8 We report a discovery of steady long-living slowly eastward moving large-scale coherent twin 9 cyclones, the equatorial modons, in the shallow water model in the equatorial beta-plane, the 10 archetype model of the ocean and atmosphere dynamics in tropics. We start by constructing an- 11 alytical asymptotic modon solutions in the non-divergent velocity approximation, and then show 12 by simulations with a high-resolution numerical scheme that such con gurations evolve into steady 13 dipolar solutions of the full model. In the atmospheric context, the modons persist in the pres- 14 ence of moist convection, being accompanied and enhanced by speci c patterns of water-vapour 15 condensation. 16 Corresponding author. Email: zeitlin@lmd.ens.fr 2 As is known, the large-scale atmosphere and ocean dynamics in tropics is largely de- 17 termined by equatorial waves. Indeed, superposition of atmospheric Outgoing Longwave 18 Radiation (OLR) data onto the spectrum of equatorial waves gives a very good correspon- 19 dence [1], except for a branch corresponding to slow eastward-moving motions associated 20 with the Madden-Julian Oscillation (MJO) [2]. The Kelvin waves, the only species of the 21 equatorial waves that can produce steady large-scale eastward-moving structures, have much 22 faster velocity than MJO [3]. This fact motivates a search for coherent dynamical structures 23 with slow eastward propagation which, thus, should be necessarily nonlinear. There do exist 24 steady eastward-moving structures in a uid layer on the mid-latitude f-plane, the modons 25 which are exact dipolar solutions of the quasi-geostrophic (QG) equations [4]. Although the 26 classical modons were found analytically in the QG approximation, it was shown rst by a 27 computer-assisted analysis [5] and then by direct numerical simulations [6] that correspond- 28 ing solutions exist also in the parent rotating shallow water (RSW) model on the f- plane. 29 The QG modon solutions were extended to the full sphere [7, 8] in the framework of equiva- 30 lent barotropic model, which is structurally close to QG. The pioneering paper [9] advanced 31 an idea that the MJO could be related to such a modon in spherical geometry and showed 32 that, like in the midlatitude tangent plane [6], the equivalent barotropic modon persists in 33 the RSW on the sphere. Yet, the archetype model for understanding dynamics in the tropics 34 is RSW in the equatorial beta-plane, e.g. [10]. We, thus, look for eastward-moving modons 35 in this model. However, it encounters an obstacle from the very beginning, because there 36 is no consistent QG approximation at the equator, e.g. [11], so the known modon solutions 37 can not be borrowed. Our main observation is that there is a dynamical regime in RSW, 38 called long-wave approximation in oceanography [12], which corresponds to small pressure 39 variations, and gives, to the leading order, equations which do allow for modon solutions. 40 In the atmospheric context this regime is the RSW analog of the non-divergent equatorial 41 balance model [13]. The relevance of this model to large-scale tropical motions in the atmo- 42 sphere is supported by scale and data analyses [14, 15]. We construct the modon solutions 43 in this regime and, following [6], use these asymptotic solutions to initialize high-resolution 44 numerical simulations with the full RSW model, and show that coherent dipolar steady 45 eastward-moving structures do arise and persist. We then show that, in the atmospheric 46 context, inclusion of moisture, with condensation and evaporation, enhances the modons, 47 without disrupting long-time coherence, and produces speci c convective patterns. 48 3 We start with RSW equations in the equatorial beta- plane with no dissipation: 49 8 >< >:@tv+vrv+ y^z^v+grh= 0; @th+r(vh) = 0;(1) 50 wherer= (@x;@y),v= (u;v),uis zonal and v- meridional components of velocity, his 51 geopotential height (thickness), is the meridional gradient of the Coriolis parameter, and ^z 52 is a unit vertical vector. The interpretation of the model in the oceanic context is direct, while 53 in the atmospheric context it should be understood as vertically averaged primitive equations 54 in pseudo-height pressure coordinates [16], and can be extended to include water vapor with 55 condensation and related latent heat release [17], as well as surface evaporation (see below). 