Quarterly Journal of the Royal Meteorological Society. Comparison of a single column model in weak temperature gradient mode to its parent AGCM

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1 Quarterly Journal of the Royal Meteorological Society Comparison of a single column model in weak temperature gradient mode to its parent AGCM Journal: QJRMS Manuscript ID: QJ--0.R Wiley - Manuscript type: Research Article Date Submitted by the Author: 0-Oct- Complete List of Authors: Zhu, Hongyan; CAWCR, Bureau of Meteorology Sobel, Adam; Columbia University, Department of Applied Physics and Applied Mathematics Keywords: tropical dynamics, SST and precipitation, tropical circulation

2 Page of Quarterly Journal of the Royal Meteorological Society Comparison of a single column model in weak temperature gradient mode to its parent AGCM Hongyan Zhu Centre for Australian Weather and Climate Research (CAWCR). A partnership between CSIRO and the Australian Bureau of Meteorology, Australia Adam H. Sobel Department of Applied Physics and Applied Mathematics, Department of Earth and Environmental Sciences, and Lamont-Doherty Earth Observatory, Columbia University, New York, NY USA Corresponding Author Address: Dr. Hongyan Zhu, CAWCR, PO Box, Melbourne, Australia hzhu@bom.gov.au

3 Quarterly Journal of the Royal Meteorological Society Page of Abstract A single column model (SCM) version of the HadGEM is run in weak temperature gradient (WTG) mode, assuming a free-tropospheric temperature profile obtained from the same single column model in radiative-convective equilibrium (RCE) over a sea surface temperature (SST) of 0K. The resulting quasi-steady solutions are compared with climate statistics from time-dependent solutions of the full D atmospheric General Circulation Model (GCM) sharing the same physics, with the aim of evaluating the strengths and weaknesses of the WTG parameterization of large-scale dynamics. Making some allowance for unavoidable differences between quasi-steady D solutions and more fully transient D solutions, the two models produce grossly similar sensitivities of precipitation and relative humidity to local sea surface temperature. The greatest differences arise from the relatively sharp transition in the SCM between very dry and rainy states as SST is varied, while the GCM statistics vary more smoothly with SST. When a relaxation on the moisture field towards a target profile - a crude parameterization of horizontal moisture advection - is included in the SCM, this difference is reduced. The SCM is then able to produce some convection at low SST, and the increases in humidity and precipitation with SST become more gradual, as in the GCM. The RCE temperature profile used to obtain these results is colder in the upper troposphere and thus more unstable to deep convection than is the climatological tropical profile from the GCM. When the latter is used in the SCM, the precipitation as a function of SST does not change greatly, but the convection becomes considerably shallower than that in either the GCM, or the SCM with the RCE temperature profile. We speculate that some of these differences may be due to the much greater transience in the GCM solutions compared to the SCM's quasi-steady states. Key words: tropical dynamics, SST and precipitation, tropical circulation

4 Page of Quarterly Journal of the Royal Meteorological Society Introduction Horizontal temperature gradients are exceedingly weak in the tropical free troposphere, due to the large scale of geostrophic adjustment. The weak temperature gradient (WTG) approximation makes use of this dynamical constraint to parameterize large-scale tropical dynamics such that its interaction with deep convection and cloud processes can be studied in a single column model, or a limited area model with resolved convection. To capture feedbacks of large-scale tropical dynamics on single column model (SCM) physics, Sobel and Bretherton (00) proposed a method for implementing the WTG approximation in an SCM. In this formulation, vertical velocity is parameterized in such a way that adiabatic cooling (warming) balances diabatic heating (cooling) in the free troposphere, consistent with the observation that temperature tendencies and horizontal gradients are both small in the deep tropics. The same authors applied the formulation to an SCM version of the first quasi-equilibrium tropical circulation model (QTCM) (Neelin and Zeng 00, Zeng et al. 00) and compared the results to the full QTCM. The QTCM is a quasi-three dimensional model with truncated vertical structure but full horizontal structure. The comparison was far from perfect, but good enough to demonstrate that the WTG method allowed some essential aspects of large-scale dynamics to be parameterized in at least a qualitatively correct manner. Various forms of the WTG method and related parameterizations of large-scale dynamics have since been applied a number of times in studies with SCMs (Chiang and Sobel 0; Shaevitz and Sobel 0; Bergman and Sardeshmukh 0; Sobel et al. 0; Sobel and Bellon 0; Ramsay and Sobel ) as well as cloud resolving models (Mapes 0; Raymond and Zeng 0; Raymond 0; Kuang 0; Blossey et al. 0; Kuang ; Sessions et al. ; Wang and Sobel ). Bellon and Sobel () used an SCM version of the second QTCM (QTCM; Sobel and Neelin 0) to explain the existence of multiple

