A Theory for the Spinup of Tropical Depressions

David J. Raymond, Sharon Sessions, and šeljka Fuchs

Physics Department and Geophysical Research Center

New Mexico Tech

Socorro, NM 87801 USA

raymond@kestrel.nmt.edu

February 21, 2007

Summary
Recent advances in our understanding of the physics of precipitation over
tropical oceans is married with vorticity dynamics to produce a simple theory for
the development or non-development of tropical depressions. This early phase
of tropical cyclogenesis is perhaps the least-well understood part of the whole
process. The theory predicts a nite amplitude threshold for intensication,
with weaker disturbances decaying. Furthermore, only disturbances with radii
larger than a critical value develop. Rough estimates of this critical radius place
it near 1600 km, though better estimates of various parameters could change this
value signicantly.

1 Introduction
The empirically determined general conditions needed for the spinup of tropical cyclones
have been known for many years, i. e., warm sea surface temperatures, low environmental
shear, and a pre-existing disturbance to provide low-level cyclonic vorticity (Gray, 1968;
McBride and Zehr, 1981). However, in spite of the vast amount of numerical modeling work
which has been done on tropical cyclones, major questions remain.
A great deal of work has gone into trying to understand the dynamical character of trop-
ical cyclogenesis. For instance, Challa and Pfeer (1980) and Pfeer and Challa (1981) con-
sidered eddy angular momentum transfer from the environment into the core of a developing
system. Briegel and Frank (1997) and Davis and Bosart (2003) emphasized mechanisms for
the baroclinic forcing of cyclones. Simpson et al. (1997) and Ritchie and Holland (1997) dis-
cussed the role of mesoscale convective systems in cyclogenesis, while Jones (1995, 2000a,b),
Frank and Ritchie (2001), and Reasor et al. (2004) investigated the eects of wind shear.
Montgomery and Enagonio (1998) and Enagonio and Montgomery (2001) showed how nu-
merous small vortices associated with individual convective events are axisymmetrized into
a single larger cyclone vortex. Hendricks et al. (2004) and Reasor et al. (2005) emphasized

1
the role of rotating convective disturbances (vortical hot towers) which develop in the large
background vorticity characteristic of tropical cyclogenesis regions.
Somewhat less attention has been paid to the thermodynamic aspects of cyclogenesis. In
a simple model Emanuel (1989, 1995) explained the nite amplitude nature of cyclogenesis
seen in the simulations of Rotunno and Emanuel (1987) as a consequence of the boundary
layer divergence associated with convective downdrafts in weak disturbances. Only when the
middle levels became moist enough to suppress downdraft production and increase precipi-
tation eciency was a weak vortex able to amplify. A notable facet of the Emanuel model
is the abandonment of the direct linkage of Ekman pumping to convective mass uxes seen
in many early models of cyclones, e. g., Charney and Eliassen (1964), Ooyama (1964, 1969),
etc. In a model similar to Emanuel's (1989), Frisius (2006) emphasizes the importance of
a negative radial gradient of precipitation eciency in conning the convection to near the
center of the cyclone vortex, a point also touched upon by Emanuel.
Gray and Craig (1998) showed exponential growth of vortex intensity from small am-
plitude in a simple linearized model. However, they assumed that a saturated eyewall had
already formed, so their results do not necessarily contradict those of Emanuel. Zehnder
(2001) also explored the eect on tropical cyclogenesis of convective forcing based on the
thermodynamic principle of boundary layer quasi-equilibrium (BLQ) of moist enthalpy or
entropy. He found that quite dierent results were obtained using this formulation in com-
parison to those obtained from an Ekman pumping model of convective forcing. Craig and
Gray (1996) also found that a cloud-resolving model of a tropical storm did not behave in a
manner consistent with the direct forcing of convection by Ekman pumping.
Using results from the TEXMEX (Tropical EXperiment in MEXico) project, Bister and
Emanuel (1997) showed that hurricane Guillermo (1991) formed out of a tropical depression
with a mid-level vortex overlying a moist, cold core. They found that this structure resulted
from cooling by the evaporation of precipitation and hypothesized that the eventual low-level
vortex apparently developed downward from the mid-level circulation.
Raymond et al. (1998) also studied the genesis of cyclones observed in TEXMEX. Using
the form of the vorticity equation advocated by Haynes and McIntyre (1987), they noted
that cyclone development occurs when the tendency of convergence to enhance the low-level
circulation of a system defeats the tendency of surface friction to spin the system down. The
downward development of circulation from the mid-level vortex, as envisioned by some
investigators, is not supported by the Haynes-McIntyre formulation, which shows that the
vertical component of vorticity undergoes only horizontal transport. The role of the mid-
level vortex is more subtle in this view; it acts primarily through geostrophic (or nonlinear)
balance to maintain the underlying cold core, which in turn has a thermodynamic eect
on the convection occurring in this region. In particular, the cold core in conjunction with
the moistening resulting from previous rainfall tends to produce a sounding prole with
almost constant moist entropy (or equivalent potential temperature) above the boundary
layer. An example of such a sounding taken during EPIC2001 (East Pacic Investigation of
Climate, year 2001; Raymond et al. 2003, 2004) is shown in gure 1. Convection occurring in
this environment is likely to have downdrafts suppressed due to the inability of subsequent
rainfall to produce negative buoyancy by evaporation. The suppression of downdrafts means
that the convergence at low levels associated with convective updrafts is not reduced by
divergence associated with downdrafts. As a result, the low-level spinup of the system is

2
 500

undisturbed
600
cyclone Lorena

pressure (hPa)
700

800

900

1000
200 220 240 260 280 300
entropy (J/K/kg)

Figure 1: Dropsonde soundings taken in the tropical east Pacic on 29 September 2001. The
◦ ◦
thin lines represent undisturbed conditions at 95.01 W, 12.27 N while the thick lines show
◦ ◦
a sounding in the core of the developing tropical cyclone Lorena at 96.98 W, 9.97 N. In
each case the left hand line is the specic moist entropy while the right hand line is the
saturated specic moist entropy.

enhanced.
We emphasize that to cause spinup of a cyclone, low-level convergence must be associated
with heating, i. e., with the latent heat release from rainfall. Without the upward export
of low-level air by moist convection, convergence results in dry-adiabatic ascent and the
production of a cold anomaly. The consequence is hydrostatic enhancement of the surface
pressure which opposes the low pressure in the center of the vortex, resulting in spindown
(see e. g. Gill 1982, p 353).
The importance of rainfall to the spinup of cyclones leads us to consider the mechanisms
controlling its production. An approach based on thermodynamic principles was proposed
by Raymond (2000). That paper explored the consequences of the simple postulate that
mean rainfall intensity is inversely proportional to the saturation decit, i. e., the dierence
between the tropospheric saturated precipitable water and the actual precipitable water. In
conjunction with Neelin and Held's (1987) ideas about gross moist stability, this led to a
model for the evolution of the humidity and precipitation rate in a tropospheric column.
Bretherton et al. (2004) performed a test of the hypothesis of Raymond (2000) by exam-
ining the relationship between precipitation and precipitable water over the tropical oceans,
as determined by microwave measurements from SSM/I satellites. They discovered that a
strong correlation exists between these quantities, but that the precise relationship varies
between ocean basins. However, these dierences vanished when the relationship was recast
in terms of precipitation rate and saturation fraction, i. e., the ratio of precipitable water
1
to saturated precipitable water. Basically all precipitation occurs for saturation fractions
between 0.6 and 0.85, with a rapidly increasing precipitation rate as a function of saturation
1 Bretherton et al. (2004) refer to the saturation fraction as the column relative humidity.

