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 poolControl 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 forbidsWISHE acting in mean westerlies (as exist in warm pool)

But in mean westerlies, WISHE will tend to induce westward propagation, because strongest winds to westof 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 ofconvective 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 ofconvective 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 variableis column-integrated water vapor.2.Quasi-equilibrium convective physics, simple cloud-radiative feedbacks. (This allows a representation ofconvective self-aggregation as in CRMs). Gross moist stabilityis constant (or parameterized…) 3.Add a horizontal dimension (longitude). Assume wind is related diagnostically and instantaneously to heating, e.g. by simple Gill-type dynamics4. 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 weaktemperature gradient balance. Zonal mean precip is part ofthe solution. Implicitly there is a Hadley cell.

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

Most unstable wavelength is ~decay length scalefor stationary response to heating L (c/ε, where ε isdamping 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 ofglobal 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 moistureand dry static energy, under WTG approximation

± is upper tropospheric divergence. Add to getmoist 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 hidemuch implicit vertical structure.We do explicitly parameterize at this point

R = max(R0-rP, 0) with R0, r constants.

Substituting into the MSE equation and expanding the total derivative, (for sake of argument assuming rP<R0)

u is the zonal wind at a a nominal steering level for W, presumablylower-tropospheric.

“effective” NGMS (including cloud-radiative feedback)

We parameterize precipitation on saturation fraction by an exponential (Bretherton et al. 2004):

(with e.g., ad=15.6, rd=0.603), and R is the saturation fraction,R=W/W*. Here W*, the saturation column water vapor, is assumedconstant 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 structureand so the most important (implicit) assumptions about verticalstructure are buried here.

Rather than use a bulk formula for E, we go directlyto the simulations of Maloney et al. A scatter plot of E vs. U850 in the model warm pool yields the parameterization

E = 100 + 7.5u

With E in W/m2 and u in m/s.Note there is no dependence onW or SST. In practice it assures that simple model does not have very different wind-evaporation feedback than the GCM.

Intraseasonal rain variance

NorthernSummer

SouthernSummer

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, exceptthat the mean doesn’t have localized minima over land

Intraseasonal OLR variance, nov-apr

Climatological mean OLR, nov-apr

Climatological patterns resemble variance, exceptthat the mean doesn’t have localized minima over land

Wave propagation

Mean flowPerturbation flow

Enhanced sfc flux

Emanuel (87) and Neelin et al. (87) proposed that the MJOis a Kelvin wave driven by wind-induced surface fluxes (“WISHE”)

θ=θ1+Δθ

θ=θ1

cool warm

Disturbance propagation (via horizontal advection…)

Mean flowPerturbation flow(partly rotational)

Enhanced sfc flux

Instead we propose a moisture mode driven by surface flux feedbacks

θ=θ1+Δθ

θ=θ1

Warm

Mean + perturbation flow

humid dry