## 1. Introduction

Decades of observational studies have provided convincing evidence that sufficiently intense deep-layer vertical wind shear in the environment will hinder the development^{1} of a tropical cyclone (Gray 1968; McBride and Zehr 1981; DeMaria et al. 2001; Kaplan et al. 2010; Tang and Emanuel 2012). Complementary modeling studies have largely corroborated this empirical finding and have substantially advanced our knowledge of the underlying dynamics (e.g., Tory et al. 2007; Rappin and Nolan 2012; Tao and Zhang 2014). The precise quantitative impact of deep-layer shear has been shown to depend on details of the associated height-dependent wind profile, the surrounding distribution of moisture, and the sea surface temperature (ibid., Ge et al. 2013; Finocchio et al. 2016; Onderlinde and Nolan 2016). It is possible that circumstances exist under which weak-to-moderate shear can assist early development (Molinari et al. 2004; Davis and Bosart 2006; Musgrave et al. 2008; Nolan and McGauley 2012). However, it is more commonly inferred from modeling results that vortex misalignment (tilt) induced by shear plays an important role in frustrating the emergence of nearly saturated air and the generation of a robust symmetric component of convection over the central region of the lower-tropospheric circulation, which would otherwise expedite surface spinup.

A number of the aforementioned modeling studies have examined idealized scenarios in which an immature tropical cyclone on the *f* plane is exposed to a constant ambient shear flow of moderate amplitude. In this paradigm, a tilt develops pointing downshear and precesses toward an upshear orientation. Upon starting to tilt upshear, the vortex begins to realign. One might imagine that the decay of upshear tilt is primarily driven by differential advection, as in analogous simulations of certain dry adiabatic vortices that are otherwise unresilient (e.g., Fig. 8 of Reasor et al. 2004). Rios-Berrios et al. (2018) suggest that a more complex alignment mechanism involving both diabatic and adiabatic processes is more probable. In a general sense, their assessment seems consistent with some of the more realistically configured simulations of sheared tropical storm development, in which the convective generation and subsequent domination of a deep subvortex within the misaligned parent cyclone can be essential to the pertinent alignment process (Nguyen and Molinari 2015; cf. Chen et al. 2018). Regardless of the particular mechanism, the realignment of a tilted tropical cyclone during its development often appears to be a catalyst for the intensification of maximal winds.

The preceding discussion suggests that an integral part of understanding development is understanding alignment. As a first step toward improving current understanding of intrinsic vortex alignment mechanisms, it seems reasonable to remove environmental wind shear from the problem. Such removal eliminates the possibility that alignment is linked to precession of tilt into an upshear orientation. The remaining dynamics is nevertheless rich, and in need of further elucidation. Earlier studies of shear-free alignment have tended to focus on essentially adiabatic mechanisms found in simplified models where moisture effects are represented merely by the effects of reducing static stability (Polvani 1991; Viera 1995; Reasor and Montgomery 2001; Schecter and Montgomery 2003,2007). Expanding this work to the realm of cloud-resolving models is necessary to account for fundamentally distinct diabatic pathways of alignment that may occur with a more realistic treatment of moist convection. The potentially novel pathways revealed by an expanded study could very well have direct relevance to conceivable situations in nature where a substantial tilt of the vortex exists under fairly weak background shear [see, e.g., sections 4a–4b of Davis and Ahijevych (2012)]. The conditions under which such a system might rapidly transition to quasi-symmetric intensification or linger in a detrimental state of misalignment are largely unknown.

The general purpose of this paper is to begin a methodical study of the nature of shear-free development (alignment and intensification) in cloud-resolving numerical simulations of tropical cyclones with variable degrees of initial tilt. Intensification of both the maximum wind speed and kinetic energy of the primary surface circulation will be examined. An effort will be made to elucidate the variation of moist convection with tilt magnitude and the consequences on the angular momentum budget of the basic state of the vortex. Conclusions regarding the effects of tilt on the organization of convection and the temporal amplification of maximum wind speed will not stray far from those found in previous studies of developing tropical cyclones exposed to moderate deep-layer shear. On the other hand, our simulated vortices will follow somewhat distinct pathways toward the state of alignment that facilitates intensification. For example, the tilt of a strongly perturbed tropical cyclone will be shown to undergo an extended intermediate growth phase (following earlier decay) that cannot be attributed to forcing by domain-averaged shear. The importance of adiabatic and diabatic mechanisms in this and other stages of tilt evolution will be addressed.

The remainder of this paper is organized as follows. Section 2 describes our numerical experiments and our method for measuring the misalignment of a developing system. Section 3 examines the impact of misalignment on tropical cyclone development, and section 4 examines the tilt dynamics primarily in the context of an illustrative example. Section 5 recapitulates our findings. The appendixes provide supplemental details and explain a number of techniques used to analyze our simulations.

## 2. Basic methodology

The forthcoming discussions of our methodology and results require a preliminary remark on notation. In this paper, the properties of a tropical cyclone are usually described in a cylindrical coordinate system. The coordinates *r*, *φ*, and *z* respectively denote radius, azimuth, and height above sea level. The variables *u*, *υ*, and *w* respectively represent the radial, azimuthal, and vertical velocity fields. An arbitrary fluid variable is sometimes decomposed into an azimuthal mean dressed with an overbar, and a perturbation dressed with a prime. For example, the azimuthal velocity field is given by *t* is time.

### a. Numerical simulations

The numerical simulations are conducted with Cloud Model 1 (CM1-r19.5; Bryan and Fritsch 2002) in an energy-conserving mode of operation. Unless stated otherwise, the model is configured with a variant of the two-moment Morrison microphysics parameterization (Morrison et al. 2005, 2009), having graupel as the large icy-hydrometeor category and a constant cloud-droplet concentration of 100 cm^{−3}. The influence of subgrid turbulence above the surface is accounted for by an anisotropic Smagorinsky-type closure. The nominal mixing lengths are tailored for tropical cyclones on grids that are deemed insufficiently dense for a standard large-eddy-simulation scheme. The horizontal mixing length increases from 0.1 to 0.7 km as the underlying surface pressure decreases from 1015 to 900 hPa. The vertical mixing length increases asymptotically to 50 m with increasing *z*. Surface fluxes are parameterized with bulk-aerodynamic formulas. The momentum exchange coefficient *C*_{d} increases with the surface wind speed from 10^{−3} to 0.0024 above 25 m s^{−1} [compare with Fairall et al. (2003) and Donelan et al. (2004)]. The enthalpy exchange coefficient is given by *C*_{e} = 0.0012 roughly based on the findings of Drennan et al. (2007). Heating associated with frictional dissipation is activated. The computationally expensive parameterization of radiative transfer is turned off under the provisional assumption that the associated diabatic forcing is unessential to understanding the slowdown of intensification induced by misalignments of incipient tropical cyclones.

The model is set up on a doubly periodic oceanic *f* plane with a sea surface temperature of 28°C and a Coriolis parameter of *f* = 5 × 10^{−5} s^{−1} corresponding to a latitude of 20°N. The dynamical system is discretized on a stretched rectangular grid. The horizontal grid spans 2660 km in each orthogonal coordinate. The 800 × 800 km^{2} central region of the horizontal mesh has grid increments of 1.25 km. At the four corners of the horizontal domain, the grid increments are 13.75 km. The vertical grid has 73 points and extends upward to *z* = 29.2 km. The vertical grid spacing gradually increases from 50 to 400 to 750 m as *z* increases from 0 to 8 to 29 km. Rayleigh damping is imposed above *z* = 25 km.

