## 1. Introduction

There has yet to emerge a complete understanding of the mechanisms that drive a tropical cyclone (TC) toward a state of vertical alignment. It is of interest to elucidate the alignment mechanisms, and the conditions that improve their effectiveness, because tilted TCs are often weaker than their upright counterparts (e.g. Riemer et al. 2010; Frank and Ritchie 2001; DeMaria 1996). This paper explicitly demonstrates some of the merits and deficiencies of a recently advanced theory of tilt dynamics.

### a. Review of the theory at issue

The literature contains a number of articles on the tilt dynamics of nonconvective vortices in a stably stratified atmosphere. Some of these articles discuss how vortices resist tilting under sustained vertical wind shear (Jones 1995, 2000a,b, 2004; Vandermeirsh et al. 2002; Reasor et al. 2004, hereafter R04). Others focus on the decay of tilt in the absence of external forcing (Polvani 1991; Viera 1995; Reasor and Montgomery 2001, hereafter RM01; Schecter et al. 2002, hereafter S02; Schecter and Montgomery 2003, hereafter SM03; Jones et al. 2009). It has been shown that TC-like vortices commonly have mechanisms to counter tilt without diabatic processes driving a mean secondary circulation to potentially assist. It has also been shown that the effectiveness of such mechanisms can depend on details in the spatial distribution of potential vorticity (PV).

In a simple but common scenario, the vertical misalignment of PV in a nonconvective vortex is dominated by a special vortex Rossby (VR) wave, here called the principal tilt mode (PTM). The PTM is usually damped by depositing wave activity into an outer critical layer, where the mode is resonant with the fluid rotation (S02; SM03). Sensitivity of critical layer damping to details partly accounts for variable tilt dynamics in seemingly similar vortices. Nevertheless, PTMs are found to follow some basic “rules of thumb.” Weaker static stability or greater inertial stability in the vortex core tends to increase the natural precession frequency and damping rate of the PTM. These changes are thought to help reduce the excitability of the PTM and thereby enhance the resistance of the vortex to slow misalignment forcing, such as that related to ambient vertical wind shear. Note that if the PTM is very strongly damped, the weak tilt created by forcing may largely consist of ordinary continuum modes and degenerate into sheared VR waves (RM01; R04). The residual alignment mechanism may then involve the spiral windup and outward propagation of such waves.

Of course, PTMs are not always damped. Appendix A discusses various conditions under which PTMs may persist or spontaneously grow without forcing. Furthermore, there may be additional shear-flow instabilities independent of the PTM that frustrate vertical alignment (cf. Nolan et al. 2001; Smith and Rosenbluth 1990). Such instabilities are often overlooked in simplified studies by choosing a vortex in which they develop slowly or do not exist. The relevance of this choice is questionable for the study of intense TCs.

Perhaps the most pressing concern regarding the PTM paradigm of tilt dynamics is the simplified manner in which moisture has been incorporated into the theory. The PTM paradigm is largely based on a simple linear model that treats moisture merely as a local reduction of static stability (Schecter and Montgomery 2007, hereafter SM07; R04). Because reducing static stability tends to lessen the excitability of a PTM, this model has a mechanism for moisture to inhibit tilt. Such moisture-induced inhibition is qualitatively consistent with earlier computational studies such as Wang and Holland (1996). However, the explanation provided by the simple model has not been thoroughly tested. The simple model neglects how tilt is influenced by deep convective transport by the diabatically maintained symmetric secondary circulation (SSC). The simple model also neglects the effects of surface fluxes and boundary layer processes on the asymmetric convection that is coupled to the behavior of a PTM. Finally, the simple model overlooks nonlinear and stochastic elements of deep convection.

One might infer from a number of quasi-realistic TC simulations in the literature that the features of moist convection neglected by current theory have some influence on the behavior of tilt (e.g. Rogers et al. 2003; Wong and Chan 2004; Braun et al. 2006; Zhang and Kieu 2006; Braun and Wu 2007; Davis et al. 2008; Zhang and Tao 2013). The same inference might be drawn from recent efforts to understand the observed relationship between convective asymmetry and tilt in real TCs exposed to ambient vertical wind shear (e.g. Reasor and Eastin 2012; Reasor et al. 2013). However, the merits and shortcomings of current theory have not been fully clarified.

### b. Objectives and overview of the present study

The main objective of the present study is to directly test the assumptions and predictions of the simple linear theory (SLT) of tilt dynamics described above. Assessment of the SLT will be based on computational experiments with a conventional cloud model (CM), in which a TC of hurricane strength is exposed to a period of idealized misalignment forcing and then released to evolve freely with time. The primary experiment (E1) is designed to include all relevant physical processes. A second experiment (E2) carefully removes all moisture and the SSC from the vortex before the forcing is applied. A third experiment (E3) removes the SSC but includes suspended cloud droplets and the attendant reduction of static stability inside the vortex. A fourth experiment (E4) removes moisture but maintains the SSC through an artificially distributed heat source.

Comparison of the primary hurricane experiment (E1) to the nonconvective dry vortex experiment (E2) will confirm that moist convection severely inhibits the development of tilt and the excitation of the PTM. Moreover, the evolution of tilt in both experiments will be found to agree reasonably well with explicit predictions of the SLT. The outcome of E3 will support the theoretical notion that reduction of static stability by cloud water is sufficient to inhibit the excitation of a PTM. However, some discrepancies will be found between the SLT and details of asymmetric convection in the eyewall region of the E1 hurricane. Moreover, E4 will provide evidence that the SSC has a nonnegligible role in reducing tilt. Diagnosis of E1 will further illustrate how the SSC has discernible influence over misalignment at least in the eyewall.

For good measure, experiments E1 and E2 will be repeated with a slightly more intense and contracted vortex. The results will confirm various differences with the original experiments predicted by the SLT.

### c. Outline of the remaining sections

The remainder of this article is organized as follows. Section 2 describes the computational setup for each CM experiment. Section 3 presents the relevant SLT. Section 4 presents the results of the CM experiments. Section 5 summarizes the main findings of this study. The appendixes supplement the main text with some notable technical details.

## 2. Setup of the numerical experiments

### a. Configurations of the cloud model

The numerical simulations are conducted with the Regional Atmospheric Modeling System (RAMS 6.0), which is maintained and distributed to the public by ATMET LLC. RAMS is a conventional weather research model with a variety of options for parameterizing cloud microphysics, radiation, subgrid turbulent transport, and surface fluxes (Cotton et al. 2003). Certain parts of the physics modules were simplified for this particular study, as described below.

The primary experiment (E1) involves a well-developed hurricane. For this experiment, RAMS is configured with single-moment warm-rain microphysics (Walko et al. 1995) and a longwave radiation scheme that neglects the effects of condensate (Mahrer and Pielke 1977). The subgrid turbulence parameterization is anisotropic, with the vertical component based on a local Smagorinsky (1963) closure. The standard RAMS enhancement of the vertical mixing coefficient in regions of moist instability is removed to limit the impact of diffusion on vertical alignment (see section 4g). The horizontal mixing coefficient is effectively constant and barely large enough to prevent excessively strong grid-scale fluctuations in the convective core of the vortex. The ratio of momentum to scalar mixing coefficients is ⅓.

