Variance scaling in shallow-cumulus-topped mixed layers

Transcription

1 QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY Q. J. R. Meteorol. Soc. 133: (7) Publised online in Wiley InterScience ( Variance scaling in sallow-cumulus-topped mixed layers R. A. J. Neggers,* B. Stevens and J. D. Neelin Department of Atmosperic and Oceanic Sciences, University of California, Los Angeles ABSTRACT: Scaling of termodynamic variance in sallow-cumulus-topped mixed layers is studied using large-eddy simulation (LES). First, te performance of te top-down scaling (te turbulent flux at mixed-layer top divided by w ) is evaluated for transient sallow-cumulus convection over land. Te results indicate tat tis scaling fails to capture all te variance in te top alf of te mixed layer wen sallow cumulus clouds are present. A variance-budget analysis is ten performed, to derive a new scaling for te variance at mixed-layer top, wic differs from te standard top-down scaling by a factor of one Ricardson number. Te essential new features of te proposed scaling are tat te local vertical gradient is retained and tat a balance is assumed between gradient production of variance and removal by transport and dissipation, using an adjustment time-scale given by w /. Evaluation against LES for a range of different cases, including a dry convective boundary layer as well as steady-state marine and transient continental sallow cumulus, reveals a datacollapse of te newly-scaled variance, for all ours and all cases in te top alf of te mixed layer. Te corresponding vertical structure is sown to resemble a power-law function. Te results suggest tat te structure of variance in te dry convective boundary layer is similar to tat in te sub-cloud mixed layer. In transient situations, te scaling reproduces te time-development of variance at sub-cloud mixed-layer top. Te new cloud-base variance scale is ten furter interpreted in te context of statistical cloud scemes, wic depend on te variance as te second moment of te associated probability density function. Te results suggest tat te area fraction of te moist convective termals uniquely depends on te ratio of cloud-base transition-layer dept to sub-cloud mixed-layer dept. Tis puts valve - or ventilation-type closures for te cloud-base mass flux in te context of te variance budget for te sub-cloud layer. Copyrigt 7 Royal Meteorological Society KEY WORDS sallow cumulus; variance scaling; transition layer; mass-flux closure Received September 6; Revised 6 February 7; Accepted 9 February 7 1. Introduction Te parametrization of vertical transport of eat, umidity and momentum by sallow-cumulus-cloud populations as been te subject of intensive researc (see, for example, (Arakawa, 4) for a recent review). Te mass-flux approac, werein te vertical advective transport by organized updraugts is explicitly modelled, as emerged as one of te more successful metods (e.g. Ooyama, 1971; Yanai et al., 1973; Betts, 1975; Siebesma and Cuijpers, 1995). Te mass flux is defined as te product of air density, convective area fraction, and vertical velocity of te associated updraugts. Wile most bulk-closure metods parametrize te mass flux as a single entity, te area fraction and vertical velocity can also be modelled individually. For instance, te updraugt vertical velocity can be estimated from te integrated mixed-layer buoyancy flux (e.g. Grant, 1), wile te associated convective area fraction can be retained and explicitly parametrized (e.g. Breterton et al., 4; Neggers et al., 4). Te area fraction can be estimated using an assumption for te underlying probability density function (PDF) of te termodynamic state variables (Sommeria and Deardorff, 1977), as constrained by te prediction of one or more of its moments. Even te simplest of tese PDF-based approaces requires knowledge of te variance of temperature and umidity. Tis raises te question motivating tis study: wat determines te structure of te variance at te top of te sub-cloud mixed layer? Standard similarity teory for te convective mixed layer, or mixed-layer scaling, grew out of attempts to understand te similarity structure of te surface layer in te early 197s (see, for example, (Stull, 1988) for a review). Near te surface in sear-free convective layers it is often argued tat conserved scalars follow te free-convective temperature scaling of (Wyngaard et al., 1971; Kaimal et al., 1976), in wic te dimensionless variance of some scalar c scales as σ c c = α( z ) 3, (1) * Correspondence to: R. A. J. Neggers, Royal Neterlands Meteorlogical Institute (KNMI) PO Box 1, 373 AE De Bilt, Te Neterlands. Roel.Neggers@knmi.nl were c w c w. Copyrigt 7 Royal Meteorological Society

2 163 R. A. J. NEGGERS ET AL. Te overline denotes te mean value, and te prime denotes fluctuations, so tat c = c + c.valuesatz = are denoted by subscript ; values at te top of te subcloud mixed layer, z =, will be denoted by subscript. Te Deardorff, or convective, velocity scale is denoted by w. Field measurements from Minnesota reported by Kaimal et al. (1976) sow tat te potential-temperature variance conforms well to Equation (1) in te lower part of te mixed layer (z <.1) wit te constant of proportionality α = 1.8. However, tis scaling works less well for z>.1. Between.1 and, θ tends to decrease more rapidly tan te teory predicts, and θ tends to increase again above. Kaimal et al. (1976) recognized te tendency of θ to increase wit z above as a signature of entrainment: a process not accounted for in te arguments leading up to Equation (1). Subsequent work by a number of investigators (e.g. Deardorff, 1974b; Nicolls and LeMone, 198; Lenscow et al., 198; Caugey, 198) reinforced tese findings, and elped motivate te concepts of top-down and bottom-up scalar diffusion (Wyngaard and Brost, 1984). According to tese ideas, te fundamental asymmetry of te convective forcing of te convective boundary layer (CBL), wereby te buoyancy flux is positive at te surface and some negative fraction of te surface forcing at te top of te CBL, causes