An idealized semi-empirical framework for modeling the MJO

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Nov 16, 2013 (3 years and 7 months ago)

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An idealized semi
-
empirical framework
for modeling the MJO


Adam Sobel and Eric Maloney



NE Tropical Workshop, May 17 2011

A set of postulates about MJO dynamics


Not a Kelvin wave (though may have something to do with
Kelvin waves)


A moisture mode


meaning moisture field is critical


Destabilized by feedbacks involving surface turbulent fluxes and
radiative fluxes


Horizontal moisture advection important to the dynamics (wind
speeds ≥ propagation speed)

A set of postulates about MJO dynamics


Not a Kelvin wave (though may have something to do with
Kelvin waves)


A moisture mode


meaning moisture field is critical


Destabilized by feedbacks involving surface turbulent fluxes and
radiative fluxes


Horizontal moisture advection important to the dynamics (wind
speeds ≥ propagation speed)

These postulates are supported by a lot of evidence

from observations and comprehensive numerical models.

E.g…

Aquaplanet GCM simulation with warm pool




Control

No
-
WISHE


WISHE appears to destabilize the MJO in the model. 30
-
90 day, zonal
wavenumber 1
-
3 variance decreases dramatically without WISHE active


Horizontal moisture advection plays large role in propagation (not shown)

A convectively coupled Kelvin wave can only be

destabilized by WISHE if mean surface winds are

easterly (Emanuel 1987; Neelin et al. 1987)


If the MJO is
not
a Kelvin wave, then no theory forbids

WISHE acting in mean westerlies (as exist in warm pool)


But in mean westerlies, WISHE will tend to induce

westward propagation, because strongest winds to west

of convection.


How might an eastward
-
propagating moisture mode,

destabilized by WISHE in mean westerlies, work?

Our approach, conceptually:


1.
Start from a single
-
column model under the weak

temperature gradient approximation.

2.
Quasi
-
equilibrium convective physics, simple

cloud
-
radiative

feedbacks. (This allows a representation of

convective self
-
aggregation as in CRMs)

Bretherton et al. 2005

Our approach, conceptually:


1.
Start from a single
-
column model under the weak

temperature gradient approximation.

2.
Quasi
-
equilibrium convective physics, simple

cloud
-
radiative

feedbacks. (This allows a representation of

convective self
-
aggregation as in CRMs).

3.
Add a horizontal dimension (longitude). Assume wind is

related diagnostically to heating by simple Gill
-
type dynamics

(or something close to that):

Gill (1980) wind and geopotential for localized heating (at 0,0)

linear, damped, steady dynamics on equatorial beta plane

Zonal wind response

to delta function heating

Zonal wind response (red)

to sinusoidal heating (blue)

Our approach, conceptually:


1.
Start from a single
-
column model under the weak

temperature gradient approximation. Only prognostic variable

is column
-
integrated water vapor.

2.
Quasi
-
equilibrium convective physics, simple

cloud
-
radiative feedbacks. (This allows a representation of

convective self
-
aggregation as in CRMs). Gross moist stability

is constant (or parameterized…)

3.
Add a horizontal dimension (longitude). Assume wind is

related
diagnostically and instantaneously to

heating, e.g.

by simple Gill
-
type dynamics

4. Allow wind to advect moisture, and influence surface fluxes.

in results shown here E depends on u only

Compute u from a projection operator:

L depends on equivalent depth and damping rate.


We cheat sometimes
and shift
G

relative to forcing by a small amount, δ

-

ascribed to missing processes in Gill model (CMT, nonlinearity…)

Using Gill dynamics (almost):

Model is 1D, represents a longitude line at a single latitude,

where the MJO is active.


But we do
not

assume that the divergence =

u/

x.

(there is implicit meridional structure,

v/

y ≠ 0)


Relatedly, the mean state is not assumed to be in

radiative
-
convective equilibrium. Rather it is in weak

temperature gradient balance. Zonal mean precip is part of

the solution. Implicitly there is a Hadley cell.