*τ*

_{ux},

*τ*

_{uy}), sensible heat (

*τ*

_{θ}), and moisture (

*τ*

_{q}):

**u**≡ (

*u*

_{x},

*u*

_{y}) is horizontal velocity,

*θ*is potential temperature, and

*q*is the water vapor mixing ratio. The variables

*θ*

_{s}and

*θ*and the saturation mixing ratio. The plus sign subscript indicates that the variable is evaluated at the first vertical grid point above sea level. The dimensionless surface-exchange coefficients are obtained from a capped modification of Deacon’s formula,

**u**

_{+}| given in meters per second. The sea surface temperature is held constant at a low value of 23°C, which prevents eyewall replacement cycles during the experiment.

The computational domain is a periodic *f* plane at 20°N. The value of the Coriolis parameter *f* is therefore 5 × 10^{−5} s^{−1}. The fields are evolved on three nested grids spanning 567, 1235, and 4500 km in the east–west and north–south directions. The corresponding horizontal grid increments are 1.67, 5, and 15 km. The vertical mesh is the same for all grids and is stretched with height *z* over 80 increments up to *z* = 31 km. The vertical grid spacing is 60 m near the ground, 140 m at *z* = 2 km, and 500 m for *z* > 10 km. Rayleigh damping is applied near the upper boundary to eliminate vertically propagating waves that would otherwise remain artificially trapped in the system. The damping rate increases linearly with *z*, from 0 at *z* = 23 km to 0.003 s^{−1} at the model top.

The reference state of the atmosphere used by the dynamical core of RAMS in E1 is similar to the Jordan mean sounding (JMS) for hurricane season in the West Indies (Jordan 1958). The actual domain-averaged sounding that develops in the process of creating the hurricane differs from the JMS as described in appendix B.

Experiments E2–E4 (introduced in section 1b) have some basic configuration similarities with E1. The grids are equivalent to those of E1, as are the settings for turbulent transport and Rayleigh damping. The atmospheric reference states are the same, but without moisture in E2 and E4. That said, the configuration differences between E2–E4 and E1 summarized in Table 1 are essential. In each case, the radiation scheme is switched off. All surface fluxes are eliminated in experiments E2 and E3, whereas only the surface momentum flux is retained (explicitly) in E4. Experiment E2 has no moisture and no artificial representation of diabatic cloud processes. The simplified methods for modeling cloud processes in E3 and E4 are explained below.

Brief description of the CM experiments. The initial vortices in E2–E4 are designed to resemble the initial hurricane in E1. The initial vortices in E1-c and E2-c are modeled after a slightly contracted state of the hurricane found late in E1.

*θ*equation of the form

*r*,

*φ*, and

*z*denote radius, azimuth, and height in a cylindrical coordinate system. The symbol ∂ with a subscript (or two subscripts, used later) concisely denotes the first (or second) partial derivative with respect to the variable indicated by the subscript. In general,

*u*,

*υ*, and

*w*denote radial, azimuthal, and vertical velocity fields. All fields having the subscript p0 correspond to the initial axisymmetric hurricane of the primary CM experiment (E1). The heat source given by the right-hand side (rhs) of Eq. (3) is that required to maintain

*θ*=

*θ*

_{p0}(

*r*,

*z*) with a secondary circulation given by

*u*=

*u*

_{p0}(

*r*,

*z*) and

*w*=

*w*

_{p0}(

*r*,

*z*). It is used here to drive an approximately steady mean secondary circulation in a dry simulation. Unlike E3, small perturbations in E4 do not experience a reduced buoyancy restoring force through phase transitions of cloud water.

### b. Initialization

Figure 1 depicts the initial axisymmetric state of the hurricane in E1, obtained from the azimuthally averaged fields of a mature system in a preliminary RAMS simulation described in appendix B. Figure 1a shows the azimuthal velocity *υ* and the perturbation of potential temperature *θ* from its *z*-dependent value at *r* ≈ 2200 km. The vortex is seen to exhibit classic warm-core structure. The absolute maximum of *υ* is 61.2 m s^{−1} at *r*_{max} = 90 km and *z*_{max} = 0.95 km. The large size of the storm is helpful for resolving small misalignments on the computational grid used for this study. Further discussion of the storm scale is deferred to section 2d. Figure 1b shows the overturning secondary circulation in the vortex core. The maximum of *w* in the eyewall updraft is 2.9 m s^{−1}. Figure 1c shows contours of saturation pseudoadiabatic entropy *q*. The state of the eyewall is reasonably close to slantwise convective neutrality, in which the contours of *q*_{r} superposed on a plot of *q*_{r} + *q*_{c}, in which *q*_{r} and *q*_{c} respectively represent the rain and cloud droplet mixing ratios.

The dry baroclinic vortex in E2 is initialized with no secondary circulation and *υ* approximately matching that of the initial hurricane of E1. The only notable difference in the primary circulation is that *υ* does not vary with height between the sea surface and *z*_{max}. To elaborate, the initial conditions for *υ*, *θ*, and the perturbation Exner function Π (the prognostic pressure variable in RAMS) correspond to the azimuthally averaged end state of a 24-h relaxation procedure. The relaxation procedure is to nudge the velocity field toward its intended state with a damping rate of 0.5 h^{−1} while letting *θ* and Π freely adjust. At the beginning of the procedure, *υ*, *θ*, and Π are matched to the initial conditions of E1, whereas *u* and *w* are set to their intended values of zero.

The cloudy vortex in E3 is initialized with no secondary circulation and *υ* approximately matching that of the initial vortex in E2. The precise initial conditions for *υ*, *θ*, and Π correspond to the azimuthally averaged end state of a 24-h relaxation procedure analogous to that of E2, but with unnudged moisture fields (*q* and *q*_{c}) included. Following the relaxation procedure, the cloud droplet mixing ratio *q*_{c} is mostly removed outside the eyewall and outflow regions defined by the E1 hurricane. In the remaining cloud, *q*_{c} is reset to approximately 5 g kg^{−1} and the edges are smoothed. The distribution of *q* obtained from the relaxation procedure is then adjusted where necessary to ensure saturation where *q*_{c} > 0 and subsaturation where *q*_{c} = 0. The ice-liquid potential temperature (the prognostic heat variable in moist RAMS simulations) is initialized in accordance with the distributions of *θ*, Π, and *q*_{c}. Section 3b and appendix C further discuss the cloud distribution in E3 and how it theoretically affects static stability in the vortex.

The initial conditions of E4 are obtained from a distinct relaxation procedure. The relaxation period starts with *u*, *υ*, *w*, *θ*, and Π matched to the initial conditions of E1. The system then evolves for 24 h under the thermal forcing of Eq. (3) with no additional nudging. The azimuthally averaged end state is used to initialize E4.

Figure 2a demonstrates that the mass-weighted *z*-averaged relative vorticity (*ζ*) distributions in E1–E4 are initially the same to within a reasonable approximation. However, the preliminary relaxation in E2 and E3 alters the initial distribution of *θ*. The modest change of *θ* combined with the removal of secondary circulation and vertical shear below *z*_{max} affects some details of the PV distribution in the vortex core. The PV in E4 also differs somewhat from that of E1 owing to a variety of modifications affecting the relaxed state, such as the removal of water mass.

