University:
Widener University
Course:
PHYS-121 | Fundamentals of Physics I
Academic year:

2023
Views:

165

Pages:

65
Author:

Mina Lane

Quasi-equilibrium Theory of Small Perturbations to RadiativeConvective Equilibrium States • See “CalTech 2005” paper on 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: Lv q * p ⎛ ⎞ ⎛ ⎞ T s* = c p ln ⎜ − Rd ln ⎜ + ⎟ ⎟ ⎝ T0 ⎠ ⎝ p0 ⎠ T • Assume hydrostatic perturbations: ∂φ ' = −α ' ∂p • Maxwell’s relation: ⎛ ∂T ⎞ ⎛ ∂α ⎞ α'=⎜ ⎟ s *' ⎟ s *' = ⎜ ⎝ ∂s * ⎠ p ⎝ ∂p ⎠ s* • Integrate: φ ' = φb '( x, y, t ) + (T ( x, y, t ) − T ) s *' Only barotropic and first baroclinic mode survive This implies, through the linearized momentum equations, e.g. ∂u ∂φ =− + fv ∂t ∂x that the horizontal velocities may be partitioned similarly: u = ub ( x, y, t ) + (T ( x, y, t ) − T ) u *( x, y, t ); v = vb ( x, y, t ) + (T ( x, y, t ) − T ) v *( x, y, t ). Implications for vertical structure of vertical velocity ⎛ ∂u ∂v ⎞ ∂ω = −⎜ + ⎟ ∂p ⎝ ∂x ∂y ⎠ Integrate: ⎛ ∂ub ∂vb ⎞ + ω = ( p0 − p ) ⎜ ⎟− ⎝ ∂x ∂y ⎠ (( p − p )T − ∫ 0 p0 p ) ⎛ ∂u * ∂v * ⎞ + Tdp ' ⎜ ⎟. ∂y ⎠ ⎝ ∂x At tropopause: ⎛ ∂ub ∂vb ⎞ + ωt = ( p0 − pt ) ⎜ ⎟ ⎝ ∂x ∂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 = −V i∇ηb , ∂t η ≡ kˆi∇ × V + 2Ω sin θ b b Feedback of Air Motion on (virtual) Temperature • Convection cannot change vertically integrated enthalpy, k = c pT + Lv q • The neglecting surface fluxes, radiation, and horizontal advection, ∂ ∂h k dp = − ∫ ω dp, ∫ ∂t ∂p • Neelin and Held (1987): This function is negative for upward motion • Upward motion is associated with column moistening: ∂T ∂ ∂q ∫ c p ∂t dp = ∂t ∫ k dp − ∫ Lv ∂t dp Ascent leads to cooling • Yano and Emanuel, 1991: N 2 eff = (1 − ε p ) N 2 Prediction: Inviscid, small amplitude perturbations under rigid lid: Shallow water solutions with reduced equivalent depth Quasi-Linear β Plane System , Neglecting Barotropic Mode ∂u ∂s * = (Ts − T ) + β yv − ru ∂t ∂x ∂v ∂s * = (Ts − T ) − β yu − rv ∂t ∂y ∂sd ∂s * Γ d ⎛ & ⎞ ε p M − w) ⎟ = ( ⎜ Qrad + ∂t Γm ⎝ ∂z ⎠ ∂sb h = Ck | V | ( s0 * − sb ) − ( M − w )( sb − sm ) ∂t ∂u ∂v w + + =0 ∂x ∂y H Quasi-Equilibrium Assumption: ∂sb ∂s * = ∂t ∂t Gives closure for convective mass flux, M System closed except for specification of & Qrad , s0 *, sm , ε p Additional Approximations: • Boundary Layer QE (Raymond, 1995): Neglect h ∂sb , gives simpler expression ∂t for M s0 * − sb M = w + Ck | V | sb − sm • Weak Temperature Approximation (Sobel and Bretherton, 2000): Neglect ∂s * ∂t (Over-determined system, ignore momentum equation for irrotational flow) 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 M and/or ε p on s * − sm • Cloud-Radiation Feedback: Dependence of Q& rad on M or s * − sm • Ocean-Atmosphere Feedback (e.g. ENSO): Feedback between perturbation surface wind and ocean surface temperature, as represented by s0 * Simple Example: • • • • Ignore perturbations of Q& rad Ignore fluctuations of ε p Make boundary layer QE approximation Fully linearize surface fluxes: | V | = U +u* 2 Uu ' |V|'= |V| 2 Introduce scalings: First define a merdional scale, Ly : Γd ∂sd 1 − ε p Ly = Ts − T ) H ( Γm ∂z β 2 4 Then let x→a x y → Ly y a t→ t 2 β Ly aCk | V | u→ u H v→ L y Ck | V | H v