1. Introduction
Models of radiative–convective equilibrium (RCE) and rotating radiative–convective equilibrium (RRCE) have been used to understand atmospheric states dominated by moist overturning (e.g., Wing et al. 2018; Carstens and Wing 2020). The equilibrated solutions provide vertical profiles of temperature, humidity, cloud fraction, vertical mass flux, and condensation rate as a function of the external parameters, including lower-boundary temperature and radiative forcing. RCE solutions have been probed for mechanisms to account for the sensitivity to external parameters. The most idealized RCE and RRCE models are homogeneous by virtue of uniform lower-boundary temperature, background rotation, and clear-sky radiative forcing combined with Cartesian geometry and doubly periodic lateral boundaries.
RCE models develop multiscale horizontal structure. An intriguing behavior of most nonrotating RCE models is the aggregation of convection into clumps or lines (e.g., Bretherton et al. 2005; Muller and Held 2012). Efforts to explain this process have exposed crucial differences and interactions between clear and cloudy regions (e.g., Wing and Emanuel 2014; Bretherton and Khairoutdinov 2015). When background rotation is included, the self-aggregation changes to widespread genesis of long-lived warm-core vortices (Nolan et al. 2007; Held and Zhao 2008; Carstens and Wing 2020). The vortices, always cyclonic, capture most of the vertical moisture and mass flux while powerfully suppressing the unorganized convection. Because the vortices are the model’s representation of tropical cyclones (TCs), RRCE states have been called TC worlds (Khairoutdinov and Emanuel 2013).
The number of vortices in the RRCE climate is a unique statistic. Whereas cloud clusters in RCE are few and far between (Cronin and Wing 2017), RRCE can develop numerous windy, rainy vortices in the same amount of space. The average number is proportional to the model area and inversely proportional to the square of the mean distance between storms. This distance is a fundamental property of the model. Zhou et al. (2014) proposed finding the scale by shrinking a doubly periodic domain until one permanent vortex remains. In domains much larger than this scale, TCs abound and the distance between them can vary widely. Because of the homogeneity of the model, the vortices persist for weeks, allowing robust statistics via both global and storm-relative averaging.
Homogeneity simplifies not only the horizontal structure but also the vertical structure of model TCs. The present study questions the importance of vertical structure by examining global RRCE characteristics and vortex dynamics at the low extreme of vertical resolution. It is known from a long history of purpose-built models that a single storm can develop and persist on an f plane with only 3 degrees of freedom in the vertical (Ooyama 1969; DeMaria and Pickle 1988; Shapiro 1992; Zhu et al. 2001; Schecter and Dunkerton 2009). The more generic model introduced here will confirm the possibility of three-level TC worlds when the domain is large enough. Beyond this result, the intention is to learn how well it captures familiar characteristics of RRCE and individual vortices.
A numerical model that develops quasi-steady vortices with the smoothest possible vertical structure might have a more direct connection to analytical models (Emanuel 1986, hereafter E86; Holland 1997; Pearce 1998) than more complex TC models. For a bit of context, the least complex model of the extratropics is the Phillips (1954) model, which has just two levels. It becomes analytical when the equations are linearized. The surface quasigeostrophic model (Held et al. 1995) is a two-dimensional model that describes a three-dimensional flow in which the vertical structure is diagnostic. It is also studied analytically (e.g., Buckmaster et al. 2019). Models with minimal degrees of freedom should be probed for their eccentricities, as has been done by Nakamura and Wang (2013) and Hsieh et al. (2021) for the Phillips model. The intention here is to investigate three-level RRCE as a theoretical tool in its own right, but also to document any peculiarities. Three-level RRCE may deserve its own rank in a tropical model hierarchy.
With the connection to analytical models in mind, a major goal of the study is to develop an analysis that moves seamlessly from the full domain to a subspace containing a single composite vortex and from there to the embedded deep overturning circulation. Theories for the average size and intensity of vortices (e.g., Khairoutdinov and Emanuel 2013) have exploited global energy and entropy budgets simultaneously with vortex-scale constraints (Chavas and Emanuel 2014). We will rely on the same tools to understand the sensitivity to parameters including vertical resolution, external forcing, and lower boundary conditions. The doubly periodic framework will avoid lateral walls and the associated artifacts.
In the next section, we describe the idealized model carefully enough to make it straightforward to reproduce. Only generic physical parameterizations are used. The moist energy and entropy budgets are presented in detail. The model includes full dissipative heating and implicit precipitation drag. In section 3, we consider the global statistics of RRCE on two vertical grids, including the minimal grid. Then we focus on energy and entropy budgets at two more scales: the energetically isolated neighborhoods of the vortices and the overturning circuits rising through the eyewalls. For the latter two scales, we rely on vortex-centric compositing and time averaging. In section 4, the pressure intensity is deconstructed and global scaling arguments for vortex size and intensity are verified. The summary and discussion are in section 5.
2. Model description
The model is hydrostatic and formulated in sigma–pressure coordinates. The continuous form of the equations is given in appendix A with a focus on moisture and energy. The A grid defined by Zhu and Smith (2003) is chosen for staggering the variables in the vertical coordinate σ. This places the horizontal velocity and temperature at full levels and the pressure and vertical velocity at half levels. The half levels include the two boundaries, σ = 0 and σ = 1, where
The vertical differencing is based on the scheme by Simmons and Burridge (1981), which converts between kinetic energy and enthalpy exactly. This keeps global energy constant in the absence of any forcing and dissipation, including the subtle forms of dissipation due to water vapor sources and sinks. Zhu and Smith (2003) warn of a computational mode produced by the vertical staggering of temperature and geopotential height. Although vortices do develop azimuthal asymmetry in the present model, especially at coarse vertical resolution, the Simmons–Burridge differencing seems to be numerically stable. The same scheme is used by Held and Suarez (1994).
In the horizontal, we use a C grid, which staggers the horizontal velocity components in a manner that resolves the horizontal divergence at the grid scale. Temperature, pressure, and vertical velocity (with the latter two shifted to half levels) are defined at the divergence points. The strategy for choosing a horizontal resolution is to minimally resolve the eyes of the vortices. For the grid scale, Δx, we use 35 km, which is a typical eyewall radius in nature. Then the radius of maximum wind is either Δx or 2Δx. The domain is square and covers roughly the same area as the tropics between the equator and 25° latitude. Time stepping is leapfrog with a Robert filter.
Advection is formulated in flux form to conserve energy globally. Vertical advection uses centered differences. Horizontal advection uses the piecewise-parabolic method of Colella and Woodward (1984), which contains adaptive diffusion/viscosity. No other horizontal diffusion or viscosity is used. The choices for advection and viscosity are especially important in tropical cyclone models, where kinetic energy dissipation is vital to the dynamics (see Bryan and Rotunno 2009a).
a. Parameterizations
A goal of idealized modeling, including analytical and conceptual models, is to land on the minimum complexity that permits the relevant phenomenon (Held 2005). Models of organized convection can be difficult to simplify because surface fluxes, radiation, and moisture physics are all crucially important. The parameterizations used in the present model and covered in this subsection are standard and suitable for general model building. Those of Ooyama (1969), Schecter and Dunkerton (2009), and Zhu et al. (2001), for example, are developed specially for hurricane simulation. The present model cannot be simplified much further without departing qualitatively from earlier RRCE solutions. The full set of parameters and their control values are tabulated at the end of the subsection.
As an alternative to realistic radiation, the model’s lower section, σ ≥ 0.25, is cooled with a constant forcing, Qc < 0, while the upper section, σ < 0.25, is relaxed to a temperature Te on a time scale of τe. The upper section is meant to represent the lower stratosphere, where the strongest TCs detrain. It is resolved by only one full-level in the three-level model. In the control runs, Qc/cp = −3 K day−1, where cp is the heat capacity. The control settings for the Newtonian relaxation are τe = 2 days and Te = 218 K. The temperature will be adjusted to a new moist adiabat when the imposed boundary temperature changes. To prevent large-scale horizontal shear, the average velocity is continuously nudged to zero in the upper section, σ < 0.25. For this Rayleigh damping, the time scale is τm = 6 days.
The combination of Newtonian relaxation and constant cooling has been used elsewhere (Pauluis and Garner 2006; Chavas and Emanuel 2014; Jeevanjee 2017). The total cooling rate is unknown a priori because of the Newtonian nudging. The constant cooling would be more realistic if the rate decreased upward from a maximum at the lowest level (e.g., Muller and Bony 2015), but we are investigating the simplest parameterizations. The restoration time for the Newtonian relaxation is shorter than in Rotunno and Emanuel (1987) or Chavas and Emanuel (2014). With the upper atmosphere so poorly resolved, the parameterization must rapidly cool the updraft as well as maintain the stratification. For their three-level model, Zhu et al. (2001) use an even shorter τe = 0.5 days.
