Dr Víctor Avendaño Porras C: Dr Carlos Calvo

Taking a snapshot in time and considering only the largest-scale horizontal and vertical structure of the convective envelope, the MJO bares more than a passing resemblance to Matsuno’s solutions to the forced equatorial beta-plane shallow water equations (Matsuno 1966, discussed in section 4.2.1).

The deep convective nature of the positive phase of the MJO (the “stormy and wet” part of fig. 6.1) suggests a dependence on moist tropospheric instability – either the underlying cause of the MJO also drives deep convection, or it is the process of moist convection in the tropics that produces an eastward propagating disturbance. The MJO manifests as a coherent accumulation of cloud and rainfall; the issue of cloud aggregation, and specifically self-aggregation, is still a largely unknown process and often entirely absent from numerical models. Self-aggregation occurs in high- resolution simulations but can be sensitive to the resolution of the model used (Muller and Held 2012). The mechanism by which the local cloud aggregation interacts with the large-scale flow, ultimately producing a global-scale MJO-like pattern, has been investigated in fully convective cloud-resolving large-eddy simulations. The essential mechanisms required for cloud aggregation in these complex models are:

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1. Cloud-radiative feedback. The spatial modulation of longwave emission drives larger areas of cloud, separated by clear skies. The radiative feedbacks that initially promote aggregation, and those that then stabilise large areas of cloud can be different (Wing and Emanuel 2014).

2. Wind-induced surface heat exchange (WISHE). Stronger wind anomalies drive a positive feedback loop, increasing moisture and latent heat flux from the surface, promoting further convection and inducing more wind gusts (Raymond 2001).

While both processes are necessary for aggregation in cloud-resolving models, it is likely that WISHE is the more critical for propagation of the MJO (Khairoutdinov and Emanuel 2018).

The idea that the MJO is a result of interaction between the large-scale, dry dynamics, and local moist convection has led to the notion of a moisture mode. In the context of the linear shallow water model of the atmosphere, the moisture mode is an atmo- spheric wave that emerges as a result of including additional prognostic quantities to the model, such as relative humidity or total moisture content. The merit in this theory is that it permits the use of highly idealised models to potentially isolate the mechanism of MJO development and propagation.

With an implicit vertical and meridional structure, the total water vapour in an atmospheric column along the equator has been used as a moisture model (Sobel and Maloney 2012). The large-scale wind is assumed to be a balanced Matsuno–Gill response to the heating, and can therefore be diagnosed from the heating. The heating is due only to convection and precipitation, these again assumed to be in in balance with the moisture budget, so the entire system is captured by a single prognostic moisture equation. This idealised system exhibits some interesting properties, the linearised equations produce an unstable moisture mode, which propagates westward relative to the background flow. When an additional source of moist static energy is added to the equator, this simple model produces an eastward moving unstable mode, akin to the MJO (Sobel and Maloney 2013).

Majda and Stechmann (2009) propose a “skeleton of the MJO” which can be sum- marised as:

1. A wave envelope containing waves propagating slowly eastwards at a speed ∼ 5 m s−1.

2. The envelope is roughly dispersionless and potentially stationary, i.e. ∂ ω/∂ k ' 0.

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3. A equatorially-symmetric vortex quadrupole.

To this criteria Majda and Stechmann too apply an extension of the Matsuno–Gill solution. In addition to the linear shallow equations on the equatorial beta-plane, they include a moisture tracer and a tracer parameterising the activity of small-scale waves on the global scale waves, namely zonal wavenumber-1 Rossby and Kelvin waves. The wave activity is assumed to contain all the high-frequency modes for which there is unstable growth, doing so produces a neutrally-stable eastward propagating mode with properties that satisfies the skeleton criteria, although the quadrupole structure is only apparent in the lowest zonal wavenumbers.

It has been shown that in the presence of an westward equatorial background flow (such as the trade winds), the low-level wind convergence of the the MJO positive- phase produces anomalously strong westward winds ahead of it to the east along the equator. With a bulk-aerodynamic treatment of surface fluxes, this increases evapora- tion at the surface to the east of the convergence, higher saturation in the atmosphere and consequently latent heat release aloft and drives a wave that propagates against the background flow (Emanuel 1987). This theory neglects the effects of moisture convergence by the flow, and the waves elicited are highly dispersive.

Further evidence for a moisture-mode–MJO coupling comes from equatorial cloud resolving models. Parameterising the effect of convective heating as a function the vertical integral of moist static stability, cloud self-aggregation occurs under uniform heating and without rotation. When rotation is included and a zonally varying forcing is applied, a moisture-mode instability can cause the cloud-aggregation to propagate eastward (Raymond and Fuchs 2009).

The common theory running through these proposed models of MJO propagation is that a feedback exists from the local effects of convection and moisture precipitation onto the circulation at a global scale.

The shallow water model of a near-tidally locked exoplanet presented in chapter 4 demonstrated that a heat source moving eastward at a velocity s< c can elicit a maximum heating in the atmosphere further east of the centre of the forcing. If the heating is generated by a feedback from the atmosphere onto itself, a self-sustaining, moving pattern would be established. In the tropics on Earth, moisture is a large energy source capable of driving such a feedback loop. In this chapter we propose a speculative model that elicits a self-sustaining MJO signal that satisfies the skeleton criteria outlined above. We couple small-scale moisture driven processes to the large- scale dynamics through latent heat release; the large-scale is explicitly captured in the dynamics, small-scale processes are parameterised with a conditional forcing.

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