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.

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Nonlinear coupling among wave modes and vortical modes is investigated with the following question in mind: can we distinguish the wave-vortical interactions largely responsible for formation versus evolution of coherent, balanced structures? The two main case studies use initial conditions that project only onto the vortical-mode flow component of the rotating Boussinesq equations: (i) an initially balanced dipole and (ii) random initial data in the vortical modes. Both case studies compare quasi-geostrophic (QG) dynamics (involving only nonlinear interactions between vortical modes) to the dynamics of intermediate models allowing for two-way feedback between wave modes and vortical modes. For an initially balanced dipole with symmetry across thex-axis, the QG dipole will propagate along thex-axis while the trajectory of the Boussinesq dipole exhibits a cyclonic drift. Compared to a forced linear (FL) model with one-way forcing of wave modes by the vortical modes, the simplest intermediate model with two-way feedback involving vortical-vortical-wave interactions is able to capture the speed and trajectory of the dipole for roughly ten times longer at Rossby Ro and Froude Fr numbers Ro = Fr ≈ 0.1. Despite its success at tracking the dipole, the latter intermediate model does not accurately capture the details of the flow structure within the adjusted dipole. For decay from random initial conditions in the vortical modes, the full Boussinesq equations generate vortices that are smaller than QG vortices, indicating that wave-vortical interactions are fundamental for creating the correct balanced state. The intermediate model with QG and vortical-vortical-wave interactions actually prevents the formation of vortices. Taken together these case studies suggest that: vortical-vortical-wave interactions create waves and thereby influence the evolution of balanced structures; vortical-wave-wave interactions take energy out of the wave modes and contribute in an essential way to the formation of coherent balanced structures.

We present a new high-resolution, non-oscillatory semi-discrete central scheme for one-dimensional shallow-water flows along channels with non uniform cross sections of arbitrary shape and bottom topography. The proposed scheme extends existing central semi-discrete schemes for hyperbolic conservation laws and enjoys two properties crucial for the accurate simulation of shallow-water flows: it preserves the positivity of the water height, and it is well balanced, i.e., the source terms arising from the geometry of the channel are discretized so as to balance the non-linear hyperbolic flux gradients. In addition to these, a modification in the numerical flux and the estimate of the speed of propagation, the scheme incorporates the ability to detect and resolve partially wet regions, i.e., wet-dry states. Along with a detailed description of the scheme and proofs of its properties, we present several numerical experiments that demonstrate the robustness of the numerical algorithm.

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