Editorial Type:
Article
Article Type:
Research Article

Intensification Rates of Tropical Cyclone–Like Vortices in a Model with Downtilt Diabatic Forcing and Oceanic Surface Drag

David A. Schecter aNorthWest Research Associates, Boulder, Colorado

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Online Publication:
10 Jul 2023
Print Publication:
01 Jul 2023
DOI:
https://doi.org/10.1175/JAS-D-22-0188.1
Page(s):
1787–1814
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Abstract

Tropical cyclones are commonly observed to have appreciable vertical misalignments prior to becoming full-strength hurricanes. The vertical misalignment (tilt) of a tropical cyclone is generally coupled to a pronounced asymmetry of inner-core convection, with the strongest convection tending to concentrate downtilt of the surface vortex center. Neither the mechanisms by which tilted tropical cyclones intensify nor the time scales over which such mechanisms operate are fully understood. The present study offers some insight into the asymmetric intensification process by examining the responses of tilted tropical cyclone–like vortices to downtilt diabatic forcing (heating) in a 3D nonhydrostatic numerical model. The magnitude of the heating is adjusted so as to vary the strength of the downtilt convection that it generates. A fairly consistent picture of intensification is found in various simulation groups that differ in their initial vortex configurations, environmental shear flows, and specific positionings of downtilt heating. The intensification mechanism generally depends on whether the low-level convergence σb produced in the vicinity of the downtilt heat source exceeds a critical value dependent on the local velocity of the low-level nondivergent background flow in a reference frame that drifts with the heat source. Supercritical σb causes fast spinup initiated by downtilt core replacement. Subcritical σb causes a slower intensification process. As measured herein, the supercritical intensification rate is approximately proportional to σb. The subcritical intensification rate has a more subtle scaling, and expectedly becomes negative when σb drops below a threshold for frictional spindown to dominate. The relevance of the foregoing results to real-world tropical cyclones is discussed.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: David A. Schecter, schecter@nwra.com

Abstract

Tropical cyclones are commonly observed to have appreciable vertical misalignments prior to becoming full-strength hurricanes. The vertical misalignment (tilt) of a tropical cyclone is generally coupled to a pronounced asymmetry of inner-core convection, with the strongest convection tending to concentrate downtilt of the surface vortex center. Neither the mechanisms by which tilted tropical cyclones intensify nor the time scales over which such mechanisms operate are fully understood. The present study offers some insight into the asymmetric intensification process by examining the responses of tilted tropical cyclone–like vortices to downtilt diabatic forcing (heating) in a 3D nonhydrostatic numerical model. The magnitude of the heating is adjusted so as to vary the strength of the downtilt convection that it generates. A fairly consistent picture of intensification is found in various simulation groups that differ in their initial vortex configurations, environmental shear flows, and specific positionings of downtilt heating. The intensification mechanism generally depends on whether the low-level convergence σb produced in the vicinity of the downtilt heat source exceeds a critical value dependent on the local velocity of the low-level nondivergent background flow in a reference frame that drifts with the heat source. Supercritical σb causes fast spinup initiated by downtilt core replacement. Subcritical σb causes a slower intensification process. As measured herein, the supercritical intensification rate is approximately proportional to σb. The subcritical intensification rate has a more subtle scaling, and expectedly becomes negative when σb drops below a threshold for frictional spindown to dominate. The relevance of the foregoing results to real-world tropical cyclones is discussed.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: David A. Schecter, schecter@nwra.com
Keywords: Tropical cyclones; Intensification; Idealized models

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 thermo-fluid 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 low-level spinup (e.g., DeMaria 1996; Riemer et al. 2010; Rappin and Nolan 2012; Tao and Zhang 2014; Finocchio et al. 2016; Rios-Berrios 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 three-dimensional 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 cloud-resolving 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 Ekman-like 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 full-physics 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 shallow-water vortex representing the low-level circulation of a tropical cyclone, forced by an off-center 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 on-site 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 rm. The other involved no such merger, nor any appreciable change of rm.

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 three-dimensional 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 low-level convergence required for core reformation remains consistent with the S20 shallow-water theory. Another issue to be addressed is the extent to which three-dimensionality and surface friction alter the nature of subcritical intensification and its dependence on the properties of the low-level 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 quasi-linear 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 quasi-linear models have been useful for assessing the extent to which waves induced by asymmetric diabatic forcing influence the azimuthal-mean flow of the vortex and thereby change its maximum tangential wind speed. However, quasi-linear 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 off-center 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 build-up 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 real-world 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 moist-convective heating (see below). Subgrid turbulent transport above the surface layer is represented by an anisotropic Smagorinsky-type parameterization specified in section 2a of SM20. Surface momentum fluxes are represented by a bulk-aerodynamic formula appropriate for oceanic systems, in which the drag coefficient Cd 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 perturbation-damping 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 km2 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.

The source term added to the equation for the material derivative of potential temperature (Dθ/Dt) is of the form
θ˙f≡aexp⁡{−(r˜/δrf)2−[(z−zf)/δzf±]2}T(t;δτf),
in which a is the strength parameter, r˜ is radius measured in the horizontal plane from the forcing center xf, δrf is the radial length scale of the forcing, and zf is the height of maximum forcing. The symbol δzf± represents the upper vertical length scale (δzf+) of the forcing if z > zf, or the lower vertical length scale (δzf−) if z < zf. The last factor is a ramp function of time t, defined by T ≡ max(t/δτf, 0) for t < δτf and T ≡ 1 for t ≥ δτf. Figure 1a shows θ˙f/(aT) for a case with typical vertical asymmetry about zf. In general, zf lies in the middle-to-upper troposphere, and the downward decay length (δzf−) is of comparable magnitude (see section 2c).
Fig. 1.
Fig. 1.

(a) Normalized heating distribution θ˙f/(aT) with typical vertical asymmetry characterized by δzf+=7δzf−/12. (b) Diagram showing the polar coordinates rf* and φf* of an arbitrarily placed target position for the heat source xf*. The polar coordinate system has its origin at the center xl of the red low-level vortex (LLV) and is oriented such that φf* is zero in the direction of the tilt vector xml, which points from xl to the center xm of the blue midlevel vortex (MLV).

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

The forcing center is governed by the following prognostic equation:
dxfdt=−xf−xf*τf,
in which τf is a relaxation time and xf*(t) is a moving target for xf that usually lies in the vicinity of the midlevel vortex center xm. Without exception, xf is initialized to xf* at t = 0. In general, xf* is specified by its radius rf* and azimuth φf* in a polar coordinate system (Fig. 1b) whose origin is at the low-level vortex center xl, and whose orientation continuously changes to keep the zero azimuth along the direction of the evolving tilt vector xml ≡ xm − xl.1 The trajectories of xm and xl are tracked while the simulation runs. The reader may consult appendix A for details on the computations of xl and xm.
A subset of simulations includes additional forcing on the right-hand side of the horizontal velocity (u) tendency equation of the form
Fs≡∂us∂te^s+fusz^×e^s.
The purpose of Fs is to generate and sustain an ambient shear flow coaligned with the fixed unit vector e^s in the horizontal plane. The shear flow is given by
us(z,t)=Us2tanh⁡(z−zlδzl)[1+tanh⁡(zu−zδzu)]T(t;δτs),
in which Us is an adjustable maximum wind speed, zl = 5 km is the center of the primary shear layer, δzl = 2.5 km is the half-width of the primary shear layer, and zu = 21 km is the upper altitude at which the shear flow decays toward zero with increasing height over a length scale δzu of 1 km. The last factor T is the temporal ramp function defined previously, but with δτf replaced by δτs =1 h. Figures 2a and 2b respectively illustrate the dependencies of us on height and time. Along with Fs, Rayleigh damping of the form Fd≡−(u−use^s)ϒd(r⌣;rd,δrd)/δτd is added to the right-hand side of the equation for ∂u/∂t in the periphery of the simulation domain to prevent sheared-away structures from re-entering the system as a result of periodic boundary conditions. The dependence of the damping on radius r⌣ from the domain center is given by ϒd = 0 for r⌣≤rd, and ϒd={1−cos⁡[πmin⁡(r⌣−rd,δrd)/δrd]}/2 for r⌣≥rd. In all simulations with applied shear flows, rd = 1230 km, δrd = 100 km, and δτd = 300 s. Note that the present methodology used for imposing the ambient shear flow excludes the coupled horizontal potential temperature gradient that would be found in nature to maintain thermal wind balance (cf. Nolan 2011). Note also that the invariant vertical structure of the ambient shear flow used for the present study clearly limits sensitivity tests to those involving variations of the magnitude Us and orientation e^s of the velocity field. Efforts to ascertain the sensitivities of vortex intensification to structural details of the shear flow, akin to those previously conducted with cloud-resolving models (e.g., Finocchio et al. 2016; Onderlinde and Nolan 2016; Gu et al. 2019; Fu et al. 2019), will be deferred to a future time.
Fig. 2.
Fig. 2.

(a) The normalized velocity us/(UsT) (solid curve) and (b) the time factor T of the shear flow [Eq. (4)] that is applied after t = 0 to a subset of simulations. (c) The time factor T˜ that is substituted for T in Eq. (4) for the preparatory shear flow that creates the initial tilt of each tropical cyclone–like vortex (see section 2b).

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

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.

At t = t−, the system is initialized with an axisymmetric baroclinic vortex in a stably stratified atmosphere (Fig. 3). The vertical vorticity of this “original vortex” has the following form:
ζ(r,z)={ζoe−(r/ro)2cos⁡[π(z−zo)2δzo±]−ζc(z)}H(rb−r)H(zo+δzo+−z),
in which r is radius from the vortex center, ro = 91 km, zo = 3 km, δzo±≡δzo+ (δzo−) for z > zo (z < zo), δzo+=11km, and δzo− has an effectively infinite value of 332.2 km. The Heaviside step function is defined by H(x) ≡ 1 for x > 0 and 0 for x < 0. The small vorticity correction −ζc brings the azimuthal velocity υ(r,z)=∫0rdr′r′ζ(r′,z)/r to zero at r = rb = 750 km. The maximum azimuthal velocity υmo occurs at the radius rmo = 100 km and the altitude zo. The υ field varies minimally below zo but gradually decays above zo until reaching zero at z = 14 km. Most simulations are prepared with ζo = 8.837 × 10−4 s−1, in which case υmo = 25 m s−1. The vertical distributions of pressure and θ outside the vortex (r > rb) match those of the Dunion (2011) moist tropical sounding. Within the vortex, the aforementioned fields are adjusted to satisfy gradient and hydrostatic balance conditions consistent with υ.
Fig. 3.
Fig. 3.

