Controls on the total dynamical heat transport of the atmosphere …paoc.mit.edu › paoc › papers › tht.pdf · 2007-11-25 · Stone (1978) argued that HT ought to be rather insensitive

Controls on the total dynamical heat

transport of the atmosphere and oceans

Daniel Enderton ∗

Massachusetts Institute of Technology, Cambridge, Massachusetts

John Marshall

Massachusetts Institute of Technology, Cambridge, Massachusetts

∗Corresponding author address: Daniel Enderton, Massachusetts Institute of Technology, Department of

Earth, Atmosphere and Planetary Sciences, 77 Massachusetts Avenue 54-1511A, Cambridge, MA 02139.

E-mail: [email protected]

ABSTRACT

The degree to which total meridional heat transport is sensitive to the details

of its atmospheric and oceanic components is explored. A coupled atmosphere,

ocean and sea ice model of an aqua-planet is employed to simulate very different

climates — some with polar ice caps, some without — even though they are driven

by the same incoming solar flux. Differences arise due to varying geometrical

constraints on ocean circulation and its ability to transport heat meridionally.

Results are discussed in the context of the study of Stone (1978) who argued that

total heat transport should be independent of the detailed dynamical processes

responsible for that transport, but depend only on the solar constant, the size

of Earth, the tilt of Earth’s axis, and the mean planetary albedo. It is found

that in a warm climate in which there is no ice, Stone’s result is useful guide.

In cold climates with significant polar ice caps, however, meridional gradients

in albedo significantly affect the absorption of solar radiation and need to be

included in any detailed calculation or discussion of total heat transport. Since

the meridional extent of polar ice caps is sensitive to details of atmospheric and

oceanic circulation, these cannot be ignored. Finally, what has been learned is

applied to a study of total heat transport estimated from the Earth Radiation

Budget Experiment (ERBE) data.

1

1. Introduction

The transport of energy from low latitudes towards the poles is a key aspect of the

climate system, warming high latitudes and cooling low latitudes. It is hence intimately

connected with the pole-equator temperature gradient, perhaps the most fundamental of all

climate parameters. Estimates show that the meridional distribution of poleward total heat

transport1, HT , is remarkably antisymmetric about the equator, peaking at ∼ 5.5 PW at

35◦ in both hemispheres (Trenberth and Caron 2001; Wunsch 2005) — see Fig. 1. The

ocean heat transport, HO, makes a large contribution to the total in the tropics, whereas

the atmospheric heat transport, HA, dominates in middle and high latitudes.

Recently, attention has been focused on understanding what sets the partition of heat

transport between the atmosphere and ocean and its meridional distribution (Held 2001;

Czaja and Marshall 2006; Marshall et al. 2007). Temporal changes in this partition have

long been thought to be a productive way of considering climate variability (Bjerknes 1964;

Marshall et al. 2001). There is also the possibility that decreased northern hemisphere HO

from a weakened Atlantic meridional overturning circulation might trigger abrupt change,

raising the question of how HA might compensate such changes (Broecker 1997; Bryden et al.

2005; Wunsch and Heimbach 2006). Important to all of these discussions is an understanding

of what influence the detailed mechanisms of atmosphere and ocean heat transport have

in setting HT , whether it be the ocean’s meridional overturning circulation, the Hadley

circulation, or atmospheric eddies, for example.

1Here the common phrase “heat transport” is used synonymously with “energy transport”, although it

should be remembered that heat is not a fluid property that can be transported: see Warren (1999) for a

discussion of terminology.

2

Stone (1978) argued that HT ought to be rather insensitive to the detailed structure of

the atmosphere and ocean circulation and instead depend only on the solar constant, the size

of Earth, the tilt of Earth’s axis, and the mean planetary albedo. In Section 2 we revisit the

question making use of coupled atmosphere-ocean-ice models of climates with various ocean

basin geometries that emphasize different modes of ocean heat transport. These coupled

calculations show that HT is rather robust under a variety of heat transport partitions

provided that the planetary albedo does not change. The HT is not constant, however,

between climates with significantly different extents of ice cover and hence different mean

and pole to equator gradients in planetary albedo.

In Section 3 our results are discussed in the context of a simple two-box (equator-pole)

model of HT and the Stone (1978) model for HT derived from truncated Legendre polynomial

expansions of the pertinent radiation budget variables. These simple models illustrate the

role of mean and pole to equator gradients in incoming solar shortwave radiation, outgoing

longwave radiations, and planetary albedo. In Section 4 our results are used to revisit

estimates of HT from the Earth Radiation Budget Experiment (ERBE). Finally, in Section

5, we conclude.

2. Coupled model employing bathymetric constraints

on ocean circulation

A series of fully coupled atmosphere-ocean-ice calculations have been carried out, one in

the total absence of land and several others with meridional ocean ridges as shown in Fig. 2.

3

Aqua is an entirely ocean-covered planet whilst Ridge has a thin, 180◦ arc of land running

from pole to pole. There is no orography in any of the calculations presented here — ridges,

when present, do not protrude into the atmosphere. Akin to the study of Smith (2004), these

idealized configurations allow us to investigate zonal versus gyral ocean circulations and their

associated heat transports. Equatorial Passage (EqPas hereafter) is identical to Ridge but

with a gap between 20◦S and 20◦N which allows for zonal flow in the tropics and gyral flow

in middle to high latitudes (Hotinski and Toggweiler 2003). Lastly, Drake, with a gap in

the ridge south of 40◦S, breaks the hemispheric symmetry (Toggweiler and Bjornsson 2000).

