Formulas/Constants, Physics/Oceanography 4510/5510 B Atmospheric Physics II N A = 6.02 10 23 molecules/mole (Avogadro s number) 1 mb = 100 Pa 1 Pa = 1 N/m 2 Γ d = 9.8 o C/km (dry adiabatic lapse rate) k = 1.38 10 23 J/K γ = c p /c v γ = 1.4 (dry air) R d = 287.05 J/(kg K) (mean gas constant for dry air) c v = 717.5 J/(kg K) (dry air) c p = 1004.5 J/(kg K) (dry air) M d = 28.94 g/mole (Mean molecular weight of dry air) pv = NkT (where N is the number of molecules, and the Boltzmann s constant k = 1.38 10 23 J/K.) p = ρr d T (R d is the gas constant for dry air.) For a mixture of gases, R is replaced by < R >, where < R > is the the mean R of the various constituents, weighted by number fraction. R d = 287.05 J/(kg K) (ideal gas constant for dry air) R v = 461.5 J/(kg K) (ideal gas constant for water vapor) ǫ = M v /M d = 0.622 e = ρ v R v T latent heat of vaporization: l v = 2.5 10 6 J/kg latent heat of fusion: l f = 3.34 10 5 J/kg density of water ρ l = 1000 kg/m 3 specific heat of of liquid water c l = 4219 J/(kgK) Water vapor M d = 18.015 g/mole (Mean molecular weight of water vapor) Mean molecular weight of air: M = f d M d + f v M v, where f d and f v are the molar fractions of dry air and the vapor (f d + f v = 1). Approximate expression for dependence of saturation vapor pressure (in hpa) on temperature (T in Kelvin): = exp[21.653 (5420/T)] dew point: e = e s (T d ) specific humidity: q = ρ v /ρ = w/(1 + w) ǫ(e/p) mass mixing ratio: w = ρ v /ρ d = ǫe/p d = ǫ(e/(p e)) ǫ(e/p) ǫ = R d /R v = 0.622 e = w ǫ+w p c v,w = 1390 J/(kg K) (specific heat at constant volume of water vapor, 273 K) c p,w = 1850 J/(kg K) (specific heat at constant pressure of water vapor, 273 K) Specific moist enthalpy of an air parcel (per kg dry mass): h m = (c pd + c l w t )T + l v w. moist enthalpy of an air parcel of dry mass m d : H m = m d h m You can assume that all specific heats are independent of temperature. Chapter 6: Cloud Microphysics g v = g w + 2σ rρ l
Curvature effect: equilibrium vapor pressure e s (r,t) of a pure water droplet of radius r and temperature T: 2σ e s (r,t) = exp( rr v ρ l T ) where σ is the surface tension, ρ l the density of water, R v gas constant for water vapor, T temperature, and the Clausius-Clapeyron saturation vapor pressure for water. Approximate form of Kelvin equation: e s (r,t) = exp( 0.00108 ) r where r is the radius of a drop in microns. G = 4πσr 2 4 3 πr3 ρ l [R v Tln(e/e s )] critical size r at which a pure water droplet becomes stable as a function of the ambient vapor pressure e and temperature T: Raoult s Law: r = 2σ R v ρ l Tln(e/) N H2 0 e s (f H2 O,T) = f H2 O = N H20 + N X where N X is the number of moles of X, and f H2 O is the molar fraction of water in the liquid. Raoult s Law for a curved droplet (Kohler curve): e s (r,t) = f H2 O where e s (r,t) is the saturation vapor pressure for a pure water droplet of radius r. Kohler Curve: f H2 O = moles water moles water + moles dissolved ions = [ 2σ ][ im x M w ] 1 exp 1 + ρ l R v Tr M x ( 4 3 πr3 ρ m x ) where M w is the molecular weight of water (g/mole), σ the surface tension of water, m x the mass of dissolved material (in kg), M x the molecular weight of the solute(g/mole), i the number of ions the solute dissolves into, ρ the density of the solution (kg/m 3 ). This can also be written: = [ 2σ ][ im x M w ] 1 exp 1 + ρ l R v Tr m w M x where m w is the mass of water in the droplet. For dilute aerosols, m w 4 3 πr3 ρ l, where ρ l is the density of liquid water. Approximate form of Kohler Curve 1 + a r b r 3
where a = (2σ)/(ρ l R v T) and b = (3im x M w )/(4πρ l M x ). Rate of change of droplet radius r due to condensational growth: dr dt = 1 r Dρ v ( ) [e( ) e(r)] ρ l e( ) where D is the molecular diffusion constant, r the radius of the drop, ρ v the density of the vapor away from the drop, and ρ v the density of the vapor at the droplets surface. e( ) refers to the ambient water vapor pressure e in the cloud (i.e. as long as you are a few droplet radii away from a cloud droplet), and e(r) refers to the water vapor pressure at the surface of the droplet. Cloud droplets are usually large enough so that one can ignore the curvature effect, so that e(r), i.e. just the Clausius Clapeyron saturated vapor pressure. In this case, one can write: where the saturation ratio is defined: r dr dt = G ls S = (e )/ G l = Dρ v( ) ρ l r(t) = Cloud Droplet Terminal Fall Speed