3
 200
25
210 EPIC soundings

EPIC P-3 radar rainrate (mm/d)

IR brightness temperature (K)
220 ECAC soundings
20
230 EPIC dropsondes
240
15
250
260
10
270
280
5
290
300
0
0 0.2 0.4 0.6 0.8 1
saturation fraction

Figure 2: Saturation fraction calculated from various soundings as a function of the infrared
satellite brightness temperature coincident with the sounding in time and averaged over a
50 km square centered on the sounding position. The rainfall rate inferred from airborne
radar measurements corresponding to the given brightness temperatures is given on the
right hand scale. This correspondence should be considered approximate, especially at high
brightness temperatures.

fraction within this range. Almost no cases were found with saturation fractions exceeding
0.85.
One might expect this type of relationship to be valid for long-time averages, but to
break down on shorter time scales. However, the authors found that the relation held even
for daily time scales, suggesting that it reects a fundamental physical process such as the
deleterious eect of dry air entrainment on updrafts. The robust nature of this correlation
is supported by gure 2, which shows saturation fraction from individual soundings in the
tropical east Pacic and the SW Caribbean as a function of the coincident satellite infrared
brightness temperature (considered as a proxy for precipitation) in a 50 km square centered
on the sounding. In this gure EPIC soundings were launched from the ship Ron Brown
◦ ◦
at 95 W, 10 N in the east Pacic intertropical convergence zone, whereas EPIC dropson-
des were deployed from 470 hPa by the National Center for Atmospheric Research's C-130
◦ ◦
aircraft (Raymond et al., 2004). Only dropsondes deployed in the range 6.5 N to 12 N
were used in order to limit measurements to regions of warm sea surface temperature. The
ECAC soundings were launched from the Mexican research vessel Justo Sierra in the dry
SW Caribbean during the ECAC project (Experimento Climático en las Albercas de Agua
Caliente de las Américas; Magaña and Caetano, 2005).
The correlation between low infrared brightness temperature and humidity could simply
be a consequence of higher humidity breeding more clouds. However, over warm ocean
regions low brightness temperature generally corresponds to high stratiform cloudiness with
no necessary correlation with higher humidity at low to middle levels where most of the
atmospheric moisture resides. On the other hand, area-averaged precipitation rates inferred

4
from airborne radar data and satellite-derived infrared brightness temperature are highly
correlated in the east Pacic intertropical convergence zone, with a correlation coecient
of −0.90 (Raymond et al. 2003). The scale on the right side of gure 2 indicates rainfall
values corresponding to brightness temperatures in this linear regression. Thus we argue
that gure 2 represents a valid, if somewhat noisy relationship between saturation fraction
and precipitation rate.
More recent work by Back and Bretherton (2005) shows that some of the variance in
precipitation rate at xed saturation fraction can be explained by variations in surface wind
speed, with higher wind speeds corresponding to higher precipitation rates. This result
indicates that the original hypothesis of Raymond (2000) that precipitation is a unique
function of saturation fraction is perhaps too simple. However, numerical simulations of
convection, made in a context which we explain later, also support a strong relationship
between rainfall rate and tropospheric humidity (Derbyshire et al., 2004; Raymond and Zeng,
2005). Furthermore, Derbyshire et al. show that many parametric treatments of convection
in global models do not reproduce the sensitivity to moisture shown by the cumulus ensemble
models.
The purpose of this paper is to marry the above ideas about precipitation and saturation
fraction to the dynamics of tropical cyclones. The feedback loop which governs cyclone
spinup or spindown is envisioned to be as follows:

1. The existing circulation produces surface uxes of water vapor, moist entropy, and
momentum which depend on the strength of the circulation and other environmental
factors such as sea surface temperature. The moist entropy ux modies the prole of
moist entropy, and hence that of saturation fraction.

2. Convection and rainfall are considered to be governed by the saturation fraction in the
convective environment inside the cyclone. Determining the dependence of convection
on this environment is central to the theory of spinup.

3. The relative strengths of the spinup tendency due to low-level vorticity convergence
induced by convective heating and the spindown tendency due to surface friction govern
whether the cyclone vortex intensies or decays.

We examine here only the case of weak systems such as tropical depressions in order to
avoid the complications of non-linearity. In this limit we do not expect to see the warm
core which is the traditional signal that a tropical storm has formed. However, this phase of
intensication is arguably less-well understood than the later phases of tropical cyclogenesis.
The paper is organized as follows: Section 2 updates the theory of Raymond (2000)
to account for the results of Bretherton et al. (2004), while section 3 links the updated
thermodynamic results of section 2 with the vorticity dynamics of the lower troposphere
as discussed by Raymond et al. (1998). In section 4 numerical results from the model of
Raymond and Zeng (2005) are used to close the theory presented in section 3 and the theory
is applied to an axially symmetric vortex in section 5. Conclusions are presented in section
6.

5
2 Thermodynamic issues
We rst examine the relevant theory governing water vapor and specic moist entropy. In
this theory the gross moist stability (GMS) plays a central role. The concept of GMS depends
on the existence of a thermodynamic variable conserved in moist processes. Unfortunately,
there is no perfectly conserved variable in this context. Neelin and Held (1987) used the
moist static energy in their pioneering work on GMS, whereas Raymond (2000) used equiv-
alent potential temperature. In this work we use the specic moist entropy, as it is easier to
deal with than equivalent potential temperature. It also has somewhat better-dened con-
servation properties than moist static energy, though given the simplied treatment here,
this advantage is not realized.
An approximate form of the specic moist entropy suitable for the present work is

s = sd + Lr/TR , (1)

where sd (T, p) is the specic entropy of dry air, L is the latent heat of condensation (assumed
constant), TR is a constant reference temperature, and r is the water vapor mixing ratio.
The specic moist entropy obeys the equation

∂s ∂ωs ∂Fe
+ U · ∇s + ∇ · (ui s) + =G+g (2)
∂t ∂p ∂p
where g is the acceleration of gravity, G is the irreversible generation of entropy per unit
mass of air, and Fe is the vertical ux moist entropy due to diusion, eddy transport,
and radiation. We have split the horizontal velocity u into a solenoidal barotropic part U
(i. e., independent of height and divergence-free) and a baroclinic part ui , and have taken
advantage of the fact that ∇ · U = 0. We dene U more precisely later.
We now dene the pressure integral operator

