## Abstract

Rotating radiative–convective equilibrium is studied by extracting the column physics of a mesoscale-resolution global atmospheric model that simulates realistic hurricane frequency statistics and then coupling it to rotating hydrostatic dynamics in doubly periodic domains. The parameter study helps in understanding the tropical cyclones simulated in the global model and also provides a reference point for analogous studies with cloud-resolving models.

The authors first examine the sensitivity of the equilibrium achieved in a large square domain (2 × 10^{4} km on a side) to sea surface temperature, ambient rotation rate, and surface drag coefficient. In such a large domain, multiple tropical cyclones exist simultaneously. The size and intensity of these tropical cyclones are investigated.

The variation of rotating radiative–convective equilibrium with domain size is also studied. As domain size increases, the equilibrium evolves through four regimes: a single tropical depression, an intermittent tropical cyclone with widely varying intensity, a single sustained storm, and finally multiple storms. As SST increases or ambient rotation rate *f* decreases, the sustained storm regime shifts toward larger domain size. The storm’s natural extent in large domains can be understood from this regime behavior.

The radius of maximum surface wind, although only marginally resolved, increases with SST and increases with *f* for small *f* when the domain is large enough. These parameter dependencies can be modified or even reversed if the domain is smaller than the storm’s natural extent.

## 1. Introduction

Rotating radiative–convective equilibrium, achieved in a doubly periodic box on an *f* plane with horizontally homogeneous forcing and boundary condition, is an informative idealized framework for studying the interactions between moist thermodynamics, radiation, and rotating dynamics. It can be studied in nonhydrostatic models in which deep convection is partly resolved and also in lower-resolution hydrostatic models with parameterized convection. In the latter case, one can think of this framework as one of a hierarchy of idealized settings in which to study the implications of the assumptions made in GCMs.

Tropical cyclones (TCs) form within these rotating radiative–convective equilibria, and they provide simulations in which very long-lived mature TCs emerge. If run at the resolution and with the column physics (including boundary layer, radiation, microphysics, moist convection, and cloud modules) of a global comprehensive model, this idealized framework can help us evaluate the TC simulations in such models. Global models are moving to high-enough resolution that aspects of their TC simulations are becoming more realistic, and these global models are one of the tools being used to try to predict the impact of climate change on TC statistics. Being able to study the model-generated TCs in this idealized geometry, eventually comparing to analogous simulations at cloud-resolving resolutions, should provide valuable information as to the limitations and strengths of the TC simulations in global models.

Held and Zhao (2008, hereafter HZ08) describe such a rotating radiative–convective equilibrium in which the column physics of a particular atmospheric model, the Geophysical Fluid Dynamics Laboratory (GFDL)’s Atmospheric Model, version 2 (AM2), is coupled to rotating hydrostatic dynamics in a doubly periodic box. A similar equilibrium is achieved in Nolan et al. (2007) and Nolan and Rappin (2008) with higher cloud-resolving resolution in a study of tropical cyclogenesis. Schecter and Dunkerton (2009) and Schecter (2011) set up an idealized three-layer hurricane model under such a framework, with simplified radiative and convective parameterizations. The variability and predictability of hurricanes in rotating radiative–convective equilibrium is studied by Hakim (2011), Hakim (2013), and Brown and Hakim (2013) using both axisymmetric and three-dimensional models.

Besides their various choices for column physics and resolution, a major difference among these studies is that HZ08 consider a sufficiently large domain for multiple TCs to coexist while others use small domains within which only one TC develops. In those small-domain simulations, whether with a doubly periodic domain or a rigid-wall box, the extent of TCs is explicitly set by the domain size. Because of this constraint, both the state of equilibrium achieved and its parameter sensitivity may be interfered with by the domain size. We refer to our doubly periodic small-domain simulation as “lattice” equilibrium, in which duplicated TCs exist simultaneously at the grid points in a lattice that can be visualized by duplicating the doubly periodic domain along its boundaries. The large-domain simulations avoid such finite-domain effects and can be regarded as “turbulent” equilibria in the sense that multiple TCs interact with respect to each other in more complex ways. The extent of TCs is thus controlled by internal processes of the turbulent equilibrium instead of domain size. This allows for analysis of size and intensity of TCs as a function of environmental parameters, independent of domain size. Neither configuration is realistic in the sense that appropriate decay mechanisms (poleward drift and interactions with land and extratropical systems) and suppression factors (wind shear, subsidence, and dry-air entrainment) are not present, resulting in very long-lived storms.

