## Abstract

Three sets of idealized, cloud-resolving simulations are performed to investigate the sensitivity of tropical cyclone (TC) structure and intensity to the height and depth of environmental vertical wind shear. In the first two sets of simulations, shear height and depth are varied independently; in the third set, orthogonal polynomial expansions are used to facilitate a joint sensitivity analysis. Despite all simulations having the same westerly deep-layer (200–850 hPa) shear of 10 m s^{−1}, different intensity and structural evolutions are observed, suggesting the deep-layer shear alone may not be sufficient for understanding or predicting the impact of vertical wind shear on TCs. In general, vertical wind shear that is shallower and lower in the troposphere is more destructive to model TCs because it tilts the TC vortex farther into the downshear-left quadrant. The vortices that tilt the most are unable to precess upshear and realign, resulting in their failure to intensify. Shear height appears to modulate this tilt response by modifying the thermodynamic environment above the developing vortex early in the simulations, while shear depth modulates the tilt response by controlling the vertical extent of the convective vortex. It is also found that TC intensity predictability is reduced in a narrow range of shear heights and depths. This result underscores the importance of accurately observing the large-scale environmental flow for improving TC intensity forecasts, and for anticipating when such forecasts are likely to have large errors.

## 1. Introduction

Despite tremendous advances in our understanding of tropical cyclones (TCs) and the processes underlying their intensification, TC intensity prediction remains exceptionally difficult. Some of the challenges associated with intensity prediction may stem from the limited intrinsic predictability of a TC. In the absence of environmental influences, measures of TC intensity and structure, such as the maximum tangential wind speed and the radius of maximum wind, are estimated to have predictability limits between two and three days (Hakim 2013; Brown and Hakim 2013). But this does not mean that we are unable to accurately forecast TC intensity beyond three days. In fact, Judt et al. (2016) found the low-wavenumber components of Hurricane Earl’s (2011) wind field to be predictable well beyond the intrinsic limits found by Brown and Hakim (2013). They surmised that this extended predictability comes from the significant control of the large-scale environment on the processes governing TC structural and intensity change.

One aspect of the TC environment known to affect the predictability of structure and intensity is the vertical wind shear. Emanuel et al. (2004) showed that the presence of vertical wind shear heightens TC sensitivity to other environmental conditions such as midtropospheric humidity. Consistent with this finding, Zhang and Tao (2013) and Tao and Zhang (2014) demonstrated in a series of cloud-resolving simulations that the timing of rapid intensification becomes increasingly sensitive to the distribution of convection in environments with stronger deep-layer (200–850 hPa) vertical shear. On the other hand, in quantifying the predictability limits of large-scale variables relevant to tropical cyclogenesis, Komaromi and Majumdar (2014) found the environmental vertical wind shear to be predictable beyond one week. Thus, it remains unclear whether a sheared TC inherits this extended predictability. This study seeks to improve our understanding of the interactions between TCs and vertically sheared environmental wind profiles in order to better diagnose the environmental influences on TC intensity and structure, and their predictability.

### a. Intensity and structural impacts of vertical wind shear

Vertical wind shear is prevalent throughout the tropics, and is a leading cause of TC intensity change (DeMaria and Kaplan 1994). In particular, strong vertical wind shear is known to inhibit TC genesis and intensification. Numerous studies have attributed the detrimental impacts of strong vertical wind shear to ventilation of the TC inner core. Gray (1968) argued that vertical wind shear impedes intensification by ventilating latent heat released in convection away from a developing storm. Frank and Ritchie (2001) built upon this argument, proposing a top-down weakening mechanism by which shear-induced asymmetries in upper-level potential vorticity expose the warm core to ventilation by the environmental flow. Ventilation can occur via two pathways: one in which low-entropy downdrafts penetrating the boundary layer are drawn into the eyewall updraft by the low-level inflow (Riemer et al. 2010), and another in which eddy fluxes transport dry midlevel air directly into the eyewall convection (Simpson and Riehl 1958; Cram et al. 2007; Tang and Emanuel 2010). DeMaria (1996) proposed an alternative to the ventilation hypothesis, whereby the adiabatic midlevel warming response to vortex tilting by the ambient vertical wind shear increases the static stability of the inner core. Alternatively, Wu and Braun (2004) attributed shear-induced weakening to eddy momentum fluxes opposing the symmetric overturning circulation. To date, none of these explanations have been invalidated, and the destructive effect of strong vertical wind shear likely owes to a cooperation of the proposed mechanisms.

Vertical wind shear also affects TC structure through its tendency to organize convection. Observational studies of sheared TCs at various stages of development have consistently shown vertical wind shear to produce azimuthal wavenumber-1 asymmetries in precipitation and winds (Chen et al. 2006; Uhlhorn et al. 2014), which arise due to asymmetric convection initiating in the downshear-right quadrant (Corbosiero and Molinari 2002; Black et al. 2002; Reasor et al. 2013; DeHart et al. 2014). These observed asymmetries in convection can be viewed as a balanced response to vortex tilt. In a set of idealized adiabatic simulations with sheared barotropic and baroclinic vortices, Jones (1995, 2000) found maximum updraft velocities to the right of the direction in which the vortices were tilted by the sheared flow. She explained the phase relationship between tilt and vertical velocity as being a consequence of the adiabatic response to a tilted vortex, wherein air parcels moving cyclonically around the midlevel vortex ascend along raised isentropes downtilt, and descend along lowered isentropes uptilt. A similar relationship between vortex tilt and vertical motion was observed in the dry simulations of Frank and Ritchie (1999). However, they found that including moist physics caused convection to occur preferentially to the left of the shear vector. This change is thought to occur when lifting due to latent heat release and differential vorticity advection becomes more significant than the isentropic lifting mechanism described by Jones.

