Entrainment processes in the diurnal cycle of deep convection over land

Transcription

1 Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. 8: 9, July A Entrainment processes in the diurnal cycle of deep convection over land A. J. Stirling* and R. A. Stratton Met Office, Exeter, UK *Correspondence to: A. J. Stirling, Met Office, FitzRoy Road, Exeter EX PB, UK. alison.stirling@metoffice.gov.uk In this article we show that entrainment during the developing stages of deep convection over land is much higher than convection at equilibrium. A series of idealised cloud-resolving model simulations are performed for a range of environmental conditions, and these show that the interaction with the environment via the entrainment and detrainment rates gradually decreases as the day progresses, reverting to the values found at equilibrium. The entrainment and detrainment rates are themselves found to depend on the environmental humidity and stability, and are also strongly linked to cloud size, suggesting that the representation of the horizontal growth of clouds in convective parametrizations is important for the representation of the diurnal cycle. We propose a simple new entrainment and detrainment formulation to take account of these findings, and show that this improves the representation of developing convection in a single-column model, providing a more gradual transition towards deep convection. Copyright c British Crown copyright, the Met Office. Key Words: cloud-resolving model; convective parametrization; detrainment Received March ; Revised September ; Accepted 8 November ; Published online in Wiley Online Library December Citation: Stirling AJ, Stratton RA.. Entrainment processes in the diurnal cycle of deep convection over land. Q. J. R. Meteorol. Soc. 8: 9. DOI:./qj.88. Introduction A long-standing problem in general circulation models (GCMs) has been the representation of the diurnal cycle of convection. Studies such as Yang and Slingo () showed that the modelled peak in the diurnal harmonic of convective precipitation over continental land occurs between three and six hours too early compared with satellite observations. This early onset affects both the radiative and water budgets in these models, and as such has important consequences for climate modelling. In several respects, the early onset of convection is no mystery, and is linked to the absence of various physical processes in convection parametrizations. One of these relates to the time-scale over which convective instability is removed. The timing and duration of convection can affect the state of the free troposphere on the following day, and therefore the timing of convective initiation. A common problem in GCMs is that convective instability is removed too quickly (e.g. Willett and Milton, ). This is in part by design due to the auxiliary role of convection in ensuring numerical stability in models, and the consequent requirement of a closure that is linked to the removal of convective available potential energy (CAPE) rather than an intrinsic convective time-scale (e.g. Martin et al., ). Closures have also tended to neglect the existence of inversions at the top of the convective mixed layer which provide an energy barrier (convective inhibition or CIN) to the vertical ascent of plumes from the boundary layer. The vertical scales over which these inversions exist are often small compared with the vertical grid scale, and can be poorly represented. As vertical resolutions have improved, changes to the closure have been proposed to take account of CIN (e.g. Kuang and Bretherton, ; Fletcher and Bretherton, ). Vertical velocity equations have also been incorporated to represent the gradual increase in boundarylayer turbulent kinetic energy (e.g. Neggers et al., 9). Copyright c British Crown copyright, the Met Office.

2 A. J. Stirling and R. A. Stratton Parametrizations tend to neglect sub-grid variability in humidity in the boundary layer, however, this has been shown to promote the development of deep convection over the ARM Southern Great Plains site (Zhang and Klein, ). The subsequent development of cold pools resulting from the outflow from convective downdraughts increases the thermodynamic variability and dynamic lifting in the boundary layer (e.g. Tompkins, ; Devine et al., ), and so can prolong the localised availability of CAPE and boundary-layer kinetic energy, enabling continued convective activity. Rio and Hourdin (8) have taken account of this effect by proposing a dynamic lifting wake term in their closure, with beneficial impacts on the later stages of the diurnal cycle. Even in low-cin environments with high CAPE, there is evidence (e.g. Grabowski et al., ; Pereira and Rutledge, ) that deep convection can still take many hours to develop. The importance of the environment in controlling the transition to deep convection has been highlighted by Zhang and Klein (), who found that the relative humidity (RH) at heights of km played a key role in determining whether convection deepens over the ARM Southern Great Plains Site. Grabowski et al.() and Khairoutdinov and Randall () postulated that development can only proceed once the cloud has reached sufficient horizontal size to protect a central core from entraining environmental air, and that this cloud size was controlled by moist static energy (m.s.e.) variability in the boundary layer, which increases with the onset of rain, and subsequent development of cold pools. There has long been the suggestion that entrainment rates in mass flux parametrizations should depend inversely on the size of the cloud (e.g. Simpson, 97), and some dependence has been included in the Kain Fritsch parametrization (e.g. Kain, ). The more recent Wagner and Graaf scheme () has also included this effect, and found improvements to representation of the diurnal cycle as a result. Del Genio and Wu () performed a modelling study of the Tropical Warm Pool International Cloud Experiment (TWP-ICE) during the monsoon break period and found evidence that the entrainment rate weakens with time as convection over land deepens, suggesting that entrainment in non-equilibrium scenarios needs to be linked to a physical property of the convection as it evolves. They suggested that the entrainment might be affected by the boundary-layer