Parametrization of non-convective condensation processes May 1987

Transcription

1 Parametrization of non-convective conensation processes May 987 By M. Tietke European Centre for Meium-Range Weather Forecasts Table of contents. Thermoynamics of moist air 2. Clou physical processes 2. Conensation of water vapour 2.2 Release of precipitation 3. Parametrization of non-convective conensation an precipitation processes in numerical moels 3. Scheme without clou stage 3.2 A clou parameterization scheme REFERENCES. THERMODYNAMICS OF MOIST AIR A brief review of the general thermoynamics of moist air is given here. Air can be consiere as a mixture of ieal gases. The equation of state is where is pressure, is ensity, is absolute temperature an is the gas constant ρ ρ () (2) an are the heat capacities at constant pressure an at constant volume, respectively. Equation () is also vali for moist air, provie that the actual temperature is replace by the virtual temperature ρ v (3) v The virtual temperature is given by v + η ( ) (4) where η an ε is the ratio of the gas constants for ry air, an for water vapour ε ε v is the specific humiity ρ vap ρ ( ρ vap is the ensity of water vapour an ρ is the ensity of moist air). The equation of state for moist air is vali since it is assume that water vapour also behaves ECMWF, 2002

2 Parametrization of non-convective conensation processes as an ieal gas vap ρ vap vap (5) where the subscript vap refers to water vapour. The first law of thermoynamics is is the specific entropy, is the specific internal energy ( ), is the specific enthalpy ( ), is the heating rate by external sources an is the specific volume ( -- ) ρ From (6) we get (6) ln + ln + const ln ln + const For ry aiabatic processes, i.e. at , the potential temperature? is conserve θ , θ (7) (8) The specific entropy can be expresse in terms of potential temperature as lnθ + const (9) The ry aiabatic lapse rate is (0) where is the gravity of the earth If the exchange of heat through conensation processes is consiere, the thermoynamic equation (6) becomes ( sat ) () sat where is the latent heat an is the saturation mixing ratio. The Clausius Clapeyron equation gives the slopes of the curves for the saturation water vapour pressure ). sat ( ) vap water sat (Fig. (2) 2 ECMWF, 2002

3 Parametrization of non-convective conensation processes vap water where an are the specific volumes for water vapour an for water, respectively. Integration of equation (2) gives (assuming that const, an ) the Tetens formula (Murray (967) water «vap vap v sat sat 0sat exp! " (3) where 0sat mb K " K Figure. Phase iagram for water is the (, ) plane. The inset shows the vapour ice equilibrium (lower curve) an the vapour liqui equilibrium (upper curve) at subfreezing temperatures. 2. CLOUD PHYSICAL PROCESSES As was one for the thermoynamics, a short summary of the principles of clou physics shall be given here. There are essentially two kins of process associate with clous that must be consiere in numerical moels: the formation of clous an the release of precipitation. The formation of clous is ue to conensation processes, i.e. a change of phase from water vapour to water roplets or to ice crystals an vice versa. Precipitation, on the other han, is release by transformation of small clou roplets into larger rain rops Fig. 2. ECMWF,

4 Parametrization of non-convective conensation processes Figure 2. Schematic iagram on conensation an precipitation processes 2. Conensation of water vapour We consier here the chemical equilibrium of liqui water an of water vapour. Measurements show that, for the case of pure liqui water an of pure water vapour, the equilibrium saturation vapour pressure over convex surfaces is higher than over plane surfaces accoring to # exp -- (4) # #$ where is the saturation vapour pressure of a roplet of raius, an is the saturation vapour pressure over a plane surface. The fraction increases with ecreasing roplet raius, as shown in Table. TABLE. FRACTION #$ FOR DIFFERENT SIZES OF DROPLETS AT 0 C ( µm) #$ Thus a pure roplet of raius 0 2 µm nees 2.8% supersaturation in orer to be in equilibrium with the surrouning water vapour. Since newly forme roplets are necessarily very small, large amounts of supersaturation are neee to start the conensation of water vapour in pure air. In the atmosphere, however, relative humiities above 0% are rarely observe. The reason is that various kins of small particles that serve as conensation nuclei are always present in the atmosphere. Due to the hygroscopic effect of these particles, a smaller equilibrium vapour pressure is neee for the very small roplets. We can, therefore, always assume that conensation takes place whenever the relative humiity excees 00%. The roplets generate by conensation processes are very small. The size spectrum of clou roplets in the presence of conensation nuclei ranges from 0. um to 0 pum. Fig. 3 shows roplet spectra of various kins of clous. As can be seen, non-precipitating clous as Sc an Cu have a rather narrow istribution. This istribution can be explaine by roplet growth ue to conensation, which is large for the smallest roplets an ecreases for the larger roplets, thus leaing to a narrow spectrum. 4 ECMWF, 2002

