Tropical wave theory, 1
Imagine the atmosphere has a rigid lid at the tropopause, z = H. (This is
not true, but for some purposes is an adequate approximation.) Then
there will be vertical modes. For simplicity assume a Boussinesq
atmosphere. Since w = 0 at surface and lid, the modes must look like (forconstant stratification)nπzw ∼ sin().
H
Since the temperature equation (linearized) looks like∂T
+ wN 2 = Qc + QR ,∂t
T must have the same vertical structure as w , peaking in mid-troposphere.
By mass conservation then we will have u, v ∼ cos( nπzH ). For dry
equations, convective heating is a ”forcing” on the right-hand side.
Observations often show a single peak in Qc as a function of height, so we
expect n = 1 to be important. However, stratiform heating will also drive
an n = 2 component.
Adam Sobel (Columbia)

Lecture 4

December 21, 201324 / 71

Tropical wave theory, 2

If Qc is given, then we have a problem in dry dynamics to solve for the
response. There is a large literature on this problem. We particularly study
the linear, Boussinesq problem on the equatorial β-plane, where the
Coriolis parameter is approximated by the first term in a Taylor expansion
around the equator:df= βy ,
f =y
dy
where

2Ω cos λ,ay is distance from the equator (positive north), λ latitude in radians, a theradius of the earth.
β=Adam Sobel (Columbia)

Lecture 4

December 21, 201325 / 71

Tropical wave theory, 3
Having assumed a single vertical mode, some imposed heating Q(x, y , t),and linear damping, we can write the equations as∂u∂p− βyv = −
− u
∂t
∂x
∂p
∂v
+ βyu = −
− v
∂t
∂x
∂p ∂u ∂v++
= Q − T p∂t∂x∂y
The last equation results from a combination of hydrostatic balance, massconservation, and temperature equations. These are isomorphic to the(linear) shallow water equations on the equatorial β plane with a mass
source and linear damping on mass and momentum.
Here T can be thought of as radiative damping. These have beennondimensionalized by free gravity wave speed c p= NH/π for velocity,
length by equatorial
deformation
radiusL=c/β, time byβ√Lβ /c = 1/ βc.
Adam Sobel (Columbia)

Lecture 4

December 21, 201326 / 71

Tropical wave theory, 4

We can first study the linear modes of the unforced, undamped
system (Matsuno, 1966):∂u∂p− βyv = −
∂t
∂x
∂v
∂p
+ βyu = −∂t∂x∂p ∂u ∂v
+
+
=0
∂t
∂x
∂y

Adam Sobel (Columbia)

Lecture 4

December 21, 201327 / 71

Tropical wave theory, 5Kiladis et al., Rev. Geopys., 2009 (following Matsuno, 1966, J. Meteor.
Soc.Adam
Japan).
Sobel (Columbia)
Lecture 4
December 21, 201328 / 71

Tropical wave theory, 6

Kiladis et al., Rev. Geopys., 2009 (following Matsuno, 1966, J. Meteor.
Soc. Japan).

Adam Sobel (Columbia)

Lecture 4

December 21, 201329 / 71

Tropical wave theory, 7Kiladis et al., Rev. Geopys., 2009 (following Matsuno, 1966, J. Meteor.
Soc. Japan).

Adam Sobel (Columbia)

Lecture 4

December 21, 201330 / 71

Tropical wave theory, 8Kiladis et al., Rev. Geopys., 2009 (following Matsuno, 1966, J. Meteor.
Soc. Japan).

Adam Sobel (Columbia)

Lecture 4

December 21, 201331 / 71

Slides on observed convectively coupled waves here.