vertical velocity which is given a kick in the presence of cold pools. While entrainment is evidently an important parameter in mass flux parametrizations, there is limited theoretical basis for its formulation in the moist convective case. The entrainment parameter used for deep convection in the Met Office Unified Model (UM) compares well with the entrainment of moist static energy in cloud-resolving models (CRMs) in cases of radiative convective equilibrium (e.g. Swann, ), however a key challenge remains to capture the gradual transition from shallow to deep convection. In this article we analyse the behaviour of entrainment and detrainment in developing convection for a range of idealised environmental conditions, and consider the role of cloud area in determining entrainment rates. The layout is as follows: section contains a description of the CRM formulation used, a summary of the deep convective parametrization currently used in the UM, and Copyright c British Crown copyright, the Met Office. the definitions of entrainment and detrainment used for this analysis. In section, we show how convection evolves under different environmental conditions, comparing the CRM behaviour with the UM single-column model. We present the entrainment and detrainment rates measured from the CRM and explore the influence of environmental conditions on these values. In section, we investigate the dependence of entrainment and detrainment rates on cloud area, and provide a simple suggestion for how cloud area could be represented in a parametrisation. In section, we propose a new formula for the entrainment and detrainment rates in convective parametrizations for evolving convection, and test its performance in a single-column model. Finally we discuss the conclusions, implications and limitations of this approach in section.. Methods.. LEM configuration The Met Office Large-Eddy Model (LEM), version. (e.g. Gray et al., ; Petch et al., 8 and references therein) was used to perform a set of three-dimensional idealised simulations of the first few hours of developing convection over land. The runs were initialised at dawn, (taken to be local time) with forcing applied via surface fluxes which increased sinusoidally with time. The Bowen ratio and surface flux amplitudes were taken from the GEWEX (Global Energy and Water Cycle Experiment) Cloud System Study (GCSS) Large-scale Biosphere Atmosphere (LBA) case (Grabowski et al., ), reaching a maximum of 7 W m in sensible heat and W m in latent heat at mid-day. For simplicity, no mean horizontal winds were applied. The simulations used periodic lateral boundary conditions, with a horizontal resolution of m and domain size of km in the horizontal, and 8 levels spanning km in the vertical. Sensitivity runs were performed to test the behaviour at increased resolution ( m) and increased domain size ( km), and were not found to alter the results significantly... Initial conditions The simulations were given a prescribed initial moisture and temperature profile, and the shape of these was devised to allow a range of different initial RHs for a given environmental stability profile, and conversely, a range of stabilities for a given RH profile. The initial profiles for RH were similar to those of Derbyshire et al.(; hereafter D). Below km, the RH was set to 8%. Above km a constant value ranging between % and 9% was applied (Figure ). The gradient of virtual potential temperature was also set to be a constant function of height above km (Figure ). The potential temperature and water vapour mixing ratios were then deduced from the humidity and virtual potential temperature stability profiles imposed. A grid of simulations spanning a range of different RHs and stabilities was set up, and these are summarised in Table. The initial environmental conditions below km were set to be the same for each run in order to isolate the Q. J. R. Meteorol. 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3 Entrainment in the Diurnal Cycle 7.. Unified Model mass flux scheme 8 RH / % 9 Figure. Initial conditions for relative humidity and virtual potential temperature for the simulations listed in Table. Table. Table of simulations used: environmental virtual potential temperature stability ( θ v / z) versus relative humidity. θ v Stability (K km ) RH (%)... 9 r9 r9 r9 7 r7 r7 r7 r r r r r r r r r Each simulation has an identifying label rxy, where x denotes the initial RH of x%, and y denotes the stability. tropospheric influence on developing convection from the strong role of the boundary layer in controlling convective development. As a result, the initial convective inhibition was J kg for all runs. The atmospheric conditions were applied uniformly in the horizontal in the LEM, apart from an initial uniform random perturbation in temperature of ±. K, which was applied to each grid point below m... SCM set-up To illustrate the behaviour of the convective parametrization, a single column of the Met Office UM (SCM) was used. The set-up is described in detail in Stratton and Stirling (), and so only a brief account is provided here. The surface forcing and initial conditions applied were the same as those used for the LEM. The radiation scheme in the SCM was turned off, and hourly mean radiative forcing increments as measured from the CRM were applied. A time step of min was used, with a CAPE closure time-scale of h. A summary of the key elements of the convection scheme used is provided below. Copyright c British Crown copyright, the Met Office. The convection scheme in the Met Office UM is based on the Gregory Rowntree (99) mass flux scheme. The height at which convection terminates is determined by the parcel reaching neutral buoyancy, or the mass flux approaching zero, whichever happens lower down. The success of mass flux schemes depends on their ability to represent the convective fluxes of temperature and moisture. Clearly this depends on the extent to which the convective flow can be represented with a top-hat vertical velocity function, and how the boundaries of this top hat (representing the updraught) are defined. In idealised pop-corn cases without shear or large-scale organisation such as those considered here, the mass flux approximation represents the temperature and moisture fluxes to within less than %, with very