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A Skeleton Model for the MJO with Rened
Vertical Structure
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Sulian Thual (1) and Andrew J. Majda (1)
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5 (1) Department of Mathematics, and Center for Atmosphere Ocean Science, Courant Institute of
6 Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012 USA
7 Corresponding author:
8 Sulian Thual, 251 Mercer Street, New York, NY 10012 USA ,
sulian.thual@gmail.com
Abstract
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The Madden-Julian oscillation (MJO) is the dominant mode of variability in the
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tropical atmosphere on intraseasonal timescales and planetary spatial scales. The skele-
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ton model is a minimal dynamical model that recovers robustly the most fundamental
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MJO features of (I) a slow eastward speed of roughly
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relation with
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model depicts the MJO as a neutrally-stable atmospheric wave that involves a simple
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multiscale interaction between planetary dry dynamics, planetary lower-tropospheric
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moisture and the planetary envelope of synoptic-scale activity.
dω/dk ≈ 0,
5 ms−1 , (II) a peculiar dispersion
and (III) a horizontal quadrupole vortex structure.
This
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Here we propose and analyse an extended version of the skeleton model with ad-
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ditional variables accounting for the rened vertical structure of the MJO in nature.
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The present model reproduces qualitatively the front-to-rear vertical structure of the
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MJO found in nature, with MJO events marked by a planetary envelope of convec-
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tive activity transitioning from the congestus to the deep to the stratiform type, in
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addition to a front-to-rear structure of moisture, winds and temperature. Despite its
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increased complexity the present model retains several interesting features of the origi-
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nal skeleton model such as a conserved energy and similar linear solutions. We further
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analyze a model version with a simple stochastic parametrization for the unresolved
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details of synoptic-scale activity. The stochastic model solutions show intermittent
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initiation, propagation and shut down of MJO wave trains, as in previous studies, in
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addition to MJO events with a front-to-rear vertical structure of varying intensity and
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characteristics from one event to another.
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1 Introduction
32 The dominant component of intraseasonal variability in the tropics is the 40 to 50 day intraseasonal
33 oscillation, often called the Madden-Julian oscillation (MJO) after its discoverers (Madden and
34 Julian, 1971; 1994). In the troposphere, the MJO is an equatorial planetary-scale wave, that is
35 most active over the Indian and western Pacic Oceans and propagates eastward at a speed of
36 around 5 ms−1 . The planetary-scale circulation anomalies associated with the MJO signicantly
37 aect monsoon development, intraseasonal predictability in midlatitudes, and the development of
38 El Niño events in the Pacic Ocean, which is one of the most important components of seasonal
39 prediction.
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In addition to the above features, the MJO in nature propagates eastward with an interesting
41 vertical structure. Observations reveal a central role of three cloud types above the boundary
42 layer in the MJO: lower-middle troposhere congestus cloud decks that moisten and precondition
43 the lower troposphere in the initial phase, followed by deep convection and a trailing wake of
44 upper troposphere stratiform clouds. Observations also reveal that the MJO envelope consists
45 of a complex front-to-rear (i.e. tilted) vertical structure as seen on all main dynamical elds
46 such as heating, winds, temperature, and moisture (Kikuchi and Takayabu, 2004; Kiladis et al.,
47 2005; Tian et al., 2006). In addition to such climatological features, the front-to-rear structure of
48 individual MJO events is often unique, with complex dynamic and convective features within the
49 MJO envelope (e.g. westerly wind bursts, etc) that vary from one event to another.
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Interestingly, the planetary front-to-rear structure of the MJO is also observed for various
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convective events at synoptic scale and mesoscale. The MJO being a planetary envelope of such
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convective events, a theoretical issue of interest is to understand how the MJO characteristics
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are related to the ones of such convective events, and inversely (Moncrie, 2004; Mapes et al.,
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2006; Majda and Stechmann, 2009a; Stechmann et al., 2013). Khouider and Majda (2006; 2007)
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developed a systematic multicloud model convective parametrization highlighting the nonlinear
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dynamical role of the three cloud types, congestus, stratiform, and deep convective clouds, and
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their dierent heating vertical structures. The multicloud model reproduces key features of the
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observational record for mesoscale and synoptic-scale convectively coupled waves, as well as a
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realistic MJO envelope analog for an intraseasonal parameter regime (Majda et al. 2007). The
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multicloud model has also been used as a cumulus parametrization in GCM experiments with
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consequent improvement of the simulated MJO variability (Khouider et al., 2011; Ajayamohan
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et al., 2013; Deng et al., 2014). As another example, the role of synoptic scale waves in producing
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key features of the MJO's vertical structure has been elucidated in multiscale asymptotic models
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(Majda and Biello, 2004; Biello and Majda, 2005, 2006; Majda and Stechmann, 2009a; Stechmann
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et al., 2013).
