1. Introduction
Tropical cyclone intensification theory has a long and venerable history (Montgomery and Smith 2014; Emanuel 2018) but has largely focused on simplified models in which the vortex is vertically aligned and the internal moist convection is either purely or statistically axisymmetric. While such a focus has facilitated progress toward understanding the thermofluid dynamics of intensification, it manifestly neglects an entire dimension of the problem. The author of the present article contends that a comprehensive conceptual understanding of tropical cyclone intensification must take into account the common reality of vortex misalignment (tilt) and the associated asymmetric distribution of moist convection. Such violation of the traditional theoretical assumption of axisymmetric structure can be especially pronounced during the prehurricane phases of intensification (e.g., Fischer et al. 2022), when the vortex seems most prone to having considerable tilt in association with exposure to a moderate degree of transient or sustained environmental vertical wind shear (e.g., Jones 1995; Reasor et al. 2004).
The effects of tilt on tropical cyclone intensification have been examined to some extent in the past but have not been fully elucidated. Numerous studies have suggested that an appreciable tilt will generally slow or even neutralize lowlevel spinup (e.g., DeMaria 1996; Riemer et al. 2010; Rappin and Nolan 2012; Tao and Zhang 2014; Finocchio et al. 2016; RiosBerrios et al. 2018; Schecter and Menelaou 2020, hereinafter SM20; Fischer et al. 2021; Schecter 2022, hereinafter S22). On the other hand, tilted systems with sufficiently strong downtilt convection have been known to occasionally exhibit core reformation followed by rapid intensification (e.g., Molinari et al. 2004; Molinari and Vollaro 2010; Nguyen and Molinari 2015; Chen et al. 2018; Alvey et al. 2022). For the common scenario of slow spinup, there does not yet exist a comprehensive quantitative theory for the dependence of the intensification rate on relevant parameters of the tilted system. Moreover, there may be a number of distinct slow modes of asymmetric intensification that have not yet been discovered or explicitly recognized. Although a quantitative condition for fast spinup initiated by core reformation has been proposed, there are still questions as to whether the underlying theory is adequate (see below). The essential purpose of the present study is to advance our current quantitative understanding of the distinct intensification mechanisms available to tilted tropical cyclones, their conditions of applicability, and their operational time scales.
The approach adopted herein is to consider a simplified fluid dynamical system that facilitates experimental control over the convection that drives intensification. In particular, this study considers a dry threedimensional vortex that is misaligned and subjected to parameterized diabatic forcing that generates deep convection concentrated downtilt of the surface vortex center for basic consistency with observations (cf. Reasor et al. 2013; Stevenson et al. 2014; Nguyen et al. 2017). The specifics of the heating distribution and the coupling of its center to the continuously changing tilt vector of the vortex are varied so as to cover a range of possibilities that are potentially relevant to tropical cyclones in nature and in cloudresolving simulations under a variety of environmental conditions. A standard oceanic surface drag parameterization is generally implemented, but its role is limited to that of an agent of kinetic energy dissipation; the regulatory influence of Ekmanlike pumping on the heating distribution is not directly incorporated into the model. Indeed, the model under present consideration cannot answer questions about what regulates the local spatiotemporal properties of the heating distribution, nor what regulates the relationship between the heating center and the tilt vector. Such issues can only be investigated through observational and fullphysics modeling studies and have been extensively (albeit incompletely) addressed elsewhere (see many of the previous references, along with, e.g., Zawislak et al. 2016; Onderlinde and Nolan 2016; Gu et al. 2019; Alvey et al. 2020; Rogers et al. 2020; Alland et al. 2021a,b). The questions to be answered herein are limited to those concerning how intensification varies with the parameters characterizing the nature of the asymmetric internal heating.
Schecter (2020, hereinafter S20) provided some preliminary insights into what to expect from the present study. To elaborate, S20 considered a shallowwater vortex representing the lowlevel circulation of a tropical cyclone, forced by an offcenter mass sink representing downtilt convection. The mechanism and time scale of vortex intensification expectedly varied with the velocity convergence generated by (and occupying the same space as) the mass sink. The prevailing intensification mechanism was largely determined by whether the magnitude of convergence exceeds a critical value dependent on the spatial extent of the mass sink, the drift velocity of the mass sink, and the contribution to the local flow velocity from the larger scale cyclonic circulation. Supercritical convergence horizontally trapped fluid undergoing vorticity amplification inside the mass sink, whereas subcritical convergence allowed the fluid to escape and recirculate around the broader cyclone. When having supercritical strength, a convergence zone displaced from the central region of the cyclone generally induced onsite reformation of the vortex core followed by fast intensification. The process notably resembled the initiation of fast spinup through core reformation that—as mentioned earlier—is occasionally seen in real and realistically simulated tropical cyclones. Vortices possessing subcritical convergence zones were found to follow one of two slower pathways of development. One of the slower modes of intensification entailed a gradual merger of the vortex center with the convergence zone, coinciding with a gradual reduction of the radius of maximum azimuthal velocity r_{m}. The other involved no such merger, nor any appreciable change of r_{m}.
