1. Introduction

It is well established that a collapse of the Atlantic thermohaline circulation (THC) leads to significant cooling over the North Atlantic and northwestern Europe (e.g., Manabe and Stouffer 1994; Vellinga and Wood 2002). The oceanic forcing of the atmosphere occurs through decreased sea surface temperature (SST), especially in regions where convective activity ceases to occur (Vellinga and Wood 2002), and the increase of sea ice coverage. As a result, the atmospheric temperature anomaly peaks in the subpolar North Atlantic and Arctic, suggesting a locally forced heat flux response.

However, the global extent of (pre)historical abrupt climate change necessitates significant changes in tropical climate. And indeed, the atmospheric response to a THC collapse is far reaching (Vellinga and Wood 2002; Zhang and Delworth 2005), that is, well into the tropics, although the signal in SST and surface air temperature is much weaker there. Assuming that Bjerknes compensation occurs for ocean heat transport changes that result from a THC collapse, Seager and Battisti (2007) argued for a tropical source for abrupt climate change that was independent of the THC. Their theory was built on a study of Lee and Kim (2003), in which the relation between tropical SST and latitude and strength of the subtropical jet and midlatitude eddy-driven jet was investigated, suggesting hysteresis behavior and multiple equilibria for these jets. Inspired by this theory, van der Schrier et al. (2007) demonstrated in a simple coupled climate model (CCM) how (sub)tropical SST anomalies indeed force changes in the midlatitude eddy-driven jet.

While smaller fluctuations in meridional overturning circulation (MOC) imply ocean and atmosphere heat transport changes that are subject to the Bjerknes compensation (Shaffrey and Sutton 2006; van der Swaluw et al. 2007), for larger fluctuations the Bjerknes compensation no longer applies and a rearrangement of the earth’s energy budget is necessary to accommodate the change in net meridional energy transport by the coupled ocean–atmosphere system. This holds true in particular for a THC collapse (Vellinga and Wu 2008). Consequently, Vellinga and Wu not only found changes in surface radiation balance and top of the atmosphere (TOA) radiative fluxes that extend into the tropic belt and even into the Southern Hemisphere (SH), they also noticed a significantly changed Hadley circulation. This suggests that the Lee and Kim (2003) theory and the THC theory for abrupt climate change act together. A link between the two could be provided by a theory for the response of the Hadley circulation to a THC collapse.

An important first step toward a theory for the Hadley circulation was taken by Schneider (1977) and Held and Hou (1980, hereafter HH80). They developed a theory that was built on angular momentum conservation. The axisymmetric model predicts, among others, the width of the Hadley cell, the meridional transport, and the distribution of the zonal velocity. Not all aspects of the Hadley circulation were quantitatively described, but the most important features appeared qualitatively correct. Since HH80, the theory for the Hadley circulation was extended further to account for, for example, asymmetric heating and seasonality (Lindzen and Hou 1988, hereafter LH88), narrowness of the heating branch (Hou and Lindzen 1992), moist convection (Fang and Tung 1996), and baroclinic eddy fluxes (Held 2000; Walker and Schneider 2005, 2006).

Because the large-scale tropical temperature response to a THC collapse mainly consists of a dipole pattern associated with reduced cross-hemispheric heat transport (Fig. 1, see also Vellinga and Wood 2002), the theory of LH88 is relevant for this case. Figure 1 shows the heating and cooling pattern associated with a THC collapse comparing two 5-member ensembles of the Ensemble Simulations of Extreme Weather Events under Nonlinear Climate Change (ESSENCE) project (Sterl et al. 2008). The heating pattern is antisymmetric, with respect to the equator, and can be compared, albeit smaller in amplitude, with the differential heating associated with the seasonal cycle. In Fig. 2, a schematic of the asymmetric Hadley circulation is depicted that is consistent with this signal.

One of the main results of LH88, namely, that seasonally varying heating amplifies the annually averaged circulation, has been criticized a number of times. Both a lack of stationarity (Fang and Tung 1999) and vertical diffusion of momentum and eddy fluxes (Walker and Schneider 2005) counteract this amplification, making it doubtful whether the averaged summer and winter solutions differ markedly from the solution for an annually averaged forcing. Because eddies affect the boundary conditions at the poleward edge of the Hadley cell (Walker and Schneider 2006), it can be argued that they compromise the axisymmetric theory for predicting other aspects of the circulation as well. However, changes in relative intensity of the Hadley cells and the latitude of the surface mass flux convergence are predicted by the axisymmetric theory. Also, an eddy-permitting model was shown to recover the sensitivity of the Hadley cell to displacements of the latitude of maximum heating, as predicted by LH88, although it did not reproduce the nonlinear amplification of the annually averaged circulation when seasonal forcing is allowed (Walker and Schneider 2005).