In this study, both lattice and turbulent equilibria are achieved in doubly periodic domains, with column physics and resolution of a 25-km global atmospheric model, which simulates many aspects of tropical cyclone statistics realistically (Zhao et al. 2009). The details of model formulation are presented in section 2. We start by describing turbulent equilibrium achieved in a large domain (2 × 10^{4} km on a side) in section 3. The parameter sensitivity of TC structure and intensity in this turbulent equilibrium is studied by varying sea surface temperature, ambient rotation rate *f*, and surface drag coefficient *C*_{D}. In section 4, we explore the variation of lattice equilibrium as a function of domain size and show how the natural extent of TCs in turbulent equilibrium is related to this domain size dependency of lattice equilibrium. A discussion and conclusion are offered in section 5.

One of our goals is to map out some of these parameter dependencies in the hope of encouraging the study of other GCM’s column physics in this idealized setting. We believe that understanding these parameter dependencies and how they differ between parameterized-convection hydrostatic and cloud-resolving nonhydrostatic models will be helpful in evaluating the simulations of TC statistics in global models in the future. To the extent that reliable cloud-resolving simulations are available, this idealized rotating radiative–convective equilibrium framework could also be used as a novel test for the performance of convection schemes.

## 2. Model formulation

The column physics used here is that of the GFDL High-Resolution Atmospheric Model (HiRAM). HiRAM has been used in Zhao et al. (2009) to realistically simulate the observed global climatology and interannual variability of hurricane frequency at 50-km resolution, although few storms are simulated higher than category 2 as measured by the maximum surface winds, especially in the North Atlantic (Zhao and Held 2010). Multiple simulations of the period 1979–2008, running over observed sea surface temperatures, with both 50- and 25-km-resolution versions of this model have been placed in the Climate Model Intercomparison Project, version 5 (CMIP5), archive. The seasonal and interannual variability of tropical storm genesis is not fundamentally different in the 25- and 50-km models. A 25-km version of the model with modified surface flux formulation and microphysical scheme has shown impressive skill for seasonal forecasts in the Atlantic (Chen and Lin 2012).

HiRAM was developed starting from an earlier lower-resolution model, AM2 (GAMDT 2004). The prognostic cloud fraction scheme is changed to a simpler diagnostic scheme assuming a subgrid-scale distribution of total water. The relaxed Arakawa–Schubert convective closure (Moorthi and Suarez 1992) in AM2 has been replaced by a modified shallow convection scheme (Bretherton et al. 2004; Zhao et al. 2009), which results in a larger fraction of precipitation occurring at resolved scale rather than through the parameterization. All other physical schemes are as in AM2. Surface fluxes are computed using Monin–Obukhov similarity theory, with a gustiness component (1 m s^{−1}) to account for contribution from subgrid wind fluctuation (Beljaars 1995). Oceanic roughness lengths are prescribed according to Beljaars (1995). For vertical diffusion, a K-profile scheme designed for cloud-topped boundary layers, with parameterized entrainment rates at the top of the boundary layer (Lock et al. 2000), is used. Stability functions with thresholds dependent on Richardson number are adopted for stable layers. We refer the reader to GAMDT (2004) and Zhao et al. (2009) for further details.

The doubly periodic model has the same vertical level altitudes as the GCM with the lowest level at about 35 m. The horizontal resolution is 25 km. As in HZ08, there is no diurnal or seasonal cycle in the doubly periodic model, and the radiative forcing is configured using an equatorial annual-mean solar zenith angle. Sea surface temperatures are homogeneously prescribed over a saturated (ocean) surface. Aerosols are absent in the atmosphere, while the stratospheric ozone is fixed at an observed tropically averaged vertical profile.