*z*< 10.5 km and

*r*<

*r*

_{b}= 750 km, where it is given by

*ζ*

_{o}= 2.4 × 10

^{−4}s

^{−1},

*z*

_{c}= 3 km,

*r*

_{1}= 70 km,

*r*

_{2}= 400 km, and

*h*= 7.5 km (5.6 km) for

*z*>

*z*

_{c}(

*z*<

*z*

_{c}). The correction

*δζ*guarantees zero net circulation and amounts to the areal average of the preceding term over a horizontal disc of radius

*r*

_{b}. The azimuthal velocity

*υ*is obtained by inverting

*ζ*, whereas

*u*and

*w*are set to zero. The ambient (

*r*>

*r*

_{b}) pressure

*p*and virtual potential temperature

*θ*

_{υ}are given by the Dunion (2011) moist tropical sounding (MTS); the perturbations to

*p*and

*θ*

_{υ}linked to the vortex are iteratively adjusted to satisfy hydrostatic and gradient balance. The relative humidity with respect to liquid water (subscript

*l*) is given by

*β*-scale core of the vortex, which has been observed in developing systems shortly before genesis [see section 3b of Davis and Ahijevych (2013)]. To elaborate, for

*r*appreciably greater than

*r*

_{3}= 325 km, the relative humidity profile corresponds to that of the Dunion MTS [RH

_{d}(

*z*)]. As

*r*approaches 0, the profile gradually increases to RH

_{0}(

*z*). The value of RH

_{0}holds steady at 0.88 with increasing

*z*below 5.5 km, whereupon it decays linearly up to 10 km and then matches RH

_{d}. Figure 1 depicts both the primary circulation and the moist thermodynamic structure of the initial condition.

The misalignment experiments are initialized with conditions derived from the control simulation 99 h into its development, shortly before one might declare the genesis of a warm core tropical cyclone. Figure 2 depicts the azimuthally averaged fluid variables at this time, henceforth labeled *t* = 0. The absolute maximum of *r* = 115.1 km and *z* = 3.7 km is 12.0 m s^{−1}; the surface maximum is 8.6 m s^{−1} at *r* = 42.9 km. An upper-tropospheric anticyclone (Fig. 2a) has emerged in the outflow of a weak secondary circulation (Fig. 2b). Furthermore, the relative humidity has broadly intensified throughout the central region of the vortex (Fig. 2c).

*τ*

_{s}= 5 h,

*δτ*

_{s}= 1 h,

*δz*

_{l}= 0.5 km,

*z*

_{u}= 21 km, and

*δz*

_{u}= 1 km. The adjustable parameters

*U*

_{s}and

*z*

_{l}respectively represent the maximum wind speed and the height at which easterly flow (

*z*<

*z*

_{l}) changes to westerly flow (

*z*>

*z*

_{l}). The Cartesian

*x*and

*y*components of the imposed velocity accelerations are respectively given by

*t*≤

*τ*

_{s}+

*δτ*

_{s}, but adds a damping term of the form −⟨

**u**⟩

_{xy}/

*τ*

_{d}to the vector horizontal velocity equation over a finite time period of 4

*τ*

_{d}thereafter. Here, ⟨

**u**⟩

_{xy}is the horizontal domain average of the horizontal velocity vector

**u**in the ES reference frame, and

*τ*

_{d}= 1.5 h. The point of the damping is to virtually remove the small amount of residual ambient shear that will have resulted from friction and the presence of the vortical system.

The DSPD method first removes all asymmetry in *φ* from the vortex. Dissipative heating is eliminated, *C*_{d} is reduced to 10^{−4}, and *C*_{e} is reduced to zero. The moisture parameterization is turned off and the water vapor is temporarily replaced with a passive tracer. The symmetric dry vortex is given one day to adjust toward a more balanced state, whereupon it is misaligned using variants of the ISPD method with adjustable values of *τ*_{s}, *δτ*_{s} and *τ*_{d} that typically correspond to slower forcing. At the end of the forcing period, the passive tracer is replaced with water vapor. The mixing ratio is reduced where necessary to 99.5% of its saturation value with respect to ice/liquid at altitudes above/below the local freezing level to prevent the minor shock of instantaneous cloud formation. All physics parameterizations of the control experiment are restored, and time *t* is reset to zero.

Table 1 lists all of the numerical experiments considered herein, except for the control run. The name of each experiment begins with the abbreviation of the misalignment method and ends with a code for two additional parameters. In this code, the symbol X represents 2*U*_{s}*τ*_{s}, and Z represents *z*_{l}; the rounded-down value (in kilometers) of each parameter is printed to the immediate right of the corresponding symbol. The value of 2*U*_{s}*τ*_{s} ranges from 100 to 400 km, and can be viewed as a target magnitude for the tilt [Eq. (4)] that is initially forced on the vortex. The actual tilt magnitude moderately varies between IS/ISPD and DSPD simulations with the same parameter code. The actual tilt magnitude also depends to some degree on *z*_{l}, which equals 5.25, 3.5, or (in one case) 1.75 km. It is found that the tilt dependence of tropical cyclone development in this study is basically consistent between groups of simulations with different values of *z*_{l}. Appendix A illustrates the diversity of vortex states within our simulation set immediately after the misalignments are created.

List of experiments (excluding the control run) and the parameters used to generate the misalignments. See text for discussion.

### b. Vortex center finding algorithm

Many of the measurements made for this study depend on a reasonable technique for establishing the center of the vortex in a given layer of the atmosphere. The technique employed herein first obtains a mass-weighted vertical average of the relative vertical vorticity distribution *ζ* in a layer defined by *h*_{−} ≤ *z* ≤ *h*_{+}. The vorticity distribution on a regular fine grid (with spacing *l*_{f} = 1.25 km) is then mapped onto a regular coarse grid (with spacing *l*_{c} = 23.75 km). The mapping involves redistributing the circulation of each fine grid cell to the larger cells associated with the four nearest coarse grid points by the method of area-weighting, and then dividing the accumulated circulation in each coarse grid cell by *ζ* while conserving the total circulation of the system. The streamfunction *ψ* and nondivergent velocity field **u**_{ψ} associated with the smoothed version of *ζ* are computed on the coarse grid and interpolated onto the fine grid. The vortex center **x**_{c} is tentatively defined to be the fine grid point where centering a polar coordinate system will maximize the peak value of **x**_{c} is made to a neighboring half grid point if such a change increases the peak value of **x**_{c} is notably limited to regions within the deepest troughs of *ψ*.

Smoothing out small features in *ζ* prior to searching for **x**_{c} is designed to reduce the probability of associating **x**_{c} with the center of an intense but transient meso-*γ*-scale vortex. Once a relatively strong tropical storm develops, the probability of such dubious centering without smoothing of *ζ* is significantly reduced in the simulations considered for this study. At such a point of development, we therefore eliminate the use of a coarse grid and the smoothing step in the center finding algorithm.

The center of the surface vortex **x**_{cs} is generally computed with *h*_{−} = 0.025 km and *h*_{+} = 1.01 km. The center of the middle-tropospheric vortex **x**_{cm} is generally computed with *h*_{−} = 7.34 km and *h*_{+} = 8.13 km. The moving reference frame having coordinates centered at **x**_{cs} will be called the surface-vortex-centered (SVC) reference frame. After misalignment, the velocity of the SVC reference frame relative to the ES reference frame is typically of order 1 m s^{−1}.

Bear in mind that the description of tilt dynamics can vary to some extent with the definition of **x**_{c} (e.g., Ryglicki and Hart 2015). The center finding technique described above is presently favored by the authors and (in essence) within the domain of conventional practice. For those interested, appendix B briefly addresses the consequences of using an alternative technique; the contents of this appendix are best read after section 4a.

## 3. Impact of misalignment on hurricane formation

### a. Hindered intensification of the maximum surface wind speed

**x**

_{c}gives the magnitude and direction of the horizontal displacement of the rotational center of the middle-tropospheric vortex from its counterpart at the surface. Our primary measure of tropical cyclone intensity is the maximum value of the azimuthally averaged tangential velocity at the lowest grid level above the ocean, denoted by

*υ*

_{m}. Needless to say, the pertinent value of

*υ*

_{m}is that measured in the SVC coordinate system.