limited sensitivity to the exact choice of updraught boundary. The mass flux, M, is derived from the mass continuity equation: M z + ρ a u = E D, () t where E and D are the mass entrainment and detrainment rates respectively. ρ a u / t is known as the storage term and represents the rate of increase in the fractional area covered by updraughts. The storage term is neglected as small, which would indeed be the case in equilibrium conditions. In developing convection, neglect of this growth term is likely to result in an underestimate of the rate of decrease in mass flux, and therefore could produce cloud tops that are too high. However, the size of this effect is greater when the cloud is developing faster, and neglect of this term tends to accelerate the convection that is already evolving rapidly, while it has a lesser impact on the cases that evolve more slowly, which is likely to be the case of interest in the diurnal cycle. A brief analysis found that this was not the controlling influence in this study, and that it could therefore be neglected. The challenge for convection schemes is therefore to characterise the entrainment and detrainment rates to produce a representative mass flux. The role of entrainment in determining the top of the convection can be seen to act in two directions to increase the mass flux gradient, and therefore raise the height at which the mass flux reaches zero, but also to decrease the plume buoyancy which leads to a lower level of neutral buoyancy. For the Gregory Rowntree scheme, the fractional entrainment rate, ɛ = E/M, for deep convection is given by: ɛ deep um ρ(z) p(z) (z) =.g, () where p is the pressure, p the pressure at the surface, and ρ the density. Detrainment also plays a dual role, and is divided into two types in the convection scheme: a mixing detrainment, in which the plume loses mass at the same buoyancy as the plume (affecting only the mass flux term and the environment increments), and a forced detrainment, where the mass is lost at the same buoyancy as the environment. The latter acts to increase the mean buoyancy of the plume p Q. J. R. Meteorol. Soc. 8: 9 ()

4 8 A. J. Stirling and R. A. Stratton by losing some of the lower end of the buoyancy distribution within the plume, and can allow parcels to ascend further. The mixing detrainment fractional rate (δ mix = D mix /M) is given by: δ mix um = ɛ um ( RH) α det, () where α det =. for deep convection. Forced detrainment is applied when the plume buoyancy excess starts to decrease with height, and follows Derbyshire et al.(). It is given by δ forced um ( log θ ) = R v det, () z where R det =.7 in the UM. The total detrainment is given by the sum of the mixing and forced detrainment terms... Measurement of entrainment A common measure of entrainment (e.g. D) can be obtained from a conservation of mass approach, tracking mass into and out of the convective updraughts using a conserved scalar variable. This is described fully in Swann (). For a scalar variable, χ, we can write the entrainment rate of χ into the convective updraughts, E χ,as χ u ( ρa u t +ρ χ u χ u) a u a u Pχ u t = M χ u () z E χ (χ u χ e) (following Eq. (7) of Swann, ), where χ u represents the mean value of χ across all the updraught points in the field (the updraught partition is defined below); χ e is the mean value of χ in the environment, (where the environment refers to all points that do not lie in the updraught partition); a u is the fractional area covered by updraughts; and χ u = w χ u /w u,wherew is the vertical velocity. M is the mass flux which is given by ρa u w u ; Pχ u represents the sources and sinks to the scalar in the updraught air due to other processes, such as radiation and microphysics; and ρ is the mean air density at that height (following the anelastic approximation). Throughout this article, we define the updraught as air moving upwards with positive buoyancy, with liquid water content greater than kg kg, however the results shown are relatively insensitive to the exact definition. Since our focus is on the early stages of deep convection, where the cloud-top height is lower than the freezing level, we use the moist static energy, h, as an approximation to a conserved variable, χ: h = θ + L v c p q v, () where θ is the potential temperature, q v the water vapour mixing ratio, L v the latent heat of vapourisation, and c p the specific heat capacity of air at constant pressure. While noting that this measure does not account for the heterogeneity in the environmental air, which Romps () showed could be important in reducing estimates of the bulk entrainment, it gives a good reconstruction of the plume-buoyancy profile which is the measure used to Copyright c British Crown copyright, the Met Office. determine cloud-top height, and therefore the growth rate of the convective cloud. Use of the moist static energy variable, h, in Eqs (7) and (8) to determine the r.h.s. of Eq. () is also found to give very good agreement with the l.h.s. of Eq. () for these runs, suggesting further that the moist static energy variable can reliably be used to infer information about the shape of the mass flux if the storage term is either known, or small. For the runs analysed in this article, the total area covered by updraughts, a u is small (a u ), and so the terms on the l.h.s. of Eq. () are found to be negligible, and E h is well approximated by.. Measurement of detrainment E h = M hu / z (h u e). (7) h Likewise, the detrainment parameter, D χ, is derived from χ e ρa e t =M χ e z +D χ (χ u χ e) (ρa e w χ e) +ρa e Pχ e, z where perturbations referred to in the term w χ e are relative to the mean values in the environment only. Since a e = a u, we find that all terms are required in the above equation, as they are all significant. The fractional entrainment and detrainment rates, ɛ and δ are given by ɛ = E χ M, (8) δ = D χ M, (9) respectively. Throughout this article, we will refer to mixing detrainment as the detrainment measured when the plume buoyancy excess increases with height. While this distinction does not take account of the buoyancy at which the plume detrains, it represents what the UM convection scheme would treat as mixing detrainment since the forced detrainment term is only applied where the plume buoyancy excess starts to decrease with height (subsection.). Next we evaluate the entrainment and mixing detrainment rate formulations, asking whether they are appropriate for the case of evolving deep convection over land.. Results.. CRM and SCM cloud evolution in different environments Figure shows how the convection develops in a subset of the simulations shown in Table. In each case, the time series is of the height of the highest cloudy point in the domain (with ice or liquid water content greater than kg kg ). There is a clear variation in the CRM development timescale as a function of environment, with the lower humidity and more stable runs taking at least h longer for the cloud top to reach the level of neutral buoyancy than runs with the highest humidity and least stable environments. This Q. J. R. Meteorol. Soc. 8: 9 ()