The homogeneous doubly periodic *f*-plane framework contains none of the synoptic variability that helps control TC genesis in the HiRAM global models, but the genesis frequency in those models is not simply a product of downscale transfer from larger scales; it is also a function of parameters in the convection scheme (Zhao et al. 2012). As one example, if one suppresses deep convection very strongly by increasing the strength of entrainment into the parameterization’s deep convective cores, the number of TCs that form in the global model is reduced dramatically. Very similar behavior is found in our homogeneous *f*-plane simulations. We hope to describe the dependence of rotating radiative–convective equilibria at this resolution on the convection scheme elsewhere.

## 3. Varying environmental parameters in turbulent equilibrium

Turbulent equilibrium is achieved using a large domain (2 × 10^{4} km)^{2}. The initial condition is an observed tropically averaged temperature profile, with moisture artificially near saturation below 100 hPa. The model is integrated 2.5 years to its equilibrium. Most statistics, such as hydrological balance, do not require this long of an integration. After the initial genesis period a qualitative equilibrium is typically achieved in a few months, but the number of TCs usually decreases slowly with time until it is finally stable. While we cannot prove that the TC number would not decrease with even longer integrations, the behavior seen in the last 6 months of the simulations is a statistically stationary number of cyclones, with the occasional collapse or merger event followed by genesis into the space left by this event. For every 3-hourly output during the last 6 months of each simulation, TCs are collected by identifying points with surface pressure *p*_{s} satisfying two criteria: 1) *p*_{s} is less than a critical value of 990 hPa, and 2) *p*_{s} is a minimum at this point within the surrounding 10 × 10 grid box.

We first look at the sensitivity to SST, with ambient rotation rate set at 20°N. Snapshots of surface wind speed for two different SSTs are compared in Fig. 1. As found in HZ08, the number of TCs decreases with SST. There are about 40 TCs within the domain for SST at 297 K but only about 15 TCs for SST at 305 K. The distribution of TC central surface pressure is shown in Fig. 2a. The peak frequency of TC central surface pressure shifts toward a lower value as SST increases, indicating consistent higher intensity at warmer SST. The mean central surface pressure intensifies by about 3 hPa K^{−1} and the mean maximum surface wind speed (the wind in the lowest model layer, at roughly 35 m) increases roughly 1.5 m s^{−1} K^{−1} (Figs. 3a and 3b). This is comparable to the sensitivity found in the global model, which indicates roughly a 4 m s^{−1} increase in the mean maximum surface wind between the case with climatological SST and that with 2-K increases in SST (Zhao and Held 2010).

Potential intensity (PI) is computed from the time- and domain-mean thermodynamic profiles, following Bister and Emanuel (2002a) (utilizing the code provided at ftp://texmex.mit.edu/pub/emanuel/TCMAX/). We choose the option of pseudoadiabatic ascent and include dissipative heating (as in the GCM, the kinetic energy dissipation in the boundary layer is returned as heat within the boundary layer). The ratio of surface exchange coefficient for enthalpy *C*_{k} to *C*_{D} is set at 0.6, to be consistent with the ratios produced in the model. A velocity reduction factor of 0.9 is applied as a way to extrapolate down through the frictional boundary layer (Bister and Emanuel 2002b). The theoretical maximum wind speed captures the slope of intensification with SST quite well except for very low SSTs (Fig. 3). The explanation for this mismatch at lower SSTs may be that over such cold SST TCs are too weak to compete with random convection and have difficulty in maintaining their mature state. Note that although the theoretical maximum wind speed always exceeds the simulated wind speed, the theoretical pressure perturbation is generally weaker than that in the simulation. Whatever the reason for this difference, the behavior of TCs in the GCM is similar: minimum central surface pressures are comparable to the PI’s theory but TCs never reach beyond category 2 in terms of maximum wind (Zhao and Held 2010).

