a. Governing equations and parameter regime

We nondimensionalize our system in two different ways: on the one hand, for convection induced by a prescribed heat flux Q at a boundary, we scale the lengths by the depth of the fluid layer H; the flow velocities by the convective velocity scale, $W^{*}={(F_{B}H)}^{1/3}$, where F_B is the buoyancy flux and is given by $F_B=g\beta Q/(\rho c_p)$, with c_p the specific heat at constant pressure, and ρ the density of the fluid, and the rotation rate by Ω. These yield a scaling of $\nabla \sim 1/H$, $t\sim H^{2/3}/F_B^{1/3}$, $p\sim {(F_{B}H)}^{2/3}$, and $T\sim F_B{}^{2/3}/{(g\beta H)}^{1/3}$. The heating itself is incorporated through the boundary conditions, detailed in the next subsection. On the other hand, for convection between two plates at fixed temperatures, we nondimensionalize T by ΔT, the temperature difference between the plates, and the velocities by the free-fall velocity scale $W_{\text{ff}}=\sqrt{g\beta \mathrm{\Delta }TH}$. This gives $t\sim \sqrt{H/(g\beta \mathrm{\Delta }T)}$ and $p\sim g\beta \mathrm{\Delta }TH$.

The 3D dry cyclones simulated by Cronin and Chavas (2019) provide some guidelines for a plausible region of parameter space to find TCs in a dry convective setup, since a parallel can be traced between RBC and the more Earthlike conditions of three-dimensional radiative–convective equilibrium. An approximate layer depth H ∼ 10⁴ m, a vertical velocity w ∼ 1 m s⁻¹, and a Coriolis parameter f ∼ 10⁻⁴ s⁻¹ yield a value of Ro_c = O(1). While realistic atmospheric conditions are characterized by Rayleigh numbers between O(10¹⁸) and O(10²²), this is far above the level of turbulence that direct numerical simulations can resolve at present, which are closer to Ra ∼ 10¹⁰ (Plumley and Julien 2019), which for Pr = 1 corresponds to Ra_f between 10^13.3 and 10¹⁵, depending on how the heat transport scales with Ra (Aurnou et al. 2020). Such forcings would be expected to yield Reynolds numbers Re ∼ 10⁴ (Fonda and Sreenivasan 2015).

We choose a value of Ra_f = 10⁹ for all simulations with at least one flux-based thermal boundary condition, and Ra = 10^7.3 for the simulation with fixed-temperature boundaries. These values are chosen such that the resulting convection is in the moderately to highly turbulent regime with flows characterized by Reynolds numbers Re ∼ 10³. We diagnose the Reynolds numbers in our simulations as $\text{Re}=R\cdot {(\Vert \mathbf{u}\Vert )}_{\text{dom}}$, where R is the parameter associated with the viscous terms in the dimensionless momentum equations, and ${(\Vert \mathbf{u}\Vert )}_{\text{dom}}$ is the domain-averaged value of the dimensionless magnitude of the simulated flow. Using a diagnostic criterion like the Reynolds number as an indication of the strength of turbulence allows us to circumvent the problem of finding an adequate theoretical relationship between Ra_f and Ra, since our goal is merely to ensure broadly similar regimes. The relatively low choice of Rayleigh and Reynolds numbers compared to what is technically feasible is an attempt to balance the trade-offs in computational cost between running too few simulations at very high levels of turbulence, and many simulations with a very weak thermal forcing. For our simulations, we confine our attention to a value of Pr = 1.0, which lies in between those typical for gases (0.7) and water (6.9) on Earth (Rapp 2016).

Four basic time scales can be defined for our system: two diffusive time scales for heat and momentum diffusivities, namely, H²/κ and H²/ν; a convective time scale, t_Conv = H/W, where W represents $W^{*}$ or W_ff for the flux-based or the temperature-based setups, respectively; and a rotational time scale, t_Rot = 2π/Ω, which can be written in terms of the convective scale as t_Rot = 4πRo_ct_Conv, from the definition of the convective Rossby number. Although time in the dimensionless equation is nondimensionalized by the convective scale, we will use the rotational time scale in our analysis, as it is analogous to a pendulum day.