5 Quarterly Journal of the Royal Meteorological Society Page of equilibria of the Hadley circulation in an axisymmetric version of the QTCM (i.e., containing latitudinal but not longitudinal structure). Apart from that, the comparison to QTCM by Sobel and Bretherton (00), and a comparison to a two-column model by Shaevitz and Sobel (0), no other comparisons between a SCM parameterized with WTG and a parent model with horizontal structure have been performed; in particular, no comparison to a fully D comprehensive climate model has been performed. As the WTG method is meant to parameterize large-scale dynamics that are fully resolved in a D model, such comparisons would seem useful for evaluating the strengths and weaknesses of that parameterization. In this study, we compare statistically steady solutions of a single column model run under the WTG approximation to statistics derived from simulations by the parent climate model with the same model physics as the SCM. We aim to investigate the degree to which the SCM can broadly capture the dependence of precipitation, relative humidity, and largescale vertical motion on sea surface temperature (SST) found in the GCM.. Model and Experiment Design We use the SCM version of HadGEM (Hadley Centre Global Environmental Model, Martin et. al. 0) developed by the UK Met Office. The results are compared to a -year climate run with the HadGEM model. The climate model is run from January of to December of with observed SST. Similar to Sobel et. al. (0), the SCM is run with fixed SST. The insolation and solar zenith angle are held constant at values of 00 Wm - and zero respectively. The surface wind speed, used in the bulk formulae for the surface fluxes, is set to a constant m s. The time and domain averaged temperature and moisture profile over the Indian Ocean and Western Pacific region ( S- N, 0 E- E) from a year climate run of HadGEM are taken as the initial conditions for the SCM, and then the

6 Page of Quarterly Journal of the Royal Meteorological Society SCM is run to a state of radiative-convective equilibrium (RCE) over an SST of 0 K, which is also approximately the time- and areally-averaged value over the above region. Sensitivity to this SST value is investigated in Section. In this calculation, the large-scale vertical velocity is set to zero. The temperature profile from the RCE calculation is then used as an input to a set of calculations in which the model is modified to implement the WTG approximation (referred to as SCM-WTG). Potential temperature Mixing ratio Figure : RCE profile of potential temperature (left panel) and water vapour mixing ratio (right panel) from the SCM. Also shown are the climatological profiles from the GCM over the Indian Ocean and Western Pacific region. Figure shows the RCE potential temperature and water vapour mixing ratio profiles. In the lower troposphere and boundary layer, the RCE temperature profile agrees well with that from the GCM climatology over the Indian Ocean and Western Pacific region, but is cooler in the upper troposphere compared to the values in the GCM. The difference between the two moisture profiles is less than g kg in the troposphere. The choice of S- N, 0 - E as the representative region from which to derive the SST for the SCM is somewhat arbitrary; it represents a very rough approximation to the most convectively active region, which we expect to set the temperature structure of the entire tropical troposphere (e.g., Wallace ; Sobel et al. 0). The choice is justified after the fact by

7 Quarterly Journal of the Royal Meteorological Society Page of the small difference between the resulting GCM and SCM profiles shown in Fig., at least in the lower and middle troposphere. The greater differences in the upper troposphere are significant and will be addressed specifically below by SCM-WTG calculations using the GCM rather than the SCM temperature profile. As in previous studies (e.g., Sobel and Bretherton 00; Sobel et. al. 0), in the SCM- WTG calculations, the temperature profile is held fixed in time in the free troposphere, defined here somewhat arbitrarily as those levels with pressures less than P bl = 0 hpa. The sensitivity of the results to the choice of P bl is also investigated below. At those freetropospheric levels, the large-scale vertical velocity is diagnosed as that which causes the vertical advection of potential temperature to precisely balance the diabatic heating computed by the model physics, consistent with the requirement of zero temperature tendency. Horizontal advection of temperature is also assumed negligible. In the nominal boundary layer, defined as levels with pressures greater than P bl, the temperature is determined prognostically, with the vertical velocity computed by linear interpolation in pressure between the diagnosed value at P bl and an assumed value of zero at the surface. At all levels, large-scale vertical advection of humidity is integrated prognostically as usual, using the large-scale vertical velocity diagnosed as described above together with the moisture field explicitly computed by the SCM. In other implementations of WTG, rather than the free-tropospheric temperature profile being held precisely constant, it is relaxed towards the reference profile on a finite time scale (e.g., Raymond and Zeng 0; Raymond 0; Sessions et al. ; Wang and Sobel ) or found by solving a vertical structure equation for both the temperature and vertical velocity perturbations, assuming a nonrotating internal gravity wave of fixed horizontal wavelength (Kuang 0; Blossey et al. 0). We do not apply any of these variants, but restrict ourselves to the Sobel and Bretherton (00) method, to limit the size of the