The tilt evolution of a sheared vortex reflects some of the mechanisms by which TCs are able to withstand even moderate vertical shear. Jones (1995) showed that the vertical coupling between the upper and lower parts of a tilted vortex causes cyclonic precession, and that the ambient vertical shear forcibly realigns vortices that are able to precess upshear. Reasor et al. (2004) demonstrated that a tilted vortex could also realign through inviscid damping of a shear-induced tilt mode. They proposed that diabatic heating could enhance this damping mechanism.

### b. Motivation

Given the importance of vertical wind shear in predicting the evolution of TC structure and intensity, accurate measurement and analysis of the environmental shear is essential. Yet despite improvements in observational sampling of TCs and their environment, defining vertical wind shear as the deep-layer shear between 200 and 850 hPa remains common practice (Velden and Sears 2014). This is not surprising, as the deep-layer shear is a leading predictor of TC intensity change (DeMaria and Kaplan 1994, 1999). Studies analyzing large samples of TCs in different ocean basins generally find that TCs begin to weaken in deep-layer shear ranging from 7 to 12 m s^{−1} (Gallina and Velden 2002; Paterson et al. 2005). Nonetheless, there are documented cases of TCs persisting or even intensifying in deep-layer shear above these thresholds (Hanley et al. 2001; Black et al. 2002; Molinari and Vollaro 2010), which suggest that deep-layer shear does not fully explain the influence of environmental flows on TC structure and intensity change. This is supported by Onderlinde and Nolan (2014), who found considerable variability in how simulated TCs exposed to the same deep-layer shear respond to differences in the environmental helicity, which is the rotation of the mean environmental wind vector with height.

Relatively little is known about how the vertical structure of a sheared environmental wind profile affects its destructive potential. Elsberry and Jeffries (1996) hypothesized that, in certain cases, the TC outflow could oppose vertical wind shear concentrated in the upper levels of the troposphere, allowing TCs to intensify despite strong deep-layer shear. Their hypothesis implies that knowing not just the deep-layer shear, but also the structure of the environmental wind profile and how it interacts with the TC circulation could improve TC intensity forecasts. Only a few studies have attempted to test this hypothesis. In their dry idealized simulations, Frank and Ritchie (1999) also shifted a layer of easterly shear to upper and lower levels, and found that low-level shear elicited stronger asymmetries in vertical velocity. How the inclusion of moist physics might have changed this result was not discussed, but recent statistical analyses of the TC intensity response to the vertical distribution of wind shear generally confirm that lower-level vertical wind shear is more detrimental to real TCs (Rhome et al. 2006; Zeng et al. 2010; Wang et al. 2015).

This study attempts to address numerous speculations about how the destructive potential of environmental vertical wind shear depends on the height and depth of a vertically sheared layer. Particular attention is given to the way in which shear height and depth affect TC structural and intensity predictability. In section 2, we describe the idealized modeling framework and the construction of the wind profiles. The intensity and structural response to independently changing the shear height and depth is described in section 3, and the joint sensitivity to shear height and depth is discussed in section 4. Section 5 contains a brief discussion including the predictability implications of the results.

## 2. Methods

We perform a large set of numerical experiments in which the deep-layer shear is fixed, while parameters controlling the height and depth of the vertically sheared layer are varied. In all simulations, we use the compressible, nonhydrostatic Advanced Research core of the Weather Research and Forecasting (WRF) Model (ARW), version 3.4.1 (Skamarock et al. 2008). The model has 40 equally spaced mass levels extending from the sea surface to 20 km altitude. The sea surface temperature is held constant at 29°C throughout the simulations. We utilize three nested grids: a stationary outer grid, and two grids that follow the 700-hPa geopotential minimum. The outer grid is a doubly periodic *f* plane at 20°N with 18-km horizontal resolution and dimensions 4320 4320 km^{2}. The nested grids have 6- and 2-km resolution with dimensions 1152 1152 km^{2} and 672 672 km^{2}, respectively. We use the WRF single-moment 6-class microphysics scheme (WSM6; Hong and Lim 2006) and the Yonsei University (YSU) planetary boundary layer scheme (Hong et al. 2006). No cumulus or radiation schemes are employed. Horizontal diffusion is computed from gradients along coordinate surfaces with constant eddy viscosity ( = 100 m^{2} s^{−1}). To determine how this choice of the eddy viscosity affects our simulations, we performed additional simulations with computed from the horizontal deformation. Aside from small-scale differences between the constant and flow-dependent cases, the basic sensitivities to shear height and depth described in section 3 are largely unaffected by the choice of eddy viscosity.

The initial TC is a cloud-free, baroclinic vortex with maximum winds of 30 m s^{−1} at a radius of 90 km and a height of 1.5 km. The radial decay rate of the tangential winds is that of a modified Rankine vortex within a radius of 600 km. Beyond this radius, we impose exponential radial decay so that the circulation in the outer domain equals zero. We follow the approach of Moon and Nolan (2010) for achieving an outward-sloping radius of maximum wind (RMW) and a realistic vertical decay rate of the tangential winds. Finally, we compute the pressure and temperature perturbations that hold this vortex in gradient wind balance.