Finally, a realistic aspect ratio Γ (horizontal to vertical length scale) for dry TCs would be O(10), taking a horizontal length scale O(10⁵) m. Given the high computational cost of turbulent flows in domains with large aspect ratio, most of our simulations use Γ = 16, unless stated otherwise.

b. Thermal boundary conditions

We explore nine different combinations of thermal boundary conditions for the simulations of rotating convection, described in Table 1. CF denotes a constant flux condition given by $\partial T/\partial z=-{({\text{Ra}}_f\text{Pr})}^{1/3}$, where the magnitude of the right-hand side represents the ratio of the scales of convective to conductive heat transport in the thermodynamic equation, i.e., $W^{*}H/\kappa$, or P in Eq. (6). This amounts to imposing a heat flux at the boundary that sustains the turbulence set by the governing parameters of the flow in the fluid interior, namely, Ra_f and Pr. Additionally, Ins refers to an insulating condition with no heat flux, i.e., $\partial T/\partial z=0$, and CT refers to constant temperature. Note that x and y represent the horizontal coordinates, and z is the vertical, oriented with the negative of the gravitational field.

Table 1.

Combinations of thermal boundary conditions used in the simulations. Conventions are detailed in the text.

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PF denotes a parameterized heat flux condition, where $\partial T/\partial z=-{({\text{Ra}}_f\text{Pr})}^{1/3}(1+C_k{U}_h)$, with C_k a dimensionless enthalpy exchange coefficient set to 1, and U_h the total horizontal wind at a height h, which we choose to be 1% of the domain depth. For reference, the choice C_k = 1 implies that the heat flux doubles when the horizontal wind speed at h is 1. We use this as an idealized way to represent a dependence of the heat flux on the strength of the flow aloft, in analogy to the surface enhancement of enthalpy fluxes by surface roughness modeled by a flux aerodynamic formula, which constitutes the basis for WISHE.

For the simulations Ins/CF, Ins/PF, CF/PF, characterized by asymmetric flux conditions, the temperature in the whole domain increases steadily in time, but temperature differences are not eliminated, which means that convection can remain active without any additional forcing, since it is the buoyancy and not the absolute temperature that has dynamical relevance. In these setups, the effective cooling is a consequence of the asymmetry in the fluxes at the top and the bottom boundaries, which cause the fluid in the interior to lose buoyancy (or to “cool”) with respect to the fluid at the bottom, while it gains less (CF/PF) or no buoyancy at all (for Ins/CF and Ins/PF) with respect to the fluid near the top. This is similar to the secular temperature drift in experiments of convection with an internal radiative heating source (Bouillaut et al. 2019). For the setup Ins/CT, an internal cooling term with magnitude −1 in the dimensionless equation [equivalent to a factor of (Ra_fPr)^1/3 times the scale of thermal diffusion] is introduced, so that the domain does not become isothermal and suppress convection. The conditions CT/CT correspond to the classical Rayleigh–Bénard model (Bénard 1901; Rayleigh 1916), where the top and bottom boundaries are held at fixed temperatures, maintaining an adverse thermal gradient.

For our numerical experiments, we hold Pr constant and equal to 1, hold Ra constant and equal to 10^7.3 for CT/CT, and Ra_f = 10⁹ for all other setups. We initially fix Ro_c to 2 to focus on the effects of the different thermal boundary conditions on the organization of convection and the formation of TCs, but subsequently explore the effects of varying Ro_c as well. It is worth noting that these nine sets of thermal boundary conditions correspond to all possible combinations from among (PF, CF, CT, Ins) that are physically distinct and able to sustain convection, because the governing equations are invariant to the combined replacement z → −z, T → −T.

For CT/CT and Ins/CT, an adverse vertical profile of temperature is prescribed at the outset with a temperature drop of 1, so that, for a large enough flux Rayleigh number, the fluid is convectively unstable. The instability is initialized as T(z) = −z − 10⁻³ × N[μ = 0, σ = 1] × (z_t − z) × (z − z_b), where z_t and z_b are the top and bottom values of z, and N[μ = 0, σ = 1] represents Gaussian noise with zero mean, and standard deviation of one. For simulations with prescribed heat fluxes, convection is initialized from a vertically isothermal profile perturbed by Gaussian noise with amplitude 10⁻⁴ and standard normal parameters.

a. Convective organization

Different combinations of thermal boundary conditions produce different levels of convective organization, as illustrated by the spatial distribution of vertically averaged temperature anomaly {T′} in Fig. 1. For these simulations, the temperature anomaly is calculated with respect to the horizontal mean at each height, and braces denote a vertical average. One way to quantify the level of organization of a field F is through the scale L_F proposed by Beucler and Cronin (2019), defined in this case as

$L_F\equiv \frac{⟨\lambda {\varphi }_F⟩}{⟨{\varphi }_F⟩},$(7)

where ϕ_F is the spatial power spectrum of F, λ is the wavelength of the spatial wavevector, and angle brackets represent the spectral average given by

$⟨A⟩\equiv \frac{{\displaystyle {\int }_{k_0}^{k_N}A(k)dk}}{{\displaystyle {\int }_{k_0}^{k_N}dk}},$(8)

where k₀ and k_N are, respectively, the smallest and the Nyquist wavenumbers.