1 Z ps
Z= Zdp (3)
g pt

for any Z(p), where ps and pt are respectively surface and tropopause pressures. Applying
this operator to (2) results in

ds
+ M δs = Fes − Fet + G (4)
dt
where Fes is the surface entropy ux due to latent and sensible heat ux plus radiation and
Fet is the upward ux of entropy out of the top of the cylinder due to radiation. The total
time derivative is dened
d ∂
≡ + U · ∇. (5)
dt ∂t
The second term in (4) requires further explanation. Let us dene the lateral mass
outow per unit time from a column of unit area as

1Z 1Z
M= ∇ · ui dp = − ∇ · ui dp, (6)
g + g −

6
where the plus and minus signs on the integrals indicate that only positive or negative
values of the integrand are included in the integration. Due to overall mass balance, the
mass owing into the column, given by the rightmost expression in (6), equals the mass
owing out. The second term in (4) can thus be written

1 Z ps 1 Z ps I
∇ · (ui s) = ∇ · (ui s)dp = sui · n̂ dl dp = M δs (7)
g pt g pt
where the line integral is around the periphery of the unit area dening the column, with n̂
being an outward unit normal. The quantity M is thus the mass current passing through
the sides of the column in a reference frame moving with the solenoidal velocity U, and
δs = sout − sin is the dierence between the mean outow and inow values of moist entropy.
The quantity δs is Neelin and Held's (1987) GMS recast in terms of the moist entropy rather
2
than the moist static energy.
An equation similar to (2) may be written for the water vapor mixing ratio r:

∂r ∂ωr ∂ ∂Fr
+ U · ∇r + ∇ · (ui r) + = − [(ω + ωt )rr ] + g . (8)
∂t ∂p ∂p ∂p
The rst term on the right side of this equation is the source of water vapor due to the
formation and evaporation of precipitation, assuming that the horizontal and time scales are
such that horizontal advection and storage of precipitation are negligible. The second term
is the source due to the convergence of the vertical eddy and diusional ux of water vapor
Fr . The role of condensed water in the form of cloud droplets or small ice crystals which
advect with the air is neglected in this approximate analysis.
Applying (3) to the moisture equation results in

dr
− M δr = Frs − R, (9)
dt
where Frs is the surface evaporation rate, R is the rainfall rate, and where δr is dened as
the mean inow minus the mean outow mixing ratio rather than the outow minus the
inow.
We assume that sd , which is a function only of temperature, is approximately constant
due to the near-invariance of the temperature prole in the tropics. It thus disappears in
the time derivative in (4), leaving the rst term on the left side of this equation equal to
(L/TR )(dr/dt). The mass ow M may then be eliminated between (4) and (9), leaving us
with
dr TR (Fes − Fet + G)
(1 + γ) = γ(Frs − R) + . (10)
dt L
Alternatively, the time derivative may be eliminated, resulting in an equation for the moisture
convergence X:
L(R − Frs ) + TR (Fes − Fet + G)
X ≡ M δr = . (11)
(1 + γ)L
2 Neelin and Held (1987) approximate ∇ · (ui s) by s∇ · ui , thus neglecting the entropy advection term
ui · ∇s, which is held to be negligible.

7
 50

40 50 km domain CEM
old conceptual model

rain rate (mm/d)

30 new conceptual model

20

10

0
0 0.2 0.4 0.6 0.8 1
saturation fraction

Figure 3: Relationship between rainfall rate and saturation fraction for the cumulus ensemble
model on a 50 km domain, the old idealized model (solid line) and the new idealized model
(dashed line).

The parameter
TR δs TR ∇ · (ui s)
γ≡ =− (12)
Lδr L∇ · (ui r)
is a dimensionless quantity closely related to the GMS. We shall refer to γ as the normalized
gross moist stability (NGMS). The advantage over the GMS is that γ is a dimensionless
parameter which relates the GMS (δs) to the moisture extracted (δr ) and ultimately to the
precipitation produced.
In a steady state, equation (10) can be solved for the net rainfall rate R − Frs , which
according to (9) also equals the moisture convergence X in this case:

TR (Fes − Fet + G)
(R − Frs )steady = Xsteady = . (13)
γL

This equation expresses the results of Neelin and Held (1987), and illustrates how important
the NGMS is to the large-scale forcing of precipitation. For a given moist entropy imbalance,
represented by the numerator on the right side of (13), the net rainfall, i. e., the rainfall minus
evaporation, is inversely proportional to the NGMS in the steady state.
As it stands, (13) is a purely diagnostic relationship. To give it prognostic value, two
additional pieces of information are needed; a relationship between the rainfall rate R and
the precipitable water r (or saturation fraction) and a way to estimate the NGMS. We now
address these issues.
So far the analysis is similar to that of Raymond (2000) except that entropy is used
in place of equivalent potential temperature. The solid line in gure 3 shows graphically
the humidity-precipitation relation assumed by Raymond (2000), expressed as a relationship
between precipitation rate and saturation fraction. This assumption clearly does not t

8
the dependence on saturation decit of the rainfall rate predicted by the cumulus ensemble
model of Raymond and Zeng (2005), which is shown by the bullets in gure 3. Nor does
it agree with the form of this curve inferred by Bretherton et al. (2004) or the related data
illustrated in gure 2. A much better t is given by the dashed line

Sc − SR
R = RR (14)
Sc − S
where R is the rainfall rate and where the saturation fraction S = r/rs with rs being the
saturation mixing ratio. The quantity Sc = 0.87 is a critical saturation fraction corresponding
to that value for which the rainfall rate goes asymptotically to innity, while RR and SR
are constant reference values of the rainfall rate and the saturation fraction. We nd that
RR = 4 mm d−1 and SR = 0.81 represent well the results of the cumulus ensemble model.
This value of RR corresponds roughly to the rainfall rate which occurs in radiative-convective
equilibrium.
In the tropics the saturated precipitable water rs is constant to the extent that the
temperature prole and the surface pressure do not change. Taking the time derivative of
(14) thus results in
dR R2 dr
= . (15)
dt rs RR (Sc − SR ) dt
Eliminating dr/dt between (10) and (15) yields a predictive equation for the rainfall rate:

dR R2 [γL(Frs − R) + TR (Fes − Fet + G)]

= (16)
dt (1 + γ)Lrs RR (Sc − SR )

Non-dimensionalizing, we arrive at the same equation as Raymond (2000),

dα
= −α3 + α2 ∆φ, (17)
dτ
where α = R/RR is the rainfall rate expressed in terms of the radiative-convective equilibrium
rate,
γLFrs + TR (Fes − Fet + G)
∆φ = (18)
γLRR
is the non-dimensional forcing, and τ = t/t0 where the scaling time t0 is dened

(1 + γ)rs (Sc − SR )
t0 = . (19)
γRR

The quantity rs /RR is just the time scale to dry out a saturated atmosphere with a steady
rain equal to the radiative-convective equilibrium rainfall rate. Numbers typical of the tropics
yield 15 d for this ratio. (For a similar calculation see Grabowski and Moncrie 2004.) The
factor Sc − SR ≈ 0.06, so the combination rs (Sc − SR )/RR ≈ 1 d. Depending on the value
of the NGMS, t0 will be somewhat greater than this value, perhaps 3 d if, as we nd later,
that γ ≈ 0.5.