We construct the environments in which the TC vortex is inserted by imposing unidirectional vertical wind shear on a quiescent basic state atmosphere. Using the point-downscaling technique of Nolan (2011), the outer domain is uniformly initialized with the thermodynamic profiles from the Dunion (2011) “moist tropical” Atlantic sounding, and then the different sheared zonal wind profiles are initialized. Adding vertical shear to the zonal winds requires meridional temperature gradients in order to maintain thermal wind balance, but such temperature gradients are inconsistent with the doubly periodic boundaries of our outer domain. Furthermore, the evolving interactions between vertical wind shear and its thermodynamic imprint on the environment make it difficult to isolate the role of vertical wind shear in the TC structural and intensity response. To overcome these issues, we use a modified version of WRF developed by Nolan and Rappin (2008) and further examined by Nolan (2011). In this modification, vertical wind shear is balanced by a pressure force added to the momentum equations equal to that which would exist if temperature gradients were present. The initial vertical wind shear is thus maintained throughout the simulations. Within this modeling framework, vertical wind shear is introduced through a piecewise zonal wind profile according to

Here *p* is pressure and and are the pressures at the bottom and top of the sheared layer, respectively. For all of the sheared simulations, the surface wind is −5 m s^{−1}, and the deep-layer zonal wind shear magnitude is 10 m s^{−1}, which is chosen to be near the deep-layer shear thresholds for weakening cited above. In (1), we define such that the zonal winds increase in log-pressure height as a cosine function minimized at and maximized at :

The cosine-shaped wind profile defined by (1) and (2) has maximum vertical wind shear when [so that the argument of the cosine in (1) is ]. Henceforth, we define the pressure level at which the vertical shear is maximized as the shear height . Setting and defining the depth of the vertically sheared layer as , we obtain the following expressions for and :

Using (2)–(4), the unidirectional wind profile in (1) is completely determined by specifying the deep-layer shear , the surface wind , and the height and depth of the sheared layer. Once each sheared environment is created by systematically varying and , the vortex and its associated temperature and pressure perturbations are added to the center of the outermost WRF grid.

In addition to a control simulation without shear or mean flow, we perform three sets of sheared simulations to demonstrate sensitivity to variations in shear height and depth . Figure 1 depicts the wind profiles for the first two sets of simulations. In the first set (Fig. 1a), is varied in increments of 5 hPa between 330 and 530 hPa, while is fixed at 350 hPa. In the second set (Fig. 1b), is varied between 50 and 650 hPa in increments of 15 hPa, while is fixed at 430 hPa. In a third set of simulations described in section 4, and are varied simultaneously.

Neglecting the temperature gradients that balance vertical wind shear has potentially important consequences for the set of simulations in which the shear depth is varied. While Nolan (2011) showed that model TCs exposed to shear without temperature gradients generally exhibit realistic structural evolutions, he found that the environment becomes somewhat less favorable for TC genesis through the absence of midlevel temperature gradients that promote convection in the left-of-shear quadrants. This behavior would be amplified if the sheared layer were to become shallower, and so its effect on the interpretation of our shear depth simulations is considered.

## 3. Results

### a. Response to and

Although a wind profile with 10 m s^{−1} of deep-layer shear would generally be considered unfavorable for intensification, changing the shear height and depth yields an array of intensifying and nonintensifying storms. Figure 2 depicts the intensity evolution for the variable shear height (Fig. 2a) and depth (Fig. 2b) simulations, where intensity is computed as the mean sea level pressure within a 16 × 16 km^{2} box centered on the location of minimum surface pressure. The shear heights and depths of selected simulations are labeled along the right axes as reference points for interpreting the results. All of the sheared simulations result in weaker TCs than the quiescent control simulation indicated by the dashed black line. For the variable shear height simulations, model TCs intensify when vertical shear is maximized above 450 hPa, and fail to intensify when vertical shear is maximized below 500 hPa. Changing the shear height for the intensifying cases affects the time at which intensification commences; as the vertically sheared layer is shifted toward the surface, the simulated storms intensify later, suggesting that lower-level shear is less favorable for TC intensification. Once intensification commences, there is remarkable consistency in the rate of intensification through the remainder of the 120-h simulation period. As a result, the contribution to the total spread among intensifying storms comes primarily from the variable time at which intensification begins, and not from the rate at which it proceeds. In the shear depth simulations (Fig. 2b), model TCs exposed to shear deeper than 200 hPa consistently intensify, while the intensity evolution of TCs in shallower shear is less certain. Intensification is delayed as the shear depth decreases, suggesting that shallow layers of vertical wind shear are less favorable for TC intensification. Within the range of between 450 and 500 hPa, and between 50 and 200 hPa separating intensifying from nonintensifying storms, the TC response to the imposed vertical wind shear becomes erratic and, in general, less predictable.

Much of the TC intensity variability within each group of simulations can be understood through the vortex tilting response. Figure 3 depicts the magnitude and direction of the midlevel (800–500 hPa) vortex tilt between 30 and 70 h for each of the variable shear height simulations. Dots represent the locations of the 500-hPa vortex center relative to the 800-hPa vortex center.^{1} The simulations are stratified based on when they reach a pressure of 980 hPa, corresponding to minimal category 2 intensity on the Saffir–Simpson scale; storms with a minimum central pressure 980 hPa by 72 h are classified as early intensifiers (Fig. 3, top); those with a minimum central pressure 980 hPa at 72 h but 980 hPa by 120 h are classified as late intensifiers (Fig. 3, middle); and those that never become stronger than 980 hPa are classified as nonintensifiers (Fig. 3, bottom). The colors of the lines correspond to Fig. 2. Circles are drawn every 10 h that are centered on the mean relative position of the 500-hPa vortex among each class of simulations. The radius of each circle encapsulates 66% of the members nearest the mean position at that time. Both early and late intensifying storms initially tilt into the downshear-left quadrant, and then precess cyclonically into the upshear-left quadrant. This precession occurs more rapidly and at tilt magnitudes smaller than 55 km for the early intensifiers, while precession is slower for the late intensifiers, with tilt magnitudes occasionally exceeding 100 km. The relationship between tilt magnitude and the rate of cyclonic precession is consistent with Jones (1995), who demonstrated that the weaker vertically penetrating flow of a strongly tilted vortex results in slower corotation. A common feature among the intensifying model TCs is the rapid alignment that occurs once an upshear-left tilt configuration is achieved. Tilting upshear is favorable for realignment due to differential advection by the vertically sheared flow. Comparing the tilt evolution with the intensity evolution in Fig. 2, vortex alignment is coincident with the onset of intensification, suggesting that tilting into the upshear-left quadrant is a precursor to intensification. Unlike the intensifying cases, nonintensifying storms, which are sheared at the lowest levels, tilt so far into the downshear-left quadrant that they are unable to precess upshear by 70 h.

