888 JOURNAL OF PHYSICAL VOLUME 34

The Thermal Structure of the Upper

GIULIO BOCCALETTI Atmospheric and Oceanic Sciences Program, Princeton University, Princeton, New Jersey

RONALD C. PACANOWSKI NOAA/Geophysical Fluid Dynamics Laboratory, Princeton, New Jersey

S. GEORGE H. PHILANDER AND ALEXEY V. F EDOROV Atmospheric and Oceanic Sciences Program, Princeton University, Princeton, New Jersey

(Manuscript received 19 November 2002, in ®nal form 3 September 2003)

ABSTRACT The salient feature of the oceanic thermal structure is a remarkably shallow , especially in the Tropics and subtropics. What factors determine its depth? Theories for the deep provide an answer that depends on oceanic diffusivity, but they deny the surface winds an explicit role. Theories for the shallow ventilated thermocline take into account the in¯uence of the wind explicitly, but only if the thermal structure in the absence of any winds, the thermal structure along the eastern boundary, is given. To complete and marry the existing theories for the oceanic thermal structure, this paper invokes the constraint of a balanced heat budget for the ocean. The oceanic heat gain occurs primarily in the zones of the Tropics and subtropics and depends strongly on oceanic conditions, speci®cally the depth of the thermocline. The heat gain is large when the thermocline is shallow but is small when the thermocline is deep. The constraint of a balanced heat budget therefore implies that an increase in heat loss in high latitudes can result in a shoaling of the tropical thermocline; a decrease in heat loss can cause a deepening of the thermocline. Calculations with an idealized general circulation model of the ocean con®rm these inferences. Arguments based on a balanced heat budget yield an expression for the depth of the thermocline in terms of parameters such as the imposed surface winds, the surface temperature gradient, and the oceanic diffusivity. These arguments in effect bridge the theories for the ventilated thermocline and the thermohaline circulation so that previous scaling arguments are recovered as special cases of a general result.

1. Introduction downward diffusion of heat. An expression for the depth of the thermocline can then be obtained by assuming The thermocline is so remarkably shallow in the Trop- geostrophic, hydrostatic, incompressible motion in ac- ics and subtropics that the average temperature of the cord with the balance , even in the western equatorial Paci®c where surface temperatures are at a maximum, is barely fuzyzxϭϪg␥T , f␷ ϭ g␥T , and ␤␷ ϭ fw z, (1) above freezing. The oceanic circulation that maintains where u and ␷ are the horizontal velocities, w is the this thermal structure has two main components, a shal- vertical velocity, T is the temperature (we shall ignore low wind-driven circulation and a deep thermohaline salinity), g is the acceleration of gravity, ␥ is the co- circulation. Traditionally, theoretical models of the ther- ef®cient of thermal expansion of water, f is the Coriolis mocline are classi®ed according to their focus on one parameter, and ␤ is its meridional gradient. The ther- or the other of these components. modynamic equation is vertical advection diffusion: Theories that focus on the thermohaline circulation in a closed basin (e.g., Bryan 1987) assume that cold wTzzzϭ ␬T . (2) water, originating from high latitudes and ¯ooding the Equations (1) and (2) result in an estimate for the depth abyss, rises uniformly everywhere and is heated by the of the thermocline: D ϭ [␬ fL21/3/(g␤␥⌬T)] , (3) Corresponding author address: Giulio Boccaletti, 54-1423, Dept. of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of where ⌬T is the static stability of the ocean, L is the Technology, 77 Massachusetts Ave., Cambridge, MA 02139-4307. horizontal dimension, and D is the depth of the ther- E-mail: [email protected] mocline.