9
 Let us now attempt to understand what this analysis is telling us. In a steady state, the
dimensionless precipitation rate equals the dimensionless forcing:

α = α0 = ∆φ (steady state). (20)

However, in the non-steady case the rainfall rate relaxes to a steady state value as long as
the NGMS γ > 0. Assuming that the rainfall rate is not too far from its equilibrium value,
the time constant for this relaxation may be obtained by linearizing α in (17) about its
steady-state valueα0 : α = α0 + α0 . The resulting linearized evolution equation is

dα0
= −α02 α0 , (21)
dτ
which implies a dimensional e-folding time for relaxation of t0 /α02 . Thus the relaxation time
becomes smaller as the steady-state rainfall rate becomes larger. For instance, if the rainfall
2
rate is three times the radiative-convective equilibrium value then α0 = 3 and t0 /α0 =
(3/9) d = 8 h.
Moisture relaxation times of 8 h −3 d are much less than those predicted by Raymond
(2000) and Grabowski and Moncrie (2004). This is because the precipitation rate is much
more sensitive to the saturation fraction at high saturation fractions than in the earlier
models. These numbers compare well with the results of Sobel and Bretherton (2003) and
Bretherton et al. (2004), who found relaxation times of 0.5 − 2.5 d.

3 Vorticity dynamics
The vorticity equation in isobaric coordinates as expressed by Haynes and McIntyre (1987)
and used by Raymond et al. (1998) in the study of east Pacic tropical storms is

∂ζa  
+ ∇ · uζa + k̂ × F∗ = 0 (22)
∂t
where the absolute vorticity is dened

∂v ∂u
ζa = f + ζ = f + − , (23)
∂x ∂y
f being the Coriolis parameter, and where

∂u
F∗ = F − ω . (24)
∂p

The viscous and turbulent force per unit mass acting on the air is F = (Fx , Fy , 0). This
force arises from the vertical eddy transport and diusion of horizontal momentum and its
vertical integral must be proportional to the surface stress T:

F = T. (25)

10
 We now divide the horizontal velocity into barotropic solenoidal and baroclinic parts as
∗ ∗
in section 2, u = U + ui , and use the vector identity ∇ · (k̂ × F ) = −k̂ · (∇ × F ) to rewrite
(22) as
dζa ∂Fy∗ ∂Fx∗
+ ∇ · (ui ζa ) − + = 0. (26)
dt ∂x ∂y
Within a weak disturbance such as a tropical depression, the absolute vorticity ζa is unlikely
to deviate much from the environmental value in the region ζae , which means that we can
approximate the second term in the above equation by ∇ · (ui ζa ) ≈ ζae ∇ · ui . In the absence
of signicant relative vorticity we would have ζae ≈ f where f is the Coriolis parameter, but
we wish to retain the possibility that the disturbance is developing in a high vorticity region.
We wish to consider the behavior of the vorticity at low levels in an incipient cyclone,
as the development of a low-level circulation is the central feature of tropical cyclogenesis.
We also wish to relate the low-level convergence to the rainfall rate. One way to meet both
of those goals is to make a weighted average of (26) in pressure with the weighting function
being the ambient mixing ratio prole r0 (z). Multiplying (26) by r0 , integrating in pressure,
and dividing by gr0 results in

dζ˜a ζae X ∂ F̃y∗ ∂ F̃x∗

= + − (27)
dt r0 ∂x ∂y
where the moisture-weighted average

Z ps Z ps
1 Z ps
Z̃ ≡ r0 Zdp r0 dp = r0 Zdp (28)
pt pt gr0 pt

for any variable Z and where the moisture convergence X is approximated by −∇ · (ui r0 ).
In computing the moisture convergence term we have assumed that ζae does not vary
much with height over the lower troposphere where the mixing ratio is large. For a vortex
of horizontal size L this is a reasonable assumption if the vortex is in a balanced state and
if the Rossby penetration depth Lf /N is greater than the scale height of the water vapor
mixing ratio. (N is the typical Brunt-Väisälä frequency.) In the tropics we typically have
penetration depths of order L/300, so systems with a horizontal scale exceeding 900 km will
have penetration depths exceeding the typical 3 km scale height of water vapor.
∗
Let us ignore the contribution of the term −ω(∂u/∂p) to F for now (see the end of
section 5) and just consider the part due to the eddy and diusive transport of momentum.
We note that
1 Z ps r̂0 Z ps r̂0 T
F̃ = r0 Fdp ≡ Fdp = (29)
gr0 pt gr0 pt r0
where r̂0 is the pressure average of r0 weighted by the prole of the frictional force. Techni-
cally r̂0 is a tensor, but as long as the vertical structure of Fx and Fy are similar, then this
quantity can be represented as a scalar. If surface friction is deposited in the lowest layer
of the atmosphere, then r̂0 equals the surface mixing ratio. For deeper distributions, r̂0 is
correspondingly less and the spindown tendency due to surface friction is weaker. Equation
(27) thus simplies to
!
dζ̃a ζae X r̂0 ∂Ty ∂Tx
= + − . (30)
dt r0 r0 ∂x ∂y

11
 We now identify U as the solenoidal part of ũ. Since (28) weights the vertical average
strongly toward low levels where the mixing ratio is largest, we think of U as the non-
divergent part of the low-level ow. We also note that

∂Uy ∂Ux
ζ˜a = f + − (31)
∂x ∂y
where U = (Ux , Uy ). Since U is divergence-free, we can dene a stream function ψ such that

∂ψ ∂ψ
Ux = − Uy = . (32)
∂y ∂x
Combining this with (31) results in an inversion relation

∇2 ψ = ζ̃a − f (33)

between the low-level vorticity and stream function.

If convection is active in the lowest few kilometers of the atmosphere, then there will
be a tendency to homogenize vertically the horizontal velocity in this layer. In addition,
relaxation toward a balanced state will have a similar eect on the vorticity if the lateral
dimensions of the incipient cyclone are suciently large, as discussed above. To some degree
of approximation, U = ũ then becomes not only a weighted average of the low-level wind,
but approximately equal to the actual wind throughout this layer, or at least to the non-
divergent part of it. Thus, we can approximate bulk surface uxes using U = |U| for the
near-surface wind. For surface evaporation and moist entropy ux we therefore assume

Frs = ρs C∆rU (34)

Fes = ρs C∆sU (35)

where ρs is the surface air density, C ≈ 0.001 is an exchange coecient, ∆r is the dierence
between the saturated mixing ratio at the sea surface temperature and pressure and the
actual mixing ratio in the atmospheric boundary layer, and ∆s is equivalent dierence for
moist entropy. Similarly, the surface stress is

T = −ρs CU U. (36)

In the present work we assume that the surface sensible heat ux is small over the oceans,
which means that the evaporation rate and moist entropy ux are approximately related
by LFrs ≈ TR Fes and therefore L∆r ≈ Tr ∆s. We also assume that ∆r, ∆s, and C are
independent of wind speed.
Equation (30) tells us that the vorticity tendency averaged over the layer of high moisture
values is composed of two competing components, a part associated with moisture conver-
gence, which tends to increase the vorticity, and a part associated with surface friction, which
tends to decrease it. This by itself is no surprise. However, the analysis of the previous sec-
tion provides a way for computing the moisture convergence X. In particular, (11) gives
us X in terms of the rainfall rate and surface uxes as long as the evolutionary time scale
of the incipient cyclone is long compared to the moisture adjustment time scale, which we
earlier estimated to be in the range 8 h −3 d. The rainfall rate is given by (14) in terms of
the precipitable water r, and the evolution of the precipitable water is governed by (9) or
(10). The system is thus closed if the NGMS and the surface and tropopause moist entropy
uxes are known.