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While theory and simulation of the MJO remain dicult challenges, they are guided by some
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generally accepted, fundamental features of the MJO on intraseasonal-planetary scales that have
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been identied relatively clearly in observations (Hendon and Salby, 1994; Wheeler and Kiladis,
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1999; Zhang, 2005). These features, referred to here as the MJO's skeleton features, are (I) a slow
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eastward phase speed of roughly
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(III) a horizontal quadrupole structure. Majda and Stechmann (2009b) introduced a minimal dy-
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namical model, the skeleton model, that captures the MJO's intraseasonal features (I-III) together
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for the rst time in a simple model. The model is a coupled nonlinear oscillator model for the
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MJO skeleton features as well as tropical intraseasonal variability in general. In particular, there
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is no instability mechanism at planetary scale, and the interaction with sub-planetary convective
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processes discussed above is accounted for, at least in a crude fashion. In a collection of numer-
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ical experiments, the non-linear skeleton model has been shown to simulate realistic MJO events
5 ms−1 ,
(II) a peculiar dispersion relation with
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dω/dk ≈ 0,
and
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with signicant variations in occurrence and strength, asymmetric east-west structures, as well
as a preferred localization over the background state warm pool region (Majda and Stechmann,
2011). A stochastic version of the skeleton model has also been developped that includes a simple stochastic parametrization of the unresolved synoptic-scale convective/wave processes (Thual
et al., 2014; Thual et al., 2015). In addition to the above features (I-III), this model reproduces
additional realistic features such as the intermittent generation of MJO events and the organization
of MJO events into wave trains with growth and demise. More recently, it has been shown that the
skeleton model reproduces realistic solutions for a realistic background state of heating/moistening
(Ogrosky and Stechmann, 2015), in addition to providing an essential theoretical estimate of the
intensity of MJO events in observations (Stechmann and Majda, 2015).
In the present article, we propose and analyze an extended version of the skeleton model with
a rened vertical structure. While previous work on the skeleton model has focused essentially on
the MJO dynamics with a coarse vertical structure, we will show that the present skeleton model
reproduces qualitatively the front-to-rear vertical structure of MJO events found in nature. This
includes MJO events marked by a planetary envelope of convective activity transitioning from the
congestus to the deep to the stratiform type, in addition to a front-to-rear structure of moisture,
winds and temperature. This is achieved by considering the evolution of the planetary envelope
of congestus and stratiform activity in the dynamical core of the skeleton model, in addition to
a secondary slaved circulation for the detailed evolution of lower and middle level moisture and
the second baroclinic mode dry dynamics. In addition to this, the present model conserves many
interesting properties of the original skeleton model such as the above features (I-III), intermittent
MJO wave trains, a converved positive energy, etc.
The article is organized as follows. In section 2, we recall the design and main features of
the skeleton model, and introduce the skeleton model with rened vertical structure used here.
In section 3 we analyze the solutions of the latter model, including the linear solutions and the
numerical solutions for simulations with a stochastic parametrization of synoptic scale activity.
Section 4 is a discussion with concluding remarks.
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2 Model Formulation
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2.1 Nonlinear Skeleton Model
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The skeleton model has been proposed originally by Majda and Stechmann (2009b) (hereafter
MS2009), and further analyzed in Majda and Stechmann (2011) (hereafter MS2011). It is a
minimal non-linear oscillator model for the MJO and the intraseasonal-planetary variability in
general. The design of the skeleton model, already presented in those previous publications,
is recalled here briey for completeness. Some slight notation changes are considered here for
consistency with the next sections.
The fundamental assumption in the skeleton model is that the MJO involves a simple multiscale interaction between (i) planetary-scale dry dynamics, (ii) lower-level moisture q and (iii) the
planetary-scale envelope of synoptic-scale convection/wave deep activity, ad. The planetary envelope ad in particular is a collective (i.e. integrated) representation of the convection/wave activity
occurring at sub-planetary scale (i.e. at synoptic-scale and possibly at mesoscale), the details of
which are unresolved. It is assumed that environment moisture inuences the tendency (i.e. the
growth and decay rates) of the envelope of deep activity:
∂t ad = Γd qad ,
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(1)
where Γd > 0 is a constant of proportionality: positive (negative) low-level moisture anomalies
create a tendency to enhance (decrease) the envelope of synoptic activity.
In the skeleton model, the q − ad interaction parametrized in Eq. (1) is further combined
with the linear primitive equations projected on the rst vertical baroclinic mode. This reads, in
non-dimensional units,
∂t u1 − yv1 − ∂x θ1 = 0
yu1 − ∂y θ1 = 0
∂t θ1 − (∂x u1 + ∂y v1 ) = Had − sθ1
∂t q + Q(∂x u1 + ∂y v1 ) = −Had + sq
∂t ad = Γd qad ,
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(2)
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with periodic boundary conditions along the equatorial belt. The rst three rows of Eq. (2)
describe the dry atmosphere dynamics, with equatorial long-wave scaling as allowed at planetary
scale. The un, vn, θn are the zonal, meridional velocity, and potential temperature anomalies,
respectively, for the baroclinic mode n = 1. The fourth row describes the evolution of low-level
moisture q. The fth row is the above Eq. (1). All variables are anomalies from a radiativeconvective equilibrium, except ad. This model contains a minimal number of parameters: Q is the
background vertical moisture gradient, Γd is a proportionality constant. The H is irrelevant to
the dynamics (as can be seen by rescaling ad) but allows us to dene a heating/drying rate Had
for the system in dimensional units. The sθ1 and sq are external sources of cooling and moistening,
respectively, that need to be prescribed in the system (see hereafter).