*r*–

*θ*profiles of dry isentropic PV, defined by

*ζ*

_{θ}≡ [∂

_{r}(

*rυ*) − ∂

_{φ}

*u*]/

*r*→ ∂

_{r}(

*rυ*)/

*r*,

*σ*≡ −∂

_{θ}(

*p*/

*g*), and

*g*= 9.8 m s

^{−2}is the gravitational acceleration. The

*r*and

*φ*derivatives are here taken at constant

*θ*, and the

*φ*derivative vanishes owing to axisymmetry of the initial vortex. The lower bound of

*θ*on each plot corresponds to the maximum of

*θ*at

*z*= 30 m. The upper bound corresponds to the minimum of

*θ*at

*z*= 14 km. It seems doubtful that subtle PV differences in E1–E4 change tilt dynamics as much as the principal configuration differences summarized in Table 1, but they are worth keeping in mind.

### c. Forcing applied to create tilt

*τ*. The acceleration vector

*A*= 1 m s

^{−1},

*n*= 2,

*H*= 12.5 km, and

*τ*= 2

*π*/

*f*= 35.1 h. Setting

*H*to the approximate depth of the vortex ensures that the top half and bottom half of the vortex experience opposite horizontal forcing. By equating

*τ*to one inertial period, the environmental air mass theoretically returns to a state of rest once the forcing stops (in the absence of frictional dissipation). In other words, the residual ambient shear flow is minimized and the vortex evolves freely for

*t*>

*τ*. Low-frequency forcing relative to the angular velocity of the vortex should also prevent the excitation of substantial inertia–gravity waves in its core.

One positive aspect of the vortex perturbation procedure used here is that it facilitates study of both forced and freely evolving tilt in the same experiment. Although the applied forcing

### d. Sensitivity experiments

To help understand the generic and peculiar aspects of the primary CM experiments, E1 and E2 will be repeated with modified initial conditions. The modified experiments are labeled E1-c and E2-c. The initial condition of E1-c consists of the *φ*-averaged fields of E1, averaged over the time interval 69 ≤ *t* ≤ 71 h. Figure 2b depicts the radially contracted relative vorticity distribution during this time period. The vorticity is seen to have a deeper central deficit and a greater peak value than at the beginning of E1. The absolute maximum of *υ* is 71.2 m s^{−1} at *r*_{max} = 72 km and *z*_{max} = 0.95 km. The initial condition for E2-c is obtained as for E2, but with *υ* (above *z*_{max}) corresponding to the initial state of E1-c.

Note that both the original and contracted vortices have uncommonly large values of *r*_{max} and aspect ratios (*r*_{max}/*H*) that are somewhat exaggerated. Nevertheless, they are dynamically similar to real hurricanes in having Rossby numbers that satisfy Ro ≡ *υ*_{max}/(*r*_{max} *f* ) ≫ 1 and dry Froude numbers that satisfy Fr ≡ *πυ*_{max}/*NH* ~ 1. Here, *υ*_{max} is the maximum azimuthal wind speed and *N* ≈ 0.01 s^{−1} is the dry Brunt–Väisälä frequency.

## 3. Simple linear theory

The following presents specific theoretical results required to compare the SLT to the CM experiments. The results are preceded by a brief description of the perturbation equations on which the SLT is based.

### a. The linearized primitive equations

The theory at issue neglects vertical variation and secondary circulation in the basic state of the vortex and treats the tilt as a small hydrostatic perturbation (cf. R04; Schecter and Montgomery 2004, hereafter SM04). A Boussinesq approximation is used to facilitate calculations. The perturbation equations are formulated using the pressure-based pseudoheight of Hoskins and Bretherton (1972, hereafter HB72) as the vertical coordinate *z*. For the purpose of comparing theoretical results to the CM experiments, it is assumed that *z* and *w* ≡ *Dz*/*Dt* approximately equal the genuine height and vertical velocity (HB72). Here and elsewhere *D*/*Dt* denotes the material derivative.

The basic state of the vortex satisfies gradient-wind and hydrostatic balance. It is characterized by the azimuthal velocity

*π*/

*H*for consistency with the misalignment force in Eq. (5). The perturbations of

*u*and the geopotential

*ϕ*have the same form as

*υ*′. The pseudovertical velocity perturbation has the form

*θ*′. It follows that

*w*′ =

*θ*′ = 0 at

*z*= 0 and

*z*=

*H*.

*πz*/

*H*)] extracted from the rhs of Eq. (5).

^{1}The potential temperature equation may be written

*θ*

_{ref}is the reference value of

*θ*(say 300 K). The dry Brunt–Väisälä frequency

*N*is set equal to 0.01 s

^{−1}in close agreement with the CM experiments. The function ϒ

_{b}(

*r*) is assumed to have values between 0 and 1 and theoretically accounts for the reduction of static stability (

*N*

^{2}) in cloudy air (cf. SM07; Durran and Klemp 1982). Hydrostatic balance and mass continuity take the forms

*r*

_{b}= 8000 km) are computed with backward differencing. Linear damping is applied after the forcing period in a sponge ring extending approximately 750 km inward from

*r*

_{b}.

*ζ*′ ≡ [∂

_{r}(

*rυ*′) − ∂

_{φ}

*u*′]/

*r*and divergence

*χ*′ ≡ [∂

_{r}(

*ru*′) + ∂

_{φ}

*υ*′]/

*r*. The corresponding

*r*–

*t*wave functions are denoted

Here, the asterisk denotes the complex conjugate, *Q*′ and _{b} = 1) in SM04.

### b. Tangential wind and N^{2}-reduction profiles relevant to the CM experiments

Table 2 summarizes the specific versions of the preceding linear model used to help predict and explain the outcomes of the CM experiments. All members of the subset {L1-1, L1-2, L2, L3-1, L3-2} have the same tangential wind profile. The azimuthal velocity

Versions of the linear model relevant to the CM experiments. Each version is defined by its basic-state vorticity _{b}, which depends on the dimensionless parameter *γ*.

Different versions of the linear model sharing the same wind profile are distinguished by their moisture parameterizations. For L2 and L2-c, the vortex is dry and ϒ_{b} = 1.

For L1-1 and L1-2, ϒ_{b} is based on the distribution of cloud water in E1. First, Eq. (32) of SM07 is used to estimate a two-dimensional *N*^{2}-reduction factor ϒ(*r*, *z*) (Fig. 4a). The time average appearing in the SM07 formula for ϒ is taken over the forcing period 0 ≤ *t* ≤ *τ*. The lowest values of ϒ tend to occur in regions of substantial cloud coverage. Second, ϒ is vertically averaged between 30 m and 12.3 km above sea level. The result is here called the raw estimate of ϒ_{b}. For L1-1, ϒ_{b} is equated to an analytic approximation of the raw estimate denoted *γ* = 1 [cf. Eq. (D3)].

