The Maximum Intensity of Tropical Cyclones in Axisymmetric Numerical Model Simulations

a. Governing equations

The model equations are written in cylindrical coordinates (r, ϕ, z), although, by assumption, no variation in ϕ is permitted herein. There are seven time-dependent variables: velocities in the radial, azimuthal, and vertical directions (u, υ, w); perturbation nondimensional pressure π′; potential temperature θ; mixing ratio of water vapor q_υ; and mixing ratio of liquid water q_l. The governing equations for these variables are as follows:

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Overbars refer to a one-dimensional (vertical) reference profile that is in hydrostatic balance [d/dz = −g/(c_p)], and primes refer to perturbations from this reference state. The definitions for π and θ are customary: π ≡ (p/p₀)^R_d/c_p and θ ≡ T/π, where p₀ is a reference pressure, c_p is the specific heat of dry air at constant pressure, R_d is the gas constant for dry air, and T is absolute temperature. Virtual potential temperature includes the effects of liquid water: θ_υ ≡ θ(1 + q_υR_υ/R_d)/(1 + q_υ + q_l), where R_υ is the gas constant for water vapor. Other symbols are defined as follows: f is the Coriolis parameter; g is gravitational acceleration; the D symbols represent tendencies from turbulent motions (described below); N represents upper-level Newtonian damping used to eliminate vertically propagating gravity waves, following RE87 (p. 546); R is the term from RE87 (p. 546) that mimics radiative cooling throughout the domain; q̇_cond is the rate of condensation/evaporation between vapor and liquid; ρ_d is density of dry air, determined using the ideal gas law, ρ_d = p₀π^c_υ/R_d[R_dθ(1 + q_υR_υ/R_d)]⁻¹; and V_t is the terminal fall velocity of liquid water. The symbols Θ₃ and ϵ are associated with dissipative heating and are explained below.

The remaining undefined symbols—Π₁, Π₂, Π₃, Θ₁, and Θ₂—are associated with the conservation of mass and internal energy in moist flows. These variables can be formulated in two ways. One yields an approximate equation set that is traditionally used for nonhydrostatic cloud models and is very similar to the equations used by RE87:

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where L_υ is the latent heat of vaporization and c_υ is the specific heat of dry air at constant volume. With this formulation, hereafter referred to as the “traditional equation set,” the model cannot conserve mass and internal energy. The second formulation, derived by Bryan and Fritsch (2002) and hereafter referred to as the “conservative equation set,” allows the total mass and internal energy to be conserved in a reversible moist environment, and the appropriate formulations are

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where variables for the mixture of moist, saturated air are defined as follows:

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and where c_pυ and c_υυ are the specific heats of water vapor at constant pressure and volume, respectively, and c_l is the specific heat of liquid water. Using these variables, a governing equation for total mass can be derived by using the ideal gas law, the definitions of mixing ratios, and using (4)–(7); for the conservative equation set, this yields

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where ρ_t ≡ ρ_d + ρ_υ + ρ_l is total density [the sum of the densities of dry air (ρ_d), water vapor (ρ_υ), and liquid water (ρ_l)] and D_qt represents the tendency from turbulence (which, we note, includes the surface flux of water vapor). It can easily be shown that total mass inside the domain is conserved in the absence of surface precipitation and surface water vapor flux, and that the so-called precipitation mass sink effect [e.g., studied by Qiu et al. (1993) and Lackmann and Yablonksy (2004)] is included in this model when using the conservative equation set. We also note that the conservative equation set reduces to the traditional equation set by setting c_pυ = c_υυ = c_l = R_υ = Π₂ = Π₃ = 0.

The diffusive terms (D) in (1)–(7) are tendencies from a parameterization of unrepresented motions (i.e., turbulence). Our formulation is similar to the one used by RE87, although we utilize the deep anelastic equations (instead of the incompressible Boussinesq equations) during our derivation of these terms. These tendencies are formulated as follows:

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where χ represents one of the model’s scalars (θ, q_υ, or q_l). The stresses (τ) and fluxes (F) are parameterized as in RE87 (p. 545 for the model’s interior, and p. 547 for the surface). Assuming steady, homogeneous turbulence, we derive a turbulence kinetic energy budget for these equations:

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where ν is an eddy viscosity, subscripts h and υ refer to effects from unrepresented horizontal and vertical eddy fluxes respectively, N_m² is the squared Brunt–Väisälä frequency, ϵ is the rate of dissipation of kinetic energy at molecular scales, and S² is deformation given by

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We split the dissipation into two components: one accounting for dissipation of horizontal turbulence motions (ϵ_h), and one accounting for dissipation of vertical turbulence motions (ϵ_υ). As in RE87, we assume on dimensional grounds that ϵ_h = ν_h³/l_h⁴ and ϵ_υ = ν_υ³/l_υ⁴, where l_h and l_υ are length scales of the most energetic turbulent motions (i.e., the energy-containing eddies). From these assumptions, we derive the diagnostic formulas for eddy viscosity:

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We set ν_υ = 0 if (S_υ² − N_m²) < 0 (i.e., if the Richardson number is greater than unity).