᭧ 2004 American Meteorological Society APRIL 2004 BOCCALETTI ET AL. 889

The estimate in Eq. (3) may be relevant to a highly depth behaves as described by Pedlosky (1987), but this, diffusive ocean, but reality appears to correspond to the as we shall see, is also a very special case, with possibly limit of low diffusivity (Ledwell et al. 1993). Further- limited applicability to the real ocean. Therefore the ther- more, Eq. (3) does not re¯ect the in¯uence of the wind mocline problem is still very much open, despite these on the oceanic circulation and its thermal structure. Let advances in our theoretical understanding. us therefore turn our attention to the shallow, wind- A theory that accounts for the thermal structure of driven circulation of the ventilated thermocline. It in- the ocean must explain not only the depth of the ther- volves into the thermocline in certain sub- mocline D, but also ⌬T, the oceanic static stability, a tropical regions and the return of water to the surface critical parameter for both the diffusive estimates in Eq. layers in upwelling zones such as that of the eastern (3) and the adiabatic one in Eq. (5). Lionello and Ped- equatorial Paci®c. Theories for thermal structure main- losky (2000) showed how this ⌬T controls the character tained by such a circulation are implicit in the work of of the solution in the adiabatic ventilated thermocline Welander (1959) and were developed by Luyten et al. theory. Implicit in this parameter is recognition that a (1983) and Huang (1986). These are often referred to thermohaline process is maintaining the abyss at the ap- as adiabatic theories because of the fundamental hy- propriate temperature, but what is the relationship be- pothesis that the ¯ow in the main thermocline is con- tween the solution given in Eq. (5) and the thermohaline servative; in these theories the thermal structure is de- circulation implied by ⌬T? What determines ⌬T? termined by wind-driven processes. This paper attempts to integrate the approaches to the Equations (1) are again appropriate for this circula- thermocline problem discussed thus far by invoking the tion. However the ¯ow is assumed to be conservative constraint of a balanced heat budget. The focus of pre- so that the thermodynamic equation is now vious theories has been largely on satisfying the con- straints imposed on the ¯ow by Eqs. (1). The global uT T wT 0. (4) xyϩ ␷ ϩ zϭ thermodynamic behavior of the ocean circulation is at Equations (1) and (4) can be solved analytically (Luy- best implicit in such an approach. Explicit treatment of ten et al. 1983; Huang 1986). At the upper boundary thermodynamic processes will allow us to construct a the vertical velocity is set to be the wind-forced Ekman theory for the thermal structure of the ocean and the external parameters that control it. Far from being an pumping w ϭ wEk. In the simplest case of a two-layer system, the depth of the thermocline is alternative to previous theories this work purports to complement those efforts by explicitly considering the wfL2 1/2 implications of an aspect of the global circulation thus D ϭϩEk D2 , (5) g␥⌬T e far largely ignored. ΂΃ The limitations of considering simpli®ed models of where De is an arbitrary constant representing the depth ocean circulation such as those represented by Eqs. (1), of the thermocline on the eastern boundary. Although (2), and (4) will be considered in the ®nal section. In the ¯ow is assumed to be adiabatic [Eq. (4)], thermo- the next section we will present a brief analysis of the dynamic processes are implicit in the solution. Different oceanic heat budget, followed by a statement of the values of the parameters will lead to different thermo- hypothesis and an outline of the rest of the work. cline depths and therefore different oceanic heat content. Diabatic processes must be present to accomodate the 2. Surface heat ¯uxes and the constraint of a conversion of water associated with different thermo- balanced heat budget cline depths. Because these thermodynamic processes are not taken into account, the solution is not complete Figure 1 shows the climatological surface annual and De remains an arbitrary constant. mean heat ¯ux. Attempts to represent both the wind-driven and the Shortwave radiation and latent heat loss are the dom- thermohaline circulations in one coherent picture have inant terms in the balance (Oberhuber 1988; da Silva et been made by Robinson and Stommel (1959) and by al. 1994). Over most of the domain, these two terms Salmon (1990). The culmination of those efforts leads are in approximate local balance so that over the course to the proposal by Samelson and Vallis (1997) of a two- of a year the net ¯ux across the ocean surface is zero. thermocline limit: the adiabatic theories are appropriate There are, however, notable exceptions to this local in the upper part of the thermocline, where ventilation equilibrium as shown in Fig. 1. Large amounts of heat is dominant, and the diffusive theories are appropriate are lost on a yearly average in the neighborhood of the in the lower part of the thermocline, where there is no western boundary currents, where those currents sepa- ventilation. rate from the coast. There the ocean loses large amounts Adding diffusion to a primarily adiabatic theory does of heat when cold dry winds blow from the continent not however resolve the issue of arbitrary background over the warm waters carried by the currents, inducing strati®cation [see, however, Tziperman (1986) and de- a large latent heat loss to the atmosphere. Szoeke (1995) for speci®c cases]. Samelson and Vallis Heat gain is concentrated in the tropical bands of the (1997) proposed that the eastern boundary thermocline Paci®c and Atlantic in the eastern side of the 890 JOURNAL OF VOLUME 34

FIG. 1. Annual mean net surface heat ¯ux (from Oberhuber 1988). Contour interval: 25 W m Ϫ2. basin along the equator and along the eastern coasts. in the Tropics exceeds 100 W mϪ2, and yet the cli- These are upwelling zones, where low sea surface tem- matological position of the mean thermocline has hardly peratures inhibit evaporation, so that latent heat loss to changed in the last 50 years (Harper 2000). This fact the atmosphere does not balance insolation. The ocean suggests that a very tight connection must exist between then gains heat as described, for example, in Bunker regions of heating and cooling, tight enough to maintain (1976). the heat content of the upper ocean as approximately The surface temperatures in those regions therefore stationary. control the amount of heat entering the ocean. The sur- The vertical thermal structure of the ocean is directly face temperatures are in turn the result of the vertical coupled to the heating in the Tropics and along the bound- thermal structure of the ocean: cold water upwells be- aries, so the constraint of a balanced heat budget must cause the thermocline is suf®ciently shallow to allow constrain the depth of the thermocline. In the next sections the entrainment of cold water into the mixed layer. Re- we shall turn to a general circulation model of the ocean gions of upwelling are therefore special in that the ver- to investigate more carefully the relationship between a tical thermal structure is directly related to the heat bud- balanced heat budget and the vertical thermal structure of get. This is particularly evident during El NinÄo events the ocean. The model is described in detail in section 3, when the interannual modulation of thermocline depth and the simulations are described in section 4. modi®es the distribution. The A theory to try to explain the behavior of the model cold tongue disappears and promptly the heat gained by is derived in section 5. A discussion of the theory is the ocean is reduced (Weare 1984; Philander and Hurlin provided in section 6. Conclusions are presented in sec- 1988). tion 7. In steady state, the entrainment of cold water in the tropical and boundary mixed layer must be maintained 3. The model by a subsurface ¯ow fed by the subduction of cold, surface waters in higher latitudes (Wyrtki 1981). Re- The model is very idealized and provides a concep- gions of upwelling, where the heat is gained, and mid- tually simple setting in which to test the hypothesis. A latitude regions, where the heat is lost, must therefore small tropical basin is forced by uniform easterly winds be connected if the ocean is in steady state. and an idealized thermal boundary condition. The basin Notice that, if the regions of heating and cooling were extends from 16ЊNto16ЊS and is 40Њ wide and 5000 not balanced, the unbalanced heating would have to m deep. The choice of such a small basin allows a large result in a change in heat content and therefore in a number of cases to be integrated. change in depth and structure of the thermocline. In the The wind stress is 0.05 Pa, and, because it is constant, absence of oceanic heat transport, a heating of 50 W the complication of a full gyre circulation and boundary mϪ2 results in a deepening of the thermocline of about current is largely avoided. Winds force upwelling within 40.0 m yrϪ1 (the temperature difference across the ther- a Rossby radius of the equator and drive surface Ekman mocline is assumed to be 10ЊC)! The maximum heating ¯ow toward the southern and northern boundaries. Be- APRIL 2004 BOCCALETTI ET AL. 891

