Precipitating convection in cold air: Virtual potential temperature structure

QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY Q. J. R. Meteorol. Soc. 133: 25 36 (2007) Published online in Wiley InterScience (www.interscience.wiley.com).2 Precipitating convection in cold air: Virtual potential temperature structure A. L. M. Grant* Department of Meteorology, University of Reading, UK ABSTRACT: Simulations of precipitating convection are used to illustrate the importance of the turbulent kinetic energy (TKE) budget in determining the virtual potential-temperature structure of the convecting atmosphere. Two sets of simulations are presented: in one the surface temperature was increased to simulate cold air flowing over a warmer surface and in the second a cooling profile, representing cold-air advection, was imposed. It is shown that the terms in the TKE budgets for both sets of simulations scale in the same way, but that the non-dimensional profiles are different. It is suggested that this is associated with the effects of sublimation of ice. It is shown that the magnitudes of the transport and precipitation terms in the virtual potential temperature budget are determined by the scaling of the TKE budget. Some implications of these results for parametrizations of moist convection are discussed. Copyright 2007 Royal Meteorological Society KEY WORDS similarity theory; turbulent kinetic-energy budget Received 24 April 2006; Revised 18 September 2006; Accepted 5 October 2006 1. Introduction Cloud-resolving models (CRMs) have become important tools for studying precipitating convection. The level of detail available from the simulated flows is unlikely to be matched by direct observations, and CRMs will play an important part in future developments of parametrizations for numerical weather prediction (NWP) and climate models. Evaluation of CRM simulations against available observations, through exercises such as the Global Energy and Water-cycle Experiment (GEWEX) Cloud Systems Study (GCSS) (Randall et al., 2003), provide the link between the real world and the model parametrizations. Analysis of CRM data has tended to focus on diagnosing those properties of cumulus ensembles, such as the mass flux, which can be compared directly with outputs from existing parametrizations. However, by focusing on the cumulus ensemble, constraints that might arise from global features of the convecting atmosphere may be missed. An alternative, and complementary, approach is to study the statistical moments of the whole convective flow, and the budgets that determine them. This is common in studies of cloudy boundary layers, but has not been widely used in studies of deep precipitating convection. An exception is the study of Khairoutdinov and Randall (2002). This study looked at similarity in the turbulent kinetic-energy (TKE) budget and the second- * Correspondence to: A. L. M. Grant, Department of Meteorology, University of Reading, PO Box 243, Reading, Berkshire RG6 2BB, UK. E-mail: a.l.m.grant@reading.ac.uk and third-order statistical moments obtained from a simulation of deep convection during an intensive observing period of the Atmospheric Radiation Measurement (ARM) program over the Southern Great Plains of the USA. Using scales based on the average buoyancy flux, originally suggested by Deardorff (1980), they showed that the five episodes of convection during the period of the simulation exhibited dynamical similarity. The present paper considers a series of idealized simulations that are intended to represent convection that occurs in cold-air outbreaks. In all of the simulations the 0 C level is close to cloud base so that precipitation processes are dominated by ice microphysics. Results for two sets of simulations, which use alternative methods of forcing the CRM, are presented. The simulations are used to study the virtual potential temperature structure of the convecting atmosphere and to highlight the importance of the TKE budget in the problem of parametrizing precipitating convection. Symbols used in this paper are defined in the Appendix. 2. Simulations The simulations were carried out using the Met Office CRM. This is an anelastic model which uses a Smagorinsky Lilly scheme for the subgrid turbulence closure. Condensation occurs when grid-point humidity exceeds the saturation value. The formulation of the model has been described by Shutts and Gray (1994). The microphysics scheme used in the model represents conversions between vapour, cloud liquid and ice, rain, snow and graupel. In addition to carrying the mixing ratios Copyright 2007 Royal Meteorological Society

26 A. L. M. GRANT of the ice species, the scheme also carries equations for particle concentration. The microphysics scheme has been described by Swann (1998). All of the simulations used in this study were three dimensional with a horizontal resolution of 250 m and a domain size of 32 32 km 2. The vertical resolution was variable with the lowest level at 50 m above the surface and the distance between levels increasing by 3% per level. The subgrid contributions to fluxes are small, except close to the surface. The lateral boundary conditions were periodic in the x and y directions. Two sets of simulations were carried out. The first set was a series of Lagrangian simulations (which will be referred to as the Ln simulations), in which the surface temperature was increased at a uniform rate to represent an atmospheric column moving over a progressively warmer surface. A problem with using this Lagrangian forcing, rather than imposing a fixed cooling rate, is that the simulations do not reach a steady state. However, the simplicity of the forcing means that no assumptions about the detailed vertical variation in the budgets of potential temperature and humidity are required. For these simulations, averages were calculated over eight hours after a spin-up period of 10 15 h. The actual length of the spin-up period depended on the forcing. Over the averaging period the height of significant levels such as the 0 C and 38 C levels only change by a few hundred metres, even in the most strongly forced simulations. The second set of simulations (which will be referred to as the En simulations) were forced by an imposed cooling and a fixed surface