The relative vertical vorticity (colors) and potential temperature (contours) of the original balanced vortex for simulations with ζo = 8.837 × 10−4 s−1. The right edge of the plot shows the atmospheric pressure p at r = 125 km for selected values of z.

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

The vortex is subsequently tilted by a transient shear flow generated by a forcing term F˜s on the right-hand side of the ∂u/∂t equation that is similar to Fs [Eqs. (3) and (4)], but with the time-factor T replaced by
T˜(t˜ ;τ˜s,δτs˜)≡{t˜/δτ˜s0≤t˜<δτ˜s,1δτ˜s≤t˜<τ˜s,1−(t˜−τ˜s)/δτ˜sτ˜s≤t˜<τ˜s+δτ˜s,0t˜≥τ˜s+δτ˜s,
in which t˜≡t−t−. The equation for T˜ implies that the shear flow accelerates from zero to its maximum value over the ramping period δτ˜s, holds steady until t˜=τ˜s, and then decelerates until terminated at t˜=τ˜s+δτ˜s (Fig. 2c). The nearly negligible domain-averaged shear flow that may exist beyond the termination time in practice is then damped by replacing F˜s with −⟨u⟩xy/τ˜sd until t≡t−+t˜=0. In the preceding expression for the damping rate, 〈…〉xy has been used to denote the horizontal average of the angle-bracketed variable. In general, the tilting procedure smoothly separates the lower vortex from the upper vortex over a transition layer between roughly 2.5 and 7.5 km above sea level.

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 low-level 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 ζo 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 core-scale tilt. Following a 6-h adjustment period after the preparatory shear flow subsides, at which point the clock reads t = 0, the tilt magnitude (|xml,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 (υbm) to 17.2 m s−1, and the radius at which it occurs (rbm) to 85.0 km. Note that both υbm and rbm are measured in a polar coordinate system whose origin is at the low-level 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 xf in the horizontal plane. The heating center is driven toward its target location—the midlevel vortex center—on a time scale τf of 1 h. There is no sustained shear flow to influence the intensification process that may commence when the diabatic forcing begins.

Table 1.

Reference group parameters.
Table 1.

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 right-most 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 2Us = 2.5 m s−1 and e^s rotated by an angle φe of 0° (−90°) from the direction of the initial tilt vector xml,0. In other words, the vortices in SH2P5‖ (SH2P5⊥) are exposed to a modest level of shear parallel to (clockwise perpendicular to) the initial tilt. Simulations in groups SH5‖ and SH5⊥ are similar to those in their SH2P5-counterparts, except for having stronger shear flows with 2Us = 5 m s−1. Simulations in group RFOUT are distinct from those in the reference group in having their heating centers shifted outward of the midlevel vortex center, by letting rf* equal 1.5 times the tilt magnitude. Simulations in groups PHIFM45 and PHIFP45 are distinct in having their heating centers shifted 45° clockwise and counterclockwise, respectively, from the direction of the tilt vector. Simulations in group ZFUP distinctly have their altitudes of maximal heating shifted 2.25 km upward. Simulations in group WEAKV have relatively weak original vortices, characterized by a 40% reduction of ζo. Simulations in group WEAKV-TLTX3 are similar to those in WEAKV, but their initial vortices have much larger tilts.

Table 2.

Features distinguishing the nonreference groups from the reference group.
Table 2.

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 free-slip, whereas CD0+ homogenizes and reduces Cd 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

Figure 4 illustrates how the time t2 required for υbm to double varies with the magnitude a of the diabatic forcing in the reference group (black diamonds). The doubling period is normalized to a certain time scale τσ that increases with decreasing a (inset). Specifically, τσ is the inverse of the mean boundary layer convergence in the neighborhood of the diabatic forcing. The aforementioned boundary layer convergence is defined by σb ≡ −∇H ⋅ ub, in which ∇H is the horizontal gradient operator, and ub is the vertical average of u over the lowest 1.2 km of the troposphere.4 The computation of τσ generally involves taking the spatial average of σb over a circular disc of radius δrf centered at xf, where the applied heating is maximized in the horizontal plane. Using the divergence theorem, the disc average can be written σbf = −2ubf/δrf, in which ubf is the azimuthally averaged radial component of ub (in a coordinate system centered at xf) along the periphery of the disc. For the present analysis, the computation of τσ also involves taking a time average of σbf that begins at tα = 0 and extends to tβ = t2. To summarize,
τσ≡−(tβ−tα)/∫tαtβ2ubfδrfdt.
Fundamentally, τσ is a characteristic time scale for horizontal fluid contraction near the surface in the vicinity of the diabatic forcing. One may also view τσ as the time scale for the amplification of vertical vorticity resulting from such contraction.
Fig. 4.
Fig. 4.

Main plot: normalized length of time required for υbm to double vs the normalized heating magnitude in the reference group (black diamonds) and in similar simulations with the surface drag severely reduced or eliminated (white diamonds). The dashed vertical lines at a = a0 (left line) and a = ac (right line) mark the boundaries between the domains of (from left to right) spindown, subcritical intensification, and supercritical intensification in the reference group. Inset: anticorrelation between the convergence time scale (in the vicinity of diabatic forcing) and the heating magnitude.

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

The data in Fig. 4 show that when the heating magnitude a exceeds a critical value, given by ac ≈ 0.0275 K s−1, the normalized intensification time scale t2/τσ has a nearly constant value between 3 and 4. In other words, the υbm-doubling period is directly proportional to τσ. Below the critical value, t2/τσ rapidly grows and diverges as a decreases toward a0 ≈ 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 < a0, the vortex decays.

Although surface friction markedly exacerbates the subcritical slowdown of intensification, there is clear evidence that the normalized growth of t2 with decreasing a (below ac) 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 t2 no longer diverges as a approaches a0 from the right, decreasing a from ac toward zero still causes multifold growth of t2/τσ. In other words, a less efficient intensification mechanism appears to emerge as a drops below ac 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 ac. Each panel shows near-surface streamlines superimposed over a contour plot of relative vertical vorticity ζ in a pertinent subregion of the low-level 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 u˜≡u−dxf/dt. Each column corresponds to a distinct simulation from the reference group, with a increasing from left to right. The heating rates of the left and middle simulations are subcritical, whereas that of the right simulation is supercritical. Both subcritical cases show confluence of streamlines with peak convergence somewhat downstream of xf. The confluence coincides with amplification of vertical vorticity, but the fluid which carries the enhanced vorticity (and remains near the surface) eventually leaves the convergence zone to potentially recirculate around the broader cyclone. When the heating rate is supercritical, the streamlines develop a point of attraction inside the convergence zone. The bulk of fluid entering the convergence zone cannot escape in the horizontal plane, and the vorticity of that which remains near the surface continuously amplifies.

Fig. 5.
Fig. 5.

(a),(b) Horizontal streamlines superimposed over relative vorticity ζ near the surface (z = 0.7 km) in the reference group simulation with a = 2ac/11, at t = 1.5 h [in (a)] and 2.5 h [in (b)]. The streamlines are in a reference frame that moves with the heating center. The large × is located at the heating center xf, and the small × is located at the convergence center xσ, which is defined in appendix A. (c),(d) As in (a) and (b), but for the reference group simulation with a = 4ac/11. (e),(f) As in (a) and (b), but for the reference group simulation with a = 12ac/11 and at t = 1.0 h [in (e)] and 1.1 h [in (f)].

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

Figures 6–8 provide broader perspectives of the near-surface vorticity evolution and wind speed intensification in each of the foregoing simulations, as viewed from an Earth-stationary 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 low-level vortex center (white +) and the heating center (large black ×) decays at a variable rate. Henceforth, this distance will be represented by the variable ℓ ≡ |xf − xl|. As ℓ progressively decays, the radius of maximum wind speed contracts and the vortex intensifies. The process resembles that found for the shallow-water vortices forced by stationary or slowly precessing subcritical mass sinks in S20. One caveat is that the location of the diabatic forcing (analogous to the mass sink) in the present simulation is explicitly linked to the location of the midlevel vortex center. Therefore—unlike a shallow-water system—the reduction of ℓ over time (indicating alignment) involves both low-level and midlevel vortex dynamics.

Fig. 6.
Fig. 6.

Subcritical vortex intensification in the reference simulation with a = 2ac/11, viewed in an Earth-stationary reference frame with a domain-centered coordinate system. (top) Sequential snapshots (shown from left to right) of the streamlines and magnitude of the horizontal velocity field u at z = 50 m. (bottom) Corresponding sequential snapshots of relative vertical vorticity ζ (normalized to ζc = 10−5 s−1) at z = 0.7 km, displayed using a logarithmic color map for all grid cells with log10|ζ/ζc| ≥ 0. Grid cells with log10|ζ/ζc| < 0 are white. In all plots, the relatively large and small black × symbols are located at the heating center and convergence center, respectively. The white + is located at the low-level vortex center.

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

Figure 7 corresponds to the subcritical system with intermediate forcing. Although the near-surface 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 xf. 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.

Fig. 7.
Fig. 7.

As in Fig. 6, but for subcritical vortex intensification in the reference simulation with a = 4ac/11.