The formulation of N_m² for unsaturated air is given by N_m² = (g/θ_υ)∂θ_υ/∂z and for saturated air is given by

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where q_s is the mixing ratio at saturated equilibrium, and Γ_m is the moist-adiabatic lapse rate, which for the conservative equation set is

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To improve energy conservation, dissipative heating can be included in this model. This effect is excluded in most nonhydrostatic numerical models (including the RE87 model), and this option is available herein by setting ϵ = 0 in (4)–(5). To include this effect with the traditional equation set we use Θ₃ = 1/(c_pπ) and for the conservative equation set we use Θ₃ = c_υ/(c_pc_υmπ). For ϵ, we use the turbulence kinetic energy balance equation, (15), for the interior of the model domain. At the surface, we utilize the specifications for surface stress and surface fluxes and, using the same assumptions as Bister and Emanuel (1998), we find the dissipation rate at the surface is

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where C_D is the drag coefficient; Δz is the vertical grid spacing; u₁ and υ₁ are the radial and azimuthal velocities, respectively, at the lowest model level; and F_z^θ and F_z^q_υ are the surface fluxes for θ and q_υ, respectively. Our formulation for dissipation rate is slightly different from that in previous studies (e.g., Bister and Emanuel 1998; Zhang and Altshuler 1999) because we account for the buoyancy flux [third term on the left side of (15) and the second term of (20)], which typically increases ϵ in tropical cyclone eyewalls.

The only significant term associated with energy conservation that has been excluded from this model is one related to sedimentation of liquid water entropy [e.g., last term on the right side of (4.6) in Ooyama (2001)]. This term is excluded herein, which is a traditional assumption in numerical models, because of uncertainties about the best way to formulate the term realistically and yet be computationally efficient [see, e.g., Walko et al. (2000) for a discussion of the difficulty in incorporating this effect realistically]. We are currently investigating this effect further and plan to report our findings in a future study.

b. Numerical methods

The time integration scheme is third-order Runge–Kutta using split-explicit integration for the acoustic modes, following Wicker and Skamarock (2002). To improve the stability and accuracy of the split-explicit time integration method, we include a weak three-dimensional divergence damper on the acoustic time steps, and we integrate θ on the small time steps, following Skamarock and Klemp (1992).

We write the advection terms in flux form for a variable α as follows:

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The first two terms on the right side are discretized using the fifth-order flux-form scheme by Wicker and Skamarock (2002). To conserve total water, we apply a positive-definite scheme [described in Bryan et al. (2006)] to the advection of q_υ and q_l.

Unlike RE87, we use a closed boundary condition (a rigid wall) at the external lateral boundary, even though an open boundary condition option is available in this code. As compared with an open boundary condition, which is difficult to constrain over long-term integrations, we find a closed boundary produces more stable (i.e., steady) solutions for long-term (>10 day) simulations. To prevent reflection of gravity waves, we apply Newtonian damping to u, υ, and w near this boundary. By comparing it with simulations that use open boundary conditions, we find no significant differences for t < 10 days, but generally more steady conditions in simulations with the closed domain for t > 10 days.

The method to determine q̇_cond was described by Bryan and Fritsch (2002, p. 2920). As shown in their study, this numerical model does not precisely conserve total mass and energy, partly because of numerical reasons. However, for simulations using the conservative equation set, we find the artificial loss of mass and energy to be several orders of magnitude lower than for simulations using the traditional equation set. At the end of the simulations (typically at t = 12 days), the artificial loss of mass and energy is less than 0.05% of the initial values.

c. Methodology

The initial conditions are identical to those used by RE87. For the base-state environment, we use their “model neutral” sounding. For our higher-resolution simulations, we interpolate their thermodynamic profile (dots in Fig. 1) to the new grid (lines in Fig. 1), and conditions below their lowest model level are extrapolated downward, as shown in this figure. The sea surface temperature (T_s) is 26.13°C for all simulations.

The domain is the same size (1500 km × 25 km) as that used by RE87. Certain settings in the numerical model are varied, depending on the test being conducted, including the grid spacing (Δr for the radial direction, Δz for the vertical direction), the time step (Δt), and a priori settings for the turbulence parameterization (specifically, l_h and l_υ). A summary of model configurations used in this article is presented in Table 1, and further details are provided at appropriate locations later in the text.

Unless stated otherwise, all simulations use the conservative equation set and include dissipative heating, and the surface exchange coefficient for entropy (C_E) is equal to that for momentum (C_D). (We reiterate that the surface fluxes and stress are formulated the same way as RE87.) The tendency term in the potential temperature equation that mimics radiative cooling (R) is capped at 2 K day⁻¹ following Experiment J by RE87, which is the same methodology used in all simulations by Persing and Montgomery (2003, hereafter PM03). For all simulations in this article, the parameterization of microphysical processes is identical to that in RE87, and we use V_t = 7 m s⁻¹ unless stated otherwise. Ice processes are neglected for the sake of simplicity and because we find they have a small effect on maximum simulated intensity.

In all of our simulations, an approximately steady state is achieved by ∼6 days into the simulation. We quantify the intensity of the simulated tropical cyclones by υ_max, which is the maximum value of υ (from any grid point in the domain) averaged every time step between t = 8 and 12 days. Typically, υ_max is located at the top of the boundary layer (at ∼1 km). For other analyses herein, we compute average fields using hourly output from t = 8 to 12 days.