The skeleton model depicts the MJO as a neutrally-stable planetary wave. In particular,
the linear solutions of the system of equations (2) exhibit a MJO mode with essential observed
features, namely a slow eastward phase speed of roughly 5 ms−1, a peculiar dispersion relation
with dω/dk ≈ 0 and a horizontal quadrupole structure (MS2009; MS2011, see also hereafter).
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2.2 Skeleton Model with Rened Vertical Structure
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We now introduce an extended version of the skeleton model with a rened vertical structure.
Previous work on the skeleton model has focused essentially on the MJO dynamics associated to
the envelope of deep convective activity ad (MS2009, MS2011). In order to produce intraseasonal
events with rened vertical structure (e.g. MJO events with a front-to-rear structure) we consider
here additional envelopes of convective activity, namely congestus activity ac and stratiform activity
as . Their evolution is dened as follows:
∂t ac = Γc (q + βs rs − βc rd )(ac − c rc )
∂t ad = Γd (q + βc rc − βd rs )ad
(3)
∂t as = Γs (q + βd rd − βs rc )(as − s rs ) ,
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where rd = ad − ad, rc = ac − ac and rs = as − as are convective activity anomalies to the RCE
state. All parameters are positive and 0 ≤ c, s ≤ 1.
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The physical assumptions for the above interactions are similar to the ones of the original skeleton model. Each convective activity grows/decays (with respective rates Γd, Γc, Γs) depending on a
combination of moisture anomalies ( q) and anomalies of the other convective activities ( rc, rd, rs).
The environment moisture q favors the growth/decay of deep convective activity ad (Kikuchi and
Takayabu, 2004; MS2009), and by extension we assume that it also favors the growth/decay of congestus and stratiform activity, ac and as. In addition, an environment with enhanced/suppressed
convective activity of a certain type may favor the growth/decay of another type of convective
activity, with transition rates βc, βd, βs . For example, enhanced congestus activity ac (rc ≥ 0)
favors the growth of deep activity ad, while inversely enhanced deep activity favors the decay of
congestus activity, both with transition rate βc. For instance, those convective activities are intimately linked at subplanetary scale, as they form coherent synoptic and mescoscale structures
(e.g. convectively coupled waves, etc) where the progressive transition from congestus to deep to
stratiform convection is observed (Biello and Majda, 2005; Khouider and Majda, 2006; Stechmann
et al., 2013). By extension, we assume here that the planetary envelopes of convective activity inuence the growth of each other, which is akin to a stretched building block hypothesis (Moncrie,
2004; Mapes et al., 2006). Finally, we introduce some pseudo-dissipation coecients for congestus
and stratiform activity, c and s respectively. While in Eq. (3) deep activity ad is assumed to grow
exponentially, due to the pseudo-dissipation coecients the congestus and stratiform activity ac
and as are assumed to grow at a weakened rate (see e.g. Khouider and Majda, 2006). For c, s = 1
in particular the growth of congestus and stratiform activity would be linear.
Similar to the original skeleton model, the interactions parametrized in Eq.(3) are further
combined with the linear primitive equations projected on the rst vertical baroclinic mode. This
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170 reads, in non-dimensional units,
∂t u1 − yv1 − ∂x θ1 = 0
yu1 − ∂y θ1 = 0
∂t θ1 − (∂x u1 + ∂y v1 ) = H(ad + ac + as ) − sθ1
∂t q + Q(∂x u1 + ∂y v1 ) = −H(ad + ac + as ) − sq
(4)
∂t ac = Γc (q + βs rs − βc rd )(ac − c rc )
∂t ad = Γd (q + βc rc − βd rs )ad
∂t as = Γs (q + βd rd − βs rc )(as − s rs ) .
171 Note that for ac , as = 0 we retrieve the original skeleton model (MS2009; MS2011).
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2.3 Secondary Circulation
173 In addition to the above dynamical core of the skeleton model with rened vertical structure, we
174 consider a slaved secondary circulation in order to grasp more details of the vertical structure.
175 First, we consider the dry dynamics of the second baroclinic mode:
∂t u2 − yv2 − ∂x θ2 /2 = 0
yu2 − ∂y θ2 /2 = 0
(5)
∂t θ2 − (∂x u2 + ∂y v2 )/2 = H(ξ2c ac − ξ2s as ) − sθ2 ,
176 where un , vn , θn are the zonal, meridional velocity, and potential temperature anomalies, respec177 tively, for the baroclinic mode n = 2. The sθ2 is an external source of cooling, and ξ2c , ξ2s are
178 projection coecients (see hereafter).
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Second, we may consider a more rened vertical structure of moisture. We assume here that
q is
180 a vertically integrated estimate of moisture that decomposes into q = ql +αqm , where ql is moisture
181 anomalies at bottom level (here z = π/4) and qm at middle level (z = π/2). This denition slightly
182 extends the one of the original skeleton model (for which α = 0, MS2009; MS2011), in agreement
183 with recent studies showing that moisture leading the MJO convective core may sometimes have
184 contributions from dierent levels (Stechmann and Majda, 2015). The parameter α ≥ 0 measures
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185 the middle level moisture contribution to q . We assume that the evolution of lower and middle
186 tropsopheric moisture anomalies is as follows:
∂t qm = M (ac − as ) − sqm
∂t ql + Q(∂x u1 + ∂y v1 ) = −H(ad + ac + as ) − αM (ac − as ) −
sql
(6)
,
q
187 where sqm and sl are external sources of moistening. In addition to the main drying and moisture
188 convergence occuring at lower level, congestus and stratiform activity favors moisture exchange
189 between the lower and middle level at a rate M (e.g. through detrainment or downdrafts, Khouider
190 and Majda, 2006). Noteworthy, the above equations sum up to the moisture budget of q = ql +αqm
191 in Eq. (4).