Let *t*_{ITC} denote the time during the evolution of an incipient tropical cyclone (ITC) when *υ*_{m} first reaches a modest pre–tropical storm value of 12.5 m s^{−1}. Furthermore, let *t*_{CAT1} denote the time when *υ*_{m} first reaches a value of 32.5 m s^{−1}, which approximately corresponds to the threshold wind speed of a category-1 (CAT1) hurricane. Henceforth, the time interval between the aforementioned events will be called the hurricane formation period (HFP). The duration of the HFP is given by *τ*_{hf} = *t*_{CAT1} − *t*_{ITC}. The time-averaged magnitude of the tilt vector over the HFP will be denoted by “tilt_{hf}.” It is worth remarking that the value of tilt_{hf} is closely linked to the maximum tilt magnitude applied within the first 6 h of simulation time (tilt_{0}). A linear regression yields tilt_{hf} = −15.066 + 0.364tilt_{0} (km) and the Pearson correlation coefficient (PC) is 0.88.^{2}

Figure 3 shows that *τ*_{hf} reliably grows with tilt_{hf}. The linear correlation is quantitatively robust in that PC = 0.92. The remainder of section 3 elaborates upon this central result. Sections 3b and 3c examine modifications of the tropical cyclones that coincide with increasing tilt magnitudes and slower development. Section 3d examines how the changes in vortex structure and moist convection associated with enhanced misalignment affect the angular momentum budget. Section 3e investigates how increasing the tilt magnitude affects the growth of alternative kinetic-energy-based measurements of vortex intensity.

### b. Reorganization of convection and enlargement of the surface vortex

The slowdown of hurricane formation in a misaligned system coincides with the reorganization of convection and enlargement of the surface vortex. Figure 4a demonstrates that greater values of tilt_{hf} correspond to greater values of the time-averaged precipitation radius *r*_{p}. By definition, *r*_{p} is the radius in the SVC coordinate system at which the azimuthally averaged 2-h precipitation (surface rainfall) distribution is maximized. Figure 4b shows that the time average of the radius *r*_{m} at which *υ*_{m} occurs grows commensurately with *r*_{p}.

In addition to moving outward, the precipitation grows increasingly asymmetric with enhanced tilt. Figure 5 depicts the tilt_{hf} dependence of the 2-h precipitation asymmetry during the HFP. The total (area integrated) 2-h precipitation in a 400-km circular disc centered at **x**_{cs} is split into individual contributions from 4 quarter circles. The quarter circles are centered in azimuth at *φ* = 0°, 90°, 180°, and 270° (−90°), with *φ* = 0 corresponding to the downtilt direction (i.e., the direction of Δ**x**_{c}). Each absolute contribution to the 2-h precipitation is then divided by the total to form a fractional contribution. The plotted precipitation probability is the time average of the fractional contribution over the HFP. As tilt_{hf} increases from 10 to 60 km, the probability of precipitation in the downtilt quadrant dramatically grows from slightly above 25% to approximately 60%. The probability of precipitation in the uptilt quadrant (centered at 180°) decays to a value substantially less than 10%. The precipitation probabilities in the quadrants centered at −90° and 90° also diminish, but the former decays less than the latter.

For illustrative purposes, Figs. 6 and 7 depict the asymmetric structure of the vortex in simulation DSPD-X400Z5 at a time during the HFP when |Δ**x**_{c}| = 240 km. DSPD-X400Z5 is among a handful of simulations most worthy (in our view) of detailed examination, because the misalignment coincides with severely hindered development of the tropical cyclone. The structure of the vortex in DSPD-X400Z5 is similar to that found in earlier studies of real-world and simulated tropical cyclones that are tilted by moderate environmental wind shear prior to achieving hurricane status (e.g., Rappin and Nolan 2012; Nguyen et al. 2017). Figure 6 shows the misalignment of quasi-circular lower-tropospheric (lw) and middle-tropospheric (md) streamlines in the slowly moving SVC reference frame, superimposed on a complementary plot of the magnitude of the local shear velocity, defined by **u**_{md} − **u**_{lw}.^{3} The lower- and middle-tropospheric flows specifically correspond to elevations of 1.2 and 7.7 km above sea level. The green ray represents the *φ* = 0 axis and therefore points exactly downtilt.

Figure 7a shows the 2-h precipitation field rotated such that downtilt is now directly to the right. Consistent with Fig. 5, much of the precipitation is seen in the downtilt quadrant, whereas the uptilt quadrant is relatively quiet. Figure 7b shows the surface streamlines superimposed on the boundary layer equivalent potential temperature *θ*_{eb}, defined here as the vertical average over the lowest 1 km of the troposphere. The distribution of *θ*_{eb} has relatively low values at and downwind of where low-entropy downdrafts are expected in association with strong convective activity. Note that the deficit of *θ*_{eb} in the vicinity of the active precipitation region coincides with a pronounced cold pool having surface values of ordinary potential temperature *θ* down to 3.3 K below the domain average (not shown). Figure 7c shows the horizontal distribution of relative humidity averaged between *z* = 2.3 and 7.7 km, taking values with respect to ice/liquid at altitudes above/below the freezing level. Humidification that could facilitate vigorous deep convection has failed to develop uptilt. Figure 7d depicts a lower-middle-tropospheric flow pattern suggesting that an influx of relatively dry air from the outer part of the vortex and a weak meso-*α*-scale downdraft (in conjunction with subsidence warming) contribute to maintaining low relative humidity uptilt.

Although more than one factor may contribute to the predominant downdraft between *φ* = 0 and 180°, its magnitude notably agrees with an estimate that assumes middle-tropospheric downgliding of unsaturated air along a nearly frozen virtual potential temperature isosurface. Figure 7e depicts one such isosurface in the CM1 simulation, which compares favorably to that of a system with equivalent *ζ* that has adjusted to a state of nonlinear balance (Fig. 7f, see appendix C). The downgliding velocity is estimated as *w*_{dg} ~ *δzU*/*L* = −0.01 m s^{−1}, in which *δz* ~ −5 × 10^{2} m is the vertical displacement along a streamline path that traverses the interval 0° ≤ *φ* ≤ 180°, *U* ~ 10 m s^{−1} is the characteristic horizontal wind speed, and *L* ~ 5 × 10^{5} m is the horizontal pathlength.

### c. Variation of moist-thermodynamic parameters

The convective and structural modifications associated with misalignment coincide with changes to several bulk moist-thermodynamic parameters in the surface-centered core of the developing system. Figure 8a verifies that strong negative correlations (PC = −0.92 ± 0.01) exist between tilt_{hf} and spatiotemporal averages of the relative humidity distribution measured as in Fig. 7c. The time averaging covers the entire HFP. The spatial averaging is over a cylindrical volume defined in the SVC coordinate system by 2.3 ≤ *z* ≤ 7.7 km and 0 ≤ *r* ≤ *R*, in which *R* is either 100 or 250 km. Qualitatively similar anticorrelations have been verified for *R* = 25 and 400 km (not shown). The association of enhanced tilt with drier air above the surface vortex is notable in view of prior studies suggesting that such low humidity alone can hinder the onset of rapid intensification (e.g., section 3 of Schecter 2016).

One might reasonably ask whether slower development in a strongly misaligned system also coincides with a reduction in the rate at which the vortex extracts the sum of latent and sensible heat from the sea surface. Figure 8b addresses the preceding question by showing the relationship between the tilt magnitude and the parameterized surface enthalpy flux. The enthalpy flux *F*_{k} is defined as in Eq. (3) of Zhang et al. (2008). As for relative humidity, the plotted value of *F*_{k} is a spatiotemporal mean taken within a variable radius *R* of the surface vortex center during the HFP. A strong anticorrelation (PC = −0.93) exists between the mean value of *F*_{k} and tilt_{hf} when *R* = 100 km; a qualitatively similar result is found for *R* = 25 km (not shown). The anticorrelation becomes far less convincing for broader averaging discs, as shown for the case in which *R* = 250 km, where PC = −0.44.

Note that slower development with enhanced tilt is not associated with a reduction of convective available potential energy (CAPE) in the central region of the surface vortex. Figure 8c shows the spatiotemporal mean value of the 500-m mixed layer CAPE, calculated under the assumption of undiluted pseudoadiabatic ascent with liquid-only condensate. The averaging is identical to that of *F*_{k}. For *R* = 100 km, the mean CAPE reliably *grows* (PC = 0.95) with increasing values of tilt_{hf}; the same is true when *R* is reduced to 25 km (not shown). When *R* is extended to 250 km, no meaningful correlation between the mean CAPE and tilt_{hf} can be established (PC = −0.26).

