Precipitation-driven cold pools play an important role in organizing tropical convection. Previous studies of tropical convection in the radiative–convective equilibrium (RCE) setup found that cold pools tend to collide with each other and trigger new convection. It remains unclear why most cold pools do not have enough space to dissipate without collision. We explain it as the smaller mean cold pool radius Req compared to its maximum potential radius Rmax. The latter denotes the radius needed for a cold pool’s buoyancy deficit to be dissipated by surface heating. Applying an energy balance constraint leads to an analytical solution for their ratio Rmax/Req, which depends on the Bowen ratio, surface precipitation–evaporation ratio, and rain sedimentation efficiency. The theory predicts that in the regime of marine tropical convection where the Bowen ratio is much smaller than one, Req cannot reach Rmax, and cold pools must collide frequently. This prediction is supported by large-eddy simulations using varying rain evaporation rates. In Part II, we combine the energy balance constraint with a convective life cycle model to obtain a theory of the mean cold pool radius Req. 

The cold pool is a crucial component of tropical convection. However, what controls the mean cold pool size remains unclear. This two-paper series presents a theory of the mean cold pool radius in idealized quasi-equilibrium convection (Req). Part I derives an energy balance constraint between Req and the maximum potential radius of a cold pool (Rmax), showing that Req cannot reach Rmax. Cold pools must be densely packed and collide frequently. This Part II derives another constraint between Req and Rmax based on a cold pool survival competition hypothesis. A convective life cycle model with various candidate cold pool sizes is built. The type of cold pool producing the most intense next-generation cold pool is hypothesized to survive and set the spacing between convective towers. The size of the dominant cold pool type is determined by the trade-off between the mechanical lifting effect that favors a smaller cold pool, the thermodynamic forcing effect that favors a bigger cold pool, and the cloud radius feedback that also favors a bigger cold pool. Combining the energy balance and survival competition constraints, we obtain a solution for Req, which has an analytically tractable upper bound. The upper bound is set by the cold pool’s fractional entrainment rate and the free-tropospheric relative humidity: a lower fractional entrainment rate or a drier free troposphere raises the upper bound of Req. The Req predicted by the theory agrees with a set of large-eddy simulations with different rainwater evaporation rates. 

Tropical cyclone precursor vortices are inherently coupled to convection. They undergo significant changes in size and intensity before transitioning to a mature hurricane or typhoon. This paper designs a stochastic quasi-2D model to study the vortices' formation and interaction. Based on the diagnostic result of a cloud-permitting simulation, we parameterize deep convection as random pulses whose probability of occurrence depends on the spatially smoothed vorticity. The dependence of convective probability on the vorticity field represents the mesoscale feedback. The smoothing represents the spontaneous spreading of convective activity by cold pools and other processes. Simulations show that the system exhibits two stages: the vortex formation stage and the vortex interaction stage. The vortex formation stage features the stochastic nucleation of vortices and their subsequent growth via the mesoscale feedback. The growth of mesoscale vorticity magnitude undergoes a power law growth and then transitions to exponential growth. An analytical theory is proposed to capture this transition. The vortex interaction stage features vortex merging. The vortex size grows due to merging and spontaneous spreading of convective activity. When the vortex size grows sufficiently large, it is squeezed by the convection-induced convergent flow, which converts the growth in size to the growth in vorticity magnitude. This adjustment process corresponds to a bidirectional kinetic energy transfer, with the rotational wind producing an upscale energy transfer and the convergent wind producing a downscale energy transfer. This quasi-2D model provides a simple framework for understanding the multiscale interaction in tropical cyclogenesis.

A step not fully understood in tropical cyclogenesis is the development of a surface cyclone, which is often preceded by a midlevel cyclone. This paper presents a single-column model to study the role of the transfer of tangential momentum in generating an initial surface cyclone. To isolate momentum transfer factors from thermodynamic factors, diabatic heating is set to be steady. The investigation starts without considering surface friction. The momentum transfer is decomposed into the transport by the vortex-scale circulation and by convection. The convective momentum transport (cumulus friction), when parameterized as a vertical eddy diffusion, leads to a vertical spectral truncation that permits an analytical solution of the single-column model. The analytical solution reveals the dual role of cumulus friction in surface cyclone formation. Cumulus friction can enhance the downward momentum transfer, but when the eddy diffusion is too strong, the vortex-scale circulation is too damped to produce a significant barotropic cyclone. Between these two extremes lies an optimal eddy diffusivity that maximizes the growth rate of the surface cyclone. Finally, we add surface friction to the single-column model. Using scale analysis, we identify a critical vortex Rossby number above which surface friction becomes non-negligible and significantly damps the development of the surface cyclone.

Rotating Rayleigh-Bénard convection denotes the convection between a warm plate and a cold plate in a rotating environment. It is a classic model for understanding convective vortices in the atmosphere and ocean. The influence of background rotation on fluid inertia breaks the symmetry between cyclones and anticyclones. Such a symmetry breaking could be represented by vorticity skewness, which still lacks a systematic theory. Rapidly rotating convection with stress-free boundaries and unit Prandtl number is a convenient starting point. The investigation starts from the convective onset stage, where the vortices grow stationarily. Asymptotic analysis shows that the volumetric vorticity skewness S is produced by the interaction between the n=0,1, and n=1,2 vertical eigenmodes. The n=0 (barotropic) mode positively contributes to S mainly by stretching the vertical relative vorticity, an ageostrophic effect. The n=2 mode makes a minor negative contribution to S by preferentially intensifying the outflow over the inflow, a non-hydrostatic effect. The theory predicts S to be proportional to the global Rossby number defined with the volumetric standard deviation of vorticity, Ro_g. The proportional factor does not depend on the Rayleigh and Ekman numbers, agreeing with direct numerical simulations. Then, the system enters the equilibrium stage. The stretching of vertical vorticity still dominantly contributes to S. At Ro_g <~ 0.5, the emergent unsteady flow significantly suppresses the asymmetry between the inflow and outflow strength and weakens its influence on S.

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 lcp+av to parameterize the combined effect of cold pools and anvils. Using a novel diagnostic method, lcp+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 nonnegligible role. 

This two-paper series studies the tropical cyclone (TC) precursor vortices spontaneously generated in an idealized setup with uniform sea surface temperature and no background wind. We focus on the small-amplitude stage, where vortices appear in an orderly pattern, and what controls the vortex size remains unclear. In Part I, Fourier analysis shows that the vortex size is constrained by a short-wavelength cutoff and a long-wavelength cutoff. The short-wavelength cutoff is explained as a convective spreading length due to cold pools and anvil clouds. In Part II, we study the long-wavelength cutoff. Diagnostic analysis shows that a TC precursor vortex has a shallow overturning cell in the lower troposphere and a deep overturning cell in the upper troposphere, driven by different convective types. A four-layer quasigeostrophic system is established to understand how the shallow and deep cells couple. Their dynamic coupling via midlevel buoyancy is negligible due to a midlevel high-speed waveguide that maintains a weak horizontal buoyancy gradient. Their thermodynamic coupling via the cooperative moistening of the air column is dominant. Linear stability analysis shows that the long-wavelength cutoff is controlled by an effective Rossby deformation radius Le, which is a mixture of the local Rossby deformation radii of the shallow and deep cells. The most unstable wavelength corresponds to the TC precursor vortex size, which is proportional to the geometric average of Le and the convective spreading length. The theoretical predictions generally agree with cloud-permitting simulations using varying Coriolis parameters and longwave radiative feedback strengths. 

 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. 

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.