Quasi-equilibrium Theory of Small Perturbations to Radiative- Convective Equilibrium States See CalTech 2005 paper on course web site Free troposphere assumed to have moist adiabatic lapse rate (s* does not vary with height Boundary layer quasi-equilibrium applies
Basis of statistical equilibrium physics Dates to Arakawa and Schubert (1974) Analogy to continuum hypothesis: Perturbations must have space scales >> intercloud spacing TKE consumption by convection ~ CAPE generation by large scale Numerical models on the verge of simulating clouds + large-scale waves We further assume convective criticality
Implications of the moist adiabatic lapse rate for the structure of tropical disturbances Approximate moist adiabatic condition as that of constant saturation entropy: Lq v * s* = c ln T p Rd ln p T 0 p + 0 T Assume hydrostatic perturbations: φ ' = α ' p
Maxwell s relation: α Integrate: α T ' = s*' = s*' s p * p s* ( ) φ φ b xyt Txyt T s ' = '(,, ) + (,, ) *' Only barotropic and first baroclinic mode survive
This implies, through the linearized momentum equations, e.g. u φ = + t x that the horizontal velocities may be partitioned similarly: fv ( ) u= u(, xyt,) + Txyt (,,) T u*(, xyt,); b ( ) v= v( xyt,, ) + Txyt (,, ) T v*( xyt,, ). b
Implications for vertical structure of vertical velocity Integrate: ω u v = + p x y ( p ) p u 0 b vb u* v* ω = ( p0 p) + ( p0 p) T Tdp' +. x y x y
At tropopause: ω t ( p p ) u = b + 0 t x v b y This implies that if a rigid lid is imposed at the tropopause, the divergence of the barotropic velocities must vanish and the barotropic components therefore satisfy the barotropic vorticity equation: ηb = Vi ηb, t η kˆ i V + 2Ωsinθ b b
Feedback of Air Motion on (virtual) Temperature Convection cannot change vertically integrated enthalpy, k = cpt + Lvq The neglecting surface fluxes, radiation, and horizontal advection, t kdp = h ω dp, p Neelin and Held (1987): This function is negative for upward motion
Upward motion is associated with column moistening: T q c dp= kdp Lv dp t t t p Ascent leads to cooling Yano and Emanuel, 1991: N ( ) 2 = 1 ε N 2 eff p
Prediction: Inviscid, small amplitude perturbations under rigid lid: Shallow water solutions with reduced equivalent depth
β Quasi-Linear Plane System, Neglecting Barotropic Mode u s* = ( T ) s T + β yv ru t x v s* = ( T ) s T β yu rv t y s Γ s Q ( ε ) = + pm w t Γ * d d rad m z sb h = C V s * s M w s s t ( ) ( )( ) k 0 b b m
u v w + + = x y H 0 Quasi-Equilibrium Assumption: sb t = s * t Gives closure for convective mass flux, M System closed except for specification of Q, s *, s, ε rad 0 m p
Additional Approximations: Boundary Layer QE (Raymond, 1995): Neglect s h b, gives simpler expression t for M s * 0 sb M = w+ Ck V sb sm Weak Temperature Approximation (Sobel and Bretherton, 2000): Neglect s * (Over-determined system, ignore momentum equation for irrotational flow) t
Important Feedbacks: Wind-Induced Surface Heat Exchange (WISHE) Coupling of surface enthalpy flux to wind perturbations (Neelin et al. 1987, Emanuel, 1987) Moisture-Convection Feedback: Dependence of sm on and/or ε p on s s Cloud-Radiation Feedback: Dependence of Q rad on M or s * sm M * m
Ocean-Atmosphere Feedback (e.g. ENSO): Feedback between perturbation surface wind and ocean surface temperature, as represented by s * 0
Simple Example: Q rad Ignore perturbations of Ignore fluctuations of ε p Make boundary layer QE approximation Fully linearize surface fluxes: 2 2 V = U + u* V ' = Uu ' V
Introduce scalings: First define a merdional scale, L y : Then let 4 Γ 1 d s ε ( ) d p Ly = Ts T H 2 Γm z β x a x y L y a ack V t t u u β L H 2 y y 2 LC y k V ack V β Ly v v s* s* H H T T ( ) s
Separate scalings for ocean temperature and lower tropospheric entropy: 1 ε s * p s s s ( ) o b m ε p 0 s m 1 ε ε p p 2 ( ) s s b m ( ) s * s* 0 s m