The justification for deriving ϒ_{b} from ϒ is admittedly questionable, because the SM07 theory formally applies to very small perturbations in nonprecipitating vortices. The simplification from a moist-baroclinic vortex to a moist-barotropic vortex raises additional concerns. Approximate slantwise convective neutrality could necessitate lowering ϒ_{b} to a magnitude much less than its raw estimate in the eyewall region of the vortex. Such further reduction of *N*^{2} is in L1-2, where *γ* = 2.2. Figure 4b shows the raw estimate of ϒ_{b} and the two variants of

For L1-c1 and L1-c2, ϒ_{b} is modeled after the cloud coverage found in E1-c. Figures 4c and 4d show ϒ and the raw estimate for ϒ_{b} during the forcing period. Figure 4d also shows the two variants of the analytic *N*^{2}-reduction factor

For L3-1 and L3-2, ϒ_{b} is modeled after the cloud coverage in E3. Here, the removal of secondary circulation from the basic state lets mid- to upper-tropospheric cloud water expand its domain during the forcing period (see appendix C). The result is a broadening of the *N*^{2}-reduction factor. Figures 4e and 4f show ϒ and the raw estimate for ϒ_{b} during the forcing period. Figure 4f also shows the two variants of the analytic *N*^{2}-reduction factor

### c. Two misalignment modes of special interest

In all cases considered, the linear perturbations generated by misalignment forcing are largely controlled by two discrete modes: the PTM and the inner wobble mode (IWM). The natural behavior of each mode is seen when *ω* ≡ *ω*_{R} + *iω*_{I} can be found by a variety of methods. One method is to solve for the eigenfrequencies of an ordinary differential equation (in *r*) for the wave function, with radiative outer-boundary conditions (cf. SM04). If the PTM is a damped quasimode, its wave equation must be solved along a complex radial contour. An alternative method is to initialize the unforced linear system with a quasi-balanced perturbation, in which *ω* from the time series of (say)

Figure 5 shows the radial and azimuthal velocity wave functions of the PTM and the IWM of the dry vortex in L2. The structural contrast seen here is typical. The IWM is essentially confined to *r* less than the radius of maximum wind (RMW), whereas the PTM extends to the periphery of the vortex. Figure 6 shows the complex frequencies of the PTMs and IWMs for all versions of the linear model listed in Table 2. The IWM oscillation frequencies are invariably close to the maximum of

In contrast to many earlier studies, the PTMs of the primary vortex (in L1-1, L1-2, L2, L3-1, and L3-2) have positive growth rates. When the vortex is completely dry, the growth rate of the PTM exceeds that of the IWM. The appreciable positive growth rate of the dry PTM coincides with a positive value of *r*_{*} = 312 km. Note that for wavenumber-1 perturbations, the critical radius is obtained from the relation _{b}*N*^{2} decreases in the eyewall region of the primary vortex, *ω*_{R} slightly increases and *r*_{*} shifts inward toward a region where

The PTMs of the contracted vortex (in L1-c1, L1-c2, and L2-c) exhibit more familiar behavior. The dry PTM of L2-c is a strongly damped quasimode with negative *r*_{*} = 194 km. Decreasing ϒ_{b}*N*^{2} increases *ω*_{R} and reduces *r*_{*} to where the negative magnitude of

The growth rates of the IWMs are clearly less sensitive to the variation of ϒ_{b}*N*^{2} considered here. A more general study of IWM growth rate variability that considers greater reduction of *N*^{2} in the eye, where the bulk of the IWM resides, is deferred to a later time.

^{2}Fig. 7a shows snapshots of

*x*) = ±1 for

Figures 7d and 7e show that sufficient reduction of *N*^{2} suppresses the inner part of the PTM, makes the core maximum of

### d. Response to forcing

Figure 8a depicts the evolution of *t* ≤ *τ*. After forcing, the low-frequency oscillations of the PTM dominate and continue to grow owing to intrinsic instability. It is reasonable to assume that the growth of the PTM would slow down considerably with time in a nonlinear model that accounts for mode–mode interactions and the leveling of *φ*-averaged PV in the vicinity of *r*_{*}.

Figures 8b and 8c illustrate how reduction of *N*^{2} in moist versions of the same vortex inhibits the excitation of the PTM. Immediately after the forcing period, the maximum amplitudes of *υ*′ in the moist vortices of L1-2 and L3-2 are less than half the maximum found in the dry vortex. At the same instant, the average amplitudes of *υ*′ between the RMW and twice the RMW are less than 0.4 times their counterpart in the dry vortex.^{3} Lesser excitability of the moist PTM seems attributable to the growth-rate reduction and structural change of the mode attending the depression of *N*^{2} in the vicinity of the eyewall. Note that despite having a smaller amplitude, the moist PTM persistently dominates *υ*′ outside the RMW for days after the forcing stops. In contrast, high-frequency IWM oscillations are not discernible until very late in the *υ*′ curve corresponding to *r* = 0.2 km. The preceding result indicates weak coupling between the IWM and the applied forcing.

Figures 9a and 9b illustrate how the linear response to misalignment forcing can change with the basic state of the vortex. Specifically, these figures depict the evolution of *υ*′ in the contracted vortices of L2-c and L1-c1. To begin with, the forcing does not appear to excite the dry PTM of L2-c as strongly as the dry PTM of L2. Once the forcing stops, the maximum amplitude of *υ*′ in L2-c is just 0.4 times the corresponding maximum in L2. It seems reasonable to assume that the PTM of L2-c is less responsive partly because of its rapid intrinsic damping. Such damping is evident in the quick decay of the post-forcing oscillations in *υ*′ that occur with the natural PTM period outside the RMW. With moisture added to the system in L1-c1, the freed PTM existing for *t* > *τ* is practically negligible. With or without moisture, fast IWM oscillations eventually dominate *υ*′ over the entire vortex. In a more realistic model, nonlinear saturation could very well prevent the IWM from dominating the perturbation outside the RMW.

## 4. Response to misalignment forcing in the cloud model

The following evaluates the predictions and assumptions of the SLT by comparison to the CM experiments. The evaluation begins by demonstrating the qualitatively correct prediction that moisture inhibits misalignment. It then proceeds to a more detailed and quantitative assessment of the predicted dry and moist tilt dynamics.

### a. Development of misalignment and vertical antisymmetry in the main CM experiments

**x**

_{c}≡ (

*x*

_{c},

*y*

_{c}) is the horizontal position vector of the rotational center at height

*z*, and

**x**

_{ca}is the

*z*average of

**x**

_{c}between

*z*

_{s}= 30 m and

*z*

_{t}= 10.7 km. The values of the Cartesian coordinates

*x*

_{c}and

*y*

_{c}specifically correspond to the center of the polar coordinate system that maximizes the peak value of the

*φ*-averaged azimuthal velocity

*z*. The height-dependent radius of maximum

*r*

_{max}and appears in the denominator of

*M*. In words,

*M*is the root-mean-square displacement of rotational centers from their vertical mean, normalized to the vertical mean of

*r*

_{max}.

Figure 10a shows *M* versus time for the four main CM experiments E1–E4. Consistent with the SLT, the moist convective vortex in E1 is substantially less responsive to the applied misalignment forcing than the dry nonconvective vortex in E2. The forcing also has trouble perturbing the nonprecipitating cloudy vortex in E3, suggesting that reduction of static stability may be sufficient to prevent substantial tilt. Whether this mechanism strongly inhibits tilt in E1 is uncertain at this point, partly because differences emerge in the *N*^{2}-reduction profiles of E1 and E3 during the forcing period (Fig. 4). Furthermore, E4 shows that *M* is considerably reduced in a dry vortex with a thermally driven SSC. It stands to reason that the presence of the SSC could help limit tilt in the E1 hurricane.