Adam Sobel (Columbia)

Lecture 4

December 21, 201332 / 71

Convectively coupled equatorial
waves
Figures from Wheeler and Kiladis (1999) Wheeler et al.
(2000), Sobel et al. (2004), Lau and Lau (1990), Reed
and Recker (1971)
Adam Sobel, Columbia
FDEPS, Kyoto

Wave number – frequency power spectra of OLR, 15S-15N, equatorially symmetricand antisymmetric components
antisymmetricsymmetric

“Background spectrum” – obtained by taking total spectrum (sym + anti-sym)and smoothing repeatedlySym & anti-sym spectra divided by background. Equivalent depths ~10-50 m means
phase speeds 10-25 m/s. Dry waves would have 50 m/s.

Spectral bands used to reconstruct dynamical signals associated with each wave typeComposite Kelvin wave: OLR, 200 hPa geopotential and wind obtained byregression against time series of Kelvin-filtered OLR at base point 0N, 90E

Theoretical Kelvin height & flow field from shallow water eqns

Kelvin vertical velocity, temperature, vertical & zonal wind wrt longitude & heightComposite n=1 equatorial Rossby wave, OLR and 850 hPa stream fn & wind

Theoretical n=1 Rossby

MRG OLR, stream fn, wind at 3 levels

Theoretical MRG

Easterly/TD-type waves are a mainly northern summer phenomenon:
wavenumber-frequency OLR spectrum for June-August (Kiladis et al. 2006)

Longitude-height composites of western Pacificeasterly waves (Reed and Recker 1971)

Meridional windTime mean zonal
Wind for 3 stations

Relative humidity

Horizontal divergenceand omega
Relative vorticity

Temperature perturbations very small compared to heating – heating is balanced
by vertical advection of potential temperature (just as in climatology)
Diabatic heating Q1
Temperature perturbations

Moisture budget: precipitation variations mostly balanced by moisture convergence(also as in climatology)

Some individual events, from KWAJEX field experiment, Kwajalein, Republic of
Marshall Islands, Jul-Sep 1999

Slides on observed convectively coupled waves here.

Adam Sobel (Columbia)

Lecture 4

December 21, 201332 / 71

Tropical wave theory, 9Instead of free waves, we can study the steady, forced-dissipative
problem in the dry system, with convective heating treated as external(Webster 1972; Gill 1980)
∂p
− u
∂x
∂p
− v
βyu = −∂x∂u ∂v+
= Q − T p∂x∂y−βyv = −Q can have any spatial structure, but we consider a zonally andmeridionally localized forcing centered on the equator. For generaity wehave allowed momentum and thermal damping coefficients  and T to bedifferent (Gill did not).
Adam Sobel (Columbia)

Lecture 4

December 21, 201333 / 71

Tropical wave theory, 10(a) Vertical velocity (contour) & low-level horizontal velocity; (b)horizontal velocity and pressure; (c) circulation in longitude-height plane.

Gill (1980, Q. J. Royal Met. Soc.).
Adam Sobel (Columbia)

Lecture 4

December 21, 201334 / 71Interaction of convection with large-scale waves -1
We are particularly interested in the way in which interactive convection— Qc a function of the resolved prognostic variables — changes the
dynamics.
The simplest case to consider is a gravity wave in two dimensions. We
imagine that there is no vertical propagation — not really true, but
adequate for qualitative discussion. (Vertical propagation will cause”leakage” of wave energy to the stratosphere.)
Assume we have a boundary layer whose thermodynamic properties are
fixed, and a troposphere with stable stratification (to dry displacements),
N > 0. Ascent cools the troposphere, increasing CAPE; descent warms,
decreasing CAPE.

Adam Sobel (Columbia)

Lecture 4

December 21, 201335 / 71Interaction of convection with large-scale waves -2

Now assume that the wave is acting against a background state in which
precipitation is constantly occurring (in a statistical sense); could be RCE(but doesn’t have to be).
If we assume the convection is in instantaneous statistical equilibrium withthe large-scale dynamical changes induced by the wave — e.g. moist
convective adjustment — convection will act to oppose the changes
in CAPE. It will heat the troposphere when there is ascent (and cool the
PBL by downdrafts), cool the troposphere when there is descent.