The control value of Qc produces stronger cooling than in RRCE models with realistic radiation. It is about 1.5 times as strong as the longwave cooling in Khairoutdinov and Emanuel (2013) and more than twice as strong as in Zhou et al. (2017), where much cooler surface temperatures are used. Rotunno and Emanuel (1987) mention that Newtonian cooling achieves approximately −2 K day−1 in the outer region of their model hurricane. The constant cooling in Chavas and Emanuel (2014) is −1 K day−1 in their control case, but ranges to −4 K day−1 in their sensitivity tests. Across this range, the gradient-wind maximum increases from about 50 m s−1 to about 100 m s−1 and the radius of maximum wind doubles to about 100 km. The proposed three-level model does not generate or sustain vortices at their control setting. However, we conducted experiments with Qc/cp = −2 K day−1 and found them to be only quantitatively different from the control. We prefer to analyze somewhat stronger forcing because it accelerates spinup and forms better-resolved vortices.
Table 1 provides a complete list of the model parameters. Not yet mentioned is the Coriolis parameter, for which the control value is f = 0.5 × 10−4 s−1. The parameters in the first three rows are the only ones that are varied in this study. Of the remaining parameters, the lapse rates have the weakest impact.
Model parameters and their control values. Only the parameters in the first three rows are varied in this study. The value given for Qc is divided by the heat capacity. The upper temperature Te will be adjusted for different surface temperatures. The vertical diffusivities will be adjusted for finer vertical grids.
There is no parameterization for convection or for horizontal diffusion. One of the functions of a convection parameterization is to reduce the sensitivity to horizontal resolution stemming from the preference of hydrostatic convection for the smallest resolved scale (e.g., Pauluis and Garner 2006; Garner et al. 2007; Jeevanjee 2017). In additional runs at doubled resolution, we found that parameterizing convection counteracts the halving of average vortex size. Some form of convective closure will be necessary at higher resolution. Vortex size also depends on horizontal diffusivity. Bryan and Rotunno (2009a) find that the radius of maximum wind of an isolated vortex decreases monotonically to a certain limiting value as the explicit horizontal diffusivity decreases to zero. They do not rule out additional dependence on computational or nonparametric diffusion of the kind used here. We are not aware of a systematic study of the sensitivity of RRCE to horizontal resolution and diffusivity. This needs to be looked at but we have not attempted it here.
Condensation takes the form of saturation adjustment based on a time-implicit evaluation of the Clausius–Clapeyron relation for saturation vapor pressure over liquid water. Condensate is removed immediately, with no reevaporation. Therefore, saturated ascent is pseudoadiabatic and precipitation efficiency is 100%. There is no freezing or melting.
b. Energy and entropy
The model is run with full dissipative heating, such that total energy (kinetic energy plus moist enthalpy) is conserved.1 For a model to conserve total energy apart from external forcing (radiation and surface fluxes), all energy that is lost by dissipation must be recaptured by internal heating. Details of the dissipative heating are covered in appendix A. The most familiar dissipation comes from damping and diffusion of momentum via the material changes in kinetic energy. This is converted directly to sensible heat in the temperature equation (e.g., Fiedler 2000).
Additional dissipation comes from the movement of relatively small amounts of kinetic and potential energy by the condensation and diffusion of water vapor. The corresponding physical processes are not explicit in the model but can be inferred from global integrals of the energy tendency. In appendix A, we identify two implicit processes, the cascade of microturbulence to dissipative heating around hydrometeors (Pauluis and Held 2002) and the tapping of macroturbulent kinetic energy to move water vapor diffusively across gradients of geopotential and resolved kinetic energy. The distribution of dissipative heating that results from this analysis seems realistic enough for an idealized model.
The accounting of heat and kinetic energy sources and sinks includes the results of the time filtering and the departure of the advective momentum tendencies from second-order centered differencing, which would be kinetic energy conserving. The net result of the energy management, including the Simmons–Burridge scheme for handling internal conversion, is a maximum difference of 0.1 W m−2 between the sum of the explicit and implicit energy source/sink terms and the actual global energy tendency. This is good enough to say precisely how much of the dissipation is computational.
The entropy budget will be used along with the energetics to characterize the equilibrated states on all three of the spatial scales mentioned in the introduction. The model includes the irreversible entropy production by parameterized turbulent water vapor flux convergence, representing surface evaporation at the lowest level and diffusive moistening at higher levels. However, the energy and entropy are simplified by ignoring the heat capacity of water substance. The computational source/sink of entropy, mainly due to vertical differencing, is derived at the end of appendix B.
3. Analysis of control runs
The model is run on two vertical grids, L3 and L10, where LN is the notation for N levels. In both configurations, the lowest full level is at σ = 0.95, close enough to the lower boundary for the aerodynamic fluxes to be valid, and the first interior half level is at σ = 0.9. The other two full levels in L3 are at σ = 0.66 and 0.2 (compared to 0.61 and 0.17 in Zhu et al. 2001) and the other half level is exactly halfway between. In L10, the spacing is uniform. The Newtonian cooling applies to two full levels (σ = 0.05, 0.15) in L10 but only one in L3.
Parameter settings are the same for L3 and L10 with two exceptions. In L10, the Newtonian relaxation temperature is lowered to Te = 200 K to give the vortices similar pressure intensities, and the heat diffusivity is decreased to Kh0 = 20 m2 s−1 to keep the average surface temperature from exceeding Ts. The ratio Km0/Kh0 = 2 is preserved. The domain has a length and width of 80° of latitude, or about 9000 km, similar to Zhou et al. (2014) but much larger than Khairoutdinov and Emanuel (2013). There is significant sensitivity to the placement of the levels in L3. The middle full level, which captures some of the return flow in the overturning circulations, must be well below middepth to permit cyclogenesis. We return to this issue in section 5.
We checked that solutions on a uniformly spaced L16 grid are similar to L10, with no notable change in output statistics or vertical structure. We therefore consider the L10 solutions to be converged with respect to vertical resolution. We are interested in both the quantitative and qualitative differences between L3 and L10. We can confirm immediately what seems to be implicit in the work of Zhu et al. (2001), that L3 is the minimal vertical grid. That is, we cannot obtain TCs or TC worlds in a two-level model. The likely reason is that L2 does not provide a level of quasi nondivergence for warm cores. The possibility that L3 behaves as a singular limit in any way was examined and eliminated by running on two intermediate grids. Those results are discussed in appendix C.
The model is considered equilibrated when the 1-day net external diabatic forcing falls to 1% of the heating. L3 equilibrates after about 30 days and L10 after about 60 days. Both are run for 120 days and the last 30 days are used for averaging. Figure 1 displays snapshots of equilibrated TC worlds in the 3-level and 10-level models at 90 days. The parameter settings are from Table 1. Shown is the total horizontal velocity at the lowest level and the mixing ratio at the surface, where the heat fluxes are computed. The average distance between vortices is about 2000 km.
Equilibrated solutions at t = 90 days in the (top) 3-level model and (bottom) 10-level model. Fields are (left) lowest-level total wind speed (m s−1) and (right) surface water vapor mixing ratio (10−2 kg kg−1). The full model domain is shown.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-22-0089.1
The filamentation of the surface moisture field in L3 reflects stronger frontogenesis than in L10. The filaments coincide with maxima in low-level convergence, precipitation rate, and temperature, as well as humidity. It would appear that removing vertical structure reduces the effective horizontal diffusivity at the solid boundary.
The distribution of vortex size and intensity superficially resembles the simulations with the more traditional RRCE model of Zhou et al. (2014), and the time dependence is also similar (W. Zhou 2022, personal communication). During model spinup, vortices generally continue to deepen after the surface pressure falls below about 980 hPa, after which they take just a day or two to reach the potential minimum pressure, which is near 910 hPa. In equilibrated solutions, new vortices struggle to reach this intensity. The stronger vortices do not weaken like the single vortices in limited-area models (e.g., Smith et al. 2014; Chavas and Emanuel 2014; Pauluis and Zhang 2017; Rousseau-Rizzi and Emanuel 2019), even after as much as 30 days. Less intense vortices can weaken further and disappear, but the fate of many is to be swept into a stronger vortex by what could be considered the model’s representation of spiral bands. Such mergers are more frequent in L10.