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

Figure 8 corresponds to the supercritical system subjected to the strongest forcing. The near-surface streamlines are seen here, as in Fig. 5e, to have formed a point of attraction near xf 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 ℓ—the low-level vortex center jumps to xf, where an intensifying subvortex becomes dominant over a length scale comparable to that of a typical hurricane eyewall. For reasons to be clarified below, the depicted evolution will be considered a proper case of “core replacement.” The subsequent intensification is unnaturally fast for a tropical cyclone, suggesting that the diabatic forcing is either unnaturally strong or would not persist for more than a brief moment in reality. Section 4a will reexamine this issue more quantitatively and put forth theoretically realizable conditions for which supercritical intensification following core replacement may operate over a longer time scale (in units of hours) under weaker forcing.

Fig. 8.
Fig. 8.

As in Fig. 6, but for supercritical vortex intensification in the reference simulation with a = 12ac/11.

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

Figure 9 shows time series of several notable vortex parameters in each of the preceding simulations. The vortex parameters include υbm, rbm, the radial offset ℓ of the diabatic forcing, and an alternative measure of the aforementioned offset given by ℓ2 ≡ |xf − xl2|. Whereas xl (in the definition of ℓ provided earlier) represents the low-level vortex center viewed on radial scales exceeding 10 km, xl2 represents the low-level vortex center viewed on radial scales exceeding 70 km, which is comparable to the original core size (see appendix A). Note that the values of υbm and rbm shown here and elsewhere are obtained from a search over the boundary layer vortex that is restricted to r ≥ 10 km, in part to ensure that the maximum wind speed measurement pertains to a well-resolved structure. The 10-km cutoff is judged to be acceptable for this study, because intensifying tropical cyclones do not usually have smaller values of rbm while at the strength of a tropical storm or low-category hurricane (e.g., Kimball and Mulekar 2004).

Fig. 9.
Fig. 9.

(a) Time series of the maximum tangential velocity in the boundary layer υbm (solid line), the radius of maximum velocity in the boundary layer rbm (long-dashed line), the distance ℓ from the heating center to the principal low-level vortex center xl (short-dashed line), and the distance ℓ2 from the heating center to the large-scale low-level vortex center xl2 (dotted line) for the subcritical reference simulation with a = 2ac/11. The plotted values of υbm are normalized to the initial value υbm0, whereas the plotted values of rbm, ℓ, and ℓ2 are normalized to the radial length scale of the heating distribution δrf. The secondary time axis shows t normalized to τσ defined with the averaging of σbf between ti and te [see section 3d(1)], which are marked on the bottom of the plot. (b) As in (a), but for the subcritical reference simulation with a = 4ac/11. (c) As in (a), but for the supercritical reference simulation with a = 12ac/11.

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

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 rbm, ℓ, and ℓ2 are virtually equivalent. After the 1-h ramping of the heat source, the vortex undergoes a 2-h adjustment to a state in which the aforementioned length scales have dropped by approximately 30%. Subsequently, rbm steadily decays and υbm continuously grows. Although ℓ and ℓ2 eventually decay toward the rbm curve, the onsets of their decays are delayed. Figure 9b corresponds to the simulation having intermediate forcing. The early contractions of the radial length scales are more pronounced, and those of ℓ and ℓ2 are not as uniform. Furthermore, the time scale of the dynamics is shorter whether viewed in units of hours or τσ. Otherwise, the plotted time series do not radically differ from those of the other subcritical system with relatively weak forcing.

Figure 9c corresponds to the supercritical simulation. In contrast to the preceding cases, the early drops of rbm and ℓ are virtually discontinuous (occurring almost entirely over an interval shorter than τσ) and terminate at lengths appreciably smaller than δrf. The discontinuous drops of rbm and ℓ occur once the tangential wind speed of the small-scale vortex emerging in the vicinity of diabatic forcing exceeds that of the large-scale parent cyclone, and xl immediately jumps to a location inside the forcing region. During this jump, the large-scale vortex center xl2 essentially holds position. Over time, the large-scale center gradually rejoins the small-scale center, through a process that presumably involves the continual convergence of outer absolute vorticity toward xl combined with axisymmetrization mechanisms similar to those found in nondivergent vortices. The major discontinuous separation and subsequent convergence of xl and xl2 are reflected in the major discontinuous splitting and gradual rejoining of ℓ and ℓ2.

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 forcing5 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 δrf. To avoid ambiguity in terminology, the supercritical mode of core reformation will be called “core replacement.”6

S20 derived a theoretical condition for the early existence of the point of attraction required to initiate core replacement. With a few simplifying assumptions, a point of attraction was found to exist in the convergence zone generated by diabatic forcing if and only if
τc/τσ>1.
In the preceding condition, τc is the time required for the local background flow to advect a fluid parcel across one-half the radial length scale of the convergence zone in a reference frame moving with the translational velocity of the convergence zone, and (as before) τσ is the local time scale for horizontal fluid contraction. We hypothesize that the condition in Eq. (8) applies not only to the shallow-water systems of S20 but is also required for core replacement in the three-dimensional systems under present consideration if τσ and τc are appropriately calculated. The formula for τσ will be given by Eq. (7). The formula for τc will be given by
τc≡(tβ−tα)/∫tαtβ2|uc−dxf/dt|δrfdt,
in which uc≡υ¯b2φ^2+⟨ub⟩xy, υ¯b2 is the azimuthal mean tangential component of ub evaluated at the radius ℓ2 in a polar coordinate system centered at xl2, φ^2 is the azimuthal unit vector at xf in the same coordinate system, and 〈ub〉xy is the domain average of ub. In the preceding formulation, uc neglects the presumably subdominant radial (r^2) velocity field of the large-scale cyclone but keeps 〈ub〉xy because of its potential importance in simulations with a substantial environmental shear flow. The end points of the time-averaging intervals (tα and tβ) used to evaluate τσ and τc must of course be chosen to have relevance for the intensification period under consideration and will be specified below.

d. Comprehensive analysis of the intensification rate

Heretofore, the focus has been on simulations from the reference group. The present goal is to demonstrate the similarity between intensification in the reference group and in all other simulations having the standard parameterization of oceanic surface drag. Rather than revisit the υbm-doubling period, which does not exist when a vortex decays, the new focus will be on the intensification rate (IR) given by
δυbmδt≡υbm(te)−υbm(ti)te−ti,
in which ti and te are the start and end times of a judiciously chosen intensification period.

1) Boundaries of the intensification period

The default and most common value for ti is set to tid ≡ 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 tid. Specifically, ti is reset to when the ratio of ℓ to ℓ2 (which starts at 1) is first seen to have precipitously fallen to a value less than 0.45. For all applicable simulations considered herein, this event coincides with virtually discontinuous drops of rbm and ℓ to values comparable to δrf or smaller. The foregoing reset of ti guarantees that the measured IR starts promptly after core replacement. Modifications to ti are also made for simulations in groups TLTX2, TLTX3, RFOUT, WEAKV, and WEAKV-TLTX3 that do not involve core replacements. Simulations from the aforementioned groups differ from others in having ℓ initially greater—sometimes much greater—than rbm. After an adjustment period, the time series of ℓ and rbm converge so as to better resemble the states of their counterparts from other groups at t = ti. Accordingly, should the event occur after tid, the start time ti is reset to when the ratio of ℓ to rbm drops below 1.05.

The default end time te is the solution of the following equation: te=ti+20τσie, in which τσie is given by the right-hand side of Eq. (7) with tα = ti and tβ = te. If the time t3 at which υbm becomes 3 times as large as its value at ti is smaller than the default end time, the end time is reset to t3. This reset generally prevents the intensification interval from overlapping the final phase of vortex development that is characterized by steady υbm.

The beginning and end of the intensification period of each reference simulation in Fig. 9 are marked by the ticks labeled ti and te 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 WEAKV-TLTX3 that intensifies without undergoing core replacement. Here the initial adjustment preceding ti involves a roughly 50% reduction of ℓ and a roughly 50% growth of rbm. Note that while ℓ at the start of the intensification period may be smaller than its initial value, it is still considerably larger than ℓ at ti in comparable reference simulations (e.g., Fig. 9a). Forthcoming analysis (the inset of Fig. 14a) will show the same to be true for all simulations devoid of core replacement events in WEAKV-TLTX3 and other groups (TLTX2 and TLTX3) whose constituent systems are initialized with relatively large tilts.

Fig. 10.
Fig. 10.

As in each panel of Fig. 9, but for a simulation from group WEAKV-TLTX3 with a = 0.0035 K s−1; ℓ2 is excluded from the plot because of its near equivalence to ℓ. The dotted line is an imaginary extension of the decay trend for υbm seen prior to ti.

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

The reader may have some concern that—for systems with applied shear—the orientation of the tilt vector relative to e^s during the intensification period (ti ≤ t ≤ te) differs considerably from its initial setting, which would render that initial setting irrelevant. For subcritical systems, the time-averaged angle between the tilt vector and e^s (−φe) during the intensification period is 15.3° ± 31.8° for SH2P5‖, 68.2° ± 19.2° for SH2P5⊥, 8.5° ± 16.3° for SH5‖, and 49.3° ± 8.3° for SH5⊥. Here, each angle is given as a group mean ± 1 standard deviation. The preceding measurements suggest that while the shear-relative tilt angles in SH2P5‖ and SH2P5⊥ (or SH5‖ and SH5⊥) are somewhat closer to each other than initially intended, the difference generally remains pronounced during the intensification period. For supercritical systems, the intensification period starts after core replacement creates an aligned vortex that rapidly intensifies and becomes virtually immune to moderate shearing. The author has difficulty imagining how at this point the orientation of the minimal tilt vector could be important.

