Diffusion and compensating subsidence as limiting cases of a single flux parameterization. David Randall

Transcription

1 ! Revised September 014 9:49 AM! 1 Diffusion and ompensating subsidene as limiting ases of a single flux parameterization David Randall The tendeny of the average of a onservative variable h due to a vertial eddy flux is given by ( ). t z ρ w h (1) In this note we onsider two different forms of the vertial eddy flux and the relationship between them. A diffusive vertial flux of a onservative variable h satisfies ρ w h = K z () where K is a positive diffusion oeffiient. For positive K the flux given by () is always down-gradient. With the use of () Eq. (1) beomes paraboli: t z K z (3) In ontrast for a single loud type the tendeny of h due to deep onvetion is given by t M z + δ ( h h ) (4) where QuikStudies for Graduate Students in Atmospheri Siene

2 ! Revised September 014 9:49 AM! M ρσ 1 σ ( ) ( ) w (5) is the positive onvetive mass flux δ is the detrainment mass flux per unit height h is the inloud value of h σ is the frational area overed by the onvetive updrafts and w! is the environmental vertial veloity (Arakawa and Shubert 1974). Eq. (4) is hyperboli. It applies when σ 1. (6) We will derive a single formula that redues to either (3) or (4) in the appropriate limits. The disussion follows Randall et al. (199) and Lappen and Randall (001). The starting point is the equation that governs the variane of h i.e. ρ h t ( ) ρ = ρ w h z z ρ w h h h τ dis (7) where h τ is the rate at whih h dis is dissipated. In writing (7) we have negleted additional terms involving advetion by the mean wind gradient prodution assoiated with horizontal fluxes and horizontal transport by triple moments involving the horizontal wind omponents. We now simplify (7) for the ase of a steady-state and use the mass-flux model to rewrite the remaining terms. With the mass flux model the vertial flux of h satisfies ρ w h = ρσ ( ) w 1 σ ( )( h h " ) (8) where h! is the environmental value of h whih satisfies h = σ h + ( 1 σ )h!. (9). Similarly the triple moment is given by ρ w h h = ρσ 1 σ ( )( 1 σ ) w ( )( h h " ). (10) QuikStudies for Graduate Students in Atmospheri Siene

3 ! Revised September 014 9:49 AM! 3 Lappen and Randall (001) showed that with the mass-flux model the dissipation term is given by ρ h = ( ε + δ ) h h! τ dis ( ) (11) where ε 0 is the rate at whih mass is flows from the environment into the updrafts and δ 0 is the rate at whih mass flows from the updrafts into the environment. Both ε and δ have dimensions of mass per unit volume per unit time. Conservation of mass for the onvetive updrafts is expressed by ( z ρσ w ) = ε δ. (1) Using (8) we an rewrite (10) and (11) as ρ w h h = 1 σ ( ) M (13) and ρ h ρ w h = ( ε + δ ) τ dis M (14) respetively. Substitution of (13) and (14) into the steady-state (or time-averaged) version of (7) gives 0 = ρ w h z z 1 σ ( ) M ε + δ M. (15) For the ase σ 1/ Eq. (15) redues to 0 = z ε + δ M ρ w h QuikStudies for Graduate Students in Atmospheri Siene (16)

4 ! Revised September 014 9:49 AM! 4 or ρ w h = M ε + δ z. (17) This has the form of () with K = M ε + δ > 0. (18) Eq. (17) is the diffusive limit of (15). The flux given by (17) is always downgradient. In the limit σ 0 Eq. (15) an be expanded and simplified to 0 = z M ( ) + ρ z ρ w h M M z ε + δ M (19) In the same limit (1) an be written as M z = ε δ. (0) Using (0) Eq. (19) an be further simplified to Substitution from (8) gives 0 = z 1 M ( ) δ ρ z ρ w h M (1) 0 = z 1 M ( ) δ h h z ρ w h M () where we have used h! h whih is appropriate in the limit σ 0. Finally () an be rearranged to QuikStudies for Graduate Students in Atmospheri Siene

5 ! Revised September 014 9:49 AM! 5 ( ) = M z ρ w h ( ) z + δ h h (3) whih is onsistent with (1) and (4). A flux that satisfies (3) an be be either upgradient or downgradient. We have shown that a parameterization based on (15) an produe either (loal) diffusion or (non-loal) penetrative onvetion depending on the value of σ.wyngaard and Weil (1991) obtained a similar result. The diffusive downgradient flux satisfying (17) is obtained when prodution is balaned by dissipation and (1) is paraboli. A (possibly upgradient) flux satisfying (3) is obtained when prodution is balaned by transport and (1) is hyperboli. Referenes Arakawa A and W. H. Shubert 1974: Interation of a umulus loud ensemble with the largesale environment Part I. J. Atmos. Si Lappen C.-L and D. A. Randall 001: Towards a unified parameterization of the boundary layer and moist onvetion. Part I. A new type of mass-flux model. J. Atmos. Si Randall D. A. Q. Shao and C.-H. Moeng 199: A seond-order bulk boundary-layer model. J. Atmos. Si Wyngaard J. C. and J. C. Weil 1991: Transport asymmetry in skewed turbulene. Phys. Fluids QuikStudies for Graduate Students in Atmospheri Siene