Tangential and total wind speeds are maximized at the lowest full level in L3, but the tangential component is slightly greater at the second full level in L10. Thus, L10 just manages to resolve the hurricane boundary layer. The circulation is mostly rotational at low levels, but divergence becomes comparable at the outflow levels, σ = 0.2 (L3) and σ = 0.15 (L10). There is anticyclonic circulation on the synoptic scale at the outflow levels, with mostly cyclonic circulation persisting at the scale of the inner radius.
a. Global scale
Table 2 lists some global statistics of the control solutions, starting with the number of vortices and the domain-maximum intensity. The TC count n is based on a detection threshold of 960 hPa. L10 is more sensitive to the threshold than L3. A stricter threshold of 950 hPa subtracts one vortex from L3 and three from L10. The time-averaged, domain-averaged kinetic energy,
Statistics from the 30-day-averaged control runs vs. vertical resolution, with L3 results on the left. The maximum velocity and minimum pressure achieved during the averaging period are denoted Vmax and pmin, respectively.
The model is driven by surface heating Qheat and radiative cooling Qcool combined with dissipative heating Qdiss, which is due to frictional drag and precipitation drag. For the total external heating and cooling, we write Q+ = 〈Qheat〉 and Q− = 〈Qcool〉, respectively, where the angle brackets denote mass-weighted integration. The parts played by the forcing and dissipation in the energy budget are covered in detail in appendix A. The energy numbers in the table are divided by the model area. Consistent with the higher precipitation rate in L3, the heating is about 25% larger than in L10 (sensible heating is relatively small). Since the model is run with full dissipative heating, equilibrated solutions have |Q+ + Q−| ≪ Q+, so that the radiative cooling is closely approximated by the negative of the heating. In L3, about half of the cooling is the Newtonian relaxation. In L10, the relaxation is one-sixth of the total cooling.
A different conception of the efficiency comes from expressing ε in terms of effective heating, cooling, and dissipation temperatures in accordance with the entropy budget (see appendix B). The effective heating and cooling temperatures are defined as T+ = Q+/〈Qheat/T〉 and T− = Q−/〈Qcool/T〉, respectively. A larger contrast between T+ and T− drives a stronger circulation for the same external heating. In the model, the cooling temperatures are representative of the middle atmosphere, which is considerably warmer than the outflow levels (227 K in L3 and 212 K in L10). The heating temperature in L3 is warmer than in L10 mainly because the vortices are warmer. Warmer vortices also raise the effective dissipation temperature, defined as TD = QD/〈Qdiss/T〉.
The dissipative heating boosts the efficiency via the factor TD/T−. The Gibbs penalty more than compensates for this, with the result that ε < εC in L3 as well as L10. From the perspective of the entropy budget, the similarity between the L3 and L10 efficiencies depends on the Gibbs penalty being much larger in L10. The computational residual due to nonconservation of entropy (see appendix B) is εcmp = 0.004 in L3 and negligible in L10.
Dissipation is due to friction and precipitation drag, the former arising from viscosity and damping and the latter from loss or gain of potential and kinetic energy through phase change. In a model without explicit condensate, precipitation drag is an interpretation of an energy source/sink. Horizontal viscosity is entirely due to advection. Vertical profiles of the different forms of dissipation are plotted in Fig. 2. Viscous dissipation is concentrated at the lower boundary and eyewall. Roughly one-half (one-third) of the dissipation is due to vertical viscosity in L3 (L10). The proportions are approximately reversed for the horizontal viscosity. In L3, deceleration from vertical viscosity peaks at around 6.0 × 10−4 m s−2 compared to 2.5 × 10−4 m s−2 for the horizontal viscosity. The values for L10 are smaller but in the same proportion. Swapping Δz (the vertical grid interval) for Δx in the scaling of the Laplacian leads to an estimate of Khoriz ≈ 5000 m2 s−1 for the effective horizontal viscosity, two orders of magnitude larger than externally imposed vertical viscosity.
Vertical profiles of 30-day horizontally averaged dissipation expressed as heating,
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-22-0089.1
Rayleigh damping contributes about 3% of the total dissipation, while precipitation drag contributes 13% of the PDI in L3 and 11% in L10. In Pauluis and Zhang (2017), the precipitation drag is more significant, at around 20%. Their model recirculates condensate and may have less diffusive moistening at large z. The effective temperature of the dissipative heating from precipitation drag, which is applied at the point of condensation, is 245 K in L3 and 249 K in L10.
b. Vortex scale
A composite vortex is created by selecting the top 5 vortices ranked by pressure intensity, shifting their centers to the same origin and averaging. The pressure intensity of an individual vortex varies by about 30 hPa over time, which is similar to the difference between the strongest and weakest vortex in the top five. Therefore, compositing acts in a similar way to time averaging to reduce swings in vortex intensity and other variables.
Figure 3 shows the 30-day-averaged composite pressure velocity, ω, at the level where the descent rate is greatest, namely, σ = 0.66 in L3 and σ = 0.25 in L10. This is an erratic field in both solutions. The time average shows L3 to have a simpler radial structure. For each solution, a ring is drawn at the radius of regional energy balance rQ, defined as the radius within which the net external forcing vanishes, or
Pressure velocity (ω; Pa s−1) from the 30-day-average composite vortex and its near-environment in (left) L3 and (right) L10. The levels of strongest descent are shown, namely, σ = 0.66 in L3 and σ = 0.25 in L10. Rings are drawn at the energy radii: rQ = 11.6° in L3 and rQ = 11.9° in L10.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-22-0089.1
Listed in Table 3 are statistics from within the cylindrical volume obtained by integrating the composite fields out to r = rQ. The averaging considerably weakens the pressure and velocity intensities,
As in Table 2, but for the 30-day-averaged composite vortex. The intensity statistics represent extremes of the azimuthal averages. All other statistics are area averages inside the energy radius rQ.
As expected from the global analysis, the efficiency improvement from dissipative heating is strongly opposed by the Gibbs penalty εG, especially in L10. For the entropy budget, the regional domains are not perfectly isolated. The net flux of entropy across r = rQ (the volume integral of the flux divergence) subtracts 0.011 from the efficiency in L3. The analogous contribution is negligible in L10, perhaps fortuitously. The computational error is negligible (<0.001) in both solutions.
We next perform azimuthal averaging to obtain radial profiles of the composite vortex and its immediate environment and show the results in Fig. 4. The radial mass flux is plotted in Fig. 4a. The profiles show the total inflow or outflow at each radius. Considerably more mass is overturned in L3, as indicated by the peak flux of 4.6 × 1010 kg s−1 compared to 3.6 × 1010 kg s−1 in L10.
Radial profiles of (a) radial mass flux (1010 kgs−1), (b) lowest-level vertical velocity (10−4 s−1), (c) midlevel tangential velocity (m s−1), (d) surface pressure (hPa), (e) lowest-level temperature (K), and (f) surface relative humidity from the azimuthally and 30-day-averaged composite vortices in L3 (red) and L10 (black). Vertical lines are drawn at the radius used as the eyewall in the circuit analysis.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-22-0089.1
The low-level vertical motion is shown in Fig. 4b. In L3, there is descent (
Figure 4c shows the tangential wind speed at σ = 0.66 in L3 and σ = 0.65 in L10, above the boundary layer. At this level, the azimuthally averaged radial component (not shown) is nowhere greater than 2 ms−1. The L3 wind profile is broader than L10, with a smaller maximum. The smaller maximum means stronger vertical wind shear, consistent with the warmer core. The greater width is the main reason for the larger domain-averaged kinetic energy (Table 2). Surface heat fluxes (not shown) are also broader. The maximum midlevel wind occurs at the radius 2Δx, which is indicated by the vertical line. In the next subsection, r = 2Δx will be used diagnostically as the eyewall radius.
The center of the vortex is a minimum in both relative and absolute humidity. The radial profile of relative humidity at the surface (where the air–sea flux is calculated) is shown in Fig. 4f. The dryness at the core reduces the moist entropy, but, as we see next, entropy is still maximized at the center because of the low pressure and high temperature (Figs. 4d,e).
Figure 5 shows vertical cross sections of the kinetic energy K, mass streamfunction χ, and moist entropy s for the azimuthally averaged composite vortices. The kinetic energy (Figs. 5a,d) is computed on the model grid before averaging. This field is clearly broader in L3. There the volume average of K is 15% larger than in L10 (Table 2), despite being 15% smaller inside a radius of 2°. One can use the variation of K along a trajectory to form a qualitative idea of the energy dissipation, although, on horizontal segments, a significant variation of pressure can complicate this picture, as discussed in section 4a. At low levels, the tilt of the kinetic energy field suggests overshooting of the inflow. Horizontal averages of K (not shown) have significant peaks at the outflow levels.