2) Similarity of the IR curves

Figure 11 shows the dependence of a nondimensional measure of the IR on a criticality parameter that can be viewed as a nondimensional measure of the strength of diabatic forcing. The nondimensional IR is given by (δυbm/δt)×τσie/υi. The velocity that appears in the denominator of the scaling factor is defined by
υi≡(3δrf/2)∬Ad2x[ζb(x,ti)+f]/A,
in which ζb≡z^⋅∇H×ub and the integral is over the area A of a circular disc of radius 3δrf centered at xf(ti). Use of the preceding scaling velocity helps to reduce the IR spread in systems having the same criticality parameter but different vortex strengths or forcing locations at the start of intensification.7 The criticality parameter is defined by the ratio τc/τσie [cf. Eq. (8)], in which the time scale τc for advection across the forcing region is given by Eq. (9) with tα = ti and tβ = ti + (te − ti)/3. Note that the averaging interval used to compute τc is confined to an early phase of intensification. Extending the interval to a later phase—when ℓ2 is smaller and the vortex is stronger—could substantially decrease the value of the criticality parameter. The time scale τσ for convergence in the neighborhood of the steady diabatic forcing is generally less sensitive to the end-point tβ used for its evaluation. Appendix C tabulates basic statistics for υi and τc for the simulations under present consideration. The fractional variations of υi and τc within a given simulation group are generally small relative to those of τσ, but their characteristic values may differ considerably between two simulation groups.
Fig. 11.
Fig. 11.

Nondimensional IR plotted against the criticality parameter for all simulations with standard oceanic surface drag. See section 3d(2) for details.

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

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 small-scale and large-scale vortex centers, coinciding with discontinuous drops of rbm and ℓ to values comparable to δrf or smaller. In some cases (symbols with the lightest shading) the small-scale core quickly escapes the forcing region and weakens relative to the large-scale circulation so as to revert into a subdominant subvortex. The preceding scenario generally coincides with ℓ becoming greater than 2δrf. In other cases (symbols with medium shading) there is no sign of the small-scale core returning to subdominant status, but its center at some point in time obtains a position where δrf < ℓ < 2δrf. Similar behavior was also seen in two WEAKV-TLTX3 simulations with criticality parameters measurably greater than unity. By contrast, ℓ promptly becomes and remains smaller than δrf after core replacement in the multitude of all other (darkly shaded) supercritical simulations.

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 μ[τc/τσie−(τc/τσie)0]. Linear regressions for data with τc/τσie<0.9 give slopes and points of zero IR of μ = 0.73 ± 0.13 and (τc/τσie)0 = 0.15 ± 0.06, respectively. Here, each parameter is expressed as a mean ± 1 standard deviation of the results obtained for each simulation group. Pearson correlation coefficients close to unity (0.983 ± 0.021) verify that the linear model is generally an appropriate working assumption.8

In the supercritical parameter regime, the combination of roughly constant values for the normalized IR and υi (Table C1) in a given simulation group implies that δυbm/δt≈cg/τσie following a core replacement event, in which cg is a group-specific constant. This means that to a good approximation, the dimensional IR is directly proportional to the boundary layer convergence in the vicinity of the diabatic forcing. In the subcritical parameter regime, the IR scaling factor is normally well described by a relation of the form τσie/υi∝(τσie/τc)χ, in which χ = 1.1 ± 0.1 according to linear regressions of log-transformed data for each simulation group.9 It follows that for the data considered herein, one might reasonably approximate the subcritical variation of dimensional IR with the criticality parameter by the nonlinear relation δυbm/δt≈kg(τc/τσie)[(τc/τσie)−(τc/τσie)0], in which kg is a group-specific constant and χ has been set to unity.

3) Sawyer–Eliassen-based analysis of low-level 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 SE-based analysis presented below is conducted in a reference frame that moves with the low-level vortex. The cylindrical coordinate system (with radius r and azimuth φ) is centered on xl. 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.

SE theory assumes that the basic state of the vortex approximately maintains thermal wind balance during its evolution. The preceding assumption leads to a diagnostic equation for the streamfunction Ψ(r, z) of the mean secondary circulation. This so-called SE equation is of the form L[Ψ]=∑α∈{h,e,T}Fα, in which L is a linear differential operator and Fα is one of several source terms. For the present analysis, the source terms are formally attributable to applied heating (h), resolved eddy forcing (e), and subgrid turbulent transport (T). Linearity of the SE equation allows the solution for Ψ to be written ∑αΨα, in which L[Ψα] = Fα. Since the velocity field of the mean secondary circulation is obtained from a linear operation on Ψ, it too can be decomposed into the following sum of three parts:
(u¯w¯)=theory(u¯hw¯h)+(u¯ew¯e)+(u¯Tw¯T).
Each component (u¯α,w¯α) on the right-hand side of Eq. (11) can be viewed as the secondary circulation that would be required to maintain thermal-wind balance under the imaginary situation in which only the forcing connected to Fα exists. The T component generally has separate contributions from turbulent momentum transport (friction) and turbulent heat transport, but the author has verified that the former dominates the latter in the lower troposphere for all of the illustrative cases considered below. Therefore, the T component is here viewed as being predominantly attributable to friction. The reader may consult appendix D for further details on the SE equation and its solution.
Let us now consider the following azimuthally averaged azimuthal velocity equation:
∂υ¯∂t=−u¯η¯−w¯∂υ¯∂z+E¯υ+T¯υ,
in which η ≡ ζ + f, ζ≡z^⋅∇H×u, Eυ is resolved eddy forcing (see appendix D), and Tυ accounts for parameterized subgrid turbulent transport. Substituting Eq. (11) into Eq. (12) yields
∂υ¯∂t=theoryΓh+Γe+ΓT,
in which
Γh≡−u¯hη¯−w¯h∂υ¯/∂z,
Γe≡−u¯eη¯−w¯e∂υ¯/∂z+E¯υ, and
ΓT≡−u¯Tη¯−w¯T∂υ¯/∂z+T¯υ.
Figure 12 shows the partial accelerations on the right-hand side of Eq. (13), and the secondary circulations regulating their advective terms, for several subcritical simulations belonging to the reference group. Each image focuses on the lower-tropospheric dynamics within 130 km of the vortex center during an early stage of the intensification period. The acceleration associated with eddy forcing (Γe) tends to be negative in the vicinity of the strongest cyclonic winds near the surface but is generally small relative to at least one of the other components of ∂υ¯/∂t. When the diabatic forcing is weak such that τc/τσie=0.12, the usually (but not invariably) opposite accelerations associated with heating (Γh) and turbulent transport (ΓT) alternate in having greater magnitude as the altitude increases near the z-dependent radius of maximum υ¯ (rzm). As τc/τσie grows to 0.37, the positive acceleration associated with heating becomes appreciably stronger than the action of turbulent transport.
Fig. 12.
Fig. 12.

SE-based analysis of intensification in several subcritical reference simulations. (a) Contributions to the mean secondary circulation (streamlines) and to ∂υ¯/∂t (colors) formally attributable to downtilt heating in the reference simulation with a = 4ac/55 and τc/τσie=0.12, during an early 6-h interval of the IR measurement period. (b) As in (a), but for contributions primarily attributable to subgrid turbulent transport. (c) As in (a), but for contributions attributable to asymmetric eddy forcing. Local streamline thickness is proportional to the local magnitude of the partial secondary velocity field and is scaled uniformly in (a)–(c). The amber line traces the 6-h time average of the z-dependent radius of maximum υ¯ in the lower troposphere. (d)–(f) As in (a)–(c), but for the reference simulation with a = 2ac/11 and τc/τσie=0.20. (g)–(i) As in (a)–(c), but for a 3-h early interval of the IR measurement in the reference simulation with a = 4ac/11 and τc/τσie=0.37.

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

Figure 13 more clearly demonstrates the rising dominance of diabatic forcing over frictional spindown by showing the ratio λ ≡ (Γh + ΓT)/Γh. When λ is close to 1, Γh is dominant; otherwise ΓT has comparable or greater magnitude. For the case of weakest diabatic forcing (Fig. 13a), λ generally falls well below unity—or is even negative—in the neighborhood of rzm; the only exception occurs in a thin vertical layer near z = 0.75 km. For the case of strongest forcing (Fig. 13b), λ generally lies between 0.75 and 1 in the neighborhood of rzm; moreover, |λ − 1| ≡ |ΓT/Γh| < 0.25 over an extensive region of the inner core of the low-level vortex.10

Fig. 13.
Fig. 13.

The ratio λ ≡ (Γh + ΓT)/Γh for the reference simulations with (a) τc/τσie=0.12 and (b) τc/τσie=0.37. The dotted, solid, and dashed white lines respectively correspond to λ = 0.75, 1 and 1.25. The amber line traces the z-dependent radius of maximum υ¯, averaged over the time period of the SE analysis.

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

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 low-level 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 τc/τσie might be partly attributable to smaller values of ℓ.

Figure 14a shows two distinct measurements of ℓ versus the criticality parameter. The inset shows ℓ at the start of the intensification period (t = ti), whereas the main graph shows the time average of ℓ over the entire intensification period (ti ≤ t ≤ te). First consider the subcritical simulations for which the base-10 logarithm of τc/τσie (the abscissa of each plot) is appreciably negative. The inset reveals that for many simulation groups, there is virtually no variation of the initial value of ℓ among subcritical systems; therefore, the initial value of ℓ is not a robust indicator of normalized IR in the subcritical parameter regime. On the other hand, the main plot shows that in a given simulation group, the time average of ℓ tends to decay with growth of the criticality parameter. Such reduction of the time average of ℓ could conceivably contribute—alongside the diminishing relative influence of frictional damping—to the attendant growth of normalized IR. In the supercritical parameter regime, ℓ likewise decays as the criticality parameter grows, but the decay cannot be firmly linked to any major variation of normalized IR (Fig. 11). Such insensitivity of the normalized IR may be connected to the following two facts: after core replacement, ℓ is generally smaller than the radial length scale δrf of the diabatic forcing, and the measurement radius of υbm (i.e., rbm) usually reduces to the enforced 10-km minimum.

Fig. 14.
Fig. 14.

(a) Main plot: base-10 logarithm of the mean value of ℓ during the IR measurement period vs the base-10 logarithm of the criticality parameter τc/τσie. Inset: as in the main plot, but for ℓ at the start of the intensification period (t = ti). (b) The mean radius of maximum wind speed in the boundary layer vs the mean value of ℓ during the IR measurement period. The dashed slanted line corresponds to rbm = ℓ. The dotted horizontal line corresponds to the minimum accepted value for rbm (10 km). The gray triangle is the region of parameter space where the nominal inner core of the low-level vortex lies entirely within the core of the heat source (ℓ + rbm ≤ δrf). The symbols are as in Fig. 11.

