Turbulence modelling in large-eddy simulations of the cloud-topped atmospheric boundary layer

Transcription

1 Center for Turbuence Research Annua Research Briefs Turbuence modeing in arge-eddy simuations of the coud-topped atmospheric boundary ayer By M. P. Kirkpatrick 1. Motivation and obectives This paper discusses turbuence modeing in arge-eddy simuations of the coudtopped atmospheric boundary ayer. Whie our primary focus is on simuations of stratocumuus couds, most of the discussion is aso reevant to other types of coud. Stratocumuus couds were chosen because of the important roe they pay in the Earth s cimate, and because the fuid dynamics associated with these couds has a number of features which researchers have found difficut to mode accuratey. Marine stratocumuus couds cover extensive areas off the west coasts of the arge continents in the subtropics. Their presence in these regions is the resut of strong static stabiity due to ow sea-surface temperatures and to atmospheric subsidence associated with the descending branch of the Hadey circuation. Due to their high abedo, stratocumuus couds have a significant effect on the Earth s radiative heat budget. From anaysis of sateite data, Kein & Hartmann 1993) cacuated top-of-the-atmosphere vaues of the order of 100 Wm 2 for the net coud radiative forcing over stratocumuus decks. Randa et a. 1984) estimated that the goba cooing resuting from a 4% increase in area coverage by marine stratocumuus couds woud offset the expected warming from a doubing of atmospheric carbon dioxide. In addition to their roe in the Earth s radiation budget, stratocumuus couds aso affect the dynamics of the atmosphere and oceans. Mier 1997), for exampe, found that stratocumuus couds provide a negative-feedback mechanism which reduces the intensity of tropica convection and damps the tropica atmospheric circuation. Simiary, stratocumuus couds over oceans in the subtropics reduce sea-surface temperatures in these regions by owering the net surface heat fux. Most atmospheric genera circuation modes GCMs) underpredict the amount of subtropica marine stratocumuus Jakob 1999). In couped atmosphere-ocean modes, this can ead to positive sea-surface temperature biases of up to 5K. Such modeing errors have been shown to have a significant infuence on both the predicted circuation Nigam 1997) and the goba radiation budget Singo 1990). The use of arge-eddy simuation LES) to study the panetary boundary ayer dates back to the eary 1970s, when Deardorff 1972) used a three-dimensiona simuation to determine veocity and temperature scaes in the convective boundary ayer. In 1974 he appied LES to the probem of mixing-ayer entrainment Deardorff 1974) and in 1980 to coud-topped boundary ayers Deardorff 1980). Since that time the LES approach has been appied to panetary boundary ayer probems by numerous authors see for exampe Moeng 1986; Mason & Derbyshire 1990; Schumann & Moeng 1991a,b; Brown et a. 1994; Saiki et a. 2000; Stevens & Bretherton 1999; Stevens et a. 2001) The popuarity of the LES technique in atmospheric research is due in part to the difficuties invoved with obtaining sufficient fied data to deveop and test theories concerning the structure and dynamics of the panetary boundary ayer. Large-eddy simuations provide three-dimensiona time-evoving veocity and scaar fieds at a resoution imited

2 240 M. P. Kirkpatrick ony by computationa resources. As such, LES is often used to isoate particuar physica processes of interest such as entrainment across the inversion Stevens et a. 2000) or transition from one type of coud to another Wyant et a. 1997). It is aso used to generate databases of different atmospheric fow regimes in order to evauate, refine and deveop parameterisation schemes for use in arge-scae modes eg. Lappen & Randa 2001). At the other end of the spectrum, LES is used as a patform on which to deveop reiabe modes of coud microphysics and radiation Ackerman et a. 2000). In spite of an increasing reiance on LES as a too for deveoping and testing coud theories and modes, there is sti considerabe uncertainty concerning the fideity of the simuations themseves. Whie LES has been shown to be reativey robust for simpe cases such as simuations of a cear, convective boundary ayer Mason 1989), mode intercomparisons for more compex cases have shown arge variations in predictions of important statistics and buk parameters. For exampe, in the 1995 Goba Energy and Water Cyce Experiment GEWEX) Coud System Studies GCSS) mode