Q. J. R. Meteorol. SOC.(1997), 123, pp. 1945-1960

An analytical study of cumulus onset By THOMAS HAIDEN* N O M E R L Environmental Technology Laboratory, USA (Received 20 May 1996; revised 3 December 1996) SUMMARY

A slab model of the well-mixed daytime boundary layer is combined with a surface-layer parametrization based on Monin-Obukhov similarity to investigate analytically basic aspects of cumulus onset. Cumulus clouds are defined to occur when the equilibrium height of surface-layer air equals or exceeds its lifting condensation level (LCL). For a linearized version of the model, a closed solution of onset time is derived as a function of atmospheric and surface parameters. It is found that the effect of the various parameters on the timing of cumulus onset can be encompassed by two non-dimensional ‘cumulus’ numbers, which describe the relationship between differential variations of equilibrium height and LCL, one for the mixed layer and one for the surface layer. The most important parameter, appearing in both these numbers, is the ratio between the potential-temperature gradient above the mixed layer and the dry-adiabatic lapse rate of dew-point depression (% 0.008 K m-’). Results from both the linearized and the nonlinear model indicate that cumulus onset is delayed by increasing the Bowen ratio in cases of moderate-to-high stability, whereas the opposite is found for less stable conditions. Qualitative aspects of the model results are discussed with reference to observed relations between cumulus formation and surface characteristics. KEYWORDS:

Bowen ratio Convective boundary layer Equilibrium level Lifting condensation level

Parametrization

1. INTRODUCTION Cumulus clouds start to form when parcels of air originating in the convective boundary layer (CBL) begin to rise past their lifting condensation level (LCL). For a given LCL, cumulus onset is largely determined by the time evolution of CBL depth, which, together with the overlying stratification, controls the height to which buoyant parcels are able to rise. The first cumulus clouds, however, usually appear well before the mean depth of the CBL reaches the mean LCL (Boers et al. 1984). They are generated by individual parcels of surface-layer air which, because of their buoyancy, manage to penetrate a certain distance into the stable air above the CBL. The problem of predicting cumulus onset thus becomes a problem of predicting characteristics of both the surface-layer and the mixed-layer. Wilde et al. (1985) suggested a method of estimating cumulus cloud amount based upon probability functions, which describe the likelihood of CBL depth and LCL deviating from their average values. They introduced the terms ‘entrainment zone’ and ‘LCL zone’ to denote the two layers of finite vertical extent that contain the majority of CBL-top heights and LCLs. In their model, cumulus onset occurs when the upper tail of the CBL height distribution begins to overlap the lower tail of the LCL distribution. Cloud amount is calculated as an integral over the product of both distributions, implicitly assuming that the temperature and moisture of rising parcels are not significantly correlated. With LCL and CBL height-distribution parameters derived from in situ measurements and laboratory experiments, respectively, this ‘overlap’ method showed considerable skill in predicting fair-weather cumulus amounts and onset times. However, as Wilde et al. (1985) note, a possible weakness of the approach is that correlations between moisture and temperature of cloud-initiating parcels (Coulman and Warner 1977; Stulll993) are not taken into account. Wetzel (1990), hereafter WE, further elaborated on this problem by arguing that warmer parcels of air rising into lower portions of the LCL zone may still not reach their own LCL, which will be located in the upper part of the LCL zone because of higher parcel temperatures. Thus, there could be instances where CBL depth and LCL zones significantly * Corresponding author, present address: Central Institute for Meteorology and Geodynamics, Hohe Warte 38, A-1 190 Vienna, Austria. 1945

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overlap, yet no clouds form. WE compared the method of Wilde et al. (1985) with two different versions of a parcel approach, in which surface-layer air is lifted to its equilibrium height while subject to prescribed fractional mixing with mixed-layer air. With such an approach, which relates LCL and equilibrium height of the same parcel, the problem of spurious overlap does not occur. WE found that, on the six days considered, the parcel methods were in consistently better agreement with observations than the overlap method for both onset time and cloud amount. The differences were most notable when the CBL was growing through layers of weak stability, and rapid cloud build-up was observed to occur as a result of individual buoyant parcels penetrating far above mean CBL height. One problem of the parcel approach, however, is the determination of the mixing parameter. The values found empirically by WE may not be generally applicable, and would have to be parametrized in terms of surface characteristics and CBL variables. The objective of this study is to provide some insight into the mechanisms determining the sensitivity of cumulus onset to atmospheric and surface parameters by using the parcel approach of WE. A closed solution to the mixed-layer growth equations is combined with Monin-Obukhov similarity theory to predict the time evolution of temperature and humidity in the surface layer. Simple initial profiles of potential temperature and mixing ratio, as well as time-dependent surface fluxes, are prescribed. In a linearized analysis, two non-dimensional numbers are derived that indicate the specific contributions of the mixed layer and the surface layer to cumulus onset. Nonlinear model results are discussed in terms of the heat and water budgets of the CBL. The question whether cumulus forms first over dry or moist surfaces is addressed, and model predictions are qualitatively compared with observational findings (Rabin et al. 1990) and numerical results (Chen and Avissar 1994).

