VII : NON-CONVECTIVE PRECIPITATION

Transcription

1 V : NON-CONVECTVE PECPTATON (largescale precipitation, grid-resolvable scale precipitation, explicit moisture scheme, cloud scheme) a. Purpose emove supersaturation after deep and shallow conveciton, and feedback to largescale b. cheme : (LGCL, CLOUD#), Hong et al. (1998). Number of prognostic water species(ncld) CPU* CLD 1 - diagnostic cloud with evaporation 1% CLD 2 - Zhao and Carr(1997) 12% CLD 3 - Dudhia(1989) 145% CLD 5 - utledge and Hobbs(1983) 18% MF employs diagnostic cloud for grid-resolvable scale precipitation as of October A hierachy of prognostic cloud shcmes implemented in the M (Hong et al.1998) is undertesting withing computer resources. Microphysical processes are described for the prognostic cloud/ice water, snow/rain water substance below. 5

2 c. scheme (CLD3) Governing equations The prognostic perturbation equations for water vapor (q v ), cloud water/ice (q ci ) and snow/rain (q rs ) species are, respectively, m u t x v y. σ q σ v 2 * v * v v vdif hdif impl expl v F F F F qv qv qv qv base t, (1) t m u x v y. σ q σ ci 2 * ci * ci ci hdif exp l ci F F qci qci base t, (2) t m u x v y. σ q σ rs 2 * rs * rs rs hdif exp l rs F F qrs qrs base t g p s ρ q V rs t σ, (3) The corresponding thermodynamic equation is σ σ σ κ Tρ Tρ Tρ. T κ ρ 2 * * m u v Q 2 * Q * κt t x y σ ρ m u v t x base T vdif hdif rad impl expl ρ F F F F F T T T T T t Q y, (4) Cloud fraction : andall (1995) Cf H 1 exp 1 1q H l microphysics (NCLD3,NCLD5) from NCEP office note The microphysics treatment of the source and sink terms are based on Lin et al.(1983), utledge and Hobbs(1983) for NCLD5, and Dudhia(1989) for NCLD3. NCLD5 employs five prognostic species including water vapor(q), cloud water(qc), cloud ice(qi), snow(qs), and rain(qr) and NCLD3 intoduces three prognostic species including water vapor(q), cloud water/ice(qci), and rain/snow(qrs). The reader is referred to original papers for the detailed physical explanations. ubscript stands for rain and for snow drops, respectively. 51

3 Equtation will be given for only rain when its formula for both are identical except for the parameter related to water phase. The definitions of precipitate size and fall velocity are described in section a and, respectively. The source and sink terms for NCLD5 are described in section c, and the differences between NCLD3 and NCLD5 are pointed out section d. ource terms for the water continuity variables are described in section e. Definitions of variables and constants used below are given in appendix B. a. Precipitate size distributions The rain and snow particles are assumed to follow the size distribution derived by Marshall and Palmer(1948), and Gunn and Marshall(1958), respectively. The size distributions for both rain and snow are formulated according to an inverse exponential distribution and its formula for rain can be expressed by λ, (A1) N ( D ) N exp( D ) for rain, where N is the intercept parameter of the rain distributions. The slope parameter of the size distributions for rain (λ ) is determined by multiplying (A1) by drop mass (A4) and integrating over all diameters and equating the resulting quantities to the appropriate water contents (ρq ). This may be written as, (A2) λ πρ N w ρq 1 / 4. b. Mass-weighted fall speeds All particles in the precipitating fields of rain and snow are assumed to fall at their massweighted fall speeds, and defined as V N ( D ) M ( D ) V ( D ) dd D. (A3) N D ( D ) M ( D ) dd where the mass distribution is assumed as M ( D D πρ D ) πρ. (A4)

4 The terminal falling velocity distribution of rain is suggested by Liu and Orville(1969) and given by (A5) ( b ) D a D ρ 1/ 2 ρ V, where the density factor in the right hand side of (A5) allows for the change in fall speed with air pressure. From (A1), (A3) and (A5), V may be written as (A6) Γ(4 b ) 6 ρ ρ 1/ 2 V λ b a. c. ources and sinks of the water continuity variables 1) Condensation and evaporation of water vapor (Pcon) When water vapor is supersaturated with respect to water, the condensation is determined as, Pcon 1 ( q q ) / t v sw, (A7) 2 2 L q / ( c T ) v sw v pm where qv, qsw and T used in this process are updated values by other microphysical processes. n other words, Pcon removes the additional supersaturated water vapor after other slower processes are taken into account. by f the air is subsaturated and q c is greater than q min, evaporation of cloud water is given [ ] Pcon min Pcon, q / t c. (A8) 53