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2.4 Properties
193 The skeleton model with rened vertical structure has properties similar to the ones of the original
194 skeleton model. The present model assumes balanced external sources of cooling and moistening,
195 sθ1 = sq , such that the background state satises H(ad + ac + as ) = sθ1 . Note in addition that
q
196 sθ2 = H(ξ2c ac − ξ2s as ) and sq = sl + αsqm . The system in equation (4) conserves a vertically
197 integrated moist static energy
∂t (θ1 + q) − (1 − Q)(∂x u1 + ∂y v1 ) = 0 ,
(7)
198 and further conserves a total positive energy (as there are no dissipative processes):
∂t [E1 + Ec + Ed + Es ] − ∂x (u1 θ1 ) − ∂y (v1 θ1 ) = 0
E1 = 12 u21 + 12 θ12 + 12 (Q(1 − Q))−1 (q + Qθ1 )2
Ec = H(QΓc )−1 (1 − c )−2 (ac − c ac − ac log(ac ))
Ed = H(QΓd )−1 (ad − ad log(ad ))
Es = H(QΓs )−1 (1 − s )−2 (as − s as − as log(as )) .
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(8)
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Note that the above condition of a conserved energy imposes several aspects of Eq.(4), e.g. a balanced heating/drying ±H(ad +ac +as) in the dry dynamics and moisture equations, as well as skewsymmetric terms βd, βc, βs in the amplitude equations (that cancel in the energy budget). Note
also that for c = 1 the term Ec in the energy budget becomes quadratic, Ec = 12 H(acQΓc)−1rc2.
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2.5 Vertical and Meridional Structure
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The reconstructed elds for the present skeleton model read {u, v}(x, y, z, t) = n{un, vn}Fn(z)
√
√
and θ(x, y, z, t) = n{θn, sn}Gn(z), for n = 1, 2, with Fn(z) = 2cos(nz) and Gn(z) = 2sin(nz),
for 0 ≤ z ≤ π. The total convective activity reconstructs as a(x, y, z, t) = a∗c Fc(z) + a∗dFd(z) +
a∗s Fs (z). Convective activity is always positive. The vertical structures Fd (z), Fc (z) and Fs (z) are
shown in Figure 1. We consider half-sinusoids Fd = G1, Fc = G2 for 0 ≤ z ≤ π/2 and Fs = −G2
for π/2 ≤ z ≤ π, as in Khouider and Majda (2008). While a∗d, a∗c and a∗s are convective activity
associated to the structures Fd(z), Fc(z) and Fs(z), the ad, ac and as from Eq. (4) are convective
activity projected on the rst baroclinic mode, with relationships ad = ξda∗d, ac = ξca∗c , and
as = ξs a∗s . In the remainer of this article, for more clarity we will rather consider a∗d , a∗c and a∗s for
presenting results in dimensional units. The projection coecients ξd, ξs and ξs have values ξd = 1
and ξc, ξs = 4/3π (with in addition ξ2c, ξ2s = 3π/8 for the second baroclinic mode). Dierent values
of those projection coecients could however be considered assuming a more complex localization
of heating/drying by convective activity (Khouider and Majda, 2008).
To obtain the skeleton model in its simplest form, it is useful to further truncate the system
from Eq. (4) to the rst meridional structures (MS2009). We consider the parabolic cylinder
functions:
√
√
n
φnm (y) = m √ Hm ( ny) exp(−ny 2 /2)
(9)
2 m! π
for the baroclinic mode n ≥ 1 and meridional mode m ≥ 0, along with H0(y) = 1, H1(y) = 2y,
H2 (y) = 4y 2 −2. We assume that all convective activity and moisture have a meridional prole φ10 ,
i.e. {ac, as, ad, q, sθ1, sθ2} = {Ac, As, Ad, Q, S1θ , S2θ }φ10. A suitable change of variables is to introduce
Kn and Rn , which are the amplitudes of the equatorial Kelvin wave and of the rst Rossby wave
for the baroclinic mode n, respectively. For instance, the meridional structure φ10 only excites K1
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225 and R1 for n = 1, and for simplicity we assume that it only excites K2 and R2 for n = 2. The
226 evolution of the Kelvin and Rossby wave amplitudes reads (for n = 1, 2):
∂t Kn + ∂x Kn /n = −nSn0
∂t Rn − ∂x Rn /3n = −(4/3)Sn0 ,
(10)
227 where S10 = H(Ad + Ac + As ) − S1θ and S20 ≈ H(ξ2c Ac − ξ2s As ) − S2θ . The variables of the dry
228 dynamics component can be reconstructed using:
√
un = (Kn /2 − Rn /4)φn0 + (Rn /4 2)φn2
√
vn = (∂x Rn − nSn0 )(3 2n)−1 φn1
√
θn = −(Kn /2 + Rn /4)φn0 − (Rn /4 2)φn2 .