The LH88 framework does not correctly predict the changes in the poleward extent of the Hadley cells and the intensity of the subtropical jets. The changes at the edge of the Hadley cells are controlled by eddy fluxes and can be explained from baroclinic instability considerations (Held 2000) and results from eddy-permitting nonaxisymmetric models (e.g., Walker and Schneider 2005, 2006). This point will be addressed in section 5c. However, other aspects of the circulation can be described within the LH88 framework. Two simplifications to the LH88 model will be introduced that allow for an analytical solution for the response of the Hadley cell to weak asymmetric heating. The first simplification is the small-angle approximation that was already introduced by HH80. The second approximation is the application standard perturbation theory in which the relation between all other variables and the latitude of maximum heating is established. Together, these relations show the intricate link between changes in the THC and changes in the Hadley circulation.

2. The equal-area solution for the Hadley circulation

HH80 and LH88 constructed equal-area solutions for the inviscid Hadley circulation, with symmetric and asymmetric heating, respectively. Their models consist of a few basic elements, which are a thermally driven flow, conservation of angular momentum, geostrophic balance, and energy conservation. The thermal forcing is represented by Newtonian cooling as

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where θ is the vertically averaged potential temperature, θ_e is the radiative equilibrium temperature, and τ is the radiative relaxation time. The general expression for θ_e that allows for an off-equatorial heating maximum is given by Eq. (1b) in LH88. For vertically averaged temperature this becomes

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where ϕ₀ is the latitude of maximum heating, Δ_H is the fractional change in potential temperature from equator to pole, and θ₀ is the globally averaged value of θ_e.

Using the small-angle approximation this equation can be rewritten as

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The antisymmetric heating term can be interpreted as a heating anomaly arising from the temperature difference between the North and South Poles (James 1994),

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Here, Δ_NS is defined as the temperature difference between the two poles. The latitude of maximum heating must then obey

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When we define Δ_TC as the temperature difference between the Tropics of Capricorn and Cancer, ϕ₀ reappears as a scaled temperature difference,

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Now we consider the case that off-equatorial heating can be due to stationary SST anomalies. The diabatic forcing becomes

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We assume that the same radiative time scale applies to SST anomalies as to radiative temperature anomalies. This is true when the SST anomalies are large scale and stationary. Whether this assumption is correct will be tested in section 5, when the appropriate scaling relations are applied to the model data. Here, θ_f can then be written as

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For certain conditions, the equation for θ_f becomes similar to the equation for θ_e, namely, when the SST anomaly consists of a cross-equatorial SST gradient that linearly scales with ϕ. In that case,

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and for ϕ*₀ Eq. (6) similarly holds, which defines the off-equatorial maximum heating latitude. In that case, the forcing temperature θ_f can be written as θ_e in Eq. (3). However, now ϕ₀ is the shift in maximum heating latitude resulting from either changes in ocean circulation, or the seasonality in insolation. The implication of this similarity is that changes in cross-equatorial heat and mass transport in the ocean impose a similar (but smaller) forcing as the insolation change between summer and winter.

To arrive at a model for the response of the Hadley circulation to such SST anomalies, we proceed as in LH88. In the small-angle limit the actual temperature profile is given by

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[see expression (7) in LH88], where ϕ₁ is the dividing line between the summer and winter cell (see Fig. 2). The expression for θ_f can be reformulated to depend on ϕ₁ as

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By substituting ϕ₀ = ϕ₁ = 0 in Eqs. (10) and (11), the expressions (14a) and (14b) from HH80 are recovered.

The equal-area solutions in HH80 and LH88 result from demanding

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and

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Similar equations hold for the SH Hadley cell, with ϕ_W being replaced by ϕ_S.