5 Entrainment in the Diurnal Cycle 9 r9 r9 r r r r r r r Figure. Time series showing the height of the highest cloudy point in the domain for a range of environmental initial conditions. Dashed lines show the single-column model cloud-top height. Columns from left to right are for decreasing stability, rows from top to bottom are for RH 9%, % and % respectively. The run numbers correspond to the initial conditions in Table. Grey lines show the precipitation rates as the runs progress (with r.h.s. scales). [Correction added February after original online publication: Figure has been replaced with a corrected version.] development is consistent with the idea of an entraining plume controlling the rate of deepening of the convection. However, if we look at the predicted development using the Met Office UM operating in single-column mode, and with the same forcing applied (Figure ), two main features are evident. The first is that the convection scheme displays an intermittent behaviour, switching on and off, with increasing regularity as the stability decreases. The second is that the convection deepens very early on (generally on the first time step), with only limited sensitivity to the environment in this set-up. This discrepancy between the timing of deep convection in the SCM and CRM is the prime motivator for the rest of the work in this article... Mass flux and cloud area Figure shows how the precipitation develops during the runs. There is significant precipitation after 9 in the 9% humidity runs. Figure shows the corresponding mass flux profiles at different time intervals during the runs. The large increases in mass flux in the higher humidity runs coincide with the increase in precipitation. Figure shows how the mean cloud area evolves with time at three different heights in the simulation. Each cloud updraught is identified as a contiguous group of adjacent and diagonally adjacent grid cells that meet the updraught criteria outlined in subsection.. Initially all humidity cases evolve in a similar way, with differences only entering once Copyright c British Crown copyright, the Met Office. the high precipitation rates in the higher humidity runs have been established... Entrainment dependence on environment Figure shows the fractional entrainment rate in the CRM runs as defined in Eq. (7), with the entrainment rates used in the UM also shown (Eq. ()). In all but the driest runs with % humidity, the entrainment rate significantly exceeds the model parametrization entrainment values, with values at a height of km being up to four times as high for the highest humidity runs. The entrainment profile shows steeper variation with height (particularly over the first km) than the existing UM parametrization, and overall could be better represented by a /z shape, similar to those used to parametrise shallow convection (e.g. for BOMEX: Siebesma et al.,, as well as the deep entrainment rates found in D). Figure shows how the fractional entrainment rate varies as a function of RH and stability. There is a clear correlation of entrainment with RH, with entrainment up to six times higher in higher RH environments. Similarly, there is evidence of lower entrainment rates at higher stabilities. Figure 7 shows how the entrainment rates evolve with time for the medium stability case ( rx from Table ) with 9, and % initial humidities, and for reference, Figure 8 shows how the RH and stability of these runs changes with time. Q. J. R. Meteorol. Soc. 8: 9 ()

6 A. J. Stirling and R. A. Stratton r9 r9 r Mass Flux / kg m - s Mass Flux / kg m - s Mass Flux / kg m - s - r r r..... Mass Flux / kg m - s Mass Flux / kg m - s Mass Flux / kg m - s - r r r..... Mass Flux / kg m - s Mass Flux / kg m - s Mass Flux / kg m - s - Figure. Mass flux at 8 (solid), 9 (dotted), 9 (dashed), and (dot-dashed) local times for a range of environmental conditions. Columns from left to right are for decreasing stability, rows from top to bottom are for RH 9%, % and % respectively. The horizontal dotted grey line shows the km level, below which all simulations have the same initial conditions A (.km)/ km.. A (.km)/ km.. A (.km)/ km Figure. Time series of average cloud area at heights of. km, km, and km. Symbols show the relative humidity of the medium stability runs ( rx in Table ) with triangles, squares and plus signs representing 9%, %, and % initial RH. At a height of. km, Figure 8 shows that the RH remains almost constant throughout the run, with little variation in the stability, however the entrainment rate in Figure 7 shows a strong decrease with time. This suggests that there is some evolution in the entrainment rate that is independent of RH and stability. At and km, the entrainment rate of the highest humidity run decreases as a function of time, while the entrainment rate of the lowest humidity run increases initially before levelling off by local time, with the overall effect being one of a convergence of the entrainment rates with time. This convergence is mirrored in the evolution of the RH fields. Copyright c British Crown copyright, the Met Office... Detrainment dependence on environment Figure 9 shows profiles of the total fractional detrainment, with the corresponding formula for the UM mixing detrainment rate shown in grey (using the measured entrainment rates at 9 local time from Figure ). The total detrainment appears to be a strong function of RH with some dependency on stability at higher levels. The existing UM description significantly underestimates the detrainment in most cases. In Figure, the fractional mixing detrainment rate is shown as a function of humidity and stability, and Figure shows how this varies with time. (This has been separated Q. J. R. Meteorol. Soc. 8: 9 ()