scalars mixing into te CBL from te top (top down) to diffuse troug te CBL differently from scalar fluxes originating at te bottom (bottom up). Using tese ideas, and noting tat any scalar c can be written as a linear combination of its top-down and bottom-up components c t and c b, Moeng and Wyngaard (1984) sowed tat te scalar variance troug te dept of te dry CBL can be well represented by superimposing te scalar variance expected for pure top-down and bottom-up scalars: ( z ( z ) ( z σc = c f + c c s f s + cs ) f s. () ) In tis expression, te quantities c = w c /w c s = w c /w }, (3) are separate scalings measuring te relative contributions of te top-down and bottom-up components of te scalar, wit w c and w c denoting te turbulent flux at te surface and top of te planetary boundary layer (PBL) respectively. Te eigt variations of te component variances are carried by te functions f and f s,wic Moeng and Wyngaard (1984) deduced empirically on te basis of existing field data and large-eddy simulation (LES). We ask ow well Equation () captures te variance profiles of liquid-water potential temperature θ l and total specific umidity q t, in te sub-cloud layers of boundary layers topped by sallow cumulus. To answer tis question, we use a suite of simulations drawn from past studies, ranging from nearly-stationary maritime sallow cumulus to igly non-stationary cases of sallow cumulus over land. Te non-stationary cases are all tied to te diurnal cycle, and allow us to explore several different convective regimes witin one simulation. Tis significantly broadens te parameter space, compared to studies of more stationary conditions. Figure 1 sows ourly variance profiles, obtained from LES, of te diurnal variation of a fair-weater cumulus case over land (see Section for its description). Te variance profiles are normalized by te mixed-layer-top scale c, as defined in Equation (3). Near z = (defined as te eigt of minimum buoyancy flux), tis scale dominates te contributions from te f s and f s terms in Equation (). In oter words, use of te eigt-dependent scaling given by Equation (3) would not improve te scaling in te upper part of te mixed layer. Altoug c explains more of te variance tan does c s, te normalized variance profiles still exibit considerable spread. Tis is especially evident during te later ours of te diurnal cycle, wen convection is more intense: te top scaling c ten underestimates te variance. Te figure also illustrates a limitation of Equation (), in tat f, wic Moeng and Wyngaard (1984) set to 14(1 ) /3 for z>.9, as a singularity at z =, tus furter inting at te inappropriateness of Equation () as z approaces. Tese observations motivate furter study of te vertical structure of variance in sallow-cumulus-topped mixed layers. In tis paper, a new top-down variance scaling is presented, wic differs from tat of Equation (3) by its incorporation of te local gradient. It is found tat tis scaling better reproduces te structure, magnitude and time-development of mixed-layer variance, for bot stationary and transient sallow-cumulus cases and a transient case of dry convection. Te new cloud-base-variance scaling is ten furter interpreted in te context of statistical cloud scemes, by assuming te form of te PDFs of umidity and temperature. Te (a) 1.5 SCMS (b) σ qt / q t * SCMS σθi / θ I * Figure 1. Scaled ourly-mean vertical profiles of te variance of (a) total specific umidity q t, and (b) liquid-water potential temperature θ l, during te continental cumulus case based on te Small Cumulus Micropysics Study (SCMS). Te eigt is scaled by te mixed-layer eigt, defined as te eigt of minimum buoyancy flux, and te variance is normalized by te mixed-layer-top scale c,asdefinedin Equation (3). Te grey lines indicate te f structure function.

3 VARIANCE SCALING 1631 results suggest tat te area fraction of te moist convective termals uniquely depends on te ratio of cloud-base transition-layer dept to sub-cloud mixed-layer dept. Tis places te type of mass-flux closures tat explicitly parametrize tis area fraction also known as valve - type closures (e.g. Mapes, ; Breterton et al., 4; Neggers et al., 4) in te context of te full variance budget. In Section, te LES model, as well as te cases tat form te basis of tis study, will be described. In Section 3, te new scaling will be presented; in Section 4 it is evaluated against te LES. In Section 5 te important role of te cloud-base transition layer in te new scaling will be explored, implying a new closure formulation for te mass-flux area fraction. Te implications of te results are discussed in Section 6, and some concluding remarks are given in Section 7.. LES model and case descriptions Te LES is performed using te model of te Royal Neterlands Meteorological Institute (KNMI), as described in detail by Cuijpers and Duynkerke (1993). Te results presented in tis paper are based on a reanalysis of a suite of simulations establised over time, of wic many were performed in order to take part in te LES intercomparison studies organized by te boundarylayer working group of te Global Energy and Water Experiment Cloud System Study (GCSS) (Browning, 1993). Te details of te model and te numerical simulations, including te domain size and resolution, differ between cumulus cases, as described below. Four sallow-cumulus cases are simulated. Two of tese represent relatively steady marine conditions, wile te oter two represent more transient continental conditions. Te first marine case is based on a period of undisturbed trade-wind convection capped by a weak inversion, as syntesized from observations made during te Barbados Oceanic and Meteorological Experiment (BOMEX) (Holland and Rasmusson, 1973; Nitta and Esbensen, 1974). Furter details are given by Siebesma et al. (3). Te second marine case is similar, altoug capped by a muc stronger inversion, below wic te cumulus clouds detrain into a sallow stratocumulus layer. It is based on a syntesis of observations taken during an undisturbed period of te Atlantic Trade-Wind Experiment (Augstein et al., 1973, 1974). Furter details are given by Stevens et al. (1). Te first continental diurnal-cycle case is based on measurements taken at te Central Facility of te Soutern Great Plain site of te Atmosperic Radiation Measurement (ARM) programme (Stokes and Scwartz, 1994). Te sallow-cumulus cloud layer was observed to slowly deepen after onset. Te set-up of te LES case is described by Brown et al. (). Te second continental cumulus case is based on te Small Cumulus Micropysics Study (SCMS) (Knigt and Miller, 1998; Frenc et al., 1999; Laird et al., ). Some relevant boundary-layer measurements, and te set-up of te LES case, are described by Neggers et al. (3). Altoug similar to te ARM case, it is somewat more umid, and features a relatively ig cloud cover in te LES, wit a peak value of about 4% sortly after cloud onset. Te cloud layer also deepens relatively rapidly compared to te ARM case. Te dry CBL case is rougly based on a previous intercomparison case of LES codes for te dry CBL (Nieuwstadt et al., 1993). Several modifications are introduced in order to increase transience: prescribed surface fluxes of w θ = 1 Wm and w q = Wm, a prescribed radiative-cooling tendency of 1 Kday 1, and an initial state consisting of a 7 m-deep mixed layer wit q = 8gkg 1 and θ = 3 K, topped by a layer wit lapse rates of Ɣ q = 1.67 g kg 1 km 1 and Ɣ θ = Kkm 1. In te first 4 of simulation, te inversion rises from.7 km to 1.8 km. 3. Variance-budget analysis Te failure of c to scale te variance correctly as z approaces could reflect differences in te scaling of te dry convective layer as compared to te sub-cloud layer, but more probably reflects te inappropriateness of te formulation near te top of te dry convective layer. We argue tat a successful scaling sould reflect te coupling between te dry convective (sub-cloud) layer and te (cloud) layer above, and terefore sould incorporate information from bot layers, as well as te local stability of te interface (transition) layer separating tem (e.g. Augstein et al., 1974; Albrect et al., 1979; Yin and Albrect, ). In revisiting tis issue, we return to te full prognostic equation for te termodynamic variance. Following Deardorff (1974a), te variance budget can be written as σ c t = w c c z w c c ɛ. (4) z Here w c c is te turbulent flux of variance, and ɛ is te molecular dissipation of variance. We ave assumed orizontal omogeneity because it is enforced in LES and it simplifies te analysis. We focus on te variance budget at te mixed-layer top, because te variances at tis eigt are critical to a determination of cloud fraction. Using LES of a developing clear CBL, Deardorff (1974b) sowed tat at te top of te dry CBL te gradient-production term is te only source term, representing dry mixed-layer termals tat oversoot into te strong gradient and generate variance accordingly. Production is countered by transport and dissipation. Figure illustrates tat tis structure is similar in sallow-cumulus situations. To te extent tat is te only relevant lengt scale in te problem, one expects te variance dissipation to decay on a large-eddy time-scale (Nieuwstadt and Brost, 1986): τ w. (5)

4 163 R. A. J. NEGGERS ET AL. z [m] 6 4 BOMEX level of free convection and te velocity scale associated wit CAPE is zero or negative. A good example is te period sortly after cloud onset in te transient continental convection. A furter feature of Equation (7) is tat it is only applied locally at mixed-layer top. As a result, troug te vertical gradient c/ z, te small jump in temperature and umidity tat is often observed at sallow-cumulus cloud base (e.g. Augstein et al., 1974; Albrect et al., 1979; Yin and Albrect, ) is incorporated into te scaling. dissipation transport production σ qt budgets [g kg - s -1 ] Figure. Te steady-state umidity-variance-budget equation (4) for te BOMEX case, as sampled in LES. Production and transport are calculated, and dissipation is obtained as teir residual. Cloud base in BOMEX is at about 6 m. Because in te balance all te leading-order terms must follow te same scaling, te steady-state variance budget of Equation (4) ten becomes: w c c z σc τ, (6) were te subscript refers to te local value at mixedlayer top. Support for tese arguments is provided by Grant and Lock (4), wo sow tat mixed-layer scaling of turbulent kinetic energy still olds in te transition zone. A furter justification is provided in Appendix A, were we approac te scaling from te perspective of te transport term: in tis framework, transport and dissipation act togeter to reduce any variance tat is produced and maintained by mixed-layer termals oversooting into te transition layer. Relation (6) suggests a new local scaling c # for te variance at mixed-layer top: c# w c c. (7) z w Te remainder of tis paper explores tis scaling in more dept. In previous scaling of sallow-cumulus variance (Grant and Brown, 1999; Lenderink and Siebesma, ), te time-scale of relaxation of variance as been modelled as te ratio of cloud-layer dept to a vertical-velocity scale tat is a function of te convective available potential energy (CAPE) of te cloud layer. Tis coice may be appropriate in te cloud layer, but becomes problematic in te mixed layer in cases of forced cumulus convection (cumulus umilis), in wic te clouds never reac teir 4. LES results Te new variance scaling (7) is now evaluated using LES. Figures 3 and 4 sow ourly-averaged variance profiles normalized by c # for all te PBL cases described in Section, irrespective of te presence of clouds. Te scaling applies best in te eigt range.6 < 1, corresponding to eigts were te production of variance at te top of te PBL migt be expected to exert more influence tan, say, near-surface variance production. Tese results suggest tat a typical vertical structure exists in te scaled variance profile in te top alf of te mixed layer. To illustrate tis, te scaled variance profiles of all ours and all cases are plotted togeter in Figure 5. Because f beaves poorly near z =, we fit a new power-law function to te scaling