rayleigh damping r. The rationale for this form of eddy parameterization is twofold: (i) as shown in Fig. 3, eddy momentum flux convergence is approximately proportional to the observed zonally and annually averaged zonal wind at midlatitudes; and (ii) Rayleigh damping provides a simple representation of eddies, allowing analytic insight of the qualitative effect of midlatitude eddies on the HC.
The simplistic shallow-water model and eddy parameterization do not account for extratropical eddy-driven jets, which decouple from the subtropical jets in a nontrivial manner (e.g., Son and Lee 2005). In addition, the physical interpretation of r is not clear, in particular since the typical damping time of r (~103 days) cannot be directly related to typical eddy–mean flow response times, which are on the order of a week. This work therefore relates to the macroturbulent HC, with limited relevance to midlatitude dynamics or the general circulation of the atmosphere.
The ascent region and midlatitude eddy response to heating is known to vary between dry and moist models. However, since the only simplifying assumption used here is that eddy damping is proportional to the mean zonal wind, we expect the robust conclusions of this work to equally apply to moist and dry models.
We performed a wide parameter sweep of the steady states of the time-dependent equations [Eqs. (2)] at long times for physically realistic values of the model parameters: α, a planetary Burger’s number; Δh, the scaled equator-to-pole temperature difference at radiative–convective equilibrium; and rτ, the product of the Rayleigh coefficient of damping and thermal relaxation time.
We find that, for the range of model parameter we examined, an effective HC Prandtl number, which we define as , nonlinearly predicts the degree of relative dominance of eddy momentum and heat fluxes over the mean HC, exhibiting distinct regimes of weak (Ro ~ 1), intermediate (0 < Ro < 1), and strong (Ro ~ 0) eddy dominance (Fig. 5). We estimate Pr ~ 0.13 for Earth’s present climate.
We also find that Pr approximately predicts the relative deviation from inviscid solutions for the boundary of the ascending branch (Fig. 8), HC strength (Fig. 7), width (Fig. 8), and change in angular momentum along the subtropics (Fig. 11). Similarly, the relative change of the subtropical jet strength and equator-to-pole temperature difference from their respective inviscid solutions are found to scale with RE = rτα (Figs. 9 and 10, respectively), which we term here as a macroturbulent Rossby number, analogous to the thermal Rossby number RT = Δhα.
Turning to Fig. 5, we find that Pr predicts a weakening of the relative dominance of eddies (i.e., increased Ro) in response to increased mean temperature (H0, α), in accordance with the response found in more complex models (e.g., Levine and Schneider 2011). Similarly, since the equator-to-pole temperature difference at radiative–convective equilibrium is smaller in the summer hemisphere, Pr predicts that eddies are more dominant in the summer hemisphere than in the winter hemisphere, in accordance with more complex models as well as the observed atmosphere (e.g., Walker and Schneider 2006; Schneider and Bordoni 2008). From Figs. 7 and 8 we see that inviscid theory predicts a monotonic increase in HC strength and width in response to mean (α) or differential (Δh) heating. From Fig. 7, we deduce that a decrease in the HC strength in response to heating can occur only at intermediate or strong eddy dominance, and only if the eddy component of the HC decreases in response to the heating. These conditions are consistent with the nonmonotonic HC strength response to heating found by Levine and Schneider (2011).
To test the effect of equatorially asymmetric eddy damping (as opposed to equatorially asymmetric heating), we performed an additional parameter sweep in which a different value of r was used for each hemisphere. For Southern Hemisphere r (rSH) greater than r at the Northern Hemisphere (rNH), we found that (i) the SH HC strength increased partly at the expense of the NH HC strength; (ii) the ITCZ (the latitude where υ changes sign) was shifted to the north; and (iii) the SH subtropical jet was damped and shifted poleward. These tendencies increased monotonically with increasing rSH/rNH ratios. Hadley circulation strength and ITCZ shift comparable to those observed in the atmosphere (Fig. 2) were obtained for rSH/rNH > 3. Such strong asymmetry, however, is not observed in the annually averaged EMFC (Fig. 3), suggesting that this simplified asymmetry plays a minor role in determining the asymmetric properties of the annually averaged HC.
From Figs. 7–10 we learn that the relative deviation of the steady states of the viscous HC is strongly dependent on the values of Pr and RE, as well as on the associated Ro regime. We thus define the “weak eddy-dominance limit” as Pr ≪ 1.
In the inviscid case, inviscid solutions follow a three-region paradigm: weak temperature gradient at the ascending branch of the HC, angular momentum conservation at the descending branch, and radiative–convective equilibrium at the extratropics. We derived semianalytic solutions of the SWM in the weak eddy-dominance limit by assuming that for sufficiently low values of Pr, the three-region paradigm, which dominates the inviscid solutions, can be applied to the eddy-permitting HC. Solutions are derived as small deviations from the known three-region inviscid steady states. The modified three regions are: weak temperature gradient at the ascending branch, a smooth monotonically decreasing angular momentum at subtropics [Eq. (25)], and modified radiative–convective equilibrium at the extratropics.
Approximate expressions for the modified steady states in the weak eddy-dominance limit are derived and solved semianalytically in section 5. These expressions yield the scaling relations found in the parameter sweep of the model for the steady-state HC strength (Fig. 7) and width (Fig. 8), the boundary of the ascending branch (Fig. 8), subtropical jet strength (Fig. 9), equator-to-pole temperature difference (Fig. 10), and the change in angular momentum along the subtropics (Fig. 12). The weak eddy-dominance solutions diverge from the time-dependent solutions at long times as Pr increases and may be invalid for Pr > 1.
The weak eddy-dominance solutions can be extended to other forms of eddy parameterization, potentially more realistic than Rayleigh damping, which may better capture seasonal climatology. A simple parameterization of the gross response of eddies to variations of external parameters, for example, r(α, Δh), may further elucidate the nonlinear interaction of midlatitude eddies and the mean HC.
Our findings suggest that in the intermediate regime of eddy dominance, the properties of the HC are determined by a base three-region inviscid state, given by RT and Δh, and a deviation from that state, predicted by Pr. The steady states of the idealized model are found to be in agreement with more complex models as well as the observed atmosphere. Thus, the idealized macroturbulent HC SWM provides powerful insight into the interaction of the HC and midlatitude eddies. Given the simplicity and qualitative agreement of the model with more complex models as well as the observed atmosphere, the weak eddy-dominance solutions may therefore be regarded as a meaningful step toward a theory of the macroturbulent HC in the intermediate eddy-dominance regime.
We wish to thank Nathan Paldor, Eyal Heifetz, Yair Cohen, and Jian Lu for their useful comments and advice during the inception and preparation of this work. We also thank Isaac Held and an anonymous reviewer for their instructive suggestions. This study was partly funded by TAU Council of Higher Education and partly by the Israeli Science Foundation Grant 1370/08.
Semianalytic Solutions of the SWM
so that to leading order, in the midregion
Now that the functional forms of M, V, and h are known in the three regions, we proceed to write the equations for μA, μH, , and h0 for r > 0 by imposing the continuity of h and M and the conservation of mass. Mass conservation yields
Imposing continuity at μA, we obtain
and from continuity at μH we obtain
By setting from continuity at μA [Eq. (A4)], h0 from continuity at μH [Eq. (A6)], using Eqs. (23b) and (24) for and , respectively, and neglecting the contribution of the exponential terms to the integral in Eq. (A3), Eqs. (A3)–(A6) are reduced to