A minimal, nonlinear oscillator model is analyzed for the Madden–Julian oscillation (MJO) “skeleton” (i.e., its fundamental features on intraseasonal/planetary scales), which includes the following: (i) a slow eastward phase speed of roughly 5 m s−1, (ii) a peculiar dispersion relation with dω/dk ≈ 0, and (iii) a horizontal quadrupole vortex structure. Originally proposed in recent work by the authors, the fundamental mechanism involves neutrally stable interactions between (i) planetary-scale, lower-tropospheric moisture anomalies and (ii) the envelope of subplanetary-scale, convection/wave activity. Here, the model’s nonlinear dynamics are analyzed in a series of numerical experiments, using either a uniform sea surface temperature (SST) or a warm-pool SST. With a uniform SST, the results show significant variations in the number, strength, and/or locations of MJO events, including, for example, cases of a strong MJO event followed by a weaker MJO event, similar to the Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE). With a warm-pool SST, MJO events often begin as standing oscillations and then propagate slowly eastward across the warm pool, a behavior imitating MJOs in nature. While displaying the fundamental features of the MJO skeleton, these MJO events had significant variations in their lifetimes and regional extents, and they displayed intense, irregular fluctuations in their amplitudes. The model reproduces all of these features of the MJO skeleton without including mechanisms for the MJO’s “muscle,” such as refined vertical structure and upscale convective momentum transport from subplanetary-scale convection/waves. Besides these numerical experiments, it is also shown that the nonlinear model conserves a total energy that includes a contribution from the convective activity.

A simple stochastic model is designed and analyzed in order to further understand the transition to strong convection. The transition has been characterized recently in observational data by an array of statistical measures, including (i) a sharp transition in mean precipitation, and a peak in precipitation variance, at a critical value of column water vapor (CWV), (ii) an approximate power law in the probability density of precipitation event size, (iii) exponential tails in the probability density of CWV values, when conditioned on either precipitating or nonprecipitating locations, and (iv) long and short autocorrelation times of CWV and precipitation, respectively, with approximately exponential and power-law decays in their autocorrelation functions, respectively. The stochastic model presented here captures these four statistical features in time series of CWV and precipitation at a single location. In addition, analytic solutions are given for the exponential tails, which directly relates the tails to model parameters. The model parameterization includes three stochastic components: a stochastic trigger turns the convection on and off (a two-state Markov jump process), and stochastic closures represent variability in precipitation and in “external” forcing (Gaussian white noise). This stochastic external forcing is seen to be crucial for obtaining extreme precipitation events with high CWV and long lifetimes, because it can occasionally compensate for the heavy precipitation and encourage more of it. This stochastic model can also be seen as a simplified stochastic convective parameterization, and it demonstrates simple ways to turn a deterministic parameterization—the trigger and/or closure—into a stochastic one.

Simulations of precipitating convection would typically use a non-Boussinesq dynamical core such as the anelastic equations, and would incorporate water substance in all of its phases: vapour, liquid and ice. Furthermore, the liquid water phase would be separated into cloud water (small droplets suspended in air) and rain water (larger droplets that fall). Depending on environmental conditions, the moist convection may organize itself on multiple length and time scales. Here we investigate the question, what is the minimal representation of water substance and dynamics that still reproduces the basic regimes of turbulent convective organization? The simplified models investigated here use a Boussinesq atmosphere with bulk cloud physics involving equations for water vapour and rain water only. As a first test of the minimal models, we investigate organization or lack thereof on relatively small length scales of approximately 100 km and time scales of a few days. It is demonstrated that the minimal models produce either unorganized (‘scattered’) or organized convection in appropriate environmental conditions, depending on the environmental wind shear. For the case of organized convection, the models qualitatively capture features of propagating squall lines that are observed in nature and in more comprehensive cloud resolving models, such as tilted rain water profiles, low-altitude cold pools and propagation speed corresponding to the maximum of the horizontally averaged, horizontal velocity.

Precipitating versions of the quasigeostrophic (QG) equations are derived systematically, starting from the equations of a cloud-resolving model. The presence of phase changes of water from vapor to liquid and vice versa leads to important differences from the dry QG case. The precipitating QG (PQG) equations, in their simplest form, have two variables to describe the full system: a potential vorticity (PV) variable and a variable M including moisture effects. A PV-and-M inversion allows the determination of all other variables, and it involves an elliptic partial differential equation (PDE) that is nonlinear because of phase changes between saturated and unsaturated regions. An example PV-and-M inversion is provided for an idealized cold-core cyclone with two vertical levels. A key point illustrated by this example is that the phase interface location is unknown a priori from PV and M, and it is discovered as part of the inversion process. Several choices of a moist PV variable are discussed, including subtleties that arise because of phase changes. Boussinesq and anelastic versions of the PQG equations are described, as well as moderate and asymptotically large rainfall speeds. An energy conservation principle suggests that the model has firm physical and mathematical underpinnings. Finally, an asymptotic analysis provides a systematic derivation of the PQG equations, which arise as the limiting dynamics of a moist atmosphere with phase changes, in the limit of rapid rotation and strong stratification in terms of both potential temperature and equivalent potential temperature.

For the tropical atmosphere on planetary scales, it is common to model the circulation using strong damping. Here with new data analysis techniques, evidence suggests that damping can actually be neglected. Specifically, near the equator, the east‐west overturning circulation is in agreement with the undamped wave response to atmospheric heating. To estimate the heating, satellite observations of outgoing longwave radiation (OLR) are used. Frequently, OLR is used as a heuristic indicator of cloudiness. Here the results further suggest that OLR variations are actually proportional to diabatic heating variations, with a proportionality constant of 18 W m⁻² (K d⁻¹)⁻¹. While the agreement holds best over long time averages of years or decades, it also holds over shorter periods of one season or 1 month. Consequently, it is suggested that the strength of the Walker circulation—and its evolution in time—could be estimated using satellite data.