The supplemental notes in appendix C address the basic premises used to infer from the results of E3 and E4 that both static stability reduction by cloud coverage and the SSC may introduce viable mechanisms for inhibiting tilt. Subsection a presents evidence that the cloudy vortex in E3 does not develop an appreciable SSC and that its wavenumber-1 thermodynamics is consistent with substantial diabatic reduction of static stability. Subsection b verifies that the SSC in E4 is comparable to that in E1 and that the wavenumber-1 thermodynamics in E4 is quasi adiabatic.

Note that small departures from equilibrium at *t* = 0 and weak instabilities cause *M* to grow somewhat without applied forcing. The discrete symbols in Fig. 10a show the unforced growth of *M* during the time interval 0 ≤ *t* ≤ *τ*. The data were obtained by letting *A* → 0 in Eq. (5) and repeating experiments E1–E4. The unforced version of E1 is denoted E1 **×** 0 and likewise for E2–E4. It is verified that the initial forced growth of *M* substantially exceeds that found in each unforced experiment.

*υ*

_{1}(

*r*,

*z*,

*t*) is the complex wavenumber-1 Fourier component of

*υ*, defined in the cylindrical coordinate system centered at

**x**

_{ca}(

*t*). The weight function

*G*(

*z*) ≡ (2

*z*−

*z*

_{t}−

*z*

_{b})/(

*z*

_{t}−

*z*

_{b}) is antisymmetric about the mean height (

*z*

_{m}) in the integral over

*z*and has values of ±1 at the two end points. The notation 〈

*h*〉

_{z}represents

*h*〉

_{r}represents

*r*

_{i}= 65 km,

*r*

_{o}= 165 km,

*z*

_{b}= 2.1 km, and

*z*

_{t}= 10.7 km. The integration volume therefore covers the bulk of the eyewall updraft in E1.

Figure 10b shows that VAP behaves much like *M* in the CM experiments under consideration. However, the two quantities are distinct. For example, VAP includes contributions from antisymmetric components of *υ*_{1} that are not directly attributable to misaligned centers of rotation. One such component is the weak ambient vertical wind shear superposed on the vortex flow during the forcing period. VAP also gives less weight to misalignments that have small vertical wavelengths and no weight to misalignments in which *υ*_{1} is symmetric about *z*_{m}. Note that because VAP has no contribution from the vertically invariant component of *υ*_{1}, it does not depend on the velocity of the reference frame.

### b. Detailed response of the dry nonconvective vortex in E2

The SLT of section 3 showed that a PTM dominates the tilt generated by slow misalignment forcing in the dry barotropic analog of the simulated hurricane. It is here verified that the same is true in the dry baroclinic analog.

Figure 11 displays snapshots of the *z*-dependent rotational centers {(*x*_{c}, *y*_{c})} of the baroclinic vortex in E2. All snapshots are taken after the forcing period in a coordinate system centered at **x**_{ca}(*t*). To within a fair approximation, the displacement of (*x*_{c}, *y*_{c}) from the lowest center of rotation increases unidirectionally with height, indicating a clean tilt. The tilt precesses with an angular frequency of *ω*_{R} = 6.8 × 10^{−5} s^{−1}. The corresponding 26-h rotation period is merely 13% less than that of the dry PTM of the barotropic vortex of L2.

Figures 12a–c illustrate the basic structural similarity of the tilt mode in E2 with the corresponding PTM of the SLT (Figs. 7a–c). The plotted fields are defined using an isentropic cylindrical coordinate system centered at **x**_{ca}. The definitions involve the wavenumber-1 and wavenumber-0 components of the following Fourier expansion: _{T} over the free evolution period *τ* ≤ *t* ≤ *τ* + *T*, in which *T* ≈ 2 days.

*ζ*

_{θ,1}|〉

_{T}and divergence amplitude 〈|

*χ*

_{θ,1}|〉

_{T}. As usual,

*ζ*

_{θ}≡ [∂

_{r}(

*rυ*) − ∂

_{φ}

*u*]/

*r*and

*χ*

_{θ}≡ [∂

_{r}(

*ru*) + ∂

_{φ}

*υ*]/

*r*. Note that the partial derivatives appearing in the definitions of

*ζ*

_{θ}, and

*χ*

_{θ}are evaluated at constant

*θ*. Figure 12c essentially shows superposed contour plots of the two components of the isentropic angular pseudomomentum density

*l*= 1 disturbance (cf. C03; Schecter 2008, hereafter S08). More precisely, the plotted quantities are

*σ*

_{0}〉

_{T}and

_{θ,0}〉

_{T}smoothed with 10-km radial boxcar averaging. The isentropic density

*σ*and potential vorticity PV

_{θ}were defined in section 2b. For dry baroclinic vortices, comparing

*χ*

_{θ,1}| ≪ |

*ζ*

_{θ,1}|, and

Figures 12d and 12e show *r*–*t* Hovmöller diagrams of *θ* = 298.7 and 331.4 K (the thick *θ* contours in Fig. 12a). The primary 26-h oscillations are approximately 180° out of phase in the upper and lower troposphere and are attributable to the slow PTM. Faster secondary oscillations are found in the lower-tropospheric portion of the inner core. Their frequency seems to exceed that of the deep misalignment IWM considered in linear response theory. Such minor fluctuations are noticeable partly because the PTM fails to grow to the extent seen in the SLT. To accurately predict the saturation amplitude of the PTM would require a nonlinear theory that incorporates the evolution of basic-state PV, especially in the neighborhood of the critical radius (cf. Balmforth et al. 2001, hereafter B01; Schecter and Montgomery 2006, hereafter SM06; S08). A nonlinear theory would also help predict changes in the radial waveform of the PTM tied to changes in the radial gradient of basic-state PV that seem to have taken place in the vortex core mostly during the forcing period (Fig. 12f).

### c. Detailed response of the moist convective vortex in E1

Figure 13 compares the moist tilt dynamics of E1 to that predicted by the SLT. The SLT is specified as in L1-2, where the PTM is effectively neutral. The alternative L1-1 specification is ruled out of consideration, because there is no clear evidence in E1 of its relatively fast growing PTM.

*z*

_{2}= 9 km. The value of

*z*

_{1}is somewhat arbitrary given the approximate invariance of

**x**

_{c}in the lowest 2 km of E1, but is here set to 30 m. An orientation angle of zero corresponds to an eastward tilt. As explained in section 4a,

**x**

_{c}is determined by the velocity field in the vicinity of

*r*

_{max}, which is on the outer edge of the eyewall updraft in E1. In this sense, the present definition of tilt marginally pertains to misalignment of the outer core.

^{4}Note that the velocity field used to obtain

**x**

_{c}and tilt in the SLT is given by the initial basic-state plus the wavenumber-1 perturbation; second-order changes to the flow are unaccounted for.

^{5}Barring greater fluctuations and somewhat greater decay after the forcing period, the tilt vector of E1 evolves much like its counterpart in the SLT.