5 2) nitiation of cloud ice crystal (Pgen) When T< C and the air is supersaturated with respect to ice, the initiation rate of cloud ice from water vapor is given by [ ], (A9) v Pgen min ( M n q ) / t, ( q q ) / t where it is always, Pgen. M is the initial mass of cloud ice which corresponds to the mass of D. 3) Depositional growth of cloud ice (Pisd) The growth rate by vapor deposition of a small ice crystal is given by dm dt C ( ) / A 1 B ε, (A1) where A B L K T * T D M e L M s s w a f w si * T 1, (A11) where C 4D ε. The diffusivity and the thermal conductivity of air are, respectively, expressed by D T / p K f a µ, (A12) where dynamic viscosity of air is given by µ T / ( T 12). (A13) 54

6 Using (A17), the growth rate of cloud ice via deposition is Pisd 4 D ( 1) n 4D ( 1)( q n ) ρ ( A B ) ρ con 2 2 L / ( K T ) 1 / ( q D ) 1/ 2 s a v sw f. (A14) 4) Autoconversion of cloud water (ice) to rain (snow) (Paut) Autoconversion is the process whereby cloud water(ice crystals) droplets form raindrops (snow) through collisions with each other. Following Kessler(1969), the autoconversion rate of cloud water to rain may be written as Paut α( q q ) C C C, (A15) where α is a rate coefficient and q C the mass threshold value for autoconversion. For cloud ice, the conversion rate is given by [ ] Paut max ( q M n ) / t, i ax m, (A16) where Mmax represents the maximum allowed ice crystal mass which corresponds to the mass of a DMAX. We assume hexagonal plate-like cloud ice crystals. The diameter D of a hexagonal plate can be computed from the mass M of the plate : D 16 3 M 1 / ρ n q 1 / 2, (A17) where the ice number concentration nc is given by Fletcher(1962) s formula and is given by, exp[. ( )] / ρ. (A18) 2 n 1 6 T T 5) Collection of cloud water (ice) by rain (snow) and cloud water by snow The collection rate of cloud water by rain is parameterized by using the continuous collection equation, and is given by 55

7 dm ( D ) dt D 2. V ( D ).. q. E D V ( D ) q E π π ρ ρ C C C C. (A19) Multiplying (A19) by (A1), using (A5), and integrating over all particle sizes yields Pacr C π ρ 1 / 2 a q E N Γ( b 3) C C b 3 4 ρ λ. (A2) Collection of cloud ice by snow(pacr) is parameterized by replacing subscript by and qc by q. n addition, collection of cloud water by snow(pacrc )is also given in the same manner in (A18) by replacing subscript by. 6) Evaporation (sublimation) of rain (snow) and depositional growth of snow The evaporation of rainwater is calculated if the air is subsaturated with respect to water and if the air is above water saturation, growth by condensation occurs. The continuous growth equation is given by dm ( D ) C D ( 1) F dt w A w B w, (A21) where C 2π. The thermodynamic constants are given by, A B w w L K T * T D M e L M v v w a * T f w sw 1. (A22) The ventilation factor is given by 1/ 3 1/ c e F f f. (A23) Using (A1) and (A5), the evaporation rate of rain is 56

8 Prec 1/ 2 1/ 4 (( b ) / ) C N ( 1) f / a W f c ( b ) / ( A B ) ρ W W λ ρ µ ρ ρ Γ λ. (A24) When the air is supersaturated with respect to water, the condensational growth rate of rain is given by (A24). When the air is subsaturated with respect to ice, the sublimation of snow and the depositional growth rate of snow(pssd) is given by (A24) with the substitution of subscripts and w into and. 7) Melting (Pmlt) and freezing (Pfrz) All snow upon melting is assumed to contribute to rain. The snow melted per unit volume is given by dm 2π ( ). (A25) dt L K D T T F f a ubstituting (A23) for snow into (A25), multiplying by (A1) for snow and integrating over all snow sizes, the melting snow can be expressed by Pmlt π ρ f λ (( b ) / ) 2 N f a K T T f a c b L / 3 ( ) 2 2 ( 5 )/ 2 ρ µ ρ ρ 1/ 2 1 / 4 When T > T, instantaneous melting of cloud ice is assumed and is given by Γ λ. (A26) Pmlt q t /. (A27) When T < T ( -4 C ), homogeneous freezing of cloud water(pfrz) is assumed by Pfrz q / t. (A28) C 8) Evaporation of melting snow (Psev) This term is identical to (A24) for evaporation of snow except that the evaporation is from a liquid surface : 57