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(11)
2.6 Stochastic Skeleton Model
230 A stochastic version of the present skeleton model is designed using a formalism similar to the
231 one of Thual et al. (2014) (hereafter TMS2014). The amplitude equations (3) are replaced by
232 stochastic birth/death processes (the simplest continuous-time Markov process, see chapter 7 of
233 Gardiner, 1994; Lawler, 2006). This accounts for the irregular and intermittent contribution of
234 synoptic activity to the intraseasonal-planetary dynamics. Let ad be a random variable taking
235 discrete values ad = Δa ηd , where ηd is a non-negative integer. The probabilities of transiting from
236 one state ηd to another over a time step Δt read as follows:
P {ηd (t + Δt) = ηd (t) + 1} = λd Δt + o(Δt)
P {ηd (t + Δt) = ηd (t) − 1} = μd Δt + o(Δt)
P {ηd (t + Δt) = ηd (t)} = 1 − (λd + μd )Δt + o(Δt)
(12)
P {ηd (t + Δt) = ηd (t) − 1, ηd (t), ηd (t) + 1} = o(Δt) ,
237 where λd and μd are the upward and downward rates of transition, respectively (see TMS2014 for
238 the deduction of their values from the amplitude equations (3). Similarly, we may dene ac = Δaηc
239 and as = Δaηs and consider transition rates for those variables as well.
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The solving method for the stochastic skeleton model is similar to the one of TMS2014: we
consider here a similar split-method between dry dynamics and convective processes, and a similar
Gillespie algorithm for the stochastic processes of convective activities that is extended to include
more transition rates (see Gillespie (1977), part III.B and III.C in particular).
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2.7 Model Parameters
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The reference parameters values used in this article read, in non-dimensional units: H = 0.22
(10 Kday−1), Q = 0.9, sθ1 = sq = 0.22 (1 Kday−1), Δa = 0.001, as in the original skeleton
model (TMS2014). In addition to this, a∗d = a∗c = a∗s = sθ1(H(ξd + ξc + ξs))−1, Γd, Γc, Γs = 1
(≈ 0.18 mKday−1), βc, βd = 1, βs = 0.1, and c, s = 0.9. For the secondary slaved circulation,
M = 0.05 and α = 1. We also recall that ξd = 1, ξc , ξs = 4/3π , and ξ2c , ξ2s = 3π/8. Finally, note
that a meridional projection parameter γ ≈ 0.6 in factor of the growth rates has been considered
in both linear solutions and the stochastic code (see TMS2014, its equation 4).
While they are many interesting parameter regimes for the present skeleton model, the above
choice of parameter values will be considered for illustration of the model solutions in the remainer
of this article. The above parameters values are plausible and allow for a realistic MJO variability
simulated by the model, as shown in the next sections. Note that although all growth rates
Γd , Γc , Γs are identical, the congestus and stratiform activity have weakened growth due to strong
pseudo-dissipation rates (c, s). In addition, we assume strong transitions rates from congestus
to deep and from deep to stratiform activity ( βc, βd), but a weak transition rate from stratiform
to congestus activity (βs), that may be associated to downdrafts.
In the following sections of this article, simulation results are presented in dimensional units.
The dimensional reference scales are x, y: 1500 km, t: 8 hours, u: 50 m.s−1, θ, q: 15 K (see table
1 of Stechmann et al., 2008).
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3 Model Solutions
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3.1 Linear solutions
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The linear waves of the present skeleton model with rened vertical structure are shown in Figure
2, as computed from the reference parameter values used in this article and using the meridional
truncation from Section 2. All linear waves are neutral. We consistently retrieve the original
skeleton model solutions (MS2009): a dry Kelvin ( ≈ 55 ms−1), dry Rossby (≈ −20 ms−1), MJO
(≈ 5 ms−1) and moist Rossby (≈ −3 ms−1) mode, for which the relation dispersions are only
slightly modied. There are in addition a slow eastward and a slow westward mode, with seasonal
frequency (≤ 0.01 cpd). There is also a second baroclinic dry Kelvin and dry Rossby mode and a
standing mode associated to the secondary slaved circulation (not shown).
Figures 3 and 4 show the x-y structure and the equatorial x-z structure of the MJO mode,
respectively, for a wavenumber k = 1. As in the original skeleton model, we retrieve moisture
anomalies q leading deep convection ad, as well as a quadrupole vortex structure on θ1. In addition
to this, the present model reproduces the front-to-rear structure of the MJO observed in nature
on both heating, moisture, temperature and winds (Kiladis et al., 2005). There is a progressive
transition from congestus to deep to stratiform activity, a transition from lower to middle level
moisture anomalies, and a transition from lower to upper wind anomalies and lower to upper
temperature anomalies. This is an attractive feature of the present skeleton model.
Figure 5 shows the x-y structure of the moist Rossby mode. The moist Rossby mode structure
is similar to the one of the original skeleton model (MS2009), with in addition a front-to-rear
structure of convective activity and moisture oriented eastward. However, the convective core
of the moist Rossby mode has here positive middle level moisture anomalies qm and negative
lower level moisture anomalies ql , that approximatively compensate leading to near zero moisture
anomalies q. This moist Rossby mode may be a potential surrogate for the convectively coupled
ER n=1 wave observed in nature (Wheeler and Kiladis, 1999, Wheeler and Kiladis 2000).