The extent to which the results of S20 should carry over to the model under present consideration is not entirely obvious. To begin with, the presence of horizontal vorticity and the associated vertical differential advection in a threedimensional tropical cyclone–like vortex could substantially alter the production of vertical vorticity in the convection zone and its subsequent evolution. Moreover, the inclusion of surface drag (absent in S20) should provide an effective counterbalance to slow intensification mechanisms, and possibly cause spindown. One important issue to be addressed is whether the critical lowlevel convergence required for core reformation remains consistent with the S20 shallowwater theory. Another issue to be addressed is the extent to which threedimensionality and surface friction alter the nature of subcritical intensification and its dependence on the properties of the lowlevel convergence zone associated with downtilt convection.
Needless to say, S20 and the present study are not the first to consider the intensification of tropical cyclone–like vortices resulting from experimentally controlled diabatic forcing. This approach has been used extensively in the context of axisymmetric models and has shown inter alia that heat sources tend to more efficiently intensify vortices when situated in regions of relatively high inertial stability near or inward of the radius of maximum wind speed (Vigh and Schubert 2009; Pendergrass and Willoughby 2009). There have also been fully 3D studies of vortex intensification resulting from various forms of asymmetric diabatic forcing. Some of the aforementioned studies have focused on quasilinear dynamics (e.g., Nolan et al. 2007), while others have employed models that include stronger nonlinear effects (Dörffel et al. 2021, hereinafter D21; Päschke et al. 2012). The quasilinear models have been useful for assessing the extent to which waves induced by asymmetric diabatic forcing influence the azimuthalmean flow of the vortex and thereby change its maximum tangential wind speed. However, quasilinear models cannot be used to investigate some of the highly nonlinear processes of present interest, such as those associated with core reformation. D21 can be seen to have some features in common with the present study, in using a nonlinear model and in prescribing the asymmetric diabatic forcing in relation to the tilt of the tropical cyclone–like vortex. On the other hand, owing to its distinct theoretical objectives, the case studies of D21 used broad dipolar heating instead of predominantly positive heating concentrated downtilt of the surface center, provisionally neglected surface friction, and did not explicitly address core reformation.
There exists another simplified experimental approach for investigating the pathways of tropical cyclone intensification driven by offcenter localized convection—not necessarily associated with tilt—that merits brief discussion. Instead of directly forcing the system with a heat source, clustered vorticity perturbations representing the product of localized convection can be added to the broader cyclonic circulation at time intervals deemed consistent with natural convective pulsing. Past studies adopting this approach have paid considerable attention to how angular momentum is redistributed by vortex Rossby waves (or subvortices) following the episodes of convection that create the vorticity anomalies (Montgomery and Enagonio 1998; Möller and Montgomery 1999, 2000; Enagonio and Montgomery 2001). These studies have also examined the intensity required for a vorticity anomaly to supplant the core of a parent cyclone (Enagonio and Montgomery 2001). The present study (and S20) can be seen to complement those just described by taking a step toward elucidating the efficiency of vorticity buildup in the convergence zone associated with convection, and how that efficiency affects the pathway of intensification.
The remainder of this paper is organized as follows. Section 2 describes the model used for the present study and provides an overview of the numerical experiments. Section 3 describes the results of the numerical experiments. Differences between subcritical and supercritical intensification are illustrated. Distinct scalings for subcritical and supercritical intensification rates are presented. Section 4 relates the results of section 3 to realworld and realistically simulated tropical cyclone dynamics. Section 5 summarizes all main findings of the study.
2. Basic methodology
a. The model used to simulate tilted “tropical cyclones”
The objectives of this study are achieved primarily through numerical simulations of tropical cyclone–like vortex intensification conducted with a simplified version of release 19.5 of Cloud Model 1 (CM1; Bryan and Fritsch 2002). CM1 is a widely used nonhydrostatic atmospheric model with high precision numerics and conventional parameterizations of subgrid turbulent transport, cloud microphysics and radiative transfer. Herein, the latter two features are deactivated. The resulting dry model is forced with an adjustable source term in the potential temperature (θ) tendency equation that substitutes primarily for downtilt moistconvective heating (see below). Subgrid turbulent transport above the surface layer is represented by an anisotropic Smagorinskytype parameterization specified in section 2a of SM20. Surface momentum fluxes are represented by a bulkaerodynamic formula appropriate for oceanic systems, in which the drag coefficient C_{d} increases from a minimum value of 0.001 to a maximum value of 0.0024 as the surface wind speed increases from 5 to 25 m s^{−1}. Surface enthalpy fluxes are invariably turned off. Rayleigh perturbationdamping is applied for z > 25 km, in which z denotes height above sea level. All simulations are set up on a doubly periodic f plane with a Coriolis parameter given by f = 5 × 10^{−5} s^{−1}.
The equations of motion are discretized on a stretched rectangular grid that spans 2660 km in both horizontal dimensions, and extends upward to z = 29.2 km. The 800 × 800 km^{2} central region of the horizontal mesh that contains the tilted vortex core has uniform increments of 2.5 km; at the four corners of the mesh, the increments are 27.5 km. The vertical grid has 40 levels spaced apart by distances that increase from 0.1 to 0.7 to 1.4 km as z increases from 0 to 8 to 29 km.
b. Simulation preparation
Each simulation is conducted in two stages. The first stage occurring over the interval t_{−} ≤ t < 0 involves initialization and vertical misalignment of the vortex. The second stage occurring for t ≥ 0 involves the evolution of the vortex under the influence of diabatic forcing. This section pertains to the first stage of the simulation.
c. Simulation groups
The simulations conducted for this study can be separated into groups that are distinguished by selected parameters used to prepare and force the system. The 8–16 simulations in any particular group differ from one another only in the strength parameter a of the diabatic forcing [Eq. (1)], which usually spans two orders of magnitude (from 10^{−3} to 10^{−1} K s^{−1}).^{2} Variation of a over such a broad interval will provide a thorough picture of how the vortex intensification process in each simulation group changes with the magnitude of the lowlevel convergence generated by the heating. A wide variety of simulation groups will be considered for the main purpose of demonstrating a certain universality of this picture. The differences between each simulation group are explained below in the context of a reference group.
