Quasi-equilibrium Theory of Small Perturbations to Radiative-

Convective 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:

• Assume hydrostatic perturbations:0 0

** ln ln vp d

L qpTs c RT p T⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

' 'pφ α∂

= −∂

• Maxwell’s relation:

• Integrate:

*

' * ' * '* p s

Ts ss pαα

⎛ ⎞∂ ∂⎛ ⎞= = ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

( )' '( , , ) ( , , ) * 'b x y t T x y t T sφ φ= + −

Only barotropic and first baroclinic mode survive

This implies, through the linearized momentum equations, e.g.

u fvt x

φ∂ ∂= − +

∂ ∂

that the horizontal velocities may be partitioned similarly:

( )( )

( , , ) ( , , ) *( , , );

( , , ) ( , , ) *( , , ).b

b

u u x y t T x y t T u x y t

v v x y t T x y t T v x y t

= + −

= + −

Implications for vertical structure of vertical velocity

u vp x yω ⎛ ⎞∂ ∂ ∂= − +⎜ ⎟∂ ∂ ∂⎝ ⎠

Integrate:

( ) ( )( )0

0 0* *' .

pb bp

u v u vp p p p T Tdpx y x y

ω⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂

= − + − − − +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠∫

At tropopause:

( )0b b

t tu vp px y

ω⎛ ⎞∂ ∂

= − +⎜ ⎟∂ ∂⎝ ⎠This implies that if a rigid lid is imposed at the tropopause, the divergence of the barotropicvelocities must vanish and the barotropiccomponents therefore satisfy the barotropicvorticity equation:

,

ˆ 2 sin

bb

b b

tk

η η

η θ

∂= − ∇

∂≡ ∇× + Ω

V

V

i

i

Feedback of Air Motion on (virtual) Temperature

• Convection cannot change vertically integrated enthalpy,

• The neglecting surface fluxes, radiation, and horizontal advection,

• Neelin and Held (1987): This function is negative for upward motion

p vk c T L q= +

,hkdp dpt p

ω∂ ∂= −

∂ ∂∫ ∫

• Upward motion is associated with column moistening:

• Yano and Emanuel, 1991:

p vT qc dp kdp L dpt t t

∂ ∂ ∂= −

∂ ∂ ∂∫ ∫ ∫

Ascent leads to cooling

( )2 21eff pN Nε= −

Prediction: Inviscid, small amplitude perturbations under rigid lid: Shallow water solutions with reduced equivalent depth

Quasi-Linear Plane System , Neglecting Barotropic Mode

β

( ) *s

u sT T yv rut x

β∂ ∂= − + −

∂ ∂

( ) *s

v sT T yu rvt y

β∂ ∂= − − −

∂ ∂

( )* d drad p

m

ss Q M wt z

εΓ ∂∂ ⎛ ⎞= + −⎜ ⎟∂ Γ ∂⎝ ⎠&

( ) ( )( )0| | *bk b b m

sh C s s M w s st

∂= − − − −

∂V

0u v wx y H∂ ∂

+ + =∂ ∂

Quasi-Equilibrium Assumption:

*bs st t

∂ ∂=

∂ ∂Gives closure for convective mass flux, M

System closed except for specification of

0, *, ,rad m pQ s s ε&

Additional Approximations:

• Boundary Layer QE (Raymond, 1995): Neglect , gives simpler expression for

• Weak Temperature Approximation(Sobel and Bretherton, 2000): Neglect (Over-determined system, ignore momentum equation for irrotational flow)

bsh t∂

∂M

*st

∂∂

0 *| | bk

b m

s sM w Cs s

−= +

−V

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 on and/or on

• Cloud-Radiation Feedback: Dependence of on or

ms pεM * ms s−

radQ& M * ms s−

• Ocean-Atmosphere Feedback (e.g. ENSO): Feedback between perturbation surface wind and ocean surface temperature, as represented by 0 *s

Simple Example:• Ignore perturbations of• Ignore fluctuations of • Make boundary layer QE approximation • Fully linearize surface fluxes:

radQ&pε

2 2| | *U u= +V

'| | '| |Uu

=VV

Introduce scalings:

:yLFirst define a merdional scale,

( )42

1 pd dy s

m

sL T T Hz

εβ−Γ ∂

= −Γ ∂

Then let

( )

2

2

| |

| | | |* *

y

k

y

y k k y

s

x a x y L y

aCat t u uL H

L C aC Lv v s s

H H T T

β

β

→ →

→ →

→ →−

V

V V

Separate scalings for ocean temperature and lower tropospheric entropy:

( )

( )( )

0

2

0

1*

1

* *

po b m

p

b mpm m

p

s s s s

s ss s

s s

εε

εε

−→ −

−−→

−

Nondimensional parameters:

( )( )

( )( )( )

0

2

20

2

2

* *1( )

| | *

( )

* *1 | |( )