By introducing such geographic constraints on ocean circulation, a range of very different

climates can be realized and the total heat transport and its partition between atmosphere

and ocean studied.

a. Model

The coupled model used is the MIT General Circulation Model (Marshall et al. 1997a,b,

2004). The level of complexity of the model is chosen to provide a balance between the in-

clusion of key processes whilst maintaining the necessary computational efficiency to permit

synchronously coupled integration for many thousands of years while resolving synoptic-scale

eddies in the atmosphere. We do not use an energy balance representation of the atmosphere

with prescribed surface winds, although this is commonly adopted for computational effi-

ciency in extended coupled climate integrations (Toggweiler and Bjornsson 2000; Hotinski

and Toggweiler 2003). The atmospheric model has 5 vertical levels and employs the simple

atmospheric radiation, surface boundary layer, shallow and moist convection, cloud and pre-

4

cipitation schemes described in Molteni (2003) — see also the summary in Marshall et al.

(2007). The ocean component has 15 vertical levels spanning 5.2 km and employs the Gent

and McWilliams (1990) and Redi (1982) eddy parameterization schemes Griffies (1998).

Convection is represented through enhanced vertical mixing which is treated implicitly. A

two layer thermodynamic sea ice model based on Winton (2000) is used. The whole system

is integrated forward on the cubed sphere (Adcroft et al. 2004), at a horizontal resolution

of C32 yielding a grid of, roughly, 2.8◦ in resolution. The grid used for both atmosphere

and ocean is identical, simplifying coupling procedures. For more details see Marshall et al.

(2007).

b. Coupled model results

Our description here of the climatology of the idealized configurations shown in Fig. 2

focuses on aspects related to total heat transport and its partition. A more detailed discus-

sion can be found in Enderton et al. (2007). The climatological zonal mean temperature and

overturning circulations are shown in Fig. 3, zonal average surface air temperature in Fig.

4, and northward heat transport in Fig. 5.

All solutions have an atmospheric zonal wind structure in thermal wind balance with

the pole-equator temperature gradient and, in gross structure, are not unlike the present

climate. In the tropics there are Hadley cells and surface easterlies and weak meridional

temperature gradients. Strong westerly jets are present in middle latitudes associated with

a marked middle-latitude meridional temperature gradient.

Ocean circulations in the various configurations are markedly different from one another,

5

however. In the absence of any meridional barriers, there is no possibility of Sverdrup balance

in the Aqua ocean. Flow is predominantly zonal and reflects the sense of the surface winds

driving them: there is westward flow in the tropics and eastward flow in middle latitudes,

as discussed in detail in Marshall et al. (2007). The Ekman layer meridional mass transport

in the ocean leads to convergence and downwelling at the edge of the tropics and divergence

and upwelling along the equator. Poleward ocean heat transport in the tropics is dominated

by these Ekman-driven subtropical cells. In middle and high latitudes, Ekman transport

is largely compensated by eddy-induced circulation (there is near-zero residual transport)

and HO is much smaller: HO, reaches a maximum of 2.4 PW at ±15◦ with less than .35

PW beyond ±50◦. Critically important to the total heat transport, ice caps extend down to

∼ 61◦ in both hemispheres of Aqua. Total heat transport, HT , reaches a peak of 6.5 PW at

±36◦. Surface air temperatures peak at 28◦C in the tropics and drop to below -15◦C at the

poles.

The ocean circulation in Ridge is quite different from that of Aqua. Due to the presence

of the ridge, equatorial, subtropical, subpolar, and polar gyres are present. Due to weaker

subtropical cells and an equatorial gyre transporting heat equatorward, HO is much smaller

(1.5 PW at ±18◦) than Aqua in the tropics. At middle and high latitudes, however, there is

strong HO relative to Aqua due to the presence of subtropical and subpolar gyres: HO remains

relatively vigorous in middle latitudes with .65 PW crossing ±50◦. Of great importance to

our discussions, there is no ice present in Ridge. In the Ridge solution, HT peaks at 6.0 PW

near ±36◦. Surface air temperatures rise to 32◦C in the tropics and fall to 3◦C at the poles.

The results for EqPas are similar to that of Ridge, except in two respects. First, the

open ocean passage in the tropics eliminates the equatorial gyres and their equatorward

6

contribution to HO. Second, in the ocean there is deep overturning reaching from high

latitudes all the way to the tropics (Hotinski and Toggweiler 2003). HO rises to a peak of

2.2 PW at ±18◦ and exceeds by 0.5 PW or so that found in the Ridge up to a latitude of

about ±45◦. There is also no ice. The HT in EqPas, however, is virtually identical to that

of Ridge at all latitudes. Surface air temperatures peak at 31◦C in the tropics and are 7◦C

at the poles.