### d. Modification of the angular momentum budget

It is now appropriate to delve deeper into how the geometrical restructuring of the vortex and the coupled reorganization of convection affect the mechanism of surface spinup in the vicinity of maximal winds. A comprehensive investigation would extend beyond the scope of this article, but a limited analysis seems fitting. The following compares the *υ*_{m} but distinct temporal trends. Whereas DSPD-X400Z5 shows a slightly negative trend, the control simulation shows substantial intensification (Fig. 9).

Figure 10 depicts the time-averaged state of each system during the aforementioned 6-h analysis periods. In the azimuthal mean, the misaligned vortex with highly asymmetric convection (DSPD-X400Z5) has a shallower cyclonic circulation, a larger value of *r*_{m}, and two distinct updrafts sprouting from the lower troposphere. The virtually symmetric convection in the control simulation is characterized by a single strong updraft peaked near the relatively small radius of maximum surface wind speed.

^{4}The analysis involves decomposing the mean secondary circulation as follows:

*e*-circulation derives from resolved eddy forcing. The

*δt*is given by

*α*-circulation,

Figures 12a–d show the individual contributions to ^{−1} in the neighborhood of the maximal winds of the time-averaged vortex. Of course, other factors are equally important to the azimuthal velocity budget. Stronger positive and negative tendencies are found near the surface in association with the *e*-circulation and direct eddy forcing near the maximal surface winds are smaller but relevant.

Figures 12e–h show the contributions to ^{−1} slightly outward of the maximal winds of the time-averaged vortex. The positive and negative contributions from

Perhaps the main result from the foregoing analysis is that the restructuring of the vortex and the reorganization of convection in the misaligned system rendered the mp-circulation less effective in accelerating the maximum of *υ*_{m}.

Note that reasonable accuracy of the SE-based analysis relied partly on the small fractional difference (0.3 or less) between the gradient wind and *υ*_{m} might be compounded by limiting supergradient flow (cf. Schecter 2013). A separate study would be required to shed light on this issue.

### e. Surface kinetic energy growth

One might wonder whether increasing the tilt magnitude has the same qualitative effect on all measures of vortex intensity. Herein, we address the preceding question by comparing time series of several intensity parameters. Figure 13 (left) shows the temporal growth of *υ*_{m} for all simulations over the time scale for the virtually aligned control vortex to mature into a well-developed hurricane. Each thick curve covers the spread in a group of vortices that are initially perturbed with similar target misalignments (2*U*_{s}*τ*_{s}) and equivalent values of *z*_{l}. Consistent with Fig. 3, the time series exhibit considerable variation; enhanced tilt markedly slows the intensification of *υ*_{m}.

*R*

_{1}≤

*r*≤

*R*

_{2}is directly proportional to

*R*

_{1}= 0 and

*R*

_{2}= 75 km, so as to represent the kinetic energy of the inner circulation. The time series show variations similar to those of

*υ*

_{m}. The picture changes dramatically upon considering the kinetic energy of the outer circulation. The right panels of Fig. 13 show

*R*

_{1}= 75 km and

*R*

_{2}= 750 km. The growth trends hardly differ from one another following an early adjustment period, regardless of the magnitude of the misalignment. Thus, the detrimental effect of misalignment on surface kinetic energy growth is confined to the inner region of the surface vortex over the time period under present consideration.

## 4. Tilt dynamics

Because tropical cyclones with minimal misalignment are generally more efficient in accelerating the cyclonic surface winds, understanding how tilt decays is an important part of understanding intensification. Figure 14 shows that the evolution of tilt is generally nonmonotonic in systems with initially forced tilt magnitudes exceeding approximately 150 km.^{5} To facilitate discussion, we divide the evolution into three consecutive stages. Stage 1 involves a rapid reduction of the misalignment. Stage 2 entails partial regrowth of the tilt magnitude. Stage 3 is eventually characterized by gradual decay of the tilt magnitude, but may begin with a repetition of the preceding cycle (see, e.g., the dotted curve in Fig. 14b). The remainder of this section will examine the three stages of evolution in detail for simulation DSPD-X400Z5, which is distinguished by having the largest initial tilt. The pathways of tilt decay and transient amplification that operate in DSPD-X400Z5 are considered illustrative of many (but not all) of the possibilities. Some of the known similarities and differences with other simulations will be noted as the narrative proceeds.

Before discussing the intricacies of tilt evolution in DSPD-X400Z5, it is worthwhile to briefly consider the potential relevance of ambient wind shear that may arise over time despite our elimination of the extraneous forcing that created the initial misalignment. Figure 15 shows time series of the magnitude and angle of the ambient shear vector, defined by ⟨**u**_{m} − **u**_{s}⟩_{xy}, in which **u**_{m} (**u**_{s}) is the vertically averaged horizontal velocity in the middle-tropospheric (near-surface) layer corresponding to where **x**_{cm} (**x**_{cs}) is measured. It is found that the shear magnitude remains weak (0–0.35 m s^{−1}) and undulates over the course of the simulation (Fig. 15a). Ambient wind shear possessing one-half the maximum intensity seen here—acting in a direction parallel (antiparallel) to the tilt vector—would amplify (diminish) the tilt of the tropical cyclone at a rate of 15 km day^{−1}. Such a rate is too small to account for the tilt tendencies found at any stage of evolution. Moreover, during the final and slowest stage of alignment, the shear vector rotates anticyclonically on a time scale that is short (an inertial period) compared to the precession period of the tilt vector (Fig. 15b). It follows that extrinsic forcing by ambient shear is not only weak but inefficient.

### a. Stage 1

The first stage of tilt evolution is characterized by rapid decay of the measured misalignment. Such decay commonly coincides with the migration of **x**_{cs} toward an area of vigorous deep cumulus convection in the general direction of **x**_{cm}. Figure 16 provides a minimal depiction of the process during the first 8 h of simulation DSPD-X400Z5. At any arbitrary instant during this 8-h period, the surface center of rotation sits roughly in the middle of a 100-km scale patch of cyclonic vorticity. Figure 16a shows the configuration at *t* = 6 h. Figure 16a also illustrates how the aforementioned vorticity patch is exposed to irrotational winds that converge toward an (initially) outward moving band of convection. The irrotational velocity field **u**_{χ} has a magnitude of approximately 2 m s^{−1} in the vicinity of **x**_{cs}. One might therefore hypothesize that advection of the central vorticity patch by westerly irrotational winds has a nonnegligible role in the 75-km eastward drift of **x**_{cs} over an 8-h period. On the other hand, advection (by any part of **u**) is not the entire story. Downtilt convection also reshapes and rescales the nondivergent (rotational) velocity field of the surface vortex that determines the location of **x**_{cs} (Fig. 16b). The nondivergent winds become enhanced in the east relative to the west. Furthermore, the radius of maximum surface wind speed *r*_{m} increases from 52.5 to 70 km (131.25 km) over the first 8 h (16 h) of development. Thus, during its eastward migration, **x**_{cs} transitions from representing the center of a modest meso-*β*-scale vortex core to representing the center of a core that is 2–3 times larger.