Table 1 lists all distinguishing or previously unspecified parameters related to the preparation and forcing of systems in the reference group. The vorticity coefficient ζ_{o} of the original vortex yields winds of tropical storm intensity. The magnitude and duration of the preparatory shear flow are set to leave the vortex with a corescale tilt. Following a 6h adjustment period after the preparatory shear flow subsides, at which point the clock reads t = 0, the tilt magnitude (x_{ml}_{,0} in Table 1) is 81.8 km. By the same time, surface drag has reduced the maximum azimuthally averaged tangential velocity in the boundary layer (υ_{bm}) to 17.2 m s^{−1}, and the radius at which it occurs (r_{bm}) to 85.0 km. Note that both υ_{bm} and r_{bm} are measured in a polar coordinate system whose origin is at the lowlevel vortex center. The diabatic forcing of the vortex is peaked in the middle troposphere and is minimal (but nonzero) at the surface. The heating distribution decays over a radial length scale of 35 km from its center x_{f} in the horizontal plane. The heating center is driven toward its target location—the midlevel vortex center—on a time scale τ_{f} of 1 h. There is no sustained shear flow to influence the intensification process that may commence when the diabatic forcing begins.
Reference group parameters.
Table 2 lists all other simulation groups considered for this study, which differ from the reference group by the parameter changes that are shown in the rightmost column. Simulations in groups TLTX2 and TLTX3 are prepared with more intense preliminary shear flows that roughly double and triple, respectively, the initial tilt magnitude. Simulations in group SH2P5‖ (SH2P5⊥) each include sustained shear flows with 2U_{s} = 2.5 m s^{−1} and
Features distinguishing the nonreference groups from the reference group.
The final two simulation groups listed in Table 2 (CD0 and CD0+) have drastic reductions of surface drag. CD0 changes the bottom surface boundary condition to freeslip, whereas CD0+ homogenizes and reduces C_{d} by two orders of magnitude. Comparison of these simulation groups with the reference group (labeled REF in tables and figures) will illustrate a sharp distinction between weakly forced simulations with negligible and standard levels of surface drag.^{3} A more comprehensive analysis of how results vary with the surface drag parameterization would stray too far from the main narrative of this paper but is provided in appendix B for readers who may have some interest in the topic. Note that appendix B is best read after section 3.
3. Simulation results
a. Variation of the intensification time scale with the heating magnitude
The data in Fig. 4 show that when the heating magnitude a exceeds a critical value, given by a_{c} ≈ 0.0275 K s^{−1}, the normalized intensification time scale t_{2}/τ_{σ} has a nearly constant value between 3 and 4. In other words, the υ_{bm}doubling period is directly proportional to τ_{σ}. Below the critical value, t_{2}/τ_{σ} rapidly grows and diverges as a decreases toward a_{0} ≈ 0.002 K s^{−1} (left dashed line). The divergence reflects diabatic spinup diminishing to the point of becoming completely countered by the negative impact of surface drag [see section 3d(3)]. For a < a_{0}, the vortex decays.
Although surface friction markedly exacerbates the subcritical slowdown of intensification, there is clear evidence that the normalized growth of t_{2} with decreasing a (below a_{c}) has other contributing factors. The white diamonds superimposed on Fig. 4—taken from groups CD0 and CD0+—show that removing surface friction from the reference group does not eliminate subcritical slowdown. Although t_{2} no longer diverges as a approaches a_{0} from the right, decreasing a from a_{c} toward zero still causes multifold growth of t_{2}/τ_{σ}. In other words, a less efficient intensification mechanism appears to emerge as a drops below a_{c} regardless of whether the simulation includes surface drag.
b. Subcritical and supercritical pathways of intensification
Figure 5 illustrates the root cause for the dynamical transition across the critical heating magnitude a_{c}. Each panel shows nearsurface streamlines superimposed over a contour plot of relative vertical vorticity ζ in a pertinent subregion of the lowlevel vortex near the center of the diabatic forcing, immediately or soon after the forcing reaches full strength. The images are in a reference frame that moves with the heating center, in which the horizontal velocity field is given by
Figures 6–8 provide broader perspectives of the nearsurface vorticity evolution and wind speed intensification in each of the foregoing simulations, as viewed from an Earthstationary reference frame. Figure 6 corresponds to the subcritical system subjected to the weakest forcing. The escape of enhanced vorticity from the convergence zone and its subsequent recirculation are evident upon comparing the ζ snapshots at t = 1.5 and 4 h. As the system evolves, the distance between the lowlevel vortex center (white +) and the heating center (large black ×) decays at a variable rate. Henceforth, this distance will be represented by the variable
Figure 7 corresponds to the subcritical system with intermediate forcing. Although the nearsurface streamlines do not develop a point of attraction in the vicinity of the heating center, the streamer of enhanced vorticity leaving the area does not travel too far away. Instead, the head of the streamer shortly coalesces with the central vorticity anomaly of the original cyclone as the latter surges closer to x_{f}. The end result is a smaller vortex core whose center lies closer to the diabatic forcing. Whether the depicted evolution should be viewed as a variant of “core reformation” will be discussed shortly.