For ϕ₀ = ϕ₁ = 0, Eqs. (12) and (13) can be solved exactly (in this case ϕ_W = −ϕ_S = ϕ_H), giving a scaling relation for ϕ_H. When ϕ₀ and ϕ₁ are unequal to zero, the expression that results from substituting Eqs. (12) and (13) into Eqs. (10) and (11) no longer has an analytical solution. Substituting Eq. (13) into Eqs. (10) and (11) yields

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Similarly, Eq. (12) can be substituted into Eqs. (10) and (11) to give (see appendix A)

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Combining Eqs. (14) and (15) and defining a thermal Rossby number,

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one obtains

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When the heating profile is symmetric, Eq. (17) becomes

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from which the HH80 scaling relation,

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is recovered. When ϕ₀ and ϕ₁ are unequal to zero, an analytical solution for Eq. (17) is impossible.

3. A perturbation analysis for small asymmetric heating profiles

Here, the case with ϕ₀ < 0 and ϕ₁ < 0 will be discussed, but this choice is not essential for the derivation. To retain the largest amount of generality we write

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Together with Eq. (17), a system of two equations is obtained with four small parameters: three unknowns (ϵ_S, ϵ_W, and ϕ₁) and one forcing parameter ϕ₀. It is evident that another equation is needed, but first the expressions for ϕ_S and ϕ_W are substituted in Eq. (17). At first order, only terms that are linear in the small parameters are retained. Equation (17) becomes, at first order,

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Substituting the expressions for ϕ_W and ϕ_S gives

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It is evident that ϵ_S = −ϵ_W. We write ϵ_W = −ϵ_S = ϵ. The final relation between the small parameters becomes

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To solve Eq. (25) an extra equation is needed, which comes from demanding continuity of θ in ϕ₁. Equation (14) expresses θ(ϕ₁) − θ_f (ϕ₁) as a function of ϕ_W, ϕ₁, and ϕ₀. However, the exact same relation, with ϕ_W replaced by ϕ_S, must hold when the SH cell is considered. As a result, all terms containing ϕ_W in Eq. (14) must equal the same expression, with ϕ_W replaced by ϕ_S. We obtain

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When expanding ϕ_W and ϕ_S in ϵ, in the first-order balance one finds

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Substituting this into Eq. (25) gives

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The dividing latitude between the Northern Hemisphere (NH) and SH Hadley cells shifts with the latitude of maximum heating, and in the linear regime this shift is 6 times larger than the shift of the latitude of maximum heating.

4. The second-order expansion

a. Linear in ϕ_H but nonlinear in ϕ₀

The perturbation method can be continued with progressively higher-order expansions for ϕ_W and ϕ_S, but care has to be taken to remain consistent with the small-angle approximation. A second-order expansion, however, is still useful. Figure 4 of LH88 shows that ϕ_W and ϕ_S display significant curvature as functions of ϕ₀. Here, ϕ_S first decreases as a function of ϕ₀, consistent with the first-order linear expansion. However, for larger ϕ₀, ϕ_S increases as a function of ϕ₀. Also, the initial linear gradient of these curves is corrected by higher-order expansions in ϕ_H. Here, the second-order expansion in ϕ₀ is considered without relaxing the (linear) small-angle approximation. This expansion is not fully consistent for small ϕ₀, but it facilitates the derivation of the full second-order problem. In the full second-order problem, a nonlinear small-angle approximation is needed, which allows a higher-order expansion in ϕ_H. This affects the linear relation between ϕ₀ and all other variables.

When using a linear small-angle approximation, the form of ϕ_W and ϕ_S is anticipated to become to second order, as

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The second-order approximation of Eq. (17) is

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and a similar equation for ϕ_S holds. Substituting the expressions for ϕ_W (ϕ_S) and retaining only those terms that are second order in the small parameters ϵ, δ, ϕ₁, and ϕ₀ gives an expression for ϕ_W and ϕ_S that can be solved directly as

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A quadratic term could be added to the equation for ϕ₁, but it turns out that this term must be zero. Thus,

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b. Nonlinear in ϕ_H and ϕ₀

To be fully consistent, the sine and cosine terms that were linearized in the small-angle approximation have to be expanded to higher order too. It must be recalled that ϕ_W is (much) larger than either ϕ₁ or ϕ₀, and that we only need to retain terms that are second order in ϕ₀ and ϕ₁, or larger. When Eqs. (14) and (15) are expanded and combined together (see appendix B), one obtains