Radius–height cross sections of (a),(d) kinetic energy (J kg−1), (b),(e) mass streamfunction (1010 kg s−1), and (c),(f) moist entropy (J kg−1 K−1) in the azimuthally and 30-day-averaged composite vortices from (top) L3 and (bottom) L10. The streamfunction for L10 is multiplied by 1.25. The kinetic energy and entropy are extrapolated to the upper boundary using the imposed velocity and temperature. The blue curves in (c) and (f) are zero-dissipation pathways used as trajectories in the circuit analysis. Vertical lines are drawn at the radius used for the eyewall.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-22-0089.1
Cross sections of the moist entropy are shown in Figs. 5c and 5f. L3 has higher values in the core and at midlevels farther out. At the nominal eyewall radius, the entropy decreases upward at low levels, more noticeably in L3. This is mainly due to nonradiative cooling, specifically horizontal mixing from averaging and diffusion. The nonradiative contribution is discussed further in section 3d in the context of ventilation. The higher entropy at large radius in L3 results from faster far-field subsidence and warmer near-field temperatures.
c. Overturning circuit
The other subspace to be examined is the deep overturning circulation. This is a perspective familiar from the steady-state analytical models (E86; Holland 1997; Pearce 1998; Bister and Emanuel 1998; Bryan and Rotunno 2009b; Rousseau-Rizzi and Emanuel 2019). A closed circuit meant to approximate a trajectory is imposed on the azimuthally and temporally averaged composite fields, as shown in Fig. 6. Bilinear interpolation is used to place the fields on the circuit, after which line integrals for the forcing and dissipation (appendix B) are computed.
Overturning circuit imposed on the time-averaged composite radius–height cross sections. The sections colored red and blue are dominated by diabatic heating and cooling, respectively. The inflow path DA is located at the lowest mass level, stopping at the nominal eyewall point A. The section AA′ extends vertically to the minimum-entropy level at the eyewall radius, and A′B follows a path of decreasing entropy to BC, which is the mass level with strongest outflow.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-22-0089.1
The subcircuits Γheat and Γcool correspond to where the circulation is predominantly heated or cooled externally, respectively. The whole circuit is bounded laterally by the energy radius, r = rQ, and from above by the level with largest radial velocity, σ = 0.20 in L3 and σ = 0.15 in L10. The point A is located at the base of the eyewall, which is offset from the center by 2 grid points. The inflow section DA is placed at the lowest full level, σ = 0.95. This is not strictly a trajectory, as there is considerable vertical advection near the eyewall, but the pressure change on a sloping inflow path would be difficult to compare to that on the boundary, where the intensity is defined.
The section AA′ connects the inflow vertically to the entropy minimum at A′, closely following the streamfunction contours (Figs. 5b,e). There is no heating to account for any further ascent at this radius. Therefore, A′B slopes outward, creating a long superadiabatic section A′C. We define A′B as the path with zero energy dissipation, which is straightforward to locate on a refined grid using linear interpolation. Alternative trajectories, for example, the streamfunction contours, can contain unrealistic rates of dissipation. Dissipation can be larger than in corresponding parts of the three-dimensional analysis (see the vanishing dissipation from vertical viscosity in Fig. 2) or negative where the path is too steep. The zero-dissipation paths are drawn as blue curves in Figs. 5c and 5f, where they can be seen to roughly follow the moist isentropes. The point A′ is at σ = 0.66 in L3 and σ = 0.55 in L10.
To finish the circuit, we partition the descending section CD between CC′, on which Qext < 0, and C′D, on which Qext > 0, where Qext = Qheat + Qcool. Thus, from C′ downward, unresolved mixing of latent and sensible heat from the boundary overwhelms the radiative cooling. Although not so indicated in Fig. 6, the descending branch is much broader than the updraft. The composite fields are uniform enough in the far field to be represented by profiles at the energy radius. Linear interpolation of the model fields puts the point C′ at σ = 0.56 in L3 and σ = 0.63 in L10.
Statistics for the overturning circuit are listed in Table 4. The dissipation and external heating are similar between L3 and L10. The effective heating and cooling temperatures are obtained from the entropy budget as described in appendix B. The effective heating temperatures (third row in the table) are a few degrees cooler than the lowest model level (cf. Fig. 4e) because of the contribution from the subsiding section of Γheat and from the cooler temperatures around the eyewall in L10. The cooling temperatures can be made colder by separating a ventilational cooling from the radiational cooling, as we show at the end of this section. L3 has a warmer effective cooling temperature than L10 because of the warmer cores.
As in Table 3, but for the embedded overturning circuit. The tilde is used to distinguish properties of the circuit from those of the three-dimensional volumes. The heating rates have changed dimensions from W m−2 to kJ kg−1. The temperature
The circuit dissipation pattern departs from traditional assumptions that place almost all of it at the surface (E86; Emanuel 1997; Bister and Emanuel 1998; Emanuel 1999). The inflow/outflow dissipation ratio is about 3/2 in both solutions (a decomposition by mechanism, as in Fig. 2, is not feasible on the circuit). As a result, the effective dissipation temperatures [see (B19)] are cooler than the lowest model level, especially in L10. Restricted to Γcool, the dissipation temperature is 283 K in L3 and 265 K in L10. Updraft dissipation includes precipitation drag,
d. Circuit ventilation
Part of the cooling along a trajectory is due to mixing with the immediate environment, rather than radiation. The impact of this so-called ventilation on vortex intensity has been analyzed by Tang and Emanuel (2010, 2012). Here it includes the effect of interpolating many updrafts onto the imposed circuit. If ventilational cooling is counted separately from the radiational cooling, the external diabatic forcing becomes unbalanced and the thermodynamic efficiency calculation changes. Also, the mechanical efficiency (Pauluis and Held 2002; Tang and Emanuel 2012) is lowered.
The cooling by radiation is not known exactly because residence times are not known exactly. We simply take Γvent to be the vertical portion of the outflow, or AA′ in Fig. 5. This is where almost all of the outflow dissipation occurs. The resulting ventilation and cooling temperatures are listed in Table 4. The cooling temperature includes all of the Newtonian relaxation on the circuit. The efficiency reductions are
4. Constraints on pressure and velocity intensity
A rotating RCE model of low to moderate complexity is an efficient tool for validating predictions from analytical TC models as well as for checking theoretical scaling of TC worlds. The goal of the first subsection below is to analyze the pressure intensity in the composite vortex. The second subsection switches to a global perspective to look at velocity intensity. Unlike pressure intensity, velocity intensity in a circuit analysis would involve material time derivatives of energy and entropy and consequently too much numerical uncertainty for direct comparisons with analytical models.
a. Pressure intensity
The time and space averaging has provided a steady, axisymmetric vortex that can be directly compared to analytical models. Here we focus on the pressure intensity as determined by the energetics of the embedded overturning circuit. The circuit is obtained from extensive averaging but otherwise uses fewer assumptions than analytical models. The model pressure is, of course, already known. The intention is to quantify the separate contributions from specific bulk changes in the inflow using the energy budget. This will determine the relative importance of subtle and controversial contributions like outflow dissipation and environmental convective available potential energy (CAPE).
Consider the slightly modified partitioning of the overturning circuit in Fig. 7. The point C′ is relocated to the environmental convective detrainment level, where the entropy matches the surface entropy at D. This occurs at σ = 0.29 and 0.23 in the control solution for L3 and L10, respectively. The dashed line drawn between C′ and D and labeled Γad is the hypothetical moist adiabat, on which s is constant. Completing the full circuit with Γad instead of Γenv assumes adiabatic, as opposed to environmental, closure (Garner 2015).
Alternative partition of the overturning circuit in Fig. 6. The inflow and outflow are colored red and blue, respectively, and the environmental return branch is drawn in black. The reversed direction for Γad creates a CAPE circuit with Γenv. The point C′ is the detrainment level based on the environmental entropy at D. The dashed line indicates a hypothetical moist adiabat.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-22-0089.1
The insight that the pressure intensity is approximately proportional to Δs under convective quasi equilibrium may have originated with Malkus and Riehl (1960), who included the CAPE contribution implicitly. As those authors seemed to recognize, Δs is not easy to constrain. This prompts us to set about eliminating Δs by making its pressure, temperature, and humidity dependence explicit. Following E86, we will do this by integrating the first law separately in the inflow.