Figure 8 corresponds to the supercritical system subjected to the strongest forcing. The nearsurface streamlines are seen here, as in Fig. 5e, to have formed a point of attraction near x_{f} after an hour of development, at which time the diabatic forcing has achieved full intensity. Immediately afterward—within a period that is appreciably shorter than the advective time scale over a distance comparable to
Figure 9 shows time series of several notable vortex parameters in each of the preceding simulations. The vortex parameters include υ_{bm}, r_{bm}, the radial offset
Let us first consider the time series for the subcritical simulations. Figure 9a corresponds to the simulation having the weakest diabatic forcing. The initial values of r_{bm},
Figure 9c corresponds to the supercritical simulation. In contrast to the preceding cases, the early drops of r_{bm} and
c. Core reformation and core replacement
The term “core (or center) reformation” is widely used in tropical cyclone meteorology in reference to the occasionally observed rapid emergence of a relatively small but dominant vorticity core in an area of localized convection away from the original center of a prehurricane vortex (e.g., Molinari et al. 2004; Molinari and Vollaro 2010; Nguyen and Molinari 2015; Chen et al. 2018; Alvey et al. 2022). This fairly broad concept might be seen to encompass the initial phases of the intensification processes in both the subcritical system with intermediate forcing^{5} and the supercritical system considered in section 3b. Nevertheless, the core reformation mechanisms differ between the two cases. Most notably, the supercritical mechanism distinctly entails the appearance of a point of attraction for the streamlines in close proximity to the heating center, where the convergence of trapped fluid generates a new core with a length scale considerably smaller than δr_{f}. To avoid ambiguity in terminology, the supercritical mode of core reformation will be called “core replacement.”^{6}
d. Comprehensive analysis of the intensification rate
1) Boundaries of the intensification period
The default and most common value for t_{i} is set to t_{id} ≡ 0.8 h, which corresponds to when the diabatic forcing has achieved 80% of its ultimate strength. A modification is made if a signature of core replacement is observed after t_{id}. Specifically, t_{i} is reset to when the ratio of
The default end time t_{e} is the solution of the following equation:
The beginning and end of the intensification period of each reference simulation in Fig. 9 are marked by the ticks labeled t_{i} and t_{e} on the bottom axis of each subplot. These examples are considered typical for systems with (Fig. 9c) and without (Figs. 9a,b) a core replacement event. Figure 10 is similar to an individual subplot of Fig. 9, but for an illustrative simulation from WEAKVTLTX3 that intensifies without undergoing core replacement. Here the initial adjustment preceding t_{i} involves a roughly 50% reduction of
The reader may have some concern that—for systems with applied shear—the orientation of the tilt vector relative to
2) Similarity of the IR curves
Each simulation group in Fig. 11 is represented by a symbol with a distinct combination of size, shape, and color (see the legend). Filled symbols with the darkest shading correspond to simulations that undergo robust core replacements. Consistent with theoretical expectations, these simulations generally have criticality parameters exceeding unity. The empty (white filled) symbols correspond to simulations that show no sign of core replacement during vortex intensification. Consistent with theoretical expectations, these simulations have criticality parameters less than unity.
However, the boundary between simulations with and without core replacement appears to be less sharp than theory would suggest. A small number of simulations with criticality parameters measurably less than unity (whose symbols have relatively light shading) were flagged by an objective algorithm for exhibiting core replacement. The algorithm does not explicitly check for a point of attraction in the vicinity of diabatic forcing but does check for a pronounced splitting of smallscale and largescale vortex centers, coinciding with discontinuous drops of r_{bm} and
In the supercritical parameter regime where all measures indicate that core replacement generally occurs and is nearly always robust, the normalized IR can be viewed to have an approximately constant value of 0.8 ± 0.1. In the subcritical parameter regime, the variation of the normalized IR in each simulation group can be approximated by a linear expression of the form
In the supercritical parameter regime, the combination of roughly constant values for the normalized IR and υ_{i} (Table C1) in a given simulation group implies that
3) Sawyer–Eliassenbased analysis of lowlevel spinup
Section 3a suggested that the growth of the nondimensional intensification rate from the point of zero IR to supercriticality is not exclusively a consequence of frictional damping becoming less effective in counteracting the growing strength of diabatic forcing. Nevertheless, the diminishing importance of frictional damping is a major factor contributing to accelerated spinup that merits further discussion. Such discussion is facilitated by using the traditional framework of Sawyer–Eliassen (SE) theory (Shapiro and Willoughby 1982; Schubert and Hack 1982; Smith et al. 2005; SM20). The SEbased analysis presented below is conducted in a reference frame that moves with the lowlevel vortex. The cylindrical coordinate system (with radius r and azimuth φ) is centered on x_{l}. The variables u, υ, and w respectively represent the radial, azimuthal and vertical velocity fields in the aforementioned coordinate system. As usual, an overbar (prime) is used to denote the azimuthal mean (perturbation) of a fluid variable.