*

( )

p k

p m

y

p k y

p s m

y

s saC U WISHEH s s

ra Rayleigh frictionL

s saC Lsurface damping

H T T s s

a zonal geostropyL

εα

ε

β

ε βχ

ε

δ

−−≡

−

≡

−−≡

− −

⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠

V

V

R

Nondimensional Equations:

u s yv ut x

∂ ∂= + −

∂ ∂R

v s yu vt y

δ⎛ ⎞∂ ∂

= − −⎜ ⎟∂ ∂⎝ ⎠R

0 ms u v u s s st x y

α χ∂ ∂ ∂= + + + + −

∂ ∂ ∂

Steady System with : 0ms= =R2 2

0s sy y s y sx y

α χ∂ ∂+ − = −

∂ ∂

Similar to Gill Model, but forcing is directly in terms of SST (s0), not latent heating

For SST of the form

0 ( ) ikxs G y e⎡ ⎤= ⎣ ⎦REthere are solutions of the form

( ) ,ikxs J y e⎡ ⎤= ⎣ ⎦RE

where

2 212 2

0( )

y uik ikyJ y y e Gu e du

χ χα α α α− + −= ∫

Example:

2byG e−=

0, 2, 1.5, 1.5k bα χ= = = =

1, 2, 1.5, 1.5k bα χ= − = = =

Basic linear wave dynamics on the equatorial β plane

Omit damping and WISHE terms from linear nondimensional equations:

u s yvt x

∂ ∂= +

∂ ∂

v s yut y

δ⎛ ⎞∂ ∂

= −⎜ ⎟∂ ∂⎝ ⎠s u vt x y∂ ∂ ∂

= +∂ ∂ ∂

Fully equivalent to the shallow water equations on a β plane

Eliminate s and u in favor of v:

2 2 22

2 2 2 0v v v vy vt t x y x

δ δ δ⎧ ⎫∂ ∂ ∂ ∂ ∂

− − + − =⎨ ⎬∂ ∂ ∂ ∂ ∂⎩ ⎭

( ) ikx i tv V y e ω−=Let

2 2 22

2 0d V k k y Vdy

ωδ ω

⎛ ⎞−→ + − − =⎜ ⎟

⎝ ⎠

y→±∞Boundary conditions: V well behaved at

:nDSolution in terms of discrete parabolic cylinder functions

22 2

( ),

1, 2 , 4 2,n

yn

v D y

where D e y y−

=

⎡ ⎤= −⎣ ⎦K

provided ω satisfies the dispersion relation

2 2

2 1k k nωδ ω−

− = +

There is, in addition, another mode satisfying v=0 everywhere. From first and third linear equations:

( )

2 2

2 2 0.

.

u ut xSatisfied by u F x t

∂ ∂− =

∂ ∂= −

Eastward-propagating, nondispersiveequatorially trapped Kelvin wave

Note that this happens to satisfy derived dispersion relation when n= -1.

There are three roots of the general dispersion relation:

( )

( )

( )

2 2

2

0 : 1 0

1: 0

1 42

k kn

kFactor k

k root not allowed does not satisfy BCs

k k

ωδ ωω ωδ ω

ω

ω δ

−= − − =

−⎛ ⎞− + =⎜ ⎟⎝ ⎠

= −

= ± +

Mixed Rossby-Gravity Waves (MRG)

For n ≥ 1, two well defined limits:

2

1. | | | |:

2 1

kkkn

ω

ωδ

<<

≅ −+ +

Planetary Rossby waves

2 2

2. | | | |:(2 1)

kk n

ω

ω δ

>>

≅ + +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 yvt x

∂ ∂= +

∂ ∂

v s yut y

δ⎛ ⎞∂ ∂

= −⎜ ⎟∂ ∂⎝ ⎠s u v ut x y

α∂ ∂ ∂= + +

∂ ∂ ∂

First look for Kelvin-like modes with v=0:

2 2

2 2

02 2

:ikx i t

u st xs u ut xu u ut x x

Let u u e

k i k

ω

α

α

ω α

−

∂ ∂=

∂ ∂∂ ∂

= +∂ ∂

∂ ∂ ∂→ = +

∂ ∂ ∂=

= −

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:

11

1

p

peq

p

eq

c

s u v Mt x y

u vM ux y

M MMt

ε

εα

ε

τ

⎛ ⎞∂ ∂ ∂= + +⎜ ⎟∂ − ∂ ∂⎝ ⎠

− ⎛ ⎞∂ ∂= + +⎜ ⎟− ∂ ∂⎝ ⎠

−∂=

∂

Go back to dimensional, quasi-linear QE equations on β plane”

Quasi-Linear Plane System , Neglecting Barotropic Mode

β

( ) *s

u sT T yv rut x

β∂ ∂= − + −

∂ ∂

( ) *s

v sT T yu rvt y

β∂ ∂= − − −

∂ ∂

( )* d drad p

m

ss Q M wt z

εΓ ∂∂ ⎛ ⎞= + −⎜ ⎟∂ Γ ∂⎝ ⎠&

( ) ( )( )0| | *bk b b m

sh C s s M w s st

∂= − − − −

∂V

0u v wx y H∂ ∂

+ + =∂ ∂

Define an equilibrum updraft mass flux from boundary layer QE:

0 *| | beq k

b m

s sM w Cs s

−≡ +

−V

Relax to equilibrium over a finite time scale:

eq

convective

M MMt τ

−∂=

∂

0M ≥and enforce

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