intercomparison, Bretherton et a. 1999) compared simuations of a smoke coud beneath a temperature inversion. Radiative cooing at the top of the coud drives convection, which eads to entrainment across the inversion and to growth of the boundary ayer. The authors found that the entrainment rates and other statistics predicted by the various LES codes differed by up to a factor of two. Simiar entrainment processes occur at the top of stratocumuus couds, athough with the added compexity of atent-heat transfer due to condensation and evaporation of coud dropets. A second exampe is the recent intercomparison of simuations of trade-wind cumui by Stevens et a. 2001). Here again, important parameters such as stratiform coud fraction and the variance of tota-water mixing ratio were found to be highy sensitive to the choice of numerica method, spatia resoution and subgrid-scae turbuence mode. Buk parameters such as boundary-ayer height, entrainment rate and coud fraction are important variabes in the parameterisations used in goba circuation modes. It is therefore essentia that LES be made robust in its prediction of these variabes if it is to be used as a too for deveopment and tuning of parameterisations for arge scae modes. One of the main probem areas in arge-eddy simuations of couds is the accurate representation of processes occurring cose to an inversion. Here, strong stabe stratification reduces the size of the energetic eddies consideraby, so that they are generay poory resoved by simuations. Bretherton et a. 1999), for exampe, identify an unduation ength scae given by z u = z i /Ri where z i is the height of the inversion and Ri the Richardson number) which is of the order 5 10 m in a strong inversion. Meanwhie, the grid-ce dimensions used for arge-eddy simuations of the panetary boundary ayer are typicay m, athough, with advances in computer technoogy, highy-resoved simuations are now becoming possibe. Stevens et a. 2000), for exampe, recenty performed stratocumuus simuations at grid sizes down to 8 m in the horizonta directions by 4 m in the vertica direction. Even at this resoution, however, they found the predicted entrainment rate and entrainment efficiency to be sensitive to the subgrid mode and numerics. A second reason for the difficuties encountered in modeing processes cose to an inversion is that the stabe stratification tends to damp vertica motions, making the turbuence in this region much more anisotropic than in an unstratified environment. Consequenty turbuence modes often use one or more corrections to account for the effects of stratification. In couds, additiona buoyancy sources resut from energy transfer due to condensation and evaporation of water, and some authors eg. MacVean & Mason 1990)) recommend appying further corrections to account for these processes.

3 Turbuence modeing in arge-eddy simuations of couds 241 An aternative to using such corrections is to adopt a dynamic approach, in which the parameters in the subgrid-scae turbuence mode are computed at each point in space and time using information contained in the resoved veocity and scaar fieds. This approach removes the need to make modeing decisions concerning the coefficients and ength scaes in the subgrid mode. It aso removes the need for corrections to account for buoyancy effects since a this information is obtained directy from the resoved fow fied. The dynamic approach, first proposed by Germano et a. 1991), has been used with considerabe success for compex engineering fows see Boivin et a. 2000; Braney & Jones 2001, for exampe), however its appication to atmospheric fows has been imited. This is due in part to arguments by authors such as Mason & Brown 1999) to the effect that the dynamic procedure is inappropriate for atmospheric appications. These arguments are based on the premise that the dynamic procedure requires a fiter cut-off wavenumber in the inertia subrange. This is incorrect. The theory behind the dynamic procedure assumes ony that the same subgrid mode can be used for both the resoved fied and the test-fitered fied. It is in fact the Smagorinsky mode Smagorinsky 1963), which is widey used for atmospheric simuations even at very coarse resoutions, whose derivation assumes resoution of the inertia subrange. Bohnert 1993) tested the dynamic procedure in combination with the Smagorinsky mode for simuations of cear and coud-topped panetary boundary ayers. The simuations were performed at a Reynods number ower than that of a reaistic atmospheric