Finally, the slow eastward and westward modes also exhibit front-to-rear structures of convective activity and moisture (not shown). Those structures are, however, opposite to the direction of
13
290 propagation. This counterintuitive behaviour is due to the transition rates ( βc , βd , βs ) in the model
291 that also allow for opposite progressions of convective activity (e.g. a transition from increased
292 stratiform to decreased deep to increased congestus activity, etc, see Eq. 3). We will however
293 show that due to their seasonal nature these slow modes are less prominent than the other ones in
294 numerical simulations (see hereafter).
295
3.2 Stochastic Simulation (Homogeneous Background)
296 We now present numerical solutions of the stochastic skeleton model with rened vertical structure.
297 First, we consider a numerical simulation with a homogeneous background state, as represented by
298 the constant and zonally homogeneous external sources of cooling/moistening sθ1 and sq (identical
299 to TMS2014). We analyze the simulations output in the statistically equilibrated regime. Note
300 that for this numerical simulation the slaved secondary circulation ( ql , qm , θ2 , u2 ) is omitted as it
301 is irrelevant to the main model dynamics.
302
The present skeleton model with rened vertical structure simulates a MJO-like signal that is
303 the dominant signal at intraseasonal-planetary scale, consistent with observations (Wheeler and
304 Kiladis, 1999). Figure 6 shows the power spectra of the variables as a function of the zonal
305 wavenumber k (in 2π/40, 000 km) and frequency ω (in cpd). The MJO appears here as a sharp
306 power peak in the intraseasonal-planetary band ( 1 ≤ k ≤ 5 and 1/90 ≤ ω ≤ 1/30 cpd), most
307 prominent in u1 , q , Had and also the variables Hac and Has . This power peak roughly corresponds
308 to the slow eastward phase speed of ω/k ≈ 5 ms−1 with the peculiar relation dispersion dω/dk ≈ 0
309 found in observations. Due to the increased complexity of the present skeleton model, the MJO
310 power peak is here less prominent than the one of TMS2014, but it still remains the main peak in
311 the intreaseasonal band. The other features at intraseasonal-planetary scale are the power peaks
312 near the dispersion curve of the moist Rossby mode, the dry Kelvin and dry Rossby waves (see
313 TMS2014), in addition to the power peaks near the dispersion curves of the slow modes ( ≈ 0.01
314 cpd), that are most prominent on Hac and Has .
315
Figure 7 shows the Hovmollers diagrams of the model variables at the equator. On average,
316 the simulated MJO events propagate eastward with a phase speed of around 5 − 15 ms−1 and a
14
317 roughly constant frequency, consistent with the composite MJO features found in nature (Zhang,
318 2005). The MJO events are most visible on u1 q , and Had , while Hac and Has also have seasonal
319 variations associated to the slow eastward and westward modes, consistent with the power spectra
320 shown in Fig.6.
321
3.3 Stochastic Simulation (Warm Pool Background)
322 We now present numerical solutions of the stochastic skeleton model with rened vertical structure
323 for a numerical simulation with a warm pool background state. The external sources vary zonally to
324 be representative of a background warm pool state, with values sθ1 = sq = 0.022(1−0.6 cos(2πx/L))
325 at the equator (identical to TMS2014), and where L is the equatorial belt length. We analyze the
326 simulations output in the statistically equilibrated regime.
327
Figure 8 shows the power spectra for the experiment with a background warm pool. The power
328 spectra are overall similar to the ones for the experiment with a homogeneous background (see
329 Fig.6). However, at k = 1 the MJO power is more diuse in comparison, as seen on u1 , q , Had ,
330 Hac and Has : this is likely due to the nonlinear interaction of the MJO wave at k = 1 with the
331 background warm pool which zonal shape also corresponds to a wavenumber k = 1 (Ogrosky and
332 Stechmann, 2015).
333
Figure 9 shows the Hovmollers diagrams of the model variables at the equator. The results
334 are overall similar to the ones for the experiment with a homogeneous background (see Fig. 7).
335 However, due to the warm pool background the variability remains conned to the warm pool
336 region, which is more realistic (MS2011; TMS2014).
337
Figure 10 shows Hovmoller diagrams ltered in the MJO band (k=1-3,
ω =1/30-1/70
cpd).
338 Note that the MJO band denition is slightly dierent than the one of previous works (TMS2014,
339 where ω =1/30-1/90 cpd) in order to avoid grasping the seasonal variability associated to the slow
340 modes. As in previous work (TMS2014), we observe intermittent MJO wave trains with prominent
341 Had , q and u1 anomalies, and here in addition prominent Hac and Has anomalies. The MJO
342 events are organized into wave trains with growth and demise, i.e. into series of successive MJO
343 events following a primary MJO event, as seen in nature (Matthews, 2008; Stachnik et al., 2015).
15
344
In addition, the MJO events are irregular and intermittent, with a great diversity in strength,
345
structure, lifetime and localization.