Figure 4 depicts the tilt progression for the shear depth simulations. Similar to the shear height simulations, the early intensifiers are the first to precess into the upshear-left quadrant, with tilt magnitudes rapidly decreasing thereafter. These storms are exposed to the deepest layers of vertical wind shear. In contrast, the nonintensifying storms subjected to the shallowest sheared layers tilt into the downshear left quadrant without precession and realignment. Thus, shearing model TCs at lower altitudes or in shallow layers acts to tilt the vortex more strongly, preventing it from precessing toward the upshear alignment that is found to precede intensification.

An interesting feature in Figs. 3 and 4 is the larger spread in tilt evolutions among the late intensifiers. In these simulations, TCs are exposed to similar shear heights and depths for which we find elevated intensity sensitivity in Fig. 2, implying that much of a TC’s intensity predictability is derived from its structural predictability. The remainder of this section focuses on identifying structural differences among simulations to further explore this relationship between the structure and intensity of sheared TCs.

We can expose some of the physical processes underlying the sensitive structural response to minute changes in the environmental wind profile by comparing the inner cores of selected shear height simulations with significantly different intensity evolutions. Figure 5 depicts time–height sections of relative humidity, absolute vorticity, and perturbation potential temperature averaged within a 160 × 160 km^{2} box centered on the innermost vortex-following grid. Figure 5a depicts the highest-level ( = 330 hPa) and Fig. 5b depicts the lowest-level ( = 530 hPa) vertical wind shear simulations representing archetypal early and nonintensifying cases, respectively. Figure 5c depicts a late intensifier with = 465 hPa, and Fig. 5d depicts a nonintensifier with = 470 hPa. Each case exhibits an initial convective burst indicated by elevated relative humidity and vorticity above 5 km between 12 and 36 h. The early intensifier (Fig. 5a) remains convectively active thereafter, with an upper-level warm core developing, while the three other cases experience an intrusion of mid- to upper-level dry air between 48 and 72 h, coincident with the period during which vortex tilt is largest. This drying persists beyond 72 h for the nonintensifying cases (Figs. 5b and 5d), while a second burst of inner-core convection occurs just after 72 h in (Fig. 5c). This second burst of convection closely follows a rapid decrease in vortex tilt depicted in Fig. 3, signifying a relationship between midlevel vortex tilt and deep convection in the inner core. After 72 h, the vastly different inner-core evolutions between cases (Figs. 5c and 5d), whose shear heights differ by only 5 hPa, and the remarkable similarity of Fig. 5d to the archetypal nonintensifier (Fig. 5b) emphasizes the extreme sensitivity to shear heights between 450 and 500 hPa.

The morphology of shear-organized convection further elucidates the processes modulating the sensitivity to small changes in shear height. Figure 6 depicts simulated reflectivity at a height of 4 km, boundary layer–averaged anomalies,^{2} and horizontal winds at a height of 1.5 km for the two cases corresponding to Figs. 5c and 5d between 48 and 72 h. Both simulations exhibit an area of heavy precipitation extending northeastward from the low-level center of circulation (LLC) into the downshear-left quadrant at 48 h, which is associated with a deep convective complex organized by the westerly vertical shear. By 60 h, the convective complex for the intensifying case (Fig. 6, left) enters the upshear-left quadrant and exhibits vigorous precipitation signatures near the LLC. In contrast, the convective complex in the nonintensifying case (Fig. 6, right) remains downshear left and drifts farther from the LLC. Both storms have a sea level pressure of 995 hPa at this time, so these structural disparities precede changes in storm intensity. By 72 h, the convective complex of the intensifying case is within the upshear-left quadrant, while that of the nonintensifying case remains downshear left and becomes increasingly disorganized. This is the time when the intensifying TC begins to have a lower central pressure than the nonintensifying TC.

Figure 6 does not exhibit any obvious differences in boundary layer moisture between the two cases at the depicted times that would explain the observed differences in their convective evolution. This suggests that within the most sensitive ranges of shear height and depth, the TC response is driven not by the resolvable differences in the environment, but by small and seemingly random differences in moist convection (Zhang and Sippel 2009).

The large number of simulations performed in this study enables compositing among shear environments farther from the sensitive ranges found above in order to identify robust differences in the structural response. We first average the simulated reflectivity over the 20 lowest model levels of the innermost domain, and then composite among the 10 highest-level, lowest-level, deepest, or shallowest vertical wind shear environments. Fourier decomposition is then used to isolate the azimuthal wavenumber-1 reflectivity asymmetry within each composite. Figure 7 depicts the composite wavenumber-1 asymmetries at 24 h, when the intensities of all composited cases are similar. Even at this early time, there are differences in the phase and location of asymmetries that help explain the subsequent intensity response. Most notably, the asymmetry maximum in the favorable shear composites (deepest and highest level) is located in the upshear-left quadrant, and near the mean RMW indicated by the black circle. In contrast, the asymmetry maximum in the unfavorable shear composites (shallowest and lowest level) is located in the downshear-left quadrant and farther outside the RMW. These differences imply that more progressive cyclonic precession of the shear-organized convective complex and more active convection near or within the inertially stable TC core (e.g., Nolan et al. 2007; Vigh and Schubert 2009) are robust features of TCs in favorable shear height and depth environments. The same asymmetry at 24 h for the simulation without shear (bottom panel of Fig. 7) is considerably deamplified and almost completely out of phase with asymmetries in the shear composites. At later times (not shown), we find the no-shear asymmetry to be highly transient, consisting of spiraling features that emanate from the eyewall region.