30-day time series of the lowest-level eyewall pressure perturbation (black) and its several parts (solid color) according to the energy budget of the overturning circuit for the composite vortex. Contributions are change in relative humidity and temperature combined (red), outflow dissipation and change in mechanical energy combined (blue), evaporation/diffusion of moisture (green), and environmental CAPE (purple). Contributions from the temperature change (red dashed) and mechanical energy change (blue dashed) are isolated from the solid red and blue curves. Data are 2-day averages from (left) L3 and (right) L10.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-22-0089.1
The pressure perturbation, shown in black in Fig. 8, is the sum of the four colored solid curves. The dominant contribution is the combined temperature and relative humidity, PT + PH ≡ Pheat, shown by the solid red lines. Together, these make up the enthalpy increase, because the pressure contribution is present in the feedback factor. The temperature part of Pheat (red dashed), is quite different between L3 and L10. It contributes 8 hPa in L3 but −7 hPa in L10. The contrast reflects the different core temperatures in the respective models. This term (PT) is often taken to be negligible on the assumption that the air temperature hews closely to the ocean temperature (e.g., E86). The core temperature in L3 responds sensitively to subsidence warming (see Fig. 3c), presumably because the surface sensible heat exchange is an inefficient damping mechanism. In L10, temperatures reach a local minimum at the eyewall.
The solid blue line combines the mechanical energy (ME) variation, ΔB ≈ ΔK, with the negative of the outflow dissipation. The sum can be expressed as
In analytical work, Pext is typically assumed to be dominated by ΔK with little reduction by dissipation. For example, E86 assumes that dissipation in the outflow is only as much as needed to remove the radial part of the kinetic energy.2 However, in the model, Pext is only about half of the ME term, which is shown separately by the dashed blue line. This means that ΔK is significantly offset by dissipation. Almost all of the outflow dissipation occurs in the vertical section AA′. The traditional assumption that Pext can be closely estimated by ΔK is used by Garner (2015), where it probably exaggerates the difference between the so-called inner- and outer-eyewall pressure intensities.
The CAPE term makes up for much of the kinetic energy dissipation. It contributes about one-fourth of the pressure intensity in L10 but one-eighth in L3. A larger CAPE reflects a colder environment, which intensifies the pressure deficit beneath the warm core. The warmer far-field temperatures in L3 are reflected in the higher entropy values in Fig. 5. The evaporation/diffusion (Gibbs) term naturally weakens the intensity, just as it reduces the thermodynamic efficiency.
The ME term is strongly correlated in time with the full pressure. The other terms are uncorrelated. The pressure and ME terms (black and dashed blue, respectively, in Fig. 8) covary with nearly the same amplitude. The difference between these terms is the inflow dissipation. That is,
b. Velocity intensity and vortex number
The scaling arguments by Khairoutdinov and Emanuel (2013) for the intensity and spacing of the vortices in a TC world rely on two independent energy relationships, one involving the radiative cooling of the overturning circulation and the other involving the global energy dissipation. After some adjustments, we will check the consistency of these relationships with the model solutions.
It is useful take Vb to be the maximum velocity in the composite vortex. Then the formulas (23) and (24) represent attempts to reduce the volume integrals QD and Q− to products of mean values of the remaining factors in the integrands. Some of the values are not easily assigned. Zhou et al. (2017) point to differences in r0 to explain why experiments with similar power dissipation have dissimilar velocities. The uncertainty in the choices, including the uncertainty about covariance between factors and the shape of the radial velocity profiles, can be subsumed in r0 and rm. Thus, with Vb and the other factors assigned, we propose to solve (23) and (24) for the effective inner radii. The more these vary over time or between experiments the less useful the scaling relationships will be.
We run four new experiments to evaluate the predictions for the maximum velocity. The first two change the surface temperature to Ts = 297 K and Ts = 303 K (the control case minus and plus 3 K) and the other two alter the constant cooling rate to Qc/cp = −2 K day−1 and Qc/cp = −4 K day−1 (the control case ±1 K day−1). For the Ts experiments, we also adjust the nudging temperature, Te, by ΔTe = ΔTs + (L0/cp)Δws, where ws is the mixing ratio at (T, p, H) = (Ts, 1013 hPa, 50%). This nudges the upper atmosphere towards convective equilibrium with the surface.
The values of QD and Q+ are listed in Table 5 for the new and control experiments, along with the maximum velocity in the azimuthally averaged composite vortex. For strict consistency, the small surface sensible heating is excluded from Q+, leaving only Qlat, and the precipitation drag is removed from QD, leaving only viscous and damping dissipation. The factor Δw is obtained by setting
Globally averaged dissipative heating rate (W m−2), globally averaged surface latent heating rate (W m−2), and maximum azimuthally averaged velocity (m s−1, boldface for emphasis), from left to right in each row, from the equilibrated control experiment and four perturbed experiments for (left) L3 and (right) L10. The control appears twice, in rows 2 and 5. Precipitation drag is removed from QD.
What stands out in much of Table 5 is the insensitivity of the maximum velocity to the surface temperature even while the forcing and dissipation are varying. The uniformity of the velocities is a consequence of the uniform pressure intensities, since vortex size is also fairly uniform. Thus, the strategy of controlling for the vortex intensity helps to expose variations in the other factors.
Table 6 lists 30-day-averaged values of the effective radius fractions,
The effective radii for viscous dissipation and surface latent heat flux, left and right values in each row, normalized by the outer radius rQ in the control experiment and four perturbed experiments. The control appears twice, in rows 2 and 5. In individual experiments, results vary over time with a standard deviation of less than 0.005.
The general decrease in dissipation and heating between L3 and L10 is mainly expressed as a decrease in the effective radii. This is consistent with the slight narrowing of the L10 radial profiles in Figs. 4 and 5. In the L3 experiments, the external heating predicts the maximum velocities with a nearly constant
Given the complicated balances in the global energetics, it is significant that the effective inner radii vary by only about 10%. We should note that these variations are not necessarily useful for predicting vortex count. In the results of Khairoutdinov and Emanuel (2013), the vortex number decreases systematically with increasing Ts, whereas the present experiments exhibit nonmonotonic variations of ±2.
Finally, we run four cases to equilibrium with the Coriolis parameter decreased and increased to f = 3.3 × 10−5 s−1 and f = 6.7 ×10−5 s−1. For each setting, we use the control value, Qc = −3 K day−1, as well as Qc = −4 K day−1. Other parameters are set to their control values. Increasing f weakens the vortices, especially in L10. For this reason, the detection threshold is relaxed to 970 hPa, where the numbers are closer to converging. The results are graphed in Fig. 9. The TC counts are the nearest whole numbers to the 30-day averages.
TC count as a function of Coriolis parameter, with the constant cooling rate set to −3 K day−1 (solid) and −4 K day−1 (dashed) for the two vertical grids, L3 (red), and L10 (black). The TC detection threshold is 970 hPa.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-22-0089.1
With increasing Coriolis parameter, TC count increases no faster than linearly. Zhou et al. (2014) found a near-linear dependence in their TC-world simulations. In Khairoutdinov and Emanuel (2013), the theoretical relationship n(f) is quadratic. Their scaling assumes that the vortex size follows rQ ∝ V/f, where V is the maximum velocity (Chavas and Emanuel 2014). Allowing for the decrease of maximum velocity with increasing f would give an even steeper dependence. A linear n(f) would be consistent with a universal value of background circulation measured at the outer radius, or
5. Summary and discussion
We have investigated rotating RCE in a hydrostatic model with only three levels. Global and vortex-relative statistics are mostly unaffected by the vertical truncation, as long as cyclogenesis is possible. When we control for the average eyewall pressure, the vortices in the three-level model (L3) have slightly broader wind fields and less symmetry, but similar thermodynamic and mechanical efficiency compared to L10. They contain 15% more kinetic energy and move 25% more mass and moisture. There is a 25% stronger energy throughput, mostly due to the broader regions of high winds and surface fluxes. Superficially, the fundamentals of genesis, maintenance, and interaction of the model vortices (TCs) appear to be the same as in better-resolved TC worlds. However, the focus on equilibrated states has meant largely ignoring transient phenomena such as vortex mergers and splits, cyclogenesis and cyclolysis, and vortex instabilities.