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

It is worth remarking that, in contrast to the supercritical state of affairs, the time averages of ℓ and rbm 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 2δrf.11 It stands to reason that the decay of the time average of ℓ as τc/τσie increases toward unity in a given simulation group generally goes hand in hand with a decay of the time average of rbm.

e. Low-level vorticity production

The transition from a slow intensification mechanism to a fast intensification mechanism initiated by core replacement in the 3D model is quantitatively consistent with shallow-water theory (S20) in occurring when the convergence generated by diabatic forcing exceeds τc−1. This result was not a foregone conclusion, since unlike shallow-water dynamics, the horizontal contraction (vertical stretching) of a vortex-tube is joined by other vertical vorticity production mechanisms—most notably vortex-tube tilting—within the convergence zone of a 3D system. Specifically, the vertical relative vorticity equation in the 3D model can be written as follows:
DζDt=−η∇H⋅u+ζH⋅∇Hw−cpdz^⋅(∇Hθ×∇HΠ)+z^⋅(∇H×TH),
in which (as usual) D/Dt is the material derivative, ζH is the horizontal vorticity vector, Π≡(p/pr)Rd/cpd is the nondimensional Exner function of pressure p normalized to pr ≡ 105 Pa, Rd (cpd) is the gas constant (isobaric specific heat) of dry air, and TH is the horizontal velocity tendency associated with parameterized turbulence. The first term on the right-hand side of Eq. (15) essentially represents the effect of vortex-tube stretching, the second represents the effect of vortex-tube tilting, the third represents (positive or negative) baroclinic vorticity production, and the fourth represents vorticity production via subgrid turbulent transport.

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 shallow-water theory is adequate for predicting the critical convergence required to initiate a core replacement event in the 3D model under present consideration.

Fig. 15.
Fig. 15.

(a) Vertical vorticity tendency ζ˙ associated with vortex-tube stretching in the subcritical reference simulation with τc/τσie=0.20 (Fig. 6), averaged over 4.0 ≤ t ≤ 5.5 h and z ≤ 3.1 km. (b) As in (a), but for the vertical vorticity tendency attributable to vortex-tube tilting, baroclinicity, and parameterized subgrid turbulence combined. Black contours in (a) and (b) show the t–z average of ζ, labeled [in (b)] in units of 10−4 s−1. The × marks the time average of xf. The Cartesian (x, y) coordinate system is centered on the time average of xl. All fields are Gaussian smoothed in x and y with a standard deviation parameter of 5 km. (c),(d) As in (a) and (b), but for the supercritical reference simulation with τc/τσie=1.09 (Fig. 8), and with the time averaging over 1 ≤ t ≤ 1.5 h.

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

4. Connection to realistic tropical cyclone dynamics

a. Intensification rates

At this point, one might appropriately ask how the preceding results relate to realistic tropical cyclone dynamics. The first issue is how the IRs compare to those in nature. A combination of theoretical reasoning and cloud-resolving simulations led Wang et al. (2021, hereinafter WLX21) to the following provisional formula for the maximum potential intensification rate (MPIR) of a tropical cyclone:
dυbmdt=MPIR27256αCdhbVmax2,
in which α = 0.75, hb = 2 km, and Cd = 0.0024 for sufficiently large values of the maximum potential intensity Vmax. A preliminary analysis in WLX21 suggested that the preceding formula is reasonably consistent with observed MPIRs—for various environmentally determined values of Vmax—extracted from 6-h intensification rates.

Figure 16 shows the IRs of all intensifying vortices under present consideration, normalized to the MPIR of WLX21 with Vmax set to a value (95 m s−1) that is near the current upper-bound 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 well-established 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 6-h 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 lower-tropospheric background flow, so as to greatly increase τc, the supercriticality condition τc/τσ > 1 required for core replacement could be satisfied with a much larger value of τσ (much weaker heating). The associated IR, which scales as τσ−1 in the supercritical parameter regime, would be proportionally smaller and potentially realistic over a 6-h time period. Appendix E explains how the time scale for supercritical intensification might also lengthen upon introducing a secondary negative component to the downtilt heat source.

Fig. 16.
Fig. 16.

The IR of nondecaying vortices normalized to 3.08 m s−1 h−1, which is equal to the MPIR of WLX21 evaluated for tropical cyclones capable of achieving 95 m s−1 maximum sustained wind speeds. The symbols are as in Fig. 11.

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

b. Diabatic forcing

The diabatic forcing used for the present study was designed to roughly conform with observations and full-physics 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 cloud-resolving 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 xf 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 2-h analysis period when the system is at depression or tropical storm intensity, whereas the bottom plot corresponds to an overlapping 6-h 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).

Fig. 17.
Fig. 17.

Selected downtilt heating profiles from the cloud-resolving simulations of S22. (a) 2-h and (b) overlapping 6-h time averages of the azimuthal mean of Dθ/Dt about the central axis of downtilt heating in a misaligned tropical cyclone over an ocean whose surface temperature is 26°C [simulation T26-HRA (226 ≤ t ≤ 232 h) of Table F1]. (c),(d) As in (a) and (b), but for a system with an SST of 28°C [T28-HRA (160 ≤ t ≤ 166 h)]. (e),(f) As in (a) and (b), but for a system with an SST of 30°C [T30-HRA (61 ≤ t ≤ 67 h)].

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

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 low-level vortex center versus the tilt magnitude. Here and in all other subplots, each data point with error bars corresponds to a 6-h interval during the prehurricane evolution of a tropical cyclone. The 6-h interval is divided into three 2-h segments. The coordinates of each data point (marked by a solid symbol) correspond to the medians of the 2-h time averages of the plotted variables. The error bars extend from the minimum 2-h 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 φf of the position vector of the heating center (in a coordinate system centered at xl) 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 φf exceeding 45°.

Fig. 18.
Fig. 18.

Characteristics of downtilt heating for a number of tropical depressions and tropical storms in the S22 dataset. (a) Relationship between the tilt magnitude |xml| and the horizontal distance ℓ from the heating center to the low-level vortex center. The slanted dashed line corresponds to ℓ = |xml|. Different symbol shapes and colors correspond to simulations with different SSTs as shown in the legend. (b) Azimuthal displacement φf of the heating center from the direction of the tilt vector, plotted against ℓ. (c) Strength parameter a of the fit function for the downtilt heating distribution plotted against the vertically integrated downtilt heating density Q. (d) The radial shape parameters for the downtilt heating distribution. (e) The downward decay length δzf− of the downtilt heating distribution plotted against the height of maximum heating in the fit function. (f) As in (e), but with the upward decay length δzf+ replacing δzf−.

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

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 2-h heating distribution (exemplified in the top row of Fig. 17) to a function equivalent to the right-hand side of Eq. (1), but with T → 1 and r˜→r˜−df so as to permit a radial offset df of the heating maximum. The parameter unrelated to the fit function is Q, which corresponds to the vertical integral of the heating rate density (ρdq defined in appendix F, section b) between 1 and 16 km above sea level, averaged within a 100-km radius of the heating center. Figure 18c shows the peak value of the heating distribution given by the fit function (a) along with the coinciding values of Q. The values of a are within the range used—mostly for subcritical systems—in the present study.13 The same can be said for the values of Q, which for the reference group equal 4.9 kW m−2 × (a/10−2 K s−1). While the average (positive or negative) error bar plotted for Q is merely 0.2 times the median of Q for a given 6-h interval, the average error bar plotted for a is 0.6 times its median. It stands to reason that 2-hourly variations of details in the structure of the diabatic forcing are more substantial than such variations of the net heating rate. Substantial structural variation is corroborated by the graph of the radial shape parameters of the fit function (Fig. 18d). Note however, that the constant radial shape parameters chosen for the diabatic forcing in this study (df = 0; δrf = 35 km) are within the depicted range of possibilities. The vertical shape parameters (Figs. 18e,f) are somewhat more stable over given 6-h periods. Furthermore, the triplet (zf,δzf−,δzf+)=(7.5,6.0,3.5)km prescribed for most of the simulations herein seems to fall within the spread of the S22 dataset.

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 cloud-resolving 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 θ˙f and w at lower and middle tropospheric levels above the near-surface layer.

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 surface-vortex 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 τσ−1 in the vicinity of the downtilt heat source is above or below a critical value. The critical value τc−1 found in each simulation group agrees with shallow-water 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 τc/τσ 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 small-but-strong 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 τc/τσ [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 υi/τσ, in which υi [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 (δυbm/δt)τσ/υi is roughly constant. In the subcritical parameter regime, the normalized IR was found to decline approximately linearly with decreasing values of τc/τσ 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 τσ 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 lower-tropospheric background flow so as to substantially increase τc (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 τc/τσ, 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 low-level 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.

1

Thus, for example, the combination rf* = |xml| and φf* = 0 would imply that xf* = xm.

2

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.

3

Data from both CD0 and CD0+ are considered to verify that negligible-drag results are insensitive to minor differences in the CM1 configuration options that are used in conjunction with free-slip and semi-slip boundary conditions.

4

This definition of the horizontal boundary layer velocity field is also used to evaluate the maximum wind speed υbm that was introduced in section 2c.

5

There are several reasons why the subcritical dynamics of the system with a = 4ac/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 one-half of its initial size in terms of rbm 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 (rc 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).

6

In S20, the author reserved the term “core reformation” for its supercritical variant (“core replacement”). In hindsight, this may have been too restrictive.

7

The alternative use of υbm(ti) is found to reduce the spread less effectively.

8

A sensitivity test has been conducted with τσ redefined to be the inverse of the average of σb within a radius δrf of the convergence center xσ that is precisely defined in appendix A; the average of |xσ − xf|/δrf 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 τσie for subcritical systems. The correlation coefficient between the normalized IR and τc/τσie remains high (0.963 ± 0.020) in the realm of subcriticality, but the spread of the point of zero IR (0.18 ± 0.12) becomes noticeably greater. The value of τc/τσie separating systems with robust core replacements from those without increases to a value slightly closer to 1.