region. To tis end, te same data are plotted on logaritmic axes: see Figure 6. Here a linear relation wit slope b implies tat te variance depends on te dimensionless eigt ratio raised to te power of b (so-called similarity of te second kind ): σc ( z ) bc =.9, (8) c # were te exponents for specific umidity b q = 4and potential temperature b θ = 6 are different. Bot fits are also sown in Figure 5. We believe tat differences in te exponent reflect te different ways in wic umidity and temperature project onto buoyancy, but tis warrants furter investigation. Te variance peaks at te level of maximum gradient, wic is sligtly iger tan te level of minimum buoyancy flux, as already observed by Deardorff (1974b) for te dry CBL. Tis is reflected in te constants of proportionality used in Equation (8), wic are sligtly less tan one. Close to te surface, te new variance scaling fails, because cloud-base caracteristics become less relevant at levels furter away from it. At tese levels one expects te surface scalings to dominate. Tis motivates combining te surface and mixed-layer-top scalings into one relation (e.g. Moeng and Wyngaard, 1984): ( z ) ( σc (z) = z ) cs +.9 bc c #. (9) Figure 7 sows te time series of te variances at mixed-layer top for tree of te cases. Te performance

5 VARIANCE SCALING 1633 (a) SCMS (b) ARM (c) BOMEX (d) ATEX (e) DRY 1.5. σ qt / q t # σ qt / q t # σ qt / q t # σ qt / q t # σ qt / q t #.... Figure 3. Vertical profiles of te moisture variance, as in Figure 1(a), but for all cases and wit te variance normalized by te new scaling c #. Te eigt is scaled by te mixed-layer dept. (a) SCMS (b) ARM (c) BOMEX (d) ATEX (e) DRY σ θi / θ I # σ θi / θ I # σ θi / θ I # σ θi / θ I # σ θi / θ I # Figure 4. As Figure 3, but for te scaled liquid-water potential-temperature variance. (a) 1.5 (b) 1.5 σ qt / q t # 1.5. σ θi / θ I # 1.5. Figure 5. Scaled variance profiles for (a) moisture and (b) liquid-water potential temperature, as in Figures 3 and 4, but averaged over all cases and all ours. Te black solid line is te mean, wile te grey area indicates te standard deviation. Te tick dased line represents te power-law fit (Equation (8)). of c # is compared to tat of te surface scaling c s and top-down scaling c. Te new variance scaling c # is most successful in reproducing te time evolution as seen in LES. Te surface scaling c s tends to reac its maximum at a muc earlier time tan te scalings for mixed-layer top. Using c correctly sifts te maximum towards later times, but is still not sufficient. Bot te scalings tat fail to capture information on te local vertical gradient also fail to capture te variance minimum tat occurs in te ARM case at our 7. Figure 8 sows te time series of te local vertical gradients of umidity and temperature at mixed-layer top for te two cumulus-over-land cases. In te course of te day, tese gradients cange significantly. In general, te mixed-layer top experiences destabilization in te first ours and stabilization in te last ours of te diurnal cycle. Tis reflects te development of te coupling between te mixed layer and te cloud layer. In tis respect, te ARM and SCMS cases develop quite differently during te first ours, te ARM case being muc more stable above te mixed layer. During te 7t our in te ARM case, te mixed-layer top suddenly destabilizes; tis is accompanied by a more rapid deepening of te cloud layer (see Figure 9(b)). Tis

6 1634 R. A. J. NEGGERS ET AL. (a) SCMS ARM BOMEX ATEX (b) SCMS ARM BOMEX ATEX σ qt / q t σ # θi / θ I # Figure 6. Logaritmic scatter plot of te scaled variance profiles of all cases and all ours. Te black solid line represents a power-law fit troug te upper part of te sub-cloud layer, wic sows evidence of scaling beaviour. Te slope of te fit corresponds to te exponent, wic is 4 for specific umidity and 6 for potential temperature (see Equation (8)). (a) BOMEX (b) SCMS (c) ARM diagnosed σ qt at [g kg - ].8 1 q ts* 1 q t* q t# σ θi at [K ] diagnosed 1 θ Is* 1 θ I* θ I# Figure 7. Time series of te diagnosed variances of q t and θ l at mixed-layer top (grey), and te corresponding variance scales c s, c and c # (black), during (a) BOMEX, (b) SCMS, and (c) ARM. Some scales are multiplied by a factor of 1 for purposes of comparison. sudden weakening of te gradients is accompanied by a significant decrease in variance (see Figure 7(c)). Tis furter suggests tat te variance near is coupled to te vertical gradients at, and tat scalings tat incorporate te local gradients can be expected to beave wit more fidelity. 5. Mass-flux area fraction Te results illustrate tat te structure of te cloud-base transition layer significantly affects te local variance. Tis interface layer is now studied in more detail, wit te aim of parametrizing tose of its properties tat appear in te new variance scaling (7). Tis is necessary for te broad class of models tat do not provide tis information naturally, and it may give more insigt into te important role of te cloud-base transition layer in sallow-cumulus convection. Figure 9 sows te typical vertical structure of a sallow-cumulus-topped boundary layer. Te cloud base eigt z b is defined as te eigt of maximum cloudcore fraction, wic is defined as te area fraction of