Drake has a deep interhemispheric meridional overturning circulation akin to the present

climate with water sinking at high latitudes in the northern hemisphere and a fraction

upwelling poleward of 40◦S. This results in increased northward HO at all latitudes relative

to Ridge and an asymmetric distribution around the equator. In the northern hemisphere,

HO peaks at 1.8 PW at 18◦N and exceeds .75 PW up to 60◦N. In the southern hemisphere,

poleward HO peaks at 1.6 PW at 15◦S and is quite close to zero southward of the tropics.

In the northern hemisphere of Drake, where there are subtropical and subpolar gyres

transporting energy northward as in Ridge, there is no ice present. However, in the southern

hemisphere, where there is zonal flow akin to Aqua circumnavigating the globe, there is an

ice cap extending down to ∼ 62◦S. In the northern hemisphere, the poleward HT is nearly

identical to that in Ridge and EqPas, peaking at 6.1 PW at 36◦N. In the southern hemisphere

HT peaks at 6.4 PW at 36◦S, almost the same as the 6.5 PW found in Aqua. Surface air

temperatures rise to 31◦C in the tropics and are -16 ◦C and 3◦C at the south and north

poles, respectively.

We see, then, that circulations resulting from the varied geographical constraints on

ocean circulation yield a range of HO varying by up to a factor of 2 at any given latitude.

In those solutions in which there is no ice, the meridional distributions of HT are almost

7

identical. This is particularly evident when one compares Ridge, EqPas, and the northern

hemisphere of Drake: HO varies substantially between these solutions whereas HT does

not. The difference in HT between Aqua, the southern hemisphere of Drake, and Ridge

is associated with the presence of large ice sheets in the former two solutions. As will be

discussed later, due to the large amount of reflected solar radiation at high latitudes in Aqua

and the southern hemisphere of Drake, the disparity in absorbed incoming energy between

the equator and poles is large, leading to high HT relative to Ridge. Drake, with its marked

difference in ice cover between the hemispheres, has a HT which is asymmetric about the

equator: poleward HT is greater in the south relative to the north by ∼ 0.3 PW (with < 0.1

PW crossing the equator). These differences in HT between the coupled model runs are

explored in the following section using a two-box model and the series expansion employed

by Stone (1978).

3. Expressions for the total heat transport

To help us understand the above results, and more generally the dependence of HT on

atmosphere and ocean dynamics, we develop a two-box model of a hemisphere as well as a

series representation of the heat transport following Stone (1978). Assuming a steady state,

the variation in total heat transport with latitude, φ, is defined as

dHT

dφ= 2πR2 cos φ[S(φ)a(φ) − I(φ)] (1)

where R is the radius of the Earth, S the incoming top of the atmosphere (TOA) solar radia-

tion (W m−2), a the co-albedo (1−albedo, the fraction of solar radiation absorbed), and I the

8

TOA outgoing longwave radiation (OLR; W m−2). Employing the non-dimensionalization

of Stone (1978), we define

x = sin φ (2)

S =S0

4s(x) (3)

I =S0

4i(x) (4)

HT =πR2S0

2f(x) (5)

and rewrite Eq. 1 as

df

dx= s(x)a(x) − i(x). (6)

where s(x), i(x), and f(x) are the non-dimensionalized versions of S, I, and HT , respectively.

a. Two-box model of total heat transport

We construct a two-box model of a hemisphere in equilibrium with no cross-equatorial

HT – see Fig. 6. The hemisphere is divided into two equal-area regions separated at x =

0.5 (φ = 30◦). The incoming solar radiation (s), co-albedo (a), and outgoing long wave

radiation (i) are defined for the equatorial (E) and poleward (P ) boxes with f being the

total meridional heat transport between them. Integrating Eq. 6 from the pole (x = 1;

f = 0) to x = 0.5 yields

f = −(sP aP − iP )∆x. (7)

where ∆x = 0.5 and the overbar represents the area-averaged values for that box.

To separate sP and aP , we define sP = sPaP aP−1 such that sP aP = sP aP . Then,

9

expressing the equatorial and polar box budgets in terms of means and differences, we write:

s =1

2(sE + sP ) ∆s = sP − sE (8)

a =1

2(aE + aP ) ∆a = aP − aE (9)

i =1

2

(

iE + iP)

∆i = iP − iE . (10)

Substituting Eqs. 8-10 into Eq. 7, and noting that s a − i ≈ sa − i = 0, the non-

dimensionalized total heat transport can be written:

f = −

(

∆sa + s∆a +1

2∆s∆a − ∆i

)

∆x

2. (11)

This decomposition of key variables into global average and pole-equator differences eluci-

dates processes controlling poleward heat transport. Table 1 shows the relative contribution

to HT of each of the four terms in Eq. 11 for Aqua, Ridge, and Drake. Given that the differ-

ential heating of the climate system drives heat transport, it is not surprising that the largest

contribution to HT in Ridge is the first term representing the solar flux gradient and mean

albedo. With small gradients in co-albedo and outgoing radiation, all other contributions to

Ridge HT are minimal.

