A radiative scaling for the warm season nocturnal boundary layer (NBL) is proposed, based on the daily mean surface net longwave radiation flux. Using this scaling, a conceptual model is proposed for the NBL, with parameters estimated from multiple linear regression of model data from the European Centre reanalysis, averaged over river basins from the tropics to high latitudes. A radiative temperature scale, computed from surface net longwave radiation flux and the slope of the Stefan-Boltzmann law, primarily determines the strength of the NBL and the amplitude of the diurnal temperature range, although the length of the nighttime period and the surface wind stress play important subsidiary roles. A related radiative velocity scale or radiative conductance, the duration of the nighttime period and the ratio of the scaled surface heat flux (which increases with wind stress) to the NBL strength determine the depth of the NBL. From an observational perspective, this suggests that the diurnal temperature range may give a useful estimate of surface net longwave radiation flux. From a modeling perspective, this provides a framework for relating model physical parameterizations, especially the coupling at night between the surface, the ground and the atmosphere, to observables, the diurnal temperature range and the strength and depth of the NBL. The model is then applied to estimate the nocturnal rise in concentration of gases such as CO2 and radon that are emitted at the surface.
 This paper proposes a radiative scaling for the nocturnal boundary layer (NBL) over land, based on the 24-h mean surface net longwave radiation flux (LWnet24). The same radiative scaling is also applied to the diurnal temperature range (DTR), an important climate parameter. Worldwide, the DTR has been decreasing in recent decades [Karl et al., 1993; Horton, 1995; Easterling et al., 1997]. Coincident increases of cloud cover, which would reduce outgoing LWnet, have been noted by many authors [e.g., Henderson-Sellers, 1992; Dessens and Bucher, 1995; Jones, 1995]. Indeed, the analysis of Dai et al.  provides evidence of the close coupling of cloud cover with DTR.
 There has been extensive development of similarity theory for the stable boundary layer (BL), but the role of radiative cooling is generally not considered in this scaling [e.g., Nieuwstadt, 1984; Derbyshire, 1994; Stull, 1988]. A nighttime surface energy balance model was developed by Holtslag and De Bruin , in which the so-called isothermal net radiation [Monteith, 1981] plays an important role. Over land, however, the diurnal cycle is driven by daytime radiative heating and the nocturnal longwave cooling of the surface [e.g., Betts, 2003]. The development of mixed layer models were a major breakthrough in understanding the unstable daytime dry and cumulus BLs [Deardorff et al., 1969; Betts, 1973; Carson, 1973; Tennekes, 1973], as well as mixed stratocumulus layers [Lilly, 1968]. These recognized the role of the radiative forcing at the surface and at cloud-top in generating the turbulence, which produces nearly well-mixed layers. Surprisingly, comparatively little attention has been paid to the corresponding role of radiative forcing in determining the strength and depth of the NBL; so this is the focus of this paper. Conceptually, it is a different starting point from the traditional wind and stability framework, although we will show that wind stress remains a significant independent factor, as it affects the coupling between surface and atmosphere. This analysis will show that the strength of the NBL at sunrise (which will be defined later in section 2.3, equation (5a) as a temperature difference, ΔTN) and the DTR are directly related to a radiative temperature scale, associated with the slope of the Stefan-Boltzmann equation. We start using the framework of model data from the European Centre for Medium-range Forecasts (ECMWF) reanalysis (known as ERA-40 [Uppala et al., 2005]) to illustrate the scaling, and quantify relationships using multiple linear regression. In the model data, after scaling, ΔTN and DTR increase with the growth-time of the NBL and decrease weakly with increasing wind stress. A conceptual model for the depth, h, of the NBL is then developed in terms of a radiative velocity scale, and the scaled surface heat flux which increases steeply with wind stress. This simple model is then applied to estimate the nocturnal rise in concentration of gases such as CO2 and radon that are emitted at the surface and trapped within the NBL.
