1. Introduction
In this work, we present evidence that cyclonic vortices that share the essential dynamical and thermodynamic characteristics of dry tropical cyclones can form in direct numerical simulations of simple setups of dry, rotating turbulent convection. We also show that the combined effects of convective aggregation and rotation are necessary but not sufficient to produce spontaneous cyclogenesis in a dry convective domain, and that only some combinations of thermal boundary conditions are likely to lead to spontaneous dry tropical cyclone formation. This constitutes a step forward in an effort to bring together dry turbulence research and tropical atmospheric dynamics, particularly the study of convective organization and tropical cyclone physics.
Meteorologists define a tropical cyclone—TC henceforth—by its structure and geographical location as “a warm-core non-frontal synoptic-scale cyclone, originating over tropical or subtropical waters, with organized deep convection and a closed surface wind circulation about a well-defined center” (National Hurricane Center 2023). We take the view here that the essential features of a TC are a low pressure center, a warm convective core that covers a substantially larger area than individual buoyant thermals, and an axisymmetric, cyclonic, bottom-intensified circulation coupled with the convective region (Cronin and Chavas 2019).
The formation of TCs in the absence of latent heat fluxes and moisture was first postulated and confirmed for an axisymmetric numerical model in radiative–convective equilibrium by Mrowiec et al. (2011). Their work inferred this possibility from the formulation of potential intensity theory (Emanuel 1986; Rotunno and Emanuel 1987), which implied that mature tropical cyclones could be sustained by a strong enough total surface enthalpy flux, regardless of the relative contribution of latent and sensible sources. Subsequent work in 3D atmospheric simulations has shown that TC-like structures emerge in a wide range of regimes of moisture availability including totally dry conditions (Cronin and Chavas 2019), and that they share many characteristics with those simulated in a moist atmosphere where irreversible phase changes are suppressed (Wang and Lin 2020, 2021).
The discovery of dry TCs in simulations has started to narrow the gap separating the study of real-world TCs and traditional turbulence research. Primarily, it has shown that moisture and its irreversible phase changes, with all their associated complexity, need not always be considered to understand important aspects of tropical cyclone physics, such as the connection between convective aggregation and TC formation. This motivates asking if TC-like structures, and the closely related phenomenon of convective organization, can be studied in simplified setups of fluid motion. A quintessential example of such a setup is the Rayleigh–Bénard model of convection—RBC henceforth—formulated on the basis of experimental work by Henri Bénard (Bénard 1901) and subsequent theoretical treatment by Lord Rayleigh (Rayleigh 1916) in the early twentieth century. In its basic form, it consists of a layer of Boussinesq fluid contained between a warm, impenetrable horizontal plate at the bottom, and a cold one at the top.
The RBC setup with rotation and others derived from it have been used for decades as a paradigmatic framework for the study of astro- and geophysical convection, both in laboratory experiments and numerical simulations (e.g., Chandrasekhar 1953; Rossby 1969; Boubnov and Golitsyn 1986; Julien et al. 1996; Stevens et al. 2009; King et al. 2012; Stevens et al. 2013; Nieves et al. 2014; Kunnen 2021). Much interest has been given to the rapidly rotating regime in particular, where the time scale of rotation is substantially shorter than that of convection—that is, where the convective Rossby number, Roc, is much less than unity [for a recent review of rotating RBC, see Ecke and Shishkina (2023)]. Large-scale vortices have been shown to emerge in the highly turbulent, rapidly rotating regime for domains with anisotropic geometries (Guervilly et al. 2014; Guervilly and Hughes 2017; Julien et al. 2018). Unlike tropical cyclones, however, their structure is depth invariant, and their formation depends on the horizontal aspect ratios being greater than 1.1:1.
A distinctive feature of tropical cyclones is the high level of organization of convection. For instance, model-based studies of moist convection have shown that, when combined with rotation, aggregated convective clusters often become tropical cyclones (Bretherton et al. 2005; Nolan et al. 2007; Muller and Romps 2018; Carstens and Wing 2020). For dry RBC setups, numerical simulations and theoretical work have shown that convective aggregation happens readily in domains where the boundary conditions are prescribed as constant fluxes instead of constant temperatures. In nonrotating RBC, constant-flux boundaries lead to the formation of a convective cell that spans the whole domain, as shown by early studies of mantle convection (Hurle et al. 1967; Chapman and Proctor 1980), and more recently for internally heated/cooled convection close to the onset of convective instability (Goluskin 2016; Lepot et al. 2018). Vieweg et al. (2021) have shown through Lyapunov analysis that this aggregation toward large scales persists above the critical Rayleigh number, and well into the highly turbulent regime.