We have compared output from this new model with output from the RE87 model. We find that the RE87 model produces more intense tropical cyclones (by about 10%, as measured by υ_max). Output from our new model also tends to be steadier over time. Details are provided in the appendix. We attribute these differences primarily to improved governing equations and numerical techniques in the new model.

a. Grid spacing

Several studies have shown that numerically simulated intensity increases as the horizontal grid spacing decreases (e.g., Braun and Tao 2000; PM03; Yau et al. 2004; Davis et al. 2008; Hill and Lackmann 2009), at least for grid spacing ≥1 km. Presumably, the ability to better resolve nonhydrostatic processes in the eyewall is the primary reason for this sensitivity.

To determine a nominal grid spacing for this study, we conduct a resolution sensitivity test where we decrease the grid spacing incrementally until a converged value for υ_max is obtained (i.e., until υ_max stops changing). For these tests, we configure the model to have minimal diffusion by specifying small values for the turbulence length scales: specifically, we use l_h = 187.5 m and l_υ = 50 m. From these simulations, we find a converged value for υ_max of ∼100 m s⁻¹ and this magnitude is achieved for approximately Δr = 1000 m and Δz = 250 m (Table 2); further decreases in either Δr or Δz yield essentially the same intensity. We note that this intensity is significantly larger than the maximum observed value (∼70 m s⁻¹); we analyze and explain this discrepancy later in this article.

Regarding the horizontal grid spacing, we find, similar to previous studies, that the tropical cyclone becomes more intense as the eyewall becomes smaller and better resolved. For Δr of order 1 km or less, the eyewall is represented by at least 8 grid increments; further decreases in Δr do not change the physical width of the eyewall.

Regarding the vertical grid spacing, we find an increase in intensity as Δz increases for this model.² In this case, the boundary layer becomes poorly resolved, and becomes artificially deeper, for Δz ≥ 500 m.

Based on these results, we use Δr = 1000 m and Δz = 250 m for all results hereafter. Based on further tests (not shown), we find it sufficient to use constant Δr in the inner-core region only, with stretched grid spacing beyond. Specifically, we use Δr = 1 km for r < 64 km, and then Δr increases gradually to a maximum value of 16 km at r = 1500 km. Comparison against a simulation with Δr = 1 km everywhere yields the same results, so the stretched grid is used hereafter to reduce computational expense.

We remind readers that simulations with an axisymmetric numerical model (and any two-dimensional model) cannot explicitly produce realistic turbulence. Rather, the effects of all turbulence must be included via parameterization. (See the discussion by RE87, p. 544, for more details.) Consequently, the grid spacing used herein cannot be expected to produce a converged solution in a three-dimensional numerical model, where turbulence is parameterized much differently, and can actually be resolved given sufficient resolution [probably with grid spacing of order 100 m, following the theoretical arguments by Bryan et al. (2003)]. Results by Rotunno et al. (2009, manuscript submitted to Bull. Amer. Meteor. Soc.) confirm this conclusion.

b. Turbulence length scales

One of the most uncertain aspects of axisymmetric numerical models is the parameterization of unrepresented motions. In addition to unresolved subgrid-scale motions, axisymmetric numerical models cannot resolve any three-dimensional motions, such as mesovortices in the eye/eyewall, boundary layer roll vortices, upper-level asymmetric outflow jets, vortex Rossby waves, and so on. Any nonaxisymmetric motions must be viewed as turbulence in an axisymmetric model and must be incorporated through parameterization; see section 2b in RE87 for more details. In this model, turbulence is included via the D terms in (1)–(7). The turbulence closure (described in section 2) allows for these tendencies to be larger when the local deformation is larger and/or when the local static stability is smaller. Additionally, the closure contains two unknown length scales: one for horizontal turbulence processes (l_h) and one for vertical turbulence processes (l_υ). The D terms are proportional to l_h and l_υ, which are specified a priori in this model.

There is no quantitative theoretical guidance for how to set l_h and l_υ in an axisymmetric model. RE87 used l_h = 3000 m and l_υ = 200 m in their simulations, which they determined by trial and error, and by subjective evaluation of model output. PM03 used l_h = 750 m and l_υ = 200 m for their standard “4x” setup. A different value for l_h was used by PM03 because the RE87 model’s code specifies l_h in terms of a coefficient times the radial grid spacing (specifically, as 0.2 × Δr), and PM03 used smaller Δr as compared with RE87. Consequently, without any compensating change in the coefficient, RE87’s code has a fundamentally different representation of horizontal turbulence effects as Δr changes, where the turbulent tendencies are decreased with higher resolution. However, l_h should be interpreted as a physical parameter in axisymmetric numerical models, because three-dimensional turbulent motions cannot be resolved at any grid spacing; thus, the turbulence length scales should be kept constant for resolution sensitivity tests (as was done in section 3a).