Figure 13 more clearly demonstrates the rising dominance of diabatic forcing over frictional spindown by showing the ratio λ ≡ (Γ_{h} +
4) Anticorrelation between the mean convective displacement and the criticality parameter
Earlier studies have suggested that faster spinup will result not only from stronger diabatic forcing, but also from decreasing the distance ℓ between the heat source and the lowlevel vortex center (cf. Pendergrass and Willoughby 2009; Vigh and Schubert 2009; S20). It is therefore reasonable to wonder whether greater normalized IRs at higher values of the criticality parameter
Figure 14a shows two distinct measurements of ℓ versus the criticality parameter. The inset shows ℓ at the start of the intensification period (t = t_{i}), whereas the main graph shows the time average of ℓ over the entire intensification period (t_{i} ≤ t ≤ t_{e}). First consider the subcritical simulations for which the base10 logarithm of
It is worth remarking that, in contrast to the supercritical state of affairs, the time averages of ℓ and r_{bm} are positively correlated in subcritical systems for which core replacement never occurs (empty symbols) or unsuccessfully attempts to occur (light filled symbols) during the intensification period (Fig. 14b). In fact, the two quantities generally become nearly equal as ℓ increases beyond approximately 2δr_{f}.^{11} It stands to reason that the decay of the time average of ℓ as
e. Lowlevel vorticity production
Figure 15 compares the stretching term with the sum of all other contributions to Dζ/Dt during the early developmental stages of typical subcritical and supercritical systems. The plotted tendencies are temporally averaged over relatively short time periods (see the caption) and vertically averaged from the surface to z = 3.1 km. The figure suggests that in the vicinity of downtilt heating, the stretching term on the whole tends to be stronger than the sum of all other terms. The disparity is evidently more pronounced in the supercritical system, which happens to be in the midst of a core replacement event. The fairly dominant status of the stretching term helps explain why shallowwater theory is adequate for predicting the critical convergence required to initiate a core replacement event in the 3D model under present consideration.
4. Connection to realistic tropical cyclone dynamics
a. Intensification rates
Figure 16 shows the IRs of all intensifying vortices under present consideration, normalized to the MPIR of WLX21 with V_{max} set to a value (95 m s^{−1}) that is near the current upperbound of observations (Kimberlain et al. 2016). A sizeable subset of subcritical systems realistically have IRs below the MPIR. On the other hand, all supercritical cases exhibiting a wellestablished core replacement event have IRs that are more than 3 times as great as the MPIR. This suggests that sustained intensification associated with core replacement in our simulation set would not be realistic. At best, the intensification following one of our simulated core replacement events could last only a brief period of time (no longer than 2 h) to permit a 6h IR within natural bounds.^{12} If the vortex were much weaker to begin with, or if the diabatic forcing happened to drift at a velocity closer to that of the local lowertropospheric background flow, so as to greatly increase τ_{c}, the supercriticality condition τ_{c}/τ_{σ} > 1 required for core replacement could be satisfied with a much larger value of τ_{σ} (much weaker heating). The associated IR, which scales as
b. Diabatic forcing
The diabatic forcing used for the present study was designed to roughly conform with observations and fullphysics simulations of misaligned tropical cyclones in having deep cumulus convection concentrated downtilt of the surface vortex center. Whether the morphological details of the diabatic forcing are realistic merits further consideration. Data from the cloudresolving simulations of S22 (specified in appendix F, section a) provide a reasonable basis for comparison. Figure 17 shows the nominal heating distributions of downtilt convection in three tropical cyclones from S22 with underlying sea surface temperatures of 26°C (left column), 28°C (middle column) and 30°C (right column). To be precise, each plot shows the azimuthal mean of the material derivative of θ in a cylindrical coordinate system whose central axis passes through the downtilt heating center x_{f} that is defined by Eq. (F1) of appendix F, section b. The top plot in each column corresponds to a time average over a selected 2h analysis period when the system is at depression or tropical storm intensity, whereas the bottom plot corresponds to an overlapping 6h average. Moderate differences of intensity and spatial structure between the “short” and “long” time averages of each heating distribution demonstrate that while downtilt convection may be persistent (appendix A of S20), the steady diabatic forcing employed for this study after ramping is inexact. Moreover, the S22 heating distributions suggest that in contrast to our simplified parameterization scheme [Eq. (1)], the peak of the diabatic forcing is not constrained to lie on its central axis. In further contrast, the S22 heating distributions often have appreciable azimuthal variation around their central axes (not shown).
Figure 18 provides a more elaborate and quantitative analysis of the S22 dataset. Figure 18a shows the distance ℓ between the heating center and the lowlevel vortex center versus the tilt magnitude. Here and in all other subplots, each data point with error bars corresponds to a 6h interval during the prehurricane evolution of a tropical cyclone. The 6h interval is divided into three 2h segments. The coordinates of each data point (marked by a solid symbol) correspond to the medians of the 2h time averages of the plotted variables. The error bars extend from the minimum 2h time average to the maximum. The condition that ℓ remain comparable to the tilt magnitude (enforced herein except in RFOUT) appears to be reasonably consistent with the unconstrained results of S22. Figure 18b shows that the angle φ_{f} of the position vector of the heating center (in a coordinate system centered at x_{l}) measured counterclockwise from the direction of the tilt vector is generally negative, but reasonably close to zero as assumed for the reference group and most other simulations examined for the present study. Only a few exceptional cases coinciding with relatively small values of ℓ have magnitudes of φ_{f} exceeding 45°.