s* → aCk | V | β Ly H (Ts − T ) 2 s* Separate scalings for ocean temperature and lower tropospheric entropy: so * → sm → 1− ε p εp (s b ( ) s ( s * − s *) 1 − ε p sb − sm εp ) − sm s0 2 m 0 Nondimensional parameters: ( ( ) ) 1 − ε p aCk U s0 * − s * α≡ ε p H | V | s * − sm ra R≡ β Ly 2 χ≡ ( Rayleigh friction) ( s * − s *) H (T − T ) ( s * − s ) 1 − ε p aCk | V | β Ly εp ⎛ a δ =⎜ ⎜L ⎝ y (WISHE ) 2 0 2 s ⎞ ⎟⎟ ⎠ m 2 ( zonal geostropy ) ( surface damping ) Nondimensional Equations: ∂u ∂s = + yv − Ru ∂t ∂x ⎛ ∂s ⎞ ∂v v⎟ = δ ⎜ − yu − R ∂t ⎝ ∂y ⎠ ∂s ∂u ∂v = + + α u + s0 + sm − χ s ∂t ∂x ∂y Steady System with R = sm = 0 : ∂s ∂s 2 2 + α y − χ y s = − y s0 ∂x ∂y Similar to Gill Model, but forcing is directly in terms of SST (s0), not latent heating For SST of the form s0 = RE ⎡⎣G ( y )e ⎤⎦ ikx there are solutions of the form ikx ⎡ s = RE ⎣ J ( y )e ⎤⎦ , where J ( y) = y −ik α e χ y2 2α ∫ y 0 Gu Example: G=e − by 2 1+ ik α e 2 χ u − 2α du 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 ∂s = + yv ∂t ∂x ⎛ ∂s ⎞ ∂v = δ ⎜ − yu ⎟ ∂t ⎝ ∂y ⎠ ∂s ∂u ∂v = + ∂t ∂x ∂y Fully equivalent to the shallow water equations on a β plane Eliminate s and u in favor of v: ∂ ⎧∂ v ∂ v ∂v ∂v 2 ⎫ =0 ⎨ 2 − 2 − δ 2 + δ y v⎬ − δ ∂t ⎩ ∂t ∂x ∂y ∂x ⎭ 2 Let 2 v = V ( y )e 2 ikx −iωt 2 2 ⎛ dV k ω −k 2⎞ → 2 +⎜ − − y ⎟V = 0 dy ω ⎝ δ ⎠ 2 Boundary conditions: V well behaved at y → ±∞ Solution in terms of discrete parabolic cylinder functions Dn : v = Dn ( y ), where Dn = e −2 y 2 ⎡⎣1, 2 y, 4 y 2 − 2,K⎤⎦ provided ω satisfies the dispersion relation ω −k k − = 2n + 1 δ ω 2 2 There is, in addition, another mode satisfying v=0 everywhere. From first and third linear equations: ∂ 2u ∂ 2u − 2 = 0. 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: ω2 − k2 k − −1 = 0 n = 0: δ ω ⎛ω −k 1 ⎞ − ⎟ (ω + k ) = 0 Factor : ⎜ ω⎠ ⎝ δ ω = − k root not allowed ( does not satisfy BCs ) ( 1 ω = k ± k 2 + 4δ 2 ) Mixed Rossby-Gravity Waves (MRG) For n ≥ 1, two well defined limits: 1. | ω |<<| k |: ω≅− k 2n + 1 + k 2 δ 2. | ω |>>| k |: ω ≅ k + δ (2n + 1) 2 2 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 ∂s = + yv ∂t ∂x ⎛ ∂s ⎞ ∂v = δ ⎜ − yu ⎟ ∂t ⎝ ∂y ⎠ ∂s ∂u ∂v = + + αu ∂t ∂x ∂y First look for Kelvin-like modes with v=0: ∂u ∂s = ∂t ∂x ∂s ∂u = + αu ∂t ∂x 2 2 ∂u ∂u ∂u → 2 = 2 +α ∂t ∂x ∂x ikx −iωt Let u = u0 e : ω = k − iα k 2 2 Note: α must be < 0 for ωr > 0 and ωi > 0 As k → ∞ ωi → -α/2 Effect of Stratosphere (Yano and Emanuel, 1991) Effect of finite convective response time: ∂s 1 = ∂t 1 − ε p ⎛ ∂u ∂v ⎞ ⎜ + ⎟+M ⎝ ∂x ∂y ⎠ −ε p ⎛ ∂u ∂v ⎞ M eq = ⎜ + ⎟ + αu 1 − ε p ⎝ ∂x ∂y ⎠ ∂M M eq − M = ∂t τc Go back to dimensional, quasilinear QE equations on β plane” Quasi-Linear β Plane System , Neglecting Barotropic Mode ∂u ∂s * = (Ts − T ) + β yv − ru ∂t ∂x ∂v ∂s * = (Ts − T ) − β yu − rv ∂t ∂y ∂sd ∂s * Γ d ⎛ & ⎞ ε p M − w) ⎟ = ( ⎜ Qrad + ∂t Γm ⎝ ∂z ⎠ ∂sb h = Ck | V | ( s0 * − sb ) − ( M − w )( sb − sm ) ∂t ∂u ∂v w + + =0 ∂x ∂y H Define an equilibrum updraft mass flux from boundary layer QE: s0 * − sb M eq ≡ w + Ck | V | sb − sm Relax to equilibrium over a finite time scale: ∂M M eq − M = ∂t τ convective and enforce M ≥0 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 θe 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