12
4 Determination of NGMS
In this section we use the cloud-resolving cumulus ensemble model of Raymond and Zeng
(2005; see also Derbyshire et al. 2004; Mapes 2004) to estimate the gross moist stability of
deep convection. In this model clouds interact with their surroundings according to the ideas
of Sobel and Bretherton (2000). In particular, large-scale circulations are assumed to act so as
to disperse buoyancy anomalies over large areas and thus keep the mean virtual temperature
prole of the computational domain close to that of the surrounding environment. This
action is approximated in the model by Newtonian relaxation of the virtual temperature
prole toward that of a reference prole which represents the atmosphere surrounding the
convection. The strength of the cooling needed to do this is diagnosed from the relaxation,
and the mean vertical velocity required to produce this cooling by dry adiabatic lifting is then
inferred from the cooling rate and the dry static stability. This imagined vertical velocity
and the associated horizontal ow demanded by mass continuity do not actually exist in
the model since cyclic lateral boundary conditions are imposed. However, their eects are
included via the imposition of moisture and moist entropy source terms consistent with this
ow. Sobel et al. (2001) refer to the approximation implicit in this approach as the weak
temperature gradient (WTG) approximation and we denote the above imaginary vertical
velocity as the WTG vertical velocity. For shorthand we refer to the cumulus ensemble
model employing this approximation as the WTG model.
In a statistically averaged sense the characteristics of the convection and the precipitation
rate in the WTG model are a function of the reference proles of temperature, humidity,
and wind, the sea surface temperature, and the imposed mean surface wind. Increasing the
sea surface temperature and the surface wind both increase surface heat and moisture uxes,
and hence the precipitation rate.
We have yet to explore the response of the convection to a wide variety of reference
proles. In this paper we take as a reference prole the radiative-convective equilibrium
prole resulting from an imposed surface wind of 5 m s−1 . However, much remains to be
done in exploring the full range of possible driving conditions and the results reported here
are preliminary.
In the formal derivation of the WTG model governing equations, we split the horizontal
and vertical velocity elds into small-scale cyclic and large-scale non-cyclic parts, v = vc +vnc
and w = wc + wnc , with the small-scale, cyclic ow computed by the cumulus ensemble
model. The large-scale part represents the above-described interaction of the model with
the surrounding environment. We further assume that wnc depends only on z and t. By
mass continuity vnc is therefore linear in x and y.
The convective-scale equations for specic moist entropy s and total cloud water mixing
ratio r are

∂ρs ∂ρswc ∂ρswnc

+ ∇ · (ρsvc ) + = ρSe − ∇ · (ρsvnc ) − ≡ ρ(Se − Ee ) (37)
∂t ∂z ∂z
and
∂ρr ∂ρrwc ∂ρrwnc
+ ∇ · (ρrvc ) + = ρSr − ∇ · (ρrvnc ) − ≡ ρ(Sr − Er ). (38)
∂t ∂z ∂z
The terms involving the non-cyclic, large-scale velocities are encapsulated into Ee and Er .
These terms represent the rate at which the large-scale environment removes moist entropy

13
and moisture from the convective domain, and as we now show, are precisely what is needed
to compute the NGMS. The terms Se and Sr represent the local sources of entropy and
moisture associated with radiation, precipitation, subgrid-scale eddy uxes, etc.
Assuming that the density ρ is a function only of height and applying the pressure integral
operator (3) to the denition of Ee , we nd

1 ∂ρswnc ∂ρswnc
E e ≡ ∇ · (svnc ) + = ∇ · (svnc ) − g = ∇ · (svnc ) (39)
ρ ∂z ∂p
where the hydrostatic equation is invoked to change a z derivative to a p derivative. The
pressure average of the pressure derivative is zero since we assume that wnc = 0 at both the
surface and tropopause. A similar relation holds for Er.
Comparing with (12) and identifying vnc with ui , we note that the NGMS is simply

TR E e
γ=− , (40)
LE r
where we recall that normally E r < 0. The model of Raymond and Zeng (2005) is written in
terms of the equivalent potential temperature θe rather than the moist entropy, but using a
simplied relation between the two, s = Cp ln(θe /TR ), we easily nd that E e = Cp (Ethe /hθe i)
where Ethe is the analog to Ee in the governing equation for equivalent potential temperature
and where the angle brackets indicate a horizontal average over the domain of the cumulus
ensemble model. Using arguments of Raymond and Zeng (2005) translated to the notation
of this paper, we nd that

(hθe i − θex ) ∂ρwnc ∂hθe i

Ethe = + wnc , (41)
ρ ∂z ∂z
where θex equals the reference prole of equivalent potential temperature in layers of net
convergence (∂ρwnc /∂z > 0) and is equal to hθe i in divergent layers. The rst term in (41)
represents entrainment of environmental air into the convective region, while the second
represents the eect of mean vertical advection. A similar expression exists for Er .
We have computed the NGMS for a number of simulations with our WTG model. As in
the case of Raymond and Zeng (2005), we make two-dimensional computations on a small,
50 km domain. Expansion of the domain to 512 km changes the results very little. The
radiation scheme of Raymond (2001) is used in place of xed radiative cooling and particle
−1 −1
fall speeds above the freezing level are reduced from 5 m s to 1 m s to emulate the
behavior of snow. Also, Ethe is calculated directly via (41) rather than indirectly, with more
accurate results. Other than that, the computations are identical to those of Raymond and
Zeng (2005).
Figure 4 shows how the NGMS varies with wind speed and sea surface temperature
(SST). Values of NGMS typically range from roughly 0.4 to 0.55 in these calculations. At
−1
xed SST a minimum in NGMS occurs near a wind of 10 m s . At a xed wind speed
−1
of 7 m s the NGMS initially decreases with an increase in SST, though this decrease is
partially reversed for SST increments more than 1−2 K above the reference value.
As a check of our calculation of NGMS, we also solve (13) for γ in the steady state

TR (Fes − Fet + G) TR (Fes − Fet + G)

γsteady = = (42)
XL (R − Frs )L

14
 0.6
A SST = 303 K
0.55

NGMS
0.5
0.45
0.4
0.35
5 10 15 20
imposed wind (m/s)
0.6
B Imposed Wind = 7 m/s
0.55
NGMS