Time series of the azimuthal variance in simulated reflectivity explained by wavenumbers 1–5 are used to see how the low-wavenumber components of convection evolve. The upper panels of Fig. 8 reveal that, for the shallowest and deepest shear composites, wavenumber 1 is consistently the dominant azimuthal mode. Nevertheless, higher-wavenumber asymmetries develop in the shallow composite (middle panel) until 60 h, seemingly at the expense of wavenumber 1. The rise of higher-wavenumber asymmetries through *t* = 60 h in the nonintensifying shallow shear simulations is consistent with the observed tendency for the convective complex in such cases to become fragmented and azimuthally localized during this period (Fig. 6). Simulated reflectivity snapshots after 100 h (not shown) show a convective complex in many of the nonintensifying storms reorganizing at radii 200 km, which likely explains the return to wavenumber-1 primacy after 96 h. In contrast to the sheared composites, low-wavenumber asymmetries for the simulation without vertical wind shear (bottom panel) explain no more than 75% of the total reflectivity variance at any time. Furthermore, a sizable fraction of the low-wavenumber variance is attributed to wavenumbers 2–5. Thus, the slowly evolving wavenumber-1 asymmetries in all of the sheared composites are forced by the sheared environmental flow, and persist even as the inner cores of storms in the intensifying composites become increasingly axisymmetric. We heuristically interpret the persistence of such vortex-scale convective asymmetries as low-wavenumber structural predictability inherited from the environment (Judt et al. 2016).

At this point, it is worth discussing how neglecting temperature gradients in our shear depth simulations might impact the basic structural and intensity response. From the thermal wind relation, the midlevel meridional temperature gradients needed to support the shear depths tested here range from −1.2 K (1000 km)^{−1} for the deepest shear ( = 650 hPa) to −15 K (1000 km)^{−1} for the shallowest shear ( = 50 hPa). We repeated the deepest and shallowest simulations with “real” shear in which the shear-balancing temperature gradients are retained, and the northern and southern domain boundaries are made to be free-slip walls. For the deepest shear case, the point-downscaling simulation does not differ considerably from its real-shear counterpart in terms of reflectivity and minimum central pressure (not shown). But when we compare the shallowest shear simulations, we find considerably stronger convection within the left-of-shear semicircle of the case with real shear, consistent with Nolan (2011). This stronger convection, which precesses upshear by 36 h and becomes axisymmetric by 72 h, causes a period of rapid intensification that does not occur in the corresponding point-downscaling simulation. In light of this evidence that point-downscaling may overestimate the destructive effects of the shallowest shears, we henceforth interpret cases in which 200 hPa with some caution.

### b. Tilt response to shear

We have shown that, as early as 24 h, the intensity response to changing and is closely tied to the vortex tilt response. But it remains unclear why shallower and lower-level vertical wind shear more effectively tilt the vortex in the first place. In this section we compare extreme cases within each set of simulations to demonstrate resolvable differences in the TCs and their environments, which are used to form preliminary hypotheses for the observed tilting response.

In the shear height simulations, large differences in vortex tilt develop prior to the onset of deep convection. By 12 h, the vortex tilts 88 km in lower-level shear ( = 530 hPa) compared to only 24 km in upper-level shear ( = 330 hPa). These initial tilting differences are largely explained by the fact that lower-level shear environments have more shear between 800 and 500 hPa than upper-level shear environments, causing greater differential advection of the vortex within this layer. Lower-level shear also places stronger storm-relative flow in the layer where the midlevel minimum in environmental resides, causing low- air to be advected over the vortex as it tilts downshear. At 6 h for example, the maximum azimuthally averaged tendency due to horizontal advection in the layer from 5- to 9-km altitude is −5.4 K h^{−1} in low-level shear, and only −1.7 K h^{−1} in upper-level shear.

Thermodynamic modification of the environment above the vortex has important consequences for convection developing near the TC core. Figure 9 depicts differences in 12–24-h time averages of axisymmetric (shading) and downward fluxes of anomalies (black contour) between extreme shear height (top) and depth (bottom) simulations. Both quantities are azimuthally averaged about an axis aligned with the 800-hPa TC center. Following (23) in Riemer et al. (2010), downward fluxes are computed as , where is downward motion and is the deviation from the azimuthally averaged . The panels on the right depict differences in the storm-relative zonal flow averaged within the same time interval. In all panels, favorable (upper level/deep) shear fields are subtracted from unfavorable (lower level/shallow) shear fields. Focusing first on the top panels (lower- minus upper-level shear), we find a tongue of negative exceeding −5 K that coincides with the layer of stronger storm-relative flow in low-level shear. In addition, a region of considerably larger downward eddy fluxes extends from *z* = 5 km into the boundary layer of the low-level shear case. The largest downward fluxes primarily occur downwind of the convective complex, creating a pool of depressed boundary layer in the upshear-left quadrant that appears to erode the leading edge of the convective complex as it precesses upshear (not shown). These enhanced downward flux signatures, which are also present when comparing other combinations of lower- and upper-level shear, are likely an indication of stronger downdrafts in low-level shear.

Dry air entrainment in the middle levels is one possible cause of stronger convective downdrafts that flux low- air into the boundary layer. To estimate the extent to which entrainment might enhance downdrafts in the low-level shear simulation, we compute radial eddy fluxes (), with the sign ambiguity removed by only considering negative anomalies () and excluding positive anomalies. Figure 10 depicts snapshots at 18 h of (shading) and updrafts exceeding 1 m s^{−1} (black contour) at *z* = 4 km for the extreme shear height simulations. The inward eddy fluxes in low-level shear (Fig. 10, right) are stronger than in upper-level shear (Fig. 10, left), and are closer to the primary updraft cores. The proximity of stronger inward fluxes of low- air to the convective updrafts lends support to the idea that the enhanced entrainment of midlevel air in low-level shear is responsible for amplifying downdrafts. At the same time, the enhanced inward fluxes may directly dissipate convection by robbing updrafts of their buoyancy. We thus propose the following pathway to describe the tilt response to shear height beyond 12 h: lower-level shear causes more dry air to be advected over the surface vortex. Convection interacting with the drier midlevel environment generates stronger downdrafts that flush the boundary layer cyclonically downwind of the convective complex with low- air. The convective complex dissipates as the midlevel vortex precesses over the hostile environment created by these downdrafts and as stronger lateral mixing directly dilutes updraft buoyancy, leaving the vortex vulnerable to further tilting downshear by the storm-relative flow.