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Vertically averaged temperature anomaly, {T′} with respect to the horizontal mean for simulations with Ro_c = 2.0. Magnitudes are normalized by the standard deviation (σ) in each setup, and values of standard deviations are indicated for each simulation. The color bar saturates at ±2.5σ. Rows represent different bottom thermal boundary conditions and columns are top boundary conditions. Ins refers to a thermally insulating boundary, CF is constant flux, PF is parameterized flux, and CT is constant temperature (see Table 1 for details). Notably, simulations with two flux-based boundary conditions (Ins, CF, or PF) show a higher level of spatial organization in {T′} than those with one or two fixed-temperature conditions (CT). All figures correspond to the snapshot at 10 rotational time periods.

Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-23-0018.1

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In particular, the simulations with any combination of flux-based boundary conditions on the lower and upper boundaries (Ins, CF, PF) result in the largest scales of convective organization, with $L_{\{T^{\prime }\}}$ values of 14.9 for Ins/CF, 15.3 for CF/CF, and 15.4 for Ins/PF, CF/PF, and PF/PF, indicating that features on the length scale of the domain are dominant. Those with at least one fixed-temperature boundary condition exhibit slightly finer spatial structures, with $L_{\{T^{\prime }\}}$ values of 8.4, 11.5, and 11.9 for Ins/CT, CT/CF, and CT/PF, respectively. The classical Rayleigh–Bénard configuration, CT/CT, produces the smallest-scale spatial structure, with an $L_{\{T^{\prime }\}}$ of 1.9, only about twice the domain depth and 1/8 of the domain width.

There are also differences between the highly organized patterns of flux-only boundary conditions: CF/CF, PF/PF, and CF/PF show banded organization with a wavenumber-1 structure in x and little variation in y. In contrast, Ins/CF shows two somewhat distinct buoyant ({T′} > 0) clusters, and Ins/PF one, very distinct cluster and an approximate wavenumber-1 pattern in both x and y.

The simulations with two flux-based boundary conditions show greater persistence than those with one and, especially, two fixed-temperature boundary conditions, as measured by the temporal autocorrelation calculated with respect to the reference state at t₀ = 10 rotational time units, and plotted in Fig. 2a. This indicates that the organization of convection in PF/PF, CF/CF, Ins/PF, and Ins/CF remains steady for several rotational time periods, while it shifts significantly within less than a rotational period for CT/CT.

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(a) Time series of autocorrelation of {T′} for the different setups. Time lags are with respect to t₀ = 10 rotational time periods. Simulations with two flux-based boundary conditions exhibit higher persistence in the spatial patterns of {T′} than those with one or two fixed-temperature conditions. (b),(c) Length scales $L_{\{T^{\prime }\}}$ and $L_{U_{20}}$ for the spatial organization of the temperature anomaly and the horizontal wind field at 20% of domain height, respectively (see text for definition of L). The $L_{U_{20}}$ series corresponds to a moving mean of a half rotational period for visual clarity.

Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-23-0018.1

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Previous work has shown that in nonrotating CF/CF setups, as well as in setups with perfectly insulating conditions but internal heating or cooling, the most unstable modes at the onset of convection are those with wavenumber zero (Hurle et al. 1967; Chapman and Proctor 1980; Goluskin 2016); that is, the disturbances grow unbounded to span the entire domain. This has been shown to be modified by rotation Takehiro et al. (2002), with faster rotating rates leading to gradually larger wavenumbers for the most unstable modes. Although these studies have focused on the onset of turbulence, and their conclusions should not be extrapolated directly to the highly turbulent regime, Vieweg et al. (2022) have used Lyapunov analysis to show that the formation of a pair of counterrotating convective cells that grow to the size of the domain persists for high levels of turbulence. An exploration of marginal stability for a rotating setup with different thermal boundary conditions is beyond the scope of this work, but we note that the patterns of organization seen in our simulations suggest that flux-based boundary conditions in a moderately turbulent, moderately rotating regime also produce organization spanning the entire domain. However, the constraining effects of rotation on this large-scale organization are not seen for the combinations of Ro_c and Γ used here.