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In section 4a, it was found that the first-order expansion of ϕ_W and ϕ_S in ϕ₀ only contains antisymmetric terms, while the second-order expansion only contains symmetric terms. Equation (36), however, allows for symmetric and antisymmetric terms at both orders in ϕ₀. We have to rewrite this as

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The symmetric part of the first-order expansion of Eq. (36) becomes

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which gives

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Having β = O(ϕ_H³), we demand that ϕ₀, ϵ, and ϕ₁ are O(ϕ_H³). In reality, this only holds for a certain range of ϕ₀, but it facilitates the derivation when it is assumed that β and ϵ are each of the same order. In practice, relaxing this demand does not change the solution in a fundamental way. It only determines the amount of terms with ϕ_H that have to be retained, relative to the amount of terms with ϕ₀. Because we are only interested in a first-order nonlinear correction to the linear results of the previous paragraph, it is convenient to take the first-order symmetric and antisymmetric terms to be of equal size.

The symmetric part of the second-order expansion of Eq. (36) is (see appendix C)

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Comparing Eqs. (42) and (32) we see that, like in case of β, curvature terms enter the equation for δ², which depend on higher-order terms in ϕ_H. The antisymmetric terms in the second-order expansion of Eq. (36) give rise to the relation (appendix C)

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From the full second-order expansion of Eq. (26), one obtains (appendix C)

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Taking Eqs. (43) and (44) together then gives

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The final relations for ϕ_W, ϕ_S, and ϕ₁ become

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These relations are displayed in Fig. 3, which can be compared to Fig. 4 in LH88. For small ϕ₀ the agreement is good. The most remarkable difference is the behavior of ϕ₁, which becomes smaller for larger ϕ_H, contrary to the functional relationship displayed in LH88. This discrepancy disappears at higher order. Then, terms of the form ϕ₀²/ϕ_H and ϕ₀³/ϕ_H⁴, etc., are added to Eqs. (46)–(48). They bend the curve of ϕ₁ for smaller ϕ_H below the curve for larger ϕ_H.

In addition, the width of the solstitial winter cell W_w can be compared to the width of the equinoctial cell W_eq,

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The width of the summer cell has the same form, but a minus term between the brackets instead of this positive term.

5. Numerical solutions from a coupled climate model

In this section the analytical solutions are compared to results from the CCM simulation: the ECHAM5/Max Planck Institute Ocean Model (MPI-OM) climate model (Marsland et al. 2003; Roeckner et al. 2003). With this model, a 17-member ensemble of model simulations was performed over the 1950–2100 period, with increasing greenhouse gas concentrations, as part of the ESSENCE project (Sterl et al. 2008), plus some additional dedicated ensemble experiments. Here, two 5-member ensemble simulations are compared. In one ensemble, a freshwater anomaly of 1 Sv (1 Sv ≡ 10⁶ m³ s⁻¹) was uniformly applied over the northern North Atlantic between 50° and 70°N from 2001 onward, starting from five 1 January 2001 states of the 17-member ensemble baseline experiment. The 5-member subset of the 17-member baseline experiment is the reference ensemble. After this date, the two ensembles develop differently. In the hosing ensemble, the additional freshwater supply leads to a collapse of the THC within 20 yr. After this collapse, the ensemble mean difference between the two ensembles hardly evolves. An almost stationary anomaly pattern prevails that can be seen as the fingerprint of the THC collapse on the atmosphere (Laurian et al. 2010). At first order, this fingerprint is independent of the precise mean atmospheric state (climate). This is probably because of the fact that the THC weakens only moderately during the twenty-first century in the baseline experiment. In this study, the atmospheric fingerprint of the THC collapse is defined as the difference in ensemble mean between the two 5-member ensembles over the 2091–2100 period.