We observe smaller differences in midlevel and downward fluxes between the shallow and deep shear simulations in Fig. 9 (bottom panel). The axisymmetric vortex PV from the same two simulations, however, indicates that shallower shear produces a shallower vortex beyond 12 h. We speculate that shallower elevated shear layers increasingly act as a cap on developing convective clouds by inducing stronger mixing at the cloud tops. Convection thus becomes less effective at building the vortex above the sheared layer, resulting in a more baroclinic vortex. Jones (2000) found that when she exposed more baroclinic vortices to vertical wind shear, they generally exhibited slower cyclonic precession. She attributed this behavior to the more baroclinic vortices having a larger tilt magnitude and weaker southerly flow at the upper levels. In our shear depth simulations, we measure how the baroclinic vortex structure evolves by computing the ratio of the axisymmetric tangential wind at *z* = 12 km to that at *z* = 1 km. We refer to this ratio as due to its similarity to in Jones (2000). Smaller values of signify more baroclinic vortices in which the tangential winds rapidly decay with height. Figure 11 shows the time evolution of tilt direction (solid lines) and (dashed lines) for the shear depth (top panel) and height (bottom panel) simulations. In the shear depth simulations, the spread in begins to increase before 12 h such that vortices exposed to shallower shear become more baroclinic than those exposed to deeper shear. The more baroclinic vortices begin to exhibit slower cyclonic precession toward an upshear (90°) tilt shortly after 12 h. In the shear height simulations, the vortices in lower-level shear exhibit slower cyclonic precession than vortices in upper-level shear by 12 h. But the vortices in lower-level shear do not become more baroclinic than vortices in upper-level shear until after 24 h. We, therefore, hypothesize that, in our simulations, shear depth modulates tilting by altering the vertical structure of the vortex, while shear height modulates tilting by different means involving thermodynamic modification of the middle levels.

## 4. Joint sensitivity analysis

The approach thus far has been to evaluate the response to independently varying shear height or shear depth. A drawback of this approach is that it restricts our analysis of the TC response along single dimensions of the parameter space spanned by the two shear parameters. We can utilize the same modeling framework to evaluate the sensitivity of TC intensity to simultaneous variations in the shear parameters by constructing a two-dimensional response surface. To this end, one might consider a Monte Carlo approach that treats the shear parameters as random input variables, and uses the nonlinear model to derive a probability distribution of TC intensity. This would require implementing WRF for many combinations of and sampled from their assumed probability distributions. While conceptually appealing, such an approach is intractable for our cloud-resolving WRF configuration.

In this section, we explore an alternative technique known as polynomial chaos expansion (PCE; Ghanem and Spanos 2002; Le Maître and Knio 2010). PCE requires only a relatively small ensemble of deterministic nonlinear model realizations to establish a functional relationship between the model output and any input parameter(s) in terms of orthogonal polynomial expansions. The functional approximation serves as a surrogate to the nonlinear model, providing distributional information about the response at a greatly reduced computational cost.

Several studies have used PCE to develop surrogates for complex geophysical models, which are then used to propagate parametric input uncertainties (e.g., Webster and Sokolov 2000; Thacker et al. 2012). In this study, we also use PCE to develop a surrogate model replacing WRF, however we use this surrogate primarily for performing a joint sensitivity analysis of the simulated TC intensity response to both and . The analysis includes both a qualitative sensitivity assessment using a response surface built from multiple implementations of the surrogate model, as well as a variance-based approach to quantifying the relative importance of the shear parameters in the TC intensity response.

Letting denote the numerical simulation output as a function of time and a single input parameter *ξ*, the surrogate model has the following form:

Here are time-dependent expansion coefficients for *k*th-order polynomials, and are 1D orthogonal polynomials. The error comes from truncating the polynomial series expansion at order *K*. Most of the effort in obtaining the surrogate model is in the computation of the expansion coefficients . In the present work, we find the coefficients that minimize the norm of through Galerkin projection of the response *M* on the orthogonal basis *ψ*. This yields the following expression for the coefficients:

where *ρ* is the assumed probability density of *ξ* and angled brackets denote the inner product. We adopt a Gauss quadrature approach (Davis and Rabinowitz 1984) to approximate the integral as

Subscripts *q* denote evaluation at Gauss quadrature points, which are values of *ξ* for which , and are the corresponding quadrature weights. Each quadrature point corresponds to parameter values used to initialize a deterministic WRF simulation. Because the WRF surrogate used here is a function of two uncertain shear parameters and (standardized values of and , respectively) the orthogonal basis becomes two-dimensional such that . The basis functions are chosen by identifying their weight function with the assumed probability density of the inputs. Knowing little about the distributions of and in nature, we initially assume uniform distributions, so the basis functions are scaled Legendre polynomials (Le Maître and Knio 2010).