The sensitivity to *f* is investigated by varying *f* from 5° to 20°N with fixed SST at 301 K. Snapshots of surface wind for *f* at 5° and 20°N are compared in Fig. 4. As found in HZ08, the number of TCs increases with ambient rotation rate. There are about 23 TCs within the domain for *f* at 20°N but only about 5 TCs for *f* at 5°N. The peak frequency of TC central surface pressure shifts toward higher value as *f* increases, indicating weaker intensity at larger *f* (Fig. 2b). As *f* increases from 5° to 20°N, the mean central surface pressure (Fig. 3c) increases by 30–40 hPa, and by the same measures the mean maximum surface wind (Fig. 3d) decreases by 5–10 m s^{−1}. The difference in vertical profiles of temperature and specific humidity show a moister and warmer environment for larger ambient rotation rate (Fig. 5). The increasing air temperature and humidity at the surface corresponds to a reduced surface moisture disequilibrium. The domain-averaged evaporation and precipitation do not change substantially, as expected from the close balance between radiation cooling and latent heat flux, but the average surface wind speed increases owing to the increased density of TCs. We interpret the reduction in surface disequilibrium as determined by the need to maintain the same strength of hydrological cycle in the presence of stronger mean surface wind. The increasing of temperature is amplified upward, approximately following a moist adiabat. The pattern of temperature and humidity differences also indicates more aggressive shallow convection in the cases with larger ambient rotation rate. The potential intensity computed from this vertical profile satisfactorily captures the decreasing intensity with increasing *f* (Figs. 3c,d).

To study the sensitivity to surface drag coefficient, we bypass the Monin–Obukhov scheme and prescribe surface exchange coefficients homogeneously within the domain. The surface drag coefficient is varied from 0.6 × 10^{−3} to 2.4 × 10^{−3}, while the surface exchange coefficient for enthalpy is fixed at 1.2 × 10^{−3}. Three sets of experiments are conducted with ambient rotation rate set at 10°, 15°, and 20°N. The distribution of TC central surface pressure is shown in Fig. 6. Interestingly, this curve shifts with *C*_{D} in the opposite direction for different rotation rates. For smaller *C*_{D}, there are more intense TCs with *f* set at 10°N but more weak TCs with *f* set at 20°N. In spite of this difference, the number of TCs within the domain decreases with *C*_{D} in all three cases (as can be seen in Fig. 6).

The sensitivity of TC intensity to *C*_{D} is summarized in Figs. 3e and 3f, with ambient rotation rate chosen at 15°N. As *C*_{D} decreases from 2.4 × 10^{−3} to 0.6 × 10^{−3}, the mean central surface pressure intensifies by only about 10 hPa and the mean maximum surface wind increases by roughly 50%. The difference in vertical profiles of temperature and specific humidity are shown in Fig. 5, indicating a warmer and more humid environment for smaller *C*_{D}, similar to the cases with larger *f*. The potential intensity computed from this vertical profile shows a stronger sensitivity to *C*_{D} compared with the model results, especially for potential pressure intensity. Modifying the ambient rotation rate does not change this result. We note that the recent modification to PI theory by Emanuel and Rotunno (2011) weakens the sensitivity to *C*_{D}.

The mean radius–height structure of the strongest TC within the domain for two SSTs is shown in Fig. 7. The boundary layer inflow and upper-tropospheric outflow characteristic of tropical cyclones are apparent, with an ascent region just inside of the radius of maximum surface wind *R*_{mw}. As shown in Fig. 8, *R*_{mw} increases with SST, decreases with *C*_{D}, and increases with *f* for small *f* (but saturates and probably decreases as *f* is increased further). The strongest storms that form for a given parameter setting have the smallest *R*_{mw}. The *R*_{mw} values simulated in this 25-km model are substantially larger than those observed (Stern and Nolan 2011). However, we feel that the systematic variation seen in this model is of interest for comparison with high-resolution simulation. We return to these results in section 4, where we show that quantitatively different results are obtained in smaller domains containing only one storm.

HZ08 explored the sensitivity of the natural extent of TCs to SST and *f* in the turbulent multistorm equilibrium. With a higher resolution and a larger domain size, we hope to have a better estimate of this sensitivity. As shown in Fig. 9, the average number of TCs *n* doubles or halves as SST is perturbed by 4 K from the reference SST (301 K) and increases nearly linearly with *f*. We estimate the natural extent of TCs *r*_{0} as

where *A* is the domain area. Two simple theoretical scalings for the Rossby radius of deformation (*NH*/*f*), and the other is a natural length scale from PI theory (*V*_{p}/*f*, where *V*_{p} is the potential intensity) (Emanuel 1988). However, as shown in Fig. 9, while both of these two scalings project the right trend, neither satisfactorily captures the sensitivity. Modifying some of the choices in the computation of potential intensity, or using actual model intensities for the *V*_{p}/*f* scalings does not affect this conclusion. We will return to this topic in section 4.