FIG. 2. Time evolution from 10ЊC isothermal conditions for two different cases (the time evolution is for the thermal structure at one point at the equator). (a) Thermal structure at 20ЊE on the equator for the case with uniform restoring temperature at 25ЊC. (b) As in (a) but for the case in which the restoring temperature tapers down to 10ЊC poleward of 12ЊN and 12ЊS. cause no vorticity is imparted by the winds, convergence which the simulation is approximately in balance with of Ekman ¯ow, due to the sphericity of the earth, pro- the wind forcing and the integral of the heat budget is duces off the equator and an equatorward small compared to the absolute values of the maxima ¯ow that balances the Ekman ¯ow in the mixed layer. and minima. In such experiments our model is integrated The winds maintain an equatorial undercurrent. for 30 years. This approach exploits the dramatic time The thermal boundary condition is provided by re- scale difference between processes that affect abyssal storing surface temperatures to an imposed temperature temperatures and the processes that are responsible for distribution (Haney 1971). This formulation is the sim- determining the upper thermocline. plest possible interactive atmosphere, one with in®nite This can be seen in the results from a simple exper- heat capacity, where heat ¯uxes respond to changes in iment, which demonstrates that the heat budget, as a sea surface temperatures as expected from a linearized global constraint, is a critical factor in determining the form of the surface heat budget. The heat ¯ux is chosen strati®cation of the ocean. Assume ®rst that tempera- to be tures at the surface are restored everywhere toward T* ϭ 25ЊC. In due course the equilibrium solution will Q ϭ ␣(T* Ϫ T), (6) necessarily be an isothermal 25ЊC as the model can only where ␣ ϭ 50 W (K m2)Ϫ1, which restores surface warm up. The ®nal solution therefore will have no ther- temperature T to a speci®ed T*. mocline and the circulation will penetrate to the bottom. The model used is the Geophysical Fluid Dynamics In Fig. 2a this is seen to happen on a time scale far Laboratory (GFDL) Modular Ocean Model 4 (MOM4). longer than the remarkably short time scale for the wind- Horizontal resolution is speci®ed as ½Њ everywhere, driven circulation to come into adjustment and for a with 32 levels in the vertical direction. The vertical thermocline to appear. A thermocline is established very resolution is 10 m in the upper 200 m. rapidly, within the ®rst three years, and then the ther- The initial temperature is a uniform 10ЊC, unless oth- mocline continues to deepen steadily on a much longer erwise speci®ed. Vertical mixing is the Pacanowski and time scale. If run for a suf®ciently long time, the ther- Philander (1981) scheme solved using a maximum ver- mocline will continue to deepen and expand until the tical mixing coef®cient of 50 ϫ 10Ϫ4 m 2 sϪ1 and a entire basin asymptotically reaches a uniform temper- background diffusivity of 0.01 ϫ 10Ϫ4 m 2 sϪ1, unless ature of 25ЊC. otherwise speci®ed. Constant lateral mixing coef®cients Now consider a case in which the restoring temper- 3 2 Ϫ1 3 2 Ϫ1 are Am ϭ 2.0 ϫ 10 m s and Ah ϭ 1.0 ϫ 10 m s atures at high latitudes in the model (poleward of 12Њ) for momentum and heat, respectively. are linearly tapered down to 10ЊC at the northern and All statements are made for quasi-steady-state cir- southern boundaries. The ®nal solution, which we will culations. Ideally all solutions should be run to equilib- analyze in more detail later, is shown in Fig. 3. A clear rium. However, even in the small basin used for these thermocline is present and the circulation is largely con- experiments the time scale for equilibrium is very long ®ned to the upper ocean. In this second case (Fig. 2b) and computationally expensive, especially for simula- both warming and cooling are present, and the ther- tions with low diffusivity. mocline is locked in place immediately after the ®rst Our focus is on the sensitivity of the upper-ocean phase of the adjustment. The basin continues to warm thermal structure to surface forcing. We shall therefore up as heat is transferred to the abyss, and the process take the working de®nition of steady state as a state in continues until the boundary layer between the venti- 892 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34