Figures 18c–f contain information on the intensity and length scales of the heating distribution. All but one of the plotted parameters are obtained from a nonlinear least squares fit of the 2h heating distribution (exemplified in the top row of Fig. 17) to a function equivalent to the righthand side of Eq. (1), but with T → 1 and
The preceding considerations offer some reassurance that the form of the diabatic forcing used for the present study is not egregiously detached from reality, or at least from what might be found in a cloudresolving model. The use of a steady heating distribution may leave a somewhat incomplete picture of the dynamics, but the complications associated with moderate temporal fluctuations can be readily examined in the future (cf. S20). There may also be circumstances worthy of future study in which a purely positive heat source inadequately represents downtilt convection (cf. appendix E). In considering the potential shortcomings of the diabatic forcing, one should further bear in mind that the heating rate applied at any point in the vortex is dynamically independent of the history and instantaneous vertical velocity of the local air parcel. In principle, this could introduce some slightly unrealistic features of 3D convection in our model. That being said, analysis of several reference simulations (not shown) has suggested that a qualitatively realistic statistical correlation tends to develop between
5. Conclusions
The study at hand aimed to gather insights into the mechanisms by which a misaligned tropical cyclone may intensify when deep convection is concentrated downtilt of the surfacevortex center. The methodology involved conducting numerous simulations with a 3D nonhydrostatic model that incorporates an imposed heat source to generate downtilt convection. The simulations were divided into over a dozen groups that differed from one another in the initial vortex strength, the initial tilt magnitude, the environmental shear flow, the prescribed displacement of downtilt heating from the moving midlevel vortex center, or the vertical heating profile. Variation of vortex intensification in each simulation group was controlled by adjusting the magnitude of the heat source. The following key results were obtained:

Distinct modes of intensification occur depending on whether the boundary layer convergence
${\tau}_{\sigma}^{1}$ in the vicinity of the downtilt heat source is above or below a critical value. The critical value${\tau}_{c}^{1}$ found in each simulation group agrees with shallowwater theory (S20) in approximately equaling 2 times the magnitude of the vector difference between the drift velocity of the heating center and the local velocity of the nondivergent background flow, divided by the radial length scale of the heat source [see Eq. (9)]. If the convergence is supercritical, such that τ_{c}/τ_{σ} exceeds unity, boundary layer fluid entering the convergence zone becomes horizontally trapped, and its vertical vorticity continuously amplifies. The result is the local emergence of a smallbutstrong vorticity core that eventually dominates the parent cyclone and rapidly intensifies. If the system is subcritical, boundary layer fluid generally passes through the convergence zone, where it experiences only a transient episode of vorticity enhancement while losing some of its original mass to vertical convection. The fluid with moderately enhanced vorticity typically recirculates around the inner core of the broader cyclone. Meanwhile, if the diabatic forcing is not too weak, the inner core progressively contracts and slowly intensifies. Bear in mind that some deviation from the preceding scenario can occur at relatively large subcritical values of τ_{c}/τ_{σ} [see sections 3c and 3d(2)]. 
Quantitatively, the fast mode of supercritical intensification that follows core replacement occurs at a rate that is measured to be approximately proportional to υ_{i}/τ_{σ}, in which υ_{i} [precisely defined in section 3d(2)] is a characteristic velocity scale that increases with the initial mean absolute vorticity in the broader vicinity of the heat source. In other words, the normalized intensification rate (IR) defined by (δυ_{bm}/δt)τ_{σ}/υ_{i} is roughly constant. In the subcritical parameter regime, the normalized IR was found to decline approximately linearly with decreasing values of τ_{c}/τ_{σ} to the point of becoming negative owing to the emergent dominance of frictional spindown. A limited number of simulations with negligible surface drag have suggested (in agreement with S20) that even without frictional dissipation, the time scale of subcritical intensification normalized to τ_{σ} can exhibit multifold growth as the diabatic forcing tends toward zero (Fig. 4).

In all of the simulation groups, the strength of diabatic forcing required to induce a supercritical downtilt core replacement event would cause unrealistically fast intensification when viewed over a typical observational time scale of 6 h or longer. It stands to reason that such strong forcing would have to end shortly after core replacement in a natural tropical cyclone. In principle, supercritical conditions are possible with weaker diabatic forcing that could realistically last well beyond core replacement. In comparison with the systems considered herein, the drift velocity of the downtilt convection zone would most likely have to be closer to the local velocity of the lowertropospheric background flow so as to substantially increase τ_{c} (see also appendix E).
While this study has clearly illustrated some basic differences between subcritical and supercritical modes of asymmetric intensification, there is undoubtedly more to learn, especially on the subject of subcritical intensification. In the linear model used to describe the subcritical relationship between the normalized IR and τ_{c}/τ_{σ}, the slopes and points of zero IR obtained from the simulation groups showed some spread that is yet to be fully elucidated. One might reasonably expect to find far greater variability in nature, owing to greater diversity in the structure and propagation dynamics of downtilt convection. In theory, such diversity could even add branches to the normalized IR curve associated with distinct pathways of lowlevel spinup (cf. S20). Let it suffice to say for now that further research will be needed to obtain a truly comprehensive understanding of subcritical dynamics.
Thus, for example, the combination r_{f}_{*} = x_{ml} and φ_{f}_{*} = 0 would imply that x_{f}_{*} = x_{m}.
The upper limit of a is extended to an unnaturally high value to provide a lucid picture of the scaling of the vortex intensification rate when the diabatic forcing is relatively strong; see section 4a for a related discussion.
Data from both CD0 and CD0+ are considered to verify that negligibledrag results are insensitive to minor differences in the CM1 configuration options that are used in conjunction with freeslip and semislip boundary conditions.