9 Psev ( ( b ) / ) C N ( 1) f a W f c 1 b A B 5 2 1/ ( 5 )/ 2 ( ) ρ W W λ ρ µ ρ ρ 1/ 2 1/ 4 Γ λ. (A29) d. No-mixed phase scheme (NCLD3) This scheme is identical to NCLD5 which represents the four prognostic variables, cloud water, cloud ice, rain and snow, separately, except that the mixed phase is not allowed. Therefore, some microphysical processes in NCLD5 should be neglected or simplified. For example, the collection of cloud water by snow(pacr sc ) and evaporation of melting snow (Psev) are omitted. ince mixed phase does not exist, melting and freezing processes occur at a level instantaneously, and the discrepancies from NCLD5 are pointed out below, 1) Melting and freezing As snow falls through the C level, it immediately melts to rain. This process is given by Pmlt g V q ρ f p. (A3) Advection of ice or snow downwards or of rain or cloud upwards through this level also melts or freeze the particles, where Pfrz / Pmlt ω p ( q q ) C. (A31) n both cases, the C isotherm is taken to be at a full model level boundary. Melting occurs at the level immediately below this boundary and freezing above it. e. ource terms for the water continuity variables The source terms for the five water continuity variables(ncld5) are listed below. For water vapor qv: Fv -[Pcon Prec Pgen Pisd Pssd Psev (T> T )] For cloud water qc: FC [Pcon Psev - PautC - PacrC - PacrC - Pfrz (T> T )] Pmlt(T> T )] For cloud ice q: F [Pgen Pisd - Paut - Pacr Pfrz (T> T )] - Pmlt (T> T )] For rain q: F [Prec PautC PacrC PacrC (T>T ) - Pmlt (T>T )] For snow q: 58

10 F [Pssd Psev (T<T ) Pmlt (T> T ) Paut Pacr PacrC (T<T )] The source for T is : FT Lv/cpm [Pcon Prec Psev] Ls/cpm [Pgen Pisd Pssd] Lfcpm [Pmlt - Pmlt PacrC (T<T)] 59

11 List of symbols ymbol Description Value units Aw Thermodynamic term in Pres for rain kg -1 ms A Thermodynamic term in Pres for snow kg -1 ms a Constant in fallspeed relation for rain 842 m (1-br) s -1 a Constant in fallspeed relation for snow m (1-bs) s -1 BW Thermodynamic term in Pres for rain kg -1 ms B Thermodynamic term in Pres for snow kg -1 ms b Fall speed exponent for rain.8 b Fall speed exponent for snow.11 C Capacitance of ice crystal F cp pecific heat of air at constant pressure 15 Jkg -1 K -1 cpm pecific heat of moist air at constant pressure D f Diffusivity of water vapor in air m 2 s -1 D Diameter of hexagonal plate ice crystal m D nitial diameter of cloud ice crystals m Dcon Constant in mass and size relation of ice crystals 16.3 mkg -1/2 D aindrop diameter m D now diameter m EC ain/cloud water collection efficiency 1 EC now/cloud water collection efficiency 1 E now/cloud ice collection efficiency.1 e aturation vapor pressure over ice Nm -2 esw aturation vapor pressure over water Nm -2 6

12 F Ventilation factor for rain F Ventilation factor for snow f1 Constant in ventilation factor for rain.78 f2 Constant in ventilation factor for rain.32 f1 Constant in ventilation factor for snow.65 f2 Constant in ventilation factor for snow.44 Ka Thermal conductivity of air Jm -1 s -1 K -1 Lf Latent of fusion of water substance Jkg -1 Ls Latent of sublimation of water substance Jkg -1 Lv Latent of condensation of water substance Jkg -1 M Average mass of cloud ice crystal kg Mmax Maximum mass of cloud ice crystal kg Mi nitial mass of cloud ice crystal 1-12 kg M Mass of rain per unit volume of air kgm -3 M Mass of snow per unit volume of air kgm -3 MW Molecular weight of water M(D ) Mass of raindrop of diameter D M(D ) Mass of snowflake of diameter D kg N ntercept value in raindrop size distribution m -4 N ntercept value in snowflake size distribution m -4 n Number concentration of cloud ice crystals kg -1 n Constant in expression for ice crystal concentration.1 m -3 p Pressure Nm-2 Pcon Condensation of water vapor kgkg -1 s -1 Pisd Deposition/sublimation of cloud ice kgkg -1 s -1 Pssd Deposition/sublimation of snow kgkg -1 s -1 61