 Modeling the stable boundary layer (BL) has presented considerable difficulties in global models. In the ECMWF model, the stable BL was reformulated [Viterbo et al., 1999] to reduce cold temperature biases in winter over continental areas. Changing the stability functions introduced more diffusion at high stable Richardson numbers, which in turn increased the coupling between surface and atmosphere, and reduced the fall of surface temperature at night, driven by radiative cooling. However, more diffusion gives deeper stable BLs, which may now be too deep in the model (Beljaars, 2005, personal communication). Viterbo et al.  also addressed the effect of the thermal inertia of soil freezing in damping the response of the surface temperature in winter. The ERA-40 reanalysis model uses this same formulation of the stable BL (documentation is available at http://www.ecmwf.int/research/ifsdocs/CY23r4/index.html), although the model data presented here will be for the warm season. Since this analysis is based entirely on model output data, our conclusions do depend on the model physics parameterizations, particularly the stable BL and ground heat flux parameterizations. The model data we use is averaged over river basins (see next section). This smooths and simplifies the analysis, but this means that the evaluation of our results against observations is not an easy task, and it is left to future work.
2. Scaling of Mean Diurnal Cycle
 Model data can provide powerful insights into the coupling of physical processes, even though quantitatively there will be dependence on a given model's specific parameterizations. The ERA-40 archive [Kållberg et al., 2004] contains hourly data averaged over river basins. These were originally archived to study the hydrometeorology of river basins [see Betts et al., 2003, 2005]. In other papers [Betts, 2004; Betts and Viterbo, 2005] these data have been used to explore the interrelation of the surface fluxes, the BL, the cloud fields and surface radiation balance on the daily timescale. Betts  showed one example of the relationship of the DTR to the diurnally averaged surface net longwave radiation flux for the Madeira River. This analysis of the DTR is extended here to include other river basins across the Americas, and adapted to address the radiative scaling of the NBL. Table 1 identifies the subbasins of the Mackenzie, Mississippi, Amazon and La Plata rivers and their approximate latitude and longitude. The data period we have chosen for analysis is a seven-year period, 1993–1999. The model data are used in two forms. In this section, the monthly mean diurnal cycle is presented to illustrate the radiative scaling of the diurnal cycle. In section 3, parameters are extracted for each day to relate the strength of the NBL, the DTR and the surface heat flux to the surface net longwave radiation flux.
Table 1. ERA-40 River Basins
Latitude, Longitude (Mean)
Great Bear Lake
Great Slave Lake
2.1. Mean Diurnal Cycle
Figure 1a shows the mean diurnal cycle of 2-m temperature (T2) for the Madeira River basin for four months. January and February are in the heart of the rainy season, and both show a small amplitude diurnal cycle of temperature. Six months later, July and August are in the dry season, and the diurnal temperature range
(where Tmax, Tmin are the maximum and minimum values of T2) has about twice the amplitude. For this Amazon basin, sunrise is around 1030 UTC; slightly later in July and August, as the basin is south of the equator). Figure1b shows that the surface LWnet has a similar increase from wet to dry season. Since the NBL is created by net long-wave cooling of the surface at night, the stronger the outgoing LWnet, the greater the cooling of the surface, and the larger the amplitude of the diurnal temperature cycle [Betts, 2004]. In the dry season, the atmospheric water vapor and cloud cover are reduced, and the lifting condensation level (LCL) is higher, and all these factors contribute to a greater outgoing LWnet [Betts and Viterbo, 2005].
2.2. Radiative Temperature Scale
Betts  showed that the diurnal temperature range was related to the 24-h mean net longwave radiation flux, LWnet24, and to a related radiative temperature scale, defined as
where λ0 = 1/(4σT3) is the radiative sensitivity parameter from the differential of the Stefan-Boltzmann law, in which σ = 5.67 10 −8 W m−2 K−4 is the Stefan-Boltzmann constant. At T = 293 K, 4σT3 = 5.7 Wm−2 K−1, giving λ0 = 0.175 K/(Wm−2).
2.3. Scaled Diurnal Cycle
 The radiative temperature scale (2) can be used to scale the diurnal cycle of temperature, by defining a scaled temperature perturbation from the 24-h mean temperature, T24, as
Figure 2 shows that the transformation from T2 to Tsc collapses the widely different diurnal temperature cycles shown in Figure 1a into almost a single curve. The scaled amplitude of the diurnal temperature cycle is defined as
For this tropical basin, DTRsc ≈ 1.