boundary ayer and used simpe parameterisations for coud physics and radiation. In order to stabiise the mode, it was necessary to average the cacuated coefficient fied over horizonta panes. Nevertheess, these resuts are encouraging. The dynamic mode gave resuts comparabe to, or better, than those obtained using the standard constantcoefficient Smagorinsky mode with a Richardson-number correction. The obective of the present study is to test the dynamic procedure in arge eddy simuations of a marine stratocumuus coud deck. The simuations wi be performed at reaistic Reynods and Rayeigh numbers, with conditions matching those measured during the 2001 DYCOMS-II fied experiment. This test case has a number of the features discussed above which typicay cause difficuties in coud simuations, namey strong stabe stratification, and buoyancy sources within the coud due to radiation, condensation and evaporation of water dropets. Foowing Zang et a. 1993), we use the mixed mode as a base subgrid mode, rather than the Smagorinsky mode. The mixed mode is a combination of the scae-simiarity mode of Bardina et a. 1980) and the Smagorinsky mode. The dynamic mixed mode of Zang et a. has been found to be more stabe than the dynamic Smagorinsky mode and it is hoped that its use wi remove the need for averaging over horizonta panes. This is important in the present case, since horizonta panes cose to the coud-top contain both staby and unstaby stratified regions. Horizonta averaging of the cacuated mode parameters woud prevent the dynamic procedure from distinguishing between these two fundamentay different fow regimes. The author is currenty impementing the dynamic mixed mode in the LES code, DHARMA, written by David Stevens. This code has performed we in mode intercomparisons see Bretherton et a. 1999; Stevens et a. 2001) and has aso been used for a number of high-resoution simuations Stevens & Bretherton 1999; Stevens et a. 2000, 2002) where it was shown to scae we on massivey parae architectures. In addition, the code has the option to use either standard parameterisations for radiation and coud

4 242 M. P. Kirkpatrick microphysics, or the more compex modes of Ackerman et a. 1995), which treat the coud microphysics expicity and incude a detaied treatment of radiative transfer. In the foowing, we describe the governing equations, the numerica methods, and the parameterisations and modes used in the DHARMA code. We rewrite the governing equations in fitered form and outine a turbuence cosure based on the dynamic mixed mode of Zang et a. Finay we discuss the test case and simuations which wi be used to assess the performance of this approach to turbuence modeing in numerica simuations of couds. 2. Governing equations The basic equations governing the dynamics of the coud-topped atmospheric boundary ayer comprise equations for conservation of mass, momentum, energy and tota water. In addition, radiative heat transfer and coud microphysics must aso be modeed. Coud microphysics refers to the transitions between vapor, iquid and soid-phase water and the dynamics of the iquid and soid-phase components. The governing equations are written in the aneastic form of Ogura & Phiips 1962) in which the thermodynamic variabes such as pressure p are decomposed into an isentropic base state p 0 corresponding to a uniform potentia temperature θ 0 ) and a dynamic component. Foowing Cark 1979), the dynamic component is further decomposed into an initia environmenta deviation in hydrostatic baance p 1 and a time-evoving dynamic perturbation p 2 to give px, y, z, t) = p 0 z) + p 1 z) + p 2 x, y, z, t). 2.1) The resuting continuous equations written in Cartesian tensor notation are t + 1 ϱ 0 u ) = Π θ v2 + g i ϱ 0 x x i θ 0 +H sub + H gw θ t + 1 ϱ 0 θ u ) = Hθ sub + H ϱ 0 x θ LS + H gw θ + H coriois, 2.2) + Hθ rad, 2.3) q t t + 1 ϱ 0 q t u ) = Hq sub ϱ 0 x t + Hq LS t, 2.4) ϱ 0 u ) = 0. x 2.5) Here is the veocity component in the i direction, ϱ is the density, Π is the perturbation pressure p 2 /ϱ 0, g i is the acceeration due to gravity, q t the tota water mixing ratio and θ = θ θ 0 )/θ 0 is a scaed iquid-water potentia temperature. Tota-water mixing ratio is the sum of the iquid and vapour mixing ratios, q t = q c + q v = ϱ c + ϱ v ϱ d, 2.6) where ϱ c, ϱ v and ϱ d are the density of the condensed water, the water vapour and the dry air respectivey. Liquid-water potentia temperature is defined as θ = θ Lq c C pd π ) Here L is the atent heat of vaporisation, C pd is the specific heat at constant pressure