56 We introduce a pressure perturbation parameter :h=H(1+), whereHis unperturbed 57 thickness, and x spatial, velocity and time- scales: ( x;y)L;(u;v)V; tL=V. Under 58 hypothesis that !0;andgH=V2=O(1), henceV <<pgH, i.e. the characteristic 59 velocity is much smaller than the phase velocity of the Kelvin waves c=pgH, the non- 60 dimensional equations take the form: 61 @tv+vrv+ y^z^v+r= 0; (2) 62 (@t+vr) + (1 +)rv= 0; (3) 63 where = L2=V, and v=v0+v1+:::. In the leading order in (3) givesrv0= 0, 64 the motion is non-divergent, and u0=@y ; v 0=@x . Cross-di erentiation of the zonal 65 and meridional momentum equations results in the equation for the stream-function: 66 r2 t+J( ;r2 ) + x= 0; (4) 67 whereJdenotes the Jacobian. The modon solutions are built following [4]. They are are 68 obtained under hypothesis of steady motion with constant zonal velocity U, by supposing a 69 linear relationship between the absolute vorticity and stream-function in co-moving frame, 70 which gives inhomogeneous Helmholtz equation. It is solved by separation of variables in 71 polar coordinates in terms of Bessel functions, rst in the outer domain under condition of 72 decay, and then in the inner domain, and matching the inner and outer solutions across a 73 circle of a given radius ain the plane. The solution has the form: 74 8 >>>>< >>>>: ext=Ua K1(pa)K1(pr) sin; r>a; int=h Up2 k2J1(ka)J1(kr)r k2(1 +U+Uk2)i sin; r<a;(5) 75 4 -1 -0.5 0 0.5 1 x (Ld)-1-0.500.51y (Ld) -1-0.8-0.6-0.4-0.200.20.40.60.81 -1 -0.5 0 0.5 1 x (Ld)-1-0.500.51y (Ld) -1-0.8-0.6-0.4-0.200.20.40.60.81FIG. 1. Normalised streamlines and velocity eld of an asymptotic modon in stationary ( left) and co-moving ( right ) frames. Dashed circle: separatrix of radius a. whereJ1andK1are ordinary and modi ed Bessel functions of order one, respectively, p 76 is real,p2= =U, soU > 0, and the motion is eastward . For each pair ( a;p) there exist 77 a series of eigenvalues karising from matching conditions, of which the lowest corresponds 78 to a dipole, to be called the asymptotic modon. The next eigenvalue gives a quadrupolar 79 solution called \shielded modon", and so on. By construction, this solution can be centred 80 anywhere in the equatorial beta-plane, yet it is only when its center is at the equator that it 81 results in a cyclonic pair. The streamlines of the asymptotic modon are presented in Fig. 1. 82 First-order corrections u1; v1can be found in the next order of the asymptotic expansion. 83 84 There is no guarantee that such expansion converges, but numerical results presented below 85 suggest that there is indeed an exact solution of (1) corresponding to this asymptotic one. 86 We initialized the numerical simulations in the RSW model with the velocity eld corre- 87 sponding to the asymptotic modon solution (5) with U pgH, and at pressure. Simula- 88 tions were performed with high-resolution well-balanced nite-volume numerical scheme [18] 89 in a rectangular domain with sponges at the boundaries, which allow to mostly evacuate short 90 inertia-gravity waves. No explicit dissipation was added. The domain was chosen to be sym- 91 metric with respect to the equator and wide enough 9 Ld6:5Ld, whereLd= (gH)1=4= 1=2is 92 the equatorial deformation radius, in order to minimize the in uence of boundaries onto the 93 modons. We varied spatial resolution from 600 600 to 12001200, to check numerical con- 94 vergence. The natural units of length and time in the numerical scheme are Ldand 1= L d, 95 5 FIG. 2. Initial adjustment of the asymptotic modon with non-dimensional a= 0:5;U= 0:15, as seen in the thickness and velocity elds. Time in units of 1 = L d. respectively, which corresponds to nondimensional beta equal to one. Their numerical val- 96 ues, withgand xed for the Earth, are uniquely de ned by the value of H, the equivalent 97 depth. In the atmospheric context, for H10km,Ld3000kmand 1= L d5h. At the 98 initial stages of the simulations, cf. Fig. 2 we observed an adjustment of initial con guration 99 with emission of inertia-gravity waves and formation of a dipolar coherent structure, both 100 in pressure and velocity elds. This dipolar structure in a form of twin cyclones then moves 101 102 eastward without changing form, as follows from Fig. 3. To exclude the in uence of inertia- 103 104 gravity waves produced by the initial adjustment and partially re ected by the boundaries, 105 the modon was \nudged", i.e. the far gravity wave eld was removed, and the simulation 106 6 FIG. 3. Snapshots of hfor eastward-moving equatorial modon after initial adjustment at t= 0;10;20;30;40;50 [1=( Ld)], from top to bottom (nondimensional a= 0:5;U= 0:2). reinitialized without it. The detailed characteristics of the \exact" modon are presented in 107 Fig. 4. Its coherence can be inferred from the scatter plot of Bernoulli function vs potential 108 vorticity (PV) in the upper-right panel of the Figure, which clearly gives a line. As is known, 109 e.g. [5], Bernoulli function and PV are functionally dependent for steady moving solutions of 110 RSW equations. The phase speed of the modon depends on initial a;U. The modons keep 111 moving eastward for the long time without losing their coherence. The modon's velocity 112 can be inferred from the distance it covered with respect to initial condition. The energy 113 of the modon remains practically constant, while we detected a weak enstrophy loss, which 114 7 0 1 2 3 4 5 6 PV-1-0.500.51B -3 -2.5 -2 -1.5 -1 -0.5 X (Ld)0.840.860.880.90.920.940.960.9811.02h (y=0) -1.5 -1 -0.5 0 0.5 1 1.5 Y (Ld)0.840.860.880.90.920.940.960.9811.02h(x=xc)FIG. 4. Upper row : thickness hof the equatorial modon at t= 15 [1= L d] (left); scatter plot of Bernoulli function in the co-moving frame vs potential vorticity (PV) (right). Lower row : Zonal section of the equatorial modon across the center of each cyclone (left), and meridional