9

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.

10

Bear in mind that because SE theory neglects unbalanced dynamics, Γh and ΓT should not be expected to precisely match the accelerations induced by heating and frictional forcing imposed separately on a vortex. Nevertheless, one may provisionally assume that SE theory applied at weak-to-moderate tropical storm intensity offers a reasonable picture of the relative magnitudes of these two accelerations (SM20).

11

The few anomalous cases in this parameter regime for which the time average of ℓ substantially exceeds that of rbm correspond to sheared systems in which the diabatic forcing is too weak to prevent the gradual separation of the low-level and midlevel vortices.

12

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.

13

Repetition of the fit with df constrained to equal 0 gives a similar range of results for a.

B1

Thus, Cd = 0.005 for group CD5, Cd = 0.003 for group CD3, Cd = 0.001 for group CD1, and Cd = 0.0005 for group CD05.

E1

One might expect a stronger vorticity anomaly to develop over time in the convergence zone, where |∇⋅u˜b| is larger and much of the entering fluid is hypothetically trapped. Conversely, one might expect a weaker vorticity anomaly to develop over time in the divergence zone, where |∇⋅u˜b| is smaller and the entering fluid is untrapped.

F1

T30-HRD (marked by an asterisk) was conducted for S22 but was inadvertently left out of the list of simulations used by the analysis software.

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 AGS-1743854 and AGS-2208205. 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 xδ represent the horizontal position vector of the vortex center in a vertical layer indicated by the subscript δ. In general, xδ corresponds to the location at which one must place the origin of a polar coordinate system to maximize the peak value of υ¯δ(r) for r ≥ rc, in which υ¯δ is the vertical average of the azimuthal-mean tangential velocity in layer-δ, and rc is a specified minimal core radius. For the analysis of simulation data presented throughout the main text, xl is the vortex center in a 1.2-km-thick boundary layer adjacent to the sea surface, whereas xm is the vortex center in the midtropospheric layer defined by 7.1 < z < 8.5 km. In both cases, rc = 10 km. By contrast, xl2 is the vortex center in the 1.2-km thick boundary layer obtained with rc = 70 km.

Slightly different definitions are used for xl and xm to calculate the right-hand side of the equation for dxf/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 short-lived fluctuations of the heating center that may occur in conjunction with similar fluctuations of the tilt vector. Specifically, the layer corresponding to xl (xm) is collapsed onto the horizontal plane at z = 1.2 (7.8) km—so that no vertical averaging is necessary for the computation of υ¯δ—and rc is set to 55 km. The search for xδ(t) is also limited to a 300 × 300 km2 region centered on xδ(t − Δt), in which Δt is the time step of the simulation.

Figure A1 illustrates how the tilt vector xml ≡ xm − xl 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 low-level or midlevel vortex decreases below the 55-km cutoff value in the runtime search algorithm. Differences will of course diminish when the small-scale and medium-scale circulations become increasingly concentric in each layer.

Fig. A1.
Fig. A1.

(top) Comparison of the magnitudes of the tilt vectors computed with the vortex centers that are used for the runtime parameterization of diabatic forcing (gray) and postruntime data analysis (black) in the reference simulation with τc/τσie=0.37 (Fig. 7). (bottom) Similar comparison of the runtime tilt angle φml and postruntime tilt angle measured counterclockwise from the positive-x direction in Fig. 7.

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

The main text contains several references to the convergence center xσ 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 −u¯b(r) for r ≥ rc, in which u¯b is the azimuthally averaged radial component of ub. The value of rc is set to the minimum horizontal grid spacing of 2.5 km, but in contrast to the vortex-center-finding algorithm an effective 20-km smoothing operation is applied to the velocity field before the search for xσ begins.

APPENDIX B

Sensitivity to Cd

Section 3a (Fig. 4) addresses the consequences of eliminating surface drag on the time scale of vortex intensification but does not thoroughly examine Cd sensitivity. Figure B1 offers a more comprehensive picture of how the normalized IR varies as Cd 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 Cd: two groups with zero or near-zero surface drag (CD0 and CD0+), and four groups labeled CDX with Cd = 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 ≤ Cd ≤ 0.0024, are shown for context. Note that the values of υi (τc) for all plotted simulations have a standard deviation of only 7% (10%) of the mean. It stands to reason that υi and τc can be viewed approximately as constants in the axis labels.

Fig. B1.
Fig. B1.

(a) Nondimensional IR plotted against the criticality parameter for a number of simulation groups with different surface drag parameterizations. The symbol shading is as in Fig. 11. (b) Enlargement of the subcritical section of (a). The solid diagonal line is a linear regression for the reference group (REF).

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

Figure B1 shows that increasing Cd generally decreases the normalized IR at a fixed value of the criticality parameter τc/τσie, and increases the threshold of τc/τσie that is required for diabatic forcing to overcome frictional damping. As in the reference group, the subcritical growth of normalized IR with the criticality parameter is roughly linear for the two simulation groups with larger drag coefficients (CD3 and CD5). By contrast, the slope of the IR curve appears to markedly steepen as the criticality parameter decreases toward the point of zero IR in the two simulation groups with relatively small but finite drag coefficients (CD1 and CD05). Understanding the details of this nonlinearity is deferred to future study. The simulations with zero surface drag are exceptional in that the normalized IR appears to settle on a finite positive value as the convergence time scale tends toward infinity and the criticality parameter approaches zero. As a final remark, the variation of surface drag considered herein does not appear to have a major effect on the transition zone (at τσie≈τc) separating systems that undergo core replacement (dark-filled symbols) from those that do not (white-filled symbols).

APPENDIX C

Group Statistics for υi and τc

Table C1 first summarizes the statistics of the scaling velocity υi that appears in the expression for the normalized intensification rate of Fig. 11. The means and standard deviations are shown for both subcritical (column υi,sub) and supercritical (column υi,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 υi 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 υi 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 WEAKV-TLTX3.

Table C1.

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.

Table C1 also summarizes the group statistics of the time scale τc 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 τc can change appreciably from one simulation group to another (compare values associated with SH5⊥ and WEAKV-TLTX3), the standard deviation for a given group is usually small. The small standard deviation implies that variation of τc/τσie (the abscissa in Fig. 11) within any particular group mainly results from variation of τσie.

APPENDIX D

SE Computations

The following briefly summarizes the SE equations for each component Ψα 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 cloud-microphysical heat source.

As mentioned in the main text, the SE equation for each streamfunction is of the form
L[Ψα]=Fα,
in which L is a linear differential operator. Specifically,
L[Ψα]≡∂z(I∂zΨα+B∂rΨαρ¯r)+∂r(S∂rΨα+B∂zΨαρ¯r),
in which the baroclinicity, static stability, and modified inertial stability parameters are respectively given by
B≡−∂z(Cκ¯),
S≡−g∂zκ¯, and
I≡κ¯η¯ξ¯+BC/g.
In addition, the forcing functions for α ∈ {h, e, T} satisfy
Fh≡∂z(Cκ2θ˙f¯)+g∂r(κ2θ˙f¯),
Fe≡−∂z(CE¯κ)−g∂rE¯κ−∂z(κ¯ξ¯E¯υ), and
FT≡∂z(Cκ2Tθ¯)+g∂r(κ2Tθ¯)−∂z(κ¯ξ¯T¯υ).
In the preceding equations, C≡υ¯2/r+fυ¯, η¯≡ζ¯+f, ξ¯≡2υ¯/r+f, κ ≡ θ−1, ρ is mass density, and g is the gravitational acceleration near the surface of Earth. The variables Tθ and Tυ represent forcing by parameterized subgrid turbulence in the potential temperature equation and tangential velocity equation, respectively. The variables associated with resolved “eddies” are given by
E¯υ≡−u′ζ′¯−w′∂zυ′¯−cpdθ′∂φΠ′¯/r and
E¯κ≡−u′∂rκ′¯−υ′∂φκ′¯/r−w′∂zκ′¯.
The last term on the right-hand side of the E¯υ equation (having cpd as a coefficient) is generally subdominant. As usual, the symbol ∂x appearing in various expressions above is shorthand for ∂/∂x, in which x is a generic variable.

For all computations of Ψα, 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 Ψα = 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 Ψα can be calculated from the following formula: (u¯α,w¯α)=(−∂zΨα,∂rΨα)/(rρ¯).

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 Ψα are time averaged over the moderately short analysis period. Similar time averages are used for η¯, ∂υ¯/∂z, E¯υ, and T¯υ in the expressions for Γα that are provided in the main text [Eq. (14)]. The time averages are obtained from data sampled every 90 s over the interval 2.5 ≤ t ≤ 5.5 h for the subcritical system with τc/τσie=0.37, every 180 s over the interval 4 ≤ t ≤ 10 h for the subcritical system with τc/τσie=0.20, and every 180 s over the interval 6 ≤ t ≤ 12 h for the subcritical system with τc/τσie=0.12.

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 τc/τσ > 1, or equivalently that σbf>τc−1. This condition [with τc essentially given by Eq. (9)] was derived under the assumption that the (low-level) downtilt flow structure can be approximated by a solitary convergence zone embedded in a larger scale background flow. Such an assumption is a reasonable approximation for the simulations conducted herein, which represent downtilt convection with a purely positive Gaussian-like heat source, and also has relevance to a certain class of “realistically” simulated tropical cyclones (see appendix A of S20). On the other hand, one might imagine a scenario in which a neighboring downdraft associated with evaporative cooling creates a substantial low-level divergence zone in close proximity to the downtilt convergence zone that persists over a time scale relevant to core replacement. It is of interest to consider how this might affect the critical convergence above which core replacement should occur.

