3.1. Limitations of Using AGCMs to Downscale Tropical Cyclone Activity
 It is tempting to believe that specification of sea surface temperatures and other boundary conditions such as sea ice in a high quality model leads to an accurate estimate of the state of the atmosphere. The success of AGCMs such as the NOAA-CIRES 20th Century reanalysis in replicating key features of the atmospheric circulations reinforces this idea. Specification of the sea surface temperature circumvents the need for surface energy balance, as would be the case in a coupled system in which the sea surface temperature is predicted. In general, there is no need for energy balance at the top of the atmosphere (TOA), since the ocean can act as a net heat source or sink when sea surface temperature is specified. Here we show that failure to achieve TOA energy balance will generally lead to an incorrect estimate of potential intensity even when the sea surface temperature is correct.
 We begin with the conservation equation for the moist static energy, h:
where V is the large-scale velocity, ρ is the air density, and Frad and Fc are the energy fluxes by radiation and by small scale turbulence (including dry and moist convection). Integrating (2) through the depth of the atmosphere and assuming equilibrium (steady-state) conditions, we obtain
where Fc0 is the surface turbulent net enthalpy flux, Frad TOA is the net top-of-the-atmosphere radiative flux, Frad0 is the net surface radiative flux, and the last term is the integral over the atmosphere of the horizontal divergence of the moist static energy flux by large-scale motions. The first two terms on the right side of (3) also represent the vertically integrated radiative cooling of the atmosphere.
 Now if the ocean mixed layer is itself in thermodynamic equilibrium, then Fc0 + Frad0 = 0, i.e. there is no net energy flux through the sea surface. In that case, according to (3), the TOA radiative flux balances the net convergence of energy into the column. If, however, the sea surface temperature is specified, there is no requirement for energy balance because the sea may act as a net source or sink of energy. Symbolically, we can write
where E is the net energy source supplied by the ocean.
 The turbulent flux of enthalpy at the sea surface may be represented by a bulk aerodynamic formula:
where Ck is the enthalpy exchange coefficient appropriate to 10 m altitude, k0* is the saturation enthalpy of the sea surface, k10 is the enthalpy at 10 m, and elsewhere the subscript 10 denotes evaluation at 10 m. Combining (5) and (4) yields
 The left side of (6) represents the thermodynamic disequilibrium between the ocean and the atmosphere, the principle factor in the definition of potential intensity [Emanuel, 1986]. Thus, if one holds the surface wind speed ∣V10∣ fixed in (6), the potential intensity will vary with the net radiative flux into the ocean and the residual energy imbalance at the sea surface. In general, raising the sea surface temperature from its equilibrium value will increase both E and the net radiative flux into the sea, the latter arising from increasing atmospheric temperature and water vapor (assuming that clouds remain fixed). But, in general, this increase in back radiation is not enough to offset the imbalance term E in (4) or (6), thus both the convective flux and the sea surface thermodynamic disequilibrium increase faster for a specified increase in sea surface temperature than for an equilibrium increase (for which E = 0) brought about by, say, an increase in carbon dioxide content.
 This point is illustrated in Figure 12, which shows the potential intensity as a function of sea surface temperature produced by a radiative-convective equilibrium model [Rennó et al., 1994] run to equilibrium for specified sea surface temperature and for equilibrium sea surface temperature calculated using a slab ocean and varied by varying the concentration of CO2 in the atmosphere. In these simulations, the distribution of clouds is held fixed but water vapor is permitted to vary according to the model's own hydrological cycle. In the calculation of potential intensity, the thermodynamic efficiency is held fixed so as to focus on the changes owing strictly to changes in the surface thermodynamic disequilibrium. When the sea surface temperature is specified, the rate of increase of potential intensity is about a third larger than when the sea surface temperature is in equilibrium with a specified CO2 concentration.
Figure 12. Variation of potential intensity with sea surface temperature in a single-column model run into a state of radiative convective equilibrium. Blue curve: specified sea surface temperature; Green curve: surface energy balance enforced with SST variation brought about by varying atmospheric CO2.