To our knowledge, only a handful of studies so far have brought together the conditions of prescribed heat fluxes at the boundaries and slow rotation for 3D RBC (Dowling 1988; Takehiro et al. 2002; Vieweg et al. 2022). Importantly, they have found that even rotation rates within the slowly rotating regime—with Rossby numbers of O(10)—can constrain the aggregation to sizes that are not domain limited, provided that the rotation is not too slow. Furthermore, a couple of recent studies have shown that cyclonic vortices with eye-like structures form in axisymmetric simulations of relatively slowly rotating, nonturbulent RBC with lateral walls and constant flux thermal boundaries (Oruba et al. 2017, 2018), although it is unclear if these phenomena can arise in 3D, highly turbulent simulations in the absence of lateral confinement. It is worth noting that an unpublished dissertation by Kannan (2023) has recently explored tropical cyclogenesis within a similar framework of horizontally constrained, rotating RBC using large-eddy simulations (LES) and hydrodynamic instability analysis. However, the material is currently not publicly available for proprietary reasons, and we lack enough information about the setup, parameter space or findings to draw a proper comparison with the literature or the present work.
A different line of work has modified the original RBC system to include a simple analog for latent heating, simulating a lapse rate that is stable to dry and unstable to moist convection (Pauluis and Schumacher 2010). Although more complex than the classical RBC, this “moist Rayleigh–Bénard” system has made possible the study of aggregation (Pauluis and Schumacher 2011) and tropical cyclone formation (Chien et al. 2022) under relatively simple conditions that are one step closer to those we observe on Earth. Particularly, Chien et al. (2022) have found that TC-like structures form when the parameter determining the conditional instability is high enough, and that simulations with convective aggregation in the nonrotating case yield TC-like vortices when a strong enough rotation is introduced. However, a preexisting conditional instability is not likely the primary source of energy for real-world TCs, and a more realistic formulation should account for the fundamental energetic contribution from surface fluxes (Craig and Gray 1996). In particular, the enhancement of surface heat fluxes by the overlying flow of air, also referred to as wind-induced surface heat exchange (WISHE), has been shown to play a fundamental role in setting the strength of real-world tropical cyclones (Emanuel 1986; Zhang and Emanuel 2016; Chavas 2017).
These recent efforts share a similar objective to ours: to better understand the nature of convective organization and TC formation, and its relation to the basic equations of fluid motion. These are thus complementary approaches that strive to gain insights into different aspects of the same big problem. By limiting our study to dry convection, our approach to the question sacrifices more of the realism of the phenomena in favor of preserving greater simplicity in their conceptualization. The motivation behind this is twofold: on the one hand, dry TCs have been shown to arise in models and their dynamics are not fully understood yet. On the other, we see great inherent value in creating a dialogue between the traditional studies of dry convection in idealized settings and tropical atmospheric physics.
In this article, we use numerical simulations of the Boussinesq equations in the rotating RBC setup in the moderately rotating, moderately turbulent regime, to show the effects of different thermal boundary conditions and rotation rates on convective organization, and on the formation of TC-like vortices. Additionally, we introduce a simple parameterization that relates the magnitude of the heat flux at the bottom to the overlying flow, and show that the flow dependence of the heat fluxes—or, in other words, a wind-induced surface heat exchange mechanism—can significantly enhance the strength of the TC-like vortices. We characterize the vortex properties, and conclude by commenting on the role of flux asymmetry, rotation, and convective organization on their formation.
2. Methods
We run a suite of direct numerical simulations of 3D, dry, turbulent convection to explore the phenomena of convective organization and dry tropical cyclone formation and persistence under different combinations of thermal boundary conditions. For the simulations, we use the pseudospectral PDE solver Dedalus, version 2 (Burns et al. 2020), to solve the Navier–Stokes, continuity, and heat equations with the Boussinesq approximation and the thermal boundary conditions explained below. The basic setup consists of a 3D domain that is doubly periodic in the horizontal, bounded by impenetrable surfaces and with no-slip flow conditions at the top and bottom, and uses 512 × 512 Fourier basis functions in the horizontal, and 32 Chebyshev basis functions in the vertical, which increase in density near the boundaries.
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,
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 Roc = 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 Raf between 1013.3 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 Raf = 109 for all simulations with at least one flux-based thermal boundary condition, and Ra = 107.3 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
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, tConv = H/W, where W represents
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
Combinations of thermal boundary conditions used in the simulations. Conventions are detailed in the text.