Adam Sobel (Columbia)

Lecture 4

December 21, 201336 / 71Interaction of convection with large-scale waves -2.5
If we assume that the convective changes do not exceed the changes due
to large-scale ascent/descent — which makes sense if they are responses
to it — then the effect, if it is instantaneous, is to reduce the effective
stratification (e.g., Emanuel, Neelin and Bretherton, 1994, QJRMS).
This is one theory for the observed fact that convectively coupledwaves have slower phase speeds than their dry counterparts with
first baroclinic mode structure.
Another plausible argument is that real convectively coupled waves are notpure first baroclinic mode, but have a second baroclinic mode component
because of stratiform heating. The second mode has smaller phase speed
even in dry theory, and may control the phase speed of the combinedmode (see papers by Mapes, Kuang, Majda and colleagues).

Adam Sobel (Columbia)

Lecture 4

December 21, 201337 / 71Interaction of convection with large-scale waves -3
Simulated convectively coupled gravity wave (Sobel and Bretherton 2003,
Tellus). y -axis is σ = p/p0 . Phase speed 30 m/s, vs. dry 50.

Adam Sobel (Columbia)

Lecture 4

December 21, 201338 / 71Interaction of convection with large-scale waves -4

Actually, the convection takes a finite time to respond to the convection.
As a result the convective heating has some overlap with the cold
temperature anomaly, resulting in moist convective damping:
Adam Sobel (Columbia)

Lecture 4

December 21, 201339 / 71Interaction of convection with large-scale waves -4

If we have∂T
+ wN 2 = Q∂tmultiply by T and average over wave phase
1 ∂ 2
T + N 2 wT = QT
2 ∂t
to first order w and T are in quadrature (exactly so for Q = 0) but
shifting Q to be partly out of phase with T will make QT < 0, thus
damping temperature variance.

Adam Sobel (Columbia)

Lecture 4

December 21, 201340 / 71Interaction of convection with large-scale waves -5

At the same time, if we have something which makes Q be in phase withT , that will lead to disturbance growth. An example could be surface
fluxes.
Surface fluxes create CAPE, by adding buoyancy (increasing T and q) toboundary layer air. If convection responds instantaneously (or at least
quickly), positive Q anomalies will follow.
This led to the ”wind-evaporation feedback” (Neelin et al. 1987) or”wind-induced surface heat exchange” (WISHE; Emanuel 1987) theory of
the Madden-Julian oscillation.

Adam Sobel (Columbia)

Lecture 4

December 21, 201341 / 71Interaction of convection with large-scale waves -6

Adam Sobel (Columbia)

Lecture 4

December 21, 201342 / 71Interaction of convection with large-scale waves -7
These ideas — and really almost all early tropical dynamics theories —
treat moisture as diagnostic. Relative humidity is typically assumed
constant, or predicted in a way that slaves it tightly to the large-scale
fields. Convection thus becomes a function either of just temperature(moist convective adjustment) or large-scale convergence. The presence ofmoisture just modifies existing dry wave modes.
However, if we have truly prognostic moisture, then there should be a newmode of behavior, in a linear model. We call this a moisture mode. How
it behaves will depend on physics.
An example from Sobel and Bretherton (2003). (See also many papers byDave Raymond, Zeljka Fuchs and associates.)

Adam Sobel (Columbia)

Lecture 4

December 21, 201343 / 71Interaction of convection with large-scale waves -8
This mode is a blob of moisture that moves with the environmental flow
(here 5 m/s) and decays slowly. No radiative feedback in this simulation.

Adam Sobel (Columbia)

Lecture 4

December 21, 201344 / 71Interaction of convection with large-scale waves -9
Moisture and precpitation in the convectively coupled gravity wave.

Adam Sobel (Columbia)

Lecture 4

December 21, 201345 / 71Interaction of convection with large-scale waves -10
Moisture and precipitation in the moisture wave.