section across the center of the modon (right). a= 0:5;U= 0:2. diminishes with increasing resolution (several % loss at high resolution), not shown. The 115 116 di erence between the asymptotic and \exact" modons is clear from the comparison of their 117 relative vorticity in Figure 5, which shows that the \exact" modon is more compact and has 118 a larger peak vorticity. Although we did not explore in detail the space of parameters of the 119 120 modon solutions, we varied the parameter a, the modon's radius. We observed that while 121 fora<L dequatorial modons always emerge, for a>L d, with the same velocity scale, the 122 initial asymptotic modon transforms into a packet of equatorial Rossby waves moving west- 123 ward. We also checked the robustness of the equatorial modon to the details of initialization, 124 by either removing the outer eld of the asymptotic modon, or misaligning its axis. In all 125 8 FIG. 5. Relative vorticity of the asymptotic ( left) vs \exact" ( right ) modons. \Exact" modon is more compact and more intense. such simulations the eastward-moving equatorial modon was always emerging after leaving 126 some \debris" and/or wobbling at initial stages. This indicates that equatorial modons are 127 attracting solutions. 128 After having established existence of equatorial modons in the adiabatic environment, we 129 switched on the e ects of moisture within the diabatic atmospheric moist-convective RSW 130 (mcRSW) [17]. The equations of the model read: 131 8 >>>>< >>>>:@tv+vrv+ y^z^v=grh; @th+r(vh) = P; @tQ+r(Qv) =P+E:(6) 132 HereQis a bulk amount of water vapor in the air column, is a parameter depending on 133 the underlying strati cation, Pis the condensation sink, and Eis the evaporation source of 134 moisture, which are parameterized as follows: 135 P=QQs H(QQs); E = jvj(QsQ)H(QsQ): (7) 136 Qsis a saturation moisture threshold, is relaxation time, is a parameter regulating 137 evaporation, andH(:::) denotes the Heaviside function. As in previous test simulations [17], 138 we take a uniform initial moisture distribution Qiclose to saturation: Qi=Qs0:01 with 139 Qs= 0:9, a short relaxation time equal to several time-steps of the numerical scheme, 140 = 1, and =O(101). We performed exactly the same simulations as in the adiabatic 141 9 −0.500.5 −0.500.5 −0.500.5Y (Ld) −0.500.5 −4 −3 −2 −1 0 1 2 3 4−0.500.5 X (Ld) −10−8−6−4−20246810 −0.500.5 −0.500.5 −0.500.5Y (Ld) −0.500.5 −4 −3 −2 −1 0 1 2 3 4−0.500.5 X (Ld) −10−8−6−4−20246810FIG. 6. Steady eastward propagation of the equatorial modon, as seen in PV eld in moist- convective (left) and adiabatic (right) environments, with the same initial conditions. t= 0;10;20;30;40 [1=( Ld)], from top to bottom. ( a= 0:5;U= 0:2; = 0:05). case. They showed that the equatorial modons keep their coherence in the moist-convective 142 environment. Moreover, as is known, and con rmed in mcRSW [19, 20], condensation leads 143 to intensi cation of the cyclonic vortices. Correspondingly, the potential vorticity (PV) 144 anomalies become stronger, the size smaller, and the phase speed of the convectively-coupled 145 modon becomes higher than that of its adiabatic counterpart. A comparison of evolution 146 of the same modon in adiabatic and moist-convective environments is presented in Fig. 6. 147 148 149 A typical pattern of the associated moisture eld, and characteristic condensation patterns 150 at the front and at the rear of the modon are shown in Fig. 7. Such patterns persist 151 all along the simulation. We should stress that, although we initialized the simulations 152 with non-divergent velocity eld of the asymptotic modon, the \exact" modon develops and 153 maintains a characteristic convergence/divergence pattern (not shown), which redistributes 154 moisture in its core, and prompts condensation in speci c zones. Notice that as condensation 155 enhances cyclonic vorticity [19], the moist-convective modon intensi es, and moves faster 156 than the dry one. 157 To conclude, we established existence of long-living, slow eastward-moving dipolar coher- 158 ent structures, the modons, in the shallow-water dynamics in the equatorial beta-plane. In 159 10 FIG. 7. (a= 0:5;U= 0:2; = 0:05). Distribution of the water vapor (left) and superposition of pressure (black) and condensation (red) isopleths (right) at t= 30 [1=( Ld)]. the atmospheric context, the modons maintain their coherence in the presence of moisture 160 and are accompanied by a speci c and robust moist convection pattern, if condensation 161 and evaporation are present. The scales of these structures are conditioned by the inter- 162 pretation of the RSW model. At given gand the actual scales depend on the value of 163 the unperturbed thickness H, or of the corresponding deformation radius Ld. Although we 164 cannot make a rm link to MJO at this stage, some features of the equatorial modon do 165 resemble those of dipolar structures associated with the MJO [21]. Work is in progress in 166 this direction, in particular including improved parametrisation of moist processes [20] and 167 vertical structure, which has speci c features in MJO, cf. [22] and