2. MODELEQUATIONS AND ANALYTICAL SOLUTIONS ( a ) Model geometry Essential features of the daytime CBL over land are a superadiabatic surface layer, a well mixed interior, and a capping inversion. In mixed-layer slab models the state of the CBL is characterized by its mean depth h , height-independent mean potential temperature 0, and height-independent water-vapour mixing ratio q . Additional parameters are necessary to describe the vertical temperature and moisture structure within the capping inversion. For the purpose of this study an inversion layer of infinitesimal thickness is assumed, equivalent to zero-order discontinuities of scalars and scalar fluxes (Fig. 1). The inversion is thus defined by jumps A0 and Aq of temperature and mixing ratio. The assumption of vanishing gradients within the mixed layer applies well to potential temperature, but is not always a good approximation with regard to mixing ratio. This is mainly due to the fact that at the CBL top the water-vapour flux is usually directed upward, maintaining a non-zero moisture lapse rate, while the heat flux is directed downward (Wyngaard and Brost 1984;Mahrt 1991).The mixed-layer model thus tends to overestimate moisture content in the upper half of the CBL while underestimating it below. It is the latter bias that is most important in the present context, since it affects the source region of cloud-forming thermals. However, at least part of the moisture gradient within the CBL is accounted for by including a superadiabatic surface layer, as in the model presented below. The assessment of moisture and temperature surplus in this layer is based on similarity theory. Empirical profile functions of Businger et al. (197 1) are used to infer ground-level values of temperature and mixing ratio as a function of surface heat fluxes, friction velocity, and roughness height.

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Figure 1. Slab model of the growing CBL, which includes a parametrized surface layer, and zero-order jumps of potential temperature AB, and humidity mixing ratio A q , at the capping inversion at height h. Dotted vertical . lines give profiles lines indicate the path of an undiluted surface-layer parcel rising to equilibrium height h ~Full of mixing ratio q&), and potential temperature BO(Z). Surface values are denoted by subscript S. rn is a scaling factor, see text for further explanation.

( b ) Mixed-layer growth The entrainment heat flux at the capping inversion is assumed to be a fixed fraction aE of the surface heat flux (Driedonks 1982). No additional assumption is required for the entrainment moisture flux which is determined by the rate of rise of the CBL top and the initial moisture profile. The equations governing the time evolution of mixed-layer variables can then be written

dh A6- = a E f S ( t ) dt A6 = &(h) - 6,

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h

t

where fs = Hs/(c,,p) and gs = E s / p are surface fluxes of temperature and moisture, the latter with the dimension of a mixing-ratio flux, and 6&), qo(z) are the initial profiles of potential temperature and mixing ratio (cf. Fig. 1). Hs and Es denote surface values of sensible-heat flux and evaporation rate, respectively. cp is the specific-heat capacity of air at constant pressure, and p is air density. Variations of p across the depth of the mixed layer are neglected. The entrainment parameter aErepresents the fraction of buoyancy-generated turbulent kinetic energy that contributes to an increase of potential energy at the top of the mixed layer. Empirical evidence from both laboratory and field experiments indicates that uE on average assumes a value near 0.2, with a typical range of f0.05 (Stull 1988). In using ( 1) and (2) the shear-induced contribution to entrainment has been neglected, to keep the analysis as simple as possible. An analytical solution to the mixed-layer growth equations which includes the effect of wind shear on entrainment has been presented by Thomson (1992). However, his solution is implicit, i.e. of the form t(h), and valid strictly only for

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constant surface heat flux, whereas emphasis here is on the time evolution of mixed-layer variables forced by the diurnal cycle. Hence, in adopting (1) to (3) to obtain closed, explicit analytical solutions, it should be kept in mind that the results are restricted to cases of light wind. Specifically, if the shear contribution to entrainment as parametrized by Thomson (1992) is compared to the buoyancy contribution, it is found that the condition L < < 0. l h , where L is the Monin-Obukhov length, must be fulfilled in order for the shear term to be negligible. The initial profile of potential temperature is assumed to be linear, i.e.