Whereas the early convection-seeking drift of **x**_{cs} substantially reduces tilt in a number of other simulations, northeastward drift of **x**_{cm} largely counters such an effect in DSPD-X400Z5 (not shown). Instead, rapid reduction of tilt occurs through a sudden jump of the middle-tropospheric center of rotation to the area of deep convection (Fig. 17a). The jump apparently results from the emergence of intense middle-tropospheric vorticity anomalies within the updraft region of the mesoscale convective system (MCS; see Fig. 17c). Appendix B (Fig. B1b) illustrates the relatively strong rotational winds associated with the emergent disturbance. The sources of the vorticity anomalies are analyzed in appendix E. Notable vorticity anomalies are also found in the northern sector of the boundary layer of the MCS (Fig. 17b), but they are insufficiently strong in the aggregate to abruptly relocate **x**_{cs}.^{6}

### b. Stage 2

The subsequent regrowth of tilt in experiment DSPD-X400Z5 occurs gradually over a 2-day period of modest intensification of *υ*_{m}. The first issue we address is whether moisture has an essential role in the process. Figure 18a compares the growth of |Δ**x**_{c}| to that found in two dry adiabatic restarts at *t* = 28 h. Two methods are used for restarting the model to demonstrate that details have little consequence on the result. The first restart eliminates cloud microphysics, dissipative heating and the surface enthalpy flux without any additional modifications. The second restart also reduces *C*_{d} to 2.5 × 10^{−5} and refines the fluid variables. The velocity field is refined by zeroing *w*, the irrotational component of **u**, and the *z*-dependent horizontal mean of **u**. The nondivergent component of **u** is obtained by inverting *ζ*, adjusted to have zero vertical gradient in a 303-m layer adjacent to the sea surface. The final refinement involves enforcing conditions of nonlinear balance on *θ* and the pressure field (see appendix C). Both restarts demonstrate that the system would have a propensity to increase tilt under dry adiabatic dynamics somewhat faster and more effectively than the actual process occurs amid moist convection.

Figure 18b compares the moist and dry trajectories of **x**_{cs} and **x**_{cm}. The middle-tropospheric vortex centers of the dry CM1 simulations move northward with their moist counterpart, but drift farther to the west. Early on, the surface center of the moist system is strongly inhibited from following any dry inclination to move southwest. Such inhibition is consistent with the common attraction of surface centers toward areas of vigorous deep convection, here situated to the northeast of **x**_{cs}. Figure 18b also shows the trajectories predicted by ideal 2D fluid dynamics. The 2D results come from two separate vortex-in-cell simulations (Leonard 1980) initialized with *ζ* distributions obtained from the pertinent surface and middle-tropospheric layers of the atmosphere at *t* = 28 h. Each vortex-in-cell simulation has roughly 10^{8} vorticity elements, a rectangular mesh with 0.65-km grid spacing, and doubly periodic boundary conditions equivalent to those of the CM1 simulations. The middle-tropospheric trajectory predicted by 2D dynamics remains relatively close to that of the moist system. Such closeness may be somewhat coincidental, but reproduction of the basic northward drift suggests some relevance of the 2D model. By contrast, the 2D surface trajectory strays considerably from its moist counterpart, and ends up far west of all 3D systems.

Figures 18c and 18d show snapshots of the moist middle-tropospheric vortex near the start and end of the northward drift of **x**_{cm} that is largely responsible for the regrowth of tilt. It is seen that the drift coincides with considerable reshaping of an asymmetric vertical vorticity distribution with multiscale structure and a prominent band extending outward from the core. The process occurs amid continual 3D-adiabatic and diabatic perturbations of *ζ*. The associated irrotational winds represented by **u**_{χ} are nontrivial (Fig. 18c), but as for any predominantly vortical flow, the nondivergent component of the velocity field **u**_{ψ} is characteristically stronger (Fig. 18d). The root-mean-square (rms) value of |**u**_{ψ}| is 3.4 times the rms value of |**u**_{χ}| over the depicted area in both snapshots.^{7}

The relative strength of **u**_{ψ} combined with the qualitatively successful prediction of the vortex-in-cell simulation in the middle troposphere motivate further consideration of how nondivergent 2D dynamics may contribute to the drift of **x**_{cm}. Figure 19 splits the middle-tropospheric nondivergent velocity field into two parts during the northward drift period. The first part *t* = 31 h) or Fig. 19d (at *t* = 44 h) using a free-space Green function. The aforementioned curve is essentially a contour where *ζ* = 5 × 10^{−6} s^{−1} after Gaussian smoothing with a kernel whose decay length is 30 km in both horizontal dimensions. By design, ^{8} It is seen that negative vorticity to the east generates an anticyclonic gyre in ^{9} Such nudging does not seem incidental to the drift of **x**_{cm}, but one should bear in mind that more subtle 2D or 3D mechanisms unapparent from the preceding analysis could have equal or even greater importance.

Although regrowth of the tilt magnitude in a cloud-resolving simulation of tropical cyclone development without assistance from environmental wind shear may seem surprising, the spontaneous misalignment of a vortex is not an entirely novel phenomenon. Previous discussions of spontaneous misalignment have often dealt with tilts arising from essentially adiabatic, three-dimensional circular shear flow instabilities (Gent and McWilliams 1986; Schecter et al. 2002; Reasor et al. 2004). Jones (2000a) examined what might be viewed as the nonlinear stage of an instability contributing to the growth of tilt in a dry vortex simultaneously interacting with environmental winds. The associated dynamics happened to resemble that described above, in which an outer anticyclonic vorticity anomaly acts to drive the cyclonic core of the vortex in one layer of the atmosphere away from the core in another layer. While the similarity is intriguing, we caution against extending the analogy too far. Events leading up to the foregoing scenario in DSPD-X400Z5 are distinct in part by having a substantial diabatic element (see section 4a and appendix E). There is also evidence, provided below, suggesting that convection and its associated irrotational winds significantly modulate the drift of **x**_{cm} during the tilt amplification period.

The aforementioned evidence is found by viewing the drift of the middle-tropospheric cyclone from a perspective that is more directly connected to the measurement of **x**_{cm}. Recall that **x**_{c} is essentially the point in a specified layer where the centering of a polar coordinate system yields the largest peak value of *z* ≤ 8.13 km) at the start and end of the 6-h analysis period in two stationary coordinate systems. One coordinate system (CS_{i}) is centered on the initial location of **x**_{cm} (Fig. 20b); the other (CS_{f}) is centered on the final location of **x**_{cm} (Fig. 20a). The top panels indicate that the drift is attributable to maintenance—as opposed to appreciable amplification—of mean-vortex intensity in CS_{f} in conjunction with decay of maximal _{i}. The bottom panels of Figs. 20a and 20b show the _{f} (Fig. 20a), while it weakens the mean vortex where _{i} (Fig. 20b). The aforementioned intensification effort in CS_{f} is notably tempered by the *net* impact of the absolute vorticity influx driven by irrotational winds **x**_{cm} from the ideal 2D trajectory (Fig. 18b), and suggests that any drift that may arise from the nudging of the middle-tropospheric cyclone by ^{10}

### c. Stage 3

The final stage of tilt evolution brings the system to a state of virtual alignment. Dry adiabatic restarts cannot reproduce the sustained alignment trends found in the moist simulation (Fig. 21a). It stands to reason that diabatic cloud processes are essential. Figure 21b shows the near-surface and middle-tropospheric vortex trajectories over the period between hours 165 and 171 of development, which essentially covers the final surge to an aligned state. The trajectories are superimposed over a depiction of the attendant moist convection. Whereas **x**_{cm} shifts little within the broader updraft region of a convective complex,^{11} **x**_{cs} darts westward toward its counterpart.

Figure 22 illustrates the nature of the fluid dynamics during the westward motion of **x**_{cs}. The top row shows that the near-surface cyclone consists of multiple meso-*γ*-scale vortices immersed in diffuse, predominantly cyclonic background vorticity. The meso-*γ*-scale vortices are typically products of convection in either the core of the parent cyclone or peripheral rainbands. Some may travel far away from their points of origin over the course of a lifetime; the three prominent eastern vortices at *t* = 165 h notably emerged from the main area of convection west of **x**_{cs}. The irrotational velocity field consists of broad inflow from the outer part of the cyclone and zones of confluence near active convection. Relatively strong western confluence of **u**_{χ} may contribute significantly to the westward shift of **x**_{cs}. On the other hand, the rms wind speed of the nondivergent velocity field is 5.6 times that of **u**_{χ} in each depicted snapshot of the evolving system. Generic mixing processes typical of nondivergent 2D flows seem to have a nontrivial role in reshaping the vorticity distribution. The bottom row of the figure shows the nondivergent velocity field and helps clarify why the measurement of **x**_{cs} moves westward. The intensity distribution becomes highly skewed to the west before the strongest winds become more evenly distributed about a 100-km scale circle whose center is substantially displaced (to the west) from where the surface vortex center resided 6 h earlier.