PF denotes a parameterized heat flux condition, where
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 (RafPr)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 107.3 for CT/CT, and Raf = 109 for all other setups. We initially fix Roc 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 Roc 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] × (zt − z) × (z − zb), where zt and zb 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.
3. Results and discussion
a. Convective organization
Vertically averaged temperature anomaly, {T′} with respect to the horizontal mean for simulations with Roc = 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
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
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.
(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
Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-23-0018.1
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 Roc 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.,
Horizontal wind at 20% of domain height for simulations with Roc = 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
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
A length scale
In Ins/PF,
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
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 − Tdom (colors), where Tdom 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 − Tdom. The alignment is less clear for the core of the vortex of Ins/CF, plausibly because of its lower axisymmetry and its weaker nature.
Azimuthally averaged radius vs height profiles centered at TC center for simulations with Roc = 2.0, averaged between the ninth and the tenth rotational time period, for (left) Ins/CF and (right) Ins/PF. (top) T − Tdom (color) and total angular momentum (contours). (middle) T − Tdom (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
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.
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 CE/CD = 1 (red; see text for details). The values of υm and rm indicate the maximum azimuthal wind speed [
Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-23-0018.1
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 rm) 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 CE/CD 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 Roc 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 Roc = 2.0 persists for much longer than a rotational time period, suggesting a qualitatively different behavior, and a threshold around Roc ∼ 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 Roc.
Plan views of horizontal wind speeds at 20% of domain height for simulations with Ins/PF conditions, for various values of Roc, 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
The structure and the persistence of the vortices for the lowest Roc simulations differ markedly from those with Roc = 0.8 and, especially, Roc = 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 Roc = 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.
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
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 Roc 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 Roc = 2.0 and the small, evanescent structures of Roc = 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 FB, 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, Roc,eff = Roc(1 + CkUh)1/3, which gives a more accurate bulk characterization of the flow in the domain than Roc. 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 Uh—the horizontal wind speed at 1% of domain depth used in our simple parameterization PF. We obtain Roc,eff = 0.23, 0.47, and 0.91 for the simulations with Roc = 0.2, 0.4, and 0.8, respectively, for which Uh stays fairly constant throughout the simulation. For the simulation with Roc = 2, we obtain Roc,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 Roc 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 Roc < 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 (Roc = 2; Γ = 32), still spanning the entire domain, which indicates that the storm has likely not reached its natural length scale. Similarly, (Roc = 3; Γ = 32) also produces a domain-spanning TC-like vortex, but with a significantly narrower eye. The smaller eye for larger Roc 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: (Roc = 4; Γ = 32) did not produce any persistent, large-scale vortices within more than 20 rotational periods simulated, and neither did Roc = 2 or Roc = 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.
Vertical profiles of horizontally averaged temperature minus domain-averaged temperature,
Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-23-0018.1
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, rm. 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 up/2Ω is, where up 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 rm as up/2Ω is. This agreement between rm and RD 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
Vertical profiles of normalized, horizontally averaged temperature
Citation: Journal of the Atmospheric Sciences 80, 9; 10.1175/JAS-D-23-0018.1
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 (RD ∼ 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 Ck 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 Ck = 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 Ck = 1.0 none form within 10 rotational periods (see Fig. 16 in the supplement for a comparison between the two values of Ck).
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.
4. Conclusions
We have shown that thermal boundary conditions significantly affect the patterns of convection in simulations of a relatively highly turbulent (Raf = 109; Ra = 107.3), moderately rotating (Roc = 2.0) flow in a Rayleigh–Bénard-like setup of dry convection with a large geometric aspect ratio (Γ = L/H = 16), with Prandtl number Pr = 1.0. Particularly, we have found that setups with boundary conditions with prescribed heat fluxes produce larger-scale and more persistent organization of convection, relative to simulations with at least one fixed-temperature boundary condition.
Importantly, in simulations with asymmetric flux-based thermal boundary conditions, we find long-lived vortices that have a number of the essential characteristics of tropical cyclones: a warm core, a bottom-heavy azimuthal circulation, high levels of axisymmetry, approximate slantwise convective neutrality, and a secondary circulation of radial convergence at the bottom and divergence aloft. These structures, obtained from direct numerical simulations, are similar to the dry TCs found in previous work in 3D simulations with atmospheric models (Cronin and Chavas 2019; Wang and Lin 2020). We note that rotation and persistent, large-scale organization of convection are not sufficient conditions for the formation of dry TCs.