Here, we evaluate the sensitivity of υ_max to l_h and l_υ using fixed model parameters (listed as “default” in Table 1). Results, in terms of υ_max, are shown in Fig. 2. There is a strong sensitivity to l_h, but essentially no sensitivity to l_υ. For l_υ = 200 m (the value used by both RE87 and PM03), there is nearly a factor of 2 increase in υ_max when l_h is decreased from 3000 (the value used by RE87) to 750 m (the value used by PM03). A similar response to changes in l_h and l_υ (for fixed grid spacing) was reported by PM03 (in their Table 3).

We find from analysis of tendencies in the model’s governing equations (not shown) that with l_h less than ∼1000 m the tendency from horizontal turbulence is negligible compared to other terms in the governing equations. So, as l_h → 0 the flow becomes essentially inviscid (in the radial direction). To investigate the effects of a truly inviscid model setup, we conduct additional simulations with further decreases in l_h, including l_h = 0. Results show that υ_max (solid line in Fig. 3) asymptotically approaches 113 m s⁻¹ as l_h → 0. For our default model setup, there is additionally a flow-and scale-dependent numerical diffusion that is inherent to the fifth-order advection scheme used herein; this term prevents features from collapsing to a scale smaller than ∼6 times the grid length. To check the effects of this diffusive term, we also ran a set of simulations (not shown) that uses sixth-order advection, which has no implicit diffusion; these simulations have the same response, although υ_max asymptotically approaches a slightly lower value (103 m s⁻¹).

These results are somewhat similar to results in Emanuel (1989), who used a balanced axisymmetric model expressed in angular momentum coordinates. Starting with very small diffusion in the lateral direction (his experiment C1), he found essentially no change in υ_max for a factor-of-3 increase in l_h (his experiment A), and a slight (∼10%) decrease in υ_max for a further factor-of-3 increase in l_h (his experiment C2, see Fig. 5 in his article). The same behavior is seen in our model for l_h ≈ 100 m; K. A. Emanuel (2008, personal communication) has found that further decreases in l_h (to near zero) in his model result in a slight decrease in υ_max, although this may be attributable to numerical problems with low diffusion or to other differences in these two models.

What is notably different from the results reported by Emanuel (1989) is the maximum intensity, which is near the theoretical PI (∼60 m s⁻¹) in his study. In our case, PI is also roughly 60 m s⁻¹ (see PM03), but υ_max significantly exceeds this value when l_h < 3000 m. This ability to significantly exceed the theoretical PI for small l_h appears to be related primarily to the existence of unbalanced flow (which is not possible in Emanuel’s model). In our model, we find that υ_max is close to the maximum gradient wind speed (υ_g,max) for l_h > 1500 m (Fig. 3), but υ_max notably exceeds υ_g,max as lateral diffusion becomes small. Because of the complexity of this issue, we will address it in a separate article.

Of more interest to this study is that υ_max greatly exceeds the maximum value that has ever been observed for this environment (∼70 m s⁻¹, see the beginning of section 3). This only happens in our model when l_h is small (i.e., when the flow is essentially inviscid in the radial direction). To understand this result further, we note that radial gradients of all model fields are strongly affected by l_h. Specifically, larger l_h increases the turbulent diffusion, which yields weaker radial gradients (in both scalar and velocity fields). For example, we show in Fig. 4 the angular momentum (M ≡ rυ + fr ²/2) and the pseudoadiabatic equivalent potential temperature (θ_e) (Bryan 2008) at the top of the boundary layer. The large differences in θ_e in the eye of these simulations (i.e., for r < 10 km in Fig. 4b) are not relevant to the maximum intensity in these simulations, as discussed below. More important are the radial gradients near the location of υ_max (dots in Fig. 4), which are largest for small values of l_h; this is because the eyewall of hurricanes is strongly frontogenetic, as discussed by Emanuel (1997). For small l_h, this frontogenetic zone collapses to a small scale (approximately 8 km wide). But the diffusion terms are frontolytic; thus, as l_h is increased, and lateral diffusion becomes a significant term in the governing equations, the diffusive terms produce weaker gradients in both scalars and momentum³ (Fig. 4). Consequently, with large l_h, M from the environment has not been drawn as far into the center of circulation as it can be with smaller l_h, and thus υ_max is smaller. The weaker radial gradients of scalars are also consistent with weaker intensity because of approximate thermal-wind balance; that is, because υ ≈ 0 at the top of the troposphere, then weaker shear (consistent with a weaker radial gradient in entropy by thermal wind) must mean weaker intensity.

Based on the preceding arguments, one might wonder whether further increases in resolution would lead to even greater υ_max, because even larger gradients in M and θ_e could be resolved. Our sensitivity studies do not support such a conclusion. Using l_h = 0, υ_max is essentially the same for any Δr < 4 km (Table 3). In fact, we find that the width of the eyewall, and hence the width of the frontogenetic zone, converges to ∼8 km for Δr < 4 km (Table 3, where the eyewall is defined as w ≥ 0.5 m s⁻¹). The processes that prevent the eyewall from collapsing to an infinitesimal scale (as might be expected for zero turbulent diffusion in a frontogenetical zone) would be an interesting topic for future study, although we surmise that the finite depth of the boundary layer in this model likely plays a role in dictating this finite updraft width.