0.5
0.45
0.4
0.35
303 304 305 306
SST (K)

Figure 4: Plot of NGMS (a) as a function of imposed wind speed at xed SST, and (b) as a
function of SST at xed imposed wind speed. The calculations are done using a radiative-
−1
convective equilibrium reference prole with an imposed wind of 5 m s and an SST of
303 K.

and compare this result with that obtained using (40) and nd satisfactory agreement. This
equation also provides a way to determine the NGMS from observations. In situations of
intense precipitation the steady-state assumption is not restrictive according to our theory,
because the humidity relaxation time is typically shorter than the dynamical time scale under
these conditions.
The main lesson we take away from the above results is that radiative-convective equilib-
rium reference proles yield NGMS values of order 0.5. The simulations are of course very
limited at this point, with a small, two-dimensional domain and no wind shear. Much needs
to be done to rene and expand these calculations and to test them against observations.
In particular, more general reference proles diering from radiative-convective equilibrium
need to be explored. Until more experience is gained with the WTG model, the above results
must be regarded as tentative. However, as we shall see, the qualitative aspects of the theory
do not depend on the precise value of the NGMS.

5 Results for an axially symmetric vortex

Insight as to how the present theory works may be obtained in a highly simplied context. We
integrate (30) over a circular area A of radius a and assume axial symmetry in all quantities.
We also assume provisionally that the moisture convergence is well approximated by its

15
steady-state value (13), resulting in

dΓ̃ πa2 ζae TR hFes − Fet + Gi 2πar̂0 |T|

= − (43)
dt γLr0 r0
where Γ is the relative circulation around the periphery of the area A. The angle brackets
here indicate an average over the area. In addition we assume that the ow is primarily
tangential to the periphery, so that the tangential component of T is approximately given
2
by |T| = ρs CU .
For the purposes of this analysis we ignore the irreversible generation of entropy and
further assume that the radiative moist entropy sink is constant independent of wind speed,
which is tantamount to ignoring cloud-radiation interactions. The quantity inside the angle
brackets in (43) is therefore equal to the deviation of the surface ux from radiative-convective
equilibrium conditions in which the imposed wind speed is U0 . Using (35), we therefore have
hFes − Fet + Gi ≈ ρs C∆shU − U0 i.
The value of U0 is uncertain but is likely to be approximately equal to the average surface
−1
wind speed in the tropics. As in section 4, we assume U0 = 5 m s .
We interpret U as the surface wind speed on the periphery of the area A, so that Γ̃ =
2πaU , and assume that the mean wind speed in the interior of the area is comparable to
the wind speed on the periphery U . This assumption is clearly a serious underestimate for
a tropical storm, but perhaps is not so bad for a depression, which lacks a strong central
vortex. Under these conditions, (43) becomes

dU Uc (U − U0 ) − U 2
= , (44)
dt D
where the critical velocity Uc is given by

aζae ∆r
Uc = (45)
2γr̂0
and the parameter D is
r0
D= . (46)
ρs C r̂0
Though the NGMS was shown to change somewhat with wind speed, let us assume here
that it is constant. The numerical simulations in the previous section suggest a value near
γ = 0.5. Then, for any intensication to be possible, we must have Uc > 4U0 . In this case
dU/dt = 0 for two particular velocities,

U1,2 = [Uc ± (Uc2 − 4Uc U0 )1/2 ]/2, (47)

(whereU1 < U2 ) and dU/dt > 0 only for U1 < U < U2 , as illustrated in gure 5. If Uc < 4U0 ,
U.
decay occurs for all values of
The condition Uc > 4U0 can be recast as a condition on the system radius a:

8γr̂0 U0
a> ≡ ac . (48)
ζae ∆r

16
 U1 U2

dU/dt
unstable stable

Figure 5: Schematic illustration of growth rate for an axially symmetric system. Point U1
is a point of unstable equilibrium, while U2 exhibits stable equilibrium. The heavy arrows
indicate the direction of evolution of U with time.

The intensication of a vortex in this model thus depends on the vortex's initial size and
intensity. If the radius a < ac , than decay is inevitable. If a > ac , then for initial peripheral
tangential velocities U < U1 , there is decay, while intensication up to U = U2 occurs for
U > U1 . For U > U2 , the vortex decays to U = U2 .
−1 −3 −5 −1 −1
If γ = 0.5, U0 = 5 m s , ρs = 1.2 kg m , ζae = f = 3 × 10 s , r̂0 = 12 g kg ,
r0 = 50 kg m−2 , and ∆r = 5 g kg−1 , then ac ≈ 1600 km and D ≈ 3500 km. Furthermore,
−1
if as an example we take Uc = 6U0 = 30 m s , then U1,2 = (3 ± 3
1/2
)U0 , or 6.3 m s−1 and
23.7 m s . The specied value of ∆r is that for 30 C sea surface temperature, 29◦ C air
−1 ◦

temperature, and 80% relative humidity.

The time constant for intensication can now be determined. Equation (44) can be
rewritten in terms of U1 and U2 as

dU (U − U1 )(U2 − U )
= . (49)
dt D
The maximum growth rate occurs when U = (U1 +U2 )/2, and for this value the time constant
for growth is
!−1
1 dU 2D(U1 + U2 )
τ= = . (50)
U dt (U2 − U1 )2
Using the above determined values, we nd that τ ≈8 d. This growth time is longer than
the range of possible moisture relaxation time constants, which implies that the steady state
assumption for moisture convergence which is used in (43) is justied.
Let us return to the question of the neglect of the term −ω(∂u/∂p) in the F∗ term. By
virtue of Stokes' theorem, this term contributes only on the periphery of the area A. If
this area is large enough to bound the region of convection, then the periphery of A will
see little or no convection, and we can assume that ω ≈ 0 there. On the other hand, if
the periphery is embedded in the convection, then this term must be considered. Tropical

17
depressions tend to be cold-core at low levels, which from thermal wind considerations tells
us that ∂ut /∂p < 0 ut is the component of the velocity in the direction of a cyclonic
where
traverse around the area A. The contribution of this term will therefore be to retard spinup
in a region of upward motion (ω < 0).