## 4. Varying domain size

The sensitivity of rotating radiative–convective equilibrium to domain size is investigated by varying the domain size from (1.25 × 10^{3} km)^{2} to (1.25 × 10^{4} km)^{2}, with SST at 301 K and *f* at 10°N. The composites of surface wind for different domain sizes are shown in Fig. 10. When the domain is small, only one tropical cyclone develops. The tropical cyclone has some elasticity in the sense that it can expand to fill the domain until multiple storms finally appear. Particularly, the extent of the single TC in the (7.5 × 10^{3} km)^{2} domain is nearly 20% larger than the natural extent of TCs in the (1.25 × 10^{4} km)^{2} domain.

The behavior of TCs evolves through four regimes, as illustrated by the time series of minimum surface pressure in each simulation (Fig. 11). When the domain is very small, only a tropical depression, with its central surface pressure at about 980 hPa, develops within the domain. As the domain size increases, the depression evolves into an intermittent tropical cyclone with intensity varying widely over a cycle of approximately 50 days. Further increase of the domain size avoids this collapse to weak intensity and leads to a sustained storm. Eventually, if the domain size becomes sufficiently large, multiple storms coexist. (We have conducted preliminary simulations analogous to these with a higher-resolution nonhydrostatic model and see similar regime behavior, although the necessary domain size for a sustained storm is somewhat smaller.)

To see how the natural extent of TCs in turbulent equilibrium is related to this regime behavior, we conduct two series of experiments with 1) different *f* at 5°, 10°, and 20°N with SST at 301 K and 2) different SST at 301 and 305 K with *f* at 10°N. The time-mean central surface pressure is summarized in Figs. 12a and 12b, respectively. The intensity first increases with domain size and then levels off approximately at the beginning of the regime of a single sustained storm (marked with an open circle). This regime shifts toward large domain size as SST increases or *f* decreases—the same as the natural extent of TCs in turbulent equilibrium. For different parameter settings, the natural extent of TCs in turbulent equilibrium (marked with a vertical line segment) always falls in this regime. The following simple picture is consistent with these results: If the number of TCs increases from its equilibrium value, then the available space for a given TC must decreases into the intermittent storm regime. This TC becomes unstable and is more likely to be eliminated by competition with surrounding storms when sufficiently weak. The number of TCs then returns to its equilibrium value. On the other hand, if the number of TCs decreases, the suppression effect of TCs on its surrounding area becomes weak as TCs expand. With fewer but larger TCs randomly moving around, there is more free space for TC genesis. These two effects favor new TC genesis and bring the number of TCs back to its equilibrium.

One explanation for this regime of a sustained storm is that the available space for TCs needs to be sufficiently large to provide enough angular momentum to support the inner vortex. In addition to the two scales mentioned in section 3, a third candidate for the natural extent of TCs then comes up by relating the angular momentum at the outer radius *r*_{0} to that at *R*_{mw}. For a reduction factor *α* between these two, this relationship can be written as . Assuming that *α* is nearly invariant when SST or *f* varies, we have . This is essentially the same relationship between *R*_{mw} and *r*_{0} as in Emanuel and Rotunno (2011). As shown in Fig. 9, this scaling nicely captures the variation of *r*_{0} with both SST and *f* except at very low SSTs. As we mentioned previously, the equilibrium achieved at very low SSTs, with a large number of weak TCs, looks very different from those with higher SST. The above argument may only apply to the cases where mature storms dominate. This parameter fit is a good description of the model results. However, this relationship between *r*_{0} and *R*_{mw} does not in itself provide a theory for *r*_{0} or *R*_{mw}.