7 VARIANCE SCALING 1635 (a) SCMS (b) ARM (c) 6. SCMS (d) 6. ARM z q t at [g kg -1 km -1 ] z q t at [g kg -1 km -1 ] Figure 8. Time series of te local vertical gradients of total specific umidity and liquid-water potential temperature at mixed-layer top, during te SCMS and ARM cases. z θ I at [K km -1 ] z θ I at [K km -1 ] (a) (b) 4 ARM 3 z b cloud q z b z[m] 1 w θ v q t Time [r] Figure 9. Typical structure of te sallow-cumulus-topped mixed layer. (a) Profiles of te buoyancy flux w θ v, total specific umidity, and cloud-core fraction A (te symbols are defined in te text). (b) Time series of and z b for te ARM case. Te cloud layer is saded (grey). Te transition layer is situated between and z b. positively-buoyant clouds (also called te moist convective termals ). In prototype sallow-cumulus convection, te cloud core does most of te vertical transport (Siebesma and Cuijpers, 1995): tis justifies its use in te bulk-mass-flux approac. Following (Grant, 1), we define te transition layer as te layer of dept between and z b. Lenderink and Siebesma () were te first to apply te mass-flux approac in te variance-production term. Combined wit a bulk-gradient approac over te transition-layer dept, tis gives for Equation (7): c # = M c w, (1) were M is te cloud-base mass flux and c is te jump in te mean vertical profile c over te transition layer. Here te umidity excess of te moist convective termals over te dry environment as been assumed to be proportional to c. Figure 1(a) sows tat tis assumption is supported by LES. Tis proportionality is due to te moist convective termals still carrying properties of te mixed layer were tey originate. Teir collective volumetric mass flux M is defined as te product of teir area fraction A and teir vertical velocity, wic at cloud base scales well wit w (Neggers et al., 4). M = Aω (11) Tis eliminates w in Equation (1), yielding: c # A c. (1) Tus, retaining te convective area fraction in te mass flux (e.g. Neggers et al., 4; Neggers et al., 6) makes it appear in te cloud-base variance scaling. Tis is intriguing from te perspective of statistical parametrization of area fractions (Sommeria and Deardorff, 1977), wic itself depends on te variance as te second moment of te turbulent distribution of umidity and temperature. As a consequence, tere are two equations for te two unknowns A and σ c,sotata and σ c are defined implicitly. In oter words, scaling σ c wit c # would yield a parametrization for A. Tis implicit representation of A still requires te definition of te sape function of te associated turbulent distribution. One could use a normal distribution (e.g. Sommeria and Deardorff, 1977; Bougeault, 198). However, suc a distribution would ave global support, and it is peraps more appropriate in tis problem to use a distribution wit compact support. For example, te most extreme q t value at cloud base tat can teoretically occur tat of an undiluted updraugt can never exceed its value at its starting eigt in te mixed layer. We coose to explore tese issues using te beta distribution (e.g. Tompkins, ), wose density is expressed

8 1636 R. A. J. NEGGERS ET AL. (a) (q t - q t co )b (b) BOMEX ATEX ARM SCMS M / w DRY BOMEX ATEX GRANT (1) ARM SCMS q / Figure 1. (a) Difference between mean-state total specific umidity q t and tat of te cloud core qt co at z b, plotted against te jump in total specific umidity q t between mixed-layer top and cloud base z b. Te points represent ourly averages from all cases. Te dased line represents te least-squares linear fit. (b) Scatter plot of / against M/w. Te points represent ourly averages from all cases. Te results of (Grant, 1) are also included. Te dotted line represents te least-squares fit of Equation (17), giving p =.. by te beta function: B(p, r) = 1 x p 1 (1 x) r 1 dx, (13) were p and r are te sape parameters of te distribution. Te standard deviation of te beta distribution is related to te distribution boundaries a and b: σ b a p + r pr p + r + 1. (14) Figure 11 supports a coice for te boundaries satisfying: b a = c. (15) Assuming a non-skewed distribution (p = r), Equation (14) ten gives σ c c = 1 p + 1. (16) Tus te variance is related to te local jump in c troug te sape function of te associated PDF. Substituting tis ratio into te cloud-base variance scaling (1) finally gives A = 1 p + 1. (17) Tis relation states tat te area fraction of te cloudy, buoyant termals is uniquely determined by te ratio of transition-layer dept to sub-cloud mixed-layer dept, multiplied by a term dependent on te sape of te distribution. Equation (17) is verified by comparing te ratio M/w at z b to te dept ratio / for all cumulus cases: see Figure 1(b). Despite some scatter, te existence of a relation between A and / is supported by te LES results (cf. (Grant, 1), were tis ratio is assumed constant). Te apparent linearity of te LES data suggests tat p is constant, implying tat te sape of te PDF at cloud base is more or less case-independent. Fitting relation (17) to te LES data gives p.. (18) Te side panels in Figure 11 illustrate tat tis beta distribution resembles te LES distributions reasonably well. Some of te scatter in Figure 1(b) can be attributed to te low vertical resolution of tese LES runs, wic at 4 m was of te same order of magnitude as. Tis causes deviations in / of up to 4 m /8 m = 5 (taking a typical value for ). Additional simulations at iger vertical resolution, and using a sorter averaging time-scale, could elp clarify tis issue. In te derivation of Equation (17), te numerator c of te local vertical gradient as dropped out against σ c, but te denominator is preserved, introducing a coupling between te area fraction of te moist convective termals and te dept of te transition layer. For example, in situations of ig transition-layer stability (large gradients, or small ), A is small, corresponding to a reduced mass flux. Te transition layer tus acts as a valve in cumulus transport. On te oter and, troug te dissipation time-scale, te mixed-layer dept as entered te equation. As a result, deepening sub-cloud mixed layers imply decreasing convective area fractions: a caracteristic feature of diurnal cycles of sallow cumulus, wic is indeed often observed in nature and in LES (e.g. Brown et al., ; Neggers et al., 4). 6. Discussion 6.1. Comparison to (Moeng and Wyngaard, 1984) We now return to te (Moeng and Wyngaard, 1984) scaling c, as discussed in Section 1, and ask wy and ow te new variance scaling c # is different. Comparing