To elaborate, the westward drift of **x**_{cs} coincides with the amplification of maximal _{f}) centered at the end point of the 6-h trajectory. Various contributions to the amplification can be seen in the azimuthal velocity budget observed in CS_{f}. Figure 23a shows the budget with each term vertically averaged over the near-surface layer (*z* ≤ 1.01 km) and temporally averaged over the 6-h analysis period. The combined effect of the radial influx of absolute vorticity driven by the irrotational winds **x**_{c} are very similar. Interestingly, both within and slightly above the near-surface layer, the radial influx of vorticity driven by nondivergent winds **x**_{cs} is somehow nullified. Among other factors to consider, convection modifies both the vorticity perturbation and the coupled nondivergent winds. The importance of convection and **u**_{χ} is less concealed when restricting the analysis to the first 3 h of the drift period [165 ≤ *t* ≤ 168 h]. Figures 23c and 22d show that in a fixed coordinate system centered where **x**_{cs} resides at *t* = 168 h, the positive combination of

As a final remark, the third stage of tilt evolution coincides with fairly slow growth of *υ*_{m} (Fig. 21a). The onset of rapid intensification does not occur until nearly one full day after the final surge of alignment that brings |Δ**x**_{c}| down to approximately 20 km.

### d. Comment on alignment paradigms based on vortex Rossby wave dynamics

There exists a sizable body of literature on the potential importance of vortex Rossby (VR) wave dynamics in contributing to the alignment process (Reasor and Montgomery 2001,2015; Reasor et al. 2004; Schecter et al. 2002; Schecter and Montgomery 2003,2007; Schecter 2015). In quasi-balanced linear perturbation theory, a relatively weak tilt decomposes into a set of discrete and sheared VR waves.^{12} If stability conditions are satisfied, free alignment may occur by the outward propagation and spiral windup of sheared VR waves, or by the negative feedback that a discrete VR wave will receive upon exciting a potential vorticity (PV) perturbation in a critical layer.

A reasonable estimate of the time scale for outward propagation and spiral windup of a sheared (tropical cyclone scale) VR wave is *τ*_{υ} ≡ 2*πr*_{υ}/*υ*_{υ}, in which *r*_{υ} and *υ*_{υ} respectively denote the characteristic radius and azimuthal velocity of the vortex. The time scale for damping of a discrete VR wave is sensitive to the average value of a quantity proportional to the radial gradient of basic-state PV in the critical layer, centered on the surface where *ω* denotes the angular frequency of the wave, *n* = 1 is the azimuthal wavenumber, and *τ*_{υ}, and is ordinarily much longer when the nondimensional deformation radius ^{13} In the preceding definition of *N* is the Brunt–Väisälä frequency and *H* is the vertical length scale of the misalignment. Weaker damping with greater *ω* and consequent outward displacement of the critical layer to where the PV gradient is small. Letting *r*_{υ} = 10^{5} m and *υ*_{υ} = 10 m s^{−1} yields *τ*_{υ} = 17.5 h, which seemingly permits spiral windup to be relevant to an alignment process lasting 1–10 d. However, letting *N* = 10^{−2} s^{−1}, *H* = 5 × 10^{3} m, and *f* = 5 × 10^{−5} s^{−1} yields

As time advances, alternative (nonlinear and diabatic) mechanisms can diminish the tilt, cloud coverage may become more diffuse, and the tropical cyclone may intensify to some degree. Cloud-related reduction of *N* and growth of *υ*_{υ} both decrease *ω* of the discrete tilt wave and thereby improve the likelihood of enhanced damping via the wave-critical layer interaction. Accordingly, classical VR wave alignment mechanisms could eventually become relevant. Such potential relevance may often be obscured by concurrent processes, and was not obvious from the basic analysis of section 4c. Further study will be necessary to resolve this issue. Bear in mind that the arrested alignment found in several adiabatic restarts (Fig. 21a) did not disprove the partial relevance of VR wave dynamics, partly because the restarts did not incorporate the potentially important cloud-related reduction of *N*.

## 5. Conclusions

This paper has examined the development of tropical cyclones with variable degrees of misalignment in a cloud-resolving model set up over a moderately warm ocean on an *f* plane. The study was distinguished from its predecessors in virtually eliminating ambient vertical wind shear. Our numerical simulations showed that increasing the initial tilt of an incipient tropical cyclone from a negligible value to several hundred kilometers can extend the time scale of hurricane formation from 1 to 10 days. The dramatic slowdown of development in a strongly perturbed system was linked to an extended duration of misalignment resulting from incomplete early decay and subsequent transient growth of the tilt magnitude. Prolonged misalignment coincided with a prolonged period of asymmetric convection peaked far from the rotational center of the surface vortex **x**_{cs}. Conventional wisdom holds that such a state of affairs generally frustrates the intensification of the maximum tangential wind speed.

To elaborate, the incipient tropical cyclones were found to immediately begin a major realignment phase lasting up to 1 day. The mechanism was examined in detail for a selected simulation starting with an exceptionally large (367 km) tilt. As in other simulations, **x**_{cs} migrated toward an area of vigorous deep cumulus activity in the general direction of the middle-tropospheric rotational center **x**_{cm}. Moreover, **x**_{cm} abruptly jumped to a pronounced vorticity anomaly that formed within the same mesoscale convective system.

Setting the initial tilt magnitude to appreciably exceed the 100-km length scale of the vortex core generally prevented a direct route to alignment, and enabled substantial regrowth of the tilt magnitude over a 1–2 day time scale following its initial decay. For the aforementioned case study, a similar propensity for the restoration of misalignment was found in two dry adiabatic restarts with slightly different initializations and computational configurations. Furthermore, a nondivergent 2D fluid model correctly predicted the direction and magnitude of the drift of **x**_{cm} away from **x**_{cs} that largely accounted for the regrowth of tilt. Analysis (of the moist 3D system) suggested that an anticyclonic component of the nondivergent velocity field **u**_{ψ} associated with an external negative vorticity patch nudged the cyclonic vorticity core containing **x**_{cm} in a manner that could assist the observed drift. That being said, diabatic processes persisted and affected the evolution of the tilt vector. Deeper analysis revealed that convection and irrotational winds modulated the specific changes of **u**_{ψ} that our center-finding algorithm used to determine the motion of **x**_{cm}. Moreover, the moist system restored the misalignment more slowly and less completely than the dry adiabatic systems.

The final stage of the alignment process lasted several days in tropical cyclones initialized with relatively large tilts. In the simulation selected for detailed examination, diabatic processes were clearly essential to the ultimate reduction of tilt; the removal of moisture at different times arrested or reversed the decay trend. The final stage of alignment seemed to involve intricate mechanics, as illustrated during a late 6-h westward surge of the surface center toward an area of deep convection containing its middle-tropospheric counterpart. The surge coincided with the mixing of subvortices and filaments within the broader surface cyclone, amid convective forcing weighted to the west. The motion of **x**_{cs} was linked to a modest amplification of maximal (lower-tropospheric)

The extent to which the foregoing dynamics of a misaligned tropical cyclone depends on details of the experimental setup remains to be seen. Sensitivity to variation of the sea surface temperature and the inclusion of a radiation parameterization merits further investigation. Of equal interest would be a study on sensitivity to the basic structure of the vortex prior to the initial misalignment, which is a well-known issue in the related dry adiabatic problem (Jones 2000a,b; Schecter and Montgomery 2003; Reasor et al. 2004).

Overall, the results of our numerical experiments appear to have corroborated inferences drawn from earlier studies (with sustained shear) on how misalignment reorganizes convection and inhibits tropical cyclone development. A robust positive linear correlation was found between the time-averaged tilt of a developing system (tilt_{hf}) and the time required for the surface-maximum tangential velocity *υ*_{m} to reach the threshold wind speed of a hurricane. Relatively large values of tilt_{hf} and slower development were associated with enhanced asymmetry (downtilt bias) of the precipitation probability distribution and larger values of the characteristic precipitation radius *r*_{p} in the surface-vortex-centered coordinate system. The growth of *r*_{p} with increasing tilt_{hf} coincided with a commensurate growth of the radius *r*_{m} at which *υ*_{m} is located.

