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1 2 A Skeleton Model for the MJO with Rened Vertical Structure 3 Sulian Thual (1) and Andrew J. Majda (1) 4 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 , [email protected] Abstract 9 10 The Madden-Julian oscillation (MJO) is the dominant mode of variability in the 11 tropical atmosphere on intraseasonal timescales and planetary spatial scales. The skele- 12 ton model is a minimal dynamical model that recovers robustly the most fundamental 13 MJO features of (I) a slow eastward speed of roughly 14 relation with 15 model depicts the MJO as a neutrally-stable atmospheric wave that involves a simple 16 multiscale interaction between planetary dry dynamics, planetary lower-tropospheric 17 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 18 Here we propose and analyse an extended version of the skeleton model with ad- 19 ditional variables accounting for the rened vertical structure of the MJO in nature. 20 The present model reproduces qualitatively the front-to-rear vertical structure of the 21 MJO found in nature, with MJO events marked by a planetary envelope of convec- 22 tive activity transitioning from the congestus to the deep to the stratiform type, in 1 23 addition to a front-to-rear structure of moisture, winds and temperature. Despite its 24 increased complexity the present model retains several interesting features of the origi- 25 nal skeleton model such as a conserved energy and similar linear solutions. We further 26 analyze a model version with a simple stochastic parametrization for the unresolved 27 details of synoptic-scale activity. The stochastic model solutions show intermittent 28 initiation, propagation and shut down of MJO wave trains, as in previous studies, in 29 addition to MJO events with a front-to-rear vertical structure of varying intensity and 30 characteristics from one event to another. 31 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. 40 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. 2 50 Interestingly, the planetary front-to-rear structure of the MJO is also observed for various 51 convective events at synoptic scale and mesoscale. The MJO being a planetary envelope of such 52 convective events, a theoretical issue of interest is to understand how the MJO characteristics 53 are related to the ones of such convective events, and inversely (Moncrie, 2004; Mapes et al., 54 2006; Majda and Stechmann, 2009a; Stechmann et al., 2013). Khouider and Majda (2006; 2007) 55 developed a systematic multicloud model convective parametrization highlighting the nonlinear 56 dynamical role of the three cloud types, congestus, stratiform, and deep convective clouds, and 57 their dierent heating vertical structures. The multicloud model reproduces key features of the 58 observational record for mesoscale and synoptic-scale convectively coupled waves, as well as a 59 realistic MJO envelope analog for an intraseasonal parameter regime (Majda et al. 2007). The 60 multicloud model has also been used as a cumulus parametrization in GCM experiments with 61 consequent improvement of the simulated MJO variability (Khouider et al., 2011; Ajayamohan 62 et al., 2013; Deng et al., 2014). As another example, the role of synoptic scale waves in producing 63 key features of the MJO's vertical structure has been elucidated in multiscale asymptotic models 64 (Majda and Biello, 2004; Biello and Majda, 2005, 2006; Majda and Stechmann, 2009a; Stechmann 65 et al., 2013). 66 While theory and simulation of the MJO remain dicult challenges, they are guided by some 67 generally accepted, fundamental features of the MJO on intraseasonal-planetary scales that have 68 been identied relatively clearly in observations (Hendon and Salby, 1994; Wheeler and Kiladis, 69 1999; Zhang, 2005). These features, referred to here as the MJO's skeleton features, are (I) a slow 70 eastward phase speed of roughly 71 (III) a horizontal quadrupole structure. Majda and Stechmann (2009b) introduced a minimal dy- 72 namical model, the skeleton model, that captures the MJO's intraseasonal features (I-III) together 73 for the rst time in a simple model. The model is a coupled nonlinear oscillator model for the 74 MJO skeleton features as well as tropical intraseasonal variability in general. In particular, there 75 is no instability mechanism at planetary scale, and the interaction with sub-planetary convective 76 processes discussed above is accounted for, at least in a crude fashion. In a collection of numer- 77 ical experiments, the non-linear skeleton model has been shown to simulate realistic MJO events 5 ms−1 , (II) a peculiar dispersion relation with 3 dω/dk ≈ 0, and 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 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. 4 105 2 Model Formulation 106 2.1 Nonlinear Skeleton Model 107 108 109 110 111 112 113 114 115 116 117 118 119 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 , 120 121 122 123 124 (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 , 5 (2) 138 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). 139 2.2 Skeleton Model with Rened Vertical Structure 125 126 127 128 129 130 131 132 133 134 135 136 137 140 141 142 143 144 145 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 ) , 146 147 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. 6 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 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 7 170 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). 172 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). 179 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 8 185 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). 192 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 )) . 9 (8) 202 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. 203 2.5 Vertical and Meridional Structure 199 200 201 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 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 10 225 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 . 229 (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. 11 243 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). 244 2.7 Model Parameters 240 241 242 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 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). 12 263 3 Model Solutions 264 3.1 Linear solutions 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 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). 17 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 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. References Ajayamohan, S., Khouider, B., and Majda, A. J. (2013). Realistic initiation and dynamics of the Madden-Julian Oscillation in a coarse resolution aquaplanet GCM. Geophys. Res. Lett., 40:62526257. Biello, J. A. and Majda, A. J. (2005). A New Multiscale Model for the Madden-Julian Oscillation. J. Atmos. Sci., 62(6):16941721. Biello, J. A. and Majda, A. J. 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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