The RRCE framework isolates the fundamental dynamics of warm-core vortices by imposing uniform surface temperature, surface roughness, large-scale winds and humidity, Coriolis parameter, and radiation. The three-level version features vertically smooth, strongly interacting vortices in statistical equilibrium. This may be the closest numerical extension of steady-state analytical models. Steady states are difficult to achieve in models of isolated TCs, a circumstance that has allowed the numerical models to be used to critique analytical models (Smith et al. 2014). The contour analysis attempts to establish a connection with analytical models by interpreting the three-level composite structure as representing a simple closed trajectory. Analytical models impose angular momentum and entropy conservation between inflow and outflow without the benefit of a middle level. However, realistic modifications of the updraft (e.g., Tang and Emanuel 2010; Emanuel and Rotunno 2011) can arguably be interpreted as adding the third degree of freedom.
It has been a priority to manage the model energy and entropy accurately. Any model that equilibrates is energetically balanced, but the balance may include a substantial amount of computational dissipation. Physical interpretation sometimes depends on knowing how much of the energy source and sink is computational. Without an expression for total dissipation that captures all of the explicit and implicit energy sources and sinks, a model with full dissipative heating does not achieve zero net external heating. The implicit energy sources and sinks were identified by integrating the energy equation over the domain. The sum of these with the explicit and computational forcing and dissipation, including the advection and time filter, differs from the actual energy tendency by less than 0.1 W m−2.
Computational errors can throw the entropy budget out of balance if entropy is not a native variable (see Tang and Emanuel 2012). A significant net source or sink of entropy would invalidate an efficiency-based diagnostic for dissipation, such as used extensively in Pauluis and Zhang (2017) and Pauluis and Held (2002). The present model conserves entropy with an error of approximately 1% in the equilibrated L3 solutions, attributable mainly to vertical differencing in the temperature equation. In L10, the residual is much smaller.
Another priority has been to make the analysis of the global and regional volumes and of the embedded circuit as seamless as possible. The regional fields are composites of five vortices and their environments. To mirror the full domain, regional analysis is done inside a radius containing zero net external diabatic forcing. The resulting forcing, dissipation, and efficiency are nearly identical to the full domain. Entropy is reasonably balanced in the energy subdomains, with only a small net entropy flux through the internal boundaries.
Three-level RRCE solutions are quite sensitive to where the middle level is located. With placement near or above middepth, there are no coherent vortices, possibly because too much of the inflow is removed from the surface heating. If the middle level is too low, the heating profile is distorted. Maximum condensational heating in L10 occurs at σ = 0.55 or σ = 0.45 and is 25% stronger than the L3 maximum at σ = 0.66. We have had some concern that the irregularity of the 3-level grid could make that model singular and only coincidentally similar, statistically, to higher resolution. In particular, the shallow indirect cell that is ubiquitous in L3, especially in the larger vortices, is only infrequently present in L10. However, results from intermediate grids L5 and L7 provide some confidence that the grid hierarchy is producing continuous behavior.
The subsidence is consistent with the warmer temperatures in L3, but we have not established causality. Core subsidence is analyzed from theoretical principles by Smith (1980), who associates it with heating in or near the eyewall. The present model develops cores that are drier and several degrees warmer than in the modeling literature. This has not been fully explained. We cannot rule out a role for the model’s horizontal temperature advection and/or surface flux parameterization. The sudden appearance and disappearance of excessively warm cores in some solutions raises questions about feedbacks.
Our approach to validating analytical models starts with identifying a deep overturning circuit in the averaged fields. It is straightforward to check the consistency of energy budgets between a closed circuit and a volume by matching units. The circuit terms should be multiplied by a mass flux to compare to the volume analysis. As an example, the L3 heating rate from Table 4 is about 3.5 × 104 J kg−1. From Fig. 4, we take the mass flux to be 4.6 × 1010 kg s−1. Hence, the heating per vortex is approximately 1.6 × 1015 W, yielding about 3.0 × 1016 W across 18 vortices. From the same experiment, Table 2 shows a global heating rate of 4.4 × 102 W m−2, which yields about 3.4 × 1016 W across the domain. Thus, the energy cycle in the composite vortex is reasonably consistent with the heating in the full domain. By the same analysis, the L10 circuit captures about 2.3 × 1016 W of the 2.7 × 1016 W of total heating.
From these and other agreements between volume and circuit statistics, including the crucial dissipation statistics, it appears that the time and space averaging preserves the Lagrangian dynamics well enough to make the numerical model useful for validating analytical models. The difference between a trajectory imposed on Eulerian averages and an average over trajectories is the motivation for isentropic mass-flux analysis (e.g., Held and Schneider 1999; Pauluis 2016). The difference can be interpreted as a mixing process that generally augments explicit diffusion. This is how we interpreted ventilation of the updraft. Some of the averaging was done by compositing several of the strongest vortices. We checked that longer-term statistics of a single vortex are essentially the same, which suggests that the intensity hierarchy is not dynamically significant (the strongest vortex is not special). Isentropic analysis may be preferable to Eulerian averaging when the TCs are especially asymmetric and dissimilar in size and intensity. Pauluis and Zhang (2017) study the single-hurricane energy cycle this way, projecting model fields onto the maximum mass-flux pathway in the isentropic analysis.
The net external diabatic forcing on a closed circuit is identically zero if the circuit is a trajectory, so that the first law and Bernoulli’s law with friction applies. Identifying the vortex-scale closed trajectory uses assumptions about the ascending and descending branches. The descending branch was placed at the radius of zero volume-integrated diabatic forcing, but this is not a strong assumption, as the far field is quite uniform out to the energy radius. The thermodynamic efficiencies on the circuits are similar between L3 and L10, despite the difference in mass flux and heating. They are about 10% larger than for the global system.
The pressure analysis based on the energy cycle relates the numerical solutions to analytical models and tests the assumptions built into the latter. The decomposition quantifies the impact of the inflow enthalpy increase, the dominant cause of the pressure drop. In L3, the pressure intensity is enhanced by warming from the stable subsidence in the core. Updraft dissipation reduces pressure intensity by unburdening the outflow pressure gradient from removing all of the kinetic energy. Both model versions dissipate a significant part of their energy in the updraft via horizontal viscosity. This also increases efficiency relative to a model with inviscid outflow. Up to one-fourth of the pressure intensity is attributable to positive ambient CAPE.
The equilibrated solutions are a platform for testing scaling theories based on global and synoptic-scale constraints. The energy, entropy, and angular momentum principles do not depend on the vertical resolution. The value of global energy scaling for predicting the velocity is degraded by the variability of the horizontal scale of the significant forcing and dissipation and the uncertainty about the surface humidity. The solutions do not agree with the traditional scaling for the outer vortex size. In the RRCE context, there is a need for an alternative theory for this scale.
We suggested in the introduction that the three-level model might be interesting and robust enough to compare to the time-honored two-level model of the extratropics. Models of RRCE are not readily linearizable like the Phillips model, but linear modes are not necessarily the key to understanding turbulence. An overarching contrast is that mean fluxes of heat and momentum are vertical instead of horizonal. If RRCE models in general test the assumptions used in analytical models, the three-level version tests the importance of boundary layer structure and heating profiles. The truncation increases vortex asymmetry and allows unusually warm and dry cores. However, the storms are long lived, strongly interactive, statistically similar to well-resolved TCs, and equally sensitive to parameters. Three-level RRCE probably has more to reveal about both analytical and comprehensive models and the connections between them.
Full dissipative heating is the limiting case analyzed by Bister and Emanuel (1998). In nature, some of this heating will occur in the ocean (e.g., Kieu 2015). Keeping it in the atmosphere eliminates an energy leak and simplifies the diagnostics. The general case is analyzed in appendixes A and B.
The argument by Rousseau-Rizzi and Emanuel (2020) that outflow dissipation is negligible appears to apply only to the anticyclonic far field, where it can be associated with the restoration of angular momentum.
The full energy sink from explicit diffusion is
Acknowledgments.
The author is grateful to Ming Zhao, Isaac Held, Wenyu Zhou, Olivier Pauluis, and Yair Cohen for helpful exchanges. The figures were drafted by Catherine Raphael.
Data availability statement.
The complete model code, scripts, and parameter files are available at https://zenodo.org/record/6561254#.YoVbNRzMJhc. The diagnostic software is available from the author.
APPENDIX A
Moist Sigma-Coordinate Model Equations and Energetics
a. Energy budget
b. Dissipative heating
APPENDIX B
Moist Entropy
a. Volume budget
b. Circuit budget
c. Computational errors
The numerical model is formulated to conserve energy except for time differencing and horizontal advection. As mentioned in section 2, the computational nonconservation is converted to dissipative heat in the model’s temperature equation along with the explicit dissipation, so that the net external diabatic forcing stays below 0.1% of the external heating or cooling once the model equilibrates.