5 Turbuence modeing in arge-eddy simuations of couds 243 ) Rd for dry air and π 0 = p0 Cp p ref with p ref a reference pressure and R d the gas constant of dry air. The virtua potentia temperature θ v appearing in the buoyancy term of the momentum equations is given by [ ) ] Rd θ v = θ + θ 0 1 q v q c, 2.8) R v where R d and R v are the gas constants of dry air and water vapor respectivey. The interior forcings H are body forces which parameterise the effects of: subsidence H sub ; horizonta arge scae advective tendencies H LS ; and the Coriois force H coriois. In addition, a Rayeigh damping term H gw is appied to the top third of the domain to absorb gravity wave energy. The subsidence and arge-scae advective tendencies resut from the fact that the LES domain is not isoated, but is embedded within the goba circuation. These forcings are generay specified as functions of z/z i where z is the vertica height, and z i is the height of the inversion. As an exampe, in the case of the trade-wind cumuus intercomparison of Stevens et a. 2001), the subsidence veocity w sub was specified to vary ineary between 0 at the surface and 6.5 mm s 1 at z i. The subsidence forcings then become, H sub H sub θ H sub q t = w sub z, 2.9) = w sub θ z, 2.10) = w sub q t z. 2.11) The arge-scae advective tendencies were specified as [ ] dθ = z ) Ks 1, dt LS z i [ ] dqt = z ) s 1, 2.12) dt z i LS for z < z i. Above the inversion, the terms were ineary reduced to zero over a distance of 300 m. The Coriois term is given by H coriois = [fv, fu, 0], 2.13) with the Coriois parameter f = 2ω sin φ, where ω is the anguar veocity of the Earth and φ is the atitude. 3. Fitered equations In LES, a spatia fiter is appied to the governing equations. The appication of a spatia fiter G to a function f is defined as fx) = Gx x ; x))fx )dx, 3.1) Ω where is the characteristic width of the fiter. A box fiter is used here as it fits naturay into the finite voume discretisation. The fiter width is written in terms of the ce dimensions, = 2 x y z ) 1/3.

6 244 M. P. Kirkpatrick Fitering the equations for conservation of momentum and mass yieds t + 1 ϱ 0 u ) = Π θ v2 + g i τ i ϱ 0 x x i θ 0 x +H sub + H gw + H coriois, 3.2) ϱ 0 u ) x = ) Here it is assumed that the isentropic fieds ϱ 0 and θ 0, and the forcings H, vary sowy in space, so that extra moments resuting from the appication of the fiter to these terms may be negected. The extra term in the momentum equations is the subgrid-scae stress or SGS tensor, τ i = u u ), 3.4) which represents transport of momentum by subgrid-scae turbuence. This term must be modeed to cose the equations. The Smagorinsky eddy-viscosity mode Smagorinsky 1963) assumes that the anisotropic part of the SGS stress tensor is proportiona to the arge scae strain rate tensor, τ i 1 3 δ iτ kk = 2ν T S i, 3.5) where S i = 1 ui + u ), 3.6) 2 x x i and that the eddy viscosity ν T is itsef a function of strain rate and fiter size, ν T = C 2 S. 3.7) Here S = 2S i S i and C is the dimensioness mode coefficient. In the basic mode, C is specified a priori and is often written as the Smagorinsky coefficient C s = C. For incompressibe fows, the isotropic part of the SGS stress tensor, τ kk, is absorbed into the pressure term. For atmospheric simuations this basic mode must be modified to account for the effects of stratification. This typicay takes the form of a correction to the eddy viscosity to give ν T = C 2 S 1 Ri/P r T, 3.8) where Ri = N 2 / S 2 is a gradient Richardson number and P r T is a turbuent Prandt number. The buoyancy frequency N for dry air is defined as N 2 = g θ θ z. 3.9) In the DHARMA code, this formua is modified, foowing MacVean & Mason 1990), to incude the effects of evaporation and condensation. As discussed in the previous section, whie these corrections have been shown to give good resuts for the reativey simpe fows for which they were derived, in more compex fows the resuts are often highy sensitive to the choice of mode coefficients and ength scaes. Apart from the need to set the mode coefficient and ength scae a priori, the Smagorinsky mode has a number of other probems. a) The mode assumes that the principa axes of the SGS stress tensor are aigned with