346
Figure 11 shows the details of a selected MJO wave train. The MJO propagations with phase
5−15 ms−1 are clearly visible on all variables.
347
speed around
348
have structures in overall agreement with the MJO linear solution shown in gure 3. Congestus
349
activity
350 Has
The MJO events within this wave train
Hac and moisture q leads the deep convective activity center Had , while stratiform activity
trails behind, consistent with the MJO front-to-rear structure of convective activity found in
351
nature (Kiladis et al., 2005). This is an attractive feature of the present stochastic skeleton model
352
in generating MJO variability.
353
4 Discussion
354
We have analyzed the dynamics of a skeleton model for the MJO with rened vertical structure.
355
It is a modied version of a minimal dynamical model, the skeleton model (Majda and Stech-
356
mann, 2009b, 2011). The skeleton model has been shown in previous work to capture together
357
the MJO's salient features of (I) a slow eastward phase speed of roughly
358
dispersion relation with
359
version, that further includes a simple stochastic parametrization of the unresolved synoptic-scale
360
convective/wave processes, has been shown to reproduce qualitatively the intermittent generation
361
of MJO events and the organization of MJO events into wave trains with growth and demise, as in
362
nature (Thual et al., 2014). In addition to those features, the present skeleton model with rened
363
vertical structure also reproduces qualitatively the front-to-rear vertical structure of MJO events
364
found in nature. This includes MJO events marked by a planetary envelope of convective activity
365
transitioning from the congestus to the deep to the stratiform type, in addition to a front-to-rear
366
structure of moisture, winds and temperature. This is achieved by considering the evolution of the
367
planetary envelope of both congestus, deep and stratiform activity in the dynamical core of the
368
skeleton model, in addition to a secondary circulation for the detailed evolution of lower and mid-
369
dle level moisture and the second baroclinic mode dry dynamics. In addition to this, the present
370
model conserves many interesting properties of the original skeleton model such as a converved
dω/dk ≈ 0,
5 ms−1 ,
(II) a peculiar
and (III) a horizontal quadrupole structure. Its stochastic
16
371 positive energy, similar linear solutions, etc.
372
While the present skeleton model appears to be a plausible representation for the essential
373 mechanisms of the MJO, several issues need to be adressed as a perspective for future work. First,
374 while here we have illustrated the model solutions for an arbitrary choice of parameters, the present
375 model has many other interesting parameter regimes that should be analyzed. Another impor376 tant issue is to compare further the skeleton model solutions with their observational surrogates,
377 qualitatively and also quantitatively. The solutions of the present model may be analyzed for a
378 more realistic background of heating/moisture (Ogrosky and Stechmann, 2015), or used in order to
379 provide a new theoretical estimate of MJO events in observations (Stechmann and Majda, 2015).
380 For example, the present model could be used to analyze zonal variations of the intensity of the
381 MJO front-to-rear structure (Kiladis et al., 2005).
382
An interesting perspective for future work would also be to increase the complexity of the
383 present skeleton model with rened vertical structure. For example, the present skeleton model
384 includes a secondary circulation that accounts for the details of the moisture vertical structure
385 as well as the second baroclinic mode dynamics. Here for simplicity this circulation is slaved to
386 the main MJO dynamics. For future work, a skeleton model coupling this circulation to its main
387 dynamical core should be considered (Khouider and Majda, 2006; Biello and Majda, 2005, 2006;
388 Majda and Stechmann, 2009a; Stechmann et al., 2013). An interesting approach to achieve this is
389 may be to propose modied moisture and amplitude equations for which the system still conserves
390 a positive energy. Other aspects of the present model that could be improved are the transition
391 rates between congestus, deep and stratiform activities ( βc , βd , βc ). For simplicity those transition
392 rates have been considered constant, which however allows for some counterintuitve evolutions of
393 convective activity (e.g. as seen on the slow eastward and westward linear modes of the model).
394 Rendering those transition rates non-constant may improve the model variability (see e.g. the
395 threshold parameter Λ in Khouider and Majda, 2006). A more complete model should also account
396 for more detailed sub-planetary processes within the envelope of intraseasonal events, including
397 for example synoptic-scale convectively coupled waves and/or mesoscale convective systems (e.g.
398 Moncrie et al., 2007; Majda et al., 2007; Khouider et al., 2010; Frenkel et al., 2012).
17
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Acknowledgements: The research of A.J.M. is partially supported by the Oce of Naval
Research Grant ONR MURI N00014 -12-1-0912. S.T. is supported as a postdoctoral fellow through
A.J.M's ONR MURI Grant.
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RG2003,
486
Figures Captions:
487
Figure 1: Vertical Structures
488
489
Fd (z), Fc (z)
and Fs (z), as a function of z (0 ≤ z
≤π
from the
bottom to the tropopause).
Figure 2: Summary of the skeleton model linear stability: (a) phase speed
ω/k (m.s−1 ),
and
490
(b) frequency ω (cpd), as a function of the zonal wavenumber k (2π/40000km). The black circles
491
mark the integer wavenumbers satisfying periodic boundary conditions. This is repeated for each
492
eigenmode, from top to bottom in order of decreasing phase speed.