b. Formation of tropical cyclone–like vortices

Simulations with the most highly organized patterns of convection also exhibit large-scale patterns in horizontal winds, as shown in Fig. 3 by the contours of total horizontal winds at 20% of domain height, i.e., $\sqrt{{[u(z=0.2H)]}^2+{[\upsilon (z=0.2H)]}^2}$—denoted by U₂₀ henceforth—10 rotational time periods after initialization. In particular, a highly axisymmetric, cyclonic circulation with a well-defined eye covers a significant fraction of the domain in the Ins/PF setup, and the location of its pressure minimum is indicated with the × symbol. The color scale saturates at 3.5σ, where σ is the standard deviation of U₂₀ in each setup. This corresponds to a dimensionless wind speed of about 3.2; that is, horizontal wind speeds that are over 3.2 times the typical velocity scale for convective motions in the domain. Two smaller cyclonic vortices also form in the Ins/CF setup, each similarly marked with the × symbol. In this case, 3.5σ corresponds to 1.4 in dimensionless velocity units, indicating that the vortices are somewhat stronger than the convective velocity scale. All three vortices mentioned are found in the areas of high vertically averaged temperature anomalies of Ins/PF and Ins/CF. The storm centers are identified by applying a Gaussian filter to the map of near-surface pressure, using a binary dilation algorithm to locate contiguous regions of negative pressure anomaly, and calculating their centroid.

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Horizontal wind at 20% of domain height for simulations with Ro_c = 2.0, at 10 rotational time periods. As in Fig. 1, magnitudes are normalized by the standard deviation in each domain, length scales are indicated at the bottom, and color shades saturate at 3.5σ. Locations of near-surface pressure minima corresponding to persistent cyclonic vortices are indicated with × markers.

Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-23-0018.1

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None of the other setups produce persistent vortex structures within 10 rotational periods, but large-scale overturning circulations are seen for those with high convective organization: CF/CF, CF/PF, and PF/PF. The traditional Rayleigh–Bénard setup, CT/CT, gives rise to convective structures with an associated cyclonic circulation in the horizontal, but they only persist for a fraction of a rotational time period, and their width, as indicated by $L_{\{T^{\prime }\}}$, is constrained to within ∼2, differing markedly from the large, persistent structures seen in the Ins/CF and Ins/PF setups.

A length scale $L_{U_{20}}$, computed analogously to $L_{\{T^{\prime }\}}$, but for the horizontal wind speeds at 20% of domain height, is shown at 10 rotational periods in 3. We use the two length scales as an imperfect but helpful way to track the evolution of the scales of organization of the temperature and the horizontal wind fields (Figs. 2b,c).

In Ins/PF, $L_{\{T^{\prime }\}}$ evolves similarly to other setups with two flux-based conditions: rapid growth, and saturation at the domain scale after about 4 rotational time periods. However, the spatial scale of horizontal winds $L_{U_{20}}$ grows up to values of about 13, close to the domain length, and about twice the size of the setups with the next largest scales for U₂₀, (Ins/CF and PF/PF). The growth of large-scale horizontal flow patterns in Ins/PF and Ins/CF seems to lag the growth of the spatial organization of the thermal field by a couple of rotational periods, indicating that convective aggregation precedes the formation of large-scale, persistent vortices in those setups. At 10 rotational periods, the scales of all simulations appear to have equilibrated. We observe no qualitative difference in the formation of the TC in Ins/PF when a different amplitude (10⁻³) or different seeds are used in the initialization of the convection, suggesting that the finding is robust.

Studies of spontaneous moist cyclogenesis also describe the formation of a large-scale midlevel vortex several days ahead of the formation of a tropical cyclone (Bretherton et al. 2005; Nolan et al. 2007; Davis 2015). However, in our dry Ins/PF setup, large-scale features of the midlevel vorticity field seem to lag rather than to lead cyclogenesis (see Video 1 in the online supplement). Key differences between dry and moist TCs arise from the effects of condensate reevaporation in subsaturated regions, which influence convective organization and warm-core consolidation in the latter (Davis 2015), but recent work suggests that dry tropical cyclones exhibit similar properties and structure to moist TCs in the absence of irreversible phase changes (Wang and Lin 2020). Because of this, we expect dry and moist tropical cyclogenesis to exhibit more similarities after significant humidification of the free troposphere by aggregated convection in the latter—an important stage in its development (Nolan 2007).

The cyclones in Ins/PF and, to a lesser extent, Ins/CF, also share other essential characteristics of tropical cyclones, besides persistence and a horizontal scale substantially larger than that of single convective elements. One of them is slantwise neutrality, or the alignment between isentropes and constant-angular momentum surfaces around the TC. Initially formulated as a characteristic of TCs in a moist atmosphere (Emanuel 1986), slantwise neutrality has been also shown to be a property of dry tropical cyclones in simulations with parameterized turbulence and surface processes (Wang and Lin 2020). In a dry, Boussinesq system, slantwise neutrality requires M = M(T), where $M=rV+(1/2)f{r}^2$ is the angular momentum per unit mass. We note that the equality implied by strict slantwise neutrality should be relaxed for nonidealized storms: M ≈ M(T).