FIG. 3. Mean structure of the steady-state solution for the default case. lated thermocline and the abyss satis®es a local advec- ocean is modi®ed and what external parameters control tive±diffusive balance. However, the ventilated ther- the strati®cation. mocline is essentially unaffected by this process, and its ®nal structure (after 500 years) is not signi®cantly a. The mean state of the model different from the one obtained immediately after the ®rst few years of the adjustment. The steady-state circulation associated with the ad- The only difference between the two cases is in the justment of Fig. 2b is shown in Fig. 3. The constant thermal boundary condition at high latitudes: in one case easterly wind drives a poleward , which cooling is allowed whereas in the other it is not. Both encounters a temperature gradient poleward of 12ЊN and models are subject to the same mechanical forcing, but 12ЊS. This results in cooling of surface waters and the solutions are dramatically different, even though downwelling at the boundary. The ¯ow then returns at both models start from the same initial condition. We depth toward the equator, where it is ®rst fed into the will not dwell on the details of the adjustment, as they undercurrent and then upwelled to the surface in a large will be analyzed in detail by Boccaletti (2004, manu- cold tongue. The surface heat ¯ux re¯ects such a cir- script submitted to Dyn. Atmos. Oceans). The important culation, with a more or less zonally uniform cooling point for the purposes of this study is that a simple in high latitudes and a warming at the equator that change in the thermal boundary condition has a pro- matches the surface structure of the cold tongue. The found effect on the structure of the thermocline, and cold tongue itself is the result of outcropping of the any theory that purports to explain the structure of the equatorial thermocline, the tilt of which is approxi- thermocline should explicitly take these effects into ac- mately set by the winds. count. b. Sensitivity to the restoring temperature 4. Numerical experiments To modify the thermal structure of the solution shown In this section we investigate the conditions under in Fig. 3, the heat budget at the surface was changed which a steady-state thermal structure of the model to increase heat loss in high latitudes. All else being APRIL 2004 BOCCALETTI ET AL. 893

FIG. 4. (right) Zonally integrated heat ¯uxes, and (left) corresponding thermal structure at the equator. (top) Restoring the high-latitude temperature toward 15ЊC, (middle) restoring the high-latitude temperature toward 10ЊC, and (bottom) same as (middle) but for high diffusivity ␬ ϭ 10Ϫ4 m2 sϪ1.

®xed, the result is a shoaling of the thermocline so that is increased by increasing the temperature gradient that heating in the Tropics increases. the Ekman ¯ow encounters. Two such cases are shown in Fig. 4, where the high- The increase in high-latitude cooling provokes a re- latitude restoring temperature is set to 15ЊC (top) and sponse in the tropical regions of upwelling. The equa- 10ЊC (middle). The high-latitude cooling of the ocean torial thermocline shoals to expose more cold water, 894 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34 thereby increasing the heating to minimize the unbal- increased tilt of the thermocline and therefore a deep- ance in the heat budget. It should be emphasized that ening of the thermocline to preserve the area of the cold neither winds nor restoring temperatures have been tongue. In our experiments, however, an increase in tilt changed in the Tropics, and yet the tropical ocean un- due to the strengthening of the winds corresponds to an dergoes change. increased heat loss, determined by a strengthened Ek- Because of the colder high-latitude temperatures the man ¯ow driven by those same winds. This in turn shoaling of the thermocline is also accompanied by an should result in a shoaling of the thermocline so that increase in the static stability of the ocean (⌬T across more heat can be gained in the Tropics. the thermocline). Therefore, this behavior is also con- The combination of these two effects in the experi- sistent with the notion that the depth of the thermocline ment results in a deeper thermocline and a stronger heat represents the depth of penetration of the wind forcing transport. This is because the cooling effect due to the and is, therefore, inversely proportional to the strength upwelling of cold water leads to a stronger heating than of the strati®cation, as seen for instance in the wind- required by the heat loss so that the thermocline adjusts driven scaling. However, this would not be a satisfying by deepening overall. explanation because, as the following experiment sug- Winds allow us to explore other possibilities involving gests, the balance cannot be merely mechanical. The the heat budget. If westerlies blow over the basin instead same change in high-latitude cooling can be produced of easterlies, a very different solution results as shown in a set of simulations with higher diffusivity (␬ ϳ 10Ϫ4 in Fig. 6. The structure and the circulation are in many m2 sϪ1 rather than 0.01 ϫ 10Ϫ4 m 2 sϪ1). For this case, ways entirely reversed. The overturning streamfunction we give solutions from an experiment in which the high- now shows convergence on the equator, equatorward ¯ow latitude restoring temperature is 10ЊC, as shown in Fig. at the surface, and poleward ¯ow in the subsurface. Many 4 (bottom). of the features are symmetrically preserved, and the tilt Higher diffusivity, for the same boundary conditions, of the thermocline is opposite to that seen in the previous results in a remarkably different static stability and strat- examples. However, the thermocline is now much deeper i®cation. In this case, the thermocline is more diffuse and sharper, and the undercurrent is absent because there and deeper. Heat loss in high latitudes is comparable in is no equatorward subsurface ¯ow to feed it. both, but in the high-diffusion case it is balanced partly The heat budget perspective provides a rationale for by heating in the cold tongue and partly by heating over these results, independent of speci®c dynamical argu- a much larger area associated with diffusion across the ments. The same thermal forcing is applied, but cold water thermocline into the abyss. In contrast, in the low-dif- is now ¯owing poleward and upwelled at high latitudes. fusion case, most of the heating occurred by exposing It is then warmed in its surface equatorward ¯ow and then water that is colder than the restoring temperature to the converged at the equator, where the thermocline is a surface, in the region of the cold tongue. boundary layer between Ekman convergent ¯ow and dif- These results suggest that the partitioning of heat fusively driven upwelling. The warming now occurs at transport between the upper wind-driven circulation and high latitudes, but to balance the global heat budget the the deeper thermohaline process depends strongly on model must also cool the same water, and it can only do diffusion. Even a theory focusing on the wind-driven so at high latitudes where the restoring temperatures are circulation should include diffusive processes explicitly, cold. As a result, warm water returns poleward in bound- because they are critical at least in the high-diffusion ary currents and is cooled at the same latitude at which limit. it is warmed. The heat budget is therefore satis®ed locally, no heat transport occurs, and the thermocline is deep as no equatorward outcropping is necessary. c. Sensitivity to wind strength While the strength of the forcing is identical to the If the mechanical forcing is increased, a number of easterly case, the solution is very different because of effects are expected. First, a stronger Ekman ¯ow will the dramatically different effects of the circulation on transport more warm water poleward, resulting in a the heat budget. stronger cooling for the same restoring temperature. Also upwelling will be stronger, as an increase in wind d. Sensitivity to tropical damping strength will result in stronger equatorward subsurface return ¯ow. The global heat budget alone does not constrain The sensitivity to the strength of the winds is shown uniquely the depth of the thermocline. It only does so in Fig. 5 for two different easterly wind stresses: 0.0125 given a number of externally imposed factors. The at- and 0.05 Pa. mosphere is here represented as a restoring boundary Complicating the interpretation of the response is the condition toward a given temperature: the atmosphere fact that winds not only change the strength of the equa- has in®nite heat capacity, an assumption that is obvi- torial upwelling but also the zonal gradient (tilt) in the ously unacceptable and will have to be eventually re- equatorial thermocline. If the buoyancy budget were laxed by using a coupled model. However, even in this ®xed, a strengthening of the winds would result in an simpli®ed context, the atmospheric role is parameterized APRIL 2004 BOCCALETTI ET AL. 895

