A Theory for the Response of Tropical Moist Convection to Mechanical Orographic Forcing

Abstract

Spatial patterns of tropical rainfall are strongly influenced by mountains. Although theories for precipitation induced by convectively stable upslope ascent exist for the midlatitudes, these do not represent the interaction of moist convection with orographic forcing. Here, we present a theory for convective precipitation produced by the mechanical interaction of a tropical ridge with a basic state horizontal wind. Deviations from this basic state are represented as the sum of a 'dry' perturbation, due to the stationary orographic gravity wave, and a 'moist' perturbation that carries the convective response. The moist component dynamics are subject to the weak temperature gradient approximation; they are forced by the dry mode's influence on lower-tropospheric moisture and temperature. Analytical solutions provide estimates of the precipitation profile, including peak precipitation, upstream extent, and rain shadow extent. The theory can be used with several degrees of complexity depending on the technique used to compute the dry mode, which can be drawn from linear mountain wave theory or full numerical simulations. To evaluate the theory, we use a set of convection-permitting simulations with a flow-perpendicular ridge in a long channel. The theory makes a good prediction for the cross-slope precipitation profile, indicating that the organization of convective rain by orography can be quantitatively understood by considering the effect of stationary orographic gravity waves on a lower-tropospheric convective quasiequilibrium state.

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A theory for the response of tropical moist convection to mechanical orographic forcing
Quentin Nicolas∗
Department of Earth and Planetary Science, University of California, Berkeley, California
William R. Boos
Department of Earth and Planetary Science, University of California, Berkeley, California
Climate and Ecosystem Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California
ABSTRACT
Spatial patterns of tropical rainfall are strongly influenced by mountains. Although theories for precipitation induced by convectively stable
upslope ascent exist for the midlatitudes, these do not represent the interaction of moist convection with orographic forcing. Here, we
present a theory for convective precipitation produced by the mechanical interaction of a tropical ridge with a basic state horizontal wind.
Deviations from this basic state are represented as the sum of a ‘dry’ perturbation, due to the stationary orographic gravity wave, and a
‘moist’ perturbation that carries the convective response. The moist component dynamics are subject to the weak temperature gradient
approximation; they are forced by the dry mode’s influence on lower-tropospheric moisture and temperature. Analytical solutions provide
estimates of the precipitation profile, including peak precipitation, upstream extent, and rain shadow extent. The theory can be used with
several degrees of complexity depending on the technique used to compute the dry mode, which can be drawn from linear mountain wave
theory or full numerical simulations. To evaluate the theory, we use a set of convection-permitting simulations with a flow-perpendicular
ridge in a long channel. The theory makes a good prediction for the cross-slope precipitation profile, indicating that the organization
of convective rain by orography can be quantitatively understood by considering the effect of stationary orographic gravity waves on a
lower-tropospheric convective quasiequilibrium state.
1. Introduction
The spatial distribution of time-mean low-latitude rain-
fall is set to first order by the latitude and intensity of
the intertropical convergence zone (ITCZ). Zonal inhomo-
geneities in sea-surface temperatures, a direct consequence
of the presence of land masses, modify this distribution by
driving the Walker circulation. These land masses also act
seasonally as strong energy sources that drive monsoon cir-
culations. On sufficiently large length scales, one might be
satisfied with this description of the principal features and
forcings of tropical precipitation; looking in more detail,
however, one sees that orography strongly modifies these
broad patterns.
Satellite-derived estimates (GPM IMERG V06B, Huff-
man et al. 2019) of climatological precipitation1for June-
August and October-December are shown in Figure 1a,b,
along with a smoothed contour of 500 m surface height.
Some of the most striking deviations from the quasi-linear
oceanic ITCZ are regions of intense rainfall located in the
vicinity of mountains, e.g. near the Northern Andes, West-
ern Ghats, Himalayas, and various ranges in the Indochi-
nese peninsula and Maritime continent. These precipita-
∗Corresponding author: Quentin Nicolas, qnicolas@berkeley.edu
1This GPM product has been shown to have little bias relative to rain
gauge measurements on seasonal time scales, even over complex terrain
(Derin et al. 2019).
tion maxima are located in regions and seasons favorable to
moist convective development, as noted by Xie et al. (2006)
for the Asian summer monsoon and Ramesh et al. (2021)
for regions experiencing an autumn monsoon. Xie et al.
(2006) note that despite their prominence, a regional atmo-
spheric model with 0.5°horizontal resolution is unable to
reproduce these features. This failure, partly attributed to
inadequate convective parameterization, was confirmed in
a more systematic study by Kirshbaum (2020). It suggests
that a main tool for evaluating climate, namely global circu-
lation models, is ill-suited to study orographic convection,
despite its importance for tropical precipitation.
Kirshbaum et al. (2018) identify two ways orography
generates or alters convective systems: mechanical forc-
ing, whereby a mountain lifts a background wind, and
thermal forcing, where surface heat fluxes from elevated
terrain produce convergence. Figure 1 displays 100 m wind
vectors from the ERA5 reanalysis (Hersbach et al. 2020).
Enhanced precipitation is mostly focused upwind of moun-
tain ranges and on their windward slopes, suggesting that
mechanical forcing is the primary mechanism at play in
the large-scale, climatological sense. To further illustrate
this point, we plot cross-sections of surface elevation and
precipitation for summer and autumn along a latitude line
spanning India, Southeast Asia, and the Philippines. We
observe a clear shift of rainfall maxima from the western
1
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2AMS JOURNAL NAME
slopes in summer to the eastern slopes in autumn, consis-
tent with the seasonal wind reversal in these regions.
Orographic precipitation in midlatitude, convectively
stable flows has been extensively studied (for a review, see
e.g. Roe 2005). Smith and Barstad (2004; hereafter SB04)
developed an analytical theory, using linear mountain wave
dynamics, that efficiently reproduces rainfall rates and spa-
tial organization in such cases. It accounts for the effects
of cloud latent heating through use of a moist stability pa-
rameter. However, because it does not account for moist
convective dynamics, this theory may not be applicable to
most tropical cases.
Orographic convection has been the focus of field ex-
periments and numerous studies in the past two decades
(Houze 2012; Kirshbaum et al. 2018). However, most have
focused on synoptic scales, with little attention devoted to
understanding what sets climatological time-mean precip-
itation rates. Chu and Lin (2000) and Chen and Lin (2005)
studied initial value problems, where the fate of condition-
ally unstable flow was qualitatively examined as a function
of moist nondimensional mountain height and convective
available potential energy (CAPE). Miglietta and Rotunno
(2009) extended their work to study the dependence on a
more exhaustive set of parameters, noting that precipitation
increased both with the ratio of mountain height to the level
of free convection, and with the aspect ratio of the moun-
tain. The ratio of advective to convective time scales was
shown to control the shape of the cross-mountain rainfall
profile, with wider mountains having their precipitation
profile shifted upstream. Subsequent studies (Miglietta
and Rotunno 2012, 2014) noted the importance of vertical
wind shear in producing large rain rates in the presence
of deep convection; soundings with strong flow at lower
levels and weak flow aloft produced higher precipitation
rates. Focusing on larger spatial and temporal scales, Wang
and Sobel (2017) simulated rainfall over isolated tropical
islands in both thermally and mechanically forced settings.
Their flat-island cases showed that surface roughness gra-
dients alone can produce substantial mechanical forcing for
precipitation, and that the transition from thermal to me-
chanical forcing can cause a non-monotonic dependence
of mean precipitation on upstream wind speed.
Few studies attempted to formulate theories for oro-
graphically forced moist convection. Kirshbaum and Smith
(2009) took inspiration from trade-wind flows over Do-
minica to develop an analytical model for the orographic
enhancement of precipitation from shallow convection.
They used a ‘slice’ method separating saturated updrafts
from unsaturated descending air, computing the impact
of an imposed mean ascent on vertical velocities. Two
elements hamper application of this theory to the time-
mean effects of orography on deep convection. First, the
mean ascent is assumed be uniform in height, whereas
deep convection spans at least one vertical wavelength of
an orographic gravity wave. Second, the theory contains
multiple unconstrained parameters (e.g., updraft area frac-
tion, an ‘entrainment’ parameter representing dissipative
effects on cloudy updrafts) that would vary greatly on time
scales larger than those of a single event. To the best of
our knowledge, Cannon et al. (2014) is the only analytical
work focused on orographic deep convection, having devel-
oped an area-averaged model with detailed microphysics.
The precipitation rate is computed level-wise by bringing
parcels to their level of neutral buoyancy, then accounting
for detrainment and evaporation due to compensating de-
scent before applying a CAPE-dependent weighting. The
model leads to a high number of equations to be solved,
hindering physical insights, and its “bulk” nature obscures
key questions such as how far upstream precipitation is
enhanced and how long the rain shadow is.
A fruitful line of development in the theory of tropical
dynamics has been the quasiequilibrium (QE) view of con-
vection, dating back to Arakawa and Schubert (1974) and
outlined by Emanuel et al. (1994). This view has its roots in
the observation that CAPE often varies slowly compared
to its generation mechanisms (i.e. tropospheric radiative
cooling, surface enthalpy fluxes, and large-scale ascent),
with CAPE anomalies consumed as convective activity re-
sponds rapidly to any change in the generation mechanisms.
An ensemble of convective motions is thus in near statisti-
cal equilibrium with the large-scale forcings, with convec-
tive motions setting the vertical temperature profile, tying it
directly to the subcloud-layer moist static energy. QE the-
ories seem most relevant for relatively slow forcings such
as the seasonal cycle, which evolves on a time scale orders
of magnitude longer than that of the convective response.
This motivates our use of QE theory here to describe time-
mean patterns of orographic precipitation, which in some
regions (e.g. South Asia) constitutes a substantial fraction
of the regional total precipitation (Xie et al. 2006), that has
in turn been understood using QE frameworks (e.g. Nie
et al. 2010). Specifically, we use QE theory to describe
the statistical average effect of mountains on time-mean
convective precipitation, rather than formulating a theory
of event-wise convective triggering by orographic ascent.
While early QE theories employed CAPE-based convec-
tive closures, we leverage recent developments that incor-
porate observed relationships between precipitation and
lower-tropospheric temperature and humidity (Derbyshire
et al. 2004; Raymond et al. 2015; Ahmed et al. 2020).
We discuss further details and possible caveats, such as
whether Eulerian or Lagrangian time scales are relevant
for evaluating the validity of QE, later in the context of
results from our idealized model.
Here we pose two key questions: what sets the mean
precipitation rate of mechanically forced orographic con-
vection? How far upstream and downstream does orogra-
phy influence tropical precipitation? Unlike some classic
QE closures that only consider near-surface temperature
and moisture anomalies, we use a convective closure that

3
Fig. 1. Annual-mean GPM IMERG V06B precipitation (shading), 500 m surface height level (thin brown contours), and wind vectors 100
m above the surface averaged over (a) June-August and (b) October-December from 2014 to 2020. (c) Cross-sections of surface elevation and
precipitation along the thick blue line shown on panels a and b at 15°N.
is sensitive to lower free tropospheric anomalies, and ac-
counts for the influence of stationary orographic gravity
waves on those anomalies. This links classic stationary
wave theory with modern QE closures for convection, and
provides nonlinear expressions for precipitation as well as
a linearized theory that can be forced by the Fourier trans-
formed terrain. Convection-permitting simulations in a
long channel are used to test theoretical rain rates. We use
the results of these simulations to evaluate the convective
time scales used in the theory and the possible influence
of spatial modulations of surface evaporation and radiative
cooling downwind of the ridge.
2. Theory
This section presents an analytical theory for the pre-
cipitation over a tropical mountain in a background wind,
based on a QE convective closure. Its aim is to account for
the main features of time-mean precipitation around the
ridge (peak value, spatial extent of upstream enhancement,
rain shadow length), as a function of large-scale flow char-
acteristics and ridge shape. The theory is based loosely on
the Quasiequilibrium Tropical Circulation Model (QTCM)
of Neelin and Zeng (2000), but employs the moisture-
temperature (𝑞−𝑇) convective parameterization proposed
by Ahmed et al. (2020). This closure was derived from
the empirical relationship between precipitation and lower-
free-tropospheric buoyancy and parameterizes precipita-
tion as a response to both temperature and moisture pertur-
bations, with different sensitivities. The reason we chose
to use this closure instead of a more classic CAPE-based
parameterization will be expanded upon in section 3b.
Our main hypothesis, assessed in later sections, is
that mountains alter convection by modulating lower-
tropospheric temperature and moisture. In the presence
of a background wind, an orographic gravity wave is ex-
cited that carries thermodynamic perturbations. Forced
ascent upwind of the ridge cools and moistens the lower
free troposphere, enhancing convection. The opposite oc-
curs downstream where subsidence prevails. In addition to
these orographically induced thermodynamic variations,
we will show that the solution depends on the convec-
tive adjustment time scales and on an advective length
scale, with the latter being the product of the background
wind speed with a time scale for relaxation to radiative-

4AMS JOURNAL NAME
convective equilibrium (RCE). The mountain wave may
also modify the tropospheric static stability and wind shear,
but we focus on lower-tropospheric temperature and mois-
ture anomalies because these have been found to exert a
strong control on deep convection (Derbyshire et al. 2004;
Raymond et al. 2015; Ahmed et al. 2020).
a. Modal decomposition
Throughout this paper, we consider a horizontally infi-
nite low-latitude domain with surface elevation ℎ(𝑥)and a
uniform and constant background horizontal wind 𝑢
𝑢
𝑢0. As
explained in Appendix A, linearizing the dynamics allows
the flow to be described as the sum of a basic state and
two perturbation modes: a dry mode, due solely to the
orographic gravity wave, and a moist mode, that repre-
sents the moist convective part of the flow. The dry mode
only influences the moist mode as a forcing for convective
heating, while the moist mode does not influence the dry
mode. The latter assumption is not entirely justified, as we
will show in section 3c that the orographic gravity wave
feels a lower effective stability due to the presence of moist
convection, in a convection-permitting numerical model.
Steady-state thermodynamic and moisture equations for
the moist mode are
𝑢
𝑢
𝑢0·∇
∇
∇𝑇𝑚+𝜔𝑚
𝜕𝑠0
𝜕 𝑝 =𝑄𝑐−𝑅, (1a)
𝑢
𝑢
𝑢0·∇
∇
∇𝑞𝑚+𝜔𝑚
𝜕𝑞0
𝜕 𝑝 =𝑄𝑞+𝐸 , (1b)
where 𝑠0(𝑝)and 𝑞0(𝑝)are, respectively, the reference dry
static energy and moisture vertical profiles, with the zero
subscript denoting a property of the basic state, and 𝜔𝑚,
𝑇𝑚, and 𝑞𝑚are the pressure velocity, temperature, and
moisture perturbations of the moist mode. 𝑄𝑐and 𝑄𝑞de-
note convective heating and moistening, while 𝑅and 𝐸are
radiative cooling and evaporation rates. Here, and subse-
quently, temperature and moisture are in energy units (i.e.
they have, respectively, absorbed the heat capacity of air at
constant pressure 𝑐𝑝, and the latent heat of vaporization of
water 𝐿𝑣). This linearized formulation neglects horizon-
tal variations in static stability and moisture stratification
caused by orography, which in the simulations presented in
Section 3 are of order 20% downstream of the ridge (with
local increases up to 30% above the ridge).
We now perform a unimodal vertical truncation of the
moist mode. We only include a deep convective mode,
with any shallow temperature anomaly induced by the oro-
graphic gravity wave represented by the dry mode. This
treatment likely renders our theory most appropriate for
deep tropical regions, where deep convection is observed
to generate most rainfall (Tan et al. 2013; Houze et al.
2015), in contrast with the midlatitudes or trade wind re-
gions (e.g. Kirshbaum and Smith 2009). Following Sobel
et al. (2001), we also employ the weak temperature gradient
(WTG) approximation for the moist mode, which allows
us to solve for that mode without the momentum equa-
tions (remember that horizontal temperature gradients in-
duced by the orographic gravity wave are carried by the dry
mode). Using notation from Neelin and Zeng (2000), we
write 𝜔𝑚(𝑥, 𝑦 , 𝑝)=𝜔1(𝑥, 𝑦 )Ω1(𝑝)and vertically average
(1a)-(1b), yielding
−𝜔1𝑀𝑠=h𝑄𝑐i−h𝑅i,(2a)
𝑢
𝑢
𝑢0·∇
∇
∇h𝑞𝑚i +𝜔1𝑀𝑞=h𝑄𝑞i+h𝐸i,(2b)
with h·i =𝑝𝑠
𝑝𝑡{·}d𝑝/𝑝𝑇. Here 𝑝𝑇=𝑝𝑠−𝑝𝑡is the tro-
pospheric depth, and 𝑝𝑠and 𝑝𝑡are, respectively, surface
and tropopause pressures (the tropospheric mass per unit
area is 𝑝𝑇/𝑔=8000 kg m−2henceforth.). The gross dry
stability and gross moisture stratification are respectively
defined by 𝑀𝑠=hΩ1𝜕𝑠0/𝜕 𝑝iand 𝑀𝑞=hΩ1𝜕 𝑞0/𝜕 𝑝i. The
quantity 𝑀=𝑀𝑠−𝑀𝑞is known as the gross moist stabil-
ity (GMS, see e.g. Neelin and Held 1987; Raymond et al.
2009).
We now introduce the energy constraint h𝑄𝑐i=−h𝑄𝑞i
and employ the 𝑞−𝑇convective closure,
h𝑄𝑐i=𝑞0
𝐿
𝜏𝑞−𝑇0
𝐿
𝜏𝑇+
,(3)
where (·)+=max(·,0).𝜏𝑞and 𝜏𝑇are the moisture
and temperature adjustment time scales, diagnosed re-
spectively as approximately 11 hours and 3 hours by
Ahmed et al. (2020). 𝑞0
𝐿and 𝑇0
𝐿are total deviations (the
sum of perturbations in both the moist and dry modes)
from the reference profiles of moisture and temperature,
with (·)𝐿=𝑝𝐿𝑡
𝑝𝐿𝑏 {·}d𝑝/(𝑝𝐿𝑏 −𝑝𝐿 𝑡 )denoting a lower-free-
tropospheric average, where 𝑝𝐿𝑏 =900 hPa and 𝑝𝐿𝑡 =700
hPa. Decomposing these anomalies into contributions
from the moist and dry modes (subscripts 𝑚and 𝑑, re-
spectively, see also Appendix A) gives
𝑞0
𝐿=𝑞𝑚𝐿 +𝑞𝑑 𝐿 and 𝑇0
𝐿=𝑇𝑚𝐿 +𝑇𝑑𝐿 =𝑇𝑑 𝐿 ,(4)
where the last equality comes from WTG (if ∇𝑇𝑚=0, one
can add any horizontally uniform nonzero 𝑇𝑚(𝑝)to the
reference profile 𝑇0(𝑝), hence resulting in 𝑇𝑚=0). The
heating term (3) will have two contributions: one from
the dry perturbations 𝑞𝑑 𝐿 and 𝑇𝑑 𝐿 (hereafter referred to as
“the dry forcing for convection”), and a moist convective
response carried by 𝑞𝑚𝐿 . Because we assumed a horizon-
tally and temporally invariant vertical profile for 𝑞𝑚, the
quantities 𝑞𝑚𝐿 and h𝑞𝑚iare proportional. Using the no-
tation of Ahmed et al. (2020), 𝑞𝑚𝐿 /𝜏𝑞=h𝑞𝑚i/˜𝜏𝑞, where
˜𝜏𝑞'0.6𝜏𝑞.
b. A general precipitation equation
We can now obtain an equation for precipitation, which
is related to the convective heating by 𝑃=𝑝𝑇h𝑄𝑐i/𝑔in

5
units of energy per unit area per unit time (dividing by
𝜌𝑤𝐿𝑣, where 𝜌𝑤is the density of liquid water, yields a
physical precipitation rate, in m s−1). Using (3), (4) and
the definition of ˜𝜏𝑞, we obtain
𝑃=𝑝𝑇
𝑔h𝑞𝑚i
˜𝜏𝑞+𝑞𝑑𝐿
𝜏𝑞−𝑇𝑑𝐿
𝜏𝑇+
.(5)
We then eliminate 𝜔1in (2a)-(2b) and use (5) to obtain an
equation for 𝑃that involves only the various imposed ther-
modynamic parameters, incident wind, and perturbations
in the dry mode:
𝑢
𝑢
𝑢0·∇
∇
∇𝑃=−𝑀
𝑀𝑠˜𝜏𝑞(𝑃−𝑃0)+ 𝑝𝑇
𝑔𝑢
𝑢
𝑢0·∇
∇
∇𝑞𝑑𝐿
𝜏𝑞−𝑇𝑑𝐿
𝜏𝑇H (𝑃),
where 𝑃0=𝑝𝑇
𝑔
𝑀𝑠h𝐸i−𝑀𝑞h𝑅i
𝑀(6)
and Hdenotes the Heaviside function. We henceforth
drop the 𝑦dependence, simplifying (6) to
d𝑃
d𝑥=−𝑃−𝑃0
𝐿𝑞+𝑝𝑇
𝑔
d
d𝑥𝑞𝑑𝐿
𝜏𝑞−𝑇𝑑𝐿
𝜏𝑇H (𝑃),
where 𝐿𝑞=𝑢0˜𝜏𝑞
𝑀𝑠
𝑀.
(7)
Note that in a state of RCE, h𝐸i=h𝑅i, hence the basic
state precipitation 𝑃0=𝑝𝑇h𝐸i/𝑔, which is the column-
integrated evaporation rate. (7) is essentially a forced
equation with 𝑃relaxed towards 𝑃0on the length scale
𝐿𝑞. Hence, if the dry forcing h𝑞𝑑i/𝜏𝑞−h𝑇𝑑i/𝜏𝑇is felt on
a distance significantly shorter than 𝐿𝑞, the latter will dom-
inate in setting the length of the downstream rain shadow.
𝐿𝑞can be understood as a Lagrangian convective length
scale, whereby a column traveling at velocity 𝑢0undergoes
moisture adjustment on a time scale ˜𝜏𝑞. It is also inversely
proportional to the relative GMS 𝑀/𝑀𝑠, which measures
the efficiency with which a column exports energy through
divergent flow, and thus returns to an equilibrium state.
𝐿𝑞is vanishingly small, and the dry forcing is felt on
equally small distances, in the limit of zero wind (no ad-
vection), instantaneous convective adjustment, or infinite
GMS (allowing for instantaneous return to equilibrium).
We estimate a ratio 𝑀/𝑀𝑠=5(see section 4c) so that for
𝑢0=10 m s−1,𝐿𝑞'1000 km. As suggested in the pre-
ceding paragraph, 𝐿𝑞places a lower bound on the rain
shadow length; such an extensive rain shadow may seem
unrealistic, but in real cases large-scale processes and flow
detouring around topography can shorten the rain shadow.
A time-varying cross-slope wind, including episodes of
reversed flow, would further shorten the time-mean rain
shadow. Note however that precipitation is reduced over
about 1000 km downwind of the Western Ghats (Figure
1c). This is consistent with the very broad rain shadow
downwind of Sri Lanka during the Indian summer mon-
soon (e.g. Biasutti et al. 2012).
c. A linear theory
Equation (7) can be solved for the precipitation field
as a function of moisture and temperature perturbations
induced by “dry” mountain flow. These perturbations can
in turn be obtained with several degrees of complexity,
from a full mountain wave simulation to a linear solution
with uniform background stratification. Here, in the spirit
of the linear model of SB04, we employ linear mountain
wave theory to obtain a closed expression relating mountain
shape to precipitation in the Fourier domain.
The linearized thermodynamic and moisture equations
of the dry mode (see Appendix A) read
𝑢0d𝑇𝑑
d𝑥+𝑤𝑑
d𝑠0
d𝑧=0, 𝑢0d𝑞𝑑
d𝑥+𝑤𝑑
d𝑞0
d𝑧=0,(8)
where we used height coordinates and dropped the 𝑦de-
pendence. Hence, the dry forcing for convection in (7)
becomes
d
d𝑥𝑞𝑑𝐿
𝜏𝑞−𝑇𝑑𝐿
𝜏𝑇=𝑤𝑑 𝐿
𝑢01
𝜏𝑇
d𝑠0
d𝑧−1
𝜏𝑞
d𝑞0
d𝑧,(9)
where we have assumed that d𝑠0/d𝑧and d𝑞0/d𝑧do not
depend on 𝑧. This assumption may seem crude for the
moisture profile, which usually decays rapidly in height,
but it is accurate given an appropriate choice for 𝑤𝑑𝐿 ;
it furthermore retains the first-order physical picture that
lower-tropospheric ascent leads to moistening. We define
the constant 𝜒as the sum of two terms both contributing
to enhanced convection,
𝜒=𝑝𝑇
𝑔1
𝜏𝑇
d𝑠0
d𝑧−1
𝜏𝑞
d𝑞0
d𝑧.(10)
These two terms represent, respectively, lower tropospheric
cooling (due to adiabatic ascent), and moistening (also due
to ascent along a vertically decreasing moisture profile,
d𝑞0/d𝑧 < 0). To obtain a linear equation, the nonlinear
Heaviside function in (7) has to be dropped; this amounts
to allowing negative 𝑄𝑐, which only influences the solution
in the downstream region, where drying and warming by
the dry mode predominate. We now substitute (9) and
(10) into the linearized (7) and take the Fourier transform,
yielding
𝑖𝑘 ˆ
𝑃0(𝑘)+ ˆ
𝑃0(𝑘)
𝐿𝑞
=ˆ𝑤𝑑𝐿 (𝑘)
𝑢0
𝜒, (11)
where 𝑃0=𝑃−𝑃0,𝑘is the horizontal wavenumber and the
Fourier transform is denoted with a hat. Linear, Boussinesq
mountain wave theory expresses ˆ𝑤𝑑as (e.g., Smith 1979)
ˆ𝑤𝑑(𝑘 , 𝑧)=𝑖𝑘 𝑢0ˆ
ℎ(𝑘)𝑒𝑖𝑚 (𝑘)𝑧
and 𝑚(𝑘)=√𝑙2−𝑘2if 𝑘2< 𝑙2
𝑖√𝑘2−𝑙2if 𝑘2> 𝑙2,(12)

6AMS JOURNAL NAME
where 𝑙=𝑁/𝑢0and 𝑁is the Brunt-Väisälä frequency,
assumed positive and constant with height. 𝑁is related
to the basic state lapse rate by 𝑁2' (𝑔/𝑇𝑟)d𝑠0/d𝑧, with
𝑇𝑟a reference temperature taken as 𝑐𝑝×300 K. We have
also assumed that the density of air is uniform, given that
perturbations are averaged over the lower troposphere only.
Combining (11) and (12) gives the final relationship:
ˆ
𝑃0(𝑘)=𝑖𝑘 𝜒
𝑖𝑘 +1/𝐿𝑞
ˆ
ℎ(𝑘)𝑒𝑖𝑚 (𝑘)𝑧𝐿.(13)
After solving for 𝑃0, negative values of precipitation are
avoided by applying the (·)+operator to 𝑃0+𝑃0. The above
linear expression for convective orographic precipitation is
meant to represent time-mean rain rates, as opposed to
single-event or extreme precipitation. It depends on two
parameters (𝜏𝑞and 𝜏𝑇) and a number of physical quantities:
𝑢0,𝑁, the moisture lapse rate d𝑞0/d𝑧, the mountain profile
ℎ(𝑥), and, through 𝐿𝑞, the relative GMS 𝑀/𝑀𝑠. We now
explore sample solutions with idealized mountain profiles
to illustrate predictions of this theory.
d. Example profiles
We now illustrate the behavior of (13), obtaining quanti-
tative estimates for the influence of the convective response
times, the location and magnitude of the maximum precip-
itaiton, and the upstream distance over which precipitation
is enhanced. For the mountain shape, we choose the classic
Witch-of-Agnesi profile ℎ(𝑥)=ℎ0𝑙2
0/(𝑥2+𝑙2
0), where 𝑙0is
the mountain half-width and ℎ0is the maximum height.
The basic state wind 𝑢0is set to 10 m s−1, and the rel-
ative GMS, which influences the Lagrangian convective
length scale 𝐿𝑞, is set to 𝑀/𝑀𝑠=0.2, a representative
value for tropical regions.. We set 𝑁=0.01 s−1, or equiv-
alently d𝑠0/d𝑧'3 J kg−1m−1or 3 K km−1. The lower
tropospheric average in (13) is taken, for convenience, be-
tween two constant height (rather than constant pressure)
surfaces at 𝑧=1000 m and 3000 m. The moisture lapse
rate is computed as an average over the same layer of the
profile 𝑞0(𝑧)=𝑟𝑞sat (𝑇𝑠)𝑒−𝑧/𝐻𝑚where 𝑟=0.8,𝑇𝑠=300 K
and 𝐻𝑚=2500 m, which yields d𝑞0/d𝑧'8.1J kg−1m−1.
Finally, the equilibrium precipitation 𝑃0=4mm day−1.
Using a fast Fourier transform (FFT), we compute solu-
tions with two different mountain heights and varying 𝜏𝑇
and 𝜏𝑞(thick solid curves in Fig. 2). The reference case
uses 𝑙0=50 km, ℎ0=1000 m, and 𝜏𝑇=3hours and 𝜏𝑞=11
hours (with the latter two computed by Ahmed et al. 2020,
yielding 𝐿𝑞=1188 km). Precipitation is significantly en-
hanced starting 1500 km upstream and peaks near the
steepest slope of the ridge’s windward side, with a 7-fold
enhancement compared to the undisturbed 𝑃0. Downwind,
the rain shadow is about 1000 km long and precipitation
overshoots 𝑃0before slowly relaxing back towards it. This
behavior can be understood as follows: precipitation is
suppressed immediately downstream of the mountain be-
cause of the warm and dry lower free-troposphere created
by the gravity wave, then humidity builds in response to
the reduced convective drying (in the presence of the ongo-
ing surface evaporation) past its RCE level to compensate
for the warm anomaly. The latter dissipates as the flow
progresses further downwind, allowing for a convective
overshoot.
A second case shows that increasing the convective time
scales by 50% mainly produces a proportional decrease
in the upstream precipitation perturbation. This occurs
because, in the linear solution (13), 𝑃0is proportional to
𝜒, which is in turn inversely proportional to the time scales;
convection responds more weakly to the dry gravity wave
as the time scales are increased, as specified in the simple
convective closure (3). Additionally, the increase in 𝜏𝑞
results in an increase in 𝐿𝑞, which primarily lengthens the
rain shadow. The third example in Fig. 2 (with ℎ0=500 m)
confirms that, in this linear framework, halving ℎ0, while
keeping other parameters constant, exactly halves 𝑃0.
Linear flow over a Witch-of-Agnesi ridge admits ap-
proximate analytical solutions, allowing us to diagnose the
various scales involved in the upstream precipitation re-
sponse. We perform an inverse Fourier transform of (11)
and then integrate, yielding
𝑃0=𝜒𝑥
−∞
𝑤𝑑𝐿 (𝑥0)
𝑢0
𝑒−𝑥−𝑥0
𝐿𝑞d𝑥0
=𝜒𝜁 𝐿(𝑥) − 𝜒𝑥
−∞
𝜁𝐿(𝑥0)
𝐿𝑞
𝑒−𝑥−𝑥0
𝐿𝑞d𝑥0
(14)
after integrating by parts and noting that 𝑤𝑑(𝑥0, 𝑧)=
𝑢0𝜕 𝜁 (𝑥0,𝑧 )/𝜕𝑥0, where 𝜁(𝑥0, 𝑧 )is the vertical displace-
ment at 𝑥0of a streamline originating upstream at 𝑧. For
mesoscale ridges satisfying 𝑙0𝑁/𝑈1,𝜁𝐿can be approx-
imated as (Queney 1948)
𝜁𝐿(𝑥)=ℎ0
𝑐𝑙2
0−𝑠𝑙0𝑥
𝑥2+𝑙2
0
,(15)
where 𝑐=(cos(𝑁 𝑧/𝑈))𝐿and 𝑠=(sin(𝑁 𝑧/𝑈))𝐿. The
first term on the right-hand side of (14) is proportional
to the lower-tropospheric averaged vertical displacement;
it scales like the cold and moist anomaly created by the
mountain wave, to which precipitation would be propor-
tional in the absence of any moist mode anomalies. This
term represents a distinct influence of the mountain wave
on precipitation compared to that portrayed in the tradi-
tional literature on orographic precipitation: this influence
is a function of vertical displacement, not vertical velocity.
The second term acts as a damping: the moist mode re-
sponds to enhanced precipitation with a negative humidity
anomaly (h𝑞𝑚i<0) that develops on a length scale 𝐿𝑞. In
the limit 𝑙0/𝐿𝑞→0, this term becomes negligible as the
moist mode responds too slowly (in a Lagrangian sense)

7
−2000 −1000 0 1000 2000 3000
Distance
wes of moun ain op (km)
0
10
20
30
40
mm day−1
P wi h
h
0= 1000 m,
τT
= 3 hr,
τq
= 11 hr
P wi h
h
0= 1000 m,
τT
= 4.5 hr,
τq
= 16.5 hr
P wi h
h
0= 500 m,
τT
= 3 hr,
τq
= 11 hr
h
(
x
)/
h
0
0
1
nondimensional heigh
Fig. 2. Precipitation predicted by the linear theory (13) with a Witch-of-Agnesi mountain profile and varying mountain heights and adjustment
time scales (solid lines). The dashed line shows the precipitation obtained when neglecting the damping term in (14), for the parameters given in
the first line of the legend. The thin horizontal line shows the basic-state precipitation 𝑃0.
to the perturbation induced by the gravity wave. However,
even for the small value of 𝑙0/𝐿𝑞'0.05 used here, this
second term provides a sizeable precipitation reduction
(compare the dashed line, which represents the solution
without this second term, with the solid line of the same
color in Figure 2).
Because the second term in (14) varies on large scales
of order 𝐿𝑞, the location of the precipitation maximum,
𝑥max, will be set to first order by the location of maximum
vertical displacement 𝜁𝐿for cases where 𝑙0/𝐿𝑞1. One
obtains
𝑥max =−𝑙01+𝑐2
𝑠2−𝑐
𝑠.(16)
With the above parameters, 𝑥max lies 76 km upstream of
the mountain peak, i.e. slightly upstream of the steepest
slope 𝑥=−𝑙0/(3)(note that with this value of 𝑁/𝑈,𝑠 > 0
and 𝑐 < 0). The amplitude of the precipitation maximum,
𝑃max, will be more affected by the moist-mode damping;
neglecting that damping provides an upper bound,
𝑃max ≤𝑃0+𝜒ℎ0𝑠/[2(1+𝑐2/𝑠2−𝑐/𝑠)].(17)
This expression overestimates the true precipitation max-
imum by about 30% for the above cases (dashed line in
Figure 2).
We can also obtain from (14) an order of magnitude
for the upstream extent of the precipitation enhancement,
defined as the location 𝑥𝑢where 𝑃0exceeds a threshold 𝛿𝑃.
Far upstream of the ridge (i.e. 𝑥 −𝑙0), approximating
𝜁𝐿' −ℎ0𝑙0𝑠/𝑥gives
𝑃0(𝑥) ' − 𝜒𝑠ℎ0𝑙0
𝐿𝑞𝐿𝑞
𝑥−𝑒−𝑥/𝐿𝑞Ei(𝑥/𝐿𝑞)
' − 𝜒𝑠ℎ0𝑙0
𝐿𝑞𝐿𝑞
𝑥+ln 1−𝐿𝑞
𝑥
≤𝜒𝑠 ℎ0𝑙0
2𝐿𝑞𝐿𝑞
𝑥2
.
(18)
Where Ei is the exponential integral function, and the
approximation used in the second equality is given in
Abramowitz and Stegun (1964). The inequality on the
last line is valid for 𝑥≤0. Hence, an upper bound on the
upstream extent is
|𝑥𝑢| ≤ 𝐿𝑞𝑙0𝜒𝑠ℎ0
2𝛿𝑃 .(19)
The right-hand side of (19) is the geometric mean of two
terms, one scaling the upstream extent of the orographic
enhancement in the absence of damping and the other one
being 𝐿𝑞. The larger 𝐿𝑞, the less effective the damping,
hence the farther upstream a given value of 𝛿𝑃 is attained;
in the absence of any moist damping (i.e. 𝐿𝑞/𝑙0→ ∞),
𝑥𝑢=𝑙0𝜒𝑠 ℎ0/𝛿𝑃. With the above values for 𝜒,𝑠, and 𝑙0,
and for ℎ0=1000 m and 𝛿𝑃 =1mm day−1, (19) gives
|𝑥𝑢| ≤ 1746 km. The very good agreement with the true
value, of about 1700 km, is due to cancellation between
the two approximations made in (18); (19) overestimates
the true |𝑥𝑢|by about 20% in the two other cases shown in
Figure 2.
e. Summary
The theory presented in this section uses a quasiequi-
librium closure to to solve for the convective precipitation
forced by a dry orographic gravity wave, through its mod-
ulation of lower tropospheric temperature and moisture.
The most general equation is (6); it retains the nonlinearity
of the convective closure and applies to flows with two
horizontal dimensions. Dropping a horizontal dimension
leads to (7) and introduces 𝐿𝑞, the Lagrangian convective
length scale on which precipitation converges to its equilib-
rium value 𝑃0; this is also the rain shadow length scale in
cases where the dry forcing is felt on small distances. This
nonlinear theory lacks a closure for the dry perturbations
induced by the orographic gravity wave.
Treating the orographic gravity wave linearly, and ne-
glecting the nonlinearity of the convective closure, yields

8AMS JOURNAL NAME
equation (13); this is a closed theory relating mountain
shape to the spatial profile of convective precipitation, in
the spirit of SB04. The theory depends on a number of
physical quantities and the time scales of the convective
closure, 𝜏𝑇and 𝜏𝑞. Whilst (13) can be solved numeri-
cally with a FFT, approximate analytical expressions are
obtained in the case of a Witch-of-Agnesi terrain, yielding
expressions for the location of the precipitation maximum
and the upstream extent of the precipitation enhancement.
For some typical tropical parameter values and a 1000 m
high, 200 km wide ridge, precipitation is enhanced about
2000 km upstream of the mountain, the peak precipitation
occurs slightly upstream of the steepest slope on the up-
wind side, and the rain shadow is around 1000 km wide.
The peak precipitation magnitude depends on parameters
that are relevant both to the dry dynamics (wind speed,
static stability, mountain height) and the moist dynamics
(convective adjustment time scales, relative GMS).
3. Numerical simulations
This section presents the framework we use to test the
theory: a set of convection-permitting simulations in which
a constant horizontal background flow in a long channel
encounters a ridge. In addition to comparing simulated
precipitation from this model with our theory, we evaluate
the validity of the QE and WTG approximations.
a. Simulation setup
We use a three-dimensional idealized version of the
Weather Research and Forecasting model (WRF-ARW,
version 4.1.5, Skamarock et al. 2019), which is fully com-
pressible and nonhydrostatic. The domain is periodic in 𝑥
and 𝑦directions, 9810 km long and 198 km wide with a
3 km horizontal grid spacing. A single 𝑦-invariant ridge
is present. It uses 60 hybrid terrain-following/pressure
vertical levels stretching from the surface to 10 hPa, cor-
responding to about 28 km. The domain length is chosen
so the flow fully recovers to an undisturbed state after
encountering the mountain ridge, before circling back in
the periodic domain. Comparisons of its vertical struc-
ture 3000 km upstream (or, equivalently 6810 km down-
stream) of the ridge with a flat, ocean-covered simulation
(not shown) shows no appreciable difference, confirming
the domain length is sufficient. The 𝑦dimension is large
enough for several convective clouds of sizes 𝑂(1−10)
km to develop in that cross-stream direction. The 3 km
grid spacing is a compromise between the need for real-
istic simulation of convection and the computational cost
of long time integrations. This resolution has been widely
used in large-domain convection-permitting simulations
(Satoh et al. 2019), including geometries similar to ours
(Wing and Cronin 2016; Wang and Sobel 2017). Kirsh-
baum (2020) found that in idealized cases of mechanically
forced orographic convection (though at smaller spatial
scales and time scales) with interactive surface fluxes (i.e.,
their MECH-FLX simulations), a resolution of 2 km gave
similar along-stream precipitation profiles to runs at much
smaller grid spacing 𝑂(100 m). Zhang and Smith (2018)
found that resolutions of 2 km vs 6 km made little differ-
ence in simulating orographic convection over the Western
Ghats.
Surface elevation is
ℎ(𝑥)=ℎ0
21+cos 𝜋𝑥
𝑙0 if |𝑥|< 𝑙0,
0otherwise, (20)
where 𝑙0=100 km (which yields a similar topography to a
Witch-of-Agnesi ridge of half-width 50 km). Hereafter, 𝑥
runs from −4405 km to 4405 km with the mountain at 𝑥=0.
Such one-dimensional terrain forces all parcels to ascend
the ridge, rather than detouring around it, but provides a
first step towards better understanding the interaction of
deep convection with orography. The mountain’s surface
(i.e., where |𝑥|< 𝑙0) is covered with land. To obtain a closed
surface energy budget and avoid having to choose a surface
temperature lapse rate, the land surface is parameterized
with the NoahMP scheme (Niu et al. 2011; Yang et al.
2011), using four soil levels and a no-flux bottom boundary
condition at the lowest level. The rest of the domain is
ocean-covered with fixed sea-surface temperature of 300
K.
The Coriolis force is applied to deviations from a uni-
form geostrophic wind 𝑢0=−10 m s−1(i.e., an easterly
wind), with fixed Coriolis parameter 𝑓=4.97 ·10−5s−1
(corresponding to 20°N). This is equivalent to imposing a
background meridional pressure gradient, which maintains
a constant background geostrophic flow. Microphysics are
computed using the single-moment Thompson et al. (2008)
scheme, surface layer mixing employs the MM5 similarity
theory (Jiménez et al. 2012), and boundary layer fluxes
are parameterized with the Mellor-Yamada-Janjić scheme
(Mellor and Yamada 1982; Janjić 2002). The model is run
without a turbulence scheme (although the surface layer
and PBL schemes do parameterize turbulent mixing), but
comparing the first 20 days of simulation with a run having
nonzero turbulent diffusion does not show any apprecia-
ble difference. Radiation is computed interactively every
minute with the RRTMG scheme (Iacono et al. 2008). All
simulations have a diurnal cycle of insolation but no sea-
sonal cycle (the solar declination angle is fixed to 0◦, a
state of perpetual equinox).
We perform two “control” simulations, one with moun-
tain height ℎ0=1000 m and the other with ℎ0=500 m. The
first case is close to several mountain ranges of South Asia
(Western Ghats, Annam Range, Arakan Yoma) and yields
a nondimensional mountain height 𝑁 ℎ0/𝑢0'1.2. This
suggests moderate flow blocking by the ridge and possible
flow splitting if the mountain were not infinite in 𝑦(Smith
1989), indicating a limitation of our setup. This limitation

9
has implications for lee-side flow: Epifanio and Durran
(2001) showed that downstream temperature perturbations
are higher with a 𝑦-invariant ridge than when the flow is
allowed to split, even with mountains of high aspect ratio.
Upstream flow could also be decelerated more rapidly for
the infinite ridge. However, our use of rotation prevents
the development of an upstream-propagating bore (Pierre-
humbert and Wyman 1985), and instead sets up a barrier
jet (see also section 3b). Convection may also reduce
the nonlinear effects that occur for a high nondimensional
mountain height by lowering the effective stability felt by
the flow, as explained in section 3c.
The ℎ0=500 m case is perhaps a better test for the linear
theory and more realistic because it exhibits reduced block-
ing (𝑁 ℎ0/𝑢0=0.6) and thus is farther from the limit of a
high, infinitely long ridge; in this regime, high aspect ratio
and 𝑦-infinite ridges have similar lee-side flows (Epifanio
and Durran 2001). Both simulations are integrated for 200
days with statistics collected after spinup of 50 days.
An additional simulation with ℎ0=1000 m is run with
latent heating turned off (the “𝐿𝑣=0simulation”) to as-
sess the effect of the mountain on the flow in the absence
of moist convection. It is initialized with mean temper-
ature and moisture soundings from the control and run
for 100 days, with radiation and surface fluxes turned off.
Water can still condense and fall, and virtual temperature
effects are retained. The temperature profile warms in the
boundary layer by about 3 K in this run, but sees little
change aloft; the vertical profile of 𝑁, which controls the
mountain-induced dry gravity wave (Durran 2003), is little
altered.
b. Precipitation and CAPE
Meridionally averaged and time averaged (from days
50 to 200) precipitation and CAPE from the ℎ0=500 m
and 1000 m runs are shown in Figure 3. Here, and sub-
sequently, the 𝑥-axis is oriented so the background wind
flows left to right; i.e. East is on the left, and West on the
right. 𝑃𝑛denotes precipitation in the 𝑛m run. Both CAPE
and precipitation are nearly constant more than 4000 km
downstream of the mountain peak and more than 1500 km
upstream. This, together with the absence of mean upward
motion in that region (not shown), indicates a state of RCE
far from the mountain. This supports our claim that the pe-
riodic domain is long enough for the flow to recover from
the disturbance imparted by the ridge.
We now discuss some key features of the precipitation
profiles. Upstream-averaged precipitation (i.e., from 2000
to 5000 km upwind of the peak) is about 𝑃0=4.5mm
day−1for both runs. The orographic enhancement exceeds
1 mm day−1starting 670 km and 720 km upstream for
𝑃500 and 𝑃1000, respectively. The length scale of this
upstream enhancement is an order of magnitude larger
than typically observed in midlatitudes (e.g., SB04) or
in shallow-convective tropical flows (e.g., Kirshbaum and
Smith (2009)). It is consistent with observational profiles
from Figure 1. Section 2 suggests this is due to the sensi-
tivity of deep convection to thermodynamic perturbations
from the mountain wave that are felt far upstream; section
4 will examine this hypothesis in greater detail. Both runs
have pronounced rainfall peaks (about 12 mm day−1and
20 mm day−1, respectively) on the upwind mountain slope,
about 55 km upstream of the peak (slightly upwind of the
maximum slope). In the rain shadow region, the negative
anomalies are smaller in the ℎ0=500 m run than in the
ℎ0=1000 m run, as expected from the linear theory. Both
runs return to the background value 𝑃0around the same
location, 2000 km downstream. Unlike the linear runs
with a Witch-of-Agnesi profile, there is no clear overshoot
past 𝑃0downstream. This is mostly due to the different
mountain shapes employed, as we will see that this over-
shoot vanishes when applying (13) to the cosine terrain
shape (20) in part 4c. Both runs display a small rainfall
peak immediately downstream of the ridge. This feature is
likely related to nonlinear dynamics, perhaps involving a
hydraulic jump causing strong upward motion at this loca-
tion (see Kirshbaum et al. 2018). We do not devote further
attention to it, given its small amplitude and limited spatial
extent.
Under the assumptions of linearity made in section 2c,
changing the mountain height should scale the orograph-
ically modified precipitation proportionately, so 𝑃1000 −
𝑃0=2(𝑃500 −𝑃0). Figure 3 shows this test of linearity,
in which 𝑃1000 is approximated by (2𝑃500 −𝑃0)+, along-
side the simulated 𝑃1000. There is remarkable agreement,
with the prediction lying within the uncertainty bounds of
𝑃1000 at nearly all locations. This provides confidence in
the relevance of linear theory to mechanically forced oro-
graphic convection (we compare with our theory in detail
in Section 4).
The presence of a barrier jet in our simulations (not
shown) raises the question of its influence on precipitation
(e.g. Neiman et al. 2013). Consistent with theory (Pierre-
humbert and Wyman 1985), it extends to a Rossby defor-
mation radius, about 2000 km, upstream of the mountain.
No significant departure from the background precipitation
is detectable this far upstream. Good agreement on the up-
stream extent of precipitation between the linear theory and
the simulations (see section 4c) is another indication that
barrier jet dynamics have little effect on rainfall in these
runs. Barrier jets might, however, have greater importance
for more nonlinear flows, with stronger jets leading to en-
hanced or shifted rainfall patterns (as shown for the Sierra
Nevada mountains by Neiman et al. 2013).
Examination of the diurnal cycle of precipitation in the
ℎ0=1000 m run can serve as a probe of the importance of
thermal forcing for convection. Between the mountaintop
and 200 km upstream, the diurnal cycle of precipitation
has an amplitude of 20% of the time-mean value. This

10 AMS JOURNAL NAME
Fig. 3. Time and meridional mean precipitation (a) and CAPE (b) in the ℎ0=500 m and 1000 m runs. Thin horizontal lines indicate the
upstream-averaged (between 𝑥=−2500 km and -3000 km) values. The magenta line in panel a shows the result of doubling the orographically-
modified part of the precipitation (𝑃−𝑃0) in the ℎ0=500 m run. Shadings show the interquartile ranges, as computed from binning each quantity
into meridional and 10-day means at each longitude. Note that the topographic shape is different here from the Witch-of-Agnesi used in Figure 2,
which was employed for purposes of analytical tractability.
is smaller than in the RCE part of the domain, where the
relative amplitude is 25%, and suggests a small role for
island surface fluxes in producing rainfall. For compari-
son, Wang and Sobel (2017) simulated thermally forced
convection over isolated islands and reported relative am-
plitudes of the diurnal cycle of around 80%.
We also show CAPE in Figure 3 to illustrate the difficulty
of using CAPE-based closures in theory for orographic pre-
cipitation. Despite the cooling effect of the gravity wave
on the lower free troposphere upstream of the mountain
(analyzed in more detail in Section 4c), CAPE gradually
decreases starting 500 km upstream and drops to almost
zero above the mountain. This can be understood as a pro-
gressive consumption of CAPE by enhanced convection,
triggered by reduced convective inhibition (CIN; Ahmed
et al. (2020) highlight the similarity of the 𝑞−𝑇closure
to CIN-based convective parameterizations). Downstream
of the ridge, reduced convection allows CAPE to build and
even overshoot its upstream value, much like the linear
theory for precipitation (Figure 2). CAPE-based closures
typically diagnose precipitation as the ratio of CAPE to a
convective time scale; the latter would have to vary spa-
tially here to accommodate the absence of proportional-
ity between CAPE and P. The time scale would have to
decrease upstream of and above the mountain (e.g., due
to orography “triggering” convection by mechanical forc-
ing) and increase downstream. The challenge of such an
approach is that the rainfall profile is highly sensitive to
spatial variations of the time scale, rendering derivation of
a physically based closure difficult.
c. Vertical motion
Figure 4 shows time and meridionally averaged vertical
velocity in the ℎ0=1000 m control run and the 𝐿𝑣=0run.
Interaction of the basic state flow with the mountain pro-
duces a gravity wave in the 𝐿𝑣=0run that influences the
flow between about 500 km upstream and 2000 km down-
stream of the mountain, beyond which the wave amplitude
decays to less than 0.01 m s−1. Vertical motion in the con-
trol run strongly resembles that in the 𝐿𝑣=0run from 200
km upstream to 1000 km downstream of the ridge2. Further
upstream (between 𝑥=−1000 km and −200 km), deep as-
cent is visible in the time-mean, suggesting that enhanced
precipitation in this region is due to a feedback of moist
convection on the mountain-induced low-level ascent. Lee
waves are more prominent in the 𝐿𝑣=0run, and 𝑤has
slightly greater amplitude above the mountain, indicating
higher flow nonlinearity in that run. This is consistent
with the mountain wave in the control run experiencing a
lower effective static stability due to latent heat release (see
examples of Lalas and Einaudi (1973) for fully saturated
atmospheres or Lapeyre and Held (2004) and O’Gorman
(2011) for more general, unsaturated flows).
2A simple scale analysis explains the prominence of the “dry”
gravity wave in the moist simulation. Dry vertical motion scales as
𝑤dry ∼𝑢0dℎ
d𝑥'0.15 m s−1, while the diabatic 𝑤scales as 𝑤diabatic =
𝑄𝑐/d𝑠0
d𝑧=(𝑔𝑃 /𝑝𝑇)/(𝑇𝑟𝑁2/𝑔) ' 0.024 m s−1, using 𝑃=20 mm
day−1(multiplied by 𝜌𝑤𝐿𝑣),𝑇𝑟=𝑐𝑝×300 K and 𝑁=0.01 s−1. Hence,
gravity wave dynamics are expected to dominate over the ridge unless
time-mean 𝑃increases by an order of magnitude.

11
Fig. 4. Time and meridionally-averaged vertical velocities from the ℎ0=1000 m control run (a) and 𝐿𝑣=0run (b). Note the nonlinearity of the
color scale.
−300 −200 −100 0 100 200 300
Distance west of mountain top (km)
0
50
100
150
Along-ridge distance (km)
−1.0
−0.5
0.0
0.5
1.0
m s−1
Fig. 5. Instantaneous vertical velocity at 500 hPa in the vicinity of the mountain, from the ℎ0=1000 m run, day 150, 19h.
The similarity between the dry and moist 𝑤above the
ridge raises the question of how much precipitation the dry
wave would produce without moist convective feedback.
The 𝐿𝑣=0simulation does produce precipitation in its
initial times, before moisture has been depleted (there are
no surface fluxes in that run), though in a very different
form (it is of stratiform type, and focused on the upwind
slope only). Including this “dry mode precipitation” in the
theory would require also decreasing 𝑞𝑑𝐿 , which would
have a compensating effect on the moist mode, leaving
total precipitation nearly unchanged.
To illustrate the dynamics producing rainfall over the
ridge, we plot the instantaneous vertical velocity at 500
hPa at a rainy time in Figure 5. Vertical motion over
the upwind slope is composed of isolated deep convective
cells (extending to 200 hPa, not shown) surrounded by cold
pools, qualitatively similar to cells observed upstream over
ocean. This justifies our approach of developing a theory
for orographic precipitation based on the behavior of an
ensemble of convective motions. Past the mountaintop,
convective motions are absent, consistent with the lack of
precipitation there.
d. Thermodynamic equation
We now evaluate the degree to which the linearization
we employed as our starting point in (1a)-(1b), as well as
WTG, are valid approximations.
Figure 6a shows vertically averaged terms from the ther-
modynamic equation (A1a) and its linearized version (1a),
computed using time and meridional mean quantities (de-
noted with an overbar). Diabatic heating is computed as a
residual and plotted with precipitation (or rather 𝑔𝑃/𝑝𝑇)
for comparison. The diagnosed diabatic heating underes-
timates precipitation immediately downstream of its peak,
but shows very good agreement upstream. The linearized
terms (i.e., replacing uand 𝑠by u0and 𝑠0) closely match
the “full” terms except between 30 km upstream to 100
km downstream of the peak. Much of the precipitation is
concentrated upwind of this region, where it is accurately
matched by the diabatic heating diagnosed from linearized
terms. Between 𝑥=−30 km and 𝑥=+100 km, the small

12 AMS JOURNAL NAME
−30
−20
−10
0
10
20
30
K
day 1
(a)
⟨
u⋅ ∇
T
⟩
⟨
ω
∂
s
/∂
p
⟩
⟨
Qc
⟩
⟨
R
⟩
⟨
u0⋅ ∇
T
⟩
⟨
ω
d
s
0/d
p
⟩
⟨
Qc
, linearized⟩
⟨
g
/
pT
⟩
P
400 200 0 200 400
Distance west of mountain top ⟨km⟩
20
10
0
10
20
K day 1
(b)
⟨
u0⋅ ∇
T
⟩
⟨
ω
d
s
0/d
p
⟩
residualdry
⟨
u0⋅ ∇
T
dry⟩
⟨
ω
dryd
s
0/d
p
⟩
Fig. 6. Vertically averaged thermodynamic budget terms. Panel a shows ‘full’ (nonlinear) terms for the ℎ0=1000 m control run, as well as their
linearized version (obtained by fixing uand 𝑠to their basic state values). The simulated precipitation rate is shown in blue for comparison. Panel b
focuses on the linearized heat advection and adiabatic cooling terms, comparing the ℎ0=1000 m control and 𝐿𝑣=0runs. The vertical black lines
show the extent of the mountain.
diabatic heating indicates that the disagreement between
linearized and full terms is unlikely to be due to moist
effects, and rather due to the nonlinear part of the dry
mountain wave.
We now turn to validation of the WTG assumption.
The good match between linearized temperature advec-
tion terms from the ℎ0=1000 m simulation (i.e., h𝑢
𝑢
𝑢0·
∇
∇
∇(𝑇𝑑+𝑇𝑚)i) and from the 𝐿𝑣=0simulation (h𝑢
𝑢
𝑢0·∇
∇
∇𝑇𝑑i),
shown in Figure 6b, ensures that |𝑢
𝑢
𝑢0·∇
∇
∇𝑇𝑚||𝑢
𝑢
𝑢0·∇
∇
∇𝑇𝑑|.
The smallness of the dry residual h𝑢
𝑢
𝑢0·∇
∇
∇𝑇𝑑+𝜔𝑑d𝑠0/d𝑝i
indicates that the dry linearized thermodynamic budget
holds. Taken together, these indicate that h𝜔𝑚d𝑠0/d𝑝i '
h𝑄𝑐i+h𝑅i, which is exactly the WTG approximation.
4. Comparing theory and simulations
Precipitation profiles from the convection-permitting
numerical simulations are now used as a first test of our
theory. We compare these profiles against both the linear
theory (13) and the nonlinear one (7), with the dry forcing
for the latter extracted from the simulations. We will show
that while precipitation profiles are well-captured by the
theory, their upstream amplitude is overestimated. This
issue is addressed by modifying the adjustment time scales
to values appropriate for seasonal means. A last adjust-
ment incorporates the downstream modulation of surface
evaporation and radiative cooling to (7).
a. Temperature and moisture deviations; validity of QE
We first examine lower-tropospheric temperature and
moisture perturbations of the dry mode, which drive con-
vection in (7), and compare these to predictions of the
linear theory (obtained by combining (8) and (12)).
Figure 7a, b shows𝑇0
𝐿and 𝑞0
𝐿from the ℎ0=1000 m con-
trol and 𝐿𝑣=0runs. These are time-averaged, meridional-
averaged, and pressure-averaged (875 hPa to 700 hPa) de-
viations from a mean sounding 3000 km upstream of the
mountaintop. Under our WTG assumptions, 𝑇0
𝐿=𝑇𝑑𝐿 , so
we also plot 𝑇𝑑 𝐿 as predicted by linear mountain wave
theory with 𝑁=0.01 s−1. The agreement is generally
good, especially given our neglect of nonlinearity of the
dry mountain wave, shown to be important above the peak
in Figure 6. However, upstream of the mountain, 𝑇0
𝐿in the
control run is about a third smaller in magnitude than in
the 𝐿𝑣=0run, which is consistent with the dry mode feel-
ing a reduced static stability in the control run with latent
heat release (see section 3c). Including this effect in the
theory would challenge our assumption that the dry mode
is unaffected by the moist mode, and is left as potential
improvement for future work.
An equally important modulator of convection is 𝑞0
𝐿,
which is shown in Figure 7b. We do not expect the control
and 𝐿𝑣=0runs to produce similar distributions of 𝑞𝐿:𝑞𝑚𝐿
is expected, upstream, to be reduced by the precipitation
forced by the dry mode, which happens as expected. Lin-
ear theory (with the same moisture lapse rate as in section
2d) matches 𝑞𝑑𝐿 estimated from the 𝐿𝑣=0run upstream
of the mountain peak (except its peak value is too high),
and overestimates orographic wave-induced drying down-
stream. Moisture depletion at the beginning of the dry run
(see section 3c) may explain this discrepancy, as 𝑞𝑑 𝐿 is
evaluated from the last 50 days of the 𝐿𝑣=0run.

13
−1
0
1
2
K
(a)
T
′
L
,
h
0= 1000 m control run
T
′
L
,
h
0= 1000 m,
Lv
= 0 run
TdL
, linear theory
h
(
x
)
4
2
0
2
K
(b)
q
′
L
,
h
0= 1000 m control run
q
′
L
,
h
0= 1000 m,
Lv
= 0 run
qdL
, linear theory
h
(
x
)
2000 1000 0 1000 2000
Distance west of mountain top (km)
10
5
0
5
K
(c)
(
θ
*
e
,
L
)
′
,
h
0= 1000 m control run
(
θe
,
b
)
′
,
h
0= 1000 m control run
h
(
x
)
0
1
height (km)
0
1
height (km)
0
1
height (km)
Fig. 7. Profiles of lower-tropospheric averaged temperature (a) and moisture (b) perturbations relative to 3000 km upstream in the ℎ0=1000
m control and 𝐿𝑣=0simulations, as well as perturbations computed from the linear theory ((8) and (12)). Panel c shows perturbations of lower-
tropospheric averaged saturation equivalent potential temperature and boundary layer averaged equivalent potential temperature, in the ℎ0=1000
m control run. Gray shading shows the convectively decoupled region where 𝑃1000 <2mm day−1.
Altogether, these results suggest that linear theory (13)
will estimate upstream distribution of precipitation nearly
as accurately as the nonlinear theory (7) forced by the
simulated 𝑞𝑑𝐿 and 𝑇𝑑 𝐿 . One might expect the linear the-
ory to yield a higher magnitude, due to an overestimated
temperature decrease, if a reduced effective static stabil-
ity is not used. Several elements are expected to affect
predictions of the linear theory in the rain shadow: the
overestimated subsidence-induced drying, and the neglect
of the constraint that convective heating be non-negative.
The thermodynamic perturbations displayed in Figure
7a, b also allow evaluation of the validity of the QE hy-
pothesis, which is often expressed as
𝛿𝜃𝑒, 𝑏 ∝𝛿𝜃∗
𝑒(𝑝)(21)
where 𝜃𝑒,𝑏 is the equivalent potential temperature below
cloud base, 𝜃∗
𝑒(𝑝)is the saturation equivalent potential
temperature at a level that is in QE with sub-cloud base
air, and 𝛿expresses a variation in space or time (e.g.
Emanuel et al. 1994). Past observational tests of QE have
assessed these variations, evaluating them only in regions
where convection is not suppressed and vertically averag-
ing the right-hand side of (21) over the layer of interest (e.g.
Brown and Bretherton 1997). Figure 7c shows time-mean
horizontal deviations of 𝜃𝑒,𝑏 (averaged over the bottom
three terrain-following levels, from approximately 70 m
to 400 m above the surface) and 𝜃∗
𝑒(averaged over our
standard lower-tropospheric layer and hence denoted 𝜃∗
𝑒, 𝐿 )
from their far upstream values. Upwind of the mountain
to the location of the peak precipitation, the decrease in
𝜃𝑒,𝑏 is commensurate with that in 𝜃∗
𝑒, 𝐿 , indicating that QE
holds to a good approximation there. Between the moun-
taintop and 𝑥=1000 km, where convection is suppressed
(gray shaded region in Fig. 3c), one does not expect QE to
hold, and 𝜃𝑒,𝑏 is greatly reduced compared to 𝜃∗
𝑒, 𝐿 ; the two
variables then converge as convective activity is recovered
farther downstream.
Although this correspondence between variations in 𝜃𝑒,𝑏
and 𝜃∗
𝑒, 𝐿 in convecting regions indicates that a lower-
tropospheric QE relation holds in the time mean in our
simulations, it also seems worthwhile to discuss some rel-
evant time scales. QE is based on the idea that the evolution
of the forcing (here, cooling and moistening of the lower
troposphere by the orographic wave) occurs on time scales
much longer than those of the convective response. In an
Eulerian framework, the orographic mechanical forcing is
steady and we are concerned with the time mean response,
so the time scales of the forcing and convective response

14 AMS JOURNAL NAME
are extremely well separated. One could, however, take a
Lagrangian view in which air columns are advected by the
mean wind toward the mountain, so the forcing evolves on
a time scale 𝑙/𝑢0, about 7 hours if 𝑙is the 𝑒-folding length
scale of perturbations upstream of the mountain (about 250
km). That time scale is not well-separated from the con-
vective response time of order 2 to 3 hours (e.g., Tulich
and Mapes 2010; Ahmed et al. 2020). However, similarly
rapid transport of air masses into the region of precipitating
large-scale ascent occurs in disturbances that QE is often
used to describe, such as the Madden-Julian Oscillation
(Madden and Julian 1971; Haertel et al. 2017) and tropi-
cal cyclones (Emanuel 2007). In such disturbances and in
our orographic forcing, QE is not being used to describe a
single Lagrangian event, but the mean of a large ensemble
of such events. In an Eulerian framework, such a system
would be described by a fixed column subject to a steady
advective tendency, with the large-scale forcing evolving
on long, seasonal time scales.
b. Precipitation from prior theories
Before comparing the simulated mean rainfall to the the-
ory developed in section 2, we evaluate the performance
of two well-known theories for mechanically forced oro-
graphic precipitation: the “upslope” model (see e.g. Roe
2005) and the linear theory of SB04. Both assume that
condensation occurs due to upward motion in a saturated
layer. The upslope model assumes terrain-parallel flow at
all levels, while SB04 use linear mountain wave theory to
solve for 𝑤. The latter model specifies time scales for the
conversion of condensed water to raindrops and subsequent
fallout.
With the topographic shape defined in (20), terrain-
parallel flow produces ascent over the upwind slope only,
and downward motion over the lee slope only. This renders
the upslope model incapable of capturing any precipita-
tion enhancement upstream of 𝑥=−100 km, and any rain
shadow downstream of 𝑥=−100 km. It predicts peak pre-
cipitation of around 450 mm day−1, an order of magnitude
larger than the simulated value. We do not plot profiles
from the upslope model because of this poor fit.
Although convection is not specifically represented in
the SB04 model, latent heating is taken into account by
using the moist Brunt-Väisälä frequency, 𝑁𝑚, as a measure
of flow stability. However this quantity is imaginary in the
present case as the flow is potentially unstable in the lower
troposphere, with 𝛾 > Γ𝑚, where 𝛾=−𝑑𝑇 /𝑑𝑧 (resp. Γ𝑚)
is the environmental (resp. moist-adiabatic) lapse rate.
Because of the similarity of the ascent patterns with and
without latent heating (see Figure 4), we choose to use
SB04’s model by replacing 𝑁𝑚with 𝑁.
Because this theory assumes saturated flow, it is ex-
pected to overestimate precipitation if non-precipitating
times are not accounted for. So we compute the time-
and meridional-mean rain rate excluding non-precipitating
times (Appendix B provides details), which roughly dou-
bles 𝑃0(i.e. 𝑃1000 −𝑃0) compared to the full time-mean.
This suggests the SB04 prediction of 𝑃0should be divided
by 2, but a better fit to the WRF simulation is obtained
when dividing 𝑃0by 2.5 (green line in Figure 8a). We
used conversion and fallout times of 2000 s, 𝑃0=4.5mm
day−1,Γ𝑚=4.3K/km, 𝛾=5K/km and 𝑁=0.012 s−1.
While the SB04 theory captures the general shape of
precipitation upstream of the mountain, it underestimates
its upstream extent. But its largest bias lies in the lee, where
instead of reproducing the simulation’s long rain shadow, it
predicts a strong secondary precipitation maximum due to
the mountain wave ascent there. Hence, although the SB04
model is regarded as a skillful predictor of midlatitude
orographic precipitation, it has important deficiencies for
this case of tropical deep convective rainfall.
c. Precipitation from the present theory
We now assess the linear theory (13). Applying it with
the same parameters used in section 2d overestimates the
unconditional time-mean peak precipitation in the WRF
simulation by a factor of two to three (not shown), as
occurred for the SB04 theory. We hypothesize that this
occurs because increased adjustment time scales (𝜏𝑞and
𝜏𝑇) are needed when applying the theory to seasonal-scale
time means. Spatio-temporal averaging of the convective
heating term (3) necessarily includes non-convective re-
gions and times, yielding higher effective adjustment scales
(Ahmed et al. 2020). The theoretical 𝑃0is, in the upstream
region, inversely proportional to the adjustment scales be-
cause (7) is linear in that region where 𝑃 > 0. This sug-
gests that accounting for non-precipitating times requires
increasing 𝜏𝑞and 𝜏𝑇, compared to the “instantaneous” val-
ues estimated by Ahmed et al. (2020). The simulated mean
precipitation is best fit by increasing both adjustment times
2.5-fold (yielding 𝜏𝑇=7.5hours and 𝜏𝑞=27.5hours), near
the factor of 2 expected from excluding non-precipitating
times from the time-average (see above and Appendix B).
Figure 8b displays the application of (13) to the moun-
tain profile (20) used in the simulations, with ℎ0=1000 m
(Fig. 8c shows the same for ℎ0=500 m). We set 𝑃0=4.5
mm day−1to match the simulations (see section 3b). We
use the same values of d𝑞0/d𝑧and d𝑠0/d𝑧as in section
2d, and we keep 𝑀/𝑀𝑆=0.2henceforth3. Linear theory
overestimates the peak precipitation on the upwind flank
of the ridge. The fit might be improved by accounting for a
lower effective static stability (as explained in section 4a),
but we did not attempt this. Note that without adjusting
the time scales, it would be substantially higher than what
the linear theory produced for the Witch of Agnesi profile
3Indeed, following the method outlined by Yu et al. (1998), we
estimate 𝑀𝑠=2370 J kg−1and 𝑀ℎ=455 J kg−1in the ℎ0=1000 m
run (respectively 2341 J kg−1and 450 J kg−1in the ℎ0=500 m run).

15
0
5
10
15
20
25
mm day−1
(a)
P
1000
P
1000 with Smith & Barstad theory
h
(
x
)
0
5
10
15
20
25
mm day−1
(b)
P
1000
P
1000 with (7),
qdL
and
TdL
from sim.,
τT
= 7.5 hr,
τq
= 27.5 hr
P
1000 with (13),
τT
= 7.5 hr,
τq
= 27.5 hr
h
(
x
)
−2000 −1000 0 1000 2000
Distance west of mo ntain top (km)
0
5
10
15
20
25
mm day−1
(c)
P
500
P
500 with (7),
qdL
and
TdL
from sim.,
τT
= 7.5 hr,
τq
= 27.5 hr
P
500 with (13),
τT
= 7.5 hr,
τq
= 27.5 hr
h
(
x
)
0
1
height (km)
0
1
height (km)
0
1
height (km)
Fig. 8. Profiles of mean precipitation rates over 1000 m high (a and b) and 500 m high (c) mountains. Solid lines are simulated rates, dashed
lines are computed from the nonlinear theory (7) with lower-tropospheric perturbations 𝑞𝑑𝐿 and 𝑇𝑑𝐿 diagnosed from simulations (see text), and
dotted lines are profiles from the closed linear theory (13). The convective adjustment scales are taken from Ahmed et al. (2020) (a), then increased
2.5-fold (b and c). The thin horizontal lines show 𝑃0.
(Figure 2) because of the steeper ascent imposed by (20).
A more gradual ascent provides a greater distance for 𝑃to
relax back to 𝑃0(first term on the right-hand side of (7)).
As an intermediate level of complexity between linear
theory and the convection-permitting simulations, Figures
8b, c display mean precipitation computed from the gen-
eral precipitation equation (7). This required estimating
the dry forcings 𝑇𝑑𝐿 and 𝑞𝑑𝐿 .𝑇𝑑 𝐿 was estimated from the
ℎ0=1000 m and ℎ0=500 m simulations. We did not take
it from the 𝐿𝑣=0run because we suspect the dry mode in
that simulation behaves differently due to a higher effective
static stability than in the control run (see sections 3c and
4a); using a more accurate estimate offers a better test of
the theory. We estimated 𝑞𝑑𝐿 from the 𝐿𝑣=0simulation
(we used half this value for the ℎ0=500 m run, due to the
absence of an ℎ0=500 m, 𝐿𝑣=0run), because it cannot
be directly estimated from the control simulation due to the
moist and dry mode both contributing to specific humidity
anomalies (see section 4a). Downstream of the mountain,
𝑞𝑑𝐿 oscillates in a way not seen in the dry mode of the
control run (owing to the smaller amplitude of the oro-
graphic lee wave, see Figure 4), but we did not correct for
this. All other parameters are the same as above, and (7)
is integrated numerically with a backward differentiation
formula method. As expected from the profiles in Figure
7a, b, precipitation computed this way compares well with
linear theory, except for a peak that is smaller and shifted
modestly downstream, hence closer to the simulated rain-
fall.
Two effects explain the differences between precipitation
rates computed from our general precipitation equation
(7) and our linear theory (13) downstream of the moun-
tain. First, oscillations in the precipitation rate, including
a weak local maximum between 400 km and 600 km down-
wind of the mountain peak, are due to the shapes of 𝑞𝑑 𝐿
and 𝑇𝑑𝐿 in the convection-permitting simulations. Sec-
ond, the Heaviside function in (7) increases the length of
non-precipitating regions; forbidding negative values of 𝑃
reduces the recovery rate (𝑃0−𝑃)/𝐿𝑞, so that 𝑃converges
towards 𝑃0more slowly.
d. Surface evaporation and radiative cooling in the rain
shadow
Using the nonlinear theory with increased time scales
overestimates the simulated precipitation downstream of

16 AMS JOURNAL NAME
0
50
100
150
W m−2
(a)
Surface latent heat flu
Surface sensible heat flu
⟨
R
⟩
h
⟨
x
⟩/
h
0
−2000 −1000 0 1000 2000
Distance west of mountain top ⟨km⟩
0
5
10
15
20
25
mm day−1
(b)
P
1000
P
500
P
1000 with ⟨7⟩ and adapted ⟨
E
⟩ and ⟨
R
⟩
P
500 with ⟨7⟩ and adapted ⟨
E
⟩ and ⟨
R
⟩
h
⟨
x
⟩/
h
0
0
1
nondimensional height
0
1
nondimensional height
Fig. 9. (a) Time and meridionally averaged surface fluxes and radiative cooling from the ℎ0=1000 m simulation. Thin horizontal lines indicate
the upstream-averaged (between -2500 km and -3000 km) values. (b) As in Figure 8b,c, without solutions from (13), and where solutions from (7)
take into account spatial variations in surface evaporation and radiative cooling. Compare the profiles between 1000 km and 2500 km downstream
of the mountains.
the mountain (Figure 8b, c), possibly because the theory
neglects variations in surface latent heat fluxes and radia-
tive cooling (i.e., it assumes 𝑃0is uniform). Profiles of
h𝐸i,h𝑅i, and the surface sensible heat flux are shown in
Figure 9a.
The starkest deviations from the constant upstream val-
ues occur above the mountain and 100 km to 2000 km
downwind of the mountaintop. A strong reduction in sur-
face evaporation above the ridge, which is covered with
land, is largely compensated by increased sensible heat-
ing. Downstream, surface evaporation increases over a
200 km-long region before decreasing to 80% of its up-
stream value. This decrease in the region 𝑥 > 200 km can
be attributed (not shown) to increased near-surface relative
humidity, likely caused by suppressed convection in that
region. Suppressed convection also reduces the occurrence
of high clouds and, as a consequence, increases radiative
cooling by up to 10% downstream. Earlier theory (e.g.
Fuchs and Raymond 2002) parameterized this effect with
a feedback factor, setting h𝑅i0proportional to 𝑃0(where
h𝑅i0denotes radiative cooling minus its RCE value).
The downstream modulation of h𝐸iand h𝑅istrongly
decreases 𝑃0, according to (6). Accounting for this in (7)
nearly halves the computed rain rate downstream, as shown
in Figure 9b. Incorporating a closure for h𝐸iand h𝑅iinto
our theory, rather than diagnosing these from simulations,
is left for future work.
5. Summary and conclusions
We present a theory of convective precipitation forced
by the mechanical effects of orography at low latitudes.
It starts with decomposition of the flow into the sum of
a dry mode, carrying the orographic gravity wave, and a
moist mode bearing the convective response. Precipita-
tion is assumed to be produced entirely by the moist mode,
whose dynamics are vertically truncated and subject to
the WTG approximation. Convective heating responds to
lower-tropospheric temperature and moisture perturbations
carried by both modes in a quasi-equilibrium framework.
Two degrees of complexity can be employed. The first
consists of computing the dry mode perturbations with a
numerical model for use in the theory, with the theory
retaining the nonlinearity of the convective closure. The
second option neglects this nonlinearity and assumes lin-
ear mountain wave dynamics to derive a linear model of
convective orographic rainfall much in the spirit of the
SB04 midlatitude model. The linear theory provides ana-
lytical solutions to probe the sensitivity of maximum pre-
cipitation, upstream extent of precipitation enhancement,
and rain-shadow length to the physical parameters at play,
namely upstream wind, convective adjustment scales, rel-
ative GMS, static stability, and moisture lapse rate.
This theory describes time-mean rainfall in tropical oro-
graphic regions, and assumptions related to the vertical
structure of the moist mode and WTG prevent its use in
midlatitudes. Its applicability to short-term precipitating
events is also questionable, owing to the unsuitability of the
QE assumption at these time scales. It does not account for
cloud delays nor the advection of hydrometeors, although
we believe these could be added to the linear version with-
out much difficulty. Most importantly, the model is not

17
suited to the description of thermally forced orographic
convection.
The theory is tested against convection-permitting sim-
ulations in long-channel geometry. The dry orographic
gravity wave is prominent in the moist model, justifying
its consideration as a driver of terrain-generated convec-
tion. After correcting adjustment time scales to account
for the effects of seasonal averaging, the theory accurately
reproduces precipitation rates simulated by this model, es-
pecially upstream of the mountain peak. The linear version
is skillful at modeling dry temperature and moisture devi-
ations, and hence precipitation, upwind of the ridge. The
mountain alters surface evaporation and radiative cooling
far downstream, reducing rainfall there. We note that the
nondimensional mountain heights considered here avoid
strongly nonlinear flows; testing the theory in such cases
is left for future work.
This theory is envisioned as a tool for understanding
the spatial variability of rainfall in tropical orographic re-
gions where mechanical forcing prevails. Examples in-
clude South Asia and Mexico during their respective sum-
mer monsoons, or most tropical land regions subject to
an autumn monsoon (Ramesh et al. 2021). Understanding
the interaction of large-scale flow with orography in such
regions is key to comprehending past variability in tropical
rainfall, as well as predicting changes in coming decades.
The theory could also be used to probe the importance
of orography in shaping large-scale tropical circulations
through its influence on moist convection.
Acknowledgments. This material is based on work sup-
ported by the U.S. Department of Energy, Office of Sci-
ence, Office of Biological and Environmental Research,
Climate and Environmental Sciences Division, Regional
and Global Model Analysis Program, under Award DE-
SC0019367. It used resources of the National Energy
Research Scientific Computing Center (NERSC), which is
a DOE Office of Science User Facility. QN acknowledges
support from the McQuown fund at UC Berkeley. The
authors wish to thank Daniel Kirshbaum for very helpful
feedback and suggestions.
Data availability statement. Processed WRF output
and code used in producing the figures will be archived
at Zenodo, with a DOI issued once the review process
and any needed revisions are complete. Raw WRF out-
put is available from the authors upon request (qnico-
las@berkeley.edu), and GPM IMERG and ERA5 data are
publicly accessible online.
APPENDIX A
Decomposition into dry and moist modes
We describe flow over a tropical mountain as the sum of
a dry mode (representing the influence of orography in the
absence of condensation) and a moist convective one. The
dry mode affects the moist mode through convective heat-
ing, but the moist mode does not feed back on the dry mode.
Steady-state thermodynamic and moisture equations are
𝑢
𝑢
𝑢·∇
∇
∇𝑇+𝜔𝜕𝑠
𝜕 𝑝 =𝑄𝑐−𝑅, (A1a)
𝑢
𝑢
𝑢·∇
∇
∇𝑞+𝜔𝜕𝑞
𝜕 𝑝 =𝑄𝑞+𝐸, (A1b)
with notation as in section 2a. Wind, moisture and tem-
perature are decomposed as follows:
𝜔=0+𝜔𝑑+𝜔𝑚,
𝑢
𝑢
𝑢=𝑢
𝑢
𝑢0+𝑢
𝑢
𝑢𝑑+𝑢
𝑢
𝑢𝑚,
𝑞=𝑞0(𝑝) + 𝑞𝑑+𝑞𝑚,
𝑇=𝑇0(𝑝) + 𝑇𝑑+𝑇𝑚,
(A2)
where subscripts 𝑑and 𝑚denote dry and moist modes,
respectively.
We now linearize (A1a)-(A1b) about a state of uniform
horizontal wind 𝑢
𝑢
𝑢0, zero vertical velocity, and horizon-
tally uniform dry static energy 𝑠0(𝑝)and specific humidity
𝑞0(𝑝). We justify this approach based on scales estimated
from the ℎ0=1000 m control and 𝐿𝑣=0simulations. For
horizontal advection terms, the assumption |𝑢
𝑢
𝑢𝑚|  |𝑢
𝑢
𝑢0|is
well justified as the standard deviation of 𝑢
𝑢
𝑢is less than
2ms−1in the RCE region of our convection-permitting
simulations. The dry mode, on the contrary, is expected to
be nonlinear, as |𝑢
𝑢
𝑢𝑑|/|𝑢
𝑢
𝑢0| ' 𝑁 ℎ0/𝑢0. Nevertheless, we find
in the simulations that |𝑢
𝑢
𝑢𝑑|  0.3|𝑢
𝑢
𝑢0|everywhere except
right above the mountain. In section 3d, we show that such
nonlinearity in the dry dynamics does not seem to affect
the moist mode dynamics, despite its local importance in
the thermodynamic budget.
For vertical advection terms, we assume
𝜕𝑠𝑑/𝜕 𝑝, 𝜕 𝑠𝑚/𝜕 𝑝 𝜕 𝑠0/𝜕 𝑝, with a similar treat-
ment for moisture stratification. Again, this is supported
by the simulations except for the dry perturbations right
above the mountain. In height coordinates, 𝜕𝑠0/𝜕𝑧 '4
K/km, and deviations above the mountain in the 𝐿𝑣=0
simulation give |𝜕𝑠𝑑/𝜕 𝑧|.1−2K/km. Static stability
variations from the moist mode are at least two orders of
magnitude smaller than the basic state static stability (see
Neelin and Zeng 2000).
Using these approximations, (A1a)-(A1b) become
𝑢
𝑢
𝑢0·∇
∇
∇𝑇𝑑+𝜔𝑑
𝜕𝑠0
𝜕 𝑝 +𝑢
𝑢
𝑢0·∇
∇
∇𝑇𝑚+𝜔𝑚
𝜕𝑠0
𝜕 𝑝 =𝑄𝑐−𝑅, (A3a)
𝑢
𝑢
𝑢0·∇
∇
∇𝑞𝑑+𝜔𝑑
𝜕𝑞0
𝜕 𝑝 +𝑢
𝑢
𝑢0·∇
∇
∇𝑞𝑚+𝜔𝑚
𝜕𝑞0
𝜕 𝑝 =𝑄𝑞+𝐸 . (A3b)
The first two terms in each equation (the dry mode) balance
each other, expressing conservation of potential tempera-
ture and moisture in the dry perturbation. Note that 𝑄𝑞
could potentially be nonzero in the dry mode (i.e., some

18 AMS JOURNAL NAME
Fig. B10. (a) Hovmöller diagram of precipitation rate in the
ℎ0=1000 m run (in the 𝑥-𝑡plane). An example characteristic line
(𝐿175days) is shown in white. (b) Precipitation averaged over character-
istic lines. The 𝑥-axis shows the time at which a characteristic line starts,
at 𝑥0=−4905 km. The solid black line denotes the threshold defining a
precipitating characteristic (2 mm day−1).
moisture could condense and fall, even in the absence of
latent heating), but this effect is confined to the upwind
mountain slope in the 𝐿𝑣=0run (consistently with Zhang
and Smith (2018)); further discussion is provided in section
3c. Accounting for the dry mode balance in (A3a)-(A3b)
yields (1a)-(1b).
APPENDIX B
Selection of precipitating times
To justify the increase in adjustment time scales needed
to represent seasonal-mean dynamics, we show how the
exclusion of non-precipitating times leads to a doubling
of the time-mean rain rate. Rainfall in the ℎ0<