493
Figure 3: Structure
x−y
of the MJO mode for
494 Ha∗s (Kday −1 ), u1 (ms−1 ), θ1 (K ), q , ql
495
k = 1,
in dimensional units:
Ha∗d , Ha∗c
and qm (K ), u2 (ms−1 ), θ2 (K ), as a function of
x
and
and y
(1000 km). The dashed line marks x = 20, 000 km.
496
Figure 4: Structure x − z of the MJO mode at the equator for k = 1. Contours of reconstructed
497
heating Ha (Kday−1 ), moisture q (K ), potential temperature θ (K ) and zonal winds u (ms−1 ), at
498
the equator and as a function of x (1000km) and z (0 ≤ z ≤ π from the bottom to the tropopause).
499
Figure 5: Structure x − y of the moist Rossby mode for k = 1, in dimensional units (see gure
500
501
3 for denitions).
Figure 6: Zonal wavenumber-frequency power spectra (with homogeneous background): for
502 u1 (ms−1 ), θ1 (K ), q (K ), Ha∗d , Ha∗c
and
Ha∗s (Kday −1 )
taken at the equator, as a function of
503
zonal wavenumber (in
504
logarithm, for the dimensional variables taken at the equator. The black dashed lines mark the
505
periods 90 and 30 days.
506
2π/40000km)
and frequency (cpd). The contour levels are in the base 10-
Figure 7: Hovmollers x−t (with a homogeneous background): for Ha∗c , Ha∗d and Ha∗s (Kday−1 ),
507 q (K ), u1 (ms−1 ) and θ1 (K ) at the equator, as a function of zonal position (1000km) and simulation
508
509
510
511
512
time (days from reference time 19986 days).
Figure 8: Zonal wavenumber-frequency power spectra (with warm pool background). See gure
6 for denitions.
Figure 9: Hovmollers
x−t
(with a warm pool background): for
Ha∗c , Ha∗d , Ha∗s (Kday −1 ), q
(K), u1 (ms−1 ) and θ1 (K) at the equator, as a function of zonal position (1000km) and simulation
22
513 time (days from reference time 19986 days).
514
Figure 10: Hovmollers
x − t (with a warm pool background).
Same as gure 9 but for variables
515 ltered in the MJO band (k=1-3, ω =1/30-1/70 cpd).
516
Figure 11: Hovmollers
x−t
for a selected MJO wavetrain (with a warm pool background).
517 Same as gure 10 but at a dierent simulation time (in days from reference time 5996 days).
23
518
Figures:
Figure 1: Vertical Structures
bottom to the tropopause).
Fd (z), Fc (z)
and
Fs (z),
24
as a function of
z (0 ≤ z ≤ π
from the
Figure 2: Summary of the skeleton model linear stability: (a) phase speed ω/k (m.s−1 ), and (b)
frequency ω (cpd), as a function of the zonal wavenumber k (2π/40000km). The black circles
mark the integer wavenumbers satisfying periodic boundary conditions. This is repeated for each
eigenmode, from top to bottom in order of decreasing phase speed..
25
Figure 3: Structure x − y of the MJO mode for k = 1, in dimensional units: Ha∗d , Ha∗c and Ha∗s
(Kday−1 ), u1 (ms−1 ), θ1 (K ), q, ql and qm (K ), u2 (ms−1 ), θ2 (K ), as a function of x and y (1000
km). The dashed line marks x = 20, 000 km.
Figure 4: Structure x − z of the MJO mode at the equator for k = 1. Contours of reconstructed
heating Ha (Kday−1 ), moisture q (K ), potential temperature θ (K ) and zonal winds u (ms−1 ), at
the equator and as a function of x (1000km) and z (0 ≤ z ≤ π from the bottom to the tropopause).
26
Figure 5: Structure x − y of the moist Rossby mode for
for denitions).
27
k = 1,
in dimensional units (see gure 3
Figure 6: Zonal wavenumber-frequency power spectra (with homogeneous background): for u1
(ms−1 ), θ1 (K ), q (K ), Ha∗d , Ha∗c and Ha∗s (Kday−1 ) taken at the equator, as a function of
zonal wavenumber (in 2π/40000km) and frequency (cpd). The contour levels are in the base
10-logarithm, for the dimensional variables taken at the equator. The black dashed lines mark the
periods 90 and 30 days.
28
Figure 7: Hovmollers x − t (with a homogeneous background): for Ha∗c , Ha∗d and Ha∗s (Kday−1 ), q
(K ), u1 (ms−1 ) and θ1 (K ) at the equator, as a function of zonal position (1000km) and simulation
time (days from reference time 19986 days).
29
Figure 8: Zonal wavenumber-frequency power spectra (with warm pool background). See gure 6
for denitions.
30
Figure 9: Hovmollers x − t (with a warm pool background): for Ha∗c , Ha∗d , Ha∗s (Kday−1 ), q (K),
u1 (ms−1 ) and θ1 (K) at the equator, as a function of zonal position (1000km) and simulation time
(days from reference time 19986 days).
31
Figure 10: Hovmollers x − t (with a warm pool background). Same as gure 9 but for variables
ltered in the MJO band (k=1-3, ω=1/30-1/70 cpd).
32
Figure 11: Hovmollers x − t for a selected MJO wavetrain (with a warm pool background). Same
as gure 10 but at a dierent simulation time (in days from reference time 5996 days).
33