This definition of the horizontal boundary layer velocity field is also used to evaluate the maximum wind speed υ_{bm} that was introduced in section 2c.
There are several reasons why the subcritical dynamics of the system with a = 4a_{c}/11 might be seen to entail a marginal case of core reformation. As shown earlier, the vortex core in the boundary layer rapidly (over a period of 1.5 h) shrinks to onehalf of its initial size in terms of r_{bm} while relocating to a position substantially closer to the diabatic forcing. Immediately after this event, the centers of the small new core and the broader circulation linked to the original core are arguably well separated. [The measured separation distance ranges from 24 to 50 km when the defining radial length scale of the broader circulation (r_{c} of appendix A) is between 70 and 100 km.] Furthermore, the subsequent wrapping of outer vorticity around the new core (Fig. 7; t = 4 h) resembles the aftermath of a prototypical reformation event illustrated in Fig. 11 of Molinari et al. (2004).
In S20, the author reserved the term “core reformation” for its supercritical variant (“core replacement”). In hindsight, this may have been too restrictive.
The alternative use of υ_{bm}(t_{i}) is found to reduce the spread less effectively.
A sensitivity test has been conducted with τ_{σ} redefined to be the inverse of the average of σ_{b} within a radius δr_{f} of the convergence center x_{σ} that is precisely defined in appendix A; the average of x_{σ} − x_{f}/δr_{f} over the intensification period is 0.23 ± 0.03 for supercritical systems and 0.66 ± 0.11 for subcritical systems. The redefinition typically results in a moderate fractional reduction of
Group TLTX3 is excluded from the stated mean and standard deviation of χ. The regression for TLTX3 (which yields χ = 0.67) has a correlation coefficient of 0.588, indicating a poor fit. For the other simulation groups, the correlation coefficient is 0.986 ± 0.015.
Bear in mind that because SE theory neglects unbalanced dynamics, Γ_{h} and
The few anomalous cases in this parameter regime for which the time average of ℓ substantially exceeds that of r_{bm} correspond to sheared systems in which the diabatic forcing is too weak to prevent the gradual separation of the lowlevel and midlevel vortices.
The model used for this study was not designed to remain realistic long after a core replacement event under general circumstances. Following such a dramatic structural transformation of the vortex in a real system, the diabatic forcing (moist convection) is expected to eventually reorganize and diminish if abnormally intense.
Repetition of the fit with d_{f} constrained to equal 0 gives a similar range of results for a.
Thus, C_{d} = 0.005 for group CD5, C_{d} = 0.003 for group CD3, C_{d} = 0.001 for group CD1, and C_{d} = 0.0005 for group CD05.
One might expect a stronger vorticity anomaly to develop over time in the convergence zone, where
Acknowledgments.
The author thanks three anonymous reviewers for their constructive feedback and suggestions on how to improve several aspects of this article. The author also thanks Dr. George Bryan of the National Center for Atmospheric Research (NCAR) for providing the original atmospheric model (CM1) that was tailored for this study to simulate the intensification of tropical cyclone–like vortices subjected to downtilt diabatic forcing. This study was supported by the National Science Foundation under Grants AGS1743854 and AGS2208205. A number of simulations conducted for this study were made possible with resources provided by NCAR’s Computational and Information Systems Laboratory (https://doi.org/10.5065/D6RX99HX).
Data availability statement.
CM1 code modifications and input files for selected simulations, which together may be used to help reproduce the main results of this study, are available online (https://doi.org/10.5281/zenodo.7637579). Archived simulation output files are presently available to researchers upon request to the author (schecter@nwra.com).
APPENDIX A
Vortex and Convergence Centers
Let x_{δ} represent the horizontal position vector of the vortex center in a vertical layer indicated by the subscript δ. In general, x_{δ} corresponds to the location at which one must place the origin of a polar coordinate system to maximize the peak value of
Slightly different definitions are used for x_{l} and x_{m} to calculate the righthand side of the equation for dx_{f}/dt in the parameterization of diabatic forcing that is added to CM1. The redefinitions are intended partly to improve computational efficiency, and partly to reduce large shortlived fluctuations of the heating center that may occur in conjunction with similar fluctuations of the tilt vector. Specifically, the layer corresponding to x_{l} (x_{m}) is collapsed onto the horizontal plane at z = 1.2 (7.8) km—so that no vertical averaging is necessary for the computation of
Figure A1 illustrates how the tilt vector x_{ml} ≡ x_{m} − x_{l} used for the runtime parameterization of diabatic forcing in a simulation can deviate from that which would result from replacing the vortex centers with those used for the postruntime data analysis in the main text. Notable differences tend to emerge when the radius of maximum wind speed of the lowlevel or midlevel vortex decreases below the 55km cutoff value in the runtime search algorithm. Differences will of course diminish when the smallscale and mediumscale circulations become increasingly concentric in each layer.