a. The shift in rising branch

First, the relation

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is examined. We take the annual mean circulation as a base state. Figure 4 shows the zonally averaged SST anomaly associated with a THC collapse, together with a linear regression of the temperature profile to latitude. Here, Δ_TC is estimated to be 1.466°C. As in HH80, Δ_H is taken to be the temperature difference on earth when forced with annual mean solar fluxes, namely, 100°C. Then, ϕ₀ is 0.838° (noting that 0.8° equals 0.014 radians). Furthermore, ϕ_H = 32.5° = 0.569 radians. Substituting these values into Eq. (50) it follows that ϕ₁ is 4.4°. Without the nonlinear term the number would be 5.0°; the nonlinear term adds a correction of about 10%. Figure 5 shows the displacement of the dividing line between the annual mean SH and NH Hadley cells as a function of height in the CCM. Here, ϕ₁ varies between 1° and 8°, depending on the latitude, with lower values near the surface and in the stratosphere and a maximum near 400 hPa. The average value of ϕ₁ is 4.2°, the difference between the CCM estimate and the theoretical value is less than 10%. This is also true when the seasonally varying circulation is taken as a base state. The agreement, however, is somewhat fortuitous, given the large variation that ϕ₁ displays with height. Also, Eq. (50) can be applied to the seasonal cycle. For the June–September (JJAS) minus December–March (DJFM) circulation, the theory predicts 15.1° versus 14.3° observed, and for the more extreme July minus January circulation the prediction is 17.5° versus 15.3° observed. The larger numbers predicted by theory are consistent with the fact that the relation between ϕ₁ and ϕ₀ is a downward-bending curve for larger ϕ₀ (see Fig. 4 of LH88).

The width of the winter and summer cells can be estimated from Eq. (49). The difference between the two is for a large part determined by ϕ₁. Equation (49), however, significantly overestimates the difference in width because the poleward extent of the Hadley cells is not correctly captured. This issue is further discussed in section 5c.

b. The strength of the anomalous meridional circulation

The strength of the anomalous cross-equatorial cell that results from a THC collapse is 3.03 10¹⁰ kg s⁻¹ (Fig. 6). This can be compared to the solstitial circulation that features a cross-equatorial cell of 24.54 10¹⁰ kg s⁻¹. The strength of the solstitial circulation is obtained by subtracting the time-averaged January and July circulations from the annually averaged meridional mass transport. The annually mean meridional circulation is asymmetric (Fig. 6), but the yearly average of the NH and SH Hadley cells is 9.43 10¹⁰ kg s⁻¹.

The HH80 and LH88 models for the Hadley circulation systematically underpredict the strength of the Hadley circulation because of a too-broad heating pattern (Hou and Lindzen 1992). Also, the absolute magnitude of the Hadley cell cannot be assessed without accounting for the role of eddies (Walker and Schneider 2005, 2006). However, if the rectification resulting from eddies scales with the strength of the mean flow, the scaling for the strength of the flow still holds in a relative sense. Therefore, only its relative strength will be compared with the scaling that results from the expansion discussed before. The starting point for estimating the mass flux in the Hadley cell is the notion that the strength of the Hadley circulation is proportional to the difference between θ and θ_e (LH88). The relevant scaling for the symmetric case is given by Eq. (18) from HH80 as

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With the aid of Eqs. (10) and (11), this equation is reformulated for the asymmetric case:

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Here, Ψ_as is the equatorially asymmetric overturning circulation driven by off-equatorially heating. To arrive at an equation that solely depends on the forcing parameters, the functional dependence of the temperature on ϕ has to be substituted. This is possible with the use of Eq. (14), which describes the temperature deviation from the radiative equilibrium temperature. Substituting Eq. (14) into Eq. (52), and using the expansion given by Eqs. (46)–(48), one finds that for small ϕ (see appendix D),

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For ϕ₀ = 0 and ϕ₁ = 0 this equation is equal to Eq. (18) of HH80, apart from the fourth-order terms that were neglected here. The part within brackets in Eq. (53) is expressed in ϕ₀, while the integrand is expressed in terms of ϕ₁. The reason for this is that the integral bounds are exact, while the formula within brackets is based on an expansion in ϕ₀. Replacing ϕ₀ by ϕ₁, here, means that higher-order terms are introduced. These terms are significant for the solstitial circulation (for which this expansion strictly spoken is not valid), but terms of similar order were neglected in that expansion. As a result, the equation becomes unbalanced when ϕ₀ is replaced by ϕ₁.

