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Cloud-permitting simulations have shown that tropical cyclones (TCs) can form spontaneously in a quiescent environment with uniform sea surface temperature. While several mesoscale feedbacks are known to amplify an existing midlevel vortex, how the noisy deep convection produces the initial midlevel vortex remains unclear. This paper develops a theoretical framework to understand the evolution of the midlevel mesoscale vorticity's histogram in the first two days of spontaneous tropical cyclogenesis, which we call the ``stochastic spin-up stage". The mesoscale vorticity is produced by two random processes related to deep convection: the random stretching of planetary vorticity ($f$) and the tilting of random vertical shear. The mesoscale vorticity is modeled as the sum of three independent normal distributions, which include the cyclones produced by stretching, cyclones produced by tilting, and anticyclones produced by tilting. Their collective effect is calculated with the central limit theorem. The theory predicts that the standard deviation of the midlevel mesoscale vorticity is universally proportional to the square root of the domain-averaged accumulated rainfall, agreeing with simulations. The theory predicts a critical latitude below which tilting is dominant in producing mesoscale vorticity. Treating the magnitude of random vertical shear as a fitting parameter, the critical latitude is shown to be around 12$^\circ$N. Because the magnitude of vertical shear should be larger in the real atmosphere, this result suggests tilting is an important source of mesoscale vorticity fluctuation in the tropics. 

What determines the vortex size at the small-amplitude stage of spontaneous tropical cyclogenesis remains unclear. A doubly periodic domain is a standard setup for numerically studying this problem, but the convectively coupled standing waves inherent to the setup could directly trigger vortices, rendering an unrealistic path for tropical cyclogenesis. We increase the Coriolis parameter to suppress the wave and double the longwave radiative feedback to make the more realistic moisture-radiation instability dominant. Experiments show that the moisture-radiation instability has a short-wavelength cutoff due to the smoothing effect of convective dynamics, which includes the nonlocal convective triggering by cold pools and the nonlocal longwave radiative effect of anvil clouds. By approximating the spread of convective activity as a Gaussian filter on the column humidity, we derive a bulk convective spreading length $l_{cp+av}$ to parameterize the combined effect of cold pools and anvils. Using a novel diagnostic method, the $l_{cp+av}$ is shown to be around 10 km. The contribution of cold pools and anvil clouds to convective spreading is comparable in the doubled radiative feedback experiments. An extrapolation to the normal radiative feedback state shows the anvil clouds play a smaller yet non-negligible role.

 Tropical deep convection plays a key role in the tropical depression stage of tropical cyclogenesis by aggregating vorticity, but no existing theory can depict such a stochastic vorticity aggregation process. A vorticity probability distribution function (PDF) is proposed as a tool to predict the horizontal structure and wind speed of the tropical depression. The reason lies in the tendency for a vortex to adjust to an axisymmetric and monotonic vorticity structure. Assuming deep convection as independent and uniformly distributed vortex tube stretching events in the low-mid troposphere, repetitive vortex tube stretching will make the air column area shrink many times and significantly increase vorticity. A theory of the vorticity PDF is established by modelling the random stretching process as a Markov chain. The PDF turns out to be a weighted Poisson distribution, in good agreement with a randomly-forced divergent barotropic model (weak temperature gradient model), and in rough agreement with a cloud-permitting simulation. The result shows that a stronger and sparser deep convective mode tends to produce more high vorticity air columns, which leads to a more compact major vortex with a higher maximum wind.  Based on the vorticity PDF theory, a parameterization of the eddy acceleration effect on the tangential flow is proposed. 

How the cumulus clouds organize into a tropical cyclone remains poorly understood. The difficulty lies in that the deep convection is noisy at the kilometer scale, but follows the physical feedbacks at the mesoscale. We build a barotropic numerical model to understand the interaction of the stochastic and deterministic processes in the genesis of a tropical depression. Deep convection is represented as a multitude of isolated convergence forcing. The convection is assigned to distribute randomly at the small scale. At the mesoscale, convection is preferentially seeded to regions with a high spatially-filtered vertical vorticity. The preferential seeding mimics the physical feedbacks, and the filter implicitly represents the nonlocal convective triggering by gravity wave and cold pool. The result shows that the early-stage evolution is dominated by random vortex tube stretching. Subsequently, the regions where repetitive stretching occurs become vortex clusters, and induce more convection around them. The collision and coalescence between vortex clusters lead to a major vortex, which accelerates the growth by the preferential seeding. This physical picture agrees with a cloud-permitting simulation of spontaneous tropical cyclogenesis over uniform sea surface temperature. A theoretical model with approximate analytical solution is presented to depict the full evolution process. 

Deep convection is an important component of tropical atmospheric circulation. A deep convective cloud is typically shut down by its precipitation-driven downdraft, which generates a gust front in the mixed layer that triggers neighboring clouds. Too large a cloud spacing makes the gust front too weak to perform mechanical lifting, and too short a cloud spacing makes the gust front too dry to fuel convection. There needs to be a qualitative theory for what sets the spacing of precipitating clouds, which is the first step toward understanding cloud interaction. We propose to view precipitating convection as a piecewise linear oscillator with cutoff and hypothesize that this oscillator system's optimal mode determines the cloud spacing. The optimal spacing is solved as a small perturbation from the maximum potential size of a free-evolving cold pool. The theory shows that the gust front's mixing length sets an upper bound for the cloud spacing, which explains the increase and stagnancy of cloud spacing with increased rain evaporation rate in large-eddy simulations.

A pure theoretical investigation of convective rain formation processes and formation efficiency (FE) is performed using a kinematic one‐dimensional time‐dependent model with warm rain microphysics. FE is defined as the ratio of total cloud‐to‐rainwater conversion to total condensation. FE is a component of precipitation efficiency, which is an important but poorly understood parameter in idealized climate models. This model represents a cloud by a cylindrical thermal bubble rising at constant velocity. The model focuses on the interaction between auto‐conversion, collection, and lateral mixing about which no theory has been proposed. Taking the auto‐conversion threshold into account, a criterion for rain formation and a semianalytical approximate solution of FE are found. The auto‐conversion threshold limits the temporal and spatial extent of the “vigorous rain formation region” where most of the rain is produced. The collection and auto‐conversion compete with lateral mixing to determine the strength of rain formation within this region. The FE is predicted to be most sensitive to auto‐conversion threshold, fractional entrainment rate, and initial bubble water vapor density.