For Aqua, the contribution to HT by the ∆sa term remains the largest, but is smaller

relative to the Ridge because of the presence of large ice caps and hence a decreased planetary

albedo. By virtue of ice caps being located in the polar box, there is now a significant gradient

in absorption. In Aqua, the second term representing the gradient in co-albedo is a third

the magnitude of the first term. A part is compensated by a stronger (weaker) OLR in the

region of strong (weak) absorption (∆i and ∆a are of same sign), but overall, the strong

gradient in co-albedo leads to an enhanced HT . In all of the coupled calculations the ∆s∆a

term is small.

10

The Drake is similar to Ridge in the northern hemisphere (the ∆sa term dominates) and

similar to Aqua in the southern hemisphere (where terms other than ∆sa are also significant).

The two-box model also captures the non-antisymmetry of the Drake HT .

These results suggest that the dominant term in setting HT is the differential heating

of the Earth by the Sun (due to the sphericity of the planet). However, if the gradient of

radiative forcing increases, typically though ice growth leading to less absorption at high

latitudes, HT will also increase. This may be at least partially compensated by OLR in-

creasing in regions of higher absorption of incoming solar radiation. As will be discussed in

the following section, it is gradients in a and i which are most likely to be sensitive to ocean

and atmospheric circulation.

b. Series representation of heat transport

Stone (1978; Eqs. 6-18 on pages 127-132) investigated these same relationships though

a more sophisticated framework. By expanding data from Ellis and Vonder Haar (1976) for

s, a, and i in a series of even order Legendre polynomials, Stone (1978) finds that the first

two terms of each expansion capture most of the structure in the fields. That is

s(x) ≈ s0P0(x) + s2P2(x) (12)

a(x) ≈ a0P0(x) + a2P2(x) (13)

i(x) ≈ i0P0(x) + i2P2(x) (14)

where

P0(x) = 1, P2(x) =1

2(3x2

− 1). (15)

11

Fig. 7 shows nondimensionalized Aqua, Ridge, and Drake planetary co-albedo (a) and OLR

(i) data expanded in zeroth and second order Legendre polynomials. Incoming solar radiation

(s) is represented extremely well by the zeroth and second order terms and is not shown.

The constants (s0, a0, and i0), and (s2, a2, and i2) are analogous to the mean and gradient

terms, respectively, of the two-box model (Eqs. 8-10). Respectively, s0, a0, and i0 are the

mean incoming solar radiation (normalized to 1 by Eq. 3), co-albedo, and OLR. Assuming

the system is in equilibrium, OLR must balance absorbed incoming solar radiation,

i0 = s0a0. (16)

The second order coefficients, s2, a2, and i2, are a measure of the equator-pole gradients

in these fields. Note that s2 (and s0) are defined externally of the climate system. The

coefficients of i and a will be dependent, at least to some degree, on atmospheric and oceanic

dynamics. In particular, if the system were extremely efficient at transporting energy, then

i2 would approach zero and the energy reradiated to space would be uniformly distributed.

The mean co-albedo, a0 (and hence i0 by Eq. 16) would be sensitive to the dynamics to the

extent that the dynamics affect ice extent, cloud cover, or land albedo.

Using these truncated expansions and integrating Eq. 6 yields

f ≈

(

s0a0 +1

5s2a2 − i0

)

(x − 1) +

(

s2a0 + s0a2 +2

7s2a2 − i2

)

(x3− x)

2. (17)

On evaluating the coefficients, the (x − 1) term is found to be negligible compared to the

(x3−x) term, as one would expect with little to no heat transport across the equator (x = 0),

leaving

f ≈

(

s2a0 + s0a2 +2

7s2a2 − i2

)

(x3− x)

2. [4 Term HT Approximation] (18)

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The four terms in the coefficient are nearly identical to the two-box model (Eq. 11; only the

geometric factor in the gradient product term differs). Note also that the last three contain

the second order coefficients expected to be the most sensitive to the atmosphere and ocean

dynamics.

Remarkably, when Stone (1978) examined Eq. 18 he found that s0a2 + 2

7s2a2 − i2 ≈ 0

(attributed to the correlation between local absorption of shortwave radiation and emission

of longwave radiation; a2 ≈ i2); see Table 2. Thus, as a final approximation for HT , Stone

(1978) arrived at

f ≈ s2a0

(x3− x)

2. [1 Term HT Approximation] (19)

For the remainder of the paper Eqs. 19 and 18 will be referred to as the one- and four-term

approximations, respectively. The one-term approximation has at least two noteworthy

characteristics:

1. Coefficients a2 and i2, which are most likely to depend on atmosphere and ocean

dynamics, do not appear. Hence this approximation of HT is insensitive to particular

atmospheric and oceanic circulations (except as they influence the mean planetary co-

albedo, a0). This implies that ocean heat transport differences between two different

climates should be met with equal and opposite changes in atmospheric transport (e.g.

compensation).

2. The one-term approximation predicts a decrease in HT if the mean planetary co-

albedo (albedo) decreases (increases). If this decrease in planetary co-albedo were

spatially uniform, the absorbed solar flux would decrease everywhere without altering

13

the differential heating responsible for driving heat transport. Changes in co-albedo,

however, are typically non-uniform and, for example, are likely to involve growth or

retreat of ice. This suggests that other terms, such as those containing a2, the pole-

equator gradient in co-albedo, may be important.