The anomalous overturning circulation can easily be derived from Eq. (53). This circulation peaks at the equator, where the mean symmetric circulation is zero. For ϕ = 0, the anomalous cross-equatorial cell equals the asymmetric solution,

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In the CCM ensemble, the ratio between the cross-equatorial cell associated with the seasonal cycle and the cell resulting from a THC collapse is 24.54/3.03 = 8.1. For the THC collapse ϕ₁ = −4.2°, ϕ_H = 32.5°, and ϕ₀ = −0.84°. For the solstitial circulation the values are ϕ₁ = −15.3°, ϕ_H = 30.3°, and ϕ₀ = −3.3° (the cross-equatorial SST gradient was estimated from a linear regression to latitude as 5.8°C). These values imply a ratio of 7.8 for the solstitial cell compared to the THC-induced cell. This number compares to the ratio found in the CCM.

c. The poleward extent of the Hadley cell and the strength of the zonal jet

Figure 6 shows that the NH meridional cell increases in strength, but that its outer boundary recedes, while the SH meridional cell weakens and its outer boundary expands after a THC collapse. This behavior is further illustrated by Fig. 7. It is seen that the SH cell slightly expands its outer edge with 0.1°, while the NH cell outer edge shrinks with 0.6°. However, according to Eq. (48), the outer edge of the SH cell should shrink in response to the displacement of maximum heating latitude, whereas the poleward extent of the NH cell should increase. The change in the poleward extent of the Hadley cells, as predicted from theory, is contrary to what the CCM shows.

The reason for the discrepancy is that the poleward extent of the Hadley cells is controlled by baroclinic instability. Angular momentum conservation only applies as long as the vertical shears do not exceed a critical value (Held 2000). If baroclinic instability occurs at a latitude that is smaller than the latitude determined by angular momentum conservation, baroclinic instability determines the poleward extent of the Hadley cell. Lu et al. (2007) found that the Hadley cell expanded and weakened under global warming in a series of Intergovernmental Panel on Climate Change Fourth Assessment Report (IPCC AR4) models, following the Held (2000) scaling, whereas angular momentum conservation predicts the opposite behavior. This result implies that baroclinic instability indeed controls the outer boundary of the Hadley circulation in the present climate. Both in Held (2000) and in Lu et al. (2007), it is assumed that the velocity profile itself does not change, and that the change in the poleward extent of the Hadley cell is determined by changes in tropopause height and static stability. In the case of a THC collapse, this assumption does not apply. The latitudinal temperature profile changes significantly and as a result the balanced zonal wind changes (see Fig. 8). The Held (2000) scaling can still be taken as a starting point to explain this behavior. The flow becomes unstable when

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where β = ∂f /∂y. In Held (2000), it is assumed that when the angular momentum–conserving wind exceeds a critical shear, the Hadley cell is terminated. However, near the Hadley cell terminus, upper-tropospheric flows deviate substantially from angular momentum–conserving winds (Walker and Schneider 2006). Instead, we simply demand thermal wind balance,

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To equate the Eqs. (55) and (56) results in a supercriticality criterion:

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This criterion is almost the same as the one advocated by Korty and Schneider (2008), only here the meridional gradient of the vertically averaged potential temperature is used instead of the surface potential temperature. We arrive at the following scaling,

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The poleward Hadley cell extent is inversely proportional to the isentropic slopes, when the slopes increase the Hadley cell contracts and vice versa.

To calculate the change resulting from a THC collapse, temperature gradients were averaged over the latitude band of 30°–40°, which includes the zero contour of the meridional streamfunction and the subtropical jet velocity maximum. First, the fractional change in Δ_Vθ₀ is considered. With surface warming in the SH and cooling in the NH, the static stability in the SH near the outer boundary of the Hadley cell decreases by 1.2% and it increases in the NH by 2.5%. The fractional change in ∂θ/∂ϕ gives a decrease of 3.2% in the SH and an increase of 6.4% in the NH. The net result is that the SH Hadley cell should expand with 2.0% ≃ 0.6° and the NH Hadley cell should shrink with 3.9% ≃ 1.2°. When comparing with Fig. 7, it is seen that the predicted changes are overestimated. In the CCM, the poleward expansion in the SH is only 0.1°, and the contraction of the outer edge of the NH cell is 0.6°. The Korty and Schneider (2008) criterion overestimates the changes in Hadley cell extent even more. Qualitatively, both criteria predict the change in the Hadley cell extent, but quantitatively there are discrepancies. These are probably related to the neglect of changes in moisture.