*w*

^{md}. By definition,

*z*

_{c}= 5.0 km,

*z*

_{d}= 7.1 km, and the radial limits of integration are adjustable. The polar coordinate system in E1 is centered at

**x**

_{ca}, which meanders and ends up roughly 70 km from where it began. The azimuthal Fourier expansion of

*w*

^{md}is written

Figures 13c and 13d show time series of the crest amplitude and azimuth of *r*_{c} = 120 km and *r*_{d} = 200 km. Here there is fairly good agreement between E1 and the SLT.^{6} Note that in both cases, the crest azimuth of

Figures 13e and 13f show time series of the crest amplitude and azimuth of *r*_{c} = 70 km and *r*_{d} = 150 km. Figure 13f also shows the wavenumber-1 crest azimuths of the midlevel heating rate (*Dθ*/*Dt*) and column-integrated rain-mass density in the same radial segment of the E1 hurricane. It is seen that the crest azimuths of the vertical velocity, heating rate, and rain-mass waves approximately coincide. Such a result is agreeable with the moisture parameterization of the SLT, but there is also cause for concern. Specifically,

In summary, the SLT appears to predict tilt and midlevel convection in the outer core better than perturbations to convection in the eyewall. There is no obvious reason why inner-core discrepancies should not corrupt outer-core dynamics in E1. However, quasi-independent outer-core dynamics seems plausible if the bulk of the PTM is outside the eyewall updraft. This scenario is somewhat suggested by the SLT (Figs. 7d and 7e).

### d. VAP^{2} growth rate analysis

Section 4c suggested that the SLT cannot fully explain how the eyewall region of the hurricane responds to weak misalignment forcing. The eyewall region contributes substantially to the wavenumber-1 vertical antisymmetry parameter [VAP; Eq. (12)]. Evaluation of the factors controlling the growth rate of VAP^{2} may therefore help one identify significant elements of eyewall dynamics neglected by the SLT. The following analysis makes use of the Fourier expansion

^{2}is obtained directly from the

*l*= 1 component of the azimuthal velocity equation and is conveniently written

*i*(

*e*) denotes a source that is intrinsic (extrinsic) to the moist primitive equation dynamics of the system. The first intrinsic source on the rhs of Eq. (17) has a direct analog in the SLT. It is given by

*S*

_{amg}is connected to the angular momentum gradient of the symmetric flow. Specifically,

*S*

_{amg}derives from the following term in the

*υ*

_{1}-tendency equation:

*S*

_{pg}is associated with the pressure gradient force. As in many CMs, the explicit pressure gradient term

*θ*

_{υ}∂

_{φ}Π/

*r*in the

*υ*equation is here approximated by

*θ*

_{υr}∂

_{φ}Π/

*r*, in which

*θ*

_{υ}and

*θ*

_{υr}are the actual and reference-state virtual potential temperature distributions. The operator

*S*

_{trb}is defined below. The term

*S*

_{ssc}is connected to the symmetric secondary circulation. The term

*S*

_{svs}is connected to the symmetric vertical wind shear, since it would be zero if

*υ*

_{0}were independent of

*z*. The term

*S*

_{ww}accounts for nonlinear, asymmetric wave–wave interactions. The term

*S*

_{trb}is associated with subgrid turbulent transport; it is expressed here by

*D*

_{υ,1}is the wavenumber-1 component of the tendency term connected to “eddy viscosity” in the azimuthal velocity equation. The extrinsic source in Eq. (17) is simply

The instantaneous growth rate of VAP^{2} may be written *d*(VAP^{2})/*dt* by **x**_{ca}. The initial VAP^{2} growth rates are directly controlled by the applied misalignment forcing ^{2}. The magnitudes of ^{−4} s^{−1}, and ^{2} through

The VAP^{2} growth rate analysis for E1 suggests that an accurate theory for how the inner core of a mature hurricane responds to weak misalignment forcing may need to incorporate the SSC. Further inferences would be more speculative. The analysis does not overtly reveal the primary mechanism by which the SSC influences VAP^{2}, let alone direct measures of misalignment. Like reduced static stability, the SSC could independently alter the structure of the PTM and thereby affect source terms other than *l* = 0 and *l* = 1 modes commingle to a degree when the coordinate center is varied. A shift of **x**_{ca} at an arbitrary time *t* could alter source terms such as *t* = 0) location.

### e. Response of the contracted vortex

The SLT of section 3 predicted that the contracted vortices of E1-c and E2-c would have faster and less excitable PTMs than their counterparts in E1 and E2. This prediction seems qualitatively consistent with the CM experiments.

Figure 15 displays time series of the tilt vector components after the forcing periods in E1, E2, E1-c, and E2-c. Thick curves show data smoothed with 5-h boxcar averaging to highlight oscillations in the PTM frequency range. PTM signals are clearly evident in the tilt vectors of the dry vortices of E2 and E2-c. The 13.7-h oscillation period of the PTM in E2-c is 19% greater than predicted by the corresponding SLT but is still substantially less than its counterpart in E2. Nonlinear processes in E2-c prevent the continual damping of the PTM found in the SLT, but the prediction of reduced excitability relative to E2 holds.^{7} Unlike the tilt vector of E2, the tilt vector of E2-c has a prominent secondary oscillation whose 3-h period equals that of an IWM. The appearance of this signal seems agreeable with the SLT (Fig. 9a).

Although the tilt vector of the moist convective hurricane in E1 exhibits weak oscillations at the expected PTM frequency, the contracted hurricane of E1-c shows no discernible sign of a PTM. The latter result is consistent in principle with very strong PTM damping found in the SLT of the contracted vortex with moisture parameterized as in L1-c1 or L1-c2.

### f. Comment on eyewall convection in E1 and E1-c

It is worth remarking that with *A* = 1 m s^{−1} chosen for the forcing function, the precipitation rings defining the eyewalls of the hurricanes in E1 and E1-c do not severely break down. Figure 16a shows how the ring of column-integrated rain mass in E1 is maintained throughout the simulation. Ring maintenance suggests that the wavenumber-1 component of *w* is insufficiently strong in the eyewall to create a broad region of unsaturated downdraft. For comparison, Fig. 16b shows substantial desymmetrization of the ring in a similar experiment (E1 × 4) with *A* = 4 m s^{−1}. On the one hand, the SLT might be more appropriate for cases in which perturbation *w* is relatively strong compared to the SSC. On the other hand, the fundamental assumption of linear dynamics (made in the SLT) seems more problematic for cases in which the perturbation is large and cloudy only on its updraft side (cf. Patra 2004). A detailed comparison of the SLT to the behavior of large amplitude tilts in mature hurricanes is deferred to a future time.

### g. Comment on subgrid turbulent transport in the CM experiments

As noted earlier, the mixing coefficients in the CM experiments were adjusted to minimize the impact of turbulent transport on tilt dynamics without creating excessive noise. On the fine grid, the horizontal mixing coefficient for momentum *K*_{h} had an approximately constant value of 593 m^{2} s^{−1} in all simulations. The vertical mixing coefficient for momentum *K*_{υ} was determined by a Smagorinsky-type closure and varied in each numerical experiment. Figure 17 shows the azimuthally averaged values of *K*_{υ} output by the CM during the forcing periods of E1–E4. It is seen that *K*_{υ} is of order unity or less in the middle troposphere.