7 Turbuence modeing in arge-eddy simuations of couds 245 the resoved strain-rate tensor whereas anaysis of DNS resuts has shown this not to be the case. b) The mode does not predict the correct asymptotic behaviour near a soid boundary or in aminar/turbuent transitions. c) The mode does not aow SGS energy backscatter to the resoved scaes. To overcome item a) in this ist, Bardina et a. 1980) proposed a mode based on an assumption of simiarity between the unresoved scaes and the smaest resoved scaes. In their scae simiarity mode the subgrid-scae stress is given by τ a i = u u ) a, 3.10) where superscript a specifies the anisotropic part of the tensors. Comparisons with DNS resuts show that the scae simiarity mode, which does not require aignment between the SGS stress tensor and the resoved strain-rate tensor, represents the structure of the SGS stress more accuratey than does the Smagorinsky mode. The mode does not, however, dissipate sufficient energy and is usuay combined with the Smagorinsky mode to form a mixed mode. Items b) and c) were addressed by Germano et a. 1991) who proposed a dynamic procedure that cacuates the mode coefficient dynamicay at each point in space and time based on oca instantaneous fow conditions. Whie the procedure can be used with any subgrid mode, Germano et a. demonstrated the approach with the Smagorinsky mode. The resuting dynamic Smagorinsky mode has the correct asymptotic behaviour near soid boundaries and in aminar fow, and aows energy backscatter. Unfortunatey, vaues of the predicted mode coefficient tend to fuctuate consideraby and some form of averaging, usuay aong homogeneous directions, is required to avoid numerica instabiity. In the present context, such averaging is probematic since there is no homogeneous direction. The stratocumuus coud-top contains regions of both stabe and unstabe stratification within the same horizonta pane. A number of variants of the dynamic procedure have been proposed to overcome the need for averaging. The ocaised dynamic modes of Ghosa et a. 1992) and Piomei & Liu 1995) are more stabe but add to the compexity of the mode. Instead, we have chosen to adopt the approach of Zang et a. 1993) who used the dynamic procedure with the mixed mode as a base mode, rather than with the Smagorinsky mode. Zang et a. tested the dynamic mixed mode for rotating stratified fow and reported a significant reduction in fuctuations of the coefficient compared with the dynamic Smagorinsky mode. The dynamic mixed mode has the added advantage that the scae simiarity term removes the restriction of tensor aignment and provides better spectra representation of the subgrid-scae stress. The mixed mode for the subgrid-scae turbuent stress is written τ a i = 2C 2 S Si + u u ) a 2ν T S i + u u ) a, 3.11) where the first term on the right-hand side is the Smagorinsky component of the mode whie the second term represents the scae simiarity component. The dynamic procedure invoves the appication of a test fiter ) to the veocity fied. By assuming that the same subgrid mode can be used to represent the unresoved stresses for both the grid-fitered

8 246 M. P. Kirkpatrick and test-fitered fieds, an expression is derived for the required parameters, where C 2 = L i H i )M i 2M i M i, 3.12) L i = û i u û i û, 3.13) and H i = ûiu ûiû, 3.14) M i = α 2 Ŝ Ŝi S S i, α = /. 3.15) Using these reations, the momentum equations become t + 1 ϱ 0 u ) = Π θ v2 + g i + 2νT S i ) a ) u ϱ 0 x x i θ 0 x u + H sub + H gw + H coriois. 3.16) Foowing a simiar argument the spatia fiter is appied to the energy equation giving θ The SGS energy fux, t + 1 ϱ 0 θ u ) ϱ 0 x = γ + H sub θ + H LS x θ + H gw θ γ = + H rad θ. 3.17) ) u θ u θ, 3.18) is approximated using a mixed mode anaogous to that used for the momentum equations, γ = ν T θ + u θ P r θt x u θ ), 3.19) with the eddy diffusivity computed using the eddy viscosity cacuated for the veocities and a turbuent Prandt number. Substituting into 3.17), the fitered energy equation becomes θ t + 1 ϱ 0 θ u ) ν T θ = u θ ϱ 0 x x P r θt x u θ ) ) + H sub θ + H LS θ + H gw θ + H rad θ. 3.20) Finay, by anaogy, the fitered transport equation for tota water is written q t t + 1 ϱ 0 q t u ) νt q = t ) ) u q ϱ 0 x x P r qt x t u q t + H sub q t + H LS q t. 3.21) The turbuent Prandt numbers, P r θt and P r qt, in the subgrid modes for the scaar variabes are determined dynamicay using the approach of Moin et a. 1991). This procedure is simiar to that used to cacuate the Smagorinsky coefficient outined above. In this way, a coefficients and ength scaes in the subgrid-scae modes for the fow variabes are cacuated dynamicay, based on information in the resoved scaar and fow fieds, and the need for a priori specification of parameters and corrections is removed.