5 ' Parametrization of non-convective conensation processes Figure 3. The mean roplet-size istributions of various clou types. (From Diem (948).) 2.2 Release of precipitation The water roplets forme by conensation of water vapour are very small compare to rainrops in rizzle an in rain, which have a size of µm an 000 µm, respectively. Therefore, processes other than conensation must be present to form rain rops. The two processes that are believe to be the most efficient in proucing rain rops are coalescence an the Bergeron process. Coalescence occurs when water roplets are moving with ifferent fall spees ue to their ifferent sizes. By collision an fusion the larger roplets increase in size. The growth rate of a collector roplet may be prescribe as % (' ) 2ρ water % ) ρ water where is the roplet iameter, is the fall velocity of the collector relative to the small roplets, is the ensity of the roplet an is the clou water content. is the efficiency of collection. The efficiency of coalescence epens on the size spectra of the roplets an is more efficient for a broaer spectrum. Coalescence becomes efficient only if roplets with raii larger than 0 µm are present. Coalescence results in a significant broaening of the spectrum of the clou roplets. However, large rain rops are prouce only in clous of large vertical extents (epths of several km) which enables falling rops to collect enough small roplets. The coalescence process can, therefore, release precipitation in eep water clous. This precipitation process is important in low latitues, where eep clous of high water content are present, but it is not very efficient at higher latitues. Bergeron Process. The most efficient process of precipitation at higher latitues is associate with the formation of ice particles at temperatures below about 5 C. Generally, ice particles form by freezing of supercoole water roplets on various freezing nuclei. Freezing nuclei play a similar role to that of conensation nuclei for the conensation process. They initiate freezing at temperatures of aroun 5 C. Ice particles that are alreay forme grow very fast by collecting supercoole roplets because the equilibrium vapour pressure over ice is lower than (5) ECMWF,

6 - Parametrization of non-convective conensation processes over water (see Fig. ). The possibility that precipitation is initiate by the formation of ice crystals at low temperatures was first mentione by Bergeron (935). Spectra of rain rops. Spectra of rain-rop sizes observe in the atmosphere are often escribe analytically. An analytical istribution that is often referre to is that erive by Marshall an Palmer (948) from observational ata 0exp[ λ% ] where is the number of rainrops per unit volume of air, an an are empirical parameters. 0 λ 3. PARAMETRIZATION OF NON-CONVECTIVE CONDENSATION AND PRECIPITATION PROCESS- ES IN NUMERICAL MODELS In this paragraph we consier clous associate with the large-scale flow as resolve in numerical moels. Typical examples of this clou type are the clous associate with fronts. As for other processes, conensation processes an precipitation processes can only be parametrize by means of the large-scale variables given at the moel's gri points, which are available at very coarse resolution of, say, about 00 km in the horizontal an km in the vertical, an in time intervals of 0 min. The most important parameters are humiity, temperature an vertical motion. The vertical velocity etermines the conensation rate an, therefore, the supply of liqui water content. Temperature also controls the liqui water content, because the maximum amount of vapour is a function of temperature. The temperature istribution in a clou is also important for the type of precipitation rain or snow as mentione before. The parametrization of clou processes must be kept very simple, since the numerical integrations must be carrie out in a reasonable time. Two simple parametrization schemes that are use in numerical moels are escribe here. In the first scheme, only conensation of water vapour is consiere while clous are not parametrize. This scheme is frequently use in numerical moels. In the secon scheme, the clou water is preicte an the release of precipitation is parametrize in a simple way. 3. Scheme without clou stage In this scheme the clou stage is skippe an it is, therefore, assume that all the conense water vapour is immeiately converte into precipitation. This scheme is easily implemente in a numerical moel. We consier the moel equations for the large-scale values of temperature an moisture which, in the case of a gri-point moel, prescribe the time changes of the temperature an specif humiity at the gri points. +. ( ) , ( ) The terms an ( ) inclue all the aiabatic an iabatic processes, except for the conensation of water vapour. The conensation rate is the rate at which the saturation mixing ratio changes in saturate air , ( ) - - sat (6) sat ECMWF, 2002

7 - Parametrization of non-convective conensation processes In numerical moels the conensation rate is generally etermine iagnostically. Therefore, preliminary values an are first preicte by neglecting the effects of conensation / 0/ + 2 +, ( ) + 2 +, ( ) (7) If the air is foun to be supersaturate > sat ( ) + + / / an are ajuste to their saturation values, which are then the final values an at time step / + sat ( ) sat ( ) + sat ( ) / sat ( ) (8)?These corrections follow from sat , sat / ( ) where the saturation value is that for the correcte temperature which is approximate as The conensation rate is therefore given as + sat ( / ) sat ( ) + sat ( ) - 2 sat ( ) sat (9) As the assumption is mae that all the conensate water vapour immeiately falls out as rain, the precipitation rate is (units : m H2 O sec ) ρ water where is the ensity of water ρ z ρ water 0 ECMWF,