where a is the initial gradient of potential temperature. The initial profile of the mixing ratio is represented by an exponential decrease

with a scale-height h,, which on average is of the order of 2-3 km. Although highly idealized when compared to actual profiles found in the lower atmosphere, (6) and (7) provide a framework suitable for analytically studying first-order effects of stratification and moisture lapse rate on cumulus onset time. Due to the vertically integrating effect of turbulent mixing within the CBL, structural details of the initial profiles of temperature and moisture have a limited effect on the overall time evolution of CBL properties. Also, by varying the parameters of (6) and (7), profiles characterizing different climatic regimes, as determined by season, latitude, and continentality, may be generated. In the absence of a pre-existing mixed layer at initial time t = 0, the following settings are made: h(0) = A6(O) = Aq(0) = 0, and O(0) = QOsand q(0) = qOs.With these initial conditions the system (1) to (3), (6) has the similarity solution

(Heidt 1977). From (4) and (7) the time evolution of the mixing ratio is found to be

The functions F s ( t ) and G s ( t )represent cumulative values of temperature and moisture flux and are defined as

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(c) Suface-layer parametrization Similarity theory predicts the vertical gradient of potential temperature in the surface laver to be of the form

where 8, = -f s / u , is the flux temperature scale, L = & ? / ( k g f s ) the Monin-Obukhov length with a reference temperature, u* the friction velocity, g gravitational acceleration and k von K h i n ' s constant; @ " ( z / L )is a non-dimensional profile function. Adopting the Businger-Dyer formulation for the unstable case for $H(z/L) (Businger et al. 1971), after integration of (13) from roughness length zo to z , yields

e

(Benoit 1977), where x = czz/(-L) and xo = c2zo/(-L). According to the re-evaluation by Hogstrom (1987) the non-dimensional constants assume the values c1 = 0.95 and c2 = 11.6. When xo is neglected compared to x , (14) reduces to a more common and widely used relation given, for example by Pielke (1984). More recent formulations of & which have been proposed, for example by Brutsaert (1992), differ significantly from the BusingerDyer function only for very large values of ( z / L ) ,and are thus not considered separately here. An important property of (14) is that as z + 00, 8 ( z ) approaches a limiting value

This suggests that 8, may be regarded as identical to 8 defined in the mixed-layer model, representing a 'background' value on which the superadiabatic surface layer is superimposed. In this way a consistent match between the mixed layer and the surface layer can be achieved. Assuming the same profile function to be valid for both potential temperature and mixing ratio, a relationship analogous to (15) may be formulated for q , with q* = -gs/u* replacing 8,. Thus, the surface-layer contribution to temperature and moisture at z = zo can be expressed as A& = O ( z o ) - 8 = f s S , (16) 4 s = q ( z 0 ) - 4 = gss,

(17)

where

is the proportionality constant relating temperature and moisture surplusses to surface fluxes. ( d ) Estimation of cumulus onset time Within the range of temperatures and pressures usually encountered in the CBL, the LCL can be approximated by

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where rc x 0.008 K m-' is an average value of the dry-adiabatic lapse rate of dew-point depression. T is the temperature and Td the dew-point of the parcel under consideration, and zc is measured relative to the height of the parcel. The parameter rc varies in the range 0.0075 < 0.0085 at parcel temperatures and dew-points between -20 and +30 "C. In using (19) slight deviations from the strictly linear relation between LCL and dew-point depression are neglected; these are implied by more accurate theoretical approximations (for example see Davies-Jones 1983). Following WE, the height to which buoyant parcels rise is set equal to the equilibrium height hE of surface-layer air, which is assumed to have a potential temperature, On,, and a mixing ratio, q,n,lying between those of the mixed layer and those at the surface, i.e., 8, = 8

+ mA8s

qm = q

+ mAqs,

(20)