As the smallest *K* guaranteeing series convergence is not known a priori, we choose an appropriate truncation order *K* based on the TC intensity response from the simulations in section 3. We consider only the ranges of hPa and hPa where the intensity response diagnosed from simulations in section 3 is found to be relatively smooth. This is done to avoid sharp transition zones where global polynomials might become suboptimal. Nevertheless, the ability of global polynomials to at least identify the location of sharp response transitions, should they still occur over certain combinations of and , is not compromised. Figure 12 depicts the TC intensity response at *t* = 120 h over the subrange of (top) and (bottom). Overlaid are the first- through fifth-order Legendre polynomial series that fit the response in a least squares sense. Higher-order series are qualitatively superior at fitting the response to , and exhibit the highest Pearson correlation coefficients and lowest RMSE. However, the diminishing improvements in correlation and error from for the response to , and the nearly linear response over the subrange of suggest that fourth-order truncation is sufficient. For 2D expansions, the number of quadrature points determining the size of the WRF ensemble increases with truncation order as . Therefore, 25 WRF simulations are sufficient for characterizing the relationship between TC intensity and the two shear parameters. Figure 13 illustrates the 25 Gaussian quadrature points in the – parameter space (top), and the corresponding zonal wind profiles that initialize each WRF simulation (bottom).

Once the expansion coefficients are found, the response surface is constructed by using the surrogate model to compute the TC intensity for 451 × 121 combinations of (every 1 hPa between 200 and 650 hPa) and (every 1 hPa between 330 and 450 hPa). Figure 14 depicts snapshots of the TC intensity response surface between 48 and 120 h. The surface consistently depicts stronger TCs with lower surface pressure in the upper-right corner (deep and upper-level shear), and weaker storms with higher surface pressure in the lower-left corner (shallow and lower-level shear), confirming the results from individual shear parameter analyses in section 3.

The TC intensity sensitivities to combinations of shear height and depth may be subjectively deduced from the distribution and alignment of contours on the response surface. At each time, most of the response surface is characterized by diagonally oriented contours, indicating joint sensitivity to and . Where contours become tightly packed are regions of elevated sensitivity. In particular, a region of elevated sensitivity develops after 72 h over similar parameter ranges for which we found the most sensitive intensity response in section 3. This region covers nearly the entire lower-left corner of the 2D parameter space. Partial derivatives on the response surface can also be used to partition sensitivities among each shear parameter. At 96 h for example, horizontally oriented contours for 380 hPa and 400 hPa indicate where the partial derivative of the response with respect to shear depth is negligible, or in other words, where the intensity response is more sensitive to changes in shear height. This particular feature has a fairly intuitive physical interpretation; when the sheared layer is already deep enough to permit intensification, deepening the shear further has a smaller impact than shifting it higher or lower in the atmosphere.

The sensitivity can be quantitatively partitioned using a variance-based global sensitivity analysis (Alexanderian et al. 2012). The total variance is computed by summing all of the squared coefficients , save the leading coefficient corresponding to the mean. If we only take the sum along dimension *i* while fixing = 0, then the variance attributed to the single shear parameter is

Figure 15 shows the percentage of variance explained by shear depth (red) and height (blue), computed by dividing by the total variance. There is a prolonged period from 36 to 72 h during which shear depth explains approximately 60% of the variance in the TC intensity response. After 72 h, the response sensitivity is almost equally partitioned between shear height and depth. The partitioning is not as straightforward prior to 36 h, which could be due to the model establishing a moist convective vortex in this time interval.

Of course, the response surface contains errors due to truncation of the series expansion, and the reliability of the response surface hinges on these errors being acceptably small. We estimate errors using both the quadrature ensemble and an independent sample of WRF simulations consisting of the 55 sheared simulations from section 3 having and in the subrange of values used to carry out the expansion [denoted by red crisscrosses (×s) in Fig. 13]. The top panel of Fig. 16 shows the time series of RMSE computed on the independent sample (solid line) and on the quadrature points (dashed line). The RMSE is normalized by the range of intensities on the response surface (independent sample) or the quadrature points (quadrature sample) at each time, so that the depicted error is large when it is a significant fraction of the intensity spread indicating that the response surface is less reliable. As expected, relative errors computed from the independent sample are almost always larger than those from the quadrature ensemble used to compute the expansion coefficients. The error from the independent sample peaks at 42% at 28 h, which is primarily due to the small intensity spread on the response surface (5.2 hPa) at this time. Beyond 36 h, however, the errors hover between acceptably small values of 10% and 20%.

The reliability of the response surface also depends on the convergence of PC expansions. We estimate convergence using another variance-based metric indicating the percentage of variance explained by the highest-order polynomial in the series. Here is defined as

where is the variance at truncation order *K*. If the expansion converges, should decrease as *K* increases. The bottom panel of Fig. 16 depicts through , which offer convincing evidence of convergence beyond 36 h. However, prior to 36 h, is occasionally greater than , indicating that the expansion does not converge at these times. Not surprisingly, these are also the times when the relative errors are largest.

The large errors and the lack of convergence prior to 48 h may be caused by the delayed intensity response to the shear parameters evident in Fig. 2. There is evidence in nature of TCs exhibiting a delayed response to vertical shear. DeMaria and Kaplan (1999) found the negative correlation between intensity change and deep-layer shear averaged within a forecast interval increases as the length of the interval is extended to 60 h, while Onderlinde and Nolan (2014) showed that this negative correlation continues increasing for forecast intervals out to 120 h. Based on these results, the response surface appears to be reliable within the range of forecast times during which we realistically expect TC intensity to respond to vertical wind shear.

## 5. Discussion and conclusions

We use cloud-resolving WRF simulations to examine how TCs respond to varying the height and depth of environmental vertical wind shear. The wide range of TC intensity evolutions in our simulations, despite fixed deep-layer (200–850 hPa) shear, provides further evidence that a TC is sensitive to more aspects of the environmental wind profile than what is captured by the deep-layer shear.