The consequences of tilt-related modifications to tropical cyclone structure on the mean-vortex spinup mechanism were briefly examined in the context of a Sawyer–Eliassen (SE) model. The SE-based analysis provided a reasonably accurate decomposition of the early azimuthal velocity tendency in the vicinity of *υ*_{m}, in selected systems with low and high degrees of misalignment. Although no single factor completely explained slower intensification in the strongly misaligned system, greater tilt was linked to weaker positive azimuthal velocity forcing near *υ*_{m} by the component of the mean secondary circulation attributed to heating by microphysical cloud processes.

One notable aspect of the present simulation set was minimal system-to-system variation in the growth of kinetic energy contained outside the core of the primary surface circulation over the time scale required for an aligned vortex to mature. In other words, the detrimental effect of tilt on surface kinetic energy growth was confined to the inner region of the surface vortex.

We conclude with a final remark on the relevance of this paper to forecasting. Although a strong correlation was found between tilt_{hf} and *τ*_{hf}, our results suggest that an *instantaneous* measurement of the tilt magnitude may not be a good predictor of the time scale for future intensification of *υ*_{m}, even in a quiescent environment. Such is evident from the nonmonotonic evolution of the tilt magnitude. Distinct time scales would be found for equivalent magnitudes measured during different growth or decay phases of the misalignment.

## Acknowledgments

The authors thank Dr. George Bryan of the National Center for Atmospheric Research (NCAR) for providing the cloud-resolving model (CM1) that was used for the numerical experiments. The authors also thank several anonymous reviewers for their generally constructive comments on the submitted manuscript. This paper was primarily supported by the National Science Foundation under Grant AGS-1743854. Additional support came from the Natural Sciences and Engineering Research Council of Canada, and from Hydro-Quebec through the IRC program. The computational resources required to conduct the simulations were provided by NCAR’s Computational and Information Systems Laboratory (doi:10.5065/D6RX99HX), and by Compute Canada.

## APPENDIX A

### Vortex States Generated by Artificial Misalignment Forcing

Figure A1 depicts a number of selected vortices immediately after their misalignments are generated by methods explained in section 2a. Figures A1a and A1b show a vortex that is misaligned using the DSPD method, with *z*_{l} = 5.25 km and a target tilt magnitude of 400 km. The top panel shows relative vertical vorticity and relative humidity in a vertical plane parallel to the tilt vector [Eq. (4)] and passing through the surface vortex center. Both fields are 500-km centered averages over the coordinate *y*′ that measures horizontal distance perpendicular to the visualization plane. The relative humidity assumes ice (liquid) condensate above (below) the freezing level. The bottom panel shows horizontal slices of lower- and middle-tropospheric vorticity. Note that the misalignment process 1) reduces middle-tropospheric relative humidity over the central and uptilt regions of the surface vortex, and 2) reweights the negative component of the vorticity distribution toward the east in the outer part of the middle-tropospheric circulation. Figures A1c and A1d are similar to Figs. A1a and A1b, but for a DSPD experiment with a target tilt magnitude of 100 km. Here, the aforementioned consequences of the misalignment process are considerably less pronounced.

Figures A1e and A1f show the vortex from ISPD-X400Z5, in which the tilt is generated while convection is active over the first 6 h of the simulation. The misalignment is moderately smaller than its counterpart in the DSPD experiment with equivalent settings for the target tilt magnitude and *z*_{l} (Figs. A1a,b). The convection has also generated a dipole-asymmetry within the predominantly cyclonic core of the middle-tropospheric vortex. The intensity of the negative vorticity hole within the core is found to decay with decreasing values of the target tilt magnitude (not shown). Of further note, as in the aforementioned DSPD experiment, the misaligned vortex has diminished negative vorticity in the outer western region of the middle troposphere. Figures A1g and A1h are similar to Figs. A1e and A1f, but for ISPD-X300Z3.

## APPENDIX B

### Sensitivity to the Center-Finding Algorithm

The “vortex center” at a specified altitude is not uniquely defined in tropical cyclone meteorology. One might wonder how changing the definition would affect the description of the dynamics. This appendix aims to remove some of the mystery for the present study without an exhaustive sensitivity test.

*l*

_{c}in the standard center-finding algorithm of section 2b ordinarily adds noise to the time series of

**x**

_{c}without changing the main trend over any extended time period of interest. The preceding result seems obvious and unworthy of lengthy discussion. Instead, let us consider the following alternative definition of a cyclonic vortex center in an arbitrary layer:

*A*of integration covers the entire horizontal domain of the simulation,

**x**is the horizontal coordinate vector,

*ζ*is the

*z*-averaged relative vertical vorticity distribution of the pertinent layer smoothed as in section 2b with

*l*

_{c}= 23.75 km, and Θ(

*ζ*−

*ζ*

_{c}) = 1 (0) for

*ζ*>

*ζ*

_{c}(

*ζ*<

*ζ*

_{c}). The cut-off vorticity is given by

*ϵ*

_{ζ}< 1.

Figure B1a compares the tilt magnitude in DSPD-X400Z5 computed with our standard method to that obtained by replacing **x**_{cs/cm} with *ϵ*_{ζ} = 0.1 but nontrivial for *ϵ*_{ζ} = 0.02. The greatest discrepancies begin when vorticity anomalies generated by convection in the middle troposphere create a sufficiently strong meso-*β*-scale rotational flow structure far away from the prior consensus vortex center (*t* ≈ 10, 75 h). At such times, the middle-tropospheric flow cannot be said to consist of a single vortex.

Figures B1b–d show snapshots of the streamlines and magnitude of the middle-tropospheric nondivergent velocity field after an early event when the single vortex model of the middle-tropospheric flow becomes questionable. The standard location of **x**_{cm} (found as in section 2b) and the locations of *ϵ*_{ζ} = 0.1 and 0.02 are indicated on each snapshot; so too are the surface centers. The first snapshot (Fig. B1b) reveals substantial differences between the standard computation of **x**_{cm} and its alternatives. At this time, two noteworthy mesoscale vortices exist. One is the large northeastern cyclone originating from the initial conditions, and the other is a small convectively generated vortex (whose details are shown by the inset) coupled to an intensified stream of southwesterly winds.^{B1} The standard center-finding algorithm locates a point near the smaller but stronger vortex. The *ζ*-weighted averages of **x** [Eq. (B1)] return points between the cores of the two vortices. Lowering *ϵ*_{ζ} from 0.1 to 0.02 shifts

## APPENDIX C

### Helmholtz Velocity Decomposition and Nonlinear Balance

**u**

_{χ}≡ ∇

_{h}

*χ*, in which ∇

_{h}is the horizontal gradient operator and

**u**

_{a}is both irrotational and nondivergent. For this paper, the Helmholtz decomposition is generally constructed by solving the Poisson equation for

*ψ*with periodic boundary conditions, equating

**u**

_{a}to ⟨

**u**⟩

_{xy}, and extracting

**u**

_{χ}from Eq. (C1). Note that

**u**

_{a}is identical to the ambient velocity field defined in the main text, which may vary with the motion of the reference frame. Throughout this appendix, the subscript

*a*will denote the

*z*-dependent horizontal domain average (ambient part) of a fluid variable, whereas the prefix

*δ*will denote a perturbation from that average.

**u**

_{ψ}dominates

**u**

_{χ}, such that

*σ*reduces to a diagnostic equation for the pressure perturbation. The approximation amounts to neglecting all terms involving

*σ*,

**u**

_{χ},

*w*and parameterized turbulence in the aforementioned tendency equation. Such neglect yields

*x*and

*y*are orthogonal Cartesian coordinates,

*R*

_{d}(

*c*

_{pd}) is the gas constant (isobaric specific heat) of dry air, and

*p*

_{0}≡ 10

^{5}Pa. Here and elsewhere, ∂

_{α}≡ ∂/∂

*α*and ∂

_{αβ}≡ ∂

^{2}/∂

*α*∂

*β*denote the first and second order partial derivatives with respect to the generic variables

*α*and

*β*. Equation (C3) is readily solved for

*δ*Π (given

*ζ*and

*ψ*at arbitrary

*z*) under the assumptions of doubly periodic boundary conditions and zero horizontal mean. Note that Eq. (C3) uses the approximation

*ρ*

^{−1}∇

_{h}

*δp*≈

*c*

_{pd}

*θ*

_{υa}∇

_{h}

*δ*Π, in which

*ρ*is density. Note further that Eq. (C3) reduces to gradient wind balance (cyclostrophic balance if

*f*= 0) when

*ζ*is axisymmetric.