The time scales for turbulent transport in the horizontal and vertical directions are reasonably estimated by *λ*_{h} and *λ*_{υ} are the horizontal and vertical length scales of the structure of interest. For *λ*_{h} = 10 and 100 km, *τ*_{h} = 47 and 4.7 × 10^{3} h. For *λ*_{υ} = 4 km and *K*_{υ} = 1–10 m^{2} s^{−1}, *τ*_{υ} = 4.4 × 10^{3}−4.4 × 10^{2} h. Using the scalar mixing coefficients would reduce each of the previous time-scale estimates by a factor of 3. The short estimate of *τ*_{h} for *λ*_{h} = 10 km suggests that parameterized turbulence may have caused modest radial smoothing of basic-state PV over the course of each simulation. However, the large estimates of *τ*_{υ} and of *τ*_{h} with *λ*_{h} = 100 km provide some reassurance that parameterized turbulence had little direct influence on the simulated vortex-scale tilts.

## 5. Concluding remarks

This paper compared the tilt dynamics of a simulated hurricane to the predictions of a simple linear theory (SLT) that neglects the symmetric secondary circulation (SSC) and treats moisture merely as a local reduction of static stability (*N*^{2}). The primary hurricane simulation (E1) was carried out with a traditionally configured CM. Additional simulations were conducted with reduced physics and/or modified vortex structure to help identify features that enable the hurricane to resist tilting when exposed to misalignment forcing. The main results are summarized and discussed below.

The simplest CM experiment (E2) consisted of a dry nonconvective vortex closely resembling the primary hurricane. The vortex was subjected to a period of idealized misalignment forcing and then left to freely evolve with time. As predicted by the SLT, the forcing primarily excited a slowly precessing tilt mode (the PTM) with VR wave characteristics.

Also as predicted, the same misalignment forcing generated much weaker tilt in the primary hurricane experiment. According to the SLT, lesser tilt was caused by moisture-induced reduction of *N*^{2} in the vicinity of the eyewall. Such reduction of *N*^{2} theoretically limited the excitability of the PTM by neutralizing its growth rate and altering its structure, while just slightly changing its natural precession frequency. An additional CM experiment (E3) with suspended cloud water but seemingly negligible SSC supported the SLT result that reduction of *N*^{2} is sufficient to inhibit the excitation of a PTM.

In a more detailed comparison to theory, the tilt vector [Eq. (15)] in the primary hurricane experiment was found to vary with time much like its counterpart in the SLT. However, the theoretical phase relation between the tilt angle and the crest azimuth of the attendant midlevel vertical velocity wave seemed to hold only in the outer core of the simulated hurricane. The inner-core discrepancy suggested some deficiencies in how the SLT parameterizes perturbations to diabatic convection in the vicinity of the eyewall. It is not entirely clear how to reconcile these deficiencies with the successful prediction of tilt evolution. On the other hand, neglected eyewall processes are conceivably incidental if the bulk of the PTM resides in the outer core (cf. Figs. 7d and 7e).

The qualified success of the SLT was further challenged by a final CM experiment (E4) that excluded moisture but kept the SSC through an artificially distributed heat source. The reduced misalignment found in E4 supported the intuitive notion that the SSC may independently inhibit tilt. Fully understanding the mechanism will require further investigation. One might speculate that the presence of the SSC changes the complex frequency and structure of the PTM in such a way that renders it less excitable. This hypothetical mechanism for limiting the growth of outer tilt would be analogous to that which occurs by reducing *N*^{2} in the SLT. A more straightforward effect of the SSC on tilt dynamics was examined in the eyewall region of the E1 hurricane. Convective transport by the SSC seemed to oppose the growth of wavenumber-1 vertical antisymmetry (VAP) in the eyewall during the early stage of forcing, but its negative influence did not persist.

Note that the structure of the primary hurricane considered in this study was well suited to illustrate the potential importance of the PTM in governing tilt and the potential importance of moist convection in limiting tilt. As predicted by the SLT, a slightly stronger and contracted vortex was found to have a less dominant PTM that effectively resisted excitation even without moisture.

In brief summary, the SLT offers partially valid insight on how tilt develops in hurricanes exposed to misalignment forcing. A more advanced theory that properly incorporates the SSC (and the boundary layer) seems needed to clarify some unresolved issues on how moist convection in the eyewall affects tilt dynamics.

## Acknowledgments

The author thanks Dr. Paul Reasor for several discussions that helped motivate parts of this paper. The author also thanks Dr. Chun-Chieh Wu and two anonymous reviewers for their constructive comments on the original manuscript that led to several improvements. This work was supported by NSF Grants AGS-1101713 and AGS-1250533. Most of the numerical simulations were performed on NCAR/CISL supercomputers through project UNWR0001. Some preliminary numerical simulations were also carried out with SDSC resources through XSEDE project TG-ATM130028.

## APPENDIX A

### Undamped and Growing PTMs

In linear theory, the damping rate of a PTM is basically proportional to the negative radial gradient of PV in the critical layer. If the amplitude of the PTM exceeds a modest threshold, its stirring of the critical layer will flatten the local PV distribution before significant damping occurs (cf. Briggs et al. 1970; Schecter et al. 2000; B01; SM06). Moreover, a positive radial PV gradient in the critical layer would cause the PTM to initially grow (S02; SM03; SM04). Transient growth may also occur through interaction of the PTM with a preexisting disturbance (cf. Antkowiak and Brancher 2004; Nolan and Farrell 1999; Lansky et al. 1997). Finally, PTMs in extremely intense vortices with negligible skirts of outer PV can amplify with an *e*-folding time of 5–10 core rotation periods by emitting spiral inertia–gravity waves with negative angular pseudomomentum (SM03; SM04; S08; cf. Hodyss and Nolan 2008; Billant and LeDizès 2009).

## APPENDIX B

### Additional Information on the Initial Setup of E1

The preliminary simulation used to obtain the initial conditions of E1 had several stages. The first stage lasted approximately 9 days with the sea surface temperature (SST) set equal to 25°C. During this time, the TC developed an outer eyewall that caused the demise of its inner predecessor. The model was then reinitialized with a symmetrized version of the reconfigured and relatively large convective vortex. The SST was lowered to 23°C and the system was allowed to relax for approximately 2 more days. Another 2 days of adjustment with weaker diffusivity (achieved by the subgrid turbulence modification described in section 2a) produced the initial condition of E1. The lower SST was used in E1 as a precaution against another eyewall replacement event.

Figure B1 compares the initial vertical distributions of actual and saturated pseudoequivalent potential temperature in E1 to those of the JMS. Both variables are approximated as in Bryan (2008) and horizontally averaged over the entire computational domain. By the end of the preliminary simulation, which did not conserve moist air mass, the domain-averaged surface pressure was unnaturally high (*p*_{s} = 1065 hPa) and a minor temperature inversion had developed just above *z* = 2 km. While such imperfections reflected in Fig. B1 may be inadequate for modeling a real hurricane, they are not critically problematic for the purpose of this idealized study.

## APPENDIX C

### Supplemental Notes on the Reduced Physics Experiments E3 and E4

#### a. The nonprecipitating cloudy vortex experiment E3

Figure C1a illustrates the evolution of *q*_{c} in the nonprecipitating cloudy vortex experiment E3. The distribution of cloud water initially resembles that of the primary hurricane in E1 but broadens over time in the middle and upper troposphere. Such broadening accounts for the distinct ϒ distribution in Fig. 4e. Note that the low cloud band in the inner core is attributable to an aesthetic shortcoming of the initialization algorithm for E3. It is not thought to have a significant consequence on tilt dynamics.