 The diurnal cycle of the sensible heat flux, H, for the same four months is also shown in Figure 2 (right-hand-scale). In the late afternoon or evening, temperature falls from Tmaxsc as H falls. For the purpose of this paper, the NBL is defined as the period when H < 0. The NBL starts growing, when H first becomes negative at an unscaled temperature, TN, corresponding to the scaled temperature TNsc shown. At this point, the potential temperature gradient near the surface is neutral, as the transition occurs from an unstable to a stable BL. To get a better estimate of TNsc, adjacent hours were interpolated linearly to give the value of Tsc at H = 0. The NBL reaches its maximum strength at sunrise, when Tsc = Tminsc (and T2 = Tmin). The NBL is growing during the period between the vertical dotted line near 2000 UTC, and the dotted line at the morning temperature minimum. For this paper, the strength of the NBL at sunrise is defined as the difference in surface temperature from the initial formation of the NBL, in unscaled units
and in scaled units
Only the ERA-40 surface parameters are analyzed in this paper, and simple vertical profiles for the NBL will be assumed for energy balance calculations in section 4. The analysis of the ERA-40 vertical profile data is a much larger task (we have years of hourly data for 18 basins), which will be attempted later.
2.4. Scaled Temperature Diurnal Cycle Across Different River Basins
Figure 3 shows the scaled diurnal temperature cycle across the set of river basins. The location of the river basins is shown in Figure 7 of Betts .The Amazon basins in Figure 3a are very similar, have the largest amplitude, and very little seasonal variation (not shown). For the La Plata (which is a single basin in the ERA-40 archive), the amplitude is a little smaller and decreases further in the austral winter, which is rather mild at this latitude. For the Mississippi subbasins, a summer average from May to August is shown in Figure 3c, and the amplitudes are also smaller, and decrease slightly from the most southern basin (Lower Mississippi) to the Missouri basin. Finally for the high latitude Mackenzie River in Figure 3d, three representative basins are shown. The amplitude of the scaled diurnal cycle decreases still further to the most northern basin shown, Great Bear Lake, which is close to the Arctic circle (see Table 1). We conclude that the scaling by the radiative temperature (2) appears to be useful across a wide range of latitudes and climates. However, the scaled amplitude decreases as latitude increases and night-length decreases. This will be explored further using daily data in section 3.2.
2.5. Scaled Surface Energy Balance
 The surface energy balance is
where SWnet, λE and G are the surface net shortwave flux, latent heat flux and ground flux respectively. For the NBL period when H < 0, SWnet ≈ λE and both are small (SWnet is not identically zero, because H becomes <0 before sunset). This leaves the approximate energy balance for the NBL period
The sign convention is that all terms in (7) are negative. This partition of the outgoing LWnetN energy loss into fluxes from the atmosphere and the ground does depend on the model parameterizations for the stable BL and the ground heat flux. It also depends on wind stress (see section 3.3, later).
 Scaling by -LWnet24 (for consistency with equation (2)) gives
where both the scaled NBL sensible heat flux and ground heat flux are now defined positive. The ratio LWnetN/LWnet24 ≈ 0.75 for these data.
3. Model Parameters From Daily ERA-40 Surface Data
 Parameters are extracted for each basin for each day in the period 1993–1999 to study the relationship of the amplitude of the diurnal cycle, the strength of the NBL and of the nighttime surface heat flux to the surface longwave cooling, NBL growth time, surface wind stress. The original model hourly data used are from single 24-h forecasts from each 0000 UTC analysis, so there can be a small discontinuity at 0000 UTC time, as the originating analysis changes, but this factor was ignored. The three temperatures, Tmax, TN and Tmin were found, as outlined in the discussion of Figure 2, for each 24-h time period starting at 1500 UTC (corresponding to roughly mid-morning for these basins in the Americas). From (1) and (5a), these define the DTR and ΔTN. Using equations (2), (3), (4) and (5b), the scaled amplitudes DTRsc and ΔTNsc were then calculated for each 24-h period. For the NBL period (defined again as the hours when H < 0), the mean values of HN, and the other surface fluxes including the surface stress, were computed, as well as the mean 10-m wind-speed, V10N. The growth time of the NBL, τN, was computed by subtracting the time at Tmin from the earlier time at TN, found by linear interpolation to where H = 0. Only a few percent of the days were rejected from the analysis if they showed a temporal structure grossly inconsistent with Figure 2, often due to large temperature advection during the 24-h period.