There is no simple theory to predict the observed changes in ∂θ/∂ϕ. However, the axisymmetric theory demands that poleward of the Hadley cell terminus θ = θ_f . This is not strictly true, because eddies transport heat when the meridional flow is absent, but it is still worthwhile to investigate whether the changes in forcing temperature as predicted by theory are consistent with the observed changes in actual temperature. The derivative of Eq. (12) is

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In other words,

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It follows that ∂θ_f /∂ϕ increases by 2.7% in the NH; it decreases by 2.5% in the SH because of the shift in maximum heating latitude.

Next, the change resulting from a change in absorbed radiation is estimated. The THC collapse induces a large albedo response in the NH, and the signal in the SH is weak and insignificant. In the NH the decrease in absorbed shortwave radiation is 7.2%. This implies a temperature drop of 0.25 × 7.2% = 1.8%, which is equivalent to 3.1°C near the North Pole. This means that Δ_H also increases by 3.1%.

According to the analytical model, the decrease in ∂θ_f /∂ϕ in the SH is 2.5%, compared to a 3.2% decrease of actual temperature gradient in the CCM. The increase in the forcing temperature gradient in the NH is predicted to be 5.8%, compared to a 6.4% increase of the actual temperature gradient in the CCM. The analytically predicted changes in the forcing temperature are consistent with the observed changes in the actual temperature gradient near the outer boundary of the Hadley cells. The heat transport by the eddies is significant, but it does not break up the relation that exists near the terminus of the Hadley cells between the actual temperature changes and changes in the forcing temperature.

6. Conclusions and discussion

Our main conclusions are as follows:

1. The cross-hemispheric SST anomaly associated with a THC collapse is equivalent to an equatorward shift of the latitude of maximum heating. The dividing latitude between the NH and SH Hadley cells features an equatorward and southward shift that is roughly 5 times larger than the shift in latitude of maximum heating.

2. The equatorward shift of the mean Hadley circulation resulting from a THC collapse makes the Hadley circulation more symmetric. The anomalous cross-hemispheric circulation features enhanced energy transport from the SH to the NH (the total Hadley circulation features reduced energy transport from the NH to the SH). This cell can be expressed as a polynomial of the shift in heating latitude.

3. In the NH, the Hadley cell contracts, while the zonal velocities increase and the subtropical jet shifts equatorward. In the SH the opposite occurs. This behavior is explained by assuming that the outer boundary of the Hadley cell is determined by baroclinic instability. Then, its poleward extent scales with the inverse of the isentropic slope at subtropical latitudes.

The qualitative agreement between the scaling laws derived from the dry axisymmetric model of the Hadley circulation and the CCM results may come as a surprise. The role of the oceans in the Hadley circulation primarily is changing the thermal forcing by (moist) convection. Also, the altered thermal forcing has a distinct zonal pattern related to the land–sea distribution (Clement 2006). This zonal asymmetry is even further strengthened by the role of the Atlantic as the primary forcing agent when the THC collapses, whereas the other oceans play a smaller role. Therefore, the changes in atmospheric moist content are large and show a distinct zonal pattern.

Also, eddies may compromise the axisymmetric model. The zonal flow in the Hadley cells deviates substantially from the angular momentum–conserving flow because eddy momentum fluxes are large (Walker and Schneider 2006), especially near the terminus of the Hadley cell. As a result, they significantly alter the momentum balance. Indeed, the CCM deviates qualitatively from the scalings that we derived from the HH80 and LH88 theories, with regard to the poleward extent of the Hadley cell and changes in zonal velocity. Both are governed by baroclinic instability (Held 2000). When this aspect is included in the simple axisymmetric model, the CCM response is consistent with the appropriate scaling. This also applies for the widening of the Hadley cell resulting from greenhouse warming (e.g., Lu et al. 2007).

The use of the Held (2000) criterion, however, has been criticized by Korty and Schneider (2008). Therefore, Korty and Schneider advocated an alternative criterion that was based on supercriticality, namely, one that measures the depth of the eddy entropy flux. The scaling that we used consists of a modification to the Held (2000) scaling, consistent with the arguments of Korty and Schneider, but with a slightly different outcome.