8 Page of Quarterly Journal of the Royal Meteorological Society parameter space to be explored. In the concluding section we discuss to what extent our results may be sensitive to this choice. The model is integrated for months. For all results after those shown in Fig., we use averages over the last month to represent the statistically steady state of the SCM-WTG. Figure a shows the time evolution of daily precipitation for the experiments with SST of 0K, and 0K. After a transient evolution phase of a few to perhaps ten days, statistically steady states are reached in which the precipitation oscillates around the values of mmd and mmd for these two experiments respectively. While Fig. a shows daily mean values, Fig. b shows the last two days' precipitation time series with output every minutes in the experiment with SST of 0K. We see that the precipitation switches between brief periods of deep convection with a rainfall rate of mmd, and periods of very weak or zero rainfall. This is typical of the other model integrations as well, with the variations in daily mean rain being attributable in large part to the frequency of the high-precipitation events. (a) (b) Figure : The time evolution of precipitation for the SCM-WTG: (a) daily output for expts with SST=0K and 0K; (b) every minutes output for expt. with SST=0K for the last two days.

9 Quarterly Journal of the Royal Meteorological Society Page of Results. Control Experiments We carried out SCM-WTG experiments with different SSTs, ranging from K to 0.K in 0. K intervals. We compare these to results from the GCM. Given that the SCM computes unique, statistically steady solutions for each SST [ignoring the possible existence of a second, dry equilibrium state over SST values sufficiently high to support a rainy equilibrium, e.g. Sobel et al. (0), Sessions et al. (), Bellon and Sobel ()] while the GCM values have a broad spectrum of both spatial and temporal variability even at a given SST, it is not entirely obvious to what GCM statistic the SCM should be compared. We choose to use monthly mean values from single grid points in the GCM. We use single grid points because the SCM is essentially equivalent to the column physics from a single grid point. We use monthly averages based on our expectation that the SCM-WTG can capture only time mean statistics since it lacks the D dynamics that leads to transience. While the use of single grid points and monthly means is somewhat arbitrary, we show percentiles of the distribution (over grid points and individual months) at each SST to give a sense of the variability that remains after this degree of averaging; further averaging in either space or time will reduce the widths of these distributions. Figure compares the relationship between precipitation and SST in the SCM solutions to percentiles of the GCM's monthly mean precipitation binned by SST with width of 0. K. The data from the GCM are also subset by the surface wind speed in the range of. ms to.ms. Fig. shows that precipitation in SCM-WTG increases rapidly with SST once the latter rises above 0K. As explained in Sobel et. al. 0, if the ensuing heating exceeds the radiative cooling, it induces large-scale ascent, which moistens the atmosphere further by large-scale advection, leading to the maintenance of the convective state. On the other hand, for SST below 0K, the convective heating is insufficient to balance the

10 Page of Quarterly Journal of the Royal Meteorological Society radiative cooling, and the induced subsidence dries the troposphere and suppresses convection. Interestingly, the critical value which separates the dry equilibrium from the rainy equilibrium is 0K, consistent with the value which is used to calculate the RCE profile for the initial condition of SCM-WTG experiments. When the SST is larger than 0K, the precipitation rate in the SCM-WTG is between the 0 th and th percentile values from the GCM. For SST smaller than 0K, the SCM-WTG precipitation is zero. The GCM values in this SST range, though small, are significantly nonzero. Figure : Precipitation (mmd ) as a function of SST in the SCM-WTG compared to the th, 0 th and th percentiles from the GCM over the Indian Ocean and Western Pacific region. Figures a and c show vertical profiles of relative humidity and vertical velocity as functions of SST in the GCM. The boundary layer is relatively moist regardless of SST, as in observations over the tropical ocean. The relative humidity in the free troposphere increases with SST in the lower troposphere when SST is greater than K. The moistening increases with depth with increasing SST, indicating that convection grows deeper with increasing SST. The vertical velocity for SST smaller than 0K is dominated by descending motion, while ascending motion dominates when SST is larger than 0K. For SST cooler than 0K, in the presence of subsidence, nonzero humidity in the lower troposphere above the PBL is

11 Quarterly Journal of the Royal Meteorological Society Page of presumably due largely to a combination of transience, shallow convection and horizontal advection. Relative humidity (a) Relative humidity (b) Vertical velocity (c) Vertical velocity (d) Figure : Relative humidity (interval 0., > 0. is shaded) (a, b) and vertical velocity (interval 0.00 ms, >0.00ms is shaded) (c,d) as a function of SST and pressure from the climate statistics of the GCM (a,c) and the statistically steady SCM-WTG solutions (b,d). Fig. b shows that for SST less than 0K, the SCM-WTG has relative humidity near zero in the free troposphere, a considerably drier solution than the mean GCM profiles shown in Fig. a. The relative lack of convective instability at low SST and the dry free troposphere inhibit convection, the subsidence and implied moisture export balances the surface evaporation in the PBL, and there is no horizontal advection. Comparing to Fig. a, SCM- WTG is not able to simulate the gradual moistening with depth with increasing SST.