FIG. 5. (right) Zonally integrated heat ¯uxes, and (left) corresponding thermal structure at the equator for different values of wind strength: (top) ␶ ϭ 0.25 ϫ 0.05 Pa, and (bottom) ␶ ϭ 1 ϫ 0.05 Pa. through an adjustable parameter, ␣, which represents thermocline, and this is precisely what happens in this the capacity of the atmosphere to absorb heat for a given experiment. temperature difference. In the cases shown up to now This is an example of how the heat budget alone, this parameter is constant and the same everywhere. through the thermal boundary condition, can profoundly There is no reason in principle for this parameter not modify the thermal structure of the ocean. We must to change with location. An example of such a case is expect any theory for the thermal structure of the ocean given in Fig. 7. Here two cases are compared, the ®rst to explain such behavior. In the next section we will being the default case with ␣ ϭ 50 W mϪ2 KϪ1 every- attempt to construct a theory that takes into explicit where [see Eq. (6)]. The other case is one in which ␣ account all these aspects of the heat budget of the ocean. is 50 W mϪ2 KϪ1 in high latitudes and then decreases cosinusoidally to 10 W mϪ2 KϪ1 at the equator. 5. Depth of the thermocline: A simple Because nothing changes in high latitudesÐthe wind thermodynamic analysis strength and the restoring strength are all the sameÐ the cooling region is unmodi®ed. The upwelling region, The experiments of the previous section show how and therefore the thermocline structure, are, however, parameters external to the model ocean control the struc- different. Because the heat loss is the same, the same ture and depth of the thermocline through their effect heat gain must occur. However, the strength of the re- on the heat budget. In this section we shall try to con- storing is now up to 5 times as weak in the region of struct a simple theoretical model to describe these ef- upwelling. This entails that the size of the cold tongue, fects. and the temperature difference between the surface and Figure 8 shows an idealized representation of the ther- the restoring, must increase. Such an increase can only modynamics at work in this system. The ocean is sub- be accomplished, for ®xed winds, by a shoaling of the divided into two domains: a tropical box and a high- 896 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34