where 0 G m G 1 (see Fig. 1). Neglecting virtual temperature effects, equilibrium height is given by Oo(hE)= Om,where OO(z)is the initial potential-temperature profile. Cumulus onset time is defined as that time T at which h E ( ~=)z c ( t >is first fulfilled. WE interprets the parameter m as representing the degree of mixing experienced during its rise by a parcel originating from near the surface. Alternatively, m may be regarded as characterizing the source region of the parcel, located somewhere between the surface ( m = 1) and the outer edge of the surface layer ( m = 0, cf. Fig. 1). Based on the Wangara dataset, WE found a value of m near 0.1 to give the lowest r.m.s. error between observed and calculated cloud amounts for both versions of his parcel model. Stull and Eloranta ( 1983, using data from a boundary-layer experiment in Oklahoma, found actual cloud-base heights to be in good agreement with LCLs calculated from temperature and dew-point at screen level. According to the surface-layer parametrization in section 2(c), a geometrical height of 2 m corresponds to a broad range of rn, between about 0.2 and 0.8, for conditions of light to moderate wind, and for roughness lengths of order 0.1 m. Within this range, low values of m are associated with small roughness lengths and low wind speeds. From aircraft measurements of specific humidity at different levels within the CBL, Wilde et al. (1985) and Crum et al. (1987) conclude that within the cores of thermals some surface-layer air rises almost undiluted through the depth of the CBL. In both studies the flight legs used to identify surface-layer properties were flown 100-200 m above the surface, which corresponds to m between about 0.05 and 0.3 for the conditions stated above. The curvature of the temperature profile may also be considered, from which it follows that the surface-layer average of m is < 0.5. Combining these different estimates, the intermediate value of m = 0.3 is adopted in subsequent calculations. (e) Solutionfor a reference case For reference purposes, a standard set of parameters and boundary conditions is defined as follows: a! = 0.0035 K m-I, h, = 2500 m, TOS=15 "C, TOS- ( T ~ ) o=s 5 degC (giving an initial LCL of zc = 625 m), zo = 0.1 m, u , = 0.3 m s-I. The diurnal cycle of surface fluxes is represented by Hs(t) = H S M sin(ot)

E s ( t ) = E S M sin(ot),

(21)

implying a constant Bowen ratio B = H S M / ( L E S MThroughout ). the paper it is assumed that w = n/43 200 SKI,corresponding to a 12-hour period of positive radiation balance. For the reference case, settings HSM= LEsM= 300 W mP2are made, i.e. B = 1. These values have been chosen to represent mid-latitude, synoptically undisturbed conditions during the warm season over a relatively dry surface. Ground elevation is assumed to be zero.

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Figure 2. Time evolution of CBL depth h , surface-layerequilibriumheight hE, and surface-layerLCL zc as given by the analytical solution for the reference parameter values (see text).

Figure 2 shows the time evolution of mixed-layer depth, equilibrium height, and LCL, as given by the analytical solution for the reference set of parameters. According to the location of the intersection of the h E ( t )and z c(t ) curves, cumulus onset is predicted to occur at t = 2.9 h, about 2.5 hours before mean CBL depth would have reached the LCL. After t = 10.1 h the surface-layer temperature surplus mAOs becomes smaller than the inversion strength AO, and hE no longer exceeds h. At times after cumulus onset, however, the results have theoretical validity only, since cloud feedback processes, like shading and compensating subsidence, are not included (for example see Stull 1993).

3. LINEAR ANALYSIS

The almost linear increase of surface fluxes, mixed-layer depth, equilibrium height, and LCL during the first hours of insolation allows approximate closed-form expressions to be obtained for cumulus onset time as functions of atmospheric and surface parameters. In order to gain insight into the specific roles of the mixed layer and the surface layer in determining onset time, the LCLs of the two layers are analysed separately before the combined effect is evaluated. ( a ) Mixed-layer LCL By differentiating (1 l), the initial rate of change of the mixed-layer mixing ratio is found to be

where B = -(dqo/dz),,o = qOs/h,is the initial moisture lapse rate near the surface (cf. (7)). For convenience the dot has been defined to denote time derivatives at t = 0 which, due to the linearization, assume constant values. The initial rate of increase of surface fluxes, for example, is given by fs = w ~ S M ,is = W g s M if (21) is used. Linearizing the

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Figure 3. Characteristic Bowen ratio as a function of temperature T , and pressure p .

saturation-mixing-ratio curve, and using (9), (19), (22) and the definition of the Bowen ratio B , the initial rate of LCL rise can be written

where B, = cp/(LqiAT)may be regarded as a characteristic Bowen ratio, and qiATdenotes the slope of the saturation-mixing-ratio curve against temperature. (The parameter B, is very similar to ‘BML’ as defined by Schrieber et al. (1996) in a related analysis of surface-layer effects on LCL.) The second equality defines a non-dimensional mixedlayer ‘cumulus number’ CuML,which is the ratio between the rate of LCL rise and the mixed-layer growth rate. A necessary condition for cumulus clouds to be initiated by parcels of mixed-layer air is that CuML< 1, i.e. the depth of the mixed layer must increase faster than its LCL. The three terms in (23) which compose CUML, represent the effects of mixed-layer warming, surface evaporation, and entrainment of drier air at the mixed-layer top, respectively. As the mixed layer is warmed and moistened by surface fluxes of sensible heat and evaporation (the first two terms in (23)), the ratio B , / B determines whether this results in an evolution towards saturation or away from it. Moistening dominates if

The characteristic Bowen ratio B,, being inversely proportional to the slope of the saturation-mixing-ratio curve, decreases with increasing temperature and, to a lesser degree, with decreasing pressure. Figure 3 shows that at temperatures above 20 “C, B, does not exceed a value of about 0.5. Considering the numerical factor of X0.4 on the r.h.s. of (24), this means that the Bowen ratio would have to be smaller than 0.2 in order to yield a decrease of dew-point depression as the CBL grows. Thus, unless there is a significant increase of moisture with height in the initial profile leading to a positive entrainment contribution (the last term in (23)), the diurnal growth of the CBL over land surfaces during summertime will usually result in a rise of the mixed-layer LCL (CuMVIL > 0), as observed.