12 Page of Quarterly Journal of the Royal Meteorological Society Including a relaxation on the moisture field towards the RCE profile as a crude parameterization of horizontal moisture advection - changes this result, as will be discussed in Section.. When SST is larger than 0K, convection and large-scale ascent (parameterized by WTG) moisten the free troposphere in the SCM. The relative humidity in the upper troposphere is higher in the SCM (Fig. b) than in the GCM (Fig. a). Relative humidity (a) Relative humidity (b) Vertical velocity (c) Vertical velocity (d) Figure : Relative humidity (interval 0., > 0. is shaded) (a,b) and vertical velocity (interval 0.00 ms, >0.00ms is shaded) (c,d) as a function of SST and pressure in the statistically steady SCM-WTG solutions (b,d); and the time-averaged profiles in GCM over the Indian Ocean and Western Pacific region, where samples with precipitation greater than. mmd are used for SST greater than 0K, and samples with precipitation less than. mmd are used for SST less than 0K (a,c).

13 Quarterly Journal of the Royal Meteorological Society Page of In the SCM-WTG, the vertical velocity fields are consistent with the relative humidity with subsidence dominating in the dry states when SST is smaller than 0K, and ascent moistening the troposphere when SST is greater than 0K (Fig.d). Again, the vertical velocity in the upper troposphere in the latter moist regime is larger than those from the same SST values in the GCM. Much of the difference between the GCM and SCM results in Figs. and presumably is a consequence of the fact that the SCM results consist of single quasi-steady solution at each SST, while the GCM results are averages over many transient snapshots. In order to separate this fundamental difference from any other differences that may be of interest, in Fig. we subsample the GCM output according to monthly precipitation rate. In this figure when SST is smaller than 0K, we only include precipitation rate in the lower tercile, less than. mmd, and when SST is greater than 0K, we include precipitation rate in the upper tercile, larger than. mmd. (The choice to use these terciles as the thresholds for defining the dry and rainy regimes is somewhat arbitrary, but the results are not sensitive to small changes in these thresholds.) The fact that the left and right panels of Fig. both show sharp transitions at 0K is now an (intentional) artifact of the sampling procedure that allows us to focus on differences in the amplitudes and vertical structures of both fields between the two models at equal SST. The amplitudes of vertical velocity and relative humidity in the rainy regime (SST>0K) in GCM are increased comparing to those in Fig. a,c, and are more comparable to the values from the SCM. The vertical structures, however, remain significantly different in the rainy regime. The SCM vertical velocity is more topheavy than is that from the GCM, and the SCM relative humidity has a relative maximum in the upper troposphere where the GCM does not, but rather decreases monotonically from the PBL to the stratosphere.

14 Page of Quarterly Journal of the Royal Meteorological Society Potential temperature (a) Potential temperature (b) Figure : Temperature difference from the RCE-SCM temperature as a function of SST and pressure for (a) the steady state in the SCM-WTG; and (b) the time-averaged profiles in the GCM over the Indian Ocean and Western Pacific region, where samples with precipitation greater than. mmd are used for SST greater than 0K, and samples with precipitation less than. mmd are used for SST less than 0K. Figure shows contour plots of temperature versus SST and pressure for the SCM- WTG calculations, and the GCM with the same method of subsampling as in Fig. The RCE temperature profile from the SCM is subtracted from the above temperature fields in order to see the small variations. In the SCM (Fig. a), the temperature difference from the RCE profile is zero above the PBL, by construction. In the PBL, the temperature is cooler than the reference temperature for SST less than 0K, which we speculate may be due to the cooling effects of shallow convection, and slightly warmer, by about 0. K, than the reference temperature when SST is greater than 0K. In the GCM, the lower-tropospheric temperature is cooler than the reference temperature for SST smaller than 0.K. In the upper troposphere, the temperature is warmer than the reference temperature by an amount which

15 Quarterly Journal of the Royal Meteorological Society Page of increases with increasing SST; the GCM does not obey the WTG approximation strictly as does the SCM. The surface air temperature in the GCM increases with SST at a rate of approximately :, that is, K increase in air temperature for every K increase in SST. In the SCM, focusing in particular on SST>0K where deep convection is occurring, the rate of increase is somewhat smaller, around :. Compared to the GCM, the surface air temperature feels a negative feedback as SST increases - presumably by parameterized convective downdrafts - which is somewhat too strong. We believe that the central problem is the steadiness of the downdrafts, rather than their intensity, and that it results from the SCM-WTG framework rather than necessarily from any flaw in the convective parameterization. We believe this because the problem appears to be a generic in WTG simulations, having been found by Ramsay and Sobel () in another SCM with parameterized convection, and by Wang and Sobel () in a cloud-resolving model run under WTG. In those studies, the effect is even stronger; e.g., the ratio of surface air temperature increase to SST increase was closer to : in Ramsay and Sobel (). Those authors speculated that the excessive negative feedback on surface air temperature might be due to the steadiness of deep convection, which does not let the boundary layer recover at any time; in observations (and presumably most GCMs) convection is always transient to some degree. Our results here appear consistent with this speculation, in that deep convection, while frequent for SST>0K in the SCM, is nonetheless transient (Fig. ), with significant periods during each day when the boundary layer can recover. This may explain the fact that the ratio of surface air temperature increase to SST increase is greater in our results than in previous studies, although still smaller than in the GCM. While appearing to give an improvement in this aspect of the SCM solutions compared to observations, the transient character of the parameterized convection schemes