Adam Sobel (Columbia)

Lecture 4

December 21, 201346 / 71Interaction of convection with large-scale waves -11

A current theory of tropical cyclogenesis, known as ”marsupial”(Dunkerton et al. 2009, Atmos. Chem. Phys) holds that the incipient
cyclone is essentially a blob of moist air that needs to be protected byclosed streamlines against dry air advection.
Montgomery web page

Adam Sobel (Columbia)

Lecture 4

December 21, 201347 / 71Interaction of convection with large-scale waves -12
Montgomery et al. (2010, Atmos. Chem. Phys. 10, 9879-

Adam Sobel (Columbia)

Lecture 4

December 21, 2013

48 / 71Interaction of convection with large-scale waves -13
Montgomery et al. (2010, Atmos. Chem. Phys. 10, 9879-

Vertical wind shear (850-200 hPa), 925 hPa co-moving streamlines.
Adam Sobel (Columbia)

Lecture 4

December 21, 201349 / 71Self-aggregation - 1

When non-rotating radiative-convective equilibrium simulations are run incloud-resolving models (resolution ∼ 1km, can simulate large cloud
updrafts but not all turbulence), on a small domain, convection is”popcorn” in appearance — distributed randomly throughout the domain.
But if the domain is large enough, convection self-aggregates to a single
spot, which can last indefinitely. The rest of the domain becomes very dry.

Adam Sobel (Columbia)

Lecture 4

December 21, 2013

50 / 71

Bretherton et al. (2005) precipitation (P), water vapor path (WVP), total
surface heat flux, outgoing longwave radiation (OLR), day 10 ofsimulation.

Adam Sobel (Columbia)

Lecture 4

December 21, 2013

51 / 71

Bretherton et al. (2005) precipitation (P), water vapor path (WVP), total
surface heat flux, outgoing longwave radiation (OLR), day 50 of
simulation.

Adam Sobel (Columbia)

Lecture 4

December 21, 201352 / 71Self-aggregation - 2

Some notes on self-aggregation:
It has now been reproduced a number of times with different models;It appears that interactions with surface fluxes and, especially,
radiation are essential;
We believe it is captured by SCM or CRM under WTG - the wet anddry regions are the multiple equilibria!
If Coriolis is present, tropical cyclones form.
(e.g. Muller and Held 2012; Wing and Emanuel 2013)

Adam Sobel (Columbia)

Lecture 4

December 21, 201353 / 71

WTG single column model, SST and free tropospheric temperature that allow
steady deep convection; initialize with very dry troposphere and it persists

Sobel, Bellon and Bacmeister, Geophys. Res. Lett., 2007

The same thing is found in a cloud-resolving model under WTG

Sessions et al., J. Geophys. Res., 2010; different symbols represent
different simulation domain sizeWe believe these equilibria to be analogs to the rainy and dry parts of thedomain under self-aggregation

Bretherton et al. (2005)

Conserved variable budgets, 1
Following Neelin and Held (1987, Mon. Wea. Rev., and Neelin and Zeng
(2000), etc. We work with moist static energy, but could use entropy withlittle modification to the arguments.
∂s∂s+ v · ∇s + ω= Q1 = Qc + QR ,
∂t
∂p
∂q
∂q
Lv (
+ v · ∇q + ω ) = −Q2 ,∂t∂pDrop overbars; everything is implicitly grid-scale. Add the equations to geta moist static energy equation:∂h∂h+ v · ∇h + ω= Qc − Q2 + QR .
∂t∂p
Now integrate vertically between nominal (fixed) surface and tropopausepressures
Z p0
Z p0
Z p0
Z p0
∂h
∂h
dp +
v · ∇hdp +ω dp = g (Lv E + H) +QR dp.
∂p
pT ∂t
pT
pT
pT
Adam Sobel (Columbia)

Lecture 4

December 21, 201355 / 71

Conserved variable budgets, 2
Now assume that the vertical velocity has a known vertical structure which
can be separated out:ω(x, y , p, t) = Ω(p)ω̂(x, y , t).
Think of ω̂ as the vertical motion at a fixed level (e.g., 500 hPa). Then we
can write
Z p0
pT

