The relationship between greenhouse-gas forcing, global mean temperature change and sea-level rise due to thermal expansion of the oceans is investigated using upwelling-diffusion and pure diffusion models. The sensitivities of sea-level to short-time scale forcing and deep-water formation rate changes are examined. The greenhouse-gas-induced thermal expansion contribution to sea-level rise between 1880 and 1985 is estimated at 2-5 cm. Projections are made to the year 2025 for different forcing scena rios. For the period 1985-2025 the estimate of greenhouse-gas-induced warming is 0.6-1.0°C. The concomitant oceanic thermal expansion would raise sea level by 4-8 cm.
FUTURE increases in the atmospheric concentrations of the greenhouse gases (carbon dioxide, methane, nitrous oxide and chlorofluorocarbons) are expected to result in substantial global-scale warming in future decades. In response to this warming, global mean sea level should change owing to thermal expansion of the oceans and the melting (or accumulation) of land ice1-7. Prediction of these sea-level changes is of importance, because many coastal regions could be adversely affected by even a small sea-level rise. As a prerequisite to such predictions we need to be able to understand past sea-level changes and to predict the future climatic conditions that will affect sea level. Here we improve on previo us estimates of the past and future contributions to sea-level change arising from thermal expansion of the oceans.
Over the past 100 years, while global mean temperature has increased by ~0.5°C (ref. 8), sea level has risen by 10-15 cm2,5,6. The relative contributions of thermal expansion and ic e melting to this sea-level rise are uncertain and estimates vary widely, from a small expansion effect4-6 through roughly equal roles for expansion and ice melting2,9 to a dominant expansion effect10.
In principle, modelling the thermal expansion effect would appear to be exceedingly difficult, as a precise determination would require one to be able to model the three-dimensional details of oceanic temperature changes. This is beyond present capabilit ies; indeed, our knowledge of how deep-ocean temperatures have varied in recent decades and of the physical processes that control any such variations is still rudimentary. In spite of this, simple diffusion or upwelling-diffusion models of the ocean can be expected to give reasonable results for the amount of thermal expansion that might occur in response to greenhouse-gas forcing, even though such models oversimplify oceanic mixing processes. This is because the main contribution to thermal expansion is concentrated in the near-surface layers; this is where both the warming and the thermal expansion coefficient are largest.
To date, only pure diffusion (PD) models have been used to estimate the thermal expansion effect (see, for example, ref. 2), although Revelle3 has included an upwelling term in an approximate way. A pure diffusion model lead s to an isothermal steady-state ocean temperature profile. Inclusion of an advective, upwelling term (balanced by high-latitude downwelling) in an upwelling-diffusion (UD) model ensures a realistic steady-state temperature profile. For small times the d ifferences between the thermal expansion predictions of PD and UD models will be small, provided one begins with a realistic initial profile and both models are calibrated to match past observations. But for times of the order of centuries PD and UD mode ls may give noticeably different results because of the different ways in which they distribute surface heating effects vertically. Here, we use a UD model, calibrated to match past temperature changes within the limits of uncertainty in the model parame ters (compare ref. 2), to estimate the thermal expansion effect from 1880 to the present and to predict the range of possible future-expansion related sea-level changes. The results are compared with those obtained using a similarly calibrated PD model.< p>
The model's output is determined by the imposed forcing, and by internal model parameters which define the rates of land-sea and inter-hemispheric exchange, the strength of ocean mixing processes (mixed layer-depth, diffusivity and upwelling velocity), an d the sensitivity of the climate system. The latter is conveniently specified by the equilibrium CO2-doubling temperature change ([Delta]T2x), that is, the global mean surface air temperature change which would eventually result if the CO2 concentration were doubled. The parameters that most affect model output are the diffusivity (K) and the climate sensitivity. Possible feedbacks involving ocean mixing processes12,13 are not considered. We concentrate on a range of K values (0.5-2.0 cm2s-1) and [Delta]T2x vaues (1.5-4.5°C) which span the limits of current uncertainty13-15. We con sider two upwelling cases, a constant upwelling rate of 4 m yr-1 (the standard estimate based on isotope tracer studies 13,16) and time-varying upwelling. We also consider the PD case by setti ng the upwelling rate to zero.
The model computes oceanic thermal expansion using expansion coefficient (ß) data from Leyendekkers17. The expansion coefficient ß ( = ß( T, p, S) where T is temperature, p is pressure and S is salinity) v aries widely with temperature and hence with latitude and depth. Vertical variations in ß are included explicitly in the model. To account for latitudinal variations we divided each hemisphere into polar, mid-latitude and tropical zones and used t he equilibrium mixed-layer results of Manabe and Stouffer18 to relate the zonal to the hemispheric mean temperature changes (compare ref. 3). (The latitudinal distribution of these changes is similar to that for observed cha nges over the past century19.) The thermal expansion results are relatively insensitive to the details of this partitioning. The effect of S on ß is small and S is assumed equal to 35%.