The variation of *R*_{mw} with parameters (SST, *f*, and *C*_{D}) in lattice equilibrium, with domain size at 5000 km, is shown in Fig. 13. *R*_{mw} values increase with *f* and decrease with SST and *C*_{D}. This result is consistent with that of Schecter and Dunkerton (2009) in which a three-layer tropical cyclone model is used but is different from what we found in turbulent equilibrium, for SST in particular (Fig. 8). The parameter sensitivity of *R*_{mw} in lattice equilibrium reflects the parameter sensitivity of the ratio between *R*_{mw} and *r*_{0}, since in a fixed-size small domain *r*_{0} is fixed, while the parameter sensitivity of *R*_{mw} in turbulent equilibrium takes into account both the parameter sensitivity of this ratio and that of *r*_{0}. As we discussed in section 3, TC extent *r*_{0} can vary significantly with parameters in turbulent equilibrium. In this interpretation, the increase of *r*_{0} with SST dominates over the decrease of and results in increasing *R*_{mw} with SST in turbulent equilibrium (Fig. 8a), while the decrease of *r*_{0} with *f* may offset the increase of and leads to saturation or even decrease of *R*_{mw} for large *f* in turbulent equilibrium (Fig. 8b).

## 5. Conclusions and discussion

Idealized models can be used to help fill the gap between theory and comprehensive climate model efforts. In much recent work, the spherical geometry is held fixed while the column physics and/or surface boundary condition are largely simplified. Instead, we choose to retain the column physics and simplify the geometry and surface boundary conditions. By coupling the column physics to the hydrostatic dynamics in a doubly periodic domain, we can obtain a homogeneous doubly periodic rotating radiative–convective framework.

This study is along the lines of HZ08, with the column physics replaced by that of a mesoscale-resolution global atmospheric model, HiRAM, which produces a quite realistic TC climatology when run with realistic boundary condition. A large domain (2 × 10^{4} km)^{2} is used to obtain turbulent multistorm equilibria. The parameter sensitivity is then studied by varying sea surface temperature, ambient rotation rate *f*, and surface drag coefficient *C*_{D}. The mean TC intensity increases with SST and decreases with *f* and *C*_{D}. The potential intensity (Bister and Emanuel 2002a) calculated from the thermodynamic environment captures the sensitivity to SST and *f* with reasonable accuracy but overestimates that to *C*_{D}, especially in terms of central surface pressure. Note that PI theory in itself does not depend on *f* so an understanding of the dependence of the intensity on *f* requires a theory for the sensitivity of the model’s equilibrated thermodynamic profiles to *f*.

The natural extent of TCs *r*_{0} in turbulent equilibrium increases with SST and decreases with *f*. Although we have some difficulty in estimating *r*_{0} by simple theoretical scalings, we find it related to a shift in the regime behavior of lattice equilibrium, from an unsteady to a relatively stable mature storm. Such regime behavior is not unique in our model with mesoscale resolution and hydrostatic dynamics; we have also observed it in preliminary simulations using a high-resolution nonhydrostatic model.

In turbulent equilibrium, the radius of maximum surface wind *R*_{mw} increases with SST and increases with *f* for small *f* but saturates or probably decreases as *f* is increased further. However, in lattice equilibrium, *R*_{mw} decreases with SST and keeps increasing with *f*. This is because *r*_{0} is constrained by the domain size in lattice equilibrium while in turbulent equilibrium *r*_{0} varies systematically with environmental parameters. It is the relationship between *R*_{mw} and *r*_{0} that is robust to change in domain size. Because of the poor resolution of the *R*_{mw} in this model, the results on the parameter sensitivity of *R*_{mw} are necessarily tentative.

We believe that simulations of rotating radiative–convective equilibrium such as these, with the resolution and column physics of a global atmospheric model being used to study the response of TCs to climate change, can provide a needed middle ground between the global model and nonhydrostatic cloud-resolving simulations in an idealized geometry that will help in establishing the strengths and limitations of the global simulations. This preliminary descriptive study is meant to help encourage further work along these lines.

## Acknowledgments

The authors thank Ming Zhao and Shian-Jiann Lin for making HiRAM available. We also thank Bruce Wyman for his work in configuring the doubly periodic model. Wenyu Zhou is partly supported by the U.S. Department of Energy under Award DE-SC0006841 and partly by the National Oceanic and Atmospheric Administration (NOAA) Cooperative Institute for Climate Science under Award NA08OAR4320752.

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