9 VARIANCE SCALING BOMEX LES 6m q q t [g kg -1 ] θ P(θ I ) zero buoyancy line saturation curve mean vertical profile P(q t ) θ I [K] Figure 11. Joint PDF of q t and θ l at BOMEX cloud base (z b = 6 m), as obtained from LES. Te complete mean vertical profile is indicated by te solid line; its values at z b are indicated by te cross. Te saturation curve and te zero-buoyancy line at tis particular level are also sown. q sat is te point were te joint PDF intersects te saturation curve, and q zb corresponds to its intersection wit te moist part of te zero-buoyancy line. Te transition-layer jumps q and θ l are also sown. Te two side panels sow te PDF P(x) (grey lines), and te PDF of te beta distribution wit p = r =. (black lines). (a) 1 (b) (c) (d) 3 w* /1 1 w* z q t at z= 1 w q t at z= q t -flux [g kg -1 m s -1 ] q t gradient [g/kg /m] z q t at z= q t jump [g/kg] q t / z q t at q t jump dept [m] q t / z q t at Figure 1. (a) Evaluation of te validity of Equation (19) in LES, for te ARM case. All parameters in te equation are diagnosed individually. Some terms are amplified by a decimal factor, for purpose of display. (b) Comparison of te bulk gradient of q t over te transition layer to te local gradient at z =. (c) Transition-layer jump of q t, and (d) transition-layer dept: dased lines indicate te values obtained using te local gradient at z =. Equations (3) and (7), we see tat te two scales are only te same if w c c w. (19) z Figure 1(a) sows te two sides of Equation (19), confirming tat tey are indeed not te same and tat te vertical gradient (plotted in Figure 1(b)) is causing te difference in time development between te scales. Furter breakup of te gradient in Figure 1(c) and (d) reveals tat growt of te transition-layer dept is uniquely responsible for te sudden decrease in te gradient at t = 7, counteracting a strengtening of te jump c in te process. To furter illustrate tat c and c # are indeed different teoretically and do not revert to eac oter in some

10 1638 R. A. J. NEGGERS ET AL. limit, we now apply te bulk-gradient approac over te transition-layer dept (see Figure 1(b)), c c z, () and te flux-entrainment relation at, w c = E c = A Ri w c, (1) were E is te top-entrainment velocity, A is a constant of proportionality, and Ri g θ v/θ w () is te standard bulk Ricardson number, wit g θ v /θ te buoyancy jump associated wit te transition layer. If Equation (19) olds, ten substitution of Equations () and (1) would imply tat Ri. (3) However, we can prove tat Equation (3) cannot old, by studying te equilibrium mass budget of te sub-cloud mixed layer (Stevens, 6; Neggers et al., 6). In equilibrium, neglecting large-scale divergence for simplicity (typically w LS E in sallow-cumulus conditions), te cumulus mass flux M sould equal te topentrainment rate E. As bot sare dependence on w, one can equate te associated factors of proportionality in E and M. Togeter wit Equation (17), as implied by te variance scaling, tis gives: Tis gives: Ri 1. (4) g θ θ v = Bw, (5) were te constant B carries te proportionality in Equation (4). Accordingly, in equilibrium, corresponds to te dept of te layer in wic: te dry plumes involved in entrainment, and wose energy scales wit w, lose teir kinetic energy; and cumulus updraugts condense and reac positive buoyancy. Clearly Equations (3) and (4) cannot bot be true, and so Equation (19) cannot old. Finally, substituting Equation (1) into Equation (3), and Equations (), (1) and (4) into Equation (7), we see tat te new local scaling c # is indeed teoretically different from te (Moeng and Wyngaard, 1984) scaling c by a factor of one Ricardson number: c # Ri c. (6) 6.. Transient convection Combining Equations (4) and (17) relates te equilibrium area fraction of transporting cumulus updraugts to te interface Ricardson number at mixed-layer top: A = A. (7) Ri Using typical values of E = 1cms 1 and w = 1ms 1 (e.g. Stevens, 6) gives te equilibrium value A = 1, explaining te small convective cloud fractions tat are caracteristic of sallow-cumulus convection. However, wat appens wen te boundary layer is not in equilibrium? Ten E does not necessarily equal M, and A can deviate from Equation (7). In fact, as illustrated in Figure 1(b), in transient conditions A (and wit it /) canges significantly: from te dry convective limit (A ) up to A = 5. Tis suggests tat it is crucial to allow to be flexible in order to allow te system to approac equilibrium: during te equilibration process, and will adjust to values suc tat M becomes equal to E, as expressed by te equilibrium condition (7). It is interesting to interpret te role of in te equilibration process in te context of a recent study by Neggers et al. (6). Tey studied sub-cloud mixed-layer equilibration using a simple bulk model including cumulus mass flux, parametrized statistically as a function of w and te normalized saturation deficit at mixedlayer top. Tey found tat te sensitivity to saturation tus introduced in te mass flux troug A constitutes a negative feedback mecanism, wic can explain te approac towards te equilibrium state of Equation (7). If A can be parametrized as a function of /, as implied by combining variance scaling wit te massflux approac, ten sould play te same equilibrating role, acting as a valve in te cumulus mass flux. Figure 9(b) illustrates tat tis is indeed te case, sowing ow opens and closes at te onset and decay, respectively, of te cumulus-cloud layer. Tis suggests tat modelling A as a function of transition-layer dept provides an alternative way to represent te concept of moist-convective inibition in PBL scemes (e.g. Mapes, ; Breterton et al., 4). It is tempting to tink of Equation (7) as a cumulus equilibrium constraint. However, it sould be noted tat Ri is not an external variable, but is igly dependent on te small internal jump θ v between te cloud and sub-cloud layers. Te eventual equilibrium value of Ri is determined by te differential large-scale forcings in te cloud layer and te sub-cloud layer, as well as by te boundary conditions at te bottom and top of te PBL (te surface and te lower tropospere respectively). Tis topic is explored in more detail by Neggers et al. (6), and also in ongoing work by Breterton and Park (submitted to JAS, 6). Understanding wat controls Ri may tus be important in elping to determine cloud fraction, as well as in applying te c # scalings in models wit coarse vertical resolution.