16 Page of Quarterly Journal of the Royal Meteorological Society may be sensitive to details of the convective parametrization and in that sense may be to some degree an artifact.. Inclusion of moisture relaxation As discussed in the earlier section, the exclusion of horizontal moisture advection could be one reason why the transition from dry to rainy state with increasing SST is more abrupt in the SCM than the GCM. In this section, we present results in which we add a relaxation of the model's specific humidity field toward the moisture profile from the RCE simulation above the boundary layer, with a fixed relaxation time scale of days. This may be considered a very crude parameterization of horizontal advection, with the time scale representing a length scale associated with the horizontal moisture gradient divided by an advective velocity scale. The resulting relaxation is a moistening influence for SST less than 0K, and a drying for SST greater than 0K, and thus can be expected to smooth the SCM s transition from a dry to rainy state. The fixed relaxation time scale can be interpreted as advection by a constant wind, uniform with height, due to a rotational flow that is entirely decoupled from the divergent flow parameterized by WTG, with advection by the latter neglected. This is the same method employed by Sobel et al. (0) and Sobel and Bellon (0), but different than that of Raymond and Zeng (0), Raymond (0), and Sessions et al. () who parameterize horizontal advection by the divergent flow only. Figure compares precipitation from the SCM to that in the GCM in the same format as Fig., but now the SCM results include the moisture relaxation. As in Fig. (but unlike Fig. ) the GCM results are not subsampled with respect to precipitation rate, but rather all values at the given SST are included. In the SCM, there is now weak (but nonzero) precipitation when SST is smaller than 0K, instead of being zero as in the control experiment. In Fig., the precipitation rate in the SCM-WTG starts to increase around SST = 00 K, followed by a

17 Quarterly Journal of the Royal Meteorological Society Page of roughly linear increase in precipitation with increasing SST. For SST smaller than 00K, the SCM-WTG steady precipitation is close to the th percentile value from the GCM. The precipitation rate in SCM-WTG is close to the 0 th percentile value from the GCM when SST exceeds 0K. Figure : Precipitation (mmd ) as a function of SST in SCM-WTG with moisture relaxation compared to the th, 0 th and th percentile values from the GCM over the Indian Ocean and Western Pacific region. Relative humidity (a) Vertical velocity (b) Figure : (a) Relative humidity (interval 0., > 0. is shaded) and (b) vertical velocity (interval 0.00 ms, >0.00ms is shaded ) as a function of SST and pressure in steady state SCM-WTG solutions with a relaxation on the moisture field, a crude representation of horizontal moisture advection.

18 Page of Quarterly Journal of the Royal Meteorological Society The comparisons between SCM and GCM in relative humidity and vertical velocity are also improved with the moisture relaxation, as shown in Fig.. The transition from dry to moist with respect to SST is more gradual as the lower troposphere is moistened by shallow convection. Relative humidity in the free troposphere starts to increase when SST increases above 00K, and the amplitude of the upper-level maximum at high SST is reduced. The vertical velocity (Fig. b) is similar to that in the control experiment, with descending motion dominating for lower SSTs (smaller than 00.K) and top-heavy ascending motion for higher SST. At large SST, the maximum ascent is somewhat weaker than in Fig.. There is a conspicuous maximum in ascent at SST=00.K which we do not understand at present. Overall, the SCM simulations with moisture relaxation are somewhat more similar to the GCM results (without sub-sampling by precipitation as in Fig. ) than are those without the relaxation. The results are similar when the relaxation time scale is increased or decreased by one day (figures not shown).. Sensitivity to free-tropospheric temperature profile In the SCM-WTG simulations shown above, the free-tropospheric temperature profile was that computed from the RCE solution over an SST of 0K. The WTG model produced a dry state for SST smaller than 0K, and rainy states for SST greater than 0K. While we offered some justification for the choice of 0K as the value used to compute the RCE solution, there is nonetheless some arbitrariness to the choice. Further, the mean temperature profile from the GCM to which we are comparing has significant differences from the RCE profile obtained from the SCM (Fig. ). It is of interest to test the sensitivities of the SCM- WTG results to variations in the free-tropospheric temperature profile. In this section we present SCM-WTG calculations performed with free-tropospheric temperature profiles