To help to explain why υ_max is bounded in our simulations, we draw upon the analytic study of tropical cyclone structure and intensity by Emanuel (1986). By assuming gradient-wind balance, hydrostatic balance, and moist slantwise neutrality in the free atmosphere, Emanuel (1986) derived a relationship between the radial gradient of moist entropy (s) and the radial gradient of M in the eyewall [see his Eq. (13)]. If we assume that both s and M could collapse to discontinuities in the frontogenetic region of the eyewall, then we can integrate Emanuel’s Eq. (13) across the discontinuities, which yields

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which is valid only in the eyewall (at the radius of maximum winds), where Δs is the change in s across the discontinuity, T_B is temperature at the location of υ_g,max, and T_out is the “outflow temperature” [which is the temperature, at large radius, along a trajectory that passes through υ_g,max (see Bister and Emanuel 1998, p. 237)]. The important conclusion to be drawn from (22) is that υ_g,max must be finite because Δs must be finite. Using estimated values of T_B, T_out, and Δs from our weak-diffusion simulations, (22) predicts υ_g,max ≈ 100 m s⁻¹, which is consistent with υ_g,max from these simulations (Fig. 3). One might wonder what limits the value of Δs, and thus what ultimately limits υ_g,max; this could be addressed in a future study of theoretical PI.

Returning now to the dependence of model-simulated υ_max on l_h, our analysis probably explains other simulations of very intense tropical cyclones with axisymmetric numerical models. For example, Hausman et al. (2006) report υ_max exceeding 130 m s⁻¹ for T_s = 28°C. [The maximum observed surface winds for T_s = 28°C in the study by DeMaria and Kaplan (1994) was ∼70 m s⁻¹.] Hausman et al. (2006) had no parameterization for horizontal turbulence in their model (other than the implicit filter in their model’s numerics, which is similar to the high-order diffusion used in our model). Their results are consistent with ours, in the sense that essentially inviscid flow leads to very large intensities that are much greater than observed maximum intensities.

Our analysis also explains why PM03 found an increase in intensity with decreasing grid spacing using the RE87 model. As discussed at the beginning of this subsection, their changes in Δr were accompanied by changes in l_h, the latter of which we find is important for maximum simulated intensity. In contrast, PM03 implicated the existence of high-entropy air in the low-level eye (e.g., r < 10 km in Fig. 4b); this feature has since been shown to be unimportant for maximum intensity in axisymmetric numerical models [see Bryan and Rotunno (2009) for a detailed analysis], and is thus not discussed further.

One might wonder whether horizontal diffusion of scalars or momentum is more important for changes in υ_max. To investigate, we conduct a series of simulations that calculate different horizontal eddy viscosities for scalars and momentum using a specified horizontal length scale for scalars (l_h^s) and one for momentum (l_h^m). From a broad set of simulations (not shown here because of space limitations), we find that both horizontal diffusion of scalars and momentum is important. If one of these length scales is fixed, and the other is decreased gradually to zero, we again find that υ_max asymptotically approaches a constant, although this constant is different for each experiment. Overall, we find that υ_max is more sensitive to changes in l_h^m than to changes in l_h^s, although we reiterate that υ_max is sensitive to changes in either length scale.

Before proceeding to other sensitivities in this numerical model, we briefly investigate the hurricane structure as the turbulence coefficients are changed. We show in Fig. 5 the azimuthal and radial flow fields from simulations with three different configurations for turbulence. For relatively large l_h and l_υ (Fig. 5a), the value of υ_max is 48 m s⁻¹, which is less than our estimate for observed maximum intensity (70 m s⁻¹). The maximum value of radial inflow is 12 m s⁻¹, which is comparable to values reported in observed tropical cyclones (e.g., LeeJoice 2000). Overall, for these settings, the model produces features that are reasonably consistent with observational analyses of strong tropical cyclones.

Using smaller l_h, but the same l_υ (Fig. 5b), the model yields a much smaller radius of maximum winds (as compared to that from the previous model setting). In this case, υ_max is 86 m s⁻¹, which is notably larger than our estimate for observed maximum intensity (70 m s⁻¹); the large discrepancy (∼15 m s⁻¹) raises concerns about the realism of this simulation. Furthermore, consistent with larger values of azimuthal velocities, the maximum value of radial inflow is also larger. In this case, the maximum value of radial inflow is ∼25 m s⁻¹, which is about twice as large as any value reported by LeeJoice (2000), but is comparable to observations of intense tropical cyclones by Bell and Montgomery (2008) and Kepert (2006a,b). The total depth of radial inflow is ∼2 km in this simulation, which is slightly higher than, although comparable to, the structure documented in these observational studies. Overall, the structures produced by this model setting can be characterized as extreme, and perhaps unrealistic (at least for υ_max), compared to available observations.

Using the same (small) value of l_h, but much smaller l_υ, the maximum azimuthal velocity and the radius of maximum winds remain essentially unchanged (Fig. 5c). However, the depth of the radial inflow is much smaller compared to the previous case. Furthermore, the magnitude of maximum radial inflow is 40 m s⁻¹, which is much higher than has been shown in the observational studies cited above. A strong radial outflow (>20 m s⁻¹) exists at z ≈ 1.5 km, which is also much greater than has been previously documented (e.g., Bell and Montgomery 2008). These structures are clearly not representative of natural tropical cyclones. In some simulations with low l_h and l_υ (although not in this case), inertial instability exists in the eyewall, which is another unnatural feature that can develop in weak-turbulence simulations.