*g*is the gravitational acceleration,

*δθ*

_{υ}≡

*θ*

_{υ}−

*θ*

_{υa}, and

*δ*Π is the solution to Eq. (C3). In this paper, the system is said to be in a state of nonlinear balance if approximations (C2)–(C4) are satisfied.

## APPENDIX D

### Sawyer–Eliassen-Based Analysis of Vortex Spinup

*φ*average of the azimuthal velocity equation in the SVC reference frame (or a stationary reference frame) yields

*η*≡

*ζ*+

*f*is the absolute vertical vorticity, and

*θ*

_{ρ}is adequately approximated by

*θ*.

*φ*-averaged thermodynamic equation may be written for the variable

*κ*≡

*θ*

^{−1}as follows:

*θ*. It is convenient to let

*ϵ*is included to acknowledge the inexact nature of the SE solution for

The main text considers the integral of each term in Eq. (D8) over a relatively short time period (*δt* = 6 h) using data stored every 10 min. The time integrals of

_{α}= 0 at

*r*= 0,

*r*=

*r*

_{o},

*z*= 0, and

*z*=

*z*

_{t}, in which

*r*

_{o}= 919 km and

*z*

_{t}= 29.2 km. Small (and even negative) static stability in some regions of the boundary layer may cause violation of the following ellipticity condition:

*z*= 400 m (where it so happens that

*ν*is a small positive parameter. The solution presented in the main text corresponds to

*ν*= 0.001 but hardly differs from its counterpart with

*ν*= 1.0. Despite such insensitivity, one should bear in mind that the necessity of an ad hoc regularization leaves the solution (at least in the boundary layer) somewhat questionable.

## APPENDIX E

### Middle-tropospheric $\mathit{\zeta}$ Budget during Stage 1 of DSPD-X400Z5

*ζ*) is given by

*ζ*_{h}is the horizontal vorticity vector,

**T**

_{h}is the horizontal velocity tendency associated with parameterized turbulence, and all other symbols were introduced earlier. From left to right, the first and second terms on the right-hand side of Eq. (E1) correspond to the horizontal and vertical advection of

*ζ*. The third and fourth terms are associated with vortex-tube stretching and tilting. The fifth and sixth terms are associated with baroclinic and frictional vorticity generation. The horizontal advection and stretching terms are commonly combined into a single term equaling −∇

_{h}⋅

*η*

**u**. Similarly, the vertical advection and tilting terms are commonly combined into a single term equaling

Figure E1 depicts the *ζ* budget in the middle troposphere prior to the sudden jump of **x**_{cm} toward a region of vigorous cumulus activity during stage 1 of experiment DSPD-X400Z5 (see Fig. 17). Figure E1a (E1b) corresponds to the time interval between hours 5 and 6.5 (8 and 9.5) of the simulation. The far-left panel shows the sum of all terms contributing to the right-hand side of Eq. (E1) averaged over the 1.5-h analysis period. The panels to the right show similar time averages of specific contributions. Contours of *ζ* at the end of the analysis period (henceforth *ζ*_{e}) are superimposed on each plot. The analysis is conducted in the domain-centered ES reference frame.

One can readily see that the sum of all tendency terms is well-correlated with *ζ*_{e} in the displayed vicinity of vigorous convection. Moreover, the “forcing” of *ζ* by the combination of vertical advection and tilting tends to have the same sign as *ζ*_{e}; rarely does the opposite sign occur where *ζ*_{e} is substantial. The combination of horizontal advection and stretching partly reinforces and partly opposes vertical advection and tilting. The sign of the stretching term alone does not consistently match that of its combination with horizontal advection or that of all terms summed together. Note that all displayed tendencies have substantial positive *and negative* parts (see Haynes and McIntyre 1987). Considering the partially negative forcing of *ζ* associated with convection, along with the initial eastern crescent of outer negative vorticity (Fig. A1b), it is unsurprising to find a prominent anticyclonic vorticity patch east of the cyclonic core of the middle-tropospheric circulation at the start of stage 2 of the simulation (see section 4b).

## APPENDIX F

### Helmholtz-Based Decomposition of ${\partial}_{t}\overline{\upsilon}$

*u*

_{ψ}or

*u*

_{a}involves only a nominal eddy-contribution. For simplicity, we have refrained from the common practice of splitting

## REFERENCES

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.Bui, H. H., R. K. Smith, M. T. Montgomery, and J. Peng, 2009: Balanced and unbalanced aspects of tropical-cyclone intensification.

,*Quart. J. Roy. Meteor. Soc.***135**, 1715–1731, https://doi.org/10.1002/qj.502.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.Davis, C. A., and L. F. Bosart, 2006: The formation of hurricane Humberto (2001): The importance of extra-tropical precursors.

,*Quart. J. Roy. Meteor. Soc.***132**, 2055–2085, https://doi.org/10.1256/qj.05.42.Davis, C. A., and D. A. Ahijevych, 2012: Mesoscale structural evolution of three tropical weather systems observed during PREDICT.

,*J. Atmos. Sci.***69**, 1284–1305, https://doi.org/10.1175/JAS-D-11-0225.1.Davis, C. A., and D. A. Ahijevych, 2013: Thermodynamic environments of deep convection in Atlantic tropical disturbances.

,*J. Atmos. Sci.***70**, 1912–1928, https://doi.org/10.1175/JAS-D-12-0278.1.DeMaria, M., J. A. Knaff, and B. H. Connell, 2001: A tropical cyclone genesis parameter for the tropical Atlantic.

,*Wea. Forecasting***16**, 219–233, https://doi.org/10.1175/1520-0434(2001)016<0219:ATCGPF>2.0.CO;2.Donelan, M. A., B. K. Haus, N. Reul, W. J. Plant, M. Stiassnie, H. C. Graber, O. B. Brown, and E. S. Saltzman, 2004: On the limiting aerodynamic roughness of the ocean in very strong winds.

,*Geophys. Res. Lett.***31**, L18306, https://doi.org/10.1029/2004GL019460.Drennan, W. M., J. A. Zhang, J. R. French, C. McCormick, and P. G. Black, 2007: Turbulent fluxes in the hurricane boundary layer. Part II: Latent heat flux.

,*J. Atmos. Sci.***64**, 1103–1115, https://doi.org/10.1175/JAS3889.1.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.Fairall, C. W., E. F. Bradley, J. E. Hare, A. A. Grachev, and J. B. Edson, 2003: Bulk parameterization of air–sea fluxes: Updates and verification for the COARE algorithm.

,*J. Climate***16**, 571–591, https://doi.org/10.1175/1520-0442(2003)016<0571:BPOASF>2.0.CO;2.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.Ge, X., T. Li, and M. Peng, 2013: Effects of vertical shears and midlevel dry air on tropical cyclone developments.

,*J. Atmos. Sci.***70**, 3859–3875, https://doi.org/10.1175/JAS-D-13-066.1.Gent, P. R., and J. C. McWilliams, 1986: The instability of barotropic circular vortices.

,*Geophys. Astrophys. Fluid Dyn.***35**, 209–233, https://doi.org/10.1080/03091928608245893.Gray, W. M., 1968: Global view of the origins of tropical cyclones.

,*Mon. Wea. Rev.***96**, 669–700, https://doi.org/10.1175/1520-0493(1968)096<0669:GVOTOO>2.0.CO;2.Haynes, P. H., and M. E. McIntyre, 1987: On the evolution of vorticity and potential vorticity in the presence of diabatic heating and frictional or other forces.

,*J. Atmos. Sci.***44**, 828–841, https://doi.org/10.1175/1520-0469(1987)044<0828:OTEOVA>2.0.CO;2.