Figure 4 shows azimuthally averaged, height-versus-radius profiles centered at the two vortices marked with a red cross (Ins/CF in the left column, and Ins/PF in the right column). The upper panel contrasts contours of angular momentum (black dashed lines) and the rescaled temperature T − T_dom (colors), where T_dom is the domain-mean temperature. In the lower region near the vortex of the Ins/PF case, there is a relatively good alignment between lines of constant M and the isentropes—for our case, those are equivalent to lines of constant T − T_dom. The alignment is less clear for the core of the vortex of Ins/CF, plausibly because of its lower axisymmetry and its weaker nature.

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Azimuthally averaged radius vs height profiles centered at TC center for simulations with Ro_c = 2.0, averaged between the ninth and the tenth rotational time period, for (left) Ins/CF and (right) Ins/PF. (top) T − T_dom (color) and total angular momentum (contours). (middle) T − T_dom (color) and the Stokes streamfunction (contours; see text). (bottom) Azimuthal wind (colors) and Stokes streamfunction (contours). All quantities are dimensionless.

Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-23-0018.1

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For Ins/CF, the radial circulation of the chosen vortex spans radially out to about 5 dimensionless units, beyond which a combination of the secondary cyclone and the background flow are picked up. Since for Ins/PF there is only one vortex, and so only one axis of flow symmetry in the domain, the secondary circulation shows an unambiguous signal that spans out to about 8 or 9 dimensionless units, suggesting that the vortex size is domain limited. This is consistent with the finding that, for more realistic TC simulations, the vortex can form in domains that are too small to allow for its natural length scales, in which case it grows as much as the domain allows (Chavas and Emanuel 2014).

The bottom-heavy spatial structure of the azimuthal winds, consistent with the balanced flow around a warm core, is also a quintessential feature of tropical cyclones. The lowest panels of Fig. 4 show profiles of azimuthal winds (colors) and the Stokes streamfunction (black lines). The maximum azimuthal winds of both vortices are found near the surface and close to the center of the cyclones, and in Ins/PF this also corresponds to the region of strongest convection as indicated by the slanted, densely packed streamlines, demarcating the boundary between the eye and the eyewall. Additionally, an upper-level anticyclone is sustained at the top. The general structure of the winds in Ins/PF is very similar to that found in previous studies of dry TCs (Mrowiec et al. 2011; Cronin and Chavas 2019; Wang and Lin 2020). That of Ins/CF is qualitatively similar, although the strength and the degree of axisymmetry are lower.

The radial profile of winds in Ins/CF and Ins/PF at 20% of domain height—approximately the depth of the strongest azimuthal winds for both—shown in Fig. 5 also makes more apparent some of the differences between the two TCs: While the presence of both vortices is discernible from the azimuthal winds, the total winds (blue curve) within the vortex in Ins/CF are not substantially stronger than those far away from it, unlike for Ins/PF, where the total winds of the vortex are enhanced substantially by the wind-dependent heat flux parameterization. Additionally, the increase in the magnitude of the radial winds with radius near the cyclone center, which by continuity indicates the region of highest convergence and strongest convection, is sharp and well-defined in Ins/PF, extending between about 1.2 < r < 2, whereas it is less steep and broader in Ins/CF, going from r = 2 all the way to the center, consistent with a less clear eyewall region.

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Radial profiles of azimuthally averaged wind speed at 20% of domain height for vortices in (left) Ins/CF and (right) Ins/PF. Plots show total winds (blue), azimuthal winds (orange), radial winds (green), and the azimuthal wind profile based on the model by ER11 using C_E/C_D = 1 (red; see text for details). The values of υ_m and r_m indicate the maximum azimuthal wind speed [$\text{max}(\overline{u_{\text{az}}})$] and the radius of maximum winds, respectively. Dashed vertical lines indicate r_m, and the innermost location where azimuthal winds drop below 0.01.

Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-23-0018.1

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The fact that Ins/CF produces long-lived, large-scale vortices in the absence of dependence between surface heat fluxes and the flow above (i.e., without WISHE) agrees with the finding that WISHE is not required to produce large-scale, persistent (albeit weaker) TCs in realistic simulations (Montgomery et al. 2015; Chavas 2017). Importantly, the storm produced in Ins/PF, where WISHE is active, intensifies more quickly and reaches substantially stronger winds than those in Ins/CF, which is also in line with findings about real-world storms (Zhang and Emanuel 2016; Chavas 2017).