Given the important implications of (1) and consequence highlighted in (2), the cal-

culations were repeated using the coupled model results and the ERBE data set (http:

//www.cgd.ucar.edu/cas/catalog/satellite/erbe/means.html).

c. Total heat transport approximations

The total heat transport from the coupled model runs are now compared with the one-

and four-term approximations. The coupled model co-albedo and non-dimensional OLR are

expanded in zeroth and second order Legendre polynomials (Fig. 7). For Aqua, the first

two terms of the expansion capture the gross equator to pole difference in a and i. There

are, however, large differences between the coupled model output and these low order fits,

particularly in the inability to capture the sharp changes in co-albedo and OLR across the ice

edge at ∼ 61◦. Despite this, the four-term approximation does reasonably well. In contrast,

the one-term approximation is inadequate in predicting the observed behavior of HT when

large ice caps were realized, as hypothesized in section b. This is consistent with the simple

two-box model which also suggests that all four terms must be included.

On carrying out the calculations for the Ridge and EqPas solutions, there is a much

smaller difference between the one- and four-term approximation as can be seen in Fig. 7.

(These two calculations have virtually identical distributions of co-albedo, OLR, and HT ;

14

only those for Ridge are shown). The Ridge has much more shortwave radiation absorption

at high latitudes due to the absence of ice and land and hence a relatively flat profile of

co-albedo2. Similarly, the OLR profile does not have much structure. Since a and i depend

little on latitude (a2 and i2 are close to zero), it is not surprising that there are much smaller

differences between the one- and four-term approximations. The four-term approximation

models HT well, whereas the one-term approximation underestimates HT by about 0.75

PW. Just as in the two-box model, the results for the Drake are very similar to Ridge in the

northern hemisphere and to Aqua in the southern hemisphere.

4. Calculations using ERBE data

Estimates of HT are calculated using top-of-atmosphere measurements (ERBE for this

study) of incoming and outgoing radiation by integration of Eq. 1 from one pole to the

other while setting HT = 0 at the poles. The data, however, must be balanced a priori to

avoid accumulation of errors leading to a large residual heat transport at one of the poles

(Carissimo et al. 1985; Wunsch 2005). One such method is simply to remove the mean net

TOA flux evenly over the entire globe, which yields a maximum poleward heat transport

of about 5.8 PW (see Fig. 8). Using slightly more sophisticated balancing techniques,

estimates of peak HT have settled around 5.5 PW (Carissimo et al. 1985; Trenberth and

Caron 2001). Using the Stone (1978) one-term approximation of HT (discussed in Section

b) as a constraint, rather than ad hoc balancing, Wunsch (2005) obtains a profile similar to

that of Trenberth and Caron (2001). Carissimo et al. (1985) ascribes an error of ±1 PW to

2The model ocean surface albedo is a function of the zenith angle as given by Briegleb et al. (1986).

15

HT estimates, similar to that generated by the approach of Wunsch (2005). Keith (1995)

estimates an uncertainty of ±0.5 PW. The estimates of Keith (1995), Trenberth and Caron

(2001), and Wunsch (2005) use data from ERBE Barkstrom et al. (1989); Bess and Smith

(1993).

We now repeat the Stone (1978) calculation using ERBE data and so expand a and i

in Legendre functions as shown in the left and middle panels of Fig. 8. While much of

the structure is indeed captured by the first two terms in the expansion, the higher order

terms (especially for i) are not completely negligible. For example, the relatively low OLR

due to the presence of clouds in the Intertropical Convergence Zone is not captured. When

these coefficients are used to evaluate Eq. 18, the difference between the first term and four

term approximation is now ∼ 15%, whereas with the data employed by Stone (1978) it was

∼ 0% (see Table 2). In the present calculation using ERBE data, the one- and four-term

approximations differ by about 0.75 PW near 35◦N, a considerable magnitude.

Again, to compute HT from observations here, the ERBE data were balanced by sub-

tracting the mean residual from the net TOA radiation, yielding a maximum of 5.8 PW (see

Fig. 8). At all latitudes the four-term approximation more closely reflects HT than does the

one-term approximation. It should be noted, however, that the one-term approximation is

a better match to estimates based on more involved HT estimation methods which put the

peak value at 5.5 PW (Trenberth and Caron 2001; Wunsch 2005). In the case of Wunsch

(2005), this is possibly a result of having used the Stone (1978) one-term approximation as

a prior assumption in the calculation of HT . In the study of Trenberth and Caron (2001), it

is less clear because the exact details of the TOA radiation adjustments are not given.

In summary, the one-term approximation is sufficient for a rough estimate of HT in the

16

present climate to within, say, 0.5 PW. Use of the four-term approximation is required to

model HT to within a few tenths of a PW.