3.1. Diurnal Temperature Range Dependence on ΔTR
Figure 4shows the dependence of the DTR on ΔTR for two basins of the Amazon, which have a large seasonal cycle, and the summer data (June, July and August) for the subbasins of the Mackenzie, together with the regression lines through the origin. For the Amazon the regression line slope is close to 1, which is consistent with Figure 3a, and correlation is quite high (R2 = 0.84). For the Mackenzie, the slope is smaller, only 0.73, consistent with Figure 3d, and the correlation is weaker (R2 = 0.59). The basins of the Mississippi and the LaPlata (not shown) lie in between these two extremes. The corresponding plots for the dependence of ΔTN on ΔTR are very similar with smaller slopes, so it is clearly useful to explore the dependencies of the scaled amplitudes DTRsc and ΔTNsc on other variables. The regression coefficients are higher using LWnet24 to calculate ΔTR in (2), than if LWnetN were used (not shown).
3.2. Dependence of DTRsc and ΔTNsc on NBL Growth Time, τN
 One factor affecting the decrease of DTRsc from the tropics to the high latitudes is the decreasing nighttime period in summer for the development of the NBL. To show this, we binned all the daily data (in 0.1 ranges of τN) from the warm season, the months June-August for the Mackenzie, May to August for the Mississippi and the corresponding months November-February for the southern hemisphere La Plata basin, together with 12 months of data from the two Amazon basins with the largest seasonal cycle (basin 42 the Madeira, and 41 the Tapajos/Xingu, see Table 1), a total of 13731 days of model data. Figure 5 shows that in the mean the scaled diurnal temperature range DTRsc and strength of the NBL ΔTNsc increase together with the duration of the NBL growth time, τN. For the most northern basins of the Mackenzie, τN ≈ 9 h in summer, and DTRsc ≈ 0.7; while for the Amazon basins, τN ≈ 13.5 h, and DTRsc≈ 1.02, consistent with Figures 3 and 4. ΔTNsc is systematically lower than DTRsc, with the mean ratio of ΔTNsc/DTRsc = 0.90 ± 0.07 for this set of days.
3.3. Dependence on Surface Stress and Surface Wind
 Another factor affecting DTRsc, ΔTNsc and the NBL heat flux, HNsc is the strength of the coupling between the surface and the atmosphere, which will be represented here by the surface friction velocity. For the same data set of 13731 warm season days, Figure 6a shows the dependence of the NBL 10-m wind, V10N, in the model on the NBL friction velocity, defined as
where StressN is the mean surface stress for the NBL period and ρ the corresponding mean density. Not surprisingly there is a quasi-linear relationship. The variability depends on the surface roughness, which at each gridpoint in the model has a local component, depending on vegetation type, and an orographic component; and in addition a stability dependence through the model surface layer parameterization. Thus, V10N has a strong dependence on the model parameterizations, and the data will be stratified by NBL friction velocity (related to wind stress), rather than wind-speed. In fact, the multiple linear regression correlation, discussed in the next section, is higher with UstarN as an independent variable than with V10N.
Figure 6b shows the dependence of DTRsc (top right-hand-scale), ΔTNsc and the NBL heat flux, HNsc on UstarN and τN, for three values of τN, together with representative standard deviations. Only τN = 12h has sufficient data in the highest UstarN class. DTRsc and ΔTNsc both increase with τN, as shown in Figure 5, and decrease slowly with increasing friction velocity (and windspeed). In contrast, the scaled surface heat flux, HNsc, has almost no dependence on τN, but increases steeply with friction velocity, as the coupling between surface and atmosphere increases. The corresponding ground storage flux, GNsc, in equation (8) decreases with increasing UstarN (not shown).
3.4. Linear Regression Fits
 For this same set of 13731 warm season days of model data from the Amazon, LaPlata, Mississippi and Mackenzie basins, multiple linear regression was used to fit ΔTNsc, DTRsc and HNsc as functions of τN and UstarN. This gave
The R2 coefficient is higher for Hsc than for ΔTNsc and DTRsc.
Figure 7 shows the structure of these linear fits. The dependence on τN is greater for ΔTNsc and DTRsc than for HNsc, and the reverse is true for the dependence on UstarN. Note the close similarity (for regions where there was data) to Figure 6b, which is plotted with identical scales. These regression fits from model data, shown in Figure 7, will be used in the next section as input to our simple model. As mentioned earlier, UstarN as an independent variable gives higher regression coefficients than using windspeed V10N. The scaling by LWnet24, rather than LWnetN, gives higher regression coefficients for DTRsc and ΔTNsc but not for the nighttime flux HNsc.