∂h
dp +
∂t

Z p0

Z p0

∂h
v · ∇hdp + ω̂Ω(p) dp = g (Lv E + H) +
∂p
pT
pT

Z p0
QR dp.
pT

or
Z p0
pT

∂h
dp +
∂t

Z p0

Z p0
v · ∇hdp − ω̂M = g (Lv E + H) +pT

QR dp,
pT

where we have defined the gross moist stability
Z p0
∂h
M=−
Ω(p) dp.
∂p
pT
Adam Sobel (Columbia)

Lecture 4

December 21, 201356 / 71

Conserved variable budgets, 2

Analogously we can define the gross dry stability and gross moisture
stratification
Z p0∂s
Ms = −Ω(p) dp,
∂p
p
Z Tp0
∂q
Mq =
Ω(p) dp,∂p
pT
so that M = Ms − Mq

Adam Sobel (Columbia)

Lecture 4

December 21, 201357 / 71

Conserved variable budgets, 4

In steady state, if we can neglect the horizontal advection term, we canwrite the dry static energy budget
Z p0−1−g ω̂Ms = Lv P + H −QR dp;
pT

using that to substitute in for ω̂ in the MSE budget, and neglecting thesensible heat flux, we have
Lv P =
where R =Ms
(Lv E + R) − R,M

R p0

pT QR dp.

Adam Sobel (Columbia)

Lecture 4

December 21, 201358 / 71

Conserved variable budgets, 4

Ms(Lv E + R) − R,
M
Strictly, all we have done is re-label things - this is formally just a
diagnostic exercise. But if we had a theory for the parameter Ms /M,
this would be a predictive theory for steady-state precipitation interms of the boundary fluxes; R is the difference between surface andtop-of atmosphere radiative fluxes, since radiative cooling is divergence ofradiative flux.
Lv P =

Adam Sobel (Columbia)

Lecture 4

December 21, 201359 / 71

Conserved variable budgets, 5

If the vertical motion profile Ω(p) were truly constant, then the grossmoist stability would be a function of the h profile (mainly of the q profile,since the T profile ∼ constant in the tropics. This is what was assumed by
Neelin and Held (1987, Mon. Wea. Rev.)
Z p0
∂h
M=−
Ω(p) dp.
∂p
pT
But in reality, the vertical motion profile is variable, and probably has astronger effect on variations in M.

Adam Sobel (Columbia)

Lecture 4

December 21, 201360 / 71

Conserved variable budgets, 6
To think about the gross moist stability it is useful to integrate by parts
Z p0
Z p0
∂h
dΩ
M=−
Ω(p) dp = −[Ωh] +
h dp;
∂p
pT
pT dp
Since Ω is a vertical velocity we assume it to be zero at the boundaries (sofirst term vanishes) and by mass conservation note that it must have anassociated horizontal velocity profile V (p) such thatdΩ(p)ω̂(x, y , t) + V (p)∇ · v̂(x, y , t) = 0.
dp
Then

Z p0
M∼

Vh dp,
pT

where V is the divergent wind.
Adam Sobel (Columbia)

Lecture 4

December 21, 201361 / 71

Conserved variable budgets, 7
Putting the tendency and advection terms back, we can write the moist
static energy equation as
Z p0
Z p0
∂
h dp +v · ∇hdp − ω̂M = g (Lv E + H) + R
∂t pT
pT
Rp
where R = pT0 QR dp. We also assume that variations in water vapor arelarge compared to variations in temperature (or geopotential) so that
Z p0
Z p0
∂
∂W
∂
h dp ≈
Lv q dp =
,
∂t pT
∂t pT
∂t
Rp
where W = pT0 Lv q dp is column water vapor (times lv ). Since moistureis mostly located in the lower troposphere, and large-scale winds tend tohave ”first baroclinic mode” structure where the winds are coherent (samesign) through the lower troposphere, it is reasonable to write the horizontal
advection term in terms of some effective lower-tropospheric wind V,
Z p0
Z p0
v · ∇h dp ≈ V · ∇h dp ≈ V · ∇W .
pTAdam Sobel (Columbia)