11 VARIANCE SCALING Concluding remarks A new local scaling c # for te variance at mixed-layer top is introduced, as defined by Equation (7), wit te aim of addressing sortcomings in te standard top-down scaling for te dry CBL in situations were sallow cumulus clouds are present. Te novelty of c # is tat, in addition to te entrainment flux, it incorporates information on local (transition-layer) stability and te convective-mixed-layer turnover time-scale. Tis significantly improves te scaling of variance in transient cumulus situations. Te local vertical gradient canges considerably during te first ours after cloud onset, and affects te local variance troug te flux-gradient-production term. Te collapse of te data, for a range of different LES cases, supports te generality of te scaling. Indeed, te fact tat te scaling also works for dry convective layers not topped by cumulus convection provides yet anoter demonstration of te similarity between dry convective layers and te sub-cloud layer in te presence of sallow, nonprecipitating, convection. Troug te definition of variance as te second moment of a turbulent distribution, te pysics in te variance budget is sown to imply a relation between te mass-flux area fraction and te ratio of transition-layer dept to mixed-layer dept. Te transition-layer dept is sown to represent te negative feedback mecanism between cloud-base mass flux, local stability and proximity to saturation at mixed-layer top (Breterton et al., 4; Neggers et al., 6). Te mixed-layer dept in general increases during diurnal cycles, tus reducing A wit time. Tis inverse relation is consistent wit observations in nature and LES (Brown et al., ; Neggers et al., 4). Te role of te transition layer in sallow-cumulus convection as been studied extensively in te past. Te present study sows ow variance and transition-layer dept enter te problem. Our results suggest new opportunities for parametrization. For example, te variance scaling could be used in combination wit statistical cloud scemes (Neggers et al., 6). Furtermore, parametrizing te transition-layer dept could be an alternative approac to representing te impact of te transition layer: for example, by relating it to te moist instability dept scale of model updraugts rising out of te mixed layer. In a fortcoming study ( A dual mass-flux model for boundary-layer convection, in preparation for te Journal of te Atmosperic Sciences), tese concepts are explored using te PBL sceme of te ECMWF Integrated Forecasting System. Te LES cases used for evaluation in tis study are of different natures, featuring a dry CBL case as well as steady-state marine and transient continental cumulus cases, wit surface Bowen ratios ranging from about 3 to. Neverteless, tere are still different scenarios for wic it is unknown weter te variance scaling applies. For example, it is interesting to study cases wit strong vertical wind sear, as mecanical turbulence migt represent a significant source in te variance budget. Te validity of te variance scaling against observations also remains a topic for future researc. Studying te variance structure in transient sallowcumulus cases suc as ARM and SCMS as te advantage tat we gain an understanding of teir beaviour for more complex scenarios tan steady-state situations. Tis significantly expands te parameter range tat is covered. Te time-dependence of te variance introduces callenges for parametrization. Te closures as presented ere ave been directly inspired by te study of tese diurnal-cycle cases. Tis empasizes te importance of setting up as many different cases as possible, preferably based on reliable observational data. Te work of te GCSS boundary-layer working group to extract cases from observations as directly contributed to te wealt of moist-convective cases tat are now available to te boundary-layer-modelling community. Acknowledgements We tank Cris Breterton and an anonymous reviewer for teir insigtful comments on tis study. Tis researc was supported in part by National Science Foundation grants DMS , ATM-859 and ATM-3465, and National Oceanograpic and Atmosperic Administration grant NA5OAR Te LES results in tis study were obtained using te model of te Royal Neterlands Meteorological Institute (KNMI), using te supercomputer facilities of ECMWF in Reading, UK. Te LES runs were performed by te first autor wen affiliated to KNMI, at tat time being supported by te European Project for Cloud Systems Studies (EUROCS) as well as te Neterlands Organization for Scientific Researc (NWO) under grant A. Appendix: Variance transport Te vertical variance flux is decomposed into contributions from strong updraugts and teir environment, w q q = a up w q q up + (1 a up )w q q en, (A1) were te superscript up represents te average over te updraugts and en represents everyting else. Te updraugt area fraction a up represents a fixed toppercentage of te tail of te PDF of vertical velocity, ere taken to be 5%. Te dased line in Figure 13 sows tat variance transport is dominated by te strongest updraugts, so tat te environmental contribution can be neglected. Te figure also sows te so-called top-at approximation for te updraugt fraction, wic neglects sub-ensemble internal variability. Tis still captures te bulk of te total transport. Tis suggests tat w q q up z wup σ q z w σ q z, (A)

12 164 R. A. J. NEGGERS ET AL. z [m] BOMEX top 5% deceleration top 5% advection top 5% top-at top 5% total -5-5 σ q transport [g kg - s -1 ] 5 Figure 13. Various terms of te decomposition (Equations (7) and (8)) of umidity-variance transport for te BOMEX case, as sampled in LES. Te strongest updraugt fraction is defined as te top 5% of te PDF of vertical velocity. Included are te grid-box-mean variance transport (solid), te contribution from tis top 5% (dased), te associated top-at approximation (dotted), and te decomposition of te latter into an advection part (dased-dotted) and a deceleration part (dased-double-dotted). were te last approximation is justified by te smallness of te deceleration term σ q wup / z: see Figure 13. Accordingly, only te term representing vertical advection of variance by te strong updraugts is retained, teir vertical velocity w up being assumed to scale wit w. Ten, applying te bulk-gradient approac over te top alf of te mixed layer, we obtain: σ σ q w z q, (A3) τ were τ /w is an adjustment time-scale tat can be used to describe te effect of organized transport on te variance at te top of te sub-cloud mixed layer. Here we ave made use of te fact tat variance in te middle of te mixed layer is typically about an order of magnitude smaller tan at te top (see Figures 3 and 4). References Albrect BA, Betts AK, Scubert WH, Cox SK A model of te termodynamic structure of te trade-wind boundary layer. Part I: Teoretical formulation and sensitivity tests. J. Atmos. Sci. 36: Arakawa A. 4. Te cumulus parameterization problem: past, present, and future. J. Climate 17: Augstein E, Riel H, Ostapoff F, Wagner V Mass and energy transports in an undisturbed Atlantic trade-wind flow. Mon. Weater Rev. 11: Augstein E, Scmidt H, Wagner V Te vertical structure of te atmosperic planetary boundary layer in undisturbed trade winds over te Atlantic Ocean. Boundary-Layer Meteorol. 