To summarize, the intensity and structure of tropical cyclones in axisymmetric numerical models is very sensitive to the specification of turbulence intensity. This means that large uncertainty is an inherent characteristic of axisymmetric numerical model simulations, considering that there is currently no quantitative theoretical guidance with which to specify turbulence effects (particularly in the radial direction). Our analysis further reveals that unnatural features may be produced in an axisymmetric model. As an example, υ_max of 113 m s⁻¹ (Fig. 3) is clearly unnaturally high as compared to the maximum observed intensity for this sea surface temperature (∼70 m s⁻¹). Consistent with such large azimuthal velocities are unnaturally large values of radial inflow (e.g., |u| > 40 m s⁻¹).

Based on these results, we conclude that nearly inviscid axisymmetric model setups (i.e., l_h → 0) cannot reproduce realistic tropical cyclones. Furthermore, if l_h is set to a moderately large value (e.g., 3000 m, which was used by RE87), then no other setup of the numerical model that has been studied herein yields an intensity larger than the climatologically observed maximum of ∼70 m s⁻¹.

It follows that turbulence in the radial direction limits axisymmetric tropical cyclone intensity. As discussed earlier, this is because turbulent diffusion weakens the radial gradient of angular momentum in the eyewall (which prevents large values of environmental angular momentum from being drawn to small radius) and also weakens radial gradients of scalars (which consistently means weaker intensity by consideration of thermal-wind balance). Further analysis with three-dimensional models are needed to verify these conclusions, particularly because realistic three-dimensional turbulent flows (e.g., eye/eyewall mesovortices, boundary layer roll vortices, upper-level asymmetric outflow jets, etc.) may act differently than the way they are parameterized in this model.

One might wonder whether we can determine values for l_h and l_υ that yield reasonably realistic hurricanes as compared to observations. Based on the estimated observed maximum intensity of 70 m s⁻¹, as well as comparisons of maximum radial inflow to observations, it seems that l_h ≈ 1500 m and l_υ ≈ 100 m are appropriate. Additionally, observations of azimuthally average properties can be useful in this regards. In an analysis of a category-5 tropical cyclone, Montgomery et al. (2006) found the radial gradient of moist entropy in the eyewall to be −1.7 × 10⁻³ m s⁻² K⁻¹; the same value occurs in our simulations if l_h = 1500 m. Despite this encouraging comparison between model output and observations, we cannot say with confidence that these values of l_h and l_υ will be appropriate for all cases.

As discussed in section 1, a primary goal of this study is to identify the model settings that allow for maximum possible intensity in numerical models. Small values of l_h clearly lead to this goal. Thus, we retain small values of l_h for the sensitivity tests that follow, although we also present results with l_h = 1500 m as a likely appropriate setting for the comparison against observations.

c. Ratio of surface exchange coefficients

Numerical simulations (e.g., Rosenthal 1971; Braun and Tao 2000) and theoretical analysis (e.g., Emanuel 1986, 1995) have demonstrated a large sensitivity of maximum intensity to the ratio of surface exchange coefficients for entropy and momentum (C_E/C_D). Observational studies have found that this ratio varies between roughly 0.5 and 1.5, with the lowest values being appropriate for near-surface wind speeds of ∼25 m s⁻¹ (e.g., Black et al. 2007). However, an appropriate value for large wind speeds (of order 50 m s⁻¹) remains uncertain.

To investigate this sensitivity in the numerical model, we set C_E = 1.2 × 10⁻³, based on recent observational studies (e.g., Drennan et al. 2007). Because C_D seems to be a more uncertain parameter (e.g., Powell et al. 2003; French et al. 2007), we vary this parameter across a broad range of values. These lower values of both C_E and C_D (cf. the default formulations from RE87) result in much slower evolution, and sometimes a steady cyclone does not develop by t = 12 days. Consequently, we double the intensity of the initial vortex (to 30 m s⁻¹) to hasten initial development for this sensitivity test. (These changes to C_E, C_D, and initial vortex intensity were made for this subsection only.) Results for C_E/C_D between 0.25 and 2 are shown in Fig. 6.

Under the assumption of inviscid flow above the boundary layer, Emanuel (1986, 1995) derived a theoretical relationship, υ_max ∼ (C_E/C_D)^1/2. For a low value of l_h in the numerical model (solid line in Fig. 6), we find a similar result: υ_max ∼ (C_E/C_D)^0.44. For even lower values of l_h (not shown), we find even closer correspondence to theory; for l_h = 94 m, we find υ_max ∼ (C_E/C_D)^0.53. Thus, the model results trend toward the theoretical results as turbulence intensity decreases (i.e., as inviscid flow is approached).