FIG. 6. As in Fig. 3 but for westerlies case. latitude box. In each box the heat ¯ux is determined by limit of ␣ → ϱ, ⌬T ϭ⌬T* and the surface temperatures the difference in temperature between the surface of the are imposed. In general, though, that is not going to be ocean and the overlying atmosphere; that is, the case. The tropical and high-latitude regions are connected Qϩ ϭ ␣(T Ϫ T) for the tropical box and a by a poleward ¯ow, comprising two components. The Ϫ ®rst is an Ekman transport ␷ , directly related to the Q ϭ ␣(TbcϪ T ) for the high latitude box, (7) Ek wind strength ␶, as ␷Ek ϭ ␶/( fh), where h is the depth where T is the surface temperature at the Tropics, Tc is of the mixed layer within which the Ekman ¯ow occurs. the surface temperature in high latitudes, Ta is the spec- This is the wind-driven ¯ow, and by continuity there i®ed tropical atmospheric temperature, and Tb is the will be a vertical ¯ow upwelling at the equator of speci®ed atmospheric high-latitude temperature. We as- strength wEk ϭ ␷Ekh/L, where L is the horizontal scale sume that the high-latitude temperature Tc is also the of the upwelling region. temperature at the base of the thermocline, and therefore The second part of the ¯ow connecting the subtropics of the abyss. By subtracting the two equations in Eq. (7) to the Tropics is a response to the meridional density we get an expression for the static stability of the ocean gradient, resulting in a meridional boundary ¯ow ␷g. ⌬T ϭ T Ϫ Tc as a function of the heat budget: The strength of this ¯ow needs to be parameterized. We 1 shall assume that it is simply a in ⌬T ϭ⌬T* Ϫ (QϩϪϪ Q ), (8) the meridional direction, the strength of which is given ␣ by ␷g ϭ g␥D⌬T/( fL): D is the depth of the thermocline where ⌬T* ϭ Ta Ϫ Tb. Equation (8) expresses the im- and is the depth at which a boundary layer separates portant fact that the static stability of the ocean cannot the cold abyssal waters from the warm surface tropical be imposed from the outside as an external parameter, waters. This assumption is justi®ed by the work of Park but rather is calculated as part of the solution. In the and Bryan (2000). APRIL 2004 BOCCALETTI ET AL. 897

FIG. 7. (right) Zonally integrated heat ¯uxes, and (left) corresponding thermal structure at the equator. Comparison of mean solution with different tropical restoring coef®cients (see text): (top) default case with constant restoring of 50 W m Ϫ2 K Ϫ1 and (bottom) case in which the restoring strength decreases to 10 W mϪ2 KϪ1 at the equator.

We now need independent expressions for Qϩ and partly taken up by the conversion of water in the wind- QϪ. In the Tropics the integrated surface heating Qϩ is driven ¯ow: partly transferred through diffusion to the abyss and ⌬T Qϩ ϭ L2␬ ϩ Lw2 (T Ϫ T ). (9) D es The ®rst term on the rhs is a crude estimate of the heat that reaches the abyss through diffusion. The sec- ond term on the rhs of Eq. (9) is the heat taken up by the wind-driven overturning. The strength of the volume 2 transport is wEkL . Notice, however, that the ¯ow will not encounter a temperature difference ⌬T but rather on

