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
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