6: Betts AK Parametric interpretation of trade-wind cumulus budget studies. J. Atmos. Sci. 3: Bougeault P Cloud ensemble relations based on te gamma probability distribution for te ig-order models of te planetary boundary layer. J. Atmos. Sci. 39: Breterton CS, McCaa JR, Grenier H. 4. A new parameterization for sallow cumulus convection and its application to marine subtropical cloud-topped boundary layers. Part I: Description and 1D results. Mon. Weater Rev. 13: Brown AR, Clond A, Golaz C, Kairoutdinov M, Lewellen DC, Lock AP, MacVean MK, Moeng C-H, Neggers RAJ, Siebesma AP, Stevens B.. Large-eddy simulation of te diurnal cycle of sallow cumulus convection over land. Q. J. R. Meteorol. Soc. 18: Browning KA Te GEWEX Cloud System Study (GCSS). Bull. Am. Meteorol. Soc. 74: Caugey SJ Observed caracteristics of te atmosperic boundary layer. In: Atmosperic Turbulence and Air Pollution Modelling, pp , Nieuwstadt FTM, van Dop H (eds). D. Reidel. Cuijpers JWM, Duynkerke PG Large-eddy simulation of tradewind cumulus clouds. J. Atmos. Sci. 5: Deardorff JW. 1974a. Tree-dimensional numerical study of te eigt and mean structure of a eated planetary boundary layer. Boundary- Layer Meteorol. 7: Deardorff JW. 1974b. Tree-dimensional numerical study of turbulence in an entraining mixed layer. Boundary-Layer Meteorol. 7: Frenc JR, Vali G, Kelly RD Evolution of small cumulus clouds in Florida: observations of pulsating growt. Atmos. Res. 5: Grant ALM. 1. Cloud-base fluxes in te cumulus-capped boundary layer. Q. J. R. Meteorol. Soc. 17: Grant ALM, Brown AR A similarity ypotesis for sallowcumulus transports. Q. J. R. Meteorol. Soc. 15: Grant ALM, Lock AP. 4. Te turbulent kinetic energy budget for sallow cumulus convection. Q. J. R. Meteorol. Soc. 13: Holland JZ, Rasmusson EM Measurement of atmosperic mass, energy and momentum budgets over a 5-kilometer square of tropical ocean. Mon. Weater Rev. 11: Josep JH, Caalan RF Nearest neigbor spacing of fair weater cumulus clouds. J. Appl. Meteorol. 9: Kaimal JC, Wyngaard JC, Haugen DA, Coté OR, Izumi Y Turbulence structure in te convective boundary layer. J. Atmos. Sci. 33: Knigt CA, Miller LJ Early radar ecoes from small, warm cumulus: Bragg and ydrometeor scattering. J. Atmos. Sci. 55: Laird NF, Ocs HT, Rauber TM, Miller LJ.. Initial precipitation formation in warm Florida cumulus. J. Atmos. Sci. 57: Lenderink G, Siebesma AP.. Combining te massflux approac wit a statistical cloud sceme. In: Proceedings of te 14t AMS conference on Boundary Layers and Turbulence, Aspen, CO, July. Lenscow DH, Wyngaard JC, Pennell WT Mean-field and second-moment budgets in a baroclinic, convective boundary layer. J. Atmos. Sci. 37: Mapes BE.. Convective inibition, subgrid-scale triggering energy, and stratiform instability in a toy tropical wave model. J. Atmos. Sci. 57: Moeng C-H, Wyngaard JC Statistics of conservative scalars in te convective boundary layer. J. Atmos. Sci. 41: Neggers RAJ, Duynkerke PG, Rodts SMA. 3. Sallow cumulus convection: A validation of large-eddy simulation against aircraft and Landsat observations. Q. J. R. Meteorol. Soc. 19: Neggers RAJ, Siebesma AP, Lenderink G, Holtslag AAM. 4. An evaluation of mass flux closures for diurnal cycles of sallow cumulus. Mon. Weater Rev. 13: Neggers RAJ, Stevens B, Neelin JD. 6. A simple equilibrium model for sallow cumulus topped mixed layers. Teor. Comp. Fluid Dyn. : Nicolls S, LeMone MA Te fair weater boundary layer in GATE: te relationsip of subcloud fluxes and structure to te distribution and enancement of cumulus clouds. J. Atmos. Sci. 37: Nieuwstadt FTM, Brost RA Te decay of convective turbulence. J. Atmos. Sci. 43: Nieuwstadt FTM, Mason PJ, Moeng C-H, Scumann U Largeeddy simulation of te convective boundary layer: A comparison of four computer codes. In: Turbulent Sear Flows 8, Durst F, Friedric R, Launder BE, Scmidt FW, Scumann U, Witelaw JH (eds). Springer: Berlin.

13 VARIANCE SCALING 1641 Nitta T, Esbensen S Heat and moisture budget analyses using BOMEX data. Mon. Weater Rev. 1: Ooyama K A teory on parameterization of cumulus convection. J. Meteorol. Soc. Jpn 49: Siebesma AP, Cuijpers JWM Evaluation of parametric assumptions for sallow cumulus convection. J. Atmos. Sci. 5: Siebesma AP, Breterton CS, Brown A, Clond A, Cuxart J, Duynkerke PG, Jiang H, Kairoutdinov M, Lewellen D, Moeng C-H, Sancez E, Stevens B, Stevens DE. 3. A large eddy simulation intercomparison study of sallow cumulus convection. J. Atmos. Sci. 6: Sommeria G, Deardorff JW Subgrid-scale condensation in models of non-precipitating clouds. J. Atmos. Sci. 34: Stevens B. 6. Boundary layer concepts for simplified models of tropical dynamics. Teor. Comp. Fluid Dyn. : Stevens B, Ackerman AS, Albrect BA, Brown AR, Clond A, Cuxart J, Duynkerke PG, Lewellen DC, MacVean MK, Neggers RAJ, Sancez E, Siebsema AP, Stevens DE. 1. Simulations of tradewind cumuli under a strong inversion. J. Atmos. Sci. 58: Stokes GM, Scwartz SE Te Atmosperic Radiation Measurement (ARM) program: programmatic background and design of te cloud and radiation test bed. Bull. Am. Meteorol. Soc. 75: Stull RB An Introduction to Boundary Layer Meteorology. Kluwer. 666 pp. Tompkins A.. A prognostic parameterization for te subgrid-scale variability of water vapor and clouds in large-scale models and its use to diagnose cloud cover. J. Atmos. Sci. 59: Wyngaard JC, Brost RA Top-down and bottom-up diffusion of a scalar in te convective boundary layer. J. Atmos. Sci. 41: Wyngaard JC, Cote OR, Izumi Y Local free convection, similarity, and te budgets of sear stress and eat flux. J. Atmos. Sci. 8: Yanai M, Esbensen S, Cu J-H Determination of bulk properties of tropical cloud clusters from large-scale eat and moisture budgets. J. Atmos. Sci. 3: Yin B, Albrect BA.. Spatial variability of atmosperic boundary layer structure over te eastern equatorial Pacific. J. Climate 13: