%% To submit your paper:
\documentclass[draft,linenumbers]{agujournal}
%\draftfalse
\usepackage{url}

%% For final version.
%\documentclass{agujournal}

% Now, type in the journal name: \journalname{<Journal Name>}

% ie, \journalname{Journal of Geophysical Research}
%% Choose from this list of Journals:
%
% JGR-Atmospheres
% JGR-Biogeosciences
% JGR-Earth Surface
% JGR-Oceans
% JGR-Planets
% JGR-Solid Earth
% JGR-Space Physics
% Global Biochemical Cycles
% Geophysical Research Letters
% Paleoceanography
% Radio Science
% Reviews of Geophysics
% Tectonics
% Space Weather
% Water Resource Research
% Geochemistry, Geophysics, Geosystems
% Journal of Advances in Modeling Earth Systems (JAMES)
% Earth's Future
% Earth and Space Science
%
%

\journalname{JAMES}




%\usepackage[letterpaper]{geometry}
%\geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}
%\usepackage{graphicx}
%\usepackage{url}
%\usepackage{lineno}
%\linenumbers*[1]
%
%  Uncomment the following command to allow illustrations to print
%   when using Draft:
%\setkeys{Gin}{draft=false}
%
% Author names in capital letters:
%\authorrunninghead{RAYMOND AND FUCHS}

% Shorter version of title entered in capital letters:
%\titlerunninghead{MJO AND WARM POOL}

%Corresponding author mailing address and e-mail address:
%\authoraddr{Corresponding author: D. J. Raymond, Physics Department
%  and Geophysical Research Center, New Mexico Tech, Socorro, NM 87801,
%  USA.  (david.raymond@nmt.edu)}

\begin{document}

\title{The Madden-Julian oscillation and the Indo-Pacific warm pool}

%\author{David J. Raymond\thanks{Corresponding author address: \\
%Physics Department and Climate and Water Center, \\
%New Mexico Tech, \\
%Socorro, NM, USA}}

\authors{David J.\ Raymond\affil{1}\thanks{Corresponding author
    address: Physics Department, New Mexico Tech, Socorro, NM, USA},
  \v{Z}eljka Fuchs\affil{1}}

\affiliation{1}{Physics Department and Climate and Water Center,
  New Mexico Tech, Socorro, New Mexico, USA.}

\correspondingauthor{David J.\ Raymond}{david.raymond@nmt.edu}

\begin{keypoints}
\item The MJO and warm pool are assumed to interact only via linear
  superposition.
\item The joint rainfall maximum moves across the warm pool more
  slowly than the MJO.
\item The westerly wind anomaly moves at nearly the MJO phase speed.
\end{keypoints}

\begin{abstract}
  A minimal model of the interaction of the Madden-Julian oscillation
  (MJO) with the Indo-Pacific warm pool is presented.  This model is
  based on the linear superposition of the flow associated with a
  highly simplified treatment of the MJO plus the flow induced by the
  warm pool itself.  Both of these components parameterize rainfall as
  proportional to the column water vapor, which in turn is governed by
  a linearized moisture equation in which WISHE (wind induced surface
  heat exchange) plays a governing role.  The MJO component has
  maximum growth rate for planetary wavenumber $1$ and is equatorially
  trapped with purely zonal winds.  The warm pool component exhibits a
  complex flow pattern, differing significantly from the classical
  Gill model as a result of the mean easterly flow.  The combination
  of the two produce a flow that reproduces many aspects of the
  observed global flow associated with the MJO.
\end{abstract}

\section{Introduction}
\label{introduction}

The Madden-Julian oscillation (MJO) is a global-scale tropospheric
wave that moves to the east around the tropical belt with a period of
$30$-$60$ d.  It is coupled to deep convection over the Indo-Pacific
warm pool (extending normally from $\approx 60^{\circ}$ E to the
dateline) but is visible in the $200$ hPa winds at all longitudes
\citep{madden_julian1971,madden_julian1972,madden1986,zhang2005,waliser_etal2009}.
For these periods and wavenumber 1 the mean propagation speed of the
MJO is $8$-$16$ m s$^{-1}$, which is greater than its observed
propagation speed of $\approx 5$ m s$^{-1}$ over the warm pool.

Many different models for the basic physics of the MJO have been
advanced over the years, but we are still not certain about the
fundamental mechanisms underlying this phenomenon.
\citet{fuchs_raymond2017} review many of these models and advance a
maximally simplified theory for the MJO, based on the work of
\citet{fuchs_raymond2005}.  This theory is a modified version of the
linear WISHE (wind-induced surface heat exchange) theories of
\citet{emanuel1987} and \citet{neelin_etal1987}.  Instead of setting
the rainfall proportional to the surface moist entropy flux or
evaporation rate as in the original WISHE models, a linearized
moisture equation acts as an intermediary, with precipitation set
proportional to the column water vapor.  This treatment follows the
ideas of \citet{raymond2000} and \citet{sobel_etal2001} and is
consistent with the observations of \citet{bretherton_etal2004}.
Unlike the original WISHE models, the moisture mode treatment results
in maximum growth rate at planetary wavenumber $1$ with predicted
propagation speeds in the observed range.

The MJO in \citet{fuchs_raymond2017} is thus characterized as a linear
WISHE-moisture mode.  However, it is not a moisture mode instability
associated with negative effective gross moist stability in the sense
of \citet{raymond_fuchs2009}, \citet{sugiyama2009},
\citet{sobel_maloney2012, sobel_maloney2013}, and
\citet{adames_kim2016}; values of the effective gross moist stability
are always taken to be non-negative.

The model of \citet{fuchs_raymond2017} assumes a zonally symmetric
environment.  However, the tropics exhibit strong zonal variations in
sea surface temperature (SST), with prominent cool regions in the
eastern Pacific and Atlantic and an extended region of deep warm water
in the western Pacific and eastern Indian Ocean.  Even though the MJO
is a global phenomenon, this Indo-Pacific warm pool hosts the
strongest convection associated with the MJO.  It is therefore
desirable to determine how the warm pool interacts with the MJO.

Consistent with the simplified nature of the \citet{fuchs_raymond2017}
model, we pose the simplest possible model for the interaction of a
traveling MJO mode with the Indo-Pacific warm pool.  One could imagine
a complex process in this case, e.g., \citet{adames_kim2016}, and it is
unlikely that this interaction would be strictly linear.  However, we
confine ourselves to investigating the characteristics of a simple
linear superposition of the warm pool and MJO circulations in order to
see how far this idealization can be pushed.  Linear WISHE is arguably
justifiable in this case because global mean easterlies of order
$2$-$3$ m s$^{-1}$ exist in the lower tropical troposphere for all
seasons in spite of the presence of westerlies over the warm pool
\citep{peixoto_oort1992}.  Global mean forcing should be what is most
important for a global oscillator.

\section{The MJO and the warm pool}
\label{obs}

In this section the National Centers for Environmental Prediction
Final Analysis (FNL) and the Reynolds SST analysis
\citep{reynolds_marsico1993} are used to examine the interaction
between the Indo-Pacific warm pool and the MJO with minimal smoothing
and only simple averaging.  The year 2001 is chosen for examination in
detail.  That year is close to being ENSO-neutral (though trending
from weak La Ni\~{n}a to weak El Ni\~{n}o) and is one for which we
have collected considerable data.  \citet{waliser_etal2009} provide a
broader analysis based on many years of compositing.

\begin{figure}
  \centering
  \includegraphics[width=7cm]{means.pdf}
  \caption{Time (January-December 2001) and latitudinal (10$^{\circ}$
    S to 10$^{\circ}$ N) means showing the SST (top), 10 m zonal wind
    (middle; red line indicates mean value), and saturation fraction
    (bottom) as a function of longitude.}
  \label{means}
\end{figure}

Figure \ref{means} presents the annual average SST, 10 m zonal wind,
and saturation fraction as a function of longitude, with averaging in
latitude from $10^{\circ}$ S to $10^{\circ}$ N.  Mean easterlies exist
on the average, with only a small region of mean westerlies near
$80^{\circ}$ E.  Note that zonal wind convergence occurs over or to
the east of the maximum warm pool SST, as indicated by the negative
slope of the zonal wind there (see figure \ref{means}), but that the
peak westerlies occur on the west side of the SST maximum.  The
symmetric averaging in latitude is performed even though the
convection associated with the MJO follows the sun back and forth
across the equator \citep{zhang2005} in order to compare with
\citet{fuchs_raymond2017}.

\begin{figure}
  \centering
  \includegraphics[width=12cm]{fnlsmean.pdf}
  \caption{SST and 10 m winds averaged over 2001 for the tropical
  Pacific and Indian Oceans.}
  \label{fnlsmean}
\end{figure}

Figure \ref{fnlsmean} shows the annually averaged longitude-latitude
structure of the SST and the 10 m winds.  Equatorial winds are near
zero on the average over the warmest SSTs, with poleward winds at
higher latitudes on the east side of the warm pool.  The flow on the
northwest segment of the warm pool is clearly influenced by land.

Figure \ref{sst} shows the SST distribution as a function longitude
and of time, averaged over $10^{\circ}$ S to $10^{\circ}$ N in
latitude.  Also shown is the zonal wind averaged over longitude and
the same latitude range for the analyzed period.  The wind is easterly
and varies with time over the range 2-3 m s$^{-1}$.  The Indo-Pacific
warm pool extends from roughly 60$^{\circ}$ E to the dateline in the
absence of El Ni\~{n}o, which actually covers most of the Indian Ocean
as well as the west Pacific.  Worldwide zonal gradients in SST undergo
seasonal weakening in March and April (days 60-120).

\begin{figure}
  \centering
  \includegraphics[width=12cm]{sst.pdf}
  \caption{Reynolds SST and 10 m FNL zonal wind averaged over
    10$^{\circ}$ S to 10$^{\circ}$ N for the period January-December
    2001.  The vertical white stripes represent Africa and South
    America and 10 d smoothing is applied.}
  \label{sst}
\end{figure}

\begin{figure}
  \centering
  \includegraphics[width=10cm]{olr.pdf}
  \caption{Infrared brightness temperature (K) as a function of
    longitude and time averaged over $10^{\circ}$ S to $10^{\circ}$ N
    for January-December 2001.  Smoothing of $1^{\circ}$ in longitude
    and $2$ d in time was applied.}
  \label{olr}
\end{figure}

The infrared brightness temperature of outgoing longwave radiation
(OLR) is shown for the same longitude and time intervals as in figure
\ref{olr}.  Several MJO events are visible over the warm pool during
this time period, with all events starting near 60$^{\circ}$ E at the
west edge of the warm pool and disappearing near the dateline at its
east edge.  The strongest events occur in January-February and
November-December of 2001, with several weaker events in between.
Some disruption in the MJO signal occurs near $120^{\circ}$ E over the
maritime continent.

\begin{figure}
  \centering
  \includegraphics[width=10cm]{fnl_quplot.pdf}
  \caption{FNL data averaged from 10$^{\circ}$ S to 10$^{\circ}$ N as
    a function of longitude and time.  Red contours show the $0.78$
    value of saturation fraction while shading and black contours show
    the 10 m zonal wind in m s$^{-1}$.  The thick black contour indicates
    zero wind.  Smoothing of $2^{\circ}$ in longitude and $5$ d in
    time is applied.}
  \label{fnl_quplot}
\end{figure}

Figure \ref{fnl_quplot} shows the $0.78$ contour of FNL saturation
fraction (precipitable water divided by saturated precipitable water)
over the same longitude-time domain used in figure \ref{sst}.  From
\citet{bretherton_etal2004} we know that precipitation is strongly
associated with high values of saturation fraction, so its
distribution is a reliable indicator of the pattern of intense deep
convection.

Also shown in figure \ref{fnl_quplot} is the 10 m zonal wind.
Westerlies are episodic and are clearly associated with convection,
tending to occur to its west.  Numerical models such as that of
\citet{raymond2001} produce strong equatorial westerlies as a balanced
response to off-equatorial vortices produced by MJO convection.  This
and their episodic nature suggest that the westerly surges are a
result of the MJO rather than acting as a forcing mechanism for it.
Easterlies over the central and eastern Pacific are also enhanced when
strong convection occurs over the warm pool.

\begin{figure}
  \centering
  \includegraphics[width=10cm]{fnl_lonlat.pdf}
  \caption{Annual average for 2001 of surface zonal wind as a function
    of longitude and latitude.  The contour interval is $2$ m s$^{-1}$
    with the thick contour indicating a value of zero.  The wind is
    low-pass filtered with smoothing lengths of $3^{\circ}$ in
    longitude and $1.5^{\circ}$ in latitude.}
  \label{fnl_lonlat}
\end{figure}

Figure \ref{fnl_lonlat} shows the 2001 annual average zonal surface
wind as a function of longitude and latitude.  This figure shows that
the westerlies over the Indian Ocean are concentrated near
$80^{\circ}$ E longitude and $2^{\circ}$-$3^{\circ}$ N latitude,
whereas the easterlies are more broadly spread out.  Though point
values of westerlies can be strong, their magnitude averaged over
latitude and time is thus small compared to the easterlies, as
indicated by figures \ref{means} and \ref{fnl_lonlat}.

\begin{figure}
  \centering
  \includegraphics[width=10cm]{udiff.pdf}
  \caption{The baroclinic zonal wind $\Delta U$ (yellow/blue shading,
    black contours, m s$^{-1}$) and the $0.78$ contour of saturation
    fraction (red), all averaged over $10^{\circ}$ S to $10^{\circ}$ N
    are shown.  The thick black contour corresponds to $\Delta U = 0$.
    Smoothing is as in figure \ref{fnl_quplot}.}
  \label{udiff}
\end{figure}

\citet{krishnamurti_etal1985} and \citet{knutson_weickmann1987} show
that the MJO is associated with an upper tropospheric, wavenumber 1
disturbance in the divergent wind that travels continuously around the
globe.  Over the warm pool and to a lesser extent over the Atlantic,
the divergent flow is associated with deep convection, which has more
of a wavenumber 2 structure due to the suppression of convection
outside of the warm pool.  As figure \ref{udiff} shows, this signal
also appears in the baroclinic zonal wind
\begin{equation}
  \Delta U = U_{200}/3 - U_{1000} - \mbox{time and longitude means},
  \label{baroclinic-wind}
\end{equation}
where $U_{200}$ and $U_{1000}$ are respectively the zonal wind
components at 200 hPa and 1000 hPa averaged over $10^{\circ}$ S to
$10^{\circ}$ N.  The factor of 3 is included to account approximately
for the difference in air densities between these pressure levels and
$\Delta U$ has been low-pass filtered in longitude and time with
smoothing lengths of $5^{\circ}$ and 5 d respectively.  In addition,
the mean value over time is subtracted at each longitude and then the
mean value over longitude is subtracted at each time in order to
eliminate the steady Walker circulation and zonal-mean wind
fluctuations.

Eastward-propagating waves in $\Delta U$ often maintain continuity
from the warm pool eastward around the globe to the Indian Ocean in
figure \ref{udiff}.  The waves move at approximately $15$ m s$^{-1}$
and are correlated with strong convection over the warm pool.  This is
further evidence for the global nature of the MJO.  Occasional
westward-moving disturbances are seen as well, particularly in the
range $60$-$140$ d.  \citet{waliser_etal2009} also show global upper
level zonal winds correlated with the MJO in their figure 10.

\section{Governing equations}
\label{governing}

We now examine how an idealized warm pool affects the convection and
precipitation associated with the MJO of \citet{fuchs_raymond2017}.
As noted in the introduction, the interaction envisioned here results
purely from linear superposition of the MJO with the circulation
forced by the warm pool.

The Boussinesq governing equations on an equatorial beta plane are as
those given in \citet{fuchs_raymond2017} with three differences,
spatially variable sea surface temperatures, the imposition of
Rayleigh damping, and expression in the Earth's reference frame rather
than the frame moving with the mean zonal wind.  Linearized, reduced
to first baroclinic mode structure, and non-dimensionalized using
\citet{matsuno1966} scaling, they are
\begin{equation}
  \frac{Du}{Dt} - yv + \frac{\partial \Pi}{\partial x} = 0 ,
  \label{govu}
\end{equation}
\begin{equation}
  \frac{Dv}{Dt} + yu + \frac{\partial \Pi}{\partial y} = 0 ,
  \label{govv}
\end{equation}
\begin{equation}
  \Pi + b = 0 ,
  \label{govp}
\end{equation}
\begin{equation}
  \frac{Db}{Dt} - \delta = (\alpha + \varepsilon)q ,
  \label{govb}
\end{equation}
\begin{equation}
  \frac{Dq}{Dt} + (1 - \Gamma)\delta = \Lambda u + \gamma \tau - \alpha q ,
  \label{govq}
\end{equation}
\begin{equation}
  \delta = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} ,
  \label{govdelta}
\end{equation}
where $(u,v)$ are the horizontal velocity components, $\Pi$ is the
kinematic pressure perturbation, $b$ is the buoyancy, $q$ is the
mixing ratio perturbation, and $\delta$ is the horizontal divergence.
The operator
\begin{equation}
  \frac{D}{Dt} = \frac{\partial}{\partial t} +
  U_0 \frac{\partial}{\partial x} + \mu
  \label{diffop}
\end{equation}
where $U_0$ is the mean zonal wind, taken to be independent of height,
and $\mu$ is the Rayleigh damping coefficient.

The dependent variables $u$, $v$, $\Pi$, and $\delta$ have the
vertical structure $\cos(mz)$ while $b$ and $q$ have $\sin(mz)$
structure, where $m = \pi /h$ is the first baroclinic mode vertical
wavenumber, $h$ being the depth of the troposphere.
\citet{matsuno1966} scales horizontal motions with horizontal length
and time scales $X = (c/\beta)^{1/2}$ and $T = (c\beta)^{-1/2}$ where
$\beta = df/dy$ is the Rossby parameter evaluated on the equator, with
$f$ being the Coriolis parameter.  Horizontal velocities scale with
$X/T = c = N / m$, where $N$ is the Brunt-V\"{a}is\"{a}l\"{a}
frequency. From the momentum equations it is clear that the kinematic
pressure perturbation scales with $c^2$.  The hydrostatic equation
therefore requires the buoyancy $b$ to scale with $mc^{2} = Nc$, so
that
\begin{equation}
  b = \left(\frac{1}{Nc}\right) \frac{g\theta'}{T_R} .
  \label{buoyancy}
\end{equation}

The mixing ratio perturbation is first converted to the units of
acceleration to be consistent with the buoyancy and then
non-dimensionalized the same way,
\begin{equation}
  q = \left(\frac{1}{Nc}\right) \frac{gL r'}{C_P T_R} .
  \label{mixratscaled}
\end{equation}
The parameter $g$ is the acceleration of gravity, $T_R$ is a constant
reference temperature, $C_P$ is the specific heat of air at constant
pressure, and $L$ is the latent heat of vaporization for water.

The parameter $\alpha$ is the rate constant that relates $q$ to the
buoyancy production due to precipitation and $\varepsilon$ represents
the effect cloudiness on radiative heating (set to zero in this
paper).  $\alpha$, $\varepsilon$, and $\mu$ are scaled by $T^{-1}$.
$\Gamma$ is the normalized gross moist stability associated with the
convection \citep{raymond_etal2009}, assumed to be constant.  $\Gamma$
is taken to be zero in the present work.  A consequence of setting
$\Gamma , \varepsilon = 0$ is that moisture mode instability does not
exist.  The effects of positive values of $\Gamma$ and $\varepsilon$
are explored by \citet{fuchs_raymond2017}.

The term $\Lambda u$ represents the effect of WISHE on the moisture
budget, with the constant $\Lambda < 0$ for mean surface easterlies,
while $\gamma \tau$ represents the increase in surface evaporation
rate due the SST anomaly associated with the warm pool.  $\tau(x,y)$
represents the spatial distribution of the SST anomaly, normalized by
the reference temperature $T_R$.  The constants $\Lambda$ and $\tau$
are scaled by $TN$ and are given by
\begin{equation}
  \Lambda = (TN) \frac{C_D q_{ss}(1 - H)U_0}{2U_E}
\end{equation}
and
\begin{equation}
  \gamma = (TN) \frac{C_D (1 - H) q_{ss} T_R U_E}{2 \Delta T_D} ,
\end{equation}
where $C_D$ is the surface moisture transfer coefficient, $q_{ss}$ is
the non-dimensionalized saturation mixing ratio at the temperature of
the sea surface [see (\ref{mixratscaled})], $H$ is the relative
humidity of the atmospheric boundary layer measured relative to the
SST, $R_V$ is the gas constant for water vapor, and
$U_E = (U_0^2 + W^2)^{1/2}$, where we recall that $U_0$ is the mean
zonal wind and $W$ is a constant related to the gustiness correction
in the surface heat flux equation.  $U_0$, $W$, and $U_E$ are
non-dimensionalized by $c$.  The quantity $\Delta T_D$ is the
temperature interval over which the saturated vapor pressure doubles,
thus encapsulating the effect of the Clausius-Clapeyron equation.

\section{Model solutions}

We assume solutions with zonal and temporal dependence
$\exp[i(kx-\omega t)]$.  We further define
\begin{equation}
  \Omega = \omega - k U_0 + i\mu .  
  \label{Omega}
\end{equation}
The significance of $\Omega$ is that whereas
$\partial / \partial t \rightarrow -i\omega$, we have
$D /Dt \rightarrow -i\Omega$.

The general linear solution consists of the superposition of solutions
of two types, a traveling homogeneous mode with $\tau = 0$
representing the MJO and an inhomogeneous steady-state solution with
$\omega = 0$ and $\Omega = -k U_0 + i\mu \equiv \Omega^*$ representing
the warm pool flow.  We further suppose that the damping is such that
the traveling solution is neither intensifying or decaying, which
implies that $\mbox{Im}(\omega) = 0$.  As a consequence, we must have
$\mu = \mbox{Im}(\Omega^*)$.  This provides a damping constant for the
inhomogeneous solution.  $\Omega$ is computed as an eigenvalue of the
homogeneous problem.

Two distinct classes of solutions exist, those with $v = 0$ and those
with $v \ne 0$ \citep{matsuno1966}.  We examine these solutions
independently.

\begin{table}
  \centering
  \caption{Values of model parameters.}
  \begin{tabular}{ccl}
    \hline
    Dimensional parameter & Value & Comment \\
    $y_m$ & $5000$ km & north channel wall \\
    $\beta$ & $2.3 \times 10^{-11}$ m$^{-1}$ s$^{-1}$ & Rossby parameter \\
    $c$ & 48 m s$^{-1}$ & scaling velocity \\
    $N$ & $0.01$ s$^{-1}$ & Brunt-V\"{a}is\"{a}l\"{a} frequency \\
    $X$ & $1.44 \times 10^6$ m & scaling length \\
    $T$ & $3.01 \times 10^4$ s & scaling time \\
    $T_R$ & 300 K & reference temperature \\
    $U_0$ & -2 m s$^{-1}$ & mean surface zonal wind \\
    $W$ & 3 m s$^{-1}$ & gustiness factor \\
    $r_{ss}$ & $0.02$ & sea surface saturation mixing ratio \\
    $y_{wp}$ & $1500$ km & warm pool meridional scale \\
    $\tau_0$ & $1$ K & warm pool SST anomaly amplitude \\
    \hline
    Dimensionless parameter & Value & Comment \\
    $U_0$ & $-0.042$ & mean surface zonal wind \\
    $q_{ss}$ & 3.39 & sea surface saturation mixing ratio \\
    $H$ & $0.75$ & relative humidity \\
    $\alpha$ & 0.35 & precipitation rate parameter \\
    $\varepsilon$ & 0 & cloud-radiation interaction \\
    $\Gamma$ & 0 & gross moist stability \\
    $k$ & 0.23 & planetary wavenumber 1 \\
    $c_r$ & 0.32 & $v = 0$ mode phase speed \\
    $\mu$ & 0.038 & $= \Omega_i$ damping coefficient \\
    \hline    
  \end{tabular}
  \label{params}
\end{table}

\subsection{Solution with $v = 0$}

Given that $v = 0$, (\ref{govu}), (\ref{govv}), and (\ref{govp}) may
be combined to yield
\begin{equation}
  \left(\frac{\partial}{\partial
      y}+\frac{ky}{\Omega}\right)b=0
  \label{v0-1}
\end{equation}
which has the solution
\begin{equation}
  b \propto \exp\left(-\frac{ky^{2}}{2\Omega}\right) .
  \label{v0-2}
\end{equation}
Equations (\ref{govu}), (\ref{govp}), (\ref{govb}), and (\ref{govq})
combine to produce an equation expressing the $x$ and $t$ dependence
of the buoyancy:
\begin{equation}
  \left[ \frac{(\Omega + i\alpha)(k^2 - \Omega^2)}{\alpha + \varepsilon}
    -ik^2(1 - \Gamma) + k \Lambda \right]b = \Omega \gamma \tau .
  \label{v0-3}
\end{equation}
The dispersion relation for the $v = 0$ traveling mode is obtained by
setting $\tau = 0$ in (\ref{v0-3}) and the steady response to the SST
anomaly comes from replacing $\Omega$ by $\Omega^*$ and solving for
$b$.

The diagnostic equations for $u$, $\delta$, and $q$ are
\begin{equation}
  u = -\frac{kb}{\Omega}
  \label{v0-4}
\end{equation}
\begin{equation}
  \delta = -\frac{ik^2 b}{\Omega}
  \label{v0-5}
\end{equation}
\begin{equation}
  q = \frac{i(k^2 - \Omega^2)b}{\Omega (\alpha + \varepsilon)} .
  \label{v0-6}
\end{equation}

It is tempting to represent the warm pool by the inhomogeneous
solution to the $v = 0$ equations.  However, for this to work the
meridional distribution of the SST anomaly representing the warm pool
would exhibit strong phase progression with increasing latitude since
its spatial structure would have to match that of the buoyancy.  The
solution would also have zero meridional winds, which seems equally
unphysical.  For this reason we seek a more general solution of
(\ref{v-1})-(\ref{v-5}) with the warm pool SST specified.

\subsection{Solutions with $v \ne 0$}

For the $v \ne 0$ case (\ref{govu})-(\ref{govq}) may be reduced to two
coupled differential equations in $y$
\begin{equation}
  \frac{\partial v}{\partial y} = Ayv + Cb + D\tau
  \label{v-1}
\end{equation}
\begin{equation}
  \frac{\partial b}{\partial y} = Eyb +
  i\left(-\Omega + \frac{y^2}{\Omega}\right) v
  \label{v-2}
\end{equation}
where the constants $A$-$E$ and $\chi$ are defined
\begin{equation}
  A = \frac{k}{\Omega} + \frac{\chi \Lambda}{\Omega^2}
  \label{v-3}
\end{equation}
\begin{equation}
  C = \frac{ik^2}{\Omega} + \chi (1 - \Gamma) - i\Omega +
  \frac{ik\chi \Lambda}{\Omega^2}
  \label{v-4}
\end{equation}
\begin{equation}
  D = -\frac{i\chi \gamma}{\Omega}
  \label{v-5}
\end{equation}
\begin{equation}
  E = -\frac{k}{\Omega}
  \label{v-6}
\end{equation}
\begin{equation}
  \chi = \frac{(\alpha + \varepsilon)\Omega}
  {\Omega + i[\alpha \Gamma - \varepsilon (1 - \Gamma)]} .
  \label{v-8}
\end{equation}
In addition, we have three diagnostic equations:
\begin{equation}
  u = \frac{iyv - kb}{\Omega}
  \label{v-9}
\end{equation}
\begin{equation}
  (\alpha + \varepsilon)q = -\chi(1 - \Gamma)b +
  i\chi(\Lambda u + \gamma \tau)/\Omega
  \label{v-10}
\end{equation}
\begin{equation}
  \delta = -(\alpha + \varepsilon)q - i\Omega b .
  \label{v-11}
\end{equation}

The inhomogeneous solution to (\ref{v-1}) and (\ref{v-2}) cannot be
obtained analytically.  Therefore, we solve these equations
numerically subject to the condition
$\Omega = \Omega^* = -k U_0 + i\mu$.  This is done as follows: These
equations are integrated equatorward twice, starting from
$y = y_m = 5000$ km, one case with $\tau \ne 0$ and the other with
$\tau = 0$.  We assume that $\tau$ has zonal wavenumber 1 and
meridional structure represented by
\begin{equation}
  \tau(y) = \tau_0 \exp[-(y/y_{wp})^2] .
  \label{tau-warmpool}
\end{equation}
Both integrations are initialized at $y_m$ with a very small value of
$b$ and with $v = 0$.  The actual solution is a linear superposition
of these two, with the amplitude of the $\tau = 0$ case adjusted so
that $v = 0$ at $y = 0$ for the combined solution.  Inspection of
(\ref{v-2}) shows that this also guarantees
$\partial b/\partial y = 0$ at $y = 0$, as is to be expected for an
equatorially symmetric solution in $b$.  The integration scheme is 2nd
order with an integration step of $\Delta y = 10$ km.  The result is
well behaved at large $y$ for $U_0 \le 0$, i.e., it shows no tendency
to blow up with large $y$.  Equations (\ref{v-9})-(\ref{v-11}) are
used to obtain $u$, $q$, and $\delta$ and the planetary wavenumber 1
longitudinal structure is then incorporated by multiplying the
solutions by $\exp(ikx)$ and taking the real parts.

\section{Results}
\label{solutions}

We now examine the above solutions.  The MJO mode has an eastward
phase speed of $15$ m s$^{-1}$ relative to the air for planetary
wavenumber 1, or $13$ m s$^{-1}$ relative to the earth and an
e-folding growth time of roughly 9 d for the parameter values in table
\ref{params}.  The gross moist stability $\Gamma$ and cloud-radiation
interactions $\varepsilon$ are both set to zero.

\begin{figure}
  \centering
  \includegraphics[width=10cm]{model/wpwind_rain.pdf}
  \caption{Plot as a function of $x$ and $y$ of the atmospheric
    response to the warm pool with $U_0 = -2$ m s$^{-1}$.  Yellow
    contours represent the SST anomaly with contour interval $0.3$ K.
    The thicker lines represent zero values, solid thin lines positive
    anomalies and dashed lines negative anomalies.  Vectors indicate
    total winds with scaling factor $500$ km per m s$^{-1}$.
    Shading and black contours represent the rainfall anomaly with
    contour interval of 2 mm day$^{-1}$.}
  \label{wpwind_rain}
\end{figure}

Figure \ref{wpwind_rain} shows the response of the atmosphere to a
warm pool anomaly with amplitude $\pm1$ K and meridional scale
$y_{wp} = 1500$ km.  This differs from the traditional
\citet{gill1980} solution in that a mean zonal wind of $U_0 = -2$ m
s$^{-1}$ is applied.  In contrast to \citet{gill1980}, the strongest
wind anomalies are displaced well away from the equator, with light
winds in the region of the precipitation maximum, with poleward
meridional winds to the east of this maximum.  This resembles the
annual mean 10 m flow over the warm pool as shown in figure
\ref{fnlsmean}.

\begin{figure}
  \centering
  \includegraphics[width=10cm]{model/mjo_xy.pdf}
  \caption{As in figure \ref{wpwind_rain} except for the traveling
    mode with constant SST at day 22 in the simulation.  The rainfall
    contour interval is $1$ mm day$^{-1}$ and the wind scaling factor
    is 500 km per m s$^{-1}$.}
  \label{mjo_xy}
\end{figure}

Figure \ref{mjo_xy} shows an $x$-$y$ snapshot of total wind and
rainfall anomaly for the wavenumber 1, $v = 0$ traveling mode with
zero gross moist stability and no cloud-radiative feedback
($\Gamma , \varepsilon = 0$), as described by
\citet{fuchs_raymond2017}.  The mode has been adjusted in amplitude to
have roughly the same equatorial zonal wind anomaly as the warm pool
flow.  The maximum equatorial rainfall anomaly is about half that for
the warm pool.  The strongest wind and rainfall perturbations occur
within about $1000$ km of the equator and the rainfall pattern is
similar to that shown in figure 6 of \citet{waliser_etal2009}.  The
maximum east wind and strongest surface fluxes are almost a quarter of
a wavelength east of the maximum rainfall anomaly, unlike the classic
WISHE models of \citet{emanuel1987} and \citet{neelin_etal1987}.

\begin{figure}
  \centering
  \includegraphics[width=10cm]{model/evap_rain.pdf}
  \caption{Rainfall perturbation (yellow: non-negative contours 2 mm
    day$^{-1}$ interval), dimensionless surface evaporation rate
    anomaly (red-blue shading), and total zonal wind (arrows, black
    contour: zero wind).  The axis of the rainfall maximum is
    indicated by the white line, the slope of which corresponds to
    eastward propagation at $5$ m s$^{-1}$.  The horizontal green
    lines indicate cross-sections used in figure \ref{summing}.}
  \label{evap_rain}
\end{figure}

Figure \ref{evap_rain} shows the combined effects of the traveling MJO
mode and the warm pool as a function of longitude and time with mean
easterlies of $U_0 = -2$ m s$^{-1}$.  The linear combination of the
rainfall anomalies due to the warm pool and the traveling mode gives
rise to a rainfall maximum that moves to the east at $5$ m s$^{-1}$ as
the MJO passes over the warm pool.  However, the combined wind
anomaly moves much more rapidly due to the fact that the warm pool
wind anomaly is weak near the equator (see figure \ref{wpwind_rain}),
resulting in domination of the wind field there by the traveling mode.
The resulting westerly ``wind burst'', which moves with the traveling
MJO mode, starts well to the west of the rainfall maximum, but ends up
overrunning it as the MJO progresses across the warm pool.  Figure
\ref{evap_rain} also shows that the strongest values of surface
evaporation occur to the east of rainfall, with an enhancement in
evaporation as the MJO moves over the warm pool.

\section{Conclusions}

This paper explores a simple model of the interaction of the
Madden-Julian oscillation with the Indo-Pacific warm pool where most
of the convection associated with the MJO occurs.  The model takes
this interaction to be a simple linear superposition of MJO and warm
pool-related fields.

The MJO is represented by the highly simplified model of
\citet{fuchs_raymond2017}.  This model is driven by the enhancement of
surface fluxes in mean easterly flow that occurs to the east of MJO
convection as postulated by \citet{emanuel1987} and
\citet{neelin_etal1987}.  It therefore requires zonally averaged mean
surface easterlies over the latitude range of the MJO in order to
work.

FNL data show that zonal and annual mean surface easterlies of $2$-$3$
m s$^{-1}$ do indeed exist over the $10^{\circ}$ S to $10^{\circ}$ N
latitude range.  Figure \ref{sst} shows that these easterlies persist
throughout the year.  Though episodic westerlies can be intense over
limited regions, figure \ref{fnl_lonlat} demonstrates that the zonal,
meridional, and temporal coverage of westerlies is much less than that
of easterlies.  Since the MJO in this model is a planetary wavenumber
1 disturbance, the zonal average is more important than regional
variations.

Unlike earlier models, surface fluxes do not drive precipitation
directly in \citet{fuchs_raymond2017}.  Instead, the fluxes are
incorporated in a linearized moisture budget and precipitation is
driven by large values of column water vapor.  As a result,
precipitation occurs far to the west of the strongest surface
evaporation.  Thus fluxes over the eastern and central Pacific provide
the water vapor required to drive the MJO while it is over the warm
pool in this model.

The warm pool calculation used in this paper takes into account the
effect of a mean easterly flow (see figure \ref{wpwind_rain}), which
results in a warm pool circulation somewhat different from that of the
classic \citet{gill1980} solution (not shown).  In particular, the
wind perturbations in the equatorial region are much weaker in the
easterly flow case than found by Gill.

Linear superposition of the MJO mode with the circulation induced by
the warm pool results in a pattern of precipitation that reproduces
some of the key aspects of observations, as shown in figure
\ref{evap_rain}.  In particular, the combined rainfall anomaly moves
across the warm pool much more slowly than the MJO mode by itself.

\begin{figure}
  \centering
  \includegraphics[width=7cm]{model/summing.pdf}
  \caption{Schematic of rainfall rates as a function of longitude from
    figure \ref{evap_rain} at three times, $12$, $22$, and $32$ days.
    Contributions to the total rainfall from the traveling MJO mode
    and the stationary warm pool are shown separately as well as the
    sum of the two.}
  \label{summing}
\end{figure}

Figure \ref{summing} illustrates why this is so.  The maximum in the
combined rainfall (thick brown line) occurs to the east of the MJO
maximum (red line) when the MJO is to the west of the warm pool (blue
line) and to the west when the MJO is east of the warm pool.  As a
result, the combined rainfall peak moves to the east more slowly
than the MJO itself.

In contrast, the westerly wind burst begins well to the west of the
precipitation maximum but rapidly catches up to it as the
precipitation maximum moves to the east.  This is a consequence of the
weakness of the near-equatorial perturbation winds produced in the
warm pool calculation, as discussed above.  As a result, the rapidly
moving MJO mode controls the zonal wind pattern and the combined
westerlies propagate rapidly to the east.  This pattern is seen to a
certain extent in figures 5, 11, and 12 of \citet{waliser_etal2009}.

The simple linear superposition model of MJO-warm pool interaction
explains many aspects of the observed MJO over the warm pool.
However, other aspects are not so readily accounted for.  For
instance, the smaller scale transient behavior of convection over the
warm pool, including strong westerly wind bursts and associated
phenomena are not explained.  These may arise out of
radiation/moisture mode instabilities or nonlinear effects that are
intentionally suppressed in the present work.

\begin{acknowledgments}
  We thank two anonymous reviewers for their comments, which have
  significantly improved the paper.  Data sources and software
  supporting this paper may be found at
  \url{http://kestrel.nmt.edu/~raymond/data/papers/warmpool/raymond_fuchs2017.xhtml}.
  This work was supported by National Science Foundation grant
  1342001.
\end{acknowledgments}