The main text contains several references to the convergence center x_{σ} of the boundary layer velocity field. In analogy to the vortex center, the convergence center is defined to be the origin of the polar coordinate system that maximizes the peak value of
APPENDIX B
Sensitivity to C_{d}
Section 3a (Fig. 4) addresses the consequences of eliminating surface drag on the time scale of vortex intensification but does not thoroughly examine C_{d} sensitivity. Figure B1 offers a more comprehensive picture of how the normalized IR varies as C_{d} increases from zero toward the upper extreme of inferred oceanic values (see Bell et al. 2012). The plotted data primarily come from six groups of simulations configured with constant C_{d}: two groups with zero or nearzero surface drag (CD0 and CD0+), and four groups labeled CDX with C_{d} = 0.00X.^{B1} Apart from modification of the surface drag coefficient at t = 0, all of the preceding simulation groups are set up like the reference group. Data from the reference group, for which 0.001 ≤ C_{d} ≤ 0.0024, are shown for context. Note that the values of υ_{i} (τ_{c}) for all plotted simulations have a standard deviation of only 7% (10%) of the mean. It stands to reason that υ_{i} and τ_{c} can be viewed approximately as constants in the axis labels.
Figure B1 shows that increasing C_{d} generally decreases the normalized IR at a fixed value of the criticality parameter
APPENDIX C
Group Statistics for υ_{i} and τ_{c}
Table C1 first summarizes the statistics of the scaling velocity υ_{i} that appears in the expression for the normalized intensification rate of Fig. 11. The means and standard deviations are shown for both subcritical (column υ_{i}_{,sub}) and supercritical (column υ_{i}_{,sup}) systems in each simulation group. The fractional deviations from the mean are usually small within either parameter regime of a particular simulation group, suggesting that the subcritical and supercritical values of υ_{i} can be viewed as approximate constants. Differences between subcritical and supercritical means are noticeable but generally minor. On the other hand, the mean value of υ_{i} in either parameter regime can change appreciably from one simulation group to another. Such can be seen by comparing values from (for example) the groups labeled REF and WEAKVTLTX3.
Left and center data columns: scaling velocities for subcritical (sub) and supercritical (sup) systems, each expressed as a group mean ± 1 standard deviation rounded to two decimal places. Right data column: time scale for background advection across the heat source.
Table C1 also summarizes the group statistics of the time scale τ_{c} for background advection across the downtilt heat source measured during the early phase of intensification, as explained in section 3d(2). Although the mean of τ_{c} can change appreciably from one simulation group to another (compare values associated with SH5⊥ and WEAKVTLTX3), the standard deviation for a given group is usually small. The small standard deviation implies that variation of
APPENDIX D
SE Computations
The following briefly summarizes the SE equations for each component Ψ_{α} of the streamfunction of the azimuthally averaged secondary circulation, and several approximations that are used to solve them. The reader may consult appendix D of SM20 for a more thorough discussion. The only notable difference between the SE analysis of this paper and that of SM20 is the substitution of applied diabatic forcing for the cloudmicrophysical heat source.
For all computations of Ψ_{α}, ellipticity of the SE equation is restored where violated below z = 400 m by adjusting the static stability as described in SM20, with the adjustment parameter (nu) given by 0.001. The solution to the SE equation is then obtained by a straightforward numerical method that enforces the boundary condition Ψ_{α} = 0 at r = 0, r = 898 km, z = 0 and z = 29.2 km. Once the SE equation is solved, the component of the azimuthally averaged secondary circulation associated with Ψ_{α} can be calculated from the following formula:
Using a method of approximation similar to that of SM20, all azimuthally averaged variables appearing in the coefficients and forcing terms of the SE equation for Ψ_{α} are time averaged over the moderately short analysis period. Similar time averages are used for
APPENDIX E
Hypothetical Effect of a Dipolar Component to Downtilt Heating on the Critical Convergence Required for Core Replacement
As noted in the main text, S20 theorized that a point of attraction would exist and core replacement would occur in the region of downtilt convergence provided that τ_{c}/τ_{σ} > 1, or equivalently that
In the preceding notation, the critical convergence above which a point of attraction exists in the absence of a divergence zone is given by
While hardly rigorous or comprehensive, the previous considerations suggest that allowing a dipolar component to exist in the downtilt convergence field could measurably reduce the critical convergence for core replacement and thus lengthen the time scale for supercritical intensification.
APPENDIX F
CloudResolving CM1 Simulations
a. Summary of the dataset
Table F1 summarizes the subset of data from S22 that is used in section 4b as a basis for assessing the adequacy of the diabatic forcing used for this study. The leftmost column lists the simulations that are included in the dataset. The naming convention is equivalent to that found in S22. The prefix indicates whether the sea surface temperature is 26° (T26), 28° (T28), or 30°C (T30). The first two letters of the suffix indicate whether the simulation is low resolution (LR) or high resolution (HR); the former (latter) has a grid spacing equal to (half of) that used herein. The terminal letter (A, B, etc.) is used to distinguish simulations with the same SST and resolution, but different initial conditions.^{F1} The second column specifies the method used to create the initial tilt, along with the magnitude of the initial tilt vector (x_{ml}_{,0}). The initialization methods [dry separation plus damping (DSPD) and impulsive separation plus damping (ISPD)] are explained in S22. The third column shows the 6h time periods during which data are collected for Fig. 18; needless to say, time is measured from when the simulation is initialized. The last column gives the maximum azimuthally averaged tangential surface velocity of the tropical cyclone (υ_{sm}), time averaged over the analysis period to the left.
Synopsis of the cloudresolving tropical cyclone simulations analyzed in section 4b. See text for discussion.
b. Tilt vector and heating parameters
The vortex centers required to compute the tilt vector x_{ml} and heating displacement ℓ for tropical cyclones in the cloudresolving CM1 simulations are obtained by the procedure explained in SM20, which differs in only a few minor details from the procedure used for the diabatically forced tropical cyclones considered herein. Further elaboration is deemed unnecessary.
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