pT
Lecture 4

December 21, 201362 / 71

Conserved variable budgets, 8Thus our MSE equation is (absorbing all factors g , Lv into definitions offluxes on RHS)∂W
+ V · ∇W = ω̂M + E + H + R.
∂t
Now we make the so-called weak temperature gradient approximationin the dry static energy equation, neglecting tendency and advection,−ω̂Ms = P + H + R,
which we can use to eliminate ω̂ from the MSE equation∂W
M+ V · ∇W = − (P + H + R) + E + H + R;∂t
Ms
or neglecting H compared to E and R (very good approximation overocean),∂W
M+ V · ∇W = − (P + R) + E + R.
∂t
Ms
Adam Sobel (Columbia)

Lecture 4

December 21, 201363 / 71

Conserved variable budgets, 9∂WM+ V · ∇W = − (P + R) + E + R.
∂t
Ms
Some notes about this equation:
The first term on the RHS is really a vertical advection term (though
it doesn’t look like it any more).
We generally expect the factor M/Ms < 1; it may be  1, or may
even be negative in some circumstances. There is no a priori reason
to treat it as constant, but neither is there any solid theory to justify
any particular variations (and hard to constrain observationally).
This would be a closed equation if we could express P, E , R, and V
as functions of W (sounds difficult but ...).

Adam Sobel (Columbia)

Lecture 4

December 21, 201364 / 71

Wang and Sobel (2011, J. Geophys. Res.)

Adam Sobel (Columbia)

Lecture 4

December 21, 201365 / 71

Sobel et al. (2004, Mon. Wea. Rev.)

Adam Sobel (Columbia)

Lecture 4

December 21, 2013

66 / 71

CAPE seems only weakly related to precipitation, while lower-tropospherichumidity seems a clearer causal factor (Sobel et al. 2004, Mon. Wea.
Rev.)

Adam Sobel (Columbia)

Lecture 4

December 21, 201367 / 71

Wang and Sobel (2011, J. Geophys. Res.), instantaneous relationships
between column water vapor and precipitation in a cloud resolving model
under WTG.

Adam Sobel (Columbia)

Lecture 4

December 21, 201368 / 71

Bretherton et al. (2004) find an exponential relationship in satellite
microwave observations.

Adam Sobel (Columbia)

Lecture 4

December 21, 201369 / 71

Radiative cooling is also tightly coupled to precipitation, since OLR is.
(E.g., Raymond et al. 2003, J. Atmos. Sci.)

Adam Sobel (Columbia)

Lecture 4

December 21, 201370 / 71

So, we argue that the following physical parameterizations may beadequate for the purposes of understanding some large-scale phenomena
qualitatively (though not for serious numerical prediction):
P = P0 exp(

W
),
W∗

that is, precipitation a function (here exponential) of column relative
humidity, and
R = R0 + rP
that is, radiative heating increasing (or really, cooling decreasing, since theclear-sky radiative heating R0 < 0) linearly with precipitation rate.

Adam Sobel (Columbia)

Lecture 4

December 21, 201371 / 71Reference1• Bretherton, C. S., M. E. Peters, and L. E. Back, 2004: Relationships between
Water Vapor Path and Precipitation over the Tropical Oceans. J. Clim., 17,1517-1528• Bretherton, C. S., P. N. Blossey, and M. Khairoutdinov, 2005: An EnergyBalance Analysis of Deep Convective Self-Aggregation above Uniform SST. J.
Atmos. Sci., 62, 4273–4292• Dunkerton, T. J., M. T. Montgomery, and Z. Wang, 2009: Tropical
cyclogenesis in a tropical wave critical layer: easterly waves. Atmos. Chem.
Phys., 9, 5587–5646• Emanuel, K. A., 1987: An Air-Sea Interaction Model of Intraseasonal
Oscillations in the Tropics. J. Atmos. Sci., 44, 2324–2340• Emanuel, K. A., J. David Neelin, and C. S. Bretherton, 1994: On large-scale
circulations in convecting atmospheres. Q. J. R. Meteorol. Soc., 120, 1111–
1143• Gill, A. E., 1980: Some simple solutions for heat-induced tropicalcirculation. Q. J. R. Meteorol. Soc., 106, 447–462