9 Turbuence modeing in arge-eddy simuations of couds Microphysics and radiation modes The DHARMA code has the option to use either a standard parameterisation for coud microphysics, or the more compex expicit mode of Ackerman et a. 1995). The code incudes two standard parameterisations: a buk-condensation mode, in which coud water q c is found by inverting Wexer s expressions for saturated vapor pressure Wexer 1976, 1977) using the method of Fatau et a. 1992); and the parameterised microphysics of Wyant et a. 1997) which incudes a treatment of precipitation. The coud microphysics mode of Ackerman et a. expicity modes the dynamics of two types of partice: condensation nucei CN) and water dropets. Partice size distributions are defined by Cr, x, t) where Cdr is the mean number concentration per unit voume) of partices with radius between r and r + dr. A fitered partice-continuity equation is soved for each partice size, C t + Cu ) = v f C) + S n R n C + g rc) x z r C r + H sub C r min K c r, r 3 r 3 ) 1/3 )Cr )Cr 3 r 3 ) 1/3 )dr rmax r min K c r, r )Cr )dr + H LS C + x ϱ0 ν T C/ϱ 0 ) u C u C) ). 4.1) P r CT x Here v f is the partice sedimentation veocity, S n represents partice creation, R n is the partice remova rate, g r is the condensationa growth rate and K c is a coaescence kerne. The first term on the right hand side is the partice fux divergence due to sedimentation. This fux is modeed using the Stokes-Cunningham expression for Re < 10 2 and the interpoation of Beard 1976) for higher Re. The second and third terms on the right hand side represent partice creation and transitions between CN and dropets. The fourth term is the divergence in radius-space due to condensation and evaporation. The first integra represents creation of partices due to coisions of smaer partices whie the second integra represents the oss of partices due to coisions with other partices. H sub C and are the subsidence and arge-scae advective tendencies simiar to those appearing H LS C in the gas phase equations. The fina term is a turbuent diffusion fux representing the subgrid-scae stresses resuting from the fitering operation. The turbuent Prandt number P r CT is set equa to that cacuated dynamicay for q t. Ackerman et a. 1995) give further detais on modeing of condensation growth, CN activation, tota evaporation of dropets, partice coisions and new partice creation. Partice size distributions for CN and dropets are each typicay divided into 20 bins with geometricay increasing size such that the partice voume doubes between successive bins. Because time scaes for the coud microphysics are typicay smaer than those for the arge-scae dynamics, the microphysics equations are integrated over a series of smaer substeps within each time step of the fow-dynamics mode. Aso, whie the fow-dynamics mode uses tota-water mixing ratio q t and iquid-water potentia temperature θ as the thermodynamic variabes see 2.3) and 2.4)), the microphysica mode uses the concentration of water vapour G and the potentia temperature θ. The equations for

10 248 M. P. Kirkpatrick these variabes are written ϱ 0 θ) t G t + Gu ) = 4πϱ w x + x rmax r 2 g r r )Cr )dr r min ϱ0 ν T G/ϱ 0 ) P r qt x ) ) u G u G + H sub G + H LS G, 4.2) + ϱ 0θu ) ϱ0 ν T θ ) ) = u θ u θ + H sub θ + H LS θ. 4.3) x x P r θt x Here ϱ w is the density of iquid water and the integra in 4.2) represents vapour exchange with the dropets. The turbuent Prandt numbers are those cacuated for the corresponding variabes in the fow-dynamics mode. The fuxes of partices and water vapour across the ower boundary are cacuated using Monin-Obukhov simiarity functions. Here, the mode integrates the surface-ayer fux-profie reations of Businger et a. 1971) foowing the method of Benoit 1977). Latera boundaries are periodic and at the upper boundary the fux divergence is set to zero. Radiation is modeed