Supporting evidence that cloud coverage in E3 acts to substantially reduce the effective static stability relevant to tilt dynamics is provided below. The analysis is carried out in a reference frame centered at **x**_{ca}. The notation *h*^{md}(*φ*, *t*) is used to represent a generic midlevel field variable analogous to *w*^{md} [Eq. (16)] with *z*_{c} = 5.0 km, *z*_{d} = 7.1 km, *r*_{c} = 70 km, and *r*_{d} = 150 km. As usual, one may write

*θ*from standard model output fields. It is found that

_{*}therefore somewhat resembles the static stability reduction factor ϒ

_{b}in Eq. (7) of section 3. The resemblance improves to a degree as the SSC becomes negligible and the lowest-order approximation of

*w*∂

_{z}

*θ*becomes

*w*∂

_{z}

*θ*

_{0}, in which

*θ*

_{0}is the azimuthal mean of

*θ*. The value of ϒ

_{*}clearly decreases from unity as

_{*}≪ 1 is taken to suggest that condensation and evaporation are substantially reducing the cooling and warming that would otherwise occur adiabatically in the updrafts and downdrafts of the wavenumber-1 disturbance (cf. Fig. C1b). For other vortices in which

_{*}less than unity might simply indicate an acceleration (deceleration) of condensational heating in the positive (negative) regions of the wavenumber-1 vertical velocity perturbation.

Figure C1c shows the time series of ϒ_{*} obtained from hourly sampling of E3. The mean value of the time series is 0.11 and the standard deviation is 0.08. It has been verified that the mean of ϒ_{*} computed with hourly output is within 5% of the mean computed with 5-min output over the intervals 1 ≤ *t* ≤ 12 h and 24 ≤ *t* ≤ 30 h. High frequency (5 min) output was not archived over any other intervals. Note that redefining ϒ_{*} with the substitution

Needless to say, the SSC of the nonprecipitating cloudy vortex in E3 is precisely zero only at the beginning of the experiment. The SSC is defined by the wind vector (*u*_{0}, *w*_{0}), in which *u*_{0} and *w*_{0} are the azimuthally averaged radial and vertical velocity fields. The following statistics of the SSC are average values obtained from hourly snapshots taken for *t* ≤ 35 h. In the cylindrical shell of the E3 vortex defined by 65 ≤ *r* ≤ 165 km and 2.1 ≤ *z* ≤ 10.7 km, the root-mean-square (rms) radial and vertical velocities are respectively ^{−1} and ^{−1}. The corresponding rms velocities in the E1 hurricane are ^{−1} and ^{−1}. Furthermore, the shell averages of *u*_{0} and *w*_{0} in E3 are merely 0.006 and 0.01 times their counterparts in E1. The minuscule means in E3 are due to lesser wind speeds (evident in the rms measurements) and greater cancellations between positive and negative velocities.

In summary, the condition ϒ_{*} ≪ 1 in E3 seems to suggest that a substantial reduction of static stability is in effect. On the other hand, the SSC in E3 appears to be at least an order of magnitude weaker than its E1 counterpart.

#### b. The dry thermally forced vortex experiment E4

In a fixed cylindrical coordinate system whose central axis is coaligned with that of the axisymmetric thermal forcing in E4, _{*} would be unity barring subgrid turbulent transport. In practice, the value of ϒ_{*} obtained from 5-min output in a reference frame centered at **x**_{ca} is approximately 0.98 ± 0.02 during the intervals 1 ≤ *t* ≤ 12 h and 24 ≤ *t* ≤ 30 h. The condition ϒ_{*} ≈ 1 distinguishes E4 from E3 and from E1 where during the forcing period ϒ_{*} = 0.14 ± 0.08.

On the other hand, the SSC of E4 is verifiably similar to that of E1. Take the same cylindrical shell used in subsection a of this appendix to define the rms and mean values of *u*_{0} and *w*_{0}. It is found that ^{−1} in E4, compared to 5.4 and 0.7 m s^{−1} in E1. The means of *u*_{0} and *w*_{0} are 2.4 and 0.36 m s^{−1} in E4, compared to 2.6 and 0.34 m s^{−1} in E1. The preceding statistics are again averages from hourly snapshots taken for *t* ≤ 35 h.

As a final remark, the VAP^{2} growth rate budget of E4 is fairly similar to that of E1 (Fig. 14a). In particular,

## APPENDIX D

### Vorticity Profiles and Moisture Parameterizations Used in the SLT

*ζ*used in the SLT to represent the initial vortex in E1–E4 is

*r*are in units of per second and meters, respectively. The analytical functions used to represent the

*N*

^{2}-reduction factors in E1, E3, and E1-c are

*γ*are given in Table 2.

## APPENDIX E

### Computation of Complex Quasimode Frequencies

Damped PTMs are not genuine normal modes of the linear system [Eqs. (6)–(8) without forcing], because the damping mechanism requires aberrant growth of perturbation PV in the neighborhood of the critical radius (S02; SM03; SM04). One practical method for finding the complex frequency of a damped PTM (also known as a quasimode) is to examine the evolution of a quasi-balanced tilt in the absence of forcing. The procedure starts by setting the wavenumber-1 vertical vorticity perturbation *ζ*′ proportional to *ζ*′, whereas the divergent component and *w*′ are set to zero. The geopotential perturbation is initialized such that it zeroes the time derivative of horizontal flow divergence.

Figure E1 illustrates the free evolution of quasi-balanced tilts in vortices with *ω*_{R} and *ω*_{I} of the PTM.

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^{1}

In other words, *ε* = 2*π*cos(*πz*/*H*).

^{2}

These plots do not show the attendant perturbation in the skirt of the vortex, where the critical layer resides. While the vorticity and divergence perturbations are relatively small in the skirt, the angular pseudomomentum density is substantial in the neighborhood of *r*_{*}.

^{3}

Comparable amplitude reductions are found in the radial velocity perturbation *u*′.

^{4}

Circumstances may exist in which *δ***x**_{c} is not firmly tied to vertical misalignment in the very outer core. Such misalignment is of interest for how it may contribute to the enhancement of low-entropy downdrafts that could limit TC intensity (Riemer et al. 2010, 2013; Tang and Emanuel 2010).

^{5}

The reliability of measuring tilt with a linearized perturbation was tested by horizontally displacing each layer of the unperturbed circular vortex of the SLT by an amount ** ϵ**(

*z*) and approximating the new velocity field only to first order in |

**|. The tilt vector**

*ϵ**δ*

**x**

_{c}obtained from the approximate velocity field was in good agreement with the displacement vector

**(**

*ϵ**z*

_{2}) −

**(**

*ϵ**z*

_{1}) for amplitudes not exceeding about 25 km. Although the amplitude error grew with increasing displacement, the orientations of the tilt vector and displacement vector persistently agreed.

^{6}

Similar agreement was found when considering *r*_{c} = 140 km and *r*_{d} = 160 km.

^{7}

Like the dimensional magnitude of the tilt vector, the nondimensional misalignment *M* is also reduced. The mean value of *M* during the free evolution period in E2-c is approximately 44% of the corresponding mean in E2.