7 Entrainment in the Diurnal Cycle r9 r9 r ε / km ε / km ε / km - r r r..... ε / km ε / km ε / km - r r r..... ε / km ε / km ε / km - Figure. Fractional entrainment rate (derived using Eqs () and (9)) at 8 (solid), 9 (dotted), 9 (dashed) and (dot-dashed) local time for a range of environmental conditions. Columns from left to right are for decreasing stability, rows from top to bottom are for RH 9%, % and % respectively. The UM deep entrainment rate (grey solid) is overlaid. Note that below km (dotted grey line) all simulations have the same initial humidity of 8%, and the same stability, so the main differences in behaviour are expected to be seen above this level. from the forced detrainment rate by ensuring that only points for which buoyancy is an increasing function of height are plotted, where the forced detrainment parametrization does not apply.) At a height of. km, there is a wide range of detrainment values for the narrow range of RHs shown. This range appears to be linked to the strong decrease in detrainment with time, similar to that seen in the entrainment profiles, and shown in Figure. At and km, there is strong dependence on humidity, with detrainment decreasing linearly as humidity increases. This is consistent with greater evaporation occurring when the plume edges are exposed to less moist environmental air. The dependence of detrainment on humidity is clearly too weak in the UM formulation (Eq. (), dashed line), and makes a case for adjusting the relationship used. The RH dependence of the detrainment is also stronger than the RH dependence of the entrainment rate, with the detrainment values spanning about twice the entrainment rate values over the same RH range (albeit in the opposite direction). In these simulations, entrainment and detrainment have similar values when the RH is around 8%. There is little evidence of a trend in the mixing detrainment with stability, suggesting that the variation seen in Figure 9 is where the plume buoyancy is decreasing with Copyright c British Crown copyright, the Met Office. height and therefore attributed to the forced detrainment term. The detrainment rates decrease strongly over time, which is expected to be due in part to a tendency towards moister conditions over time, although the behaviour at. km, where the RH does not change significantly, suggests this is only part of the reason. We will investigate the possible link to cloud area in the next section... Discussion The plots of this section show that there is a correlation between entrainment and detrainment and RH, but what can we deduce about the causality between these parameters? Does the RH directly affect the turbulence at the cloud edges, leading to different entrainment rates, or is it an indirect consequence of the nature of the clouds forming in different humidity environments? This distinction becomes important when deciding how to parametrise entrainment, and to unravel this, we need to consider the physical processes occuring. RH is likely to influence entrainment through three processes homogeneous mixing of environmental air into the plume, inhomogeneous mixing from the plume edges towards the plume centre, and the consequent moistening of environmental air just outside the plume due to detrainment. First, consider the effects of mixing environmental air homogeneously into a field of clouds of different sizes. Q. J. R. Meteorol. Soc. 8: 9 ()

8 A. J. Stirling and R. A. Stratton ε (.km)/ km -.8 ε (.km)/ km -.8 ε (.km)/ km RH / %. 8 RH / %. 8 RH / % (d). (e). (f).... ε (.km)/ km -.8 ε (.km)/ km -.8 ε (.km)/ km dθ v /dz / K m dθ v /dz / K m dθ v /dz / K m - Figure. Entrainment dependence (a c) on relative humidity, and (d f) on stability at heights of (a, d). km, (b, e) km, and (c, f) km. In (a c), only the medium stability runs (rx) are shown. For (d f), only the % initial relative humidity runs (rx) are shown (as in Table ). This plot shows the behaviour between 8 and local time. The dashed line shows the UM entrainment rate from Eq. (). Each symbol in this and subsequent figures represents the instantaneous value, with output shown at min intervals ε (.km)/ km ε (.km)/ km ε (.km)/ km Figure 7. Entrainment as a function of local time for the medium stability case (rx runs from Table ), at heights. km, km and km. Symbols show the different initial humidities of the runs (9% triangles, % squares, and % pluses). Grey lines correspond to the parametrization suggested in Eq. (). The same amount of entrainment will have differing effects at different humidities because at low humidity smaller clouds are less likely to survive the dilution, and so there is a tendency for only the larger clouds to survive. At higher humidities, a wider spectrum of cloud sizes can be supported. Since the larger clouds would be expected to have lower dilution rates (because of their greater area to perimeter ratio; section ), the entrainment measured in a field of larger clouds would be expected to be lower. This constitutes an indirect link between entrainment and RH via cloud area. Copyright c British Crown copyright, the Met Office. Second, consider what happens if we assume there is a longer time-scale relating to the length of time air takes to reach the centre of the plume, so that the mixing takes place inhomogeneously. In drier or more stable environments, smaller amounts of air are required to make the saturated plume lose its buoyancy, and so much of the entrained air at the plume edges does not have time to reach the plume centre before it detrains. The remaining buoyant plumes in low-humidity environments are therefore comparatively less diluted than their higher-humidity counterparts. They also tend to be surrounded by larger areas of high moist static Q. J. R. Meteorol. Soc. 8: 9 ()