8 Parametrization of non-convective conensation processes 3.2 A clou parameterization scheme Most numerical moels consier the conensation processes as escribe before. The clou phase is skippe by assuming that the conense water vapour falls immeiately out as precipitation. Although the release of latent heat seems the most important process for the large-scale flow, the formation of clous an the release of precipitation may also be important as they are a part of the hyrological cycle. Consequently there have been several attempts to parametrize clous an precipitation processes, notable schemes are those by Kessler (969) an Ogura an Takahashi (97). Kessler's scheme is esigne for warm clous, since it consiers the release of precipitation to be mainly ue to coalescence, whereas Ogura an Takehashi esigne their scheme for cumulus clous. The scheme escribe here takes into account the two main processes important for the release of precipitation coalescence an the Bergeron process. This scheme has recently been propose by Oklan while staying at the EC- MWF. The avantage of this scheme it that it is rather inexpensive for computer calculations an that it takes the main precipitation processes into account. As the scheme consiers clou water as a further variable, the prognostic equations are, therefore cl , ( ) (- ) an are the mixing ratios for water vapour an clou water, respectively. cl an rain are the evaporation rates of clou water an rain water an 4 rain is the rate of release of precipitation ue to conversion from clou water to rain water. The parametrization of conensation is one in the same way as escribe before, an the evaporation of clou water ue to increase of the saturation mixing ratio is similarly calculate. cl The release of precipitation is parametrize as follows. Two types of precipitating clous are consiere. (a) Clous with low temperatures at clou top: < with crit 5 K. For this clou type it is assume that the total liqui clou water is immeiately release as precipitation. (b) Clous with temperatures at top above : > but with a high clou water content cl > crit, with crit 2 mm H2 O, where cl , ( ) (- ) cl , ( cl) ρ water cl cl rain ( 4 rain rain) top crit rain rain crit top crit cl top cl ρ z ρ water base It is further assume that clou water in excess of this critical value is instantaneously remove as precipitation. Besies the conensation processes an precipitation processes, the following processes are also consiere: (a) Evaporation of rain rain (b) Collection of clou water in non precipitating clou layers by precipitation from above ( ) coll l These processes are parameterize follows Kessler (969). 4 rain 8 ECMWF, 2002

9 Parametrization of non-convective conensation processes " where,, an are constants. rain - sat ( 4 rain) coll - 3:9 2 cl ( ) 87 REFERENCES Arakawa, A. an W.H. Schubert, 974: Interaction of a cumulus clou ensemble with the large-scale environment, Part I. J.Atmos.Sci. 3, Bates, J.R., 977: Parameterization of convective processes, ECMWF Seminar Proc., Bergeron, T., 935: On the physics of clou an precipitation. Proc. 5th Assembly UGGI Lisbon, Vol. 2. Betts, A.K., 974: Further comments on a comparison of the equivalent potential temeprature an the static energy, J.Atmos.Sci., 3, Cho, H.R., 975: Cumulus clou population an its parameterization. Pure an Applie Geophysics, 3, Diem, M., 948: Mesungen er Grow von Wolkenelementen, II. Met. Rsch, 26. Kessler, E., 969: On the istribution an continuity of water substance in atmospheric circulation. Met. Monogr., Vol 0, No. 32, Amer.Met.Soc., Boston, Mass. Kuo, H.L., 965: On formation an intensification of tropical cyclones through latent heat release by cumulus convection. J.Atmos.Sci., 22, Kuo, H.L., 974: Further stuies of the parameterization of the influence of cumulus convection on a large-scale flow. J.Atmos.Sci., 3, Kurihara, Y., 973: A scheme of moist convective ajustment. Mon.Wea.Rev., 0, Manabe, S., F. Smagorinsky an R.F. Strickler, 965: Simulate climatology of a general circulation moel with a hyrological cycle. Mon.Wea.Rev., 93, No. 2, pp Marshall, J.S., an W. Mck Palmer, 948: The istribution of rainroplet spectra an large roplets by conensation in cumulus clous. Quart.J.Met.Soc., 00, Murray, F.W., 967: On the computation of saturation vapour pressure. J.Appl.Meteor. 6, Ogura, Y., an T. Takahashi, 97: Numerical simulation of the life cycle of a thunerstorm cell. Mon.Wea.Rev., 99, Ooyama, K., 97: A theory on parameterization of cumulus convection, J.Met.Soc., Japan, 49, Sunquist, H., 977: Atmospheric conensation an moelling its non-convective regime. ECMWF - Seminar Proc., ECMWF,

10 Parametrization of non-convective conensation processes 0 ECMWF, 2002

11 Parametrization of non-convective conensation processes ECMWF, 2002