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(b ) Su@ace-layerLCL and equilibrium height Moving from the interior of the CBL towards the surface (from m = 0 to m = l), both temperature and mixing ratio rise above their mixed-layer values. With regard to cumulus formation, the question is whether this is accompanied by an increase or decrease of LCL and, in particular, which air has the lowest LCL. Due to the similarity of the surface-layer profiles of temperature and mixing ratio, m drops out of the relation, and it suffices to consider the surface values At?, and Aqs. The LCL of surface-layer air will be lower than the mixed layer LCL if

Here, (16), (17), and the definition of B have been used. Based on the limited vertical extent of the surface layer, the difference between ATs and A& may be neglected in comparison with A&. In terms of B,, defined above, (25) is equivalent to the condition B < B,. Thus, the Bowen ratio must not exceed the values shown in Fig. 3 in order for the surface layer to have a LCL lower than that of the mixed layer. Like the condition CuML< 1, which was found necessary for the mixed-layer LCL to descend as h increases, in general this will not be fulfilled for typical summer conditions over land surfaces. However, the air next to the surface will have the highest equilibrium level, which may more than compensate for the higher LCL. To determine the net effect, differentials of zc and hE are compared within the surface layer. Using (6), (19), and the definition of equilibrium height, this gives

Keeping mixed-layer variables fixed, and linearizing the saturation-mixing-ratio curve,

where use has been made of (25), and a surface-layer cumulus number CusL has been defined closely analogous to CuMLin (23). Equation (27) states that, as long as the stratification above the CBL is less than rc,the parcel equilibrium height will rise more strongly than the LCL as the surface is approached, regardless of how little surface evaporation there may be. Hence, the parcel closest to the surface has the greatest potential to rise to its LCL. If B < B,, zc even decreases with increasing hE,so that in this case the surface parcel is favoured regardless of the stratification above the CBL. (c ) Cumulus onset time Linearized onset time ?, defined by the intersection of h E ( t )and zc(t) when both are extrapolated according to their initial tendencies, is given by

where, as before, the dot indicates a time derivative at t = 0. Expressed in terms of potential temperature and mixing ratio, this can be written

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Figure 4. Cumulus onset time (hours) as a function of the amplitudes of sensible-heat flux HSMand latent-heat flux L E ~ MAll . other parameters are fixed at their reference values. The cross marks the position of the standard case. see text.

Substituting mixed-layer solutions (8) to (1 l ) , surface-layer relations (16) to (18), and using the above definitions of mixed-layer and surface-layer cumulus numbers, leads to

The contributions of the mixed layer and the surface layer to cumulus onset can readily be identified. Both are strongly controlled by stability, a , and by their respective cumulus numbers, which contain the combined influence of initial profiles and surface fluxes. Another important parameter appearing in both terms is the rate of increase of surface fluxes, represented by fs.

4. NONLINEAR MODEL RESULTS Without linearization, cumulus onset time is obtained by combining the full set of algebraic equations (8) to (1 1) and (16) to (20), with the initial and boundary conditions (6) and (21), and searching (for example with a bisection algorithm) for the first t = t which fulfills h E ( t ) = z c ( t ) . Figure 4 shows the resulting onset time as a function of the amplitudes of sensible- and latent-heat flux (cf. (21)), with all other parameters fixed at their reference values. The cross on Fig. 4 marks the location of the standard case depicted in Fig. 2. As expected, t strongly increases as fluxes become smaller. No clouds are predicted to form as HSMand LESM drop to near 100 W m-’. The decrease of t with increasing sensible-heat flux is mainly due to stronger overshooting (the distance between hE and h), i.e. a surface-layer effect. The corresponding mixed-layer effect of enhanced growth of CBL depth is largely compensated by a steeper rate of temperature rise, and thus LCL. In the more unstable case (a=0.002 K m-’, Fig. 5(a)), t is generally smaller because the mixed layer can grow rapidly without much rise in temperature and LCL, and a given