9 Entrainment in the Diurnal Cycle RH (.km) / %; Γ / K km - 8 RH (.km) / %; Γ / K km - 8 RH (.km) / %; Γ / K km Figure 8. Time series of relative humidity and stability at heights of. km, km, and km. Symbols show the relative humidity of the medium stability runs ( rx in Table ) with triangles, squares and plus signs representing 9%, %, and % initial RH. Lines show times the stability of the ( rx ) runs, with solid, dotted and dashed lines representing the least, medium and most stable runs respectively. r9 r9 r9 δ / km - δ / km - δ / km - r r r δ / km - δ / km - δ / km - r r r δ / km - δ / km - δ / km - Figure 9. Fractional detrainment rates (using Eqs (8)) and (9)) at 8 (solid), 9 (dotted), 9 (dashed), and (dot-dashed) local times for a range of environmental conditions. Columns from left to right are for decreasing stability, rows from top to bottom are for RH 9%, % and % respectively. The UM mixing detrainment formula from Eq. () using the measured entrainment rates at 9 from Figure is ahown as a grey solid line. Note that below km all simulations have the same initial humidity of 8%, and same stability, so the dominant differences in behaviour are expected to be above this level. energy where the outer parts of the plume have detrained. The high detrainment early on acts to moisten the lower- RH environments quite rapidly, and so as the simulation progresses, more environmental air can reach the plume core before detraining. Figure shows the moist static energy field for runs at low and high humidities, and shows that at low humidity there are many failed plumes, as might be expected via the first effect, but also that the surviving plumes are surrounded by a ring of detrained air, as would be expected by the Copyright c British Crown copyright, the Met Office. second effect. Figure shows that early on the average cloud area is the same for the different humidity runs, suggesting that any selection effect towards larger clouds in the lowest humidity runs is offset by detrainment at the edges of these clouds, bringing them down in size. We therefore expect that the first effect is not the dominant one in influencing the entrainment values early on. The rings of high m.s.e. outside the low-humidity plumes in Figure suggest there may also be some recirculation of air that has already been detrained back into the plume. Q. J. R. Meteorol. Soc. 8: 9 ()

10 A. J. Stirling and R. A. Stratton δ (.km)/ km - δ (.km)/ km - δ (.km)/ km - 8 RH / % 8 RH / % 8 RH / % (d) (e) (f) f) δ (.km)/ km - δ (.km)/ km - δ (.km)/ km dθ v /dz / K m dθ v /dz / K m dθ v /dz / K m - Figure. Mixing detrainment (a c) versus RH and (d f) versus stability between 8 and local time. Entrainment rates shown in grey are repeated from Figure for reference. Symbols are consistent with Figure. δ (.km)/ km - δ (.km)/ km - δ (.km)/ km Figure. Mixing detrainment as a function of time at heights. km,. km and. km. Symbols are as in Figure 7. This would tend to make the measure of entrainment using m.s.e. lower than the actual mass entrained into the plume, as the entrained air would have higher m.s.e. than the environment, reducing the impact of the environmental air on the gradient of the plume m.s.e. (Eq. (7)). However, this cannot account for the factor of four difference in entrainment between the highest and lowest RH cases, as use of the m.s.e. measure of entrainment in the continuity equation (Eq. ()) still provides a good prediction of the mass flux gradient. In each of these cases, the environment impacts on the entrainment rate through the patterns created by detrainment of unbuoyant air from the plume, and so the Copyright c British Crown copyright, the Met Office. correlations observed are arguably an indirect effect from the detrainment. A consequence of the lower entrainment at lower humidity is that the plume buoyancy excess profiles are similar irrespective of the humidity of the environment, and it is the vertical decrease in cloud area that differentiates the runs most strongly. Representation of a humidity-dependent detrainment rate is therefore very important in providing the correct shape of the mass flux profile early on.. Cloud area dependence Figures 7 and showed that there is strong evolution of the entrainment and detrainment rates at low levels, even Q. J. R. Meteorol. Soc. 8: 9 ()

11 Entrainment in the Diurnal Cycle y / km x / km x / km MSE / K Figure. Horizontal slice showing the moist static energy fluctuations at a height of km and local time 8 for two simulations with different initial humidity: started at % RH (r), and at 9% RH (r9). Black contours show the active convective plumes. though the RH at these levels remains almost constant. Figure showed how the cloud area increases with time in the runs, and in this section we explore the link between cloud updraught area and the evolution of entrainment and detrainment with time. Figure shows how the entrainment and detrainment rates depend on the average area per cloud at different heights for RHs greater than 7%. This limited RH range is to reduce the effects of differing environments on the results, but the results show similar trends in other parts of the environment parameter space (not shown). The entrainment rates at all levels are a clear function of cloud area, with the entrainment rate decreasing as the cloud size increases. If the mixing into the cloud at this height is purely local in nature, this can be interpreted as a geometric effect that as the cloud grows, the ratio of cloud edge to cloud area decreases, and therefore environmental dilution into the cloud is weakened. The grey line in Figure illustrates how a simple (/r) geometrical dilution would behave, and is similar to the results found in the simulations. The detrainment rates show a similar pattern to the entrainment rates. This could be directly linked to the reduced entrainment measured for larger clouds the larger the cloud the less environmental air per unit volume of cloud is mixed in, and so the buoyancy of the cloud stays higher, and less of it is detrained. Since the cloud size tends to grow with time, the behaviour is consistent with the reversion of the entrainment and detrainment rates to smaller values as the deep convective activity matures, and lends itself to potential parametrization in GCMs. We make some preliminary suggestions about how this might be done in the next section.. Revisions to the convection scheme In this section, we suggest simple revisions to the parametrization of the entrainment and detrainment profiles within the UM. The aim here is to seek the simplest possible formulation that takes an overall account of the