\begin{thebibliography}{}
\bibitem[{\textit{Adames and Kim}}(2016)]{adames_kim2016} Adames,
  A. F., and Kim, D., (2016). The MJO as a dispersive, convectively
  coupled moisture wave, \textit{J. Atmos. Sci.} 73, 913-941.
\bibitem[{\textit{Bretherton et al.}}(2004)]{bretherton_etal2004}
  Bretherton, C. S., Peters, M. E., and Back, L. E.,
  (2004). Relationships between water vapor path and precipitation over
  the tropical oceans, \textit{J. Climate} 17, 1517-1528.
\bibitem[{\textit{Emanuel}}(1987)]{emanuel1987} Emanuel, K. A.,
  (1987). An air-sea interaction model of intraseasonal oscillations in
  the tropics, \textit{J. Atmos. Sci.} 44, 2324-2340.
\bibitem[{\textit{Fuchs and Raymond}}(2005)]{fuchs_raymond2005} Fuchs,
  \v{Z}., and Raymond, D. J., (2005). Large-scale modes in a rotating
  atmosphere with radiative-convective instability and WISHE,
  \textit{J. Atmos. Sci.} 62, 4084-4094.
\bibitem[{\textit{Fuchs and Raymond}}(2017)]{fuchs_raymond2017} Fuchs,
  \v{Z}., and Raymond, D. J., (2017). A simple model of intraseasonal
  oscillations, \textit{J. Adv. Model. Earth Syst.} ,
  doi:10.1002/2017MS000963.
\bibitem[{\textit{Gill}}(1980)]{gill1980} Gill, A. E., (1980). Some
  simple solutions for heat-induced tropical circulation,
  \textit{Quart. J. Roy. Meteor. Soc.} 106, 447-462.
\bibitem[{\textit{Knutson and Weickmann}}(1987)]{knutson_weickmann1987}
  Knutson, T. R., and Weickmann, K. M., (1987). 30-60 day atmospheric
  oscillations, \textit{Mon. Wea. Rev.} 115, 1407-1436.
\bibitem[{\textit{Krishnamurti et al.}}(1985)]{krishnamurti_etal1985}
  Krishnamurti, T. N., Jayakumar, P. K., Sheng, J., Surgi, N., and
  Kumar, A., (1985). Divergent circulations on the 30-50 day time scale,
  \textit{J. Atmos. Sci.} 42, 364-375.
\bibitem[{\textit{Madden}}(1986)]{madden1986} Madden, R. A.,
  (1986). Seasonal variations of the 40-50 day oscillation in the
  tropics, \textit{J. Atmos. Sci.} 43, 3138-3158.
\bibitem[{\textit{Madden and Julian}}(1971)]{madden_julian1971}
  Madden, R., and Julian, P. R., (1971). Detection of a 40-50 day
  oscillation in the zonal wind in the tropical Pacific,
  \textit{J. Atmos. Sci.} 28, 702-708.
\bibitem[{\textit{Madden and Julian}}(1972)]{madden_julian1972}
  Madden, R., and Julian, P. R., (1972). Description of global-scale
  circulation cells in the tropics with a 40-50 day period,
  \textit{J. Atmos. Sci.} 29, 1109-1123.
\bibitem[{\textit{Matsuno}}(1966)]{matsuno1966} Matsuno, T.,
  (1966). Quasi-geostrophic motions in the equatorial area,
  \textit{J. Meteor. Soc. Japan} 44, 25-43.
\bibitem[{\textit{Neelin et al.}}(1987)]{neelin_etal1987} Neelin,
  J. D., Held, I. M., and Cook, K. H., (1987). Evaporation-wind feedback
  and low-frequency variability in the tropical atmosphere,
  \textit{J. Atmos. Sci.} 44, 2341-2348.
\bibitem[{\textit{Peixoto and Oort}}(1992)]{peixoto_oort1992} Peixoto,
  J. P., and Oort, A. H., (1992). \textit{Physics of Climate}, American
  Institute of Physics, New York, 520 pp.
\bibitem[{\textit{Raymond}}(2000)]{raymond2000} Raymond, D. J.,
  (2000). Thermodynamic control of tropical rainfall,
  \textit{Quart. J. Roy. Meteor. Soc.} 126, 889-898.
\bibitem[{\textit{Raymond}}(2001)]{raymond2001} Raymond, D. J.,
  (2001). A new model of the Madden-Julian oscillation,
  \textit{J. Atmos. Sci.} 58, 2807-2819.
\bibitem[{\textit{Raymond et al.}}(2009)]{raymond_etal2009} Raymond,
  D. J., Sessions, S., Sobel, A., and Fuchs, \v{Z}., (2009). The
  mechanics of gross moist stability, J, Adv, Model, Earth Syst, , 1,
  \textit{art} 9, 20 pp.
\bibitem[{\textit{Raymond and Fuchs}}(2009)]{raymond_fuchs2009}
  Raymond, D. J., and Fuchs, \v{Z}., (2009). Moisture modes and the
  Madden-Julian oscillation, \textit{J. Climate} 22, 3031-3046.
\bibitem[{\textit{Reynolds and Marsico}}(1993)]{reynolds_marsico1993}
  Reynolds, R. W., and Marsico, D. C., (1993). An improved real-time
  global sea surface temperature analysis, \textit{J. Climate} 6,
  114-119.
\bibitem[{\textit{Sobel et al.}}(2001)]{sobel_etal2001} Sobel, A. H.,
  Nilsson, J., and Polvani, L. M., (2001). The weak temperature gradient
  approximation and balanced tropical moisture waves,
  \textit{J. Atmos. Sci.} 58, 3650-3665.
\bibitem[{\textit{Sobel and Maloney}}(2012)]{sobel_maloney2012} Sobel,
  A., and Maloney, E., (2012). An idealized semi-empirical framework for
  modeling the Madden-Julian oscillation, \textit{J. Atmos. Sci.} 69,
  1691-1705.
\bibitem[{\textit{Sobel and Maloney}}(2013)]{sobel_maloney2013} Sobel,
  A., and Maloney, E., (2013). Moisture modes and the eastward
  propagation of the MJO, \textit{J. Atmos. Sci.} 70, 187-192.
\bibitem[{\textit{Sugiyama}}(2009)]{sugiyama2009} Sugiyama, M.,
  (2009). The moisture mode in the quasi-equilibrium tropical
  circulation model, Part I, \textit{J. Atmos. Sci.} 66, 1507-1523.
\bibitem[{\textit{Waliser et al.}}(2009)]{waliser_etal2009} Waliser,
  D., Sperber, K., Henden, H., Kim, D., Maloney, E., Wheeler, M.,
  Weickmann, K., Zhang, C., Donner, L., Gottschalck, J., Higgins, W.,
  Kang, I.-S., Legler, D., Moncrieff, M., Schubert, S., Stern, W.,
  Vitart, F., Wang, B., Wang, W., and Woolnough, S., (2009). MJO
  simulation diagnostics, \textit{J. Climate} 22, 3006-3030.
\bibitem[{\textit{Zhang}}(2005)]{zhang2005} Zhang, C.,
  (2005). Madden-Julian oscillation, \textit{Rev. Geophys.} 43,
  doi:10.1029/2004RG000158.
\end{thebibliography}

\end{document}