The modeled TC intensity response to changing shear height and depth is intimately connected to the structural response. Shallow layers of vertical wind shear positioned lower in the atmosphere tilt the TC vortex farther into the downshear-left quadrant. The balanced response to vortex tilting is the development of an asymmetric convective complex that remains collocated with the midlevel vortex. Cyclonic precession of the tilted vortex and the attendant convection into the upshear quadrant is found to precede realignment and intensification. When the shear is maximized lower than 500 hPa or is shallower than 200 hPa, it tilts model TCs so far into the downshear-left quadrant that they are unable to precess upshear and intensify. A joint sensitivity analysis is conducted using a TC intensity response surface constructed from expansions in a 2D orthogonal polynomial basis. This analysis and the resulting response surface generally confirm the sensitivities deduced from the individual parameter analyses, while portraying detailed structures of the response in a shear depth-height parameter space. Within the selected parameter ranges, shear depth explains most of the TC intensity variance between 36 and 72 h, suggesting that TC intensity forecasts at such lead times may be more sensitive to errors representing the depth of environmental vertical wind shear.

Although the results support the hypothesis of Elsberry and Jeffries (1996) that upper-level shear is more favorable for TC intensification, we offer an alternative explanation as to why this may be the case. In our simulations, the height of vertical shear modulates a boundary layer flushing mechanism similar to that articulated in Riemer et al. (2010). Specifically, lower-level shear enhances the advection of thermodynamically unfavorable air from the midlevel environment over the initial vortex. In turn, more destructive downdrafts flush the boundary layer with low- air downwind of the asymmetric convective complex. Wang et al. (2015) speculated that a similar chain of events may explain why low-level shear is more detrimental to western Pacific typhoons. Upper-level shear is more favorable for TC intensification in our simulations because this boundary layer flushing mechanism is less active, which allows the boundary layer downwind of the convective complex to remain relatively warm and moist. The convective complex thus remains intact as the midlevel vortex cyclonically precesses over the favorable low-level environment, ultimately allowing the model TC to align and intensify. Unlike the simulations of Riemer et al. (2010), in which boundary layer flushing occurred outside of the eyewall region, the flushing being modulated in our simulations is maximized near or within the RMW, implicating local downdrafts associated with inner-core convection (Molinari et al. 2013). We also note that, while previous studies identify vertical shear interacting with asymmetric distributions of dry air as a possible cause of TC intensity change (Riemer and Montgomery 2011; Molinari et al. 2013), our proposed mechanism operates in an initially axisymmetric thermodynamic environment and is driven by the minimum typically found in the middle levels of moist tropical environments.

Our results do not invalidate the kinematic explanation of Elsberry and Jeffries in which the TC outflow opposes upper-level wind shear; indeed performing similar simulations with a model configuration that better resolves the storm outflow layer is worthwhile. Combined with the results presented here, such simulations would improve our ability to anticipate intensity changes during TC–trough interactions, as the structure and intensity of upper-level troughs can affect both the height and depth of the vertical shear that they induce on a nearby TC (Molinari et al. 1998; Hanley et al. 2001).

A notable feature in our simulations is the narrow range of shear heights and depths where model TCs respond unpredictably to the imposed shear, which we attribute to seemingly random differences in moist convection. One might reasonably conclude that the predictability of TCs in these sensitive shear regimes is no longer controlled by the environment, but is born of the smallest scales of motion. However, any initial differences in our simulations necessarily arise from differences in the imposed large-scale flow by design; even tiny differences in the specification of the large-scale flow lead to potentially significant differences at the smallest scales of motion. One can imagine these differences as errors that grow upscale and manifest themselves as the vastly different TCs we observe. But even though the errors grow out of the small scales, they originate at the large scales, emphasizing the need to accurately observe the large-scale flow in order to mitigate error growth at the TC scale (Durran and Gingrich 2014; Durran and Weyn 2016). Using observing system simulation experiments (OSSEs) to understand how improved observations of the large-scale 3D flow field around a TC might improve intensity and structural forecasts is a subject of ongoing research.

Additional work is needed to apply our idealized results to real TCs. As discussed, the point-downscaling technique may not realistically simulate shallow vertical wind shear, so it is worth repeating the suite of simulations in a modeling framework that permits shear-balancing temperature gradients. Furthermore, initializing cloud-free vortices in sustained shear allows substantial modifications to occur in the absence of moist convection early in the simulations. A more realistic approach would be to initialize each vortex in a quiescent environment and gradually introduce the different vertical shears. Such an approach would guarantee a more homogeneous ensemble of TC-like vortices by the time the different vertical wind shears are imposed. Finally, the present study intentionally isolates the influence of vertical wind shear from that of other environmental factors. The results may thus depend on the selected environmental moisture profile and SST, the direction and magnitude of deep-layer shear, and the initial vortex structure (e.g., Trier et al. 2000). Nevertheless, this study represents a necessary step toward redefining vertical wind shear to account for its vertical distribution. A forthcoming study will attempt to estimate distributions of shear height and depth in nature. Such distributions will enable us to deem which values of and are most realistic, and to explore how the response of real TCs over the realistic ranges of these two parameters compares with the numerical results presented in this study.

## Acknowledgments

We thank Rob Rogers and Scott Braun for insightful discussions during this study, and three anonymous reviewers whose feedback greatly improved the manuscript. The first author gratefully acknowledges support from the AMS and the University of Miami graduate fellowships, as well as the National Defense Science and Engineering Graduate Fellowship (NDSEG) Program. We performed all simulations on supercomputers at the University of Miami’s Center for Computational Science.

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## Footnotes

^{1}

Throughout this study, the vortex center at any vertical level is estimated from the centroid of the signed squared absolute vorticity . This method was found to locate the centers of weak vortices more reliably than the regular vorticity centroid.

^{2}

The anomalies are computed with respect to the average within the inner domain, which differs by no more than 0.2 K between the two depicted simulations at each time. Therefore, the anomaly field highlights similar areas of reduced/elevated as the absolute (not shown).