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As in Fig. 4 but for an initial potential-temperature gradient of (a) 0.002 K m-', and (b) 0.005 K m-'.

temperature surplus in the surface layer can produce stronger overshooting (the difference between equilibrium height and mixed-layer depth). In contrast to the reference case, the sensitivity o f t is now smaller with regard to L EsM than it is to HSM.This is also due to more rapid CBL growth, which enhances the effect of entrainment on the CBL water budget relative to that of surface evaporation. Quite different behaviour is found in the stable case (a = 0.005 K m-', Fig. 5(b)), where cumulus onset is delayed by high values of sensibleheat flux. High stability reduces overshooting and causes any increase in sensible-heat input to result in increased rates of LCL rise, which more than compensate for faster CBL growth. A distinction between situations where the CBL moisture budget is dominated by entrainment drying, and situations where it is dominated by evaporation moistening, has been suggested by Mahrt (199 l),based on observations of boundary-layer moisture fluctuations. These observations also show a positive correlation between surface-layer temperature and humidity fluctuations on scales smaller than a few kilometres. According to Mahrt (1991) the correlation is positive because, on this scale, local fluctuations are mainly caused by advection due to turbulent eddy motion. The negative correlation between temperature and humidity found on larger-scales seems to be energy controlled, due to variations of Bowen ratio. The present model contains both mechanisms. A small-scale positive correlation is implicitly assumed in the source-level concept by using the same parameter m for both temperature and moisture. The negative correlation on larger scales can be taken into account by varying the Bowen ratio for a fixed total heat flux. To investigate Bowen-ratio effects, we consider four cases by combining two different stabilities (2 and 5 K krr-') with two initial surface temperatures (5 and 25 "C), deviating symmetrically from their respective reference values. In what follows we always keep the dew-point depression constant when changing the initial temperature. With warm,unstable conditions a high Bowen ratio is favourable for cumulus onset (Fig. 6) because it enhances mixed-layer growth and overshooting, while at the same time the reduced evaporation does not have as much effect on the CBL moisture budget, which in this case is determined mainly by entrainment. In unstable conditions but at lower temperatures a given amount, or a lack, of moisture input has a stronger effect on CBL water content. Hence, in the cold,

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Figure 6. Cumulus onset time 5, as a function of Bowen ratio B , for combinations of two different temperatures ( 5 "C, 25 "C) and stabilities (0.002 K m-', 0.005 K m-I).

unstable case, the rapid vertical development associated with high B is counteracted by the reduced evaporation rate, giving a small net effect. In the case of high stability, on the other hand, the gain in CBL growth rate and overshooting associated with a high Bowen ratio is more than compensated for by the loss of moisture input. This effect is particularly pronounced at high temperatures where, due to the steeper slope of the saturation curve, a larger evaporationflux is needed to balance a given rate of increase of dew-point depression caused by CBL warming. In the limit as B + 0 onset time always goes to zero, because there is evaporation into an infinitesimally thin CBL leading to instant saturation. However, in the real atmosphere, buoyancy associated with virtual temperature would still generate a mixed layer of finite extent, leading to small yet non-zero onset times. Most of the nonlinear model results described in this section are captured by the linear solution (30). The linear model, however, shows a smaller overall sensitivity of t with regard to initial conditions and model parameters. This is because it is based upon a linear extrapolation using the initial tendencies of h E ( t )and z c ( t ) , while the nonlinear model takes into account the decrease of the slope of these functions with time. An intersection between more gently sloped h E ( t )and z c ( t ) curves means higher sensitivity of onset time to initial conditions and model parameters. Quantitative differences between the linear and the nonlinear model increase with onset time, and thus become more pronounced for stable and dry conditions. The linear solution (30) predicts onset times which differ from those of the nonlinear model by less than half an hour in the unstable case (Fig. 5(a)). In the reference case (Fig. 4) and stable case (Fig. 5(b)), differences may reach several hours. Unlike the nonlinear solution, t does not increase beyond the duration of the insolation period within the range of values of HSMand L ESMshown in the diagrams. The effect on t of varying Bowen ratio, shown in Fig. 6, is essentially reproduced by the linear model. The nonlinear model exhibits a high sensitivity of onset time to surface-layer parameters zo, u, and m. In the reference case, for example, an increase of t by one hour can be achieved by either increasing the roughness length zo from 0.1 to 0.4 m, or by increasing friction velocity u , from 0.3 to 0.8 m s-l. Using a value of 0.2 instead of 0.3 for the parcel source-region parameter m ,also results in an onset delayed by one hour.