The winds in the innermost core (i.e., inwards from r_m) in Ins/PF are poorly represented by ER11, likely due to the frontogenetic nature of the region and the breakdown of the assumptions of the model there. For similar reasons, the apparent good agreement between the model and the simulated wind speeds in the innermost core in Ins/CF is likely a coincidence, rather than evidence of the skill of the theory. Additionally, the lack of a parameterization for the drag in our simulations, as well as the simplified form of our enthalpy-flux enhancement, make it challenging to determine the appropriate value of C_E/C_D to use for a more rigorous comparison with the model.

c. Rotation rate and TC formation

The Ins/PF setup produces the circulation with the clearest resemblance to a TC, which makes it a suitable candidate to investigate the response of TC characteristics to changes in the governing parameters. Particularly, a faster rotation rate has been shown to lead to a smaller size for equilibrated TCs on an f plane (Wang et al. 2022), but it is unclear if this scaling breaks down at a particular rotation rate or convective strength. Since the convective Rossby number quantifies the relationship between rotation and convective strength, we perform an additional suite of simulation to study the dependence of vortex characteristics on Ro_c for Ins/PF boundary conditions. The results are shown in Fig. 6. All simulations produce vortices of some sort, although only that in the simulation with Ro_c = 2.0 persists for much longer than a rotational time period, suggesting a qualitatively different behavior, and a threshold around Ro_c ∼ 1 for the formation of proper TC-like structures. The intensity and the size of these vortices is affected by the rotation rate as well, with the latter seeming to scale roughly with Ro_c.

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Plan views of horizontal wind speeds at 20% of domain height for simulations with Ins/PF conditions, for various values of Ro_c, at 10 rotational time periods from the start of the simulations. The values are dimensionless and scale with the convective scale. Colors saturate at a value of U = 2.5.

Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-23-0018.1

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The structure and the persistence of the vortices for the lowest Ro_c simulations differ markedly from those with Ro_c = 0.8 and, especially, Ro_c = 2.0. Figure 7 shows the temporal autocorrelation of the temperature anomaly field for the same suite of simulations as Fig. 6, spanning five rotational time periods, starting at least four rotational periods after the initialization. In addition to these, the figure includes two simulations with lower aspect ratios. The high autocorrelation found for the simulation with Ro_c = 2.0 shows evidence that the large, highly axisymmetric vortex that forms in it, or more precisely its associated pattern of convection and its warm core, is persistent in time and fixed in space.

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Temporal autocorrelation of vertically averaged temperature anomaly {T′} for the four simulations of Fig. 6 (solid lines), and two additional simulations (dashed lines), with domain aspect ratio of 2, and Rossby numbers of 0.2 and 0.4.

Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-23-0018.1

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We note that the temporal autocorrelation of temperature anomaly is not a perfect metric for the persistence of vortices if those are drifting in the domain. However, the small vortices for small Ro_c in fact do not persist for much longer than a rotational period (see Video 2 in the supplement). Additional simulations were run with Γ = 2 in order to test if the lack of persistence of the small vortices was associated to the size of the domain. Figure 7 shows that the autocorrelation does not exhibit a consistent increase. These simulations thus indicate that it is not enough for the domain to tightly contain a vortex—as in a “lattice equilibrium,” borrowing the terminology of Zhou et al. (2014) for a doubly periodic domain that contains exactly one vortex—for it to be persistent in time, if the convective Rossby number is too low. All of this is evidence of a fundamental difference between the TC-like structure seen for Ro_c = 2.0 and the small, evanescent structures of Ro_c = 0.2 and 0.4.

The convective Rossby numbers for Ins/PF used here are based on the same scaling used for Ins/CF. That means that, instead of redefining the buoyancy flux based on the changing bottom flux, we take a fixed reference flux F_B, which has an associated velocity scale that is constant. Because this scale does not include the enhancement of the heat flux by the flow, we cannot expect the typical dimensionless wind speeds in the domain to scale with it. There will thus be an effective convective Rossby number, Ro_c,eff = Ro_c(1 + C_kU_h)^1/3, which gives a more accurate bulk characterization of the flow in the domain than Ro_c. This effective Rossby number cannot be determined a priori, since it depends on the horizontal wind speed, which in turn depends on the heat flux. However, we can calculate it from the values of U_h—the horizontal wind speed at 1% of domain depth used in our simple parameterization PF. We obtain Ro_c,eff = 0.23, 0.47, and 0.91 for the simulations with Ro_c = 0.2, 0.4, and 0.8, respectively, for which U_h stays fairly constant throughout the simulation. For the simulation with Ro_c = 2, we obtain Ro_c,eff starting at around 2.34 and stabilizing at about 2.45 once the TC has equilibrated. These differences are not negligible, but the fact that they are small indicates that Ro_c approximately characterizes the bulk properties of our Ins/PF simulations.