In contrast, with higher values of l_h, υ_max shows a clearly different dependence. For l_h = 1500 m (dashed line in Fig. 6), υ_max varies as (C_E/C_D)^0.34. For l_h = 3000 m (dotted line in Fig. 6), υ_max varies as (C_E/C_D)^0.31 for C_E/C_D ≤ 1.25 and υ_max is independent of C_E/C_D for C_E/C_D ≥ 1.5. Given that nonzero turbulence is needed in the model to reproduce realistic hurricane structures (see section 3b), these results suggest that viscous terms are needed for an analytical theory of maximum intensity that is appropriate for natural tropical cyclones. Our results might also explain why Braun and Tao (2000) did not find close correspondence between their high-resolution three-dimensional simulations and Emanuel’s theoretical model; that is, turbulent diffusion in their model (resolved and/or parameterized) must be relatively important.

As discussed earlier, the approximate maximum intensity of observed tropical cyclones is roughly 70 m s⁻¹ for this environment. For l_h = 375 m, C_E/C_D needs to be less than 0.5 to match this intensity; this seems too low relative to observed values (e.g., Powell et al. 2003; Black et al. 2007), although it could be argued that such low values have never been observed because of difficulties measuring exchange coefficients in high wind speeds. For l_h = 1500 m, υ_max matches the maximum observed intensity for C_E/C_D ≈ 0.75; the same conclusion was drawn by Emanuel (1995). For l_h = 3000 m, the model cannot reproduce maximum observed intensity, which suggests that this specification of turbulence intensity is too extreme.

Although the formulation of surface exchange coefficients has been studied often in recent observational campaigns, these results suggest that turbulence in the radial direction is a crucially important parameter that should also be studied further. Indeed, assuming l_h = 1500 m is an appropriate value, then υ_max from this model changes by only ∼25% if the ratio C_E/C_D is doubled from 0.5 to 1.0 (Fig. 6). In contrast, υ_max can change by 100% for values of l_h that have been used in published literature (Fig. 2). Further comparison between model results and observations could be undertaken to help to constrain the value of l_h for axisymmetric models. Additional analyses of the axisymmetric structure of intense tropical cyclones in steady state—such as the analyses presented by Bell and Montgomery (2008)—would be valuable for this purpose. Additionally, a similar study with a three-dimensional numerical model should be conducted to check whether these results are sufficiently general.

d. Liquid water fall velocity

We now investigate sensitivity to the specification of terminal fall velocity for liquid water, V_t. The scheme used herein is the same as that used by RE87, and it is quite simple: if q_l exceeds 1 g kg⁻¹, then this liquid is assumed to fall at V_t, which by default is 7 m s⁻¹. This approach is somewhat unrealistic, but it allows us to document the fundamental response of υ_max to the fall velocity of condensate in an easily understandable manner.

Results are shown in Fig. 7 for simulations using l_h = 375 m (solid) and l_h = 1500 m (dashed). We note that this model’s governing equations allow for exact reversible thermodynamics when V_t = 0 (in the absence of turbulence effects). This capability is not available in most atmospheric numerical models, which typically neglect terms in the thermodynamical equation (see Bryan and Fritsch 2002).

For reversible thermodynamics (V_t = 0), the weakest intensity is produced. As V_t is increased, υ_max increases, and the intensity for large V_t is about 60% larger than the reversible case. We also ran simulations (not shown) in which condensate is immediately removed from the atmosphere when q_l exceeds 1 g kg⁻¹; this is analogous to a pseudoadiabatic process, in which condensate is assumed to immediately fall out from air parcels upon formation. In these latter simulations, υ_max is the same as simulations with large V_t (Fig. 7).

These results are generally consistent with those from previous studies (e.g., Wang 2002a; Hausman et al. 2006), in the sense that larger fall velocities yield greater intensities. We have also conducted simulations (not shown) with more complex specifications for terminal fall velocity in which V_t varies proportionally to q_l, and also simulations that incorporate ice microphysics. From these simulations, we draw the same overall conclusion concerning the correlation between υ_max and V_t (in which we use a characteristic value of V_t, such as an average value in the eyewall).

A qualitative consideration of buoyancy helps explain this result. Specifically, larger (positive) buoyancy in a column yields lower perturbation pressure at the bottom of the column;⁴ stronger near-surface winds are, of course, consistent with lower pressure. In the simulations with small V_t, there is a great deal of condensate in the column, which contributes to lower buoyancy, and is thus consistent with weaker intensity. For V_t → ∞, there is no condensate in the column, and thus buoyancy is comparatively higher, which is consistent with stronger intensity. This line of reasoning is supported by the analytic study by Emanuel (1988), who found that an assumption of pseudoadiabatic thermodynamics (V_t → ∞) yielded much stronger tropical cyclones (by ∼25 mb, in terms of minimum central pressure) than an assumption of reversible thermodynamics (V_t = 0). Emanuel (1988) similarly concluded that water loading plays a key role in reducing the intensity in the reversible case.

To provide further guidance for the development and evaluation of PI theories, we examine the total moist entropy (s) from our simulations. In the absence of turbulence effects, s should be conserved following a parcel. To determine an appropriate mathematical formulation of s, a further assumption must be made about the liquid water fall velocity. On one extreme, assuming V_t = 0, an exact expression for s in reversible conditions—hereafter s_r—can be derived:

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(e.g., Emanuel 1994), where H is relative humidity and p_d is the partial pressure of dry air. On the other extreme, if liquid water is immediately removed upon formation (i.e., for V_t → ∞), then a highly accurate expression for s in pseudoadiabatic conditions—hereafter s_p—can be derived:

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(Bryan 2008), where L₀ = 2.555 × 10⁶ J kg⁻¹ is a constant.