Figure 2 shows the modelled temperature to 1985, [Delta]T0, for various values of K and [Delta]T2x. If greenhouse-gas forcing were the sole mechanism responsible for the 1880-1985 temperature rise, the n Fig. 2 would give the range of possible [Delat]T2x values required for compatibility between model and observations. For an observed warming in the range 0.4-0.6°C, the inferred [Delta]T2x range is 1.2-2.2°C, values much lower than recent general circulation model (GCM) estimates which give [Delta]T2x at ~4°C31,35,36. However, similarly low values for the climate sensitivity have been obtained in other model-based empirical analyses (for example, 1.6°C in ref. 37).
This apparent discrepancy between observations and recent GCM results can be interpreted in a number of ways: either recent GCM experiments have overestimated the climate sensitivity to a CO2 change; and/or some additional forcing factor exists which has contributed an overall cooling effect over the past 100 yr, partly offsetting the greenhouse-gas forcing; and/or the upwelling-diffusion parameterization of ocean mixing grossly underestimates the extent of vertical mixing; and/or the 0.4-0.6&# 176;C estimate of global warming is considerably less than the true warming. GCM uncertainties and/or neglected forcings are likely to be the most important factors. For example, changes in cloud optical properties, which may have a strong negative feed back effect38-41 and are not accounted for in current GCMs, could explain at least part of the discrepancy. The available evidence for additional forcings that are of comparable magnitude to the greenhou se-gas forcing on the century timescale is debatable, but various possibilities have been hypothesized37,42-44.
A useful way to reduce speculation in interpreting Fig. 2, which we will exploit further below, is to assume only that it gives the greenhouse-gas contribution to the 1880-1985 temperature changes. If future model results or observations were to show tha t the 'correct' values for [Delta]T2x and K, say, were 3.0°C and 1 cm2s-1, then from Fig. 2, the greenhouse-gas contribution to the 1880-1985 warming (namely, [Delta]T0) would be 0.78°C. This would th en require the existence of a compensating cooling of 0.28 +/- 0.1°C due to other factors.
Figure 3 shows the modelled sea-level change ([Delta]Z0) for the period 1880-1985 due to thermal expansion of the oceans. The range of [Delta]Z0 values compatible with [Delta]T0 = 0.4-0.6 76;C is 2.3-4.8 cm. This range of values is insensitive to the above-described uncertainties surrounding the greenhouse-gas contribution to the observed warming. We demonstrate this with an example. Suppose that some other external forcing factor (X) o perating over the interval 1880-1985 has offset the greenhouse-gas forcing (G) to give a total forcing T = G -X, where X = 0.4 G. With a reduced total forcing compared with G alone, the implied [Delta]T2x to give a 0.5°C warming over the period 1880-1985 must be larger--in this case, 3.3°C for K = 1 cm2s-1 (as compared with 1.6°C for greenhouse-gas forcing alone). In spite of the very different forcing and the larger implied [Delta]T2x value, th e corresponding [Delta]Z is virtually unchanged. For G-minus-X forcing and with the model tuned to give [Delta]T0 = 0.5°C, [Delta]Z is 3.44 cm compared with 3.48 cm for the case of G alone. Thus, for century-timescale forcing, [Delta]Z i s largely determined by the 1880-1985 temperature change, independent of the magnitude of the external forcing which produced this change.
The results shown in Figs 2 and 3 and the link between [Delta]Z0 and [Delta]T0 are well approximated by the empirical expression
for [Delta]Z0 in cm, [Delta]T0 in degrees Celsius and K in cm2s-1. As [Delta]Z0 and [Delta]T0 have approximately the same [Delta]T2x dependence, the ratio [Delta]Z0/[ Delta]T0 depends only on K. A similar result is obtained with a PD model,
As the diffusivity value for a PD model fitted to tracer data is higher than for a UD model, PD estimates of [Delta]Z0 may be considerably greater than those obtained with a UD model.
To illustrate this, we consider an extreme example. The largest short-timescale perturbatlon on the overall global warming trend is the Northern Hemisphere cooling that occurred between ~1940 and 1975. (Note that the warming between 1910 and 1940, which exceeded that expected to result from greenhouse-gas forcing, can be considered as part of this perturbation.) If we simulate these changes in different ways, we can maximize the possible shorter-timescale thermal-expansion-related sea-level fluctuation s which might be superimposed on the greenhouse-gas-induced rise of 2.3-4.8 cm. As the temperature perturbations are largest in the Northern Hemisphere, we will use hemispherically specific forcing, as suggested in refs 37 and 45. The vertical ocean dif fusivily is assumed to be 1 cm2s-1, an acceptable assumption in a sensitivity study like this given the results of Figs 2 and 3.