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(The corresponding figures in the linear solution are: increase of zo to 0.7 m and u, to 1.0 m s-'; reduction of m to 0.15.) Based on CBL dynamics and common experience, it is generally assumed that strong wind suppresses or delays cumulus onset because of its detrimental effect on surface-layer heating and on coherent thermal circulations (Young 1988). However, no quantitative analysis seems to have been published in the open literature on the relationship between cumulus onset and wind speed, or cumulus onset and surface roughness.

5. DISCUSSION Rabin et al. (1990) studied the effect of landscape variability on cumulus clouds in Oklahoma using satellite data. Observations during a day in early May, with relatively dry and unstable lower tropospheric conditions, showed clouds forming first over areas with high Bowen ratio. Qualitatively, this agrees with the results of the present model for warm, unstable conditions (Fig. 6). Based on these observations, and on results from a surface energy-budget model, Rabin et al. (1990) suggested that under dry atmospheric conditions clouds generally form first over high Bowen-ratio areas, whereas in a moist environment cloud initiation would start over moister surfaces. According to the present model, however, static stability and temperature determine the sign of the Bowen-ratio effect, with atmospheric humidity merely affecting its magnitude. This can be seen by referring to the linear solution (30). Mixed-layer and surface-layer contributions each contain two counteracting effects of variations of Bowen ratio on ?. First, an increase in CBL growth rate and overshooting with increasing B, as evident from the appearance of fs in both terms. This mechanism favours cumulus onset over drier surfaces. Second, the effect of reduced mixed-layer moisture input and reduced surface-layer moisture surplus (as represented by the cumulus numbers given by (23) and (27)) promoting onset over moister surfaces. Stability atmultiplying B,/B in both cumulus numbers, controls whether enhanced growth or reduced moisture dominates the response of the CBL to an increase of Bowen ratio. Temperature exerts a similar controlling influence through the slope of the saturation curve in B,. Atmospheric humidity enters (30) in the form of the initial LCL zc(0),and via /3 in (23). Its variations do not affect the sensitivities of the cumulus numbers to B. Effects of land-surface moisture variations on cumulus convection in a twodimensional mesoscale numerical model were investigated by Chen and Avissar (1994). The lower boundary of their model domain was divided into a moist (irrigated) surface, and a dry surface where soil water was limited or absent. Thermally induced circulations developed at the boundary between the two regions. The largely undisturbed, one-dimensional response to the different surface forcings could be inferred from the solution at greater distances. Boundary-layer clouds formed over the moist surface about two hours earlier, and at a lower level, than over the dry surface. Using initial and boundary conditions adjusted to match those of the reference run SM1 in Chen and Avissar (1994), that is, Tos = 298 K, TdOs = 293 K, a = 0.004 K m-', h, = lo4 m, zO = 0.1 m, the present model likewise predicts earlier onset over the moister surface, provided the Bowen ratio is smaller than 0.2. Above this value, cumulus onset is found to be largely unaffected by variations of B . Under conditions of solar heating and light wind, fair-weather cumulus cloud formation is delayed or suppressed over and downwind of lakes (Wielicki and Welch 1986; Rabin et al. 1990). Since low Bowen ratios can be expected to occur over such surfaces, this seems to contradict the model-predicted asymptotic behaviour of very short onset times for small B (Fig. 6). However, a lake surface is also characterized by a significantly lower total (sensible plus latent) forcing as compared to the land surface, since short-wave

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radiation is absorbed across a much deeper layer. Model results for low total heat flux (lower left regions in Figs. 4 and 5) indicate delayed cumulus onset over such surfaces, provided the initial stratification is not too stable. As WE points out, several possible alterations to, and extensions of, the parcel method may be considered. For example, the heuristic multi-parcel approach of WE showed slightly higher predictive skill than the one-parcel method of WE used here. It was not adopted in the present study because it involves a large number of additional parameters that define the probability of certain combinations of temperature and humidity values of surface-layer parcels. Observed joint frequency distributions of surface-layer temperature and humidity over different surfaces have been shown to exhibit a complex pattern (Stull 1993),which has only very recently been analyzed with regard to possible parametrizations in terms of bulk surface-layer variables (Schrieber et al. 1996). An obvious shortcoming of the solutions presented is their restriction to constant stratification. (The negative exponential humidity profile may be replaced by any integrable function to yield a closed form solution, as can be seen from (4).) This ignores the ‘rapidrise phase’ of mixed-layer growth associated with the rise of the CBL top through a nearly dry-adiabatic residual layer (Segal et al. 1992). If the atmosphere is sufficiently moist, and if the moisture lapse rate within the residual layer is small, cumulus onset will tend to occur during this rapid rise, because it allows a large increase of parcel equilibrium height with a small rise in temperature and LCL. Onset time will then essentially be determined by the time the sensible-heat input needs to eliminate the stable surface layer. Within the framework of the present model this time is given by 1 w