APPENDIX C
Intermediate Vertical Grids
Here we are interested in whether there is a smooth approach to the limit of the coarsest vertical grid. Between L3 and L10, we set up two intermediate grids, L5 and L7. There is freedom to place the levels in a way that optimizes continuity across the hierarchy. Since L3 requires a highly irregular grid for cyclogenesis, L5 is more challenging to configure than L7. We obtain the best continuity by keeping the additional two levels below middepth. For L7, a further two levels are added in the upper half of the model, including one in the nudging layer. In all configurations, the lowest full level is kept at σ = 0.95 and the lowest interior half level at σ = 0.9. The Newtonian cooling applies to the top level In L3 and L5, and the top two levels in L7 and L10. Table 5 identifies the lower (or sole) level in the Newtonian nudging layer, denoted σe, along with the target temperature in the layer, Te. The heat diffusivity decreases gradually from L3 to L10, while the momentum diffusivity is set to the L10 value in all but L3. Control values are used for the other parameters.
Table C1 gives several key statistics of the equilibrated solutions, averaged over 30 days and the 5 strongest vortices, as before. The strategy in choosing the level configuration and target temperature was to induce heating rates Q+ and energies
Statistics of equilibrated solutions across a limited hierarchy of vertical grids. Full levels are at σ = 0.20, 0.66, and 0.95 in L3, σ = 0.18, 0.6, 0.7, 0.8, and 0.95 in L5, and σ = 0.06, 0.16, 0.36, 0.52, 0.68, 0.82, and 0.95 in L7. Levels are equally spaced in L10. Vortex count n is based on a detection threshold of 960 hPa. The lowest level in the relaxation layer is denoted as σe and the relaxation temperature as Te. The central temperature and vertical velocity in the time-averaged composite vortex are denoted as Tc and
The entries for the central low-level vertical velocity
REFERENCES
Bister, M., and K. A. Emanuel, 1998: Dissipative heating and hurricane intensity. Meteor. Atmos. Phys., 65, 233–240, https://doi.org/10.1007/BF01030791.
Bister, M., and K. A. Emanuel, 2002: Low frequency variability of tropical cyclone potential intensity 1. Interannual to interdecadal variability. J. Geophys. Res., 107, 4801, https://doi.org/10.1029/2001JD000776.
Bister, M., N. Renno, O. Pauluis, and K. Emanuel, 2011: Comment on Makarieva et al. ‘A critique of some modern applications of the Carnot heat engine concept: The dissipative heat engine cannot exist’. Proc. Roy. Soc. London, 467A, 1–6, https://doi.org/10.1098/rspa.2010.0087.
Bretherton, C. S., and M. F. Khairoutdinov, 2015: Convective self‐aggregation feedbacks in near‐global cloud‐resolving simulations of an aquaplanet. J. Adv. Model. Earth Syst., 7, 1765–1787, https://doi.org/10.1002/2015MS000499.
Bretherton, C. S., P. N. Blossey, and M. Khairoutdinov, 2005: An energy-balance analysis of deep convective self-aggregation above uniform SST. J. Atmos. Sci., 62, 4273–4292, https://doi.org/10.1175/JAS3614.1.
Bryan, G. H., and R. Rotunno, 2009a: The maximum intensity of tropical cyclones in axisymmetric numerical model simulations. Mon. Wea. Rev., 137, 1770–1789, https://doi.org/10.1175/2008MWR2709.1.
Bryan, G. H., and R. Rotunno, 2009b: Evaluation of an analytical model for the maximum intensity of tropical cyclones. J. Atmos. Sci., 66, 3042–3060, https://doi.org/10.1175/2009JAS3038.1.
Buckmaster, T., S. Shkoller, and V. Vicol, 2019: Nonuniqueness of weak solutions to the SQG equation. Commun. Pure Appl. Math., 72, 1809–1874, https://doi.org/10.1002/cpa.21851.
Carstens, J. D., and A. A. Wing, 2020: Tropical cyclogenesis from self‐aggregated convection in numerical simulations of rotating radiative‐convective equilibrium. J. Adv. Model. Earth Syst., 12, e2019MS002020, https://doi.org/10.1029/2019MS002020.
Chavas, D. R., and K. Emanuel, 2014: Equilibrium tropical cyclone size in an idealized state of axisymmetric radiative–convective equilibrium. J. Atmos. Sci., 71, 1663–1680, https://doi.org/10.1175/JAS-D-13-0155.1.
Chavas, D. R., N. Lin, W. Dong, and Y. Lin, 2016: Observed tropical cyclone size revisited. J. Climate, 29, 2923–2939, https://doi.org/10.1175/JCLI-D-15-0731.1.
Colella, P., and P. R. Woodward, 1984: The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys., 54, 174–201, https://doi.org/10.1016/0021-9991(84)90143-8.
Cronin, T. W., and A. A. Wing, 2017: Clouds, circulation, and climate sensitivity in a radiative-convective equilibrium channel model. J. Adv. Model. Earth Syst., 9, 2883–2905, https://doi.org/10.1002/2017MS001111.
DeMaria, M., and J. D. Pickle, 1988: A simplified system of equations for simulation of tropical cyclones. J. Atmos. Sci., 45, 1542–1554, https://doi.org/10.1175/1520-0469(1988)045<1542:ASSOEF>2.0.CO;2.
Emanuel, K. A., 1986: An air–sea interaction theory for tropical cyclones. Part I: Steady-state maintenance. J. Atmos. Sci., 43, 585–604, https://doi.org/10.1175/1520-0469(1986)043<0585:AASITF>2.0.CO;2.
Emanuel, K. A., 1988: The maximum intensity of hurricanes. J. Atmos. Sci., 45, 1143–1155, https://doi.org/10.1175/1520-0469(1988)045<1143:TMIOH>2.0.CO;2.
Emanuel, K. A., 1995: Sensitivity of tropical cyclones to surface exchange coefficients and a revised steady-state model incorporating eye dynamics. J. Atmos. Sci., 52, 3969–3976, https://doi.org/10.1175/1520-0469(1995)052<3969:SOTCTS>2.0.CO;2.
Emanuel, K. A., 1997: Some aspects of hurricane inner-core dynamics and energetics. J. Atmos. Sci., 54, 1014–1026, https://doi.org/10.1175/1520-0469(1997)054<1014:SAOHIC>2.0.CO;2.
Emanuel, K. A., 1999: Thermodynamic control of hurricane intensity. Nature, 401, 665–669, https://doi.org/10.1038/44326.
Emanuel, K. A., 2005: Increasing destructiveness of tropical cyclones over the past 30 years. Nature, 436, 686–688, https://doi.org/10.1038/nature03906.
Emanuel, K. A., and R. Rotunno, 2011: Self-stratification of tropical cyclone outflow. Part I: Implications for storm structure. J. Atmos. Sci., 68, 2236–2249, https://doi.org/10.1175/JAS-D-10-05024.1.
Fiedler, B. H., 2000: Dissipative heating in climate models. Quart. J. Roy. Meteor. Soc., 126, 925–939, https://doi.org/10.1002/qj.49712656408.
Garner, S., 2015: The relationship between hurricane potential intensity and CAPE. J. Atmos. Sci., 72, 141–163, https://doi.org/10.1175/JAS-D-14-0008.1.
Garner, S., D. M. W. Frierson, I. M. Held, O. Pauluis, and G. K. Vallis, 2007: Resolving convection in a global hypohydrostatic model. J. Atmos. Sci., 64, 2061–2075, https://doi.org/10.1175/JAS3929.1.
Held, I. M., 2005: The gap between simulation and understanding in climate modeling. Bull. Amer. Meteor. Soc., 86, 1609–1614, https://doi.org/10.1175/BAMS-86-11-1609.
Held, I. M., and M. J. Suarez, 1994: A proposal for the intercomparison of the dynamical cores of atmospheric general circulation models. Bull. Amer. Meteor. Soc., 75, 1825–1830, https://doi.org/10.1175/1520-0477(1994)075<1825:APFTIO>2.0.CO;2.
Held, I. M., and T. Schneider, 1999: The surface branch of the zonally averaged mass transport circulation in the troposphere. J. Atmos. Sci., 56, 1688–1697, https://doi.org/10.1175/1520-0469(1999)056<1688:TSBOTZ>2.0.CO;2.
Held, I. M., and M. Zhao, 2008: Horizontally homogeneous rotating radiative–convective equilibria at GCM resolution. J. Atmos. Sci., 65, 2003–2013, https://doi.org/10.1175/2007JAS2604.1.
Held, I. M., R. T. Pierrehumbert, S. T. Garner, and K. L. Swanson, 1995: Surface quasi-geostrophic dynamics. J. Fluid Mech., 282, 1–20, https://doi.org/10.1017/S0022112095000012.