The two possible causes we consider are an external forcing and a change in the rate of upwelling. Both perturbations are taken to be one-cycle sinusoidal changes spanning the period 1915-85. The variable-upwelling case simulates a change in North Atlant ic Deep (NADW) formation rate, as the North Atlantic is the main source of deep water in the Northern Hemisphere.
The effects of deep-water formation rate changes on global mean temperatures have been considered previously,13,46 but this is the first time that hemispherically specific changes have been considered. ( For evidence pointing towards recent NADWchanges, see refs 47-49. Major NADW changes are thought to have occurred on the ice-age timescale50,51.) Although NADW changes may well be an important factor in explaining recent temperature fluctuations, especially in the Northern Hemisphere, the main reason for considering them here is because their influence on the vertical ocean temperature profile is radically different from that of an external forcing change.
In both cases the model is calibrated (by varying the climate sensitivity and the amplitude of the forcing or upwelling changes) to produce similar hemispheric and global mean temperature changes. The global mean warming over the period 1880-1985 is set to 0.5°C.
For the external forcing perturbation (shown in Fig. 1), the required amplitude is 0.78 W m-2 (that is, a decrease of 1.56 W m-2 over 1932.5 to 1967.5; compare this with the greenhouse-gas-forcing increase of 1.81 W m-2 ov er 1880-1985) and the [Delta]T2x value is 1.77°C. This gives a maximum modelled Northern Hemisphere cooling of 0.20°C, during which the Southern Hemisphere warmed by 0.07°C. Global mean changes are shown in Fig. 4. Both hemisp heric and global changes agree well with observations. For the upwelling rate changes, the required amplitude is 0.91 m yr-1 (see Fig. 4) and the [Delta]T2x value is 1.76°C. The corresponding temperature perturbations lead tho se for the forcing perturbation case by a few years, but they are of the same magnitude--that is, maximum Northern Hemisphere cooling of 0.20°C with concomitant Southern Hemisphere warming of 0.07°C. Global mean changes are shown in Fig. 4.
The sea-level effects of these two perturbations are quite different (see Fig. 4), even though both produce similar surface temperature effects. For the two cases, the vertical profiles of the temperature changes agree only at the surface. Clearly, for any given surface temperature history, there is no unique history for sub-surface temperature changes, and hence no unique sea-level time series. In our analysis, however, this is a secondary effect; both perturbations are small relative to the century-t imescale influence of the greenhouse effect--deviations of <0.7 cm compared with a total change of 3.4 cm. The implied uncertainty in the overall rise to 1985 is therefore ~ +/- 20%, owing to short-timescale forcing uncertainties.
The 2.3-4.8 cm range of values for [Delta]Z obtained here is compatible with the previous estimates of Gornitz et al.2. These authors were the first to examine systematically the thermal expansion effect using a model-based approach; a number of other estimates (for instance, refs 7 and 9) are based on different interpretations of Gornitz et al. They used a PD model from ref. 42 which considered the world's oceans as a single column. They applied CO2, volcanic and so lar forcing and tuned [Delta]T2x and K to match model output with the Hansen et al.42 estimate of global mean temperature changes since 1880. Because they ignored the forcing of other greenhouse gases, the y were able to obtain a match for much higher [Delta]T2x values than obtained here. Also, because their forcing history contains shorter-timescale fluctuations than we have used, their modelled fluctuations in sea level show considerable short -timescale variability. In spite of these differences they obtain similar thermal expansion estimates to ours, that is, 2.3 cm over the period 1880-1980 and 4.1 cm over the period 1900-80 (for [Delta]T2x = 2.8°C and K = 1.2 cm2 s-1). Further support for the values calculated here comes from the empirical estimates of Barnett5,6, who judges the 1880-1980 expansion effect to be <5 cm.
It is somewhat more diffitult to account for model uncertainties. We do this by using some remarkable, robust features of the model projections. First, we use the fact that the temperature change ratio, [Delta]T1[Delta]T0 (where su bscript 0 refers to 1880-1985 and 1 refers to 1985-2025), is practically independent of the assumed values of [Delta]T2x and K52. [Delta]T1[Delta]T0 depends only on the 1985-2025 to 1880-198 5 forcing ratio, [Delta]F1[Delta]F0. The relationship is nearly linear over a wide range of values of [Delta]F1. (For the three models here, [Delta]F1 = 1.77, 2.65 and 3.97 W m-2, while [Delta]F0 = 1.81 Wm-2.) For [Delta]T2x = 1.5-4.5°C and K = 0.5-2.0 cm2s-1,we can estimate [Delta]T1 to within a few per cent using
A similar relationship results if one uses a PD rather than a UD model; indeed, equation (3) gives results which are accurate for a PD model to within a few percent. As an example, equation (3) implies that, if the greenhouse-gas contribution to [Delta]T (1880-1985) were 0.6°C, then for the intermediate-forcing case) the greenhouse-gas contribution to [Delta]T (1985-2025) would be ~1.6 x 0.6 = 0.96°C, independent of [Delta]T2x, K and the model structure.