6 Conclusions
In this paper a theory is developed for the early stages of the spinup of tropical cyclones.
It is limited to the early development stage in order to avoid complications associated with
feedback of the developing vortex on the vortex environment, i. e., non-linear eects. The
theory incorporates recent advances in our understanding of the thermodynamic forcing of
tropical convection and precipitation. Beyond that, it bears a strong conceptual resemblance
to the simple cyclone model of Emanuel (1989, 1995) though the scales are very dierent.
The key assumptions of the theory are as follows: (1) cyclogenesis results when the spinup
tendency of low-level convergence forced by latent heat release associated with precipitation
production exceeds the spindown tendency of surface friction, and (2) precipitation is gov-
erned by the surface moist entropy ux with a small lag (several hours to a few days) and is
modulated in intensity by the normalized gross moist stability.
Though numerous uncertainties exist in the details of this theory, its qualitative basis is
reasonably robust. The rst of the above points follows directly from the vorticity equation
in ux form (Haynes and McIntyre 1987). Emanuel's (1989) model makes the second of
the above assumptions and the numerical results of Craig and Gray (1996) support the
hypothesis that convection is forced by surface uxes rather than Ekman pumping.
Our focus on the role of precipitation rather than convection in cyclogenesis is appropriate
because of the closer relationship of the former to low-level convergence. It is possible to have
strong convection without signicant low-level convergence if the convection produces strong
downdrafts, as noted by Emanuel (1989, 1995). Such convection does not produce low-level
spinup. However, heavy precipitation without strong low-level convergence is considerably
less likely, as most of the moisture for intensely precipitating systems is drawn in laterally
from the moisture-rich lower troposphere.
The lag mentioned above in key assumption two is the time required for the convective
environment to moisten or dry in response to a change in the surface moist entropy ux.
Ample evidence exists that the production of precipitation is exceptionally sensitive to the
environmental relative humidity (Bretherton et al. 2004; Derbyshire et al. 2004; Raymond
and Zeng 2005), and this considerably shortens the lag compared to simple estimates not
taking this factor into account; the exact form of rainfall rate versus saturation fraction
shown in gure 3 is not essential to the theory.
The idea of relating the convective mass ux and precipitation to the surface entropy
ux via the gross moist stability is well established (Neelin and Held 1987; Raymond 2000),
though the means for estimating the NGMS in dierent circumstances are relatively new and
untested. Recent work not presented here suggests that moistening and stabilization of the
environment act to decrease the NGMS by a factor of 2 to 3 relative to the values reported
here.
More observations are needed in developing (and non-developing) cyclone precursor dis-

18
turbances to test the present theory. In particular, measurements of NGMS under a variety
of circumstances are needed, as are observations of the vertical distribution of momentum
uxes associated with surface drag. We believe that a well-tested cumulus ensemble model
run in WTG mode will eventually constitute a reliable tool for estimating the NGMS.
The theory predicts for an axially symmetric vortex that spinup can only occur if the
radius of the vortex is greater than some critical value. Smaller vortices decay according to
the theory. For parameter values typical of the tropical environment, the minimum radius
is estimated to be 1600 km. Smaller NGMS values would decrease the critical radius. For
radii greater than this value, a nite threshold for further growth exists in the tangential
velocity. Vortices weaker than this threshold decay, while those stronger than the threshold
intensify further.
The critical radius governs spinup because with other things being equal, the integrated
precipitation rate and hence the spinup tendency over the area being considered scale with
the area, whereas the frictional spindown tendency scales with the circumference, and hence
the square root of the area. Thus, increasing the radius increases the spinup-spindown ratio;
the critical radius is that for which this ratio equals unity.
The threshold for growth exists because the ratio of spinup and spindown tendencies
Uc (U − U0 )/U 2 increases with surface wind speed, exceeding unity at the threshold. Re-
call that U0 is the wind speed associated with radiative-convective equilibrium andUc is a
constant proportional to disturbance radius. The spinup tendency scales as U − U0 since
rainfall, which varies with the surface moist entropy ux and hence the wind speed, must
exceed the radiative-convective equilibrium value for moisture convergence to occur.
The surface friction increases as the square of the wind speed due to the form of the
bulk formula for surface stress. Because of this factor, a limit is eventually reached where
the frictional spindown tendency exceeds the convergence-driven spinup tendency, and the
cyclone cannot intensify beyond this limit. However, this maximum in wind speed is well
into the non-linear regime where the detailed assumptions of the theory are invalid.
One factor not explored in our axisymmetric vortex model is the tendency discovered by
Emanuel (1989) and examined further by Frisius (2006) for convection to expand away from
the center of the cyclone in a concentric ring, thus halting cyclogenesis. This phenomenon
is not addressed in our axisymmetric control volume example, but it could be accounted for
in a spatially resolved version of our model by a radial gradient in the NGMS, in analogy to
the precipitation eciency gradient in the above papers.
Relaxation of some of the assumptions behind the theory allows one to explore qualita-
tively certain behaviors not strictly encompassed by the theory as it stands. If the atmosphere
is sheared, one might expect that dierentially advected dry air could make its way into the
region of convection, with resulting suppression of precipitation. This would manifest itself
as an increase in the NGMS and a corresponding decrease the potential for intensication.
Alternatively, if the system is embedded in a region of cyclonic relative vorticity, then in-
gested air would have absolute vorticity larger than f, and the spinup would be enhanced.
The theory is thus in essential agreement with the ideas of Challa and Pfeer (1980) and
Pfeer and Challa (1981).
Also interesting would be to apply the theory to non-axisymmetric situations. Equa-
tions for humidity (10) and vorticity (30) along with associated diagnostic relations form a
closed system assuming that the NGMS can be determined. These equations could form the

19
basis for studying the intensication of easterly waves and related disturbances into trop-
ical depressions. Such an eort would be highly worthwhile, given that this may be the
least-well-understood phase of tropical cyclogenesis.
Acknowledgments. We thank Daniel Martínez of the Cuban Institute of Meteorology
for providing the soundings launched from the Mexican oceanographic research vessel Justo
Sierra during the ECAC project. Two anonymous reviewers made this a much better pa-
per via their constructive suggestions. This work was supported by U. S. National Science
Foundation Grant ATM-0352639.

7 References
Back, L. E., and C. S. Bretherton, 2005: The relationship between wind speed and precip-
itation in the Pacic ITCZ. J. Climate, 18, 4317-4328.
Bister, M., and K. A. Emanuel, 1997: The genesis of hurricane Guillermo: TEXMEX
analyses and a modeling study. Mon. Wea. Rev., 125, 2662-2682.
Bretherton, C. S., M. E. Peters, and L. E. Back, 2004: Relationships between water vapor
path and precipitation over the tropical oceans. J. Climate, 17, 1517-1528.
Briegel, L. M., and W. M. Frank, 1997: Large-scale inuences on tropical cyclogenesis in
the western North Pacic. Mon. Wea. Rev., 125, 1397-1413.
Challa, M., and R. L. Pfeer, 1980: Eects of eddy uxes of angular momentum on model
hurricane development. J. Atmos. Sci., 37, 1603-1618.
Charney, J. G. and A. Eliassen, 1964: On the growth of the hurricane depression. J.

Atmos. Sci., 21, 68-75.

Craig, G. C., and S. L. Gray, 1996: CISK or WISHE as the mechanism for tropical cyclone
intensication. J. Atmos. Sci., 53, 3528-3540.
Davis, C. A., and L. F. Bosart, 2003: Baroclinically induced tropical cyclogenesis. Mon.
Wea. Rev., 131, 2730-2747.
Derbyshire, S. H., I. Beau, P. Bechtold, J.-Y. Grandpeix, J.-M. Piriou, J.-L. Redelsperger,
and P. M. M. Soares, 2004: Sensitivity of moist convection to environmental humidity.
Quart. J. Roy. Meteor. Soc., 130, 3055-3079.
Emanuel, K. A., 1989: The nite-amplitude nature of tropical cyclogenesis. J. Atmos.