Holland, G. J., 1997: The maximum potential intensity of tropical cyclones. J. Atmos. Sci., 54, 2519–2541, https://doi.org/10.1175/1520-0469(1997)054<2519:TMPIOT>2.0.CO;2.
Hsieh, T.-L., C.-Y. Chang, I. M. Held, and P. Zurita-Gotor, 2021: Nonlinear generation of long waves and the reversal of eddy momentum fluxes in a two-layer quasigeostrophic model. J. Atmos. Sci., 78, 3525–3536, https://doi.org/10.1175/JAS-D-20-0368.1.
Jeevanjee, N., 2017: Vertical velocity in the gray zone. J. Adv. Model. Earth Syst., 9, 2304–2316, https://doi.org/10.1002/2017MS001059.
Khairoutdinov, M. F., and K. Emanuel, 2013: Rotating radiative-convective equilibrium simulated by a cloud-resolving model. J. Adv. Model. Earth Syst., 5, 816–825, https://doi.org/10.1002/2013MS000253.
Kieu, C., 2015: Revisiting dissipative heating in tropical cyclone maximum potential intensity. Quart. J. Roy. Meteor. Soc., 141, 2497–2504, https://doi.org/10.1002/qj.2534.
Malkus, J. S., and H. Riehl, 1960: On the dynamics and energy transformations in steady-state hurricanes. Tellus, 12, 1–20, https://doi.org/10.1111/j.2153-3490.1960.tb01279.x.
Muller, C. J., and I. M. Held, 2012: Detailed investigation of the self-aggregation of convection in cloud-resolving simulations. J. Atmos. Sci., 69, 2551–2565, https://doi.org/10.1175/JAS-D-11-0257.1.
Muller, C. J., and S. Bony, 2015: What favors convective aggregation and why? Geophys. Res. Lett., 42, 5626–5634, https://doi.org/10.1002/2015GL064260.
Nakamura, N., and L. Wang, 2013: On the thickness ratio in the quasigeostrophic two-layer model of baroclinic instability. J. Atmos. Sci., 70, 1505–1511, https://doi.org/10.1175/JAS-D-12-0344.1.
Nolan, D. S., E. D. Rappin, and K. A. Emanuel, 2007: Tropical cyclogenesis sensitivity to environmental parameters in radiative–convective equilibrium. Quart. J. Roy. Meteor. Soc., 133, 2085–2107, https://doi.org/10.1002/qj.170.
Ooyama, K., 1969: Numerical simulation of the life cycle of tropical cyclones. J. Atmos. Sci., 26, 3–40, https://doi.org/10.1175/1520-0469(1969)026<0003:NSOTLC>2.0.CO;2.
Pauluis, O. M., 2016: The mean air flow as Lagrangian dynamics approximation and its application to moist convection. J. Atmos. Sci., 73, 4407–4425, https://doi.org/10.1175/JAS-D-15-0284.1.
Pauluis, O. M., and I. M. Held, 2002: Entropy budget of an atmosphere in radiative–convective equilibrium. Part I: Maximum work and frictional dissipation. J. Atmos. Sci., 59, 125–139, https://doi.org/10.1175/1520-0469(2002)059<0125:EBOAAI>2.0.CO;2.
Pauluis, O. M., and S. Garner, 2006: Sensitivity of radiative–convective equilibrium simulations to horizontal resolution. J. Atmos. Sci., 63, 1910–1923, https://doi.org/10.1175/JAS3705.1.
Pauluis, O. M., and F. Zhang, 2017: Reconstruction of thermodynamic cycles in a high-resolution simulation of a hurricane. J. Atmos. Sci., 74, 3367–3381, https://doi.org/10.1175/JAS-D-16-0353.1.
Pearce, R. P., 1998: A study of hurricane dynamics using a two-fluid axisymmetric model. Meteor. Atmos. Phys., 67, 71–81, https://doi.org/10.1007/BF01277502.
Phillips, N. A., 1954: Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level, quasi-geostrophic model. Tellus, 6, 273–286, https://doi.org/10.1111/j.2153-3490.1954.tb01123.x.
Rotunno, R., and K. A. Emanuel, 1987: An air–sea interaction theory for tropical cyclones. Part II: Evolutionary study using a nonhydrostatic axisymmetric numerical model. J. Atmos. Sci., 44, 542–561, https://doi.org/10.1175/1520-0469(1987)044<0542:AAITFT>2.0.CO;2.
Rousseau-Rizzi, R., and K. Emanuel, 2019: An evaluation of hurricane superintensity in axisymmetric numerical models. J. Atmos. Sci., 76, 1697–1708, https://doi.org/10.1175/JAS-D-18-0238.1.
Rousseau-Rizzi, R., and K. Emanuel, 2020: Reply to “Comments on ‘An evaluation of hurricane superintensity in axisymmetric numerical models.’” J. Atmos. Sci., 77, 1893–1896, https://doi.org/10.1175/JAS-D-19-0248.1.
Schecter, D. A., and T. J. Dunkerton, 2009: Hurricane formation in diabatic Ekman turbulence. Quart. J. Roy. Meteor. Soc., 135, 823–838, https://doi.org/10.1002/qj.405.
Shapiro, L. J., 1992: Hurricane vortex motion and evolution in a three-layer model. J. Atmos. Sci., 49, 140–154, https://doi.org/10.1175/1520-0469(1992)049<0140:HVMAEI>2.0.CO;2.
Shapiro, L. J., and H. E. Willoughby, 1982: The response of balanced hurricanes to localized sources of heat and momentum. J. Atmos. Sci., 39, 378–394, https://doi.org/10.1175/1520-0469(1982)039<0378:TROBHT>2.0.CO;2.
Simmons, A. J., and D. M. Burridge, 1981: An energy and angular-momentum conserving vertical finite-difference scheme and hybrid vertical coordinates. Mon. Wea. Rev., 109, 758–766, https://doi.org/10.1175/1520-0493(1981)109<0758:AEAAMC>2.0.CO;2.
Smith, R. K., 1980: Tropical cyclone eye dynamics. J. Atmos. Sci., 37, 1227–1232, https://doi.org/10.1175/1520-0469(1980)037<1227:TCED>2.0.CO;2.
Smith, R. K., M. T. Montgomery, and J. Persing, 2014: On steady-state tropical cyclones. Quart. J. Roy. Meteor. Soc., 140, 2638–2649, https://doi.org/10.1002/qj.2329.
Tang, B., and K. Emanuel, 2010: Midlevel ventilation’s constraint on tropical cyclone intensity. J. Atmos. Sci., 67, 1817–1830, https://doi.org/10.1175/2010JAS3318.1.
Tang, B., and K. Emanuel, 2012: Sensitivity of tropical cyclone intensity to ventilation in an axisymmetric model. J. Atmos. Sci., 69, 2394–2413, https://doi.org/10.1175/JAS-D-11-0232.1.
Wing, A. A., and K. A. Emanuel, 2014: Physical mechanisms controlling self‐aggregation of convection in idealized numerical modeling simulations. J. Adv. Model. Earth Syst., 6, 59–74, https://doi.org/10.1002/2013MS000269.
Wing, A. A., K. A. Reed, M. Satoh, B. Stevens, S. Bony, and T. Ohno, 2018: Radiative–Convective Equilibrium Model Intercomparison Project. Geosci. Model Dev., 11, 793–813, https://doi.org/10.5194/gmd-11-793-2018.
Zhou, W., I. M. Held, and S. T. Garner, 2014: Parameter study of tropical cyclones in rotating radiative–convective equilibrium with column physics and resolution of a 25-km GCM. J. Atmos. Sci., 71, 1058–1069, https://doi.org/10.1175/JAS-D-13-0190.1.
Zhou, W., I. M. Held, and S. T. Garner, 2017: Tropical cyclones in rotating radiative–convective equilibrium with coupled SST. J. Atmos. Sci., 74, 879–892, https://doi.org/10.1175/JAS-D-16-0195.1.
Zhu, H., and R. K. Smith, 2003: Effects of vertical differencing in a minimal hurricane model. Quart. J. Roy. Meteor. Soc., 129, 1051–1069, https://doi.org/10.1256/qj.02.78.
Zhu, H., R. K. Smith, and W. Ulrich, 2001: A minimal three-dimensional tropical cyclone model. J. Atmos. Sci., 58, 1924–1944, https://doi.org/10.1175/1520-0469(2001)058<1924:AMTDTC>2.0.CO;2.