findings, and to demonstrate how these impact on the representation of the diurnal cycle. Copyright c British Crown copyright, the Met Office... Entrainment We assume that dependence of entrainment on cloud area is predominantly geometrical in nature, and can therefore be inversely linked to the cloud radius. Since the cloud radius is not predicted by the mass flux scheme, we need a simple representation for this. While the convection is still in the shallow phase, in these idealised runs over land, we find a strong correlation between the cloud area and lifting condensation level (or convective mixed layer depth), as shown in Figure, where the following relationship can be fitted: A c = Fz lcl, () where F =.9 if A c and zlcl have the same units. Clearly, as the convective precipitation intensifies, the cloud area becomes more closely linked to internal feedback processes such as the formation of cold pools in the boundary layer, and so this relationship is expected to break down. However, by this time the entrainment rates have approached their equilibrium values, and so we suggest applying this formula while it predicts a higher entrainment value, and reverting to a minimum equilibrium value given by Eq. (). Here, we suggest a simple linear relationship between the entrainment rates and RH, although we note that the RH of the environment will also be expected to affect the cloud area, and so this relationship may not be appropriate for more fully developed convection. There could equally be scope for including the RH dependence in a forced detrainment term (section. provides a discussion of causality), but we do not attempt this here. In the next section, we will be making the mixing detrainment a function of the entrainment, and the detrainment shows little dependence on stability, so for simplicity we leave out this dependence in the entrainment rate parametrization. For the early stages of developing convection over land, we suggest a parametrized form for entrainment: ɛ fit = [ ] ARH, () z z lcl where A =.. The fit is shown overlaid for the 9% and % initial RH cases in Figure 7, and shows that it captures Q. J. R. Meteorol. Soc. 8: 9 ()

12 A. J. Stirling and R. A. Stratton... ε (.km)/ km -.. ε (.km)/ km -.. ε (.km)/ km Area per cloud / km Area per cloud / km Area per cloud / km (d) (e) (f) δ (.km)/ km - δ (.km)/ km - δ (.km)/ km Area per cloud / km..... Area per cloud / km..... Area per cloud / km Figure. Dependence of (a c) entrainment and (d f) detrainment on the average horizontal area per cloud at heights (a, d). km, (b, e). km, and (c, f). km. The entrainment points shown are for environmental relative humidities above 7%. Symbols distinguish between the high, medium and low stability cases (diamonds, asterisks and pluses respectively). The grey line shows a / (Area per cloud) function for reference. Area per cloud / km z lcl / m Figure. Average cloud area at. km as a function of lifting condensation level height for clouds with tops lower than km. The solid line is a fit given by Eq. (). Symbols indicate initial relative humidity as in Figure 8. the time evolution behaviour of the entrainment rates for a range of humidities... Mixing detrainment Here we follow a similar approach to that currently used in the UM, which links the mixing detrainment rate to the product of the entrainment rate and a function of RH, as in Copyright c British Crown copyright, the Met Office. Eq. (). Currently in the UM, for deep convection, α det is set to.. Figure shows a scatter plot of δ/ɛ against RH, with the gradient corresponding to α det =. plotted from Eq. () (dotted line). If keeping the same linear relationship, a value of α det = gives a much better fit to the data (dashed line), however there is some evidence of a nonlinear relationship at low RH, of the form: δ = ɛ ( RH), () which is also shown (solid line). Finally we use the revised entrainment and detrainment rates in Eqs () and () to recalculate how the SCM represents this development phase. We revert to the original deep entrainment formulation if the fitted entrainment rate drops below the original deep values in Eq. (), and keep a forced detrainment value of R det =.7. The evolution is shown in Figure. This shows a more gradual evolution of the convection in the lower-rh environments, while allowing the convection to develop quickly in the higher-rh environments. The revised parametrization has also removed the intermittent switching on and off of the convection scheme, suggesting that the increased entrainment has slowed the rate at which instability can be removed from the atmosphere. A side effect is that the convective height now seems to be lower than the CRM results, indicating perhaps that the /z form of entrainment rate may need capping at low levels. Q. J. R. Meteorol. Soc. 8: 9 ()

13 Entrainment in the Diurnal Cycle δ/ε (.km) δ/ε (.km) δ/ε (.km) RH RH RH Figure. Ratio of detrainment to entrainment plotted against relative humidity at heights. km, km, and km. The dotted line shows the relationship used in Eq. () with α det =., and the dashed line with α det =. The solid line shows the relationship suggested by Eq. (). There may also be a case for augmenting the buoyancyincreasing effect of the forced detrainment to allow the plumes to rise higher. We leave analysis of these effects for future work.. Discussion and conclusions This article has tested some standard assumptions about the behaviour of entrainment and detrainment in the parametrization of deep convection, with a view to understanding why the diurnal cycle of convection over land is poorly represented in GCMs. We have used an idealised CRM framework that has allowed us to explore independently the roles of environmental stability and RH in controlling the early stages of convection. We find that, even under high CAPE and low CIN conditions, deep convection can take many hours to develop. This is shown to be strongly affected by the environmental conditions, suggesting that the entrainment is important in controlling the deepening of the cloud. Comparisons with SCMs with the same set-up suggest that this development phase is currently poorly captured. The CRM simulations showed the following:. The entrainment and detrainment rates in the early stages of developing convection are significantly higher than those