4. Conceptual Model for the NBL
 A conceptual model for the NBL, using LWnet24 as a scaling parameter, is now proposed.
4.1. Simplified NBL Structure
Figure 8 shows the vertical structure of the conceptual model. After the local temperature maximum, we suppose the daytime mixed layer is exactly neutral (with constant potential temperature, θ) at the moment when H changes sign from positive to negative. At this point the 2-m temperature, TN, lies on this same dry adiabat. During the nocturnal period, lasting time τN, the surface cools and the NBL is created. By dawn, after the nighttime cooling, an NBL has formed with depth h, with a surface cooling of ΔTN = TN − Tmin below the potential temperature of the preceding daytime mixed layer. The temperature profile with height will be assumed: Figure 8 shows the linear and quadratic profile (with ∂θ/∂z =0 at height h), that will be used. The radiative cooling of the residual mixed layer above h will be neglected.
4.2. Energy Balance for the NBL
 Integrating over the NBL growth period, τN, the heat storage in the NBL satisfies
where ρ is air density and Cp is the specific heat of air at constant pressure (both assumed constant). The radiative flux divergence within the NBL is an additional term in the energy budget, which will be ignored for simplicity; although it could also be related to the surface LWnet. For the linear, quadratic profiles of NBL temperature shown in Figure 8, β = 0.5, 0.333. Scaling by −LWnet24 and using (2), (3) and (5b) gives
Rearranging gives the depth of the NBL
where the radiative velocity scale (or radiative conductance) is given by
At 293K and 1000hPa, WR ≈ 0.0048m s−1, which is about 40 hPa day−1. Substituting
so that with these simple assumptions, a typical NBL depth for τN ≈ 12 hours is of order 20 hPa or 200m.
 More exactly, equation (15) shows that (for constant β) h is proportional to the scaled heat flux, Hsc, the length of the stable period, τN, and inversely to ΔTNsc. Using equations (11) and (12), the regression fits from the ERA-40 data in Figure 7, as an estimate of these dependencies, (15) can be rearranged as
Figure 9 shows the structure of the solution of (18a) at 293 K, for both linear and quadratic NBL profiles. NBL depth at sunrise increases steeply with mean nighttime UstarN, and has little dependence on NBL duration. Comparing Figures 7 and 9, increasing UstarN gives a larger sensible heat flux and a somewhat smaller NBL strength, giving a deeper NBL in the coupled system.
Monteith  noted the importance of the velocity scale (16) in surface energy transfer. He defined the reciprocal, ρCpλ0, as the resistance to longwave radiative transfer. His analysis however focused primarily on the daytime BL. Nieuwstadt  derived a rate equation for NBL depth, related to the surface heat flux and cooling rate. He did not use the surface LWnet as a scaling parameter, but instead related the radiative cooling rate to the surface cooling rate with an undetermined coefficient, noting that NBL growth solutions depended on his radiation parameterization.
4.3. Nighttime Rise of CO2 From Respiration
 This simple model can be used to estimate the nocturnal rise of other gases that are emitted at the surface and trapped within the NBL. If we confine the CO2 from respiration at night in a similar way within the NBL with a profile up to depth h, we get the storage balance
where RESPN is the nighttime respiration in ppm CO2 m s−1, and ΔCO2N is the surface increase in surface CO2 at night above the previous daytime ML. Once again, β = 0.5, 0.33 for linear and quadratic profiles. Substituting for h from (15) gives a corresponding estimate of ΔCO2N (both β and τN cancel)
Figure 10 shows that the ratio R = (ΔTNsc/HNsc) varies weakly with τN, but decreases steeply with UstarN as the NBL gets deeper. Over the range of UstarN from 0.1–0.5 m s−1 (corresponding in the model to a range of V10N from roughly 0.8 to 3.6 m s−1), ΔCO2N decreases from 237 to 47 ppm (for τN = 12 h) for a typical respiration rate, RESPN = 0.2 ppm CO2 m s−1, and WR ≈ 0.0048 m s−1 as before. For fixed respiration rate, the ratio R can be thought of as the amplifier, which increases the NBL rise of CO2 by reducing the NBL depth, h, in (18a), when the surface wind and wind stress are low.