For simplicity, suppose that the initial boundary layer velocity field in the neighborhood of downtilt convection, and in a reference frame moving with the convection, can be approximated by
u˜b=Vly^−σ+δr+2min⁡(δr+,r+)max⁡(δr+,r+)r^++σ−δr−2min⁡(δr−,r−)max⁡(δr−,r−)r^−.
Here, Vly^ is a spatially uniform velocity field representing the large-scale background flow. The second and third terms respectively account for relatively small, circular convergence and divergence zones. The variables σ+, δr+, and r+ (σ−, δr−, and r−) denote the strength, radial width, and distance from the center of the convergence (divergence) zone. The variable r^+ (r^−) is the radial unit vector of a polar coordinate system whose origin is at the center of the convergence (divergence) zone.

In the preceding notation, the critical convergence above which a point of attraction exists in the absence of a divergence zone is given by τc−1=2Vl/δr+ (S20). Figures E1a and E1b depict the streamlines of u˜b for a system having a firmly subcritical solitary convergence zone characterized by σ+ = 1.25Vl/δr+ and σ− = 0. The depiction suggests that a fluid volume entering the convergence zone will pass through, after losing some of its mass to vertical convection. Figures E1c and E1d illustrate how the local flow structure changes when a moderately weaker divergence zone with σ− = 0.75σ+ and δr− = δr+ is placed at a distance of 2.18δr+ from the center of the convergence zone, directly downwind with respect to the background flow. The modification has introduced a point of attraction near the downwind edge of the convergence zone, which could in principle enable a core replacement event. The region below the red curve in Fig. E1c provides an initial estimate of the fluid destined to become horizontally trapped in the convergence zone, where its vorticity may continuously amplify. Figures E1e and E1f illustrate what would happen to the streamlines if the convergence and divergence zones were given uniform vorticity anomalies of 1.2σ+ and −0.32σ−, respectively.E1 These figures suggest that the existence of a nominal point of attraction in the convergence–divergence dipole may not be highly fragile to the development of local vorticity anomalies over time. The same inference can be drawn from qualitatively similar streamline plots (not shown) that have been constructed for systems with 2–3 times the positive vorticity anomaly in the convergence zone, and either a proportional or zero change of the negative vorticity anomaly in the divergence zone.

Fig. E1.
Fig. E1.

(a) Streamlines in the vicinity of a subcritical solitary convergence zone (red circle) embedded in a large-scale background flow. The Cartesian coordinates x˜ and y˜ are measured from the center of the convergence zone. (b) Enlargement of (a) near the downwind edge of the convergence zone. (c),(d) As in (a) and (b), but with the addition of a moderately weaker divergence zone (blue circle). The × symbols mark stagnation points; the thick red × in (d) is the nominal point of attraction. The thick red curve in (c) corresponds to a streamline very close to the separatrix. (e),(f) As in (c) and (d), but with positive and negative vorticity anomalies respectively added to the convergence and divergence zones.

Citation: Journal of the Atmospheric Sciences 80, 7; 10.1175/JAS-D-22-0188.1

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

Cloud-Resolving 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 left-most 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 (|xml,0|). The initialization methods [dry separation plus damping (DSPD) and impulsive separation plus damping (ISPD)] are explained in S22. The third column shows the 6-h 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 (υsm), time averaged over the analysis period to the left.

Table F1.

Synopsis of the cloud-resolving tropical cyclone simulations analyzed in section 4b. See text for discussion.
Table F1.

b. Tilt vector and heating parameters

The vortex centers required to compute the tilt vector xml and heating displacement ℓ for tropical cyclones in the cloud-resolving 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.

The downtilt heating center of a cloud-resolving CM1 simulation is obtained from the following formula:
xf≡1Q+∫∫∫VdVmax⁡(ρdq,0)x,
in which q ≡ TDsd/Dt, T (in the present context) is absolute temperature, Dsd/Dt is the material derivative of the specific dry entropy sd, ρd is the mass density of dry air, x is the horizontal position vector, and
Q+≡∫∫∫VdVmax⁡(ρdq,0).
The integration volume V is centered at xm, has a radius of 250 km, and extends vertically from 1 to 16 km above sea level. Although V may extend well into the uptilt sector of the vortex, xf generally falls well within the cluster of downtilt convection owing to the relative paucity of convective latent heat release elsewhere in the tropical cyclone. In section 4b, the values of xm and ρdq that are used in the preceding formula for xf are either 2-h [Figs. 17 (top row) and 18] or 6-h [Fig. 17 (bottom row)] time averages.
The mean vertically integrated heating density appearing in Fig. 18c is given by
Q≡1A∫∫AdA∫zbotztopdzρdq.
In the preceding formula, A is the horizontal area within a 100-km radius of xf, zbot = 1 km, and ztop = 16 km.

REFERENCES

  • Alland, J. J., B. H. Tang, K. L. Corbosiero, and G. H. Bryan, 2021a: Combined effects of midlevel dry air and vertical wind shear on tropical cyclone development. Part I: Downdraft ventilation. J. Atmos. Sci., 78, 763–782, https://doi.org/10.1175/JAS-D-20-0054.1.

    • Search Google Scholar
    • Export Citation

  • Alland, J. J., B. H. Tang, K. L. Corbosiero, and G. H. Bryan, 2021b: Combined effects of midlevel dry air and vertical wind shear on tropical cyclone development. Part II: Radial ventilation. J. Atmos. Sci., 78, 783–796, https://doi.org/10.1175/JAS-D-20-0055.1.

    • Search Google Scholar
    • Export Citation

  • Alvey, G. R., III, E. Zipser, and J. Zawislak, 2020: How does Hurricane Edouard (2014) evolve toward symmetry before rapid intensification? A high-resolution ensemble study. J. Atmos. Sci., 77, 1329–1351, https://doi.org/10.1175/JAS-D-18-0355.1.

    • Search Google Scholar
    • Export Citation

  • Alvey, G. R., III, M. Fischer, P. Reasor, J. Zawislak, and R. Rogers, 2022: Observed processes underlying the favorable vortex repositioning early in the development of Hurricane Dorian (2019). Mon. Wea. Rev., 150, 193–213, https://doi.org/10.1175/MWR-D-21-0069.1.

    • Search Google Scholar
    • Export Citation

  • Bell, M. M., M. T. Montgomery, and K. A. Emanuel, 2012: Air-sea enthalpy and momentum exchange at major hurricane wind speeds observed during CBLAST. J. Atmos. Sci., 69, 3197–3222, https://doi.org/10.1175/JAS-D-11-0276.1.

    • Search Google Scholar
    • Export Citation

  • Bryan, G. H., and J. M. Fritsch, 2002: A benchmark simulation for moist nonhydrostatic numerical models. Mon. Wea. Rev., 130, 2917–2928, https://doi.org/10.1175/1520-0493(2002)130<2917:ABSFMN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Chen, X., Y. Wang, J. Fang, and M. Xue, 2018: A numerical study on rapid intensification of Typhoon Vicente (2012) in the South China Sea. Part II: Roles of inner-core processes. J. Atmos. Sci., 75, 235–255, https://doi.org/10.1175/JAS-D-17-0129.1.

    • Search Google Scholar
    • Export Citation

  • DeMaria, M., 1996: The effect of vertical shear on tropical cyclone intensity change. J. Atmos. Sci., 53, 2076–2088, https://doi.org/10.1175/1520-0469(1996)053<2076:TEOVSO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Dörffel, T., A. Papke, R. Klein, N. Ernst, and P. K. Smolarkiewicz, 2021: Dynamics of tilted atmospheric vortices under asymmetric diabatic heating. Theor. Comput. Fluid Dyn., 35, 831–873, https://doi.org/10.1007/s00162-021-00591-x.

    • Search Google Scholar
    • Export Citation

  • Dunion, J. P., 2011: Rewriting the climatology of the tropical North Atlantic and Caribbean Sea atmosphere. J. Climate, 24, 893–908, https://doi.org/10.1175/2010JCLI3496.1.

    • Search Google Scholar
    • Export Citation

  • Emanuel, K., 2018: 100 years of progress in tropical cyclone research. A Century of Progress in Atmospheric and Related Sciences: Celebrating the American Meteorological Society Centennial, Meteor. Monogr., No. 59, Amer. Meteor. Soc., https://doi.org/10.1175/AMSMONOGRAPHS-D-18-0016.1.

  • Enagonio, J., and M. T. Montgomery, 2001: Tropical cyclogenesis via convectively forced Rossby waves in a shallow-water primitive equation model. J. Atmos. Sci., 58, 685–706, https://doi.org/10.1175/1520-0469(2001)058<0685:TCVCFV>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Finocchio, P. M., S. J. Majumdar, D. S. Nolan, and M. Iskandarani, 2016: Idealized tropical cyclone responses to the height and depth of environmental vertical wind shear. Mon. Wea. Rev., 144, 2155–2175, https://doi.org/10.1175/MWR-D-15-0320.1.

    • Search Google Scholar
    • Export Citation

  • Fischer, M. S., R. F. Rogers, P. D. Reasor, and J. P. Dunion, 2021: Relationships between vortex tilt, convective structure, and intensity change in early-stage tropical cyclones. 34th Conf. on Hurricanes and Tropical Meteorology, Online, Amer. Meteor. Soc., 5B.6, https://ams.confex.com/ams/34HURR/meetingapp.cgi/Paper/373691.

  • Fischer, M. S., P. D. Reasor, R. F. Rogers, and J. F. Gamache, 2022: An analysis of tropical cyclone vortex and convective characteristics in relation to storm intensity using a novel airborne Doppler radar database. Mon. Wea. Rev., 150, 2255–2278, https://doi.org/10.1175/MWR-D-21-0223.1.

    • Search Google Scholar
    • Export Citation

  • Fu, H., Y. Wang, M. Riemer, and Q. Li, 2019: Effect of unidirectional vertical wind shear on tropical cyclone intensity change—Lower-layer shear versus upper-layer shear. J. Geophys. Res. Atmos., 124, 6265–6282, https://doi.org/10.1029/2019JD030586.

    • Search Google Scholar
    • Export Citation

  • Gu, J. F., Z. M. Tan, and X. Qiu, 2019: Intensification variability of tropical cyclones in directional shear flows: Vortex tilt–convection coupling. J. Atmos. Sci., 76, 1827–1844, https://doi.org/10.1175/JAS-D-18-0282.1.