19 Quarterly Journal of the Royal Meteorological Society Page of obtained from RCE calculations with the SCM over SSTs different from 0K (the value used above) and also with the profile taken directly from the GCM. Two sets of SCM-WTG experiments were carried out in which the free-tropospheric temperature profile is taken from RCE calculations with an SST of 00K and 0K. Figure shows the potential temperature profiles from RCE calculations with SST = 00K, 0K, and 0K, all compared to the averaged profile from the GCM. Below km, the RCE profile for SST=0K matches the GCM closely (as shown in Fig. ), though it becomes cooler than the GCM at higher levels. The SCM profiles for SST=00K and 0K differ from that at 0K in a way roughly consistent with moist adiabatic structure, i.e., larger differences aloft than at the surface. Potential temperature (a) Potential temperature (b) Figure : (a) RCE profiles of potential temperature for different SSTs in the SCM, together with the average GCM profile from the tropical Western Pacific and Indian Ocean; (b) differences between the GCM profile and the RCE profiles shown in (a). The RCE profiles shown in Fig. are used as the reference profiles for the SCM- WTG experiments. The precipitation from these experiments is shown in Fig., compared to the GCM in the same format as in Fig.. The results are similar to those in Fig. (obtained from SST=0K) but with the SCM precipitation curve shifted by K along the x- axis. When the profile from the RCE calculation with SST=00K is used, the precipitation in

20 Page of Quarterly Journal of the Royal Meteorological Society SCM-WTG starts to increase at SST=00K. The SCM precipitation rate is about mmd higher than the th percentile values from the GCM for SST=00K and 00.K, and close to the th percentile value from the GCM when SST is between 0K and 0.K (See Fig. a). For the experiment with the RCE profile calculated at SST=0K, the precipitation increase occurs at SST=0K, with the precipitation rate slightly lower than the 0 th percentile value from the GCM at that point (Fig. b). Because the initial RCE value controls the temperature profile, which controls the stability for given SST, the critical value of SST for separating dry and moist steady states in SCM-WTG is consistent with the SST value which was used to calculate the RCE profile. (a) (b) Figure : Precipitation (mmd ) as a function of SST in the SCM-WTG with freetropospheric temperature profiles from RCE solutions over (a) SST=00K and (b) SST=0K. The results are compared to the th, 0 th and th percentiles from the GCM over the Indian Ocean and Western Pacific region. We performed an additional set of SCM-WTG integrations using the freetropospheric temperature profile from the climatology over the Indian Ocean and Western Pacific region in the GCM. Figure a shows the relationship of precipitation to SST as in Figs. and. Similar to the control experiment, in Fig. precipitation begins to increase when SST reaches 0K. The precipitation rate matches well with the 0 th percentile values

21 Quarterly Journal of the Royal Meteorological Society Page of from the GCM for SST larger than 0K. Above that SST value, the precipitation rates arguably agree in a more consistent manner with the GCM than do those in Figs. or, which used RCE profiles from the SCM for the free-tropospheric temperature. However, this is not the case when we consider other variables. Fig. b shows relative humidity against SST and pressure from the SCM, in the same format as Fig.. In the rainier region (SST>0K) there is an RH maximum with height as in the SCM results in Fig., but it is lower down, around 00 hpa, with RH dropping to very low values by around 00 hpa. This is consistent with much lower-altitude maxima (compared to the control SCM-WTG experiments) in upward vertical motion and cloudiness as well (not shown). This is understandable, as the warmer upper troposphere might be expected to inhibit deep convection. What is less clear is how the GCM achieves deeper convection. We suspect that transience plays an important role, as discussed in the concluding discussion. In any case, the steady solutions from the SCM-WTG are clearly imperfect in their representation of the timeaveraged relationship between free tropospheric temperature and convection. (a) Relative humidity (b) Figure : Results from the SCM-WTG experiment with the free-tropospheric temperature profiles from the GCM. (a) Precipitation (mmd ) as a function of SST in SCM-WTG compared to th, 0 th and th percentiles from the GCM over the Indian Ocean and Western Pacific region; (b) Relative humidity (interval 0., > 0. is shaded) as a function of SST and pressure in the statistically steady SCM-WTG solutions.

22 Page of Quarterly Journal of the Royal Meteorological Society Sensitivity to boundary layer top height In the SCM-WTG, the nominal boundary layer is defined as levels with pressures greater than a fixed value, P bl. In this section, we investigate the sensitivity of the above results to the choice of P bl. Instead of setting P bl = 0 hpa, in the sensitivity experiments, we choose P bl = 00hPa and 00hpa. Figure compares the precipitation rate as a function of SST in these three experiments. The results from both sensitivity experiments are qualitatively similar to those from the control, with zero precipitation below SST=0K and rapid increases, with modest differences in the different experiments, above that. Interestingly, the calculations with the deepest boundary layer, P bl = 00hPa, have weak but distinctly nonzero precipitation rates at the lowest SSTs, -K, which is from drizzling shallow cumulus clouds. The difference among three experiments for the rainy stage is modest, within mmd, and there is no linear relationship between the precipitation rate and the boundary layer depth. Indeed, at high SST there is a weak maximum in the response to PBL depth, with the greatest rainfall at the control value of 0 hpa. We do not understand the reason for this maximum.