The convective Rossby number can be understood as the ratio of the time scale of domain rotation to the time scale of convection. The condition Ro_c < 1 thus corresponds to situations where the depth of penetration of convective thermals is constrained by rotation. While the vertical velocity is not affected by Coriolis, the horizontal convergence and divergence associated with it is, and this affects the rest of the circulation via the pressure gradient and continuity. This constraint causes a reduced ability of convective thermals and plumes to sustain a stable warm core against the destabilizing effects of the internal cooling, which we see as a likely cause for the impermanence of the vortices formed.

Additionally, for a given convective velocity scale, a larger convective Rossby number represents a slower rotation rate, which, as mentioned, is associated with a larger storm at equilibrium (Wang et al. 2022). This suggests that in order to generate storms for higher convective Rossby numbers, larger aspect ratios would likely be needed. For moist RCE simulations, Zhou et al. (2014) have shown that mature TCs only reach sizes that are domain independent if the domain aspect ratio is large enough, but that domain-size limited TCs form for significantly smaller domains.

We have run additional simulations with reduced horizontal resolution and large aspect ratios in the Ins/PF setup to test the effects of Γ on TC size and formation (see supplemental Figs. 4–7). A large, TC-like vortex forms for (Ro_c = 2; Γ = 32), still spanning the entire domain, which indicates that the storm has likely not reached its natural length scale. Similarly, (Ro_c = 3; Γ = 32) also produces a domain-spanning TC-like vortex, but with a significantly narrower eye. The smaller eye for larger Ro_c is qualitatively consistent with the model by Chavas et al. (2015) and Chavas and Lin (2016), which shows a narrowing of the eye for lower rotation rates when the outer size is fixed—a condition imposed in our case by the domain size limitation—as fluid parcels must travel further inwards to reach a given maximum tangential wind speed.

Other simulations performed also suggest a lower bound for the domain aspect ratio for a given convective Rossby number, below which no TC-like structures are observed: (Ro_c = 4; Γ = 32) did not produce any persistent, large-scale vortices within more than 20 rotational periods simulated, and neither did Ro_c = 2 or Ro_c = 3 at either Γ = 4 or Γ = 8. It remains an avenue for future research to determine whether this lack of formation is due to a dynamical threshold to the domain size that can support a TC for a given convective Rossby number or caused by insufficient resolution to simulate a narrowing eye adequately for slower rotation rates.

d. Thermal profile

A feature shared by all simulations with a flux-based top boundary condition is a horizontally averaged temperature profile with a stable layer in the upper half of the domain, as seen in Fig. 8. The strength of this bulk inversion varies between the simulations, and is most pronounced for Ins/PF, followed roughly in order by CF/PF, Ins/CF, PF/PF, Ins/CT, and CF/CF, while none of the simulations with fixed temperature at the top show it. This feature can result from the fluid in the upper layers having been more recently in contact with the bottom surface than the fluid of the middle layers. In a framework with prescribed internal cooling, the situation is equivalent to saying that the fluid in the middle layers has cooled more than that of the upper layers, relative to the bottom. This plausibly explains the thermal inversion between the middle and the upper part of the domain as resulting from more effective detrainment of buoyant air at the top than in the middle.

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Vertical profiles of horizontally averaged temperature minus domain-averaged temperature, $\overline{T}-T_{\text{dom}}$, for all nine original simulations with Ro_c = 2.0 after 10 rotational time periods. Setups with flux-based upper boundary condition (i.e., Ins/PF, CF/PF, Ins/CF, PF/PF, CF/CF, and Ins/CT) show thermal inversions in the upper half of the domain.

Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-23-0018.1

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The bulk deformation radius calculated for Ins/PF is ∼1.0, about 3 times as large as that of Ins/CF (∼0.3), and reasonably close to the scale of the radius of maximum winds, r_m. Previous work has shown that the deformation radius is not an accurate scale for either the outer radius or the radius of maximum winds of moist TCs in axisymmetric (Chavas and Emanuel 2014) or in 3D simulations on an f plane (Zhou et al. 2014; Cronin and Chavas 2019), and other scales, like u_p/2Ω is, where u_p is the potential intensity, have been proposed. However, for the driest TCs simulated by Cronin and Chavas (2019), the deformation radius is at least as much of a plausible scaling for r_m as u_p/2Ω is. This agreement between r_m and R_D thus holds in our simulations as well. These observations should be tested further before any physical mechanisms are invoked, but they suggest a possible link among the scale of the radius of maximum winds, the thermal stratification, and the convective Rossby number in fully dry TCs.