We show the distribution of s for three simulations in Fig. 8, where s_r is shown on the left and s_p is shown on the right. In all panels, the trajectory for the parcel that passes through υ_max is shown as a thick line. As expected, s_r is approximately conserved in the free atmosphere (i.e., above the boundary layer) in the simulation with V_t = 0 (Fig. 8a), as revealed by the near equivalence of the trajectory and a contour of s_r. In contrast, s_p is clearly not conserved in this case (Fig. 8b), especially when the parcel reaches midlevels and q_l is relatively large. For V_t = 7 m s⁻¹ (which is the value used by RE87 and PM03), neither s_r nor s_p are conserved following a parcel (Figs. 8c,d). Consequently, this case would be difficult to analyze analytically, because neither the reversible nor the pseudoadiabatic assumption is truly applicable. For V_t = 20 m s⁻¹, s_r is clearly not conserved following a parcel, as expected (Fig. 8e); however, s_p is conserved well along the trajectory (Fig. 8f), indicating that liquid water is removed sufficiently quickly such that the pseudoadiabatic assumption can be invoked for this case.

These examples reveal that a thermodynamical constraint that is suitable for analytical study can probably only be made for one of the extreme situations (i.e., either reversible or pseudoadiabatic thermodynamics): it seems unlikely that a conserved variable could be formulated for the V_t = 7 m s⁻¹ case. If the goal is to study the maximum possible intensity of tropical cyclones, then clearly the pseudoadiabatic assumption should be made. Of course, this state would never be realized in natural tropical cyclones, because the fall velocity of condensate is of order 5 m s⁻¹ (for liquid condensate, but smaller for snow). Consequently, there is a dilemma that is analogous to the choice for turbulence intensity (section 3b); that is, the pseudoadiabatic assumption may yield the maximum possible intensity, but this assumption is clearly not appropriate for natural tropical cyclones.

One might wonder whether the environmental sounding used for these simulations has some affect on these results. The sounding used herein was generated by RE87 to be approximately neutral to convection in their modeling study. With the different governing equations and resolution of this model, it is conceivable that a different result might be obtained for a truly moist-neutral sounding appropriate for our model. Furthermore, by changing the physics of the model from reversible (V_t = 0) to pseudoadiabatic (V_t → ∞), the fixed model atmosphere is clearly no longer neutral to convection across these tests. To investigate, we create two exactly moist-neutral soundings using the model’s equations: one for the reversible case (solid line in Fig. 9) and one for the pseudoadiabatic case (dashed line in Fig. 9). Both soundings have exactly zero CAPE under their respective thermodynamical assumption. To be as comparable as possible to the control simulations, we set the surface θ_e to be identical to that in the control sounding, and we use a similar tropopause height (15 km). Results using l_h = 375 m are listed in Table 4. Overall, the same conclusion is obtained: reversible thermodynamics yields the weakest intensity and pseudoadiabatic thermodynamics yields the strongest intensity, although for these new soundings the difference is greater. We are unsure, at this time, why the difference in intensity is much larger than that found by theoretical estimates (e.g., Emanuel 1988), which could be a topic for future study.

Finally, we note that these results reveal potential implications for intensity change forecasting, as well as for NWP model development. Specifically, modest changes in V_t can result in significant changes in tropical cyclone intensity. This is especially the case for V_t < 5 m s⁻¹ (Fig. 7) where a change in V_t of only 1 m s⁻¹ leads to a change in intensity of ∼10%. Consequently, from a physical perspective, changes in the aerosol content of a tropical cyclone’s environment should change V_t, which might then lead to significant (∼10%) changes in intensity. These microphysical aspects of tropical cyclones might be part of the reason why tropical cyclone intensity has been so difficult to predict operationally.

e. Equation set

One unique aspect of this numerical model is the equation set, which mathematically conserves internal energy for reversible moist flows. In contrast, most nonhydrostatic cloud-scale models use an approximate equation set where the heat capacities of water are neglected, which leads to a cold bias when the liquid water content is large (Bryan and Fritsch 2002). For numerical models that do include these effects (e.g., Ooyama 2001; Satoh 2003), it is unclear whether simulations of tropical cyclones are considerably different from simulations that use traditional approximate equation sets. Therefore, in this subsection we investigate the impacts of the different equation sets in our model.

Figure 10 shows a comparison of υ_max from simulations that use the conservative equation set (solid lines) and simulations that use the traditional equation set (dashed lines). For relatively large fall velocity (V_t > 5 m s⁻¹), there is essentially no difference in results. This is because the derivation of the traditional equation set is analogous to making the pseudoadiabatic assumption, where liquid water contents are low and thus the heat content of liquid water can be neglected (Bryan 2008). In contrast, for relatively small fall velocity (V_t < 5 m s⁻¹), the simulations using the traditional equation set are 10%–20% weaker than simulations using the conservative equation set. This result is consistent with the arguments provided in section 3d; that is, lower column-integrated buoyancy yields weaker intensities. In this case, the cool bias incurred by neglecting the specific heats of water leads to the weaker intensities.