for r 30µm: r 2 0 + 2G lst ν = 2 gρ l r 2 9 η where η is the viscosity of air, ρ l is the density of water, and g is the gravitational acceleration. Collision Efficiency: E(r 1,r 2 ) = y 2 (r 1 + r 2 ) 2 Growth by Continuous Collision/Coalescence. In these equations, M is the mass of the collector droplet, r 1 is the radius of the collector droplet, ν 1 is the velocity of collector droplet, ν 2 is the velocity of the cloud droplets (assumed to be a constant), w l is the LWC (in kg/m 3 ), E c is the collection efficiency, E is the coalescence efficiency, and ρ l is the density of water. dm dt = πr 2 1(ν 1 ν 2 )w l E c dr 1 dt = (ν 1 ν 2 )w l E c 4ρ l A cloud droplet is being carried upward in a cloud with updraft velocity w. r 0 is the initial cloud droplet radius, r H is the final radius, and H is the final height. The initial height is 0 (assumed cloud base). For the case ν 1 >> ν 2 :
H = 4ρ l [ rh w l r 0 w rh ν 1 E dr dr 1 ] 1 r 0 E where E is the collision efficiency, w l is the liquid water content, ν 1 is the terminal velocity of the collector drop, r 1 is the radius of the collector drop, H is the height above cloud base, and w is the cloud updraft speed. Under the above assumptions, the final radius of the droplet as it exits the cloud (H = 0) will be given by: R dr 1 R = r 0 + w r0 v 1 Marshall-Palmer Raindrop Distribution: N(D) = N 0 e ΛD, where D is the diameter, and Λ is a parameter which depends on the rain rate. Radar reflectivity Z: Z = 0 N(D)D 6 dd ICN concentration N: lnn = a(t 1 T), where a is a parameter, T 1 is the temperature at which N = 1 ICN per liter. Condensational growth of an ice crystal: dm/dt = DC/ǫ 0 [ρ v ( ) ρ vo ], where C/ǫ 0 is a shape dependent parameter. Alternatively, dm/dt = (C/ǫ 0 )G i S i, where G i = Dρ v ( ) and S i = (e( ) e si )/e si LFC CAPE = R d (T v T v )dlnp EL where EL is the equilibrium level, or level of neutral buoyancy, LFC is the level of free convection, T v is the virtual temperature of the background atmosphere, and T v is the virtual temperature of the air parcel. Cyclostrophic balance: v 2 /r = (1/ρ) ( p/ r) Rate of increase in vorticity due to stretching: d/dt lnξ w/ Kinetic energy balance: V dv dt = d dt V 2 2 = V Φ + F V Efficiency W/Q H Approximate expression for z LCL : z LCL z o = T o T do 8, (in km), where T o is the temperature at the ground and T do is the dew point temperature at the ground. θ e = θ d exp( l vw s c p T ), equivalent potential temperature (for saturated air) θ e = θ d (z LCL )exp( l v(t LCL )w c p T LCL ), equivalent potential temperature (for unsaturated air) Richardson Number: R i = Local heating due to turbulent heat flux divergence: ( u g θ T ) 2+ ( v θ t = θ w ) 2
Frictional acceleration due to turbulent momentum flux divergence: Definition of Eddy Viscosity Coefficient K: Du Dt = u w w u = K u logarithmic wind profile: u(z) = (u /k)ln(z/z 0 ) Kinematic Turbulent heat flux (K m/s): F H = w θ Dynamic Turbulent Heat flux (in W/m2): Q H = ρc p w θ Expression for net downward radiative flux: F = F S F S + F L F L Net upward transfer of energy from earth s surface: F net = F + F Hs + F Es Change in entropy : ds = dq/t Bulk Aerodynamic Formulae: Kinematic Heat Flux (in K m/s): F Hs = C H V (T s T air ) Dynamic Heat Flux (in W m/s): F Hds = ρc p F Hs Kinematic Moisture Flux (in kgv/kga m/s): F water = C E V (q sat (T s ) q air ) Kinematic Latent Heat flux (K m/s): F Es = (L v /c p )F water Dynamic Latent Heat flux (W m/s): F Eds = ρl v F water Bowen ratio: B = F Hs /F Es downward momentum flux at the surface : u 2 = C D V 2 = w u Simplified TKE equation: d(t KE/m) dt = M + B ǫ TKE = (u 2 + v 2 + w 2 )/2 M = ( u ) 2+ ( v) 2 Bulk Richardson Number: B = g T θ R B = g θ z < T > ( u) 2 + ( v) 2 Useful approximation for determining variation in θ across small layer: θ(z) = T(z) + Γ d z. Rate of change in height of inversion layer: dz i /dt = w e + w i Relationship between large scale vertical velocity at the inversion layer, and the mean pressure weighted boundary layer divergence: w i = z i [ V] Relationship between surface heat flux and entrainment velocity: w e = AF Hs / θ Definition of Ball parameter: A = F Hzi /F Hs Change in average θ of Boundary layer (under certain conditions):
d < θ > dt = F Hs F Hzi z i = (1 + A)F Hs z i Buoyancy Flux: the sum over all air parcels in cloud, the product of the mass, vertical velocity, and buoyancy. BF = p M p w p B p