The basic setups that produce TC-like vortices are characterized by an insulating top and either a constant or a parameterized heat flux condition at the bottom. These are analogous to setups with constant or parameterized heat fluxes at the top with an imposed internal cooling rate matching the domain-averaged rate of warming. The parameterized condition prescribes a low-level temperature gradient enhancement that depends on the overlying flow—a simplified analog to the bulk aerodynamic formulae for enthalpy exchange used in atmospheric models—promoting a WISHE-like mechanism of intensification that produces the strongest, largest, and most axisymmetric TC simulated. An additional simulation with parameterized heat flux at the bottom and constant heat flux at the top also gives rise to a well-defined TC-like vortex, suggesting that a strong enough asymmetry of fluxes is conducive to TC development.
Also for the setup with insulating top and parameterized-flux bottom, which produces the clearest TC-like vortices, we find a threshold in the convective Rossby number somewhere between Roc = 0.8 and 2.0. Below it, vortices are disorganized and short lived, suggesting that faster domain rotation weakens the convection, and prevents the buoyant plumes from the bottom from sustaining an organized warm core that extends to the top of the domain.
We find that the simulations resulting in TC formation are characterized by a positive average thermal stratification in the upper half of the domain. This stable layer has a corresponding bulk deformation radius, which is proportional to the convective Rossby number, and which agrees fairly well with the scale of the radius of maximum winds, particularly in the simulation with parameterized bottom fluxes. Testing whether the thermal stratification and the associated propagation of gravity waves in the upper part of the domain associated with this deformation radius are a controlling factor or merely a result of TC formation would require additional simulations with larger domain aspect ratios. Further tests would also be necessary to confirm that the radius of maximum winds for dry TCs does indeed scale with the deformation radius.
To synthesize, we have provided evidence that flux-based thermal boundary conditions promote larger spatial organization and persistence of convection in time than temperature-based conditions. We have also argued that flux asymmetry results in a persistent and localized input of buoyancy at the bottom, allowing spatially aggregated and temporally autocorrelated convection to sustain a warm core in a developing large-scale vortex more easily than when the heat fluxes at the boundaries are symmetric. This process is enhanced when a mechanism analogous to a wind-induced surface heat exchange (WISHE) is prescribed.
We cannot definitively rule out the formation of dry TC-like structures under conditions that did not produce them in our simulations, as it is possible that a stronger thermal forcing, a larger aspect ratio, a slower rotation rate, or a longer time of simulation could lead to flow transitions that depart from our results. Confirming this would require development of a full theory of the conditions for dry TC formation. That said, this work attests to the higher propensity for development of TC-like vortices of highly turbulent setups with asymmetric heat flux conditions and a rotation rate that is slow enough for a given thermal forcing (
Although recent work by Chien et al. (2022) has found TCs in the relatively simple moist-RBC system of Pauluis and Schumacher (2010), to our knowledge, this is the first time that TCs are simulated in a dry, three-dimensional, RBC-like setup. We would like to reiterate that our work seeks to promote a closer dialogue between the two distinct communities studying the fluid mechanics of dry convection, and the geophysical problems of convective aggregation and tropical cyclone formation, by showing that dry hurricanes can form under relatively simple conditions of rotating convection in familiar setups about which much has been learned in the past decades. Future questions to explore include, for instance, whether the most unstable modes of marginal stability for the different convective setups persist in the highly turbulent regime; how TC strength depends on the thermal forcing or the enhancement parameter Ck; how finite-amplitude vortices behave in domains with faster rotation than
We end by noting that not all aspects of the physics of dry TCs can be extrapolated to their moist counterparts, but even exploring that distinction may produce valuable insights about the factors that control hurricanes on Earth and potentially on other planetary contexts.
Acknowledgments.
We acknowledge valuable discussions with Kerry Emanuel, Glenn Flierl, Morgan O’Neill, Peter Molnar, Keith Julien, and Matthieu Kohl. We also thank Dan Chavas and two anonymous reviewers for useful, constructive suggestions that substantially improved the work presented here; all shortcomings are exclusively the authors’ responsibility. This work was made possible by high-performance computing support from Cheyenne (DOI:10.5065/D6RX99HX) provided by NCAR’s Computational and Information Systems Laboratory (CISL), sponsored by the National Science Foundation. We declare no conflict of interest.
Data availability statement.
The output data and the scripts used for postprocessing are publicly available in the Zenodo repository with DOI:10.5281/zenodo.7622997.
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