5. Conclusions

Stone (1978) argued that HT should be independent of the detailed dynamical processes

responsible for that transport but depend only on the solar constant, the size of Earth, the

tilt of Earth’s axis, and the mean planetary albedo: the one-term approximation. We find

that in a warm climate in which there is no ice, Stone’s result is a useful guide. In cold

climates with significant polar ice caps, however, meridional gradients in albedo significantly

affect the absorption of solar radiation and need to be included in any detailed calculation or

discussion of total heat transport. Since the meridional extent of polar ice caps is sensitive

to details of atmospheric and oceanic circulation, these cannot be ignored. The one-term

approximation does a poor job approximating HT when there is significant ice coverage. As

ice cover increases, the mean co-albedo decreases but, significantly, the pole-equator gradient

in co-albedo increases. While a reduction of the mean co-albedo implies a decreased HT ,

enhanced albedo gradients increase HT . Hence in some cases the one-term approximation

predicts that HT will decrease, whereas in fact it increases. A case in point is that of cold

climates. Extensive polar ice caps during the Last Glacial Maximum (LGM) suggest a lower

mean co-albedo relative to the present climate, but also a larger pole-equator gradient in co-

albedo. If the effect of the coalbedo gradient were important, the one-term approximation

might predict the wrong change in HT for the LGM. In fact, LGM simulations typically

yield greater HT relative to the present (Ganopolski et al. 1998; Shin et al. 2003) despite a

17

decrease in mean planetary co-albedo.

Despite its limitations, the series expansion methodology of Stone (1978) remains a useful

framework for analysis of the total heat transport. Our study suggests that the four-term

approximation is required for quantitative study, but at the cost of further complication.

The one-term approximation (Eq. 19) expresses HT in terms of s2 (equator-pole gradient in

incoming solar radiation) and a0 (the global-mean co-albedo). Given that s2 is an external

parameter and that it might be possible to estimate a0 (to a reasonable degree of accuracy)

independent of specific knowledge of the dynamics, the one-term approximation might be

usable as a predictor for HT . That is, the one-term approximation allows for the estimation

of HT without detailed knowledge of the circulation. The four-term approximation (Eq. 18)

is a much better approximation to HT , but depends on both a2 and i2 and hence on the

complex details of the circulation and radiative-convective balance. With both a2 and i2,

the four-term approximation is not suitable as a predictor for HT because all the radiation

data are required to compute a2 and i2, at which point HT can be calculated directly using

Eq. 1. Knowledge of the circulation, through its role in setting a2 and i2, is required for

estimating HT . Stone (1978) observed from the data available to him, that the s0a2, s2a2,

and i2 terms largely cancelled one-another in Eq. 18, leaving the one-term approximation.

In our study, both with models and ERBE data, we do not find such a close cancellation

and must retain further terms in the expansion.

Nevertheless, in our Ridge and EqPas calculations, it is quite remarkable how insensitive

HT is to the starkly different HO profiles brought about by the differences in ocean basin

geometry. The atmosphere is able to carry poleward whatever energy is demanded of it.

This is especially evident in the comparison of Ridge and EqPas which have HT which are

18

indistinguishable despite marked differences in HO (Fig. 5).

It is also interesting to point out the major differences in climate possible in the presence

of virtually identical HT . For example, surface air temperatures at high latitudes in Ridge

and EqPas span 5◦C (Fig. 4). This study suggests that when discussing changes in HO

relative to the present day in, for example, a paleoworld, assuming that HT does not change

is probably a good assumption if there is little change in ice cover. However, a repartitioning

of heat transport could still have an important impact on the surface climate. For example,

the possibilities of high HO in the Eocene (Barron 1987) or low northern hemisphere HO due

to a change in Atlantic meridional overturning (Broecker 1997) could be very important for

the surface climate. Thus, it should not be argued that just because the atmosphere heat

transport will compensate, a change in ocean heat transport will have little impact on the

climate.

Finally it is worth pointing out that the differences in HT between a warm climate and a

cold climate may not be that great. For example, comparing Aqua (cold) and Ridge (warm),

the peak in HT differs by only 0.5 PW, even though the climates are starkly different. Given

that estimates of the uncertainty in the present climate HT are around ±1 PW Carissimo

et al. (1985); Wunsch (2005), this suggests that getting the total heat transport correct is

a necessary, but not sufficient condition, in validating climate models (a similar point is

made by Stone 1978). Simulated climates with differences in HT that are of the order of

measurement errors can be very dissimilar.

19

Acknowledgments.

A special thanks to Peter Stone for explaining the details of his 1978 paper which mo-

tivated work, and for his helpful comments on this paper. Thanks to Carl Wunsch, David

Ferreira, and Brian Rose for their reviews and suggestions. Jean-Michel Campin assembled

the coupled model and was critical in providing technical assistance. This study was made

possible by the Polar Programs division of NSF and private donations to MIT’s Climate

Modelling Initiative.

20

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25

List of Figures

1 Estimates of the total meridional heat transport and its partition between the

atmosphere and ocean reproduced from Trenberth and Caron (2001). . . . . 28

2 Configurations of the ocean basins on the aqua-planet: light grey denotes

ocean, black land. The land, when present, comprises a thin strip running

in a 180◦ arc from pole to pole and does not protrude into the atmosphere.

Meridional gaps are introduced in the thin strip of land in EqPas and Drake.