23 Quarterly Journal of the Royal Meteorological Society Page of Figure : Precipitation rate (mmd ) as a function of SST in SCM-WTG for different values of P bl.. Conclusion and discussion The conventional evaluation of single-column models (SCMs) against observations is done with large-scale vertical velocity or large-scale vertical advective tendencies specified. In this configuration there is no feedback between the model physics and large-scale dynamics; the external specification of the vertical velocity or vertical advection strongly constrains the precipitation field, and thus prevents these models from being very useful in understanding the factors which would control the occurrence or intensity of precipitation in a more realistic setting with interacting physics and dynamics. The weak temperature gradient (WTG) approximation offers a simple but physically-based way to parameterize large-scale dynamics in an SCM, allowing some interaction between physics and dynamics. The parameterization of large-scale dynamics by WTG is, of course, imperfect. In this work, we have attempted to test the SCM-WTG framework in a perfect model setting, more comprehensive than what has been previously used for this purpose. We have compared statistically steady solutions with an SCM run under the WTG approximation to selected tropical statistics from a simulation done with the parent GCM from which the SCM was derived. We used the HadGEM climate model for this purpose. We are interested to understand the degree to which an SCM with parameterized large-scale dynamics can serve as a surrogate for the full GCM. Both successes and failures those aspects of the GCM which the SCM can and cannot capture are of interest. With the free-tropospheric temperature profile taken from an RCE solution computed with the SCM over an SST of 0K, SCM-WTG has zero precipitation for SST less than

24 Page of Quarterly Journal of the Royal Meteorological Society K, with precipitation increasing with SST above that value. This is qualitatively consistent with what has been found in similar studies with other models, both SCMs and cloud-resolving models. Above the critical SST of 0K, the SCM precipitation values are between the 0 th and th percentile values from the GCM over similar SST. For smaller SST, the SCM-WTG underestimates the precipitation, obtaining essentially zero compared to a distribution in the GCM with significant nonzero values. Including a relaxation on the moisture field towards the RCE profile - a crude parameterization of horizontal moisture advection - in the SCM improves the agreement with the GCM, particularly in bringing the precipitation above zero in the lower-sst region; it also reduces the precipitation slightly for SST>0K, so that it closely matches the 0 th percentile from the GCM. The critical SST in the SCM-WTG at which precipitation begins to occur is determined by the SST value which was used for RCE calculation that produced the free-tropospheric temperature profile. This is not surprising; the RCE profiles are roughly moist adiabatic starting from a surface temperature close to the SST, and so become warmer as the SST does. In the WTG model, this profile then is held fixed and controls the stability for a given SST. A somewhat more challenging problem is posed by the difference in vertical structure between the RCE temperature profiles and the climatological profile from the GCM's warm tropical regions. While very close to the RCE profile used for our integration in the lower troposphere, the GCM profile is significantly warmer than that RCE profile in the upper troposphere. When this profile is used in the SCM, the convection becomes significantly shallower than that in either the GCM, or the SCM with the RCE temperature profile, although the precipitation as a function of SST does not change dramatically. We do not fully understand either what makes the RCE so much more unstable than the GCM, or why the SCM-WTG is unable to produce convection as deep as that in the GCM for the same

25 Quarterly Journal of the Royal Meteorological Society Page of profile (although the two are consistent, and likely different symptoms of the same inadequacy in the ability of an SCM to represent the mean features of a D GCM solution). We suspect that these differences are related to the much more transient, as well as spatially variable, nature of convection in the GCM. The deepest convection may effectively determine the upper tropospheric temperature while being relatively infrequent, while more frequent but shallower convection plays a greater role in controlling the precipitation. The SCM is unable to have a spectrum of different depths and intensities of convection, having a unique steady state (or perhaps two, though we have not explored that here; e.g., Sobel et al. 0, Sessions et al. ) for a given SST. These results are based on only one implementation of WTG, the strict one in which the free-tropospheric temperature profile remains precisely constant (Sobel and Bretherton 00) rather than being relaxed towards the reference profile on a finite time scale (e.g., Raymond and Zeng 0; Raymond 0; Sessions et al. ; Wang and Sobel ) or obtained by solution of a vertical structure equation, assuming a nonrotating internal gravity wave of fixed horizontal wavelength (Kuang 0; Blossey et al. 0). Use of one of these other variants would almost certainly lead to quantitative differences in the results, but we suspect that the primary qualitative differences between the GCM and SCM-WTG solutions would remain present. We have argued that these differences result from the inability of the SCM- WTG to produce the degree of transience that is present in the GCM, and from the differences in the relationships between the convective heating and temperature profiles in the two model configurations. We suspect that these two symptoms are related to each other, and that both derive inherently from the difference in dimensionality. The difference in temperature profiles between the RCE and GCM solutions is certainly not a product of the WTG formulation, since the WTG approximation was not used to obtain either of those two solutions.

26 Page of Quarterly Journal of the Royal Meteorological Society Acknowledgments This work is supported by ACCSP funding in Australia. We are most grateful to Dr. Steve Derbyshire and Dr. Rachel Stratton at the UK Met office for providing help with the model. AHS acknowledges support from National Science Foundation grant NSF AGS-0, and thanks the management, scientists and administrators of the Centre for Australian Weather and Climate Research for their support and hospitality during his sabbatical in 0-0, when this work was begun. We thank Dr. Matt Wheeler, Dr. Lawrie Rikus, and two anonymous reviewers for their perceptive critiques and constructive suggestion of the manuscript. References: Bergman, J., and P. D. Sardeshmukh (0), Dynamic stabilization of atmospheric single column models, J. Climate,, 0-. Blossey, P. N., C. S. Bretherton, and M. C. Wyant (0): Understanding subtropical low cloud response to a warmer climate in a superparameterized climate model. Part II: Column modeling with a cloud-resolving model. J. Adv. Model Earth Syst.,, pp., doi:./james.0... Bellon, G. and A. H. Sobel (): Multiple equilibria of the Hadley circulation in an intermediate-complexity axisymmetric model J. Climate,, 0-. Chiang, J. C. H., and A. H. Sobel (0): Tropical tropospheric temperature variations caused by ENSO and their influence on the remote tropical climate. J. Climate, -. Kuang, Z., and C. S. Bretherton (0): A mass flux scheme view of a high-resolution

27 Quarterly Journal of the Royal Meteorological Society Page of simulation of a transition from shallow to deep cumulus convection, J. Atmos. Sci.,,,. Kuang, Z. (0) : Modeling the interaction between cumulus convection and linear waves using a limited domain cloud system resolving model, J. Atmos. Sci.,, -. Kuang, Z. () : The wavelength dependence of the gross moist stability and the scale selection in the instability of column integrated moist static energy, J. Atmos. Sci.,, -. Mapes, B.E. (0): Sensitivities of cumulus ensemble rainfall in a cloud-resolving model with parameterized large-scale dynamics. J. Atmos. Sci,, 0. Martin, G. M., Ringer, M. A., Pope, V.D., Jones, A., Dearden, C., and Hinton, T., J. (0): The physical properties of the atmosphere in the new Hadley Centre Global Environmental Model, HadGEM. Part I: Model description and global climatology. J. Climate,, - 0. Neelin, J. D., and N. Zeng (00) : A quasi-equilibrium tropical circulation model - formulation, J. Atmos. Sci,,. Ramsay, H. A. and A. H. Sobel (): The effects of relative and absolute sea surface temperature on tropical cyclone potential intensity using a single column model. J. Climate,, -. Raymond, D. J. (0): Testing a cumulus parameterization with a cumulus ensemble model in weak temperature gradient mode. Quart. J. Roy. Meteor. Soc.,,, doi:.0/qj.0.

28 Page of Quarterly Journal of the Royal Meteorological Society Raymond, D. J., and X. Zeng (0): Modelling tropical atmospheric convection in the context of the weak temperature gradient approximation. Quart. J. Roy. Meteor. Soc.,, 0- Sessions, S., D. J. Raymond, and A. H. Sobel (): Multiple equilibria in a cloud-resolving model. J. Geophys. Res.,, D0, doi:./0jd0 Shaevitz, D. A., and A. H. Sobel (0): Implementing the weak temperature gradient approximation with full vertical structure. Mon. Wea. Rev.,, -. Sobel. A., G. Bellon, and J. Bacmeister (0): Multiple equilibria in a single-column model of the tropical atmosphere. Geophys. Res. Lett.,, L, doi:./0gl0. Sobel A. H., I. M. Held, and C. S. Bretherton (0): The ENSO signal in tropical tropospheric temperature. J. Climate,, 0-0. Sobel, A. H., and C. S. Bretherton (00): Modelling tropical precipitation in a single column, J. Climate,,. Sobel A. H., and G. Bellon (0): The effect of imposed drying on parameterized deep convection, J. Atmos. Sci.,, -. Sobel, A. H. and J. D. Neelin (0): The boundary layer contribution to intertropical convergence zones in the quasi-equilibrium tropical circulation model framework. Theoretical and Computational Fluid Dynamics,, -0.

29 Quarterly Journal of the Royal Meteorological Society Page of Wang, S., and A. H. Sobel () : Response of convection to relative SST: cloud-resolving simulations in D and D. J. Geophys. Res., in press. Wallace, J. M. (): Effect of deep convection on the regulation of tropical sea surface temperature. Nature,, 0-. Zeng, N., J. D. Neelin, and C. Chou (00): A quasi-equilibrium tropical circulation model - implementation and simulation, J. Atmos. Sci.,,,. Zhou, B. and A. H. Sobel (0): Nonlinear shallow water solutions using the weak temperature gradient approximation. Theoretical and Computational Fluid Dynamics,, -.

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