All linear modes are unstable due to WISHE, but westward
-

propagating

Most unstable wavelength is ~decay length scale

for stationary response to heating L (c/ε, where ε is

damping rate; here 1500 km)

Nonlinear model configuration details


1D domain 40,000 km long, periodic boundaries


Background state is uniform zonal flow


eastward at
5 m/s
; perturbation flow is added to it for advection
and surface fluxes.


In simulations shown below normalized gross moist
stability = 0.1; cloud radiative feedback = 0.1;
saturation column water vapor =70 mm; these factors
largely control stability;


Nonlinear behavior is or is not qualitatively similar to

linear, depending on wind response shift δ

Saturation fraction vs. longitude and time, for different values of δ

With a small adjustment to the wind response to heating

(westerlies a little further east) we get very nonlinear behavior

Perturbation zonal wind; total is that plus mean 5m/s.

Relative strength of easterlies and westerlies is tunable.

This semi
-
empirical model is not a satisfactory theory for the MJO,

yet. It is a framework within which the consequences of several

ideas can be explored.


Key parameters:



The gross moist stability


Cloud
-
radiative feedback


Mean state


zonal wind and mean rainfall/divergence


The quasi
-
steady wind response to a delta function heating (G)




very sensitive to small longitudinal shifts!



These can all
-

in principle
-

be derived from/tuned to diagnostics of

global models.


We see that very nonlinear behavior can emerge (and can go eastward).


Working on: mixed layer ocean coupling, variable gross moist stability…


Vertically integrated equations for moisture

and dry static energy, under WTG approximation

±

is upper tropospheric divergence. Add to get

moist static energy equation

Substitute to get

where

is the

normalized gross moist stability


Our physics is semi
-
empirical:

The functional forms chosen are key components of the model
-

and hide

much implicit vertical structure.

We do explicitly parameterize at this point




R = max(R
0
-
rP, 0) with R
0
, r constants.


Substituting into the MSE equation and expanding the total derivative,

(for sake of argument assuming rP<R
0
)


u

is the zonal wind at a a nominal steering level for W, presumably

lower
-
tropospheric.


effective


NGMS (including cloud
-
radiative feedback)

We parameterize precipitation on saturation fraction

by an exponential (Bretherton et al. 2004):

(with e.g., a
d
=15.6, r
d
=0.603), and R is the saturation fraction,

R=W/W*. Here W*, the saturation column water vapor, is assumed

constant as per WTG.


We represent the normalized GMS either as a constant or as a

specified function of W. NGMS is very sensitive to vertical structure

and so the most important (implicit) assumptions about vertical

structure are buried here.


Rather than use a bulk formula for E, we go directly

to the simulations of Maloney et al. A scatter plot of

E vs. U
850

in the model warm pool yields the parameterization

E = 100 + 7.5u

With E in W/m
2

and u in m/s.

Note there is no dependence on

W or SST. In practice it assures that simple

model does not have very different

wind
-
evaporation feedback than the GCM.



Intraseasonal rain variance

Northern

Summer

Southern

Summer

Variance of rainfall on intraseasonal timescales shows
structure on both global and regional scales

Sobel, Maloney, Bellon, and Frierson 2008:
Nature Geosci.,

1
, 653
-
657.

Intraseasonal OLR variance (may
-
oct)

Climatological mean OLR (may
-
oct)

Climatological patterns resemble variance, except

that the mean doesn

t have localized minima over land

Intraseasonal OLR variance, nov
-
apr

Climatological mean OLR, nov
-
apr

Climatological patterns resemble variance, except

that the mean doesn

t have localized minima over land

Wave propagation

Mean flow

Perturbation flow

Enhanced

sfc flux

Emanuel (
87
) and Neelin et al. (
87
) proposed that the MJO

is a Kelvin wave driven by wind
-
induced surface fluxes

(

WISHE

)


θ=θ
1
+Δθ

θ=θ
1

cool

warm

Disturbance propagation (via horizontal advection…)

Mean flow

Perturbation flow

(partly rotational)

Enhanced

sfc flux

Instead we propose a moisture mode driven by surface

flux feedbacks

θ=θ
1
+Δθ

θ=θ
1

Warm

Mean + perturbation flow

humid

dry