13 Psev Evaporation of melting snow kgkg -1 s -1 PacrC Accretion of cloud water by rain kgkg -1 s -1 PacrC Accretion of cloud water by snow kgkg -1 s -1 Pacr Accretion of cloud ice by snow kgkg -1 s -1 Pgen nitiation of cloud ice kgkg -1 s -1 PautC Autoconversion of cloud water kgkg -1 s -1 Paut Autoconversion of cloud ice kgkg -1 s -1 Prec Evaporation/condensation of rain kgkg -1 s -1 Pmlts Melting of snow kgkg -1 s -1 Pmlt Melting of cloud ice kgkg -1 s -1 qc Mixing ratio of cloud water kgkg -1 qc Mixing ratio of cloud water and cloud ice in NCLD3 kgkg -1 q Mixing ratio of cloud ice kgkg -1 q Mixing ratio of rain kgkg -1 q Mixing ratio of rain and snow in NCLD3 kgkg -1 q Mixing ratio of snow kgkg -1 qc Mixing ratio threshold for Pautc.7 kgkg -1 qs aturation mixing ratio with respect to ice kgkg -1 qstw aturation mixing ratio with respect to water kgkg -1 qv Mixing ration of water vapor kgkg -1 v Gas constant of water vapor 461 Jkg -1 K -1 c chmidt number ν / D f FC ource term for cloud water kgkg -1 s -1 62

14 F ource term for cloud ice kgkg -1 s -1 F ource term for rain kgkg -1 s -1 F ource term for snow kgkg -1 s -1 Fv ource term for water vapor kgkg -1 s -1 T Temperature K T eference temperature K T Threshold temperature for homogeneous freezing K t Time s V r Mass-weighted fallspeed of rain ms -1 V (D) Fallspeed of raindrop of diameter Dr ms -1 V s Mass-weighted fallspeed of snow ms -1 V (D) Fallspeed of snowflake of diameter Ds ms -1 x Horizontal distance m z Vertical distance m α ate coefficient for autoconversion.1 s -1 β Constant in ice crystal concentration.6 K -1 Γ Gamma function ε Permittivity of free space??? ρ Air density kgm -3 ρ eference air density 1.28 kgm -3 ρw Density of water 1 kgm -3 ρs Density of snow 1 kgm -3 λ lope of raindrop size distribution m -1 λ lope of snowflake size distribution m -1 µ Dynamic viscosity of air kgm -1 s -1 63

15 ν Kinematic viscosity of air m 2 s -1 κ d /Cp

16 d. rermarks The rognostic water substance is to represent better radiation feedback in low resolution models (GCM), whereas in high resolution model (mesoscale area) its purpose is to suppress unphysical feedback at the PBL top causing excessive rain A distinct problem in existing cloud schemes is too much ice crystals in the upper atmosphere, and too little cloud water in the lower atmosphere 65

17 A modified scheme from H (1983), Dudhia(1989), and Hong et al. (1998) improves the distribution of water substance 1) ntercept parameter for snow : Cox (1988), Houze et al. (1979) N 2 6 exp{( T T ) / 818. } : Aggregation, which leads to broadening of the snow spectra at higher temperature, is implicitly parameterized. 2) Autoconversion : TC(198) Paut. 14gE 4 / 3 c 7 / 3 1/ 3 c c c µ ρ ( N ) c w ρ q H( q q ) : Nc differs over land and ocean, q c 4 πρ ρ r N w cr c 3) Accretion of snow Pacr C π ρ 1 / 2 a q E N Γ( b 3) C C b 3 4 ρ λ Erc exp (.25*(To-T)) 4) Fall speed for snow : An integrated form of snow spectra 5) Limit of Pisd : Pisd : < (qs-q)/(2dt) --> alleviate numerical instability : ncrease cloud time step 66