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 Thus a fundamental limitation of using AGCMs to deduce tropical cyclone activity is that errors in the surface energy balance will lead to errors in potential intensity, which has a strong effect on both the intensity and frequency of tropical cyclones. For example, AGCMs run with specified increases of SST without a concomitant increase in surface radiative forcing may overpredict the increase in potential intensity and thereby overpredict the response of tropical cyclone activity to warming. Thus the underprediction of the response of tropical cyclones to changing climate, so evident in Figure 8, is probably not related to the lack of surface energy balance in AGCMs.
3.2. Other Considerations
 It is evident from (6) that the sensitivity of air-sea thermodynamic disequilibrium to changing radiative forcing is highly sensitive to surface wind speed as it is represented in surface flux formulations. This is worrisome, as it is common, for example, to add “gustiness factors” to the calculation of mean wind in models, and this varies from model to model [Fairall et al., 2003]. (This problem applies equally to coupled models and to AGCMs.) This may help explain why the tropical mean potential intensity varies by more than 30% among the seven 20th century climate simulations used in the downscaling study by Emanuel et al. . This problem is compounded by differences in simulated mean surface winds among the various models, which is particularly acute in regions of low mean surface wind speeds, as is evident from (6).
 Another problem arises from the difficulty that GCMs experience in simulating trends in temperature in the upper troposphere and lower stratosphere [Cordero and Forster, 2006]. This affects the potential intensity by influencing the altitude, and therefore the temperature, at which air flowing up through the eyewall of a hurricane attains neutral buoyancy with respect to its environment; this is known as the “outflow temperature” [Emanuel, 1986]. For the downscaling simulations described here, the outflow temperature is calculated in the course of finding the potential intensity using the algorithm discussed in Bister and Emanuel . While the outflow temperature is often confused with the temperature at a fixed altitude, it is also a function of the entropy of parcels ascending in the eyewall, and therefore of the sea surface temperature.
 Unfortunately, upper air observations were sparse or nonexistent during much of the period covered by the 20th century reanalysis used here, so that it is not possible to compare the outflow temperatures calculated from this reanalysis with observations. Instead, we compare the outflow temperatures of two different reanalyses that assimilated all available upper air observations during the period 1980–2001: the NCAR/NCEP reanalysis [Kalnay and co-authors, 1996] and the ERA-40 reanalysis [Uppala and co-authors, 2006]. Figure 13a compares the time series of ERA-40-derived outflow temperature averaged over the Atlantic Main Development Region during August–October with that calculated from the NCAR/NCEP reanalysis over the same region and period. Beginning in the early 1990s, there is a large downward trend in the NCAR/NCEP-derived outflow temperature, resulting in a decline of almost 5 degrees over about 12 years. But there is no trend in the ERA-40-derived outflow temperature during this period. Figure 13b compares the potential intensities over this region and period; while the ERA-40 series shows no trend, there is a significant upward trend in the NCAR/NCEP-derived potential intensity. Figure 13c compares a measure of the air-sea thermodynamic disequilibrium between the two products, showing that this contribution to the potential intensity is not responsible for the differing trend in potential intensity; in fact, the trend in ERA-40 is significantly larger.
Figure 13. Comparison of ERA-40-derived (green) with NCAR/NCEP reanalysis-derived (blue) outflow temperatures (a), potential intensity (b), and air-sea thermodynamic disequilibrium (c) during the years 1980–2006. Values are averaged over the Atlantic genesis region, August–October. In (c), the metric is saturation entropy of the sea surface minus saturation entropy at 600 hPa.
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 Figure 14 compares the annual tropical cyclones numbers downscaled from the NCAR/NCEP and ERA-40 reanalyses to best-track data for the period 1980–2001. The trend of the NCEP-downscaled counts is very nearly equal to the observed trend, while the ERA-40-downscaled trend is much less; moreover, the NCEP-downscaled counts are highly correlated with the observed frequency of events r2 = 0.65, while the ERA-40-downscaled counts are uncorrelated with the best-track data. The evidence presented here points to the lack of trend in outflow temperature in the ERA-40 reanalysis as the principal culprit in the underprediction of the downscaled storm frequency trend. This leads to an underprediction of the trend in potential intensity (Figure 13b), which influences the downscaled storm counts.
Figure 14. Comparison of Atlantic annual storm counts from best-track data (blue) to those from the NCAR/NCEP (green) and ERA-40 (red) reanalyses. Linear trends are superimposed.
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