used in the Met Office UM, and this is likely to be the reason why parametrized convective cloud deepens too quickly. These values gradually decrease as the convection deepens, which is consistent with the behaviour found in a WRF study of a TWP-ICE period by Del Genio and Wu ().. Both the entrainment and detrainment rates are sensitive to the environmental conditions, with entrainment rates higher in higher humidity environments, and lower in higher stability environments. The entrainment behaviour we find here differs from the CRM findings of D, who found that convection in equilibrium conditions was relatively insensitive to RH. It is likely therefore that the size of this effect reduces as the cloud deepens and the horizontal size of the cloud grows, or indeed that, where simulations are relaxed back to a given RH, the environmental humidity has time to influence the cloud size, leading Copyright c British Crown copyright, the Met Office. to smaller clouds at lower RH. Depending on the extent of this effect, this may boost the entrainment seen in the low-rh runs compared with what we observe here. We note further that Bechtold et al.(8) have implemented an entrainment scheme which increases as the environmental humidity decreases, with considerable improvements to the ECMWF GCM representation of the Madden Julian Oscillation. This may simply be an alternative way of inhibiting the convection in low-rh environments, (where our CRM results achieve this through higher detrainment rates), but it does suggest that caution should be applied about the universality of the relationship we find between entrainment and RH, and that studies of convection in different environments particularly over the sea would merit further investigation. The detrainment dependence on RH measured here is found to be about four times stronger than is currently used in the UM, and there is evidence from Derbyshire et al.() that this is also the case for deep convection at equilibrium. We have not investigated the buoyancy of the detrained air to determine whether the term we call mixing detrainment is detraining with the average plume buoyancy. It is likely that it actually contains an element of differential detrainment much as the forced detrainment term, in which the detrainment acts to enhance the plume buoyancy.. Entrainment and detrainment are strong functions of cloud size. In the early stages of convection, this can be interpreted as a geometric effect that the surface area to volume ratio decreases with increasing cloud size, and is consistent with Simpson s (97) early suggestions that the entrainment rate should be an inverse function of cloud radius. It also backs up the suggestion by Khairoutdinov and Randall () that convective cloud needs to reach a certain size to overcome the stifling effects of the entrained environmental air. This area dependence is likely to have the most significant effect in representing the early stages of developing convection. Q. J. R. Meteorol. Soc. 8: 9 ()

14 8 A. J. Stirling and R. A. Stratton r9 r9 r r r r r r r Figure. Time series of cloud-top height for a range of environmental initial conditions, with dashed lines showing results from the single-column model. Columns from left to right are for decreasing stability, rows from top to bottom are for RH 9%, % and % respectively. The run numbers correspond to the initial conditions in Table. These runs use the fitted entrainment rate in Eq. () and mixing detrainment rate in Eq. ().. In the idealised scenario considered here, the cloud area can initially be linked to the lifting condensation level. This relationship opens up the possibility of including a simple representation for cloud area in a parametrization, however there are clear limitations to this approach as, in the late afternoon, the relationship will break down, with feedbacks of the convection onto larger scales via cold pools starting to control the size of the convection. There is therefore a need to revert to the original deep formulation as the entrainment rates decrease towards this value. Note also that, over sea, the relationship is unlikely to hold because the convection is closer to a state of equilibrium, where inhomogeneity in the boundary layer and perhaps upper-level forcing is likely to be more important than the mixed-layer depth. Exploration of a more general relationship for cloudbase area would be a useful line of enquiry for convection parametrizations.. We have proposed an entrainment and detrainment rate formulation that could be used in convective parametrizations to represent the early stages of evolution of deep convection. We have tested these revised entrainment and detrainment rates in a single-column model framework, and find that the rate of growth of convection is much better represented, suggesting a possible route Copyright c British Crown copyright, the Met Office. to improving the representation of the diurnal cycle in GCMs. In an accompanying paper (Stratton and Stirling, ), we test the use of an area-dependent entrainment rate in a GCM and show that the representation of the diurnal cycle over land can be significantly improved using this approach. Acknowledgements We would like to thank Andy Brown, Steve Derbyshire, Roy Kershaw, Adrian Lock, and Jon Petch for useful discussions in the preparation of this manuscript. S. J. and J. L. Stirling are thanked for their role in finishing this article. References Bechtold P, Köhler M, Jung T, Doblas-Reyes F, Leutbecher M, Rodwell MJ, Vitart F, Balsamo G. 8. Advances in simulating atmospheric variability with the ECMWF model: From synoptic to decadal time-scales. Q. J. R. Meteorol. Soc. : 7. Del Genio AD, Wu J.. The role of entrainment in the diurnal cycle of continental convection.j. Climate : Derbyshire SH, Beau I, Bechtold P, Grandpeix J-Y, Piriou J-M, Redelsperger J-L, Soares PMM.. Sensitivity of moist convection to environmental humidity. Q. J. R. Meteorol. Soc. : 79. Derbyshire SH, Maidens AV, Milton SF, Stratton RA, Willett MR.. Adaptive detrainment in a convective parametrization. Q. J. R. Meteorol. Soc. 7: Devine GM, Carslaw KS, Parker DJ.. The influence of subgrid surface-layer variability on vertical transport of a chemical species in a convective environment.geophys. Res. Lett. : L87. Q. J. R. Meteorol. Soc. 8: 9 ()