This deviation from the HH80 and LH88 model stresses the role of eddies in affecting the Hadley circulation. Also, the relation between ENSO and variations in the Hadley cell strength point to an important role of eddies, with ENSO affecting eddy stresses in the tropical momentum balance (Quan et al. 2004; Caballero 2007). The main difference between tropical forcing by ENSO and by a THC collapse is that the THC collapse is associated with an SST anomaly with a strong cross-hemispheric SST gradient, which imposes a similar forcing as seasonally varying insolation; whereas the ENSO-related tropical SST anomaly mainly consists of zonal contrasts, which affect the tropical stationary waves.

The latitude where the summer and winter cell divide shifts southward, and as a result the latitude of the intertropical convergence zone (ITCZ) shifts southward, in response to a THC collapse (see, e.g., Zhang and Delworth 2005). Such a shift also occurs either when large-scale NH SST cooling is applied in an atmosphere–slab ocean model (Broccoli et al. 2006), or when NH land and sea ice cover changes (Chiang and Bitz 2005). With the present theory, it is clear why all of these forcings are equivalent. Instrumental in the southward shift of the ITCZ is the decrease in the latitude of maximal heating ϕ₀, which follows from imposing a cross-hemispheric SST anomaly. Such an anomaly immediately causes a response in the tropical Hadley circulation, with a southward shift of the dividing line between the NH and SH cells ϕ₁, which is equivalent to a southward shift of the annual mean position of the ITCZ.

This southward shift in latitude of maximum heating dominates the Hadley cell response to reduced ocean heat transport when the THC collapses. Clement (2006) argues that ocean heat transport weakens the symmetric Hadley circulation in favor of the asymmetric solstitial cell. Reduced ocean heat transport resulting from a THC collapse then enhances the symmetric Hadley circulation but weakens the solstitial cell. In the CCM both cells weaken. The effect of reduced ocean heat transport in enhancing the symmetric cell is offset by the equatorward shift of the latitude of maximum heating that acts to decrease the symmetric Hadley cell.

It has often been surmised that reduced ocean heat transport associated with a THC collapse would be compensated by increased atmospheric heat transport by the Bjerknes compensation (see, e.g., Cheng et al. 2007; Vellinga and Wu 2008). There is a tendency for the atmosphere to compensate for reduced ocean heat transport in the CCM, but this compensation is far from complete and a different climate state with a different radiation balance results after the THC collapses. The anomalous meridional mass flux that arises because of this different radiation balance consists of a cross-equatorial cell. This cell is maintained by anomalous TOA net downward radiation in the SH tropics and anomalous TOA net upward radiation in the NH tropics. At midlatitudes and subpolar latitudes, the anomalous net TOA radiation is upward, consistent with reduced northward heat transport in the NH. These net changes in radiation balance are dominated by a large-scale cloud response (Laurian et al. 2010), because clear-sky long- and shortwave radiation changes almost compensate for one another (Vellinga and Wu 2008).

The enhanced northward angular momentum transport by the NH Hadley cell must be associated with enhanced angular momentum transport by the eddy-driven Ferrel cell (Cheng et al. 2007), and as a result an increase in NH transient baroclinic wave activity occurs. The NH subtropical jet intensifies but also shifts southward. This affects the NH eddy-driven jet and storm tracks as well (Lee and Kim 2003). All of these changes point to an intricate link between a THC collapse, a response in the tropical ocean and atmospheric circulation, and the extratropical atmospheric response. Note that for a large part the extratropical atmospheric response is not directly forced by extratropical SST changes, but rather is mediated by the tropical atmospheric response to a tropical SST change, most noticeably the cross-equatorial SST gradient.

Seager and Battisti (2007) discussed possible causes for abrupt climate change and advocated for an active role for the tropics as an alternative for what they called the THC theory. Although they noted that a THC collapse induces changes in the tropics, they concluded that the overall atmospheric response was too small to explain the paleorecord. Their alternative theory was based on a bifurcation in the position and strength of the eddy-driven jets (following Lee and Kim 2003). The results presented in this paper strongly suggest that this competition between the tropics and the THC is artificial. In reality, all mechanisms discussed previously appear to be intrinsically linked. The THC collapse induces tropical SST changes resulting from ocean adjustment and a wind–evaporation–SST (WES) feedback (Chiang and Bitz 2005), and this, in turn, affects the Hadley circulation. The changed Hadley circulation, being associated with a shift in the subtropical jets, impacts midlatitude storm tracks and the eddy-driven jet.