e. Boundary-condition symmetry

Among the nine basic setups studied, only four are characterized by an effective internal cooling: Ins/CF, Ins/PF, CF/PF, and Ins/CT. For Ins/CT, the cooling is prescribed explicitly to prevent the fluid from evolving toward an isothermal, quiescent state. For the other three, the effective cooling is a consequence of the flux asymmetry between the upper and lower boundaries: the fluid in the interior is constantly losing buoyancy (and in that sense is “cooling”) with respect to the fluid at the bottom, while it is not gaining as much (for CF/PF) or any buoyancy at all (for Ins/PF and Ins/CF) with respect to the fluid near the top. This leads to vertical temperature profiles that are asymmetric, in the sense that $\widetilde{T(z)}\ne \widetilde{-T(1–z)}$, where $\widetilde{T(z)}$ represents the horizontal average of temperature at height z minus the domain-average temperature. This is shown in Fig. 9. The asymmetry coefficient α is defined as $\alpha \equiv {[(1/N){\displaystyle {\sum }_i^N{({\mathrm{\Delta }}_i\tilde{T})}^2}]}^{1/2}$, where the sum is across all i ∈ {1, 2, …, N} layers of the domain, and ${\mathrm{\Delta }}_i\tilde{T}=\widetilde{T(z_i)}+\widetilde{T(1-z_i)}$.

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Vertical profiles of normalized, horizontally averaged temperature $\tilde{T}$ for all nine original simulations with Ro_c = 2.0 after 10 rotational time periods, and for additional simulation with CF/PF conditions and C_k = 2 at the bottom, denoted by CF/PF(C2). Legend indicates the bulk deformation radius for each set of boundary conditions, as well as the asymmetry coefficient (see text for details), with “N.R.” indicating simulations where no stable layer formed. $\tilde{T}$ is calculated by subtracting the domain-mean temperature T_dom from the horizontally averaged temperature at each level $\overline{T}$ and then normalizing by the difference between the maximum and the minimum values of $\overline{T}-T_{\text{dom}}$ for each setup. Setups were divided into two groups for visual clarity: (left) those with higher level of asymmetry and (right) the more symmetric profiles.

Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-23-0018.1

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The case of CF/PF is of particular interest, because despite the somewhat symmetric thermal profile (with α = 0.06) and the absence of TC-like vortices, it exhibits a strong inversion as well as the second largest deformation radius (R_D ∼ 0.4). This raises the question of whether enhancing the thermal asymmetry sightly could potentially result in TC formation—in fact, CF/PF will approach the Ins/PF case as C_k is increased to the point where the flux at the top becomes orders of magnitude weaker than the enhanced bottom fluxes. However, we find that the asymmetry need not be pushed that far in order for TCs to form: an additional simulation run with CF/PF boundary conditions, but with an enhancement coefficient C_k = 2.0 instead of 1.0 for the bottom parameterization [represented by CF/PF(C2) in Fig. 9] does produce a TC-like vortex after 3 rotational time periods. This indicates that either the asymmetry of the fluxes must be stronger than a given threshold for TCs to form at all, or that a stronger flux asymmetry accelerates their formation substantially, as for C_k = 1.0 none form within 10 rotational periods (see Fig. 16 in the supplement for a comparison between the two values of C_k).

The other setup with internal cooling that does not produce TCs, Ins/CT, is also characterized by a thermal inversion (albeit weaker than the other three) and by substantial asymmetry of the thermal profile. We are thus prevented from concluding that thermal asymmetry is a sufficient condition for TC formation in our setups in the explored parameter regime, although it is still possible that TCs would develop given a longer spinup time. It may also be the case that the thermal forcing is not strong enough, or that the homogeneous temperature boundary condition at the bottom is less conducive to the large-scale organization of convection and this in turn affects the likelihood of formation of a TC-like structure.

From our numerical experiments, it is unclear how the flux symmetry hinders TC formation. A possibility is that, when the updrafts and downdrafts are similarly strong, an incipient warm core, necessary to sustain the TC and to activate a wind-dependent heat flux enhancement (akin to the WISHE) that would power its energy cycle, is more easily disrupted, for example by accelerated heat loss to the top boundary.

A warm core is an essential part of a TC, so in general, conditions that represent a net input of heat (as is the case for asymmetric fluxes) and keep convective areas relatively persistent in space and time would be more likely to sustain TC development. Investigating if other combinations of asymmetric fluxes can produce TCs—for instance, CF/CF with constant fluxes of different values, or PF/PF with different enhancement coefficients—would help elucidate the mechanisms that disrupt TC formation, and to answer how the spatial pattern of asymmetry matters.