    • Search Google Scholar
    • Export Citation

  • Jones, S. C., 1995: The evolution of vortices in vertical shear. I: Initially barotropic vortices. Quart. J. Roy. Meteor. Soc., 121, 821–851, https://doi.org/10.1002/qj.49712152406.

    • Search Google Scholar
    • Export Citation

  • Kimball, S. K., and M. S. Mulekar, 2004: A 15-year climatology of North Atlantic tropical cyclones. Part I: Size parameters. J. Climate, 17, 3555–3575, https://doi.org/10.1175/1520-0442(2004)017<3555:AYCONA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Kimberlain, T. B., E. S. Blake, and J. P. Cangialosi, 2016: National Hurricane Center tropical cyclone report: Hurricane Patricia. NOAA/NWS Rep. EP202015, 32 pp., https://www.nhc.noaa.gov/data/tcr/EP202015_Patricia.pdf.

  • Molinari, J., and D. Vollaro, 2010: Rapid intensification of a sheared tropical storm. Mon. Wea. Rev., 138, 3869–3885, https://doi.org/10.1175/2010MWR3378.1.

    • Search Google Scholar
    • Export Citation

  • Molinari, J., D. Vollaro, and K. L. Corbosiero, 2004: Tropical cyclone formation in a sheared environment: A case study. J. Atmos. Sci., 61, 2493–2509, https://doi.org/10.1175/JAS3291.1.

    • Search Google Scholar
    • Export Citation

  • Möller, J. D., and M. T. Montgomery, 1999: Vortex Rossby waves and hurricane intensification in a barotropic model. J. Atmos. Sci., 56, 1674–1687, https://doi.org/10.1175/1520-0469(1999)056<1674:VRWAHI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Möller, J. D., and M. T. Montgomery, 2000: Tropical cyclone evolution via potential vorticity anomalies in a three-dimensional balance model. J. Atmos. Sci., 57, 3366–3387, https://doi.org/10.1175/1520-0469(2000)057<3366:TCEVPV>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Montgomery, M. T., and J. Enagonio, 1998: Tropical cyclogenesis via convectively forced vortex Rossby waves in a three-dimensional quasigeostrophic model. J. Atmos. Sci., 55, 3176–3207, https://doi.org/10.1175/1520-0469(1998)055<3176:TCVCFV>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Montgomery, M. T., and R. K. Smith, 2014: Paradigms for tropical cyclone intensification. Aust. Meteor. Oceanogr. J., 64, 37–66, https://doi.org/10.22499/2.6401.005.

    • Search Google Scholar
    • Export Citation

  • Nguyen, L. T., and J. Molinari, 2015: Simulation of the downshear reformation of a tropical cyclone. J. Atmos. Sci., 72, 4529–4551, https://doi.org/10.1175/JAS-D-15-0036.1.

    • Search Google Scholar
    • Export Citation

  • Nguyen, L. T., R. F. Rogers, and P. D. Reasor, 2017: Thermodynamic and kinematic influences on precipitation symmetry in sheared tropical cyclones: Bertha and Cristobal (2014). Mon. Wea. Rev., 145, 4423–4446, https://doi.org/10.1175/MWR-D-17-0073.1.

    • Search Google Scholar
    • Export Citation

  • Nolan, D. S., 2011: Evaluating environmental favorableness for tropical cyclone development with the method of point-downscaling. J. Adv. Model. Earth Syst., 3, M08001–, https://doi.org/10.1029/2011MS000063.

    • Search Google Scholar
    • Export Citation

  • Nolan, D. S., Y. Moon, and D. P. Stern, 2007: Tropical cyclone intensification from asymmetric convection: Energetics and efficiency. J. Atmos. Sci., 64, 3377–3405, https://doi.org/10.1175/JAS3988.1.

    • Search Google Scholar
    • Export Citation

  • Onderlinde, M. J., and D. S. Nolan, 2016: Tropical cyclone-relative environmental helicity and the pathways to intensification in shear. J. Atmos. Sci., 73, 869–890, https://doi.org/10.1175/JAS-D-15-0261.1.

    • Search Google Scholar
    • Export Citation

  • Päschke, E., P. Marschalik, A. Owinoh, and R. Klein, 2012: Motion and structure of atmospheric mesoscale baroclinic vortices: Dry air and weak environmental shear. J. Fluid Mech., 701, 137–170, https://doi.org/10.1017/jfm.2012.144.

    • Search Google Scholar
    • Export Citation

  • Pendergrass, A. G., and H. E. Willoughby, 2009: Diabatically induced secondary flows in tropical cyclones. Part I: Quasi-steady forcing. Mon. Wea. Rev., 137, 805–821, https://doi.org/10.1175/2008MWR2657.1.

    • Search Google Scholar
    • Export Citation

  • Rappin, E. D., and D. S. Nolan, 2012: The effect of vertical shear orientation on tropical cyclogenesis. Quart. J. Roy. Meteor. Soc., 138, 1035–1054, https://doi.org/10.1002/qj.977.

    • Search Google Scholar
    • Export Citation

  • Reasor, P. D., M. T. Montgomery, and L. D. Grasso, 2004: A new look at the problem of tropical cyclones in shear flow: Vortex resiliency. J. Atmos. Sci., 61, 3–22, https://doi.org/10.1175/1520-0469(2004)061<0003:ANLATP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Reasor, P. D., R. Rogers, and S. Lorsolo, 2013: Environmental flow impacts on tropical cyclone structure diagnosed from airborne Doppler radar composites. Mon. Wea. Rev., 141, 2949–2969, https://doi.org/10.1175/MWR-D-12-00334.1.

    • Search Google Scholar
    • Export Citation

  • Riemer, M., M. T. Montgomery, and M. E. Nicholls, 2010: A new paradigm for intensity modification of tropical cyclones: Thermodynamic impact of vertical wind shear on the inflow layer. Atmos. Chem. Phys., 10, 3163–3188, https://doi.org/10.5194/acp-10-3163-2010.

    • Search Google Scholar
    • Export Citation

  • Rios-Berrios, R., C. A. Davis, and R. D. Torn, 2018: A hypothesis for the intensification of tropical cyclones under moderate vertical wind shear. J. Atmos. Sci., 75, 4149–4173, https://doi.org/10.1175/JAS-D-18-0070.1.

    • Search Google Scholar
    • Export Citation

  • Rogers, R. F., P. D. Reasor, J. A. Zawislak, and L. T. Nguyen, 2020: Precipitation processes and vortex alignment during the intensification of a weak tropical cyclone in moderate vertical shear. Mon. Wea. Rev., 148, 1899–1929, https://doi.org/10.1175/MWR-D-19-0315.1.

    • Search Google Scholar
    • Export Citation

  • Schecter, D. A., 2020: Distinct intensification pathways for a shallow-water vortex subjected to asymmetric “diabatic” forcing. Dyn. Atmos. Oceans, 91, 101156, https://doi.org/10.1016/j.dynatmoce.2020.101156.

    • Search Google Scholar
    • Export Citation

  • Schecter, D. A., 2022: Intensification of tilted tropical cyclones over relatively cool and warm oceans in idealized numerical simulations. J. Atmos. Sci., 79, 485–512, https://doi.org/10.1175/JAS-D-21-0051.1.

    • Search Google Scholar
    • Export Citation

  • Schecter, D. A., and K. Menelaou, 2020: Development of a misaligned tropical cyclone. J. Atmos. Sci., 77, 79–111, https://doi.org/10.1175/JAS-D-19-0074.1.

    • Search Google Scholar
    • Export Citation

  • Schubert, W. H., and J. J. Hack, 1982: Inertial stability and tropical cyclone development. J. Atmos. Sci., 39, 1687–1697, https://doi.org/10.1175/1520-0469(1982)039<1687:ISATCD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Shapiro, L. J., and H. E. Willoughby, 1982: The response of balanced hurricanes to local sources of heat and momentum. J. Atmos. Sci., 39, 378–394, https://doi.org/10.1175/1520-0469(1982)039<0378:TROBHT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Smith, R. K., M. T. Montgomery, and H. Zhua, 2005: Buoyancy in tropical cyclones and other rapidly rotating atmospheric vortices. Dyn. Atmos. Oceans, 40, 189–208, https://doi.org/10.1016/j.dynatmoce.2005.03.003.

    • Search Google Scholar
    • Export Citation

  • Stevenson, S. N., K. L. Corbosiero, and J. Molinari, 2014: The convective evolution and rapid intensification of Hurricane Earl (2010). Mon. Wea. Rev., 142, 4364–4380, https://doi.org/10.1175/MWR-D-14-00078.1.

    • Search Google Scholar
    • Export Citation

  • Tao, D., and F. Zhang, 2014: Effect of environmental shear, sea-surface temperature, and ambient moisture on the formation and predictability of tropical cyclones: An ensemble-mean perspective. J. Adv. Model. Earth Syst., 6, 384–404, https://doi.org/10.1002/2014MS000314.

    • Search Google Scholar
    • Export Citation

  • Vigh, J. L., and W. H. Schubert, 2009: Rapid development of the tropical cyclone warm core. J. Atmos. Sci., 66, 3335–3350, https://doi.org/10.1175/2009JAS3092.1.

    • Search Google Scholar
    • Export Citation

  • Wang, Y., Y. Li, and J. Xu, 2021: A new time-dependent theory of tropical cyclone intensification. J. Atmos. Sci., 78, 3855–3865, https://doi.org/10.1175/JAS-D-21-0169.1.

    • Search Google Scholar
    • Export Citation

  • Zawislak, J., H. Jiang, G. R. Alvey III, E. J. Zipser, R. F. Rogers, J. A. Zhang, and S. N. Stevenson, 2016: Observations of the structure and evolution of Hurricane Edouard (2014) during intensity change. Part I: Relationship between the thermodynamic structure and precipitation. Mon. Wea. Rev., 144, 3333–3354, https://doi.org/10.1175/MWR-D-16-0018.1.

    • Search Google Scholar
    • Export Citation
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