4.4. Nighttime Rise of Radon From Surface Emission
 Radon is produced in the soil from decay of radium at a constant rate, and the rate of emission from the soil is approximately constant. Radon has a decay constant, Ld = 2.089 * 10−6 s−1 (corresponding to its half life of 3.8 days), so typical concentrations of radon in the free troposphere are small, about 0.4 Bq m3 at STP (standard temperature and pressure), and 1.8–3.7 Bq m3 in the daytime mixed layer [Kritz et al., 1998]. However, in the NBL, radon accumulates and the fraction that decays as the NBL grows is small, as it depends on the product τNLd ≈ 0.09, for τN =12 h. If this small nocturnal decay term is neglected, the nighttime rise of surface radon, ΔRnN, is approximately given by
Over the same range of Ustar from 0.1–0.5 m s−1, ΔRnN decreases from 24.9 to 4.9 Bq m−3 for a typical emission rate, FRn = 0.021 Bq m−2 s−1. A more exact calculation of the decay loss requires the mixed layer radon at the time when the NBL first forms and decouples from the layer above.
 This paper proposes a model for the strength and depth of the warm season NBL based on radiative scaling by the outgoing net longwave flux. This framework differs from the traditional similarity approach to modeling the stable BL in that it emphasizes the primary role of the nocturnal radiative cooling in generating and maintaining a stable structure at night. Historically, comparatively little attention has been paid to the importance of radiative forcing in determining the strength and depth of the NBL. This analysis shows that the strength of the NBL at sunrise, defined here as ΔTN, and the diurnal temperature range, DTR, are both related to the radiative temperature scale, ΔTR, derived from the LWnet and the slope of the Stefan-Boltzmann equation. Hourly reanalysis data from ERA-40, averaged over river basins, is used to illustrate the scaling, and extract quantitative relationships using multiple linear regression. When scaled by ΔTR, ΔTNsc and DTRsc increase with the length of the nighttime stable period and decrease weakly with surface wind stress. The ratio ΔTNsc/DTRsc ≈ 0.9 ± 0.07. The scaled heat flux increases steeply with wind stress, which controls the strength of the coupling between surface and atmosphere, and the partition between sensible heat flux and ground heat flux. From the NBL energy budget, a simple model for the depth, h, of the NBL was then developed in terms of a radiative velocity scale and the scaled surface heat flux, which depends primarily on wind stress. Since increasing wind stress gives a larger sensible heat flux and a somewhat smaller NBL strength, the result in the coupled system is that the depth of the NBL at sunrise increases steeply with wind stress. This simple model for NBL depth was used to estimate the amplification of the nocturnal rise in concentration of CO2 and radon, that are emitted at the surface.
 Numerically, the results are specific to this reanalysis model, ERA-40, and its stable BL and ground heat flux parameterizations, documented in Viterbo et al. . In addition, the analysis has focused on the warm season. However, since both DTR and ΔTN are observables, this analysis framework can be used to assess the model coupling parameters at the surface (to the atmosphere and to the ground). The model suggests that the DTR, an important climate parameter, is a useful linear measure of the mean surface LWnet. The analysis here used the 24-h mean LWnet as a scaling parameter for the NBL (as in Betts, 2004). The measurement of nighttime surface fluxes, such as HN is however challenging observationally.
 This paper has focused on the growth of the NBL at night in the warm season, and shown also that the DTR is closely related to the strength of the NBL. It might seem surprising that the daytime processes only appear in the analysis through LWnet24. For example, Durre and Wallace  have shown the impact of evapotranspiration in reducing the DTR during the growing season. Betts and Viterbo  showed, for the Madeira subbasin of the Amazon, that diurnally averaged LWnet24 is related in ERA-40 to both cloud cover and the mean depth of the mixed layer (which is also related to the DTR), and to mean evaporative fraction, itself linked in the model to soil water. In the model, all these processes and the entire diurnal behavior are coupled on the daily timescale, with the result that DTR and ΔTN have a close relationship to LWnet24.
 A. Betts acknowledges support from NSF under grant ATM0529797, from NASA under NEWS grant NNG05GQ88A, and from ECMWF for travel. I am grateful to A. Beljaars and P. Viterbo for helpful discussions and to the entire ERA-40 team for their assistance. Suggestions from two reviewers have significantly improved the paper.