Sci., 46, 3431-3456.

Emanuel, K. A., 1995: The behavior of a simple hurricane model using a convective scheme
based on subcloud-layer entropy equilibrium. J. Atmos. Sci., 52, 3960-3968.
Enagonio, J., and M. T. Montgomery, 2001: Tropical cyclogenesis via convectively forced
vortex Rossby waves in a shallow water primitive equation model. J. Atmos. Sci., 58,
685-705.

20
Frank, W. M., and E. A. Ritchie, 2001: Eects of vertical wind shear on the intensity and
structure of numerically simulated hurricanes. Mon. Wea. Rev., 129, 2249-2269.
Frisius, T., 2006: Surface-ux-induced tropical cyclogenesis within an axisymmetric atmo-
spheric balanced model. Quart. J. Roy. Meteor. Soc., 132, 2603-2623.
Gill, A. E., 1982: Atmosphere-Ocean Dynamics. Academic Press, New York, 662 pp.

Grabowski, W. W., and M. W. Moncrie, 2004: Moisture-convection feedback in the trop-

ics. Quart. J. Roy. Meteor. Soc., 130, 3081-3104.
Gray, W. M., 1968: Global view of the origin of tropical disturbances and storms. Mon.
Wea. Rev., 96, 669-700.
Gray, S. L., and G. C. Craig, 1998: A simple theoretical model for the intensication of
tropical cyclones and polar lows. Quart. J. Roy. Meteor. Soc., 124, 919-947.
Haynes, P. H., and M. E. McIntyre, 1987: On the evolution of vorticity and potential
vorticity in the presence of diabatic heating and frictional or other forces. J. Atmos.
Sci., 44, 828-841.
Hendricks, E. A., M. T. Montgomery, and C. A. Davis, 2004: The role of "vortical" hot
towers in the formation of tropical cyclone Diana (1984). J. Atmos. Sci., 61, 1209-
1232.

Jones, S. C., 1995: The evolution of vortices in vertical shear: I: initially barotropic vortices.
Quart. J. Roy. Meteor. Soc., 121, 821-851.
Jones, S. C., 2000a: The evolution of vortices in vertical shear: II: large-scale asymmetries.
Quart. J. Roy. Meteor. Soc., 126, 3137-3159.
Jones, S. C., 2000b: The evolution of vortices in vertical shear: III: baroclinic vortices.
Quart. J. Roy. Meteor. Soc., 3161-3185.

Magaña, V., and E. Caetano, 2005: Temporal evolution of summer convective activity over
the Americas warm pools. Geophys. Res. Letters, 32, doi:10.1029/2004GL021033.
Mapes, B. E., 2004: Sensitivities of cumulus-ensemble rainfall in a cloud-resolving model
with parameterized large-scale dynamics. J. Atmos. Sci., 61, 2308-2317.
McBride, J. L., and R. Zehr, 1981: Observational analysis of tropical cyclone formation.
Part II: Comparison of nondeveloping versus developing systems. J. Atmos. Sci., 38,
1132-1151.

Montgomery, M. T., and J. Enagonio, 1998: Tropical cyclogenesis via convectively forced
vortex Rossby waves in a three-dimensional quasigeostrophic model. J. Atmos. Sci.,
55, 3176-3207.
Neelin, J. D., and I. M. Held, 1987: Modeling tropical convergence based on the moist
static energy budget. Mon. Wea. Rev., 115, 3-12.
21
Ooyama, K., 1964: A dynamical model for the study of tropical cyclone development.
Geofísica Internacional, 4, 187-198.
Ooyama, K., 1969: Numerical simulation of the life cycle of tropical cyclones. J. Atmos.
Sci., 26, 3-40.
Pfeer, R. L., and M. Challa, 1981: A numerical study of the role of eddy uxes of mo-
mentum in the development of Atlantic hurricanes. J. Atmos. Sci., 38, 2393-2398.
Raymond, D. J., 2000: Thermodynamic control of tropical rainfall. Quart. J. Roy. Meteor.

Soc., 126, 889-898.

Raymond, D. J., 2001: A new model of the Madden-Julian oscillation. J. Atmos. Sci.,
58, 2807-2819.
Raymond, D. J., C. López-Carrillo, and L. López Cavazos, 1998: Case-studies of developing
east Pacic easterly waves. Quart. J. Roy. Meteor. Soc., 124, 2005-2034.
Raymond, D. J., and X. Zeng, 2005: Modelling tropical atmospheric convection in the
context of the weak temperature gradient approximation. Quart. J. Roy. Meteor.

Soc., 131, 1301-1320.

Raymond, D. J., G. B. Raga, C. S. Bretherton, J. Molinari, C. López-Carrillo, and š.
Fuchs, 2003: Convective forcing in the intertropical convergence zone of the eastern
Pacic. J. Atmos. Sci., 60, 2064-2082.
Raymond, D. J., S. K. Esbensen, C. Paulson, M. Gregg, C. S. Bretherton, W. A. Petersen,
R. Cifelli, L. K. Shay, C. Ohlmann, and P. Zuidema, 2004: EPIC2001 and the coupled
ocean-atmosphere system of the tropical east Pacic. Bull. Am. Meteor. Soc., 85,
1341-1354.

Reasor, P. D., M. T. Montgomery, and L. F. Bosart, 2005: Mesoscale observations of the

genesis of hurricane Dolly (1996). J. Atmos. Sci., 62, 3151-3171.
Reasor, P. D., M. T. Montgomery, and L. D. Grasso, 2004: A new look at the problem of
tropical cyclones in vertical shear ow: Vortex resiliency. J. Atmos. Sci., 61, 3-22.
Ritchie, E. A. and G. J. Holland, 1997: Scale interactions during the formation of Typhoon
Irving. Mon. Wea. Rev., 125, 1377-1396.
Rotunno, R., and K. A. Emanuel, 1987: An air-sea interaction theory for tropical cyclones.
Part II: Evolutionary study using a nonhydrostatic axisymmetric numerical model. J.
Atmos. Sci., 44, 542-561.
Simpson, J., E. Ritchie, G. J. Holland, J. Halverson, and S. Stewart, 1997: Mesoscale
interactions in tropical cyclone genesis. Mon. Wea. Rev., 125, 2643-2661.
Sobel, A. H., and C. S. Bretherton, 2000: Modeling tropical precipitation in a single column.
J. Climate, 13, 4378-4392.

22
Sobel, A. H., J. Nilsson, and L. M. Polvani, 2001: The weak temperature gradient approx-
imation and balanced tropical moisture waves. J. Atmos. Sci., 58, 3650-3665.
Sobel, A. H., and C. S. Bretherton, 2003: Large-scale waves interacting with deep convec-
tion. Tellus, 55a, 45-60.
Zehnder, J. A., 2001: A comparison of convergence- and surface-ux-based convective
parameterizations with applications to tropical cyclogenesis. J. Atmos. Sci., 58, 283-
301.

23