## 1. Introduction

Despite decades of research, hurricane formation is not fully understood. State-of-the-art, cloud-system-resolving (CSR) models are commonly used to investigate the process, but are computationally expensive and difficult to interpret. In principle, reduced models can be used to clarify the essential dynamics, and to efficiently discover new phenomena. The primary purpose of this paper is to evaluate the use of a reduced model to understand the transformation of turbulent flows into tropical cyclones.

There are many reduced models to evaluate, each having many adjustable parameters. The present study is by no means comprehensive, but focuses on a familiar three-layer model with typical settings and three alternative cumulus parametrizations. One parametrization is a minor variant of the classic convergence-based scheme of Ooyama (1969; henceforth O69), originally proposed for the purpose of understanding the intensification and steady-state maintenance of an isolated tropical cyclone. Another parametrization regulates convective activity through the principle of boundary-layer quasi-equilibrium (cf. Raymond, 1995; Emanuel, 1995a; Zehnder, 2001). A third resembles the convergence-based parametrization, but boosts convection in regions of exceptionally high instability.

The reduced model is evaluated, for each cumulus parametrization, by direct comparison to tropical cyclogenesis in the Regional Atmospheric Modeling System (RAMS; Cotton *et al.*, 2003). Although RAMS has two-moment microphysics with icy hydrometeors, the present study activates only single-moment warm rain microphysics. Furthermore, the surface flux parametrization is simplified to conform with the reduced model. While direct comparison of one reduced model to another has precedents (e.g. Zehnder, 2001; Zhu *et al.*, 2001; Zhu and Smith, 2002), direct comparison to a CSR model has been largely neglected. Another distinguishing feature of the present comparison is the starting point – a disorganized state of rotational moist-convective turbulence, as opposed to a weak axisymmetric cyclone or a synoptic-scale dipole.

In general, a reduced model is not a rigorous approximation of the full equations of motion. Therefore, it is unreasonable to expect precise agreement between a reduced model and a CSR numerical simulation. Nevertheless, CSR simulations and basic theoretical considerations have established minimal criteria that a reduced model should satisfy to have relevance. Two of the least controversial are:

(i) Tropical cyclogenesis must not occur solely through the conversion of ambient Convective Available Potential Energy (CAPE) into a warm core cyclone, but through an air–sea interaction instability (Rotunno and Emanuel, 1987; henceforth RE87). Consequently, the rate of tropical cyclogenesis should increase from zero with the surface exchange coefficient of moist entropy *C*_{E}. Furthermore, tropical cyclogenesis in a low-shear environment should accelerate with increasing sea-surface temperature (SST), all else being equal.

(ii) The intensity of a mature tropical cyclone should typically increase with the sea-surface temperature, and asymptotically tend to zero with the ratio of *C*_{E} to the surface drag coefficient, *C*_{D}.*

The literature contains ample evidence that previous versions of the reduced model used here satisfy criteria (i) and (ii) (e.g. O69; DeMaria and Pickle, 1988 (DP88); Schecter and Dunkerton, 2009 (SD09); Schecter, 2010 (S10)). Therefore, the present paper does not dwell on these issues.

On the other hand, the literature does not contain a thorough evaluation of the chaotic flow statistics of the reduced model during genesis. Such statistics include the spectral distributions of horizontal kinetic energy, relative vorticity and horizontal divergence. If the reduced model cannot reproduce the statistics of turbulence in the intermediate mesoscale of the RAMS simulation, the adequacy of the model would be questionable. Discrepancies could indicate inaccurate convective forcing by the cumulus parametrization, incorrect mechanisms of mode-to-mode energy transfer, etc. Moreover, improper modelling of 10–100 km scale ‘fluctuations’ could substantially affect the statistics of hurricane formation, such as the mean and standard deviation for the time of genesis.

Of further interest is the evolution of the thermodynamic *η*-variable prior to the intensification of an incipient hurricane. The value of *η* is a combined measure of deep convective instability and middle tropospheric moisture. The reduced model under consideration theoretically requires that *η* exceed a finite threshold *within the storm* for the possibility of intensification (O69; SD09). In our reduced simulations, the time series of *η* in an incipient hurricane spikes well above this threshold at the onset of rapid wind speed acceleration. If *η* fails to exhibit pronounced growth prior to intensification in RAMS, a significant feature of hurricane formation in the reduced model would have dubious credibility.

The final issue under present consideration is the influence of surface friction on tropical cyclogenesis. To date, the literature presents an ambiguous truth on the subject. Two frequently cited studies, based on a standard axisymmetric cloud model, suggest that increasing the surface drag coefficient *C*_{D} from zero to a realistic value decelerates vortex intensification (Craig and Gray, 1996; Gray and Craig, 1998 (GC98)). More recent studies based on axisymmetric and fully three-dimensional (3D) models present cases with the opposite behaviour (Fang *et al.*, 2009 (FTW09); MSN10). The RAMS simulations carried out for this paper show that genesis (within one month) over a relatively cool ocean requires non-zero *C*_{D}, and that increasing *C*_{D} (to a realistic value) accelerates the process. We take this result as a provisional truth for the purpose of evaluating the reduced model. It will be shown that not all cumulus parametrizations are entirely consistent with the assumed ‘truth’.

The remainder of this paper is organized as follows. Section 2 briefly describes tropical cyclogenesis in RAMS. Section 3 presents the reduced model under evaluation for its ability to simulate the same process. Section 4 compares the reduced model to RAMS. Section 5 grades the reduced model based on past and present findings.