Where there is ocean, its depth is a constant 5.2 km. . . . . . . . . . . . . . 29

3 20 year time and zonal mean potential temperature (shading) and overturning

(black contour lines). Atmosphere and ocean potential temperature contour

intervals are 20 K and 2 K, respectively. The zero overturning line is bold,

clockwise thin solid, and counter-clockwise thin dashed. Overturning contours

are every 10 Sv = 1010 kg s−1. Only the top 2 km of the 5.2 km ocean are

shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Zonal mean surface air temperature (◦C) for the coupled calculations; the

black dashed line the indicated configuration and the gray lines the other

configurations. Surface air temperature in Drake is virtually identical to Ridge

in the northern hemisphere and to Aqua in the southern hemisphere. . . . . 31

5 Total (top), oceanic (bottom left), and atmospheric (bottom right) heat trans-

port for aqua-planet configurations shown in Fig. 2. Uncertainty envelopes

for observation-based estimates of Earth’s heat transport are shown by the

shaded regions (from Wunsch 2005). . . . . . . . . . . . . . . . . . . . . . . . 32

26

6 Setup for two-box model. The hemisphere is divided into two equal-area

regions separated at x = 0.5 (φ = 30◦). The incoming solar radiation (s), co-

albedo (a), and outgoing long wave radiation (i) are defined for the equatorial

(E) and poleward (P ) boxes with f being the total meridional heat transport

between them. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7 Aqua (first row), Ridge (second row), northern (third row) and southern

(fourth row) hemisphere Drake coalbedo (left column) and nondimensional-

ized OLR (middle column) data expanded in zeroth and second order Legen-

dre polynomials (denoted “2nd order Legendre”). Coupled model total heat

transport (dashed black) and its one-term (solid red) and four-term (solid

blue) approximations are shown in the rightmost column. . . . . . . . . . . . 34

8 Same as Fig. 7 but with northern hemisphere ERBE data. To compute the

total heat transport, the ERBE data were balanced by subtracting the mean

imbalance from the net TOA radiation. . . . . . . . . . . . . . . . . . . . . . 35

27

Fig. 1. Estimates of the total meridional heat transport and its partition between the

atmosphere and ocean reproduced from Trenberth and Caron (2001).

28

(a) (b)

(c) (d)

Fig. 2. Configurations of the ocean basins on the aqua-planet: light grey denotes ocean,

black land. The land, when present, comprises a thin strip running in a 180◦ arc from pole

to pole and does not protrude into the atmosphere. Meridional gaps are introduced in the

thin strip of land in EqPas and Drake. Where there is ocean, its depth is a constant 5.2 km.

29

Aqua, θ and ψres

Pre

ssur

e [h

Pa]

0

250

500

750

1000

Dep

th [m

]

Latitude

Ice Ice

−60 −30 0 30 602000

1500

1000

500

0

Ridge, θ and ψres

Pre

ssur

e [h

Pa]

0

250

500

750

1000

300

350

400

450

Dep

th [m

]

Latitude

No Ice No Ice

−60 −30 0 30 602000

1500

1000

500

0

0

10

20

30

EqPas, θ and ψres

Pre

ssur

e [h

Pa]

0

250

500

750

1000

Dep

th [m

]

Latitude

No Ice No Ice

−60 −30 0 30 602000

1500

1000

500

0

Drake, θ and ψres

Pre

ssur

e [h

Pa]

0

250

500

750

1000

300

350

400

450

Dep

th [m

]

Latitude

Ice No Ice

−60 −30 0 30 602000

1500

1000

500

0

0

10

20

30

Fig. 3. 20 year time and zonal mean potential temperature (shading) and overturning

(black contour lines). Atmosphere and ocean potential temperature contour intervals are

20 K and 2 K, respectively. The zero overturning line is bold, clockwise thin solid, and

counter-clockwise thin dashed. Overturning contours are every 10 Sv = 1010 kg s−1. Only

the top 2 km of the 5.2 km ocean are shown.

30

−60 −30 0 30 60−20

−10

0

10

20

30

Aqua Surface Air Temperature

Latitude

° C

Aqua IceExtent

Aqua IceExtent

−60 −30 0 30 60−20

−10

0

10

20

30

Ridge Surface Air Temperature

Latitude° C

−60 −30 0 30 60−20

−10

0

10

20

30

EqPas Surface Air Temperature

Latitude

° C

−60 −30 0 30 60−20

−10

0

10

20

30

Drake Surface Air Temperature

Latitude

° C

Drake IceExtent

Fig. 4. Zonal mean surface air temperature (◦C) for the coupled calculations; the black

dashed line the indicated configuration and the gray lines the other configurations. Surface

air temperature in Drake is virtually identical to Ridge in the northern hemisphere and to

Aqua in the southern hemisphere.

31

Fig. 5. Total (top), oceanic (bottom left), and atmospheric (bottom right) heat transport

for aqua-planet configurations shown in Fig. 2. Uncertainty envelopes for observation-based

estimates of Earth’s heat transport are shown by the shaded regions (from Wunsch 2005).

32

Fig. 6. Setup for two-box model. The hemisphere is divided into two equal-area regions

separated at x = 0.5 (φ = 30◦). The incoming solar radiation (s), co-albedo (a), and outgoing

long wave radiation (i) are defined for the equatorial (E) and poleward (P ) boxes with f

being the total meridional heat transport between them.