in different ways depending on the requirements of the particuar study at hand. A simpe approach, often used for mode intercomparisons, is to parameterise radiation as the sum of two components: a cear-sky radiative-cooing component, typicay taken to be a fixed -2 K/day everywhere beow the inversion; and a coudassociated Beer s Law component. In the atter, ong-waveength radiative cooing is assumed to be proportiona to the iquid-water content and is exponentiay attenuated as the overying iquid-water path increases. The resuting radiative heat fux F r is then given by ) H F r z) = F r H)exp K a ϱ 0 q c dz, 4.4) where H is the height of the domain. A more compex approach modes radiative heat transfer foowing the method of Toon et a. 1989). The mode computes mutipe scattering over 26 soar waveengths 0.26 µm < λ < 4.3µm) and absorption and scattering over 14 infrared waveengths 4.4 µm < λ < 62µm). Backbody energy beyond those waveength domains is incuded to agree with the Stefan-Botzmann aw. An exponentia-sum formuation is used to treat gaseous absorption coefficients whie the optica properties of partices are determined through Mie cacuations in which the compex refractive index for iquid water is used as interpoated from the datasets of Painter et a. 1969), Pamer & Wiiams 1974) and Downing & Wiiams 1975). The mode uses a vaue for carbon dioxide concentration appropriate to the year of the study. Measurements of the present goba annua mean carbon dioxide concentration give a vaue of approximatey 370 ppm by voume 10% higher than in the eary 1980 s). The ozone profie is taken from the U.S. Standard Atmosphere NOAA 1976). z

11 5. Numerica method Turbuence modeing in arge-eddy simuations of couds 249 The numerica method is described in detai by Stevens & Bretherton 1996). The equations are integrated using a forward-in-time proection method based on a 2nd-order Runge-Kutta scheme simiar to that of Be & Marcus 1992). The integration proceeds as foows: advance veocities to t = n + 1/2 using expicit Euer sove a Poisson equation and do pressure correction at t = n + 1/2 advance scaars to t = n + 1 advance veocity to t = n + 1 using a modified trapezoid rue sove Poisson equation and do pressure correction at t = n + 1 The proection procedure is described in detai by Amgren et a. 1998). The spatia discretisation is performed on a staggered grid Arakawa C). Second-order centra differences are used for diffusion terms and pressure gradients whie the advection terms use a modified version of the Uniform Third-Order Poynomia Interpoation Agorithm UTOPIA) of Leonard et a. 1993). The modified scheme deveoped by Stevens & Bretherton 1996) incudes additiona transverse correction terms which improve the stabiity of the scheme whie maintaining its accuracy. For the scaar equations, the 3D fux imiter of Zaesak 1979) is used to ensure a monotonic soution. Source terms are computed using the second-order accurate method of Smoarkiewicz & Pudykiewicz 1992) and Smoarkiewicz & Margoin 1993). Stevens and Bretherton show that the overa scheme is second-order accurate in space and time, energy-conserving and stabe up to a CFL number of 1.0. At the ower boundary, surface fuxes of momentum, θ and q t are cacuated using the same simiarity reations as those used for water vapour and partice fuxes see Section 4). The atera boundaries are periodic whie the top boundary uses a rigid id. As discussed above, numerica probems due to gravity waves refecting from the top boundary are prevented by using a Rayeigh damping ayer in the upper third of the domain. 