average T Ϫ Ts, where Ts is the average temperature of the water at the base of the mixed layer of depth h. This re¯ects the fact that what matters for the heat transport in the tropical box is not the overall static stability of the ocean, but rather the temperature of the water en- trained into the mixed layer. During El NinÄo, for in- stance, the thermocline ¯attens in the east Paci®c, and the cold water at depth is invisible to the surface en- FIG. 8. Graphical representation of the model in section 5. See text trainment. The static stability has not changed, but the for explanation. temperature of the water entrained into the mixed layer 898 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34 has. This fact re¯ects the dependence of the tropical where ␤ ϳ f/L. This recovers the advective diffusive heat budget on the depth of the thermocline and not just estimate of Eq. (3). It is a special case in which the on the temperature difference across the thermocline. diffusively driven overturning circulation is responsible We choose as a simple model of such a dependence: for the heat transport. This is the limit postulated in the work of Tziperman (1986) (see also Walin 1982; de- (T Ϫ Tc) Ts ϭ T Ϫ h (10) Szoeke 1995). Notice that we also obtain an estimate D for the static stability ⌬T, which is expected to differ so that, when D ϭ h, the water entrained into the mixed from ⌬T* by a factor proportional to diffusivity. → layer is at a temperature Tc; when D ϱ, the entrained In the limit of very small diffusion the assumption water is at a temperature T. Substituting Eq. (10) into of a diffusive thermocline poses serious limitations to Eq. (9) we get the heat transport that can be effected by the ocean. While the open-ocean diffusivity can be very small ⌬T ϩ 2 (Ledwell et al. 1993), mixing in the surface mixed layer Q ϭ L (␬ ϩ whe ) , (11) D is expected to be high so that, as open-ocean diffusion which relates the tropical heat budget to the depth of decreases, the wind-driven circulation must dominate the thermocline. the heat transport. If open-ocean diffusion ␬ is identi- The heat loss in high latitudes QϪ is balanced by cally zero, then Eqs. (13) reduce to advection of warm water by the oceanic transport. We D22(⌬T) L have assumed that and that extends ␷ Ek⌬Th ϩ g␥ ϭ wh Ek ⌬T and ␷ ϭ ␷Ek ϩ ␷g ␷Ek fL D vertically over a depth h, while ␷g extends to a depth D. Therefore, 2⌬T ⌬T ϭ⌬T* Ϫ wh.Ek (15) D22(⌬T) D␣ QϪ ϭϪ␷ ⌬ThL Ϫ g␥ , (12) Ek f In the absence of any response to the density gradient (␷ ϭ 0) the solution becomes a trivial one and D ϭ where the ®rst term is due to the Ekman transport and g Lw /␷ ϭ h, the mixed layer depth. This simply means the second term is due to the geostrophic transport. Ek Ek that all water that is converted to cold temperatures must If we now assume that the heat budget is balanced, be returned warm by being exposed to the surface. In that is, that this limit the temperature difference ⌬T is irrelevant to QϩϪϭϪQ , the depth of the thermocline, because for any ⌬T the then we get from Eqs. (8), (11), and (12) a system of same mass transport will cross the temperature gradient two equations in two unknowns, D and ⌬T: twice, once through being warmed and then through be- ing cooled. When ␷g is not zero, the solution is not trivial. D22(⌬T) L The entrainment into the tropical mixed layer must bal- ␷ ⌬Th ϩ g␥ ϭ (␬ ϩ wh)⌬T and EkfL D Ek ance the heat loss associated with both midlatitude Ek- man ¯ow and . For a deep thermocline, 2⌬T D will behave as D ϭ [ fL2w h/(g␥⌬T)]1/3. ⌬T ϭ⌬T* Ϫ (␬ ϩ whEk ). (13) Ek D␣ In the limit of no diffusion the wind-driven circulation Equations (13) constitute our theory for the structure is entirely responsible for determining the background of the thermocline. Let us now consider the results of strati®cation. This limit provides a closure for the ven- previous theories. tilated thermocline problem that depends only on the It has been argued that the adiabatic solution can be winds and on the properties of the mixed layer. This interpreted as the nondiffusive limit of a diffusive so- limit contradicts the idea that the wind-driven circula- lution (Young and Ierley 1986; Salmon 1990; Samelson tion is just a ®nite-amplitude perturbation on a back- and Vallis 1997). Diffusion is then the only process ground strati®cation determined by mixing. Rather, in responsible for setting the background strati®cation, as the limit of small diffusion the wind-driven circulation proposed by Tziperman (1986). In that case the only determines its own background strati®cation and dif- thermodynamic component of the system should be dif- fusion is just a perturbation of that. fusion and for the purposes of our thermodynamic bud- For the heat transport across a circle of latitude to be get the wind-driven circulation should not cross tem- zero, the Ekman transport must be opposite in direction to the geostrophic transport so that perature gradients (i.e., ␷e, wEk⌬T ϭ 0). In this case Eq. (13) reduces to D2⌬T g␥ ϭϪ␷ h. (16) 1/3 Ek fL2 ␬ fL D ϭ and (g␤␥⌬T) This limit gives a prediction for the depth of the ther- [] 1/2 mocline, D ϭ [Ϫ fL␷ h/(g␥⌬T)] , which is akin to 2⌬T Ek ⌬T ϭ⌬T* Ϫ ␬, (14) the critical depth for the eastern boundary depth derived D␣ by Pedlosky (1987). In our simulations, this balance is APRIL 2004 BOCCALETTI ET AL. 899 realized in the westerly wind case (Fig. 6), and is pos- a ␬ 2/3 dependence is always recovered if the variation sibly true for a model such as that of Samelson and of ⌬T with ␬ is also taken into account. Equations (13) Vallis (1997) in which very little heat transport is ef- can be solved numerically to show that, for high ␬, ⌬T* fected because the mass transport in the boundary cur- Ϫ⌬T ϳ k1/3. This is the empirical dependence found rent is almost entirely recirculated equatorward in the by Park and Bryan (2000) and also con®rmed in our horizontal gyre. simulations. These results are in good agreement with the changes 6. Parameter dependence of the theory in thermal structure obtained with our numerical general circulation model and shown in Figs. 4 and 5. Figure 9 shows how the depth of the thermocline D (top), the static stability ⌬T (middle), and the heat trans- port H (bottom) depend on the externally imposed pa- 7. Discussion and conclusions rameters ␶ and ⌬T* for different values of diffusivity. The results presented here demonstrate that the con- Here H is de®ned in terms of Qϩ, the heat gained in straint of a balanced heat budget at the ocean surface low latitudes: strongly in¯uences the thermal structure of the ocean. ⌬T The depth of the tropical thermocline is such that the 2 H ϭ ␳ 0CL (␬ ϩ whe ) , (17) oceanic heat uptake in the upwelling zones of low lat- D itudes balances the heat loss in mid- and high latitudes. where C is the heat capacity of water and ␳ 0 is its den- The relationship between the heat budget and depth of sity. The two terms on the right-hand side in this equa- the thermocline in the Tropics provides a conceptual tion represent respectively the contribution of the deep closure for the background strati®cation invoked by the- thermohaline circulation, whose intensity depends on ␬, oretical models of the ventilated thermocline. It fur- and the wind-driven circulation. thermore allows us to put in a uni®ed context the pre- For low diffusivity (␬ ϭ 10Ϫ6 m 2 sϪ1), the depth of vious conceptual models of the thermocline, which are the thermocline is almost insensitive to the imposed shown to belong to particular limits of this theory. temperature gradient ⌬T* and is rather a function of The abyssal and intermediate circulations were not the wind strength. For weak winds the static stability is realistically simulated in these runs. The absence of determined by the imposed temperature gradient, while high-latitude convection and deep-water formation lim- for strong winds it deviates from ⌬T*. The heat trans- its the extent to which deep processes can alter the upper port also depends mostly on wind strength, although ocean. In the presence of deep-water formation, heating sensitivity to ⌬T*, through its effects on D and ⌬T, is must be invoked to close the deep circulation. Where also present. and how this heating occurs are still a matter of debate, As diffusion is increased, the sensitivity to the wind but it is unlikely that such questions can be answered forcing is reduced and that to the imposed temperature within the context of basin simulations alone. difference increases. This is a result of the increased In this study, the oceanic circulation that connects the role of the thermohaline circulation in the thermody- regions of heat loss and heat gain has two main com- namics of the system as that component of the circu- ponents: the deep thermohaline and the shallow, wind- lation responds to the temperature gradient. The ther- driven circulation of the ventilated thermocline. Their mocline depth D is more sensitive to ⌬T*, and the de- relative importance depends on the oceanic diffusivity. viation of the static stability from the imposed temper- Low diffusivities, which favor the wind-driven circu- ature gradient increases. This change is re¯ected in the lation, maximize the importance of upwelling zones in heat transport, which becomes more sensitive to the the oceanic adjustment to a change in the heat budget. imposed temperature gradient and less sensitive to the High diffusivities minimize the role of upwelling zones wind strength. because oceanic heat uptake occurs over a region with Last, for very high diffusion (␬ ϭ 10Ϫ2 m2 sϪ1) the a large areal extent. Observations suggest that, at least solution is almost insensitive to the winds and ther- in the main thermocline, mixing is indeed small (Led- mohaline effects dominate. The thermocline is very dif- well et al. 1993), including that the observed world is fuse and deep, and the static stability of the system is one in which upwelling zones are of paramount im- far from the imposed gradient ⌬T*. The heat transport portance. The choice of simulating a thermohaline cir- is effectively independent of wind strength as most of culation by specifying a mixing rate is driven by the it is effected by the diffusive thermohaline circulation. necessity of simplifying the problem. Clearly this is a In classic advective diffusive theory, the heat trans- limited representation of thermohaline effects in this port depends on ␬ through D alone, because ⌬T is taken study (Wunsch 2002). However, in the absence of a to be a constant. That leads to a ␬ 2/3 dependence of the satisfying closure for the observed mixing in the ocean, heat transport on diffusivity (Park and Bryan 2000). it does allow a useful comparison with the behavior of Modeling studies vary, in their simulations of heat trans- most existing models. port sensitivity, from ␬ 2/3 (e.g., Vallis 2000) to ␬1/2 (Mar- The results presented here depend critically on an otzke 1997). Recently Park and Bryan (2000) found that oceanic surface boundary condition of the form given 900 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34

FIG. 9. (top) Depth of the thermocline D (m) for three different values of diffusivity (␬ increases left to right). (middle) Static stability ⌬T (ЊC) for the same three values of diffusivity. (bottom) Heat transport H (1014 W). Parameters are shown as functions of imposed wind strength ␶ and imposed temperature atmospheric difference ⌬T*. by Eq. (6). In that equation the speci®ed temperature what happens in different coupled ocean±atmosphere T* represents atmospheric temperatures, and T is the models. For example, in some of those models the ocean oceanic surface temperature, which is determined by is simply a mixed layer so that T ϭ T* at each point, oceanic processes. The surface winds are speci®ed in- and the constraint of a balanced heat budget is met dependently. In reality and in coupled ocean±atmo- locally at each point. In such models horizontal heat sphere models, the winds and T* are related and depend transport is zero, and there are no upwelling zones that on T. The purpose of the approach taken here is to can adjust in response to nonlocal changes in the heat explore the effect of different relations between T* and budget. In the case of coupled ocean±atmosphere mod- the winds, thus shedding light on what could happen in els in which the oceanic component has low horizontal worlds different from the present, familiar one and on resolution, diffusivity is necessarily large and the roles APRIL 2004 BOCCALETTI ET AL. 901 of the wind-driven circulation, and of the upwelling therefore complementary, and attempts to combine these zones in particular, are minimized as mentioned above. two conceptual frameworks in a uni®ed picture are cur- Last, some models have, in Eq. (17), an additional, spec- rently under way. i®ed term Q* that is usually referred to as a ¯ux cor- rection term. The presence of such a term inhibits the Acknowledgments. The authors thank Robert Hall- ability of the oceanic thermal structure, and heat trans- berg, Geoff Vallis, Kirk Bryan, Isaac Held, and Anand port, to adjust to a change in the oceanic heat budget. Gnanadesikan for useful comments at different stages With the previously illustrated limitations in mind, it of this work. Author GB gratefully acknowledges is possible to speculate on the implications of this theory NASA Fellowship NGT5-30333. NOAA Grant for , as it does make predictions for the NA16GP2246, NASA Grant NAG5-12387, and NASA, heat transport and heat content of the ocean under dif- from JPL Contract 1229837, are gratefully acknowl- ferent surface conditions. For example, an increase in edged. the oceanic heat loss, as is likely in glacial climates, results in a shoaling of the thermocline so that increased heat uptake can balance the loss. At the same time pole- REFERENCES ward heat transport increases. 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