Reference2• Kiladis, G. N., C. D. Thorncroft, and N. M. J. Hall, 2006: Three-Dimensional
Structure and Dynamics of African Easterly Waves. Part I: Observations. J.
Atmos. Sci., 63, 2212–2230• Kiladis, G. N., M. C. Wheeler, P. T. Haertel, K. H. Straub, and P. E. Roundy,
2009: Convectively coupled equatorial waves. Rev. Geophys., 47, 1–42
• Lau, K.-H., and N.-C. Lau, 1990: Observed Structure and Propagation
Characteristics of Tropical Summertime Synoptic Scale Disturbances. Mon.
Weather Rev., 118, 1888–1913• Matsuno, T., 1966: Quasi-Geostrophic Motions in the Equatorial Area. J.
Meteorol. Soc. Japan, 44, 25–43• Montgomery, M. T., L. L. Lussier III, R. W. Moore, and Z. Wang, 2010: The
genesis of Typhoon Nuri as observed during the Tropical Cyclone Structure
2008 (TCS-08) field experiment – Part 1: The role of the easterly wave
critical layer. Atmos. Chem. Phys., 10, 9879–9900• Muller, C. J., and I. M. Held, 2012: Detailed Investigation of the SelfAggregation of Convection in Cloud-Resolving Simulations. J. Atmos. Sci.,
69, 2551–2565

Reference3• Neelin, J. D., and I. M. Held, 1987: Modeling Tropical Convergence Basedon the Moist Static Energy Budget. Mon. Weather Rev., 115, 3–12
• Neelin, J. D., and N. Zeng, 2000: A Quasi-Equilibrium Tropical Circulation
Model—Formulation. J. Atmos. Sci., 57, 1741–176• Neelin, J. D., I. M. Held, and K. H. Cook, 1987: Evaporation-Wind Feedback
and Low-Frequency Variability in the Tropical Atmosphere. J. Atmos. Sci.,
44, 2341–2348• Raymond, D. J., G. B. Raga, C. S. Bretherton, J. Molinari, C. López-Carrillo,and Ž. Fuchs, 2003: Convective Forcing in the Intertropical Convergence
Zone of the Eastern Pacific. J. Atmos. Sci., 60, 2064–2082• Reed, R. J., and E. E. Recker, 1971: Structure and Properties of SynopticScale Wave Disturbances in the Equatorial Western Pacific. J. Atmos. Sci.,
28, 1117–1133• Sessions, S. L., S. Sugaya, D. J. Raymond, and A. H. Sobel, 2010: Multiple
equilibria in a cloud-resolving model using the weak temperature gradientapproximation. J. Geophys. Res., 115, D12110

Reference4• SOBEL, A. H., and C. S. BRETHERTON, 2003: Large-scale waves interactingwith deep convection in idealized mesoscale model simulations. Tellus A,
55, 45–60• Sobel, A. H., S. E. Yuter, C. S. Bretherton, and G. N. Kiladis, 2004: LargeScale Meteorology and Deep Convection during TRMM KWAJEX. Mon.
Weather Rev., 132, 422–444• Sobel, A. H., G. Bellon, and J. Bacmeister, 2007: Multiple equilibria in asingle-column model of the tropical atmosphere. Geophys. Res. Lett., 34,
L22804• Wang, S., and a. H. Sobel, 2011: Response of convection to relative sea
surface temperature: Cloud-resolving simulations in two and three
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