Next we use the, fact that the ratio [Delta]Z1/[Delta]T1 is virtually independent of [Delta]T2x and depends only weakly on K. [Delta]Z1/[Delta]T1 does, however, depend on the chosen forcing scenario. This result can be expressed in the form (compare equation (1))
Equations (3) and (4) can be combined to give
Equation (5) describes the model results to better than +/- 4% for a wide range of [Delta]T2x and K values ([Delta]T2x is included implicitly through the term [Delta]T0). Equation (5) also implies that [Delta]Z, is non-ze ro even if [Delta]F1, = 0 (for this extreme case, equation (5) has a maximum error of ~20%), a result which is a necessary consequence of the oceanic lag effect and the current disequilibrium between global mean temperature (and sea level) and the greenhouse-gas forcing.
An expression similar to equation (5) may be derived for PD model results:
Thus, PD estimates of future expansion may noticeably exceed UD estimates (for K > 0.2).
Consider an example based on the UD result, equation (5). Suppose that the greenhouse-gas contribution to the 1880-1985 global warming is 0.4-0.6°C, and that the future forcing follows the intermediate scenario. The implied 1985-2025 greenhouse-gas -induced warming is 0.64-0.96°C, and the corresponding range for [Delta]Z, allowing for uncertainties in K in the range 0.5-2.0 cm2s-1 is 3.8-7.8 cm. We consider this range of values to be the best estimate of future oceanic th ermal expansion effects over the 1985-2025 interval. Details of changes as a function of time between l985 and 2025 are shown in Fig. 5.
The lower value is probably close to a lower bound, as it is unlikely that the greenhouse-gas contribution to [Delta]T (1880-1985) is <0.4°C (0.4°C would require [Delta]T2x to be l.2-l.3°C). For the low-forcing case and [Delt a]T0 = 0.4°C, [DeltaA1 is 2.9 cm (equation (5) gives 3.0 cm). To obtain an upper bound, consider first the accepted upper bound for [Delta]T2x of 4.5°C. With 0.5 ¾ K ¾ 2.0 cm2s-1 , this would imply a greenhouse-gas contribution to l880-1985 temperature changes of [Delta]T0 = 0.93-1.09°C (see Fig. 2). Such large values would require a very large additional (negative) forcing to be operating on the century timescale to be compatible with observations. A more realistic upper bound to [Delta]T0 is 0.8°C (implying [Delta]T2x in the range 2.8-3.5°C). For the high-forcing case, the corresponding value for [Delta]T1 is l .80 76;C and the maximum (K = 2.0 cm2s-1) value of [Delta]Z1 is 13.4 cm (equation (5) gives l3.7 cm). Figure 5 shows the time evolutions of these upper and lower extremes.
The [Delta]Z1 range of 3.8-7.8 cm, with extreme limits of 2.9-13.4 cm, is noticeably less than other estimates of the future thermal expansion effect. Gornitz et al.2 give a value of 20 cm for the change f rom 1980 to 2050 which can be translated to a 1985-2025 change of ~l0 cm. Hoffman et al.9 give values of 6.5, 13.1 and l8.3 cm as low, intermediate and high values for l980-2025 (equivalent to 6.0, l2.5 and 17.6 cm fo r l985-2025). More recently these authors give lower and upper bounds of 5.7-10.8 cm (l985-2025)53. These are the only other publications where time-dependent, transient-response estimates have been made which can be compa red with the present work. The differences between these estimates and the present work arise from model differences (all other work has been based on PD models which produce larger expansion effects; see equations (2) and (6)), from differences in the a ssumed future forcing scenarios, and from differences in the way spatial variations in ß have been accounted for and model parameter ranges have been chosen.
This work was supported by grants from the US Department of Energy, Carbon Dioxide Research Division and the European Economic Community Climate Programme. The National Center for Atmospheric Research is sponsored by the NSF. Comments from R.A. Warrick helped to improve this paper substantially. Radiative transfer model results were kindly provided by J.T. Kiehl.
* Environmental and Societal Impacts Group, National Center for Atmospheric Research, Boulder, Colorado 80307-3000, USAt Climatic Research Unit, University of East Anglia, Norwich, NR4 7TJ, UK
Received 26 May; accepted 17 September l987.
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