where D is the vertically integrated temperature deficit of the stable layer, and (21) has been used. Subsequent mixed-layer growth through a near-adiabatic layer cannot be realistically described by models that use (2) as an entrainment relation, because infinite growth rates would be predicted for a = 0 (see also Gryning and Batchvarova 1994). An alternative parametrization could be based on the experimental results of Deardorff et al. (1980). They indicate that, in the absence of stratification, mixed-layer growth proceeds at a rate proportional to the convective velocity scale w,, and the overshooting distance Ah = zE - h is proportional to h. Without elaborating such an approach here, its implications are noticed when interpreting the results for a = 0 obtained from the present model. From (23) it follows that for vanishing stratification and within the limits of linearization

where zco is the hypothetical initial LCL of surface air having the same potential temperature as the residual layer. It is worth noting that the Bowen ratio appearing in (23) has dropped out of the problem because of the mathematically infinite mixed-layer growth rate, which in the real atmosphere is simply a high growth-rate scaling with 20,. This is consistent with the diminishing influence of the Bowen ratio on onset time as conditions become less stable (Fig. 6). Adopting the scaling A h / h = f X 0.25 suggested by Deardorff er al. (1980), we obtain as a necessary condition for cumulus onset to occur while the CBL grows through the residual layer

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where zREsdenotes the top of that layer. From (3 1) and (33) we conclude that the effect of an adiabatic residual layer overlying a surface-based inversion (as compared to uniform stratification) is to favour cumulus onset over drier surfaces. Furthermore, onset time becomes more sensitive to the initial moisture profile (ZCO, p), and less dependent on surface evaporation.

6. CONCLUSIONS An analytical model of cumulus onset is presented, which allows the calculation of onset time as a function of atmospheric and surface parameters for simple initial profiles of temperature and humidity. Closed solutions of a simplified set of mixed-layer growth equations are combined with a surface-layer parametrization based on Monin-Obukhov similarity theory. Cumulus onset is diagnosed in terms of equilibrium height and LCL of surface-layer air. The almost linear increases of equilibrium height and LCL during the first hours of insolation allow approximations to be made in which these rates of increase, and the slope of the saturation curve, are treated as constants. The resulting closed-form expression for linearized onset time involves two non-dimensional ‘cumulus’ numbers; these represent the ratio between variations of equilibrium height and LCL, one for the mixed layer and one for the surface layer. The most important parameters appearing in both cumulus numbers are the stratification of the air above the CBL, and the Bowen ratio. Increased stability, as well as an increased humidity lapse rate in the air above the CBL, always act to delay cumulus formation. A necessary condition for the CBL to initiate cumuli is that the Bowen ratio does not exceed a certain threshold value, which decreases with increasing temperature. Based on the linear analysis it is shown that, regardless of surface evaporation, parcels closest to the surface have the highest potential of reaching their LCL, provided the gradient of potential temperature above the CBL is less than the dry-adiabatic lapse rate of dew-point depression (z0.008 K m-’). The effect of Bowen ratio on onset time is found to depend on stratification. In the case of moderate to high stability, the large rate of LCL rise associated with CBL warming can only be counteracted by surface evaporation. Hence, early cumulus onset is favoured by a small Bowen ratio. In the case of low stability, the warming associated with CBL growth is smaller, and the moisture budget of the boundary layer is dominated by entrainment, thereby reducing the sensitivity of the LCL to surface evaporation. Accordingly, in this case early cumulus onset is favoured by a high Bowen ratio, because the reduced moisture input is more than compensated by the gain in equilibrium height. Lifting a surface-layer parcel adiabatically to its equilibrium height, to check for possible cumulus formation, leads to a high sensitivity of cumulus onset to surface roughness and wind speed. In the real atmosphere, the maximum height attained by thermals is controlled not only by surface-layer buoyancy but also by mixed-layer properties, which act to reduce this sensitivity. It should also be noted that the model has been designed for CBLs over land surfaces. If applied to the boundary layer over a water surface, virtual-temperature effects would have to be included in the calculation of equilibrium height. ACKNOWLEDGEMENT The valuable comments of reviewer Andrew Crook at the National Center for Atmospheric Research and an anonymous reviewer are highly appreciated. This research was carried out while the author was a Visiting Scientist at the NOAA/ERL Environmental Technology Laboratory. Financial support was provided by Fonds zur Foerderung der wissenschaftlichen Forschung (FWF)under Grant J01057-TEC.

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