Nondimensional parameters: α R 1 ε ε p ra β L p 2 y ac k U ( s ) 0 * s* ( ) m H V s* s ( WISHE) ( Rayleigh friction) χ 1 ε ac V βl ε p ( s ) 0 * s* 2 p k y H( T ) s T s s ( * ) m 2 ( surface damping) a δ = L y 2 ( zonal geostropy)
Nondimensional Equations: u t s = + yv x Ru v t s = δ yu y Rv s u v = + + αu+ s + s χs 0 m t x y
Steady System with R = s = 0 : m s x s + αy χy 2 s= y 2 s y 0 Similar to Gill Model, but forcing is directly in terms of SST (s 0 ), not latent heating For SST of the form s = RE G( y) e ikx 0 there are solutions of the form s ikx = RE J( y) e,
where 2 2 ik χ y y 1+ ik χu α 2α α 2α J( y) = y e Gu e du 0 Example: 2 G = e by
α = 0, k = 2, b= 1.5, χ = 1.5
α = 1, k = 2, b= 1.5, χ = 1.5
Basic linear wave dynamics on the equatorial β plane Omit damping and WISHE terms from linear nondimensional equations: u t s = + x yv v t s = δ y s u v = + t x y yu
Fully equivalent to the shallow water equations on a β plane Eliminate s and u in favor of v: v v v v t t x y x 2 2 2 2 δ + δyv δ = 0 2 2 2 Let v = V( y) e ikx i t ω + = dy δ ω 2 2 2 dv ω k k 2 y V 0 2
Boundary conditions: V well behaved at y ± Solution in terms of discrete parabolic cylinder functions D n : v= D ( y), n 2 2y 2 where Dn = e 1, 2 y, 4 y 2, provided ω satisfies the dispersion relation ω 2 2 δ k k = 2n + 1 ω
There is, in addition, another mode satisfying v=0 everywhere. From first and third linear equations: 2 2 u u = 0. 2 2 t x Satisfied by u = F x t ( ) Eastward-propagating, nondispersive equatorially trapped Kelvin wave Note that this happens to satisfy derived dispersion relation when n= -1..
There are three roots of the general dispersion relation: n = 0: 2 2 ω k k 1= 0 δ ω Factor : ω k 1 ( ω + k ) = 0 δ ω ω = k root not allowed does not satisfy BCs 1 ω = ± + 2 ( ) 2 k k 4δ ( ) Mixed Rossby-Gravity Waves (MRG)
For n 1, two well defined limits: 1. ω << k : ω k 2n + 1+ k 2. ω >> k : ω 2 2 2 δ k + δ(2n+ 1) Planetary Rossby waves Inertia-gravity waves
Kelvin wave
Mixed Rossby-Gravity
Rossby
Inertia-gravity
Intraseasonal Variability Stochastic excitation of the equatorial waveguide WISHE Moisture-convection feedback Cloud-radiation feedback Ocean interaction
Wind-Induced Surface Heat Exchange (WISHE)
Add back WISHE term to linear undamped equations: u t s = + x yv v s δ = yu t y s u v = + + t x y α u
First look for Kelvin-like modes with v=0: u t s t s x u = + x 2 2 2 2 0 u u u u = + α t 2 x 2 x Let u ω = = α u e ikx iωt α = k i k :
Note: α must be < 0 for ω r > 0 and ω i > 0
As k ω i -α/2
Effect of Stratosphere (Yano and Emanuel, 1991)
Effect of finite convective response time: s 1 u v = + + t 1 ε x y p M eq M t M eq τ c M M ε u v = p + + 1 ε x y p = α u
Go back to dimensional, quasilinear QE equations on β plane
β Quasi-Linear Plane System, Neglecting Barotropic Mode u s* = ( T ) s T + β yv ru t x v s* = ( T ) s T β yu rv t y s Γ s Q ( ε ) = + pm w t Γ * d d rad m z sb h = C V s * s M w s s t ( ) ( )( ) k 0 b b m
u v w + + = x y H 0 Define an equilibrum updraft mass flux from boundary layer QE: M w+ C eq k s 0 * s V b s s b m Relax to equilibrium over a finite time scale: M t and enforce M 0 = M τ eq M convective
Numerical solution of β plane quasi-linear equations Nonlinearity retained only in surface fluxes Zonally symmetric SST specified; also symmetric about equator Background easterly wind of 2 ms -1 imposed Convection relaxed to equilibrium over time scale of 3 hours
Cloud-Radiative Feedback Set OLR proportional to difference in θ e between boundary layer and mid troposphere (Sandrine Bony)
Moisture-Convection Feedback Allow precipitation efficiency to depend on relative humidity Greater heating/upward motion in moister air upward motion moistens air Necessary for tropical cyclones Appears to excite planetary Rossby waves near equator
Possible effects of ocean response