6. Future pans The performance of the dynamic mixed mode wi be tested using a series of simuations of a nocturna marine stratocumuus coud deck. The particuar test case chosen is the DYCOMS-II fied experiment which took pace off the coast of San Diego in Juy, DYCOMS-II is an acronym for Dynamics and Chemistry of Marine Stratocumuus Phase II: Entrainment. The purpose of the experiment was to coect data for use in testing arge-eddy simuations of nocturna stratocumuus. In particuar, the experiment focused on coud-top processes invoved in entrainment. The domain to be used for the simuations has size 3.2 km 3.2 km 1.5 km and is periodic in the horizonta directions. Two sets of simuations wi be performed: one set using the dynamic mixed mode and the other using the cassica Smagorinsky mode with the standard corrections. Each set comprises a series of simuations at grid resoutions ranging from coarse resoution 32 m horizonta by 16 m vertica grid size) to very fine resoution 4 m horizonta by 2 m vertica). Previous studies eg. Stevens et a. 2000) indicate that a energetic scaes wi be resoved in the very fine resoution simuations, so that these simuations can reasonaby be used as a benchmark for comparing the performance of the subgrid-scae turbuence modes. Comparisons wi be made with a view to answering the foowing questions: 1) Does the dynamic mixed mode more accuratey represent the subgrid-scae turbuence,

12 250 M. P. Kirkpatrick in the sense that it reduces the difference between the coarser grid soutions and the benchmark soution? 2) How do the subgrid-scae stresses predicted by the dynamic mode compare with those of the standard mode? Does the dynamic mode, for exampe, predict a Richardsonnumber dependence in regions of stabe stratification simiar to that used in the standard mode? 3) How does the dynamic mode perform for simuations in which the inertia subrange is not we resoved? For coarser simuations in which the inertia subrange is not resoved, Stevens et a. 1999) found that the turbuent kinetic energy equation approach gives resuts which are ess dependent on grid resoution than those obtained using a Smagorinsky mode. A possibe future study woud test the performance of a turbuence cosure scheme in which the dynamic procedure is used to determine the coefficients in the turbuent kinetic energy equation. REFERENCES Ackerman, A. S., Hobbs, P. V. & Toon, O. B A mode for partice microphysics, turbuent mixing, and radiative transfer in the stratocumuus-topped marine boundary ayer and comparisons with measurements. J. Atmos. Sci. 52, Ackerman, A. S., Toon, O. B., Stevens, D. E., Heymsfied, A. J., Ramanathan, V. & Weton, E. J Reduction of tropica coudiness by soot. Science ), Amgren, A. S., Be, J. B., Coea, P., Howe, L. H. & Wecome, M. L A conservative adaptive proection method for the variabe density incompressibe Navier-Stokes equations. J. Comp. Phys. 142, Bardina, J., Ferziger, J. H. & Reynods, W. C Improved subgrid scae modes for arge eddy simuation. AIAA Paper Beard, K. V Termina veocity and shape of coud and precipitation drops aoft. J. Atmos. Sci. 33, Be, J. B. & Marcus, D. L A 2nd-order proection method for variabe-density fows. J. Comp. Phys. 101, Benoit, R Integra of surface-ayer profie-gradient functions. J. App. Met. 16, Bohnert, M A numerica investigation of coud-topped panetary boundary ayers. PhD thesis, Stanford University. Boivin, M., Simonin, O. & Squires, K. D On the prediction of gas-soid fows with two-way couping using arge eddy simuation. Phys. Fuids 12, Braney, N. & Jones, W. P Large eddy simuation of a turbuent non-premixed fame. Submitted to Combustion and Fame. Bretherton, C. S., Macvean, M. K., Bechtod, P., Chond, A., Cotton, W. R., Cuxart, J., Cuipers, H., Khairoutdinov, M., Kosovic, B., Leween, D., Moeng, C. H., Siebesma, P., Stevens, B., Stevens, D. E., Sykes, I. & Wyant, M. C An intercomparison of radiativey driven entrainment and turbuence in a smoke coud ; as simuated by different numerica modes. Quart. J. Roya Meteor. Soc /pt.B), Brown, A. R., Derbyshire, S. H. & Mason, P. J Large-eddy simuation of

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