temperature. The cooling was constant with height up to the 38 C level and reduced to zero between the 38 C level and the model tropopause at 10 km. The height variation of the forcing was based on an analysis of the simulation described by Kershaw and Gregory (1997). In the microphysics scheme homogeneous freezing of water occurs at temperatures below 38 C, so in these simulations there is some ambiguity about the significance of changes that occur at and above this level. The initial potential-temperature profile was constant up to the chosen cloud-base height, and linear between cloud base and the tropopause at 10 km. The cloud-base height, cloud-base temperature and surface moist-staticenergy flux were specified, and from these the humidity and temperature of the sub-cloud layer were obtained. The moist static energy at the tropopause was set to the sub-cloud layer value. A relative humidity of 70% with respect to water was assumed. To help initiate turbulence, random perturbations were added to the temperature at the first model level. During the simulations the temperature and humidity profiles changed significantly from the initial profiles. To avoid the complications due to the effects of wind shear on the convection, the mean wind in these simulations was zero. However, in the surface-transfer calculations for thermodynamic fluxes a non-zero average wind speed was assumed to reduce the air surface differences in temperature and humidity required to support the large surface fluxes. This was done by replacing the actual model wind speed at the first model level, U mod, in the flux calculations by (U 2 av + U 2 mod )1/2, where U av is the specified mean wind speed (this was not done in the momentum-flux calculations). Some parameters for the two sets of simulations are given in Tables I and II. Table I. Parameters for the Ln simulations. Heating rate CAPE m b w θ v0 z lcl z MAB z m38 z 0 C C day 1 Jkg 1 ms 1 ms 1 K m m m m 7.5 787 0.155 21 850 7945 6647 1657 5.1 750 0.114 0.126 777 7403 6410 1561 5.7 681 0.112 0.167 706 6892 5224 963 2.7 677 71 69 777 6647 5736 1041 3.0 399 95 92 925 6178 4829 474 12.0 704 45 24 777 6892 5737 1041 Symbols are defined in the Appendix. Table II. Parameters for the En simulations. Cooling rate CAPE m b w θ v0 z lcl z MAB z m38 z 0 C C day 1 Jkg 1 ms 1 ms 1 K m m m m 6.4 374 0.117 0.149 684 5500 5878 1037 3.7 349 67 74 527 5200 5725 952 2.9 325 59 49 473 4900 5725 881 4.6 311 78 0.100 578 4900 5725 952 8.3 373 0.134 09 684 5500 6025 1256

PRECIPITATING CONVECTION IN COLD AIR 27 15 15 Height (km) 10 5 Level of MAB 38 ο C Z cld Height (km) 10 5 Level of MAB 38 ο C LCL Z lcl LCL 0 0 285 290 295 300 305 310 315 295 300 305 310 q v (K) h 10 3 (J kg 1 ) Figure 1. Examples of mean profiles. The virtual potential temperature θ v ; the dotted lines mark the lifting condensation level (LCL), the 38 C level and the level of maximum adiabatic buoyancy (MAB), the dashed curve shows the virtual moist adiabat for sub-cloud layer air (using the latent heat of sublimation), and the area of the shaded region gives the convective available potential energy (CAPE). As but for the moist static energy h. 3. Mean structure Figure 1 and show profiles of the mean virtual potential temperature and the moist static energy for one simulation (the specific examples given in this paper are from the Ln simulation with a heating rate of 7.5 C day 1 ). Profiles from other simulations are qualitatively similar. Within the sub-cloud layer the gradient of θ v is positive, but small. Above cloud base the vertical gradient of θ v is approximately constant with height up to about 6 km, decreasing with height above this. Also shown in Figure 1 is the virtual moist adiabat for sub-cloud layer air at its lifting condensation level (LCL) (the virtual moist adiabat has been calculated using the latent heat of sublimation rather than the latent heat of vaporization to account for the effects of freezing). The gradient of θ v is less than the virtual moist-adiabatic lapse rate below about 8 km, and greater above. The buoyancy associated with a parcel of air rising adiabatically from the lifting condensation level is a maximum at 7.9 km for this simulation this will be referred to as the level of maximum adiabatic buoyancy (MAB). The height at which the temperature is 38 C is also marked in Figure 1. In all of the Ln simulations the level of the MAB is above the 38 C level, while for the En simulations it is below (see Tables I and II). The moist static energy, h, decreaseswith height within the sub-cloud layer, and more rapidly between cloud base and 1700 m where it is a minimum. Above the minimum, h increases almost uniformly with height up to the tropopause. 4. The turbulent kinetic energy budget For stationary homogeneous conditions, neglecting shear production the TKE budget can be written as, w b (w z E + 1ρ ) w p ɛ = 0, (1) Height (km) 15 10 5 Level of MAB 38 ο C LCL 0 10 05 00 05 10 Terms in TKE Budget (m 2 s 3 ) Figure 2. An example of the turbulent kinetic energy budget. The solid curve is the resolved buoyancy production, the dashed curve the transport term and the dot-dash curve the dissipation rate. Other details as in Figure 1. where the terms on the left-hand side are buoyancy production, turbulent and pressure transports, and dissipation by viscosity. Figure 2 shows an example of the TKE budget. The buoyancy flux decreases with height from the surface to a local minimum at the lifting condensation level. Between the LCL and the level of MAB the buoyancy flux is positive, with a local maximum at about 2 km, corresponding to the height of the minimum in h. The buoyancy flux decreases rapidly with height between the 38 C level and the level of MAB, and between 8 km and 12 km it is negative. The height variation of the transport term mirrors that of the buoyancy-flux profile, with transport being a loss term below the level of MAB and a source of kinetic energy above 8 km. Between cloud base and the 38 C level the magnitude of the transport term is 30 50% of the buoyancy term. The dissipation rate is more uniform with height than either the buoyancy or transport terms, and decreases with height above 10 km. The magnitude of the dissipation term is comparable to that of the buoyancy and transport terms.

28 A. L. M. GRANT Figure 2 shows that the level of MAB marks the boundary between the region in which TKE is generated and the region in which TKE is maintained by the transport of energy from below. In the case of the cumuluscapped boundary layer, the level of MAB corresponds to the base of the capping inversion. Grant and Lock (2004) suggested that in the inversion the cumulus transports are maintained by transport of kinetic energy from the layer below. A difference between the present simulations and the shallow cumulus case is that, above the level of MAB, in the present simulations the buoyancy flux is large and negative, while in simulations of tradewind boundary layers the buoyancy flux remains positive in the inversion. Observations (Smith and Jonas, 1995) and large-eddy simulations (LES) (Grant and Lock, 2004) show that the positive buoyancy flux arises from the generation of negatively buoyant downdraughts through mixing in the upper part of the inversion. In the present simulations, there is no liquid water in this region and the negative buoyancy flux reflects the overshooting of negatively buoyant plumes. The TKE budget shown in Figure 2 has some qualitative similarities to budgets obtained by Khairoutdinov and Randall (2002) from simulations of continental convection. The shape of their buoyancy-flux profiles are broadly similar, although in the present case there is no region of negative buoyancy flux around cloud base. The similarity between the buoyancy-flux profile and the transport profile in Figure 2 and the relatively uniform dissipation rate are also similar to the results of Khairoutdinov and Randall (2002). However, in contrast to the present simulations, the magnitude of the dissipation term in the continental simulations was much smaller than either of the buoyancy or transport terms. 5. Scaling the turbulent kinetic energy budget Khairoutdinov and Randall (2002) showed that the terms in their TKE budgets scaled with the vertically averaged buoyancy flux. They also found that the secondand third-order moments of the vertical velocity could be scaled using the velocity scale based on the average buoyancy flux, as originally proposed by Deardorff (1980). Although the results of Khairoutdinov and Randall (2002) demonstrated the dynamical similarity of the simulated precipitating convection, they did not relate the average buoyancy flux to other parameters. Grant and Brown (1999) developed a scaling for shallow non-precipitating cumulus convection, which was applied to the TKE budget of shallow convection by Grant and Lock (2004). The use of this scaling for precipitating convection is considered in this paper. Based on Grant and Brown (1999), the following are taken to be the basic parameters describing the TKE budget in the cumulus cloud layer: (1) the cloud-base mass flux, ρm b (for convenience in what follows m b will be referred to as the cloud-base mass flux), (2) the convective available potential energy (CAPE), and (3) the depth of the layer from cloud base to the level of MAB, z cld. Figure 1 illustrates the definitions of z cld and CAPE. Using these scales, the rate at which potential energy is generated through the formation of saturated air at cloud base is estimated as, m b CAPE/z cld per unit volume. In terms of the cumulus velocity scale, w, the rate of dissipation of kinetic energy by viscosity can be estimated as w 3 /z cld (Tennekes and Lumley, 1972). Assuming a balance between buoyancy production and dissipation leads to the following expression for the velocity scale, w = (m b CAPE) 1/3. (2) One independent non-dimensional group can be formed from the parameters given above and, following Grant and Brown (1999), this is taken to be m b /w.grantand Brown (1999) discuss interpretations of this parameter, but for this study it will be interpreted as the fractional area occupied by the cumulus updraughts. The dissipation rate, ɛ, can be written in a similarity form as, ( ɛ = w 3 z F ɛ, m ) b z cld z cld w, (3) where F ɛ is a similarity function that must be determined. Observations show that dissipation is concentrated in the cumulus clouds (McPherson and Isaac, 1977). Taking the number density of the cumulus updraughts to be 1/zcld 2, the linear dimension of an updraught which occupies a fractional area m b /w is l up (m b /w ) 1/2 z cld. The length-scale l up represents the largest scale within the cumulus ensemble, suggesting that the dissipation rate within the ensemble, ɛ up is w 3 /l up. The average dissipation rate, (m b /w )ɛ up, is, therefore, of order, ( mb ) 1/2 w 3 ɛ w ϒ. (4) z cld Figure 3 and show the terms in the TKE budget for the Ln simulations, between cloud base and the level of MAB, non-dimensionalized by ϒ. Comparison of the size of the error bars and the shaded region shows that ϒ is a good scale for the terms of the TKE budget. The scaling works particularly well for the dissipation rate, which shows much less variability amongst the simulations than the buoyancy or transport terms. The magnitude of the sampling error for these simulations is not known, but the variations in the buoyancy and transport terms left after scaling with ϒ do not appear to be related to variations in m b /w, and probably represent statistical sampling errors. 6. The turbulent kinetic energy budget and large-scale forcing Khairoutdinov and Randall (2002) commented that, while their cases of continental convection exhibited dynamical similarity, the same was not true of simulations of

PRECIPITATING CONVECTION IN COLD AIR 29 2 1 0 1 2 Buoyancy and dissipation 2 1 0 1 2 Transport Figure 3. Terms in the TKE budget for the Ln simulations between cloud base and the level of MAB, scaled by ϒ ; height is scaled by z cld. The buoyancy term (positive) and dissipation term (negative); the solid curve is the average of the scaled profiles (with the error bars showing ±one standard deviation), the shaded area shows ±one standard deviation when the profiles are scaled with the average value of ϒ, the dotted line shows the average position of the 38 C level, and the dashed curve shows the budget residual. As but for the transport term. 2 1 0 1 2 Buoyancy and dissipation 2 1 0 1 2 Transport Figure 4. As Figure 3, but for the En simulations. The buoyancy, dissipation and budget residual terms, and the transport term. convection during GATE. This raises the question as to whether there is a universal scaling for moist convection, as there is for the dry convective boundary layer. Figure 4 and show the buoyancy, dissipation and transport terms in the TKE budget for the En simulations. The scaling by ϒ appears to work well for the En simulations, but the shape of the non-dimensional buoyancy and transport profiles are significantly different from those of the Ln simulations, shown in Figure 3. For the En simulations, the buoyancy-flux profiles have a maximum at around z/z cld = and decrease with height above this. In contrast, the buoyancy-flux profiles for the Ln simulations are more uniform with height and larger in magnitude, although there is a local maximum around z/z cld =. In the Ln simulations the transport term is a sink for TKE between cloud base and the level of MAB, while in the En simulations the transport term is a sink for TKE below z/z cld, but acts as a source of TKE above this. The differences in the non-dimensional profiles are most pronounced in the upper part of the layer. The difference between the Ln and En simulations is in the method used to force the CRM. Since the large-scale forcing does not appear explicitly in the TKE budget the differences that have been described cannot be a direct effect of the different forcing methods. The Ln simulations are not steady in the mean, and one possibility is that the differences are a result of nonstationarity. Figures 3 and 4, however, show that the tendency term in the TKE budget is small and the convection can be considered to be in a quasi-steady state. Another possible reason for this difference is considered in section 8. 7. The virtual potential temperature budget For the present simulations the budget for θ v can be written as, Dθ v = w θ + P θv, (5) Dt z where the virtual liquid-water potential-temperature flux, w θ, includes only the liquid-water flux, since ice has a significant fall speed. In Equation 5 it has been assumed that the tendency term in the liquid-water budget is small compared with the other terms. For the present simulations, this is a reasonable assumption since the area-averaged liquid-water content is small, due to the

30 A. L. M. GRANT Height (km) 15 10 5 0 Level of MAB 38 ο C LCL 40 20 0 20 40 q v tendency ( K Day 1) Figure 5. An example of the virtual potential temperature budget, Equation (5), showing the transport term (dashed curve), the precipitation term (dash-dot curve), the sum of the transport and precipitation terms (full curve), and the average heating rate (long dashed curve). small area occupied by the cumulus updraughts. The terms on the right-hand side of Equation (5) represent the effects of turbulent transports and microphysical processes on the mean virtual potential temperature. Figure 5 shows an example of the budget of θ v.below 10 km the tendency of θ v is positive, and increases slightly with height. The residual between the transport and precipitation terms over this period is similar to the net heating and is comparable to the surface heating rate. The cooling around 12 km is due to the tropopause gradually rising under the action of the convection. The precipitation term in the budget acts to decrease θ v below about 2 km, mainly through the sublimation/evaporation of precipitation, with melting making a smaller contribution. Between 2 km and 6.9 km, the height of the 38 C level, the precipitation term acts to increase θ v, while above 6.9 km the net effect of microphysical processes is negligible. The height variation of the flux divergence mirrors the precipitation term, and acts to increase θ v below 3 km and decrease it between 3 km and 6.9 km. There is a rapid change in the flux divergence across the 38 C level which is associated with the homogeneous freezing of water. Above the 38 C level the warming is through the transport term in the budget. Individually the contributions of the turbulent transport and precipitation terms to the budget of θ v are significantly larger than the net warming. The virtual potential temperature budget can be linked to the TKE budget through the buoyancy flux. The buoyancy flux can be written as w b = g { ( ) L w θ θ + v c p π 1.61θ w q l } θ(w q i + w q p ). (6) The flux w θ which enters into the virtual potential temperature budget (Equation 5) can also be considered as a component of the buoyancy flux. Equation (6) suggests scaling w θ with ϒ, which is shown in Figure 6 for the region between cloud base and the level of MAB (the factor (g/θ v ) 1, needed to convert between the buoyancy flux and w θ, is treated as a constant and, for convenience, is not written explicitly in the scale). The w θ profile has a minimum at about z/z cld = 0.3 and the flux becomes positive above z/z cld. For all of the simulations w θ is approximately zero at cloud base. The scaling by ϒ accounts for a significant part of the variation in the magnitude of w θ amongst the simulations. The variation with height is qualitatively consistent with the entraining plume model of convection (see section 8). Figure 6 compares scaled profiles of w θ from the Ln and En simulations. Below z/z cld = the profiles are similar in shape and magnitude but differ above this, with the difference increasing with height. The cause of this difference is considered further in section 8 along with the differences in the TKE budgets. Since w θ scales with ϒ, the transport term in the virtual potential temperature budget should scale as 10 8 6 4 2 0 2 4 w q (Scaled) 10 8 6 4 2 0 2 4 w q (Scaled) Figure 6. Profiles of w θ for the Ln simulations scaled by ϒ ; other details as for Figure 3. Comparison of scaled w θ profiles for the Ln simulations (full curve) and the En simulations (dashed curve).

PRECIPITATING CONVECTION IN COLD AIR 31 40 20 0 20 40 40 20 0 20 40 dw q /dz (Scaled) p qv (Scaled) Figure 7. Terms in the virtual potential temperature budget for the Ln simulations, scaled by ϒ /z cld. The transport term, and the precipitation term. Other details as in Figure 3. 5 0 5 Condensation/Evaporation (Scaled) 5 0 5 Transport (scaled) (c) 5 0 5 Precipitation Production (Scaled) Figure 8. Terms in the liquid-water budget, for the Ln simulations, scaled by ϒ /z cld. Condensation (positive) and evaporation (negative), transport, and (c) collection of water by precipitation. Other details as in Figure 3. ϒ /z cld. Figure 7 and show the results of scaling the transport and precipitation terms in the virtual potential temperature budget by ϒ /z cld.belowthe 38 C level the scaling works well for both the transport and precipitation terms in the budget. The precipitation term changes from cooling to warming below the minimum in the flux w θ. The high variability around z/z cld =, particularly in the precipitation term, is due to the rapid changes that occur around the 38 C level. The precipitation term in the virtual potential temperature budget is composed of contributions from a number of microphysical processes, such as the collection of liquid water by precipitation, the sublimation of ice and precipitation, etc. It is not within the scope of this study to consider all of these processes in detail, and only the collection of precipitation will be considered as an example. The collection of liquid water by precipitation enters into the liquid-water budget, which can be written as, w q l = C E P acw, (7) z where C is the adiabatic condensation rate, E the evaporation, and P acw the rate of collection by precipitation. The tendency in the mean liquid-water content is assumed to be negligible. From the results in Figure 7 the term P acw in Equation (7) might be expected to scale with ϒ /z cld, as would the transport term since the liquidwater flux is a component of the buoyancy flux (see Equation 6). Figure 8 (c) show the terms in the liquidwater budget, scaled by ϒ /z cld. The condensation rate C is shown in Figure 8. The condensation rate was obtained by averaging

32 A. L. M. GRANT w q sat / z ad over points with liquid water (note that C is a net condensation rate, since adiabatic evaporation occurs in w<0). It is large just above cloud base and decreases with height upto the 38 C level. Above this, C = 0. Evaporation is approximately constant with height between cloud base and the 38 C level. The evaporation rate is less than the condensation rate and there is net condensation at all levels. This differs from shallow, non-precipitating convection in which there is net condensation in the lower part of the cloud layer and net evaporation in the upper part (e.g. Siebesma and Cuijpers, 1995), with condensation and evaporation balancing when integrated over the cloud layer. The scaling accounts for a significant proportion of the variability in the condensation rate. Figure 8 shows that liquid water is transported from the region below z/z cld = 0.3 (where the condensation rate is large) into the layer above. In addition to evaporation, the formation of precipitation acts as a significant sink for liquid water, as shown in Figure 8(c). The loss of cloud water by collection is largest around z/z cld =. In a some mass-flux schemes (Tiedtke, 1989; Gregory and Rowntree, 1990) the formation of precipitation is assumed to limit the liquid-water content of the cumulus ensemble, and water in excess of this limit is assumed to be precipitation. More complex representations of microphysical processes require a knowledge of the vertical velocity of the cumulus ensemble, in addition to the mass flux (Anthes, 1977; Donner, 1993). For the present results, this link between the cumulus dynamics and the microphysics is reflected in the scale ϒ being involved in the scaling of the liquid-water budget. 8. Discussion The simulations used in this study were designed to be dynamically similar, and the aim has been to use a similarity approach to investigate the budget of the mean virtual potential temperature, showing its relationship to the properties of the turbulent kinetic energy budget. The results give some insight into how the cumulus dynamics and precipitation processes interact to determine the budget of θ v. This study has focused on the region below the level of MAB, as the region in which the TKE and precipitation are produced. 8.1. Ice and the buoyancy flux It was shown in section 3 that the scaling of the terms in the TKE budget by ϒ did not depend on the method used to force the simulations, but that shape of the vertical profiles did. Figure 6 showed that there were differences between the profiles of w θ from the Ln and En simulations. Differences in the liquid-water and precipitation contributions to the buoyancy flux (not shown) are much smaller for the two sets of simulations, so the difference between the buoyancy-flux profiles from the Ln and En simulations is associated with the difference in the profiles of w θ. The difference between the w θ profiles is greatest close to the level of MAB, which suggests that something in the upper part of the layer in which TKE is being generated is responsible for the difference between these simulations. A significant difference between the two sets of simulations in this region is the location of the 38 C level relative to the level of MAB. For the Ln simulations the level of MAB is, on average, about 1200 m above the 38 C level, while in the En simulations it is about 600 m below the 38 C level. Deposition of vapour on to ice and sublimation of ice are important microphysical processes that occur around the 38 C level. Just above the 38 C level there is a region of net deposition of vapour onto ice that is associated with the homogeneous freezing of liquid water, and immediately below the 38 C level there is a region, approximately 600 m deep, of net sublimation of ice. In the Ln simulations sublimation and deposition processes occur in the layer in which TKE is generated, while for the En simulations they occur above it. This suggests that the differences between the TKE budgets is due to the interaction between the cumulus dynamics and the ice microphysics, and that the different methods of forcing influence the TKE budget indirectly through the location of the level of MAB relative to the 38 C level. Cohen and Craig (2004) used CRM simulations to investigate the response of convection to sudden changes in the large-scale forcing. They found that the cumulus mass flux responded to such changes on two timescales. Initially there was rapid adjustment over about one hour, which was followed by a much longer period of adjustment. In the similarity framework these timescales can be associated with the eddy turnover timescale, z cld /w, and an adjustment timescale, z cld /m b (Grant and Brown, 1999). For the Ln simulations the eddy turnover timescale is 15 30 min, while the adjustment timescale is 10 15 h. For the surface warming rates in the Ln simulations, the surface temperature increases by 2 3 K over a period of z cld /m b, which is significant compared with the change in θ v between cloud base and the level of MAB. This relatively slow adjustment might lead to the level of MAB being higher in the Ln simulations compared to the En simulations due to the virtual potential temperature in the upper part of the cloud layer lagging behind the surface warming. 8.2. Parametrization of convection The relevance of the results of this study to the parametrization of moist convection can be seen by considering their implications for parametrizations based on an ensemble plume approach. In such schemes fluxes are represented in terms of the properties of the cumulus ensemble, using the mass-flux approximation, i.e. w θ m(θ cu θ v ), (8) where θ cu is the average of θ over the cumulus ensemble. Siebesma and Cuijpers (1995) discussed various

PRECIPITATING CONVECTION IN COLD AIR 33 ways of defining the cumulus ensemble, and here averages over updraughts containing cloud water are used. This definition means that the ensemble contains just those regions of the flow in which condensation is occurring, as well as a large part of the production of precipitation. However, the ensemble is only defined below the 38 C level. Including cloud ice in the definition would extend the ensemble above the 38 C level, but since ice has a significant fall speed it is not necessarily associated with the air parcel in which it was produce. This can lead to biases in the conditional statistics. The total flux w θ scales with ϒ, so Equation 8 suggests that the appropriate scale for (θ cu θ v ) is θ ϒ /m b. Figure 9 and show the cumulus ensemble mass flux, normalized by the cloud-base mass flux, and (θ cu θ v ) scaled by θ. The mass-flux profiles for the individual simulations are approximately constant with height, and the scaling by the cloud-base mass flux accounts for much of the variation between simulations. For (θ cu θ v ) there is little variation between simulations to be explained by the scaling, but the scaling can be considered effective in as much as it does not increase this variability. These results show that the variations in w θ between simulations are associated with variations in the mass flux rather than the thermodynamic properties of the cumulus ensemble. The profile of (θ cu θ v ) reaches a minimum at z/z cld = 0.3, which can be explained qualitatively using the entraining plume model. A simple model of the thermodynamic properties of the cumulus ensemble is, cu (θ θ v ) = θ v cu ε(θ θ v ) + S θ, (9) z z where the second term on the right-hand side of Equation (9) represents the effects of entrainment on the ensemble and S θ represents the effects of microphysical processes. From Figure 8(c) the production of precipitation by collection of water is small below z/z cld = 0.3, so that S θ 0. Consequently Equation (9) implies that (θ cu θ v ) will decrease with height, since θ v increases. Above z/z cld = 0.3 precipitation production becomes significant and S θ > 0, since the removal of liquid water causes θ cu to increase, as seen in Figure 9. If S θ is large enough, the conversion of liquid water to precipitation will cause (θ cu θ v ) to increase with height above z/z cld = 0.3. While the qualitative form of the profiles shown in Figure 9 is readily explained, the quantitative behaviour is more difficult. The gradient of (θ cu θ v ) between cloud base and z/z cld = 0.3 depends on θ and z cld, which suggests that the entrainment rate in Equation (9) must be a function of the non-dimensional parameters describing the flow. Such a scaling for the fractional entrainment rate has been proposed by Grant and Brown (1999) for nonprecipitating cumulus. Similarly the scaling of the precipitation term, S θv, must be consistent with this scaling. Figure 9(c) compares the profiles of w θ estimated using the mass-flux approximation and obtained directly from the simulations. The flux profiles obtained using the mass-flux approximation are similar in shape to the total fluxes, but the magnitude is only 65% of the directly estimated flux. The proportion of the flux explained in these simulations by the mass-flux approximation is smaller than found for shallow cumulus convection 0.5 1.5 2.0 2.5 Normalized Mass Flux (c) 4 2 0 2 (q cu q v ) (Scaled) 6 4 2 0 2 m (q cu q v ) (Scaled) Figure 9. Mass-flux profiles for cloudy updraughts containing liquid water, scaled by the cloud-base mass flux. The difference between θ averaged over cloudy updraughts containing liquid water and the area-average virtual potential temperature, scaled by ϒ /m b.(c)profiles of m(θ cu θ v ) (full line) and w θ (dashed line) scaled by ϒ /m b. All plots are for the Ln simulations.

34 A. L. M. GRANT (Siebesma et al., 2003)) where, for conserved variables, between 80 90% of the flux is accounted for by the mass-flux approximation. 8.3. The cloud-base mass flux The parametrization of the cloud-base mass flux (the cloud-base closure problem) remains a significant problem in the parametrization of convection in large-scale models. In the present study the cloud-base mass flux is a key scale and so, for the results described to be useful, it is also important to understand what determines the magnitude of the cloud-base mass flux. Figure 10 is an example of a profile of w θ between the surface and 2.5 km. A feature of this profile is that the vertical gradient appears to be continuous across cloud base. Above cloud base the scaling results for w θ mean ( ) z w θ ϒ F 1, (10) z cld Height (km) 2 1 LCL 0 0.5 0.5 w q (K ms 1 ) Figure 10. Example of a profile of the flux w θ as a function of dimensional height in the region of the lifting condensation level. The full curve is the resolved flux, the dashed curve the sugrid contribution. while, assuming that w θ 0 at cloud base, for the subcloud layer ( ) w θ w3 z F 2. (11) z lcl z lcl For the gradient of w θ to be continuous across cloud base, the following condition must hold, ( mb w ) 1/2 w 3 z 2 cld w3 zlcl 2, (12) where the definition of the TKE scale (Equation (4)) has been used. This is the same condition obtained by Grant and Lock (2004) for cumulus-capped boundary layer. Equation (12) can be rearranged to give, m b { g θ v w θ v0 (CAPE) 5/6 z 2 cld z lcl } 3/4, (13) where the definition of w 3 = (g/θ v) w θ v0 z lcl has been used. Figure 11 shows a comparison of Equation (13) with the cloud-base mass fluxes diagnosed from the CRM simulations for both the Ln and En simulations. Equation (13) appears to hold reasonably well for both sets of simulations. For shallow cumulus convection, Grant (2001) suggested that m b w. Figure 11 shows that a relationship clearly exists between m b and w, but that m b is not simply proportional to w.thecurvein Figure 11 is obtained from Equation (13), taking all parameters other than w to be constant (using the average values for the simulations). The CRM results are consistent with this, suggesting that for these simulations it is the variation in the surface flux that is most important in determining the cloud-base mass flux. It is difficult to say how general the present results are and, in particular, how relevant they are to the problem of parametrizing tropical convection. Khairoutdinov and Randall (2002) found that their similarity results for convection over the Great Plains of the USA did not 0.5 0.5 m b (ms 1 ) 0.3 m b (ms 1 ) 0.3 0.1 0.1 0 1 2 3 4 m b (Param) (ms 1 ) 5 0.5 1.5 2.0 w * (ms 1 ) 2.5 Figure 11. Comparison of Equation (13) and cloud-base mass fluxes obtained from the CRM simulations. Crosses are from the Ln simulations, and diamonds from the En simulations. Cloud-base mass flux as a function of the sub-cloud layer convective velocity scale. Symbols are as in. The dotted curve shows y = 4x 9/4.

PRECIPITATING CONVECTION IN COLD AIR 35 extend to simulations of convection during GATE. The similarity approach used here was originally developed for non-precipitating convection, and its applicability to precipitating convection holds out some hope that it might apply more generally. However, the fact that the profiles of terms in the TKE budget depend on how the simulations were performed indicates that it will be necessary to understand how the height variations of these profiles are determined to apply the present results to a broad range of convective situations. The scalings for the terms in the mean virtual potential temperature budget may also apply generally, but the height variation of the profiles cannot be universal, since profiles of the heat source and moisture sink associated with convection considered in this study are particular to this convective regime; for tropical convection they have a more complex height variation (e.g. Betts, 1996). 9. Conclusions This paper has presented an analysis of data obtained from simulations of convection in cold air. The main conclusions are: The TKE budget from the simulations can be scaled using results previously applied to non-precipitating convection. The form of the non-dimensional profiles of the terms in the TKE budget depend on the way in which the simulations were forced. However, the scaling for the TKE budget is more generally applicable. It was suggested that the differences were related to the impact of the sublimation of ice on the cumulus dynamics. Scales for the transport and precipitation terms in the virtual potential temperature budget are determined by the scaling of the TKE budget. Matching of the scalings below and above cloud base leads to a relationship between the cloud-base mass flux and CAPE, surface virtual potential temperature flux and the depths of the sub-cloud and cloud layers. The same matching condition holds for shallow nonprecipitating convection. Acknowledgements This work was done under contract to the Met Office. Appendix: Symbols used in the text Horizontal space and time average cu Conditional average over a cumulus ensemble Fluctuation relative to mean 0 Surface value b Buoyancy = (g/θ v )θ v θ Potential temperature Virtual potential temperature θ v θ l Liquid-water potential temperature θ = θ l + 1θq T q Specific humidity q l Liquid-water content q T Total water content q sat Saturation specific humidity q sat / z ad Derivative of the q sat with height following a moist adiabat w Vertical velocity p Pressure ρ Density z Height above cloud base ρm Cumulus mass flux ρm b Cloud-base mass flux w Cumulus velocity scale (= (m b CAPE) 1/3 ) CAPE Convective available potential energy w Mixed-layer velocity scale z lcl Depth of sub-cloud layer z cld Depth of layer from cloud base to the level of maximum buoyancy z MAB Height of the level of maximum adiabatic buoyancy z m38 Depth of layer from cloud base to the 38 C level z 0 C Height of the 0 C level C Net adiabatic condensation rate E Evaporation rate P acw Rate of collection of cloud liquid water by precipitation P θv Precipitation term in the virtual potential temperature budget S θv Precipitation term in the ensemble-plume model ɛ Dissipation rate ε Fractional entrainment rate c p Specific heat of air at constant pressure L Latent heat of vaporization g Acceleration due to gravity π Exner function ϒ Scale for the turbulent kinetic energy budget, = (m b /w ) 1/2 w 3 /z cld θ Scale for the difference between θ in the cumulus ensemble and the area-averaged value of θ v, = ϒ /m b F 1, F 2, F ɛ Similarity functions References Anthes RA. 1977. A cumulus parameterization scheme utilizing a onedimensional cloud model. Mon. Weather Rev. 105: 270 286. Betts AK. 1996. The parameterization of deep convection: A review. In Proceeding of the Workshop on New Insights and Approaches to Convective Paramtrization. European Centre for Medium-Range Weather Forecasts, Shinfield Park, Reading, Berkshire RG2 9AX, UK. Cohen BG, Craig GC. 2004. The response time of a convective cloud ensemble to a change in forcing. Q. J. R. Meteorol. Soc. 130: 933 944. Deardorff JW. 1980. Stratocumulus-capped mixed layers derived from a three-dimensional model. Boundary-Layer Meteorol. 18: 495 527.

36 A. L. M. GRANT Donner LJ. 1993. A cumulus parameterization including mass fluxes, vertical momentum dynamics and mesoscale effects. J. Atmos. Sci. 50: 889 906. Grant ALM, Brown AR. 1999. A similarity hypothesis for cumulus transports. Q. J. R. Meteorol. Soc. 125: 1913 1935. Grant ALM. 2001. Cloud-base fluxes in the cumulus-capped boundary layer. Q. J. R. Meteorol. Soc. 127: 407 421. Grant ALM, Lock AP. 2004. The turbulent kinetic energy budget for shallow cumulus convection. Q. J. R. Meteorol. Soc. 130: 401 422. Gregory D, Rowntree PR. 1990. A mass flux scheme with representation of cloud ensemble characteristics and stability dependent closure. Mon. Weather Rev. 118: 1483 1506. Kershaw R, Gregory D. 1997. Parametrization of momentum transport by convection. Part I. Theory and cloud modelling results. Q. J. R. Meteorol. Soc. 123: 1153 1183. Khairoutdinov MF, Randall DA. 2002. Similarity of deep continental cumulus convection as revealed by a three-dimensional cloudresolving model. J. Atmos. Sci. 50: 2250 2566. MacPhearson JI, Isaac G. 1977. Turbulent characteristics of some Canadian cumulus clouds. J. Appl. Meteorol. 16: 81 90. Randall DA, Krueger S, Bretherton C, Curry J, Duynkerke P, Moncrieff M, Ryan B, Starr D, Miller M, Rossow W, Tselioudis G. Wielcki B. 2003. Confronting models with data. The GEWEX Cloud Systems Study. Bull. Amer. Meteorol. Soc. 84: 454 469. Shutts GJ, Gray MEB. 1994. A numerical modelling study of the geostrophic adjustment process following deep convection. Q. J. R. Meteorol. Soc. 12: 1145 1178. Siebesma AP, Cuijpers JWM. 1995. Evaluation of parametric assumptions for shallow convection. J. Atmos. Sci. 52: 650 666. Siebesma AP, Bretherton CS, Brown A, Chlond A, Cuxart J, Duynkerke PG, Jiang H, Khairoutdinov M, Lewellen D, Moeng C-H, Sanchez E, Stevens B, Stevens DE. 2003. A large-eddy simulation intercomparison study of shallow cumulus convection. J. Atmos. Sci. 60: 1201 1219. Smith SA, Jonas PR. 1995. Observations of turbulent fluxes in fields of cumulus clouds. Q. J. R. Meteorol. Soc. 121: 1185 1208. Swann HA. 1998. Sensitivity to the representation of precipitating ice in CRM simulations of deep convection. Atmos. Res. 127: 1239 1260. Tennekes H, Lumley JL. 1972. A First Course in Turbulence. Massachusetts Institute of Technology: USA. Tiedtke M. 1989. A comprehensive mass flux scheme for cumulus parameterization (sic) in large-scale models. Mon. Weather Rev. 117: 1779 1800.