33

0 20 40 60 800.3

0.4

0.5

0.6

0.7

0.8

Latitude

Coa

lbed

o

Aqua

Aqua Ice Extent

2nd Ord. Leg.Cpld. Model

0 20 40 60 800.5

0.6

0.7

0.8

Latitude

Non

dim

ensi

onal

OLR

Aqua

Aqua Ice Extent

2nd Ord. Leg.Cpld. Model

0 20 40 60 800

2

4

6

8

Latitude

Hea

t Tra

nspo

rt [P

W]

Aqua

0 20 40 60 800.3

0.4

0.5

0.6

0.7

0.8

Latitude

Coa

lbed

o

Ridge

2nd Ord. Leg.Cpld. Model

0 20 40 60 800.5

0.6

0.7

0.8

Latitude

Non

dim

ensi

onal

OLR

Ridge

2nd Ord. Leg.Cpld. Model

0 20 40 60 800

2

4

6

8

Latitude

Hea

t Tra

nspo

rt [P

W]

Ridge

1 Term4 TermModel

0 20 40 60 800.3

0.4

0.5

0.6

0.7

0.8

Latitude

Coa

lbed

o

Drake NH

2nd Ord. Leg.Cpld. Model

0 20 40 60 800.5

0.6

0.7

0.8

Latitude

Non

dim

ensi

onal

OLR

Drake NH

2nd Ord. Leg.Cpld. Model

0 20 40 60 800

2

4

6

8

Latitude

Hea

t Tra

nspo

rt [P

W]

Drake NH

1 Term4 TermModel

−80 −60 −40 −20 00.3

0.4

0.5

0.6

0.7

0.8

Latitude

Coa

lbed

o

Drake SH

Drake Ice Extent

2nd Ord. Leg.Cpld. Model

−80 −60 −40 −20 00.5

0.6

0.7

0.8

Latitude

Non

dim

ensi

onal

OLR

Drake SH

Drake Ice Extent

2nd Ord. Leg.Cpld. Model

−80 −60 −40 −20 0−8

−6

−4

−2

0

Latitude

Hea

t Tra

nspo

rt [P

W]

Drake SH

1 Term4 TermModel

1 Term4 TermModel

Fig. 7. Aqua (first row), Ridge (second row), northern (third row) and southern (fourth

row) hemisphere Drake coalbedo (left column) and nondimensionalized OLR (middle col-

umn) data expanded in zeroth and second order Legendre polynomials (denoted “2nd order

Legendre”). Coupled model total heat transport (dashed black) and its one-term (solid red)

and four-term (solid blue) approximations are shown in the rightmost column.

34

0 20 40 60 800.3

0.4

0.5

0.6

0.7

0.8

Coa

lbed

o

Earth

2nd Ord. Leg.Cpld. Model

0 20 40 60 800.5

0.6

0.7

0.8

Non

dim

ensi

onal

OLR

Earth

2nd Ord. Leg.Cpld. Model

0 20 40 60 800

2

4

6

8

Hea

t Tra

nspo

rt [P

W]

Earth

1 Term4 TermModel

Fig. 8. Same as Fig. 7 but with northern hemisphere ERBE data. To compute the total

heat transport, the ERBE data were balanced by subtracting the mean imbalance from the

net TOA radiation.

35

List of Tables

1 Two-box model terms for the northern hemispheres of Aqua and Ridge and

both hemispheres of Drake from Eq. 11. The last column contains the f

redimensionalized into HT by Eq. 5. EqPas results are nearly identical to

Ridge and are not shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2 Terms used in the Stone (1978) total heat transport approximations. Only

the first term is used in the one-term approximation while all four terms

(Sum column) are required in the four-term expression. While the one-term

and four-term approximations are nearly identical in the calculation of Stone

(1978), here they differ by about 15% when using ERBE data. . . . . . . . . 38

36

Two-box model Coupled model

−∆sa −s∆a −1

2∆s∆a +∆i Sum HT at 30◦ (PW) HT at 30◦ (PW)

Aqua NH 0.240 0.086 -0.016 -0.021 0.289 6.30 6.25

Ridge NH 0.249 0.023 -0.004 0.002 0.270 5.88 5.81

Drake NH 0.250 0.024 -0.004 0.007 0.277 6.03 5.86

Drake SH 0.239 0.087 -0.016 -0.026 0.285 6.21 6.11

Table 1. Two-box model terms for the northern hemispheres of Aqua and Ridge and both

hemispheres of Drake from Eq. 11. The last column contains the f redimensionalized into

HT by Eq. 5. EqPas results are nearly identical to Ridge and are not shown.

37

−s2a0 −s0a2 −2

7s2a2 +i2 Sum

N.H. (Stone) 0.319 0.192 -0.026 -0.165 0.320

S.H. (Stone) 0.319 0.217 -0.029 -0.179 0.328

N.H. (ERBE) 0.326 0.199 -0.027 -0.133 0.365

S.H. (ERBE) 0.322 0.262 -0.035 -0.183 0.365

N.H. (Model) 0.313 0.129 -0.017 -0.080 0.345

S.H. (Model) 0.318 0.124 -0.017 -0.061 0.365

Table 2. Terms used in the Stone (1978) total heat transport approximations. Only the

first term is used in the one-term approximation while all four terms (Sum column) are

required in the four-term expression. While the one-term and four-term approximations are

nearly identical in the calculation of Stone (1978), here they differ by about 15% when using

ERBE data.

38