We apply the ADCIRC model [Luettich et al., 1992; Westerink et al., 1992] to simulate the surge induced by the selected storms. For NYC, the surge-simulation mesh is an unstructured grid developed by Lin et al.  with a resolution of approximately 100 m around NYC and up to 100 km over the deep ocean. The ADCIRC parameters are set to follow those of Colle et al. , whose results were validated against observations for the NYC area. For Tampa, the simulation mesh, modified from a mesh of Blain et al. , covers the entire Gulf of Mexico and has a resolution of approximately 600 m around Tampa Bay. The ADCIRC parameters are set to follow those of Westerink et al. , whose results were validated against observations for the Gulf area. Both of the numerical meshes are confined to the open ocean and we focus on the coastal surge. To be consistent, we present the simulated peak wind speeds and surge heights (over the storm lifetime) at the same coastal locations: the Battery, NYC (74.02 W, 40.9 N) and the City of Tampa (82.46 W, 27.89 N). The presented peak wind is the 1-min. average wind at 10 m over ocean-like surface roughness. For the ADCIRC surge simulations, the 1-min. wind is adjusted to a 10-min. average by a reduction factor of 0.893 [Powell et al., 1996], and the storm surface pressure is estimated from the pressure model of Holland .
 To study the uncertainties in parametric winds and the sensitivities of surge responses, we define a control case and investigate how wind and surge estimates vary with each parameter when other parameters are controlled. The parameters used for the control case are: surface background wind parameters α = 0.55 and β = 20°, the Emanuel and Rotunno  wind profile (used by [Lin et al., 2012] in surge analysis), SWRF = 0.85 [Batts et al., 1980], NWS's expression of inflow angle [Bretschneider, 1972], and Garratt's  expression of Cd, capped at 0.0025 [Powell et al., 2003]. Each of these parameters is introduced with wind and surge sensitivity analysis in the following subsections. The wind and surge sensitivities are expressed as the deviation (%) from the control case for the varied parameter; the mean, median, and range of the deviation over all selected storms for each location and for each parameter are shown. The deviations of each case from all other cases are also calculated, and the largest mean deviation (variation), indicating the overall uncertainty range, is presented.
3.1. Surface Background Wind
 In Section 2, we suggest using the storm translation to surface background wind reduction factor, α = 0.55, and counter-clockwise rotation angle, β = 20°, to estimate the surface background wind. Here we test if the wind and surge estimates are sensitive to these two parameters varying between 0.5–0.6 and 15°–22°, respectively, which were identified as the main ranges of radial variation (Figure 2). Figure 3 compares the wind and surge estimates using (α, β) = (0.5, 20°), (α, β) = (0.6, 20°), (α, β) = (0.55, 15°), and (α, β) = (0.55, 22°) respectively, with those using control values (α, β) = (0.55, 20°). Increasing α (from 0.55 to 0.6) often increases the peak wind (with a mean deviation of 0.46% for NYC and 0.65% for Tampa) and peak surge (1.9% for NYC and 1.4% for Tampa), but in some cases it decreases the (peak) wind when they are generated from the wind fields to the left of the storms where the background wind reduces the magnitude of the total wind; vice versa for the decreasing of α (from 0.55 to 0.5). While increasing β (from 20° to 22°) often increases the wind (0.42% for NYC and 0.085% for Tampa), it increases the surge for NYC (0.6%) but decreases the surge for Tampa (−0.26%); vice versa for the decreasing of β (from 20° to 15°). This difference between the two sites may be due to their geographical features: the majority of the storms affecting NYC move northward along the Atlantic coast and the effect of β in these storms usually pushes water into New York Harbor, but storm tracks affecting Tampa have larger variations and the effect of β in storms that pass northward along the west Florida coast often moves water away from Tampa Bay.
Figure 3. Comparison of the wind and surge estimates using various values of the surface background wind reduction factor, α, and counter-clockwise rotation angle, β, over the observed ranges, with those of the control case using α = 0.55 and β = 20, for (a and b) NYC (295 storms) and (c and d) Tampa (135 storms). The wind/surge sensitivity is expressed as the deviation (%) from the control case; the mean, median, and range of the deviation over all storms for each location are displayed in the form of “Dev. = mean/median% (0.5 quantile ∼ 0.95 quantile %).” The same format also applies for latter plots.
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 Over all, the differences in the wind and surge estimates for α and β varying within the main observational ranges are relatively small, with the largest mean variation of the wind estimates to be 0.9% for NYC and 1.3% for Tampa and the largest mean variation of the surge estimates to be 4% for NYC and 2.9% for Tampa. This smallness in variation is because for extreme events the wind field is dominated by the component associated with the storm itself, and thus the relatively small changes in the background wind induce small changes in the wind and surge estimates. However, as discussed in Section 4, greater changes in the background wind parameters can still induce significant over- or under- estimation of the wind and surge for extreme events. In addition, it is noted that surge estimates are more sensitive than wind estimates. As shown in the following analysis, this difference in wind and surge sensitivities is general with respect to all wind parameters, because the (peak) wind is determined by the wind field at a particular moment at the site while the (peak) surge is induced by the cumulative nonlinear effect of the wind stress (a quadratic function of the wind speed) over space and time.
3.2. Gradient Wind Profile
 We study four gradient wind profiles, using the storm characteristics along storm tracks generated by the synthetic hurricane model: the storm central pressure deficit (ΔP), symmetrical maximum wind speed (Vm), radius of maximum wind (Rm), and outer radius (Ro). The Holland  wind profile (hereafter H80) is most widely used and is based on an empirical radial distribution of the storm pressure and the assumption of gradient wind balance. The gradient wind velocity, V, at a radius r is given by
where e is the base of natural logarithms, ρ is the air density, f is the Coriolis parameter (f = 2Ωsinφ, where Ω = 7.292 × 10−5 and φ is the latitude), and B is the Holland parameter,
Since the Coriolis force is relatively small in the region of maximum winds, the cyclostrophic approximation has often been applied, neglecting the terms associated with f in equations (1) and (2) (hereafter H80c). However, the cyclostrophic approximation may induce inaccurate wind estimation away from the region of maximum winds and thus inaccurate local wind and surge estimates. It is also important to note that the cyclostrophic form of equation (2) is derived from the cyclostrophic form of equation (1). Using the cyclostrophic form of equation (2) together with the gradient wind-balance form of equation (1), as in many previous applications, will result in an underestimation of the wind speed near the radius of maximum wind (it is easy to show that the gradient wind-balance form of equation (1) will give a value of Vm that is smaller than its input value in the cyclostrophic form of equation (2)). Holland et al.  extends H80c by allowing a variation in the wind equation exponent (1/2 in equation (1)) to fit outer wind observations. As such observational data is not always available and unavailable for synthetic storms, this profile is not applied here for sensitivity analysis. However, out of curiosity, we use another wind profile [Emanuel and Rotunno, 2011; see below for descriptions] to estimate the radius of the 12-m/s wind outside Rm and apply it to the Holland et al.  profile; then the two profiles give statistically similar wind and surge estimates (not shown).
 The NWS currently uses the Sea, Lake, and Overland Surges from Hurricanes (SLOSH) model [Jelesnianski et al., 1992] for hurricane storm surge simulations and forecasting. The SLOSH wind profile (hereafter S92) is expressed as
This wind profile was first used by Jelesnianski  to form a simple algebraic formulation of the wind speed for surge analysis. Although wind estimates from the entire SLOSH wind model have been compared with observations [Houston et al., 1999], the wind-profile component of the model has not been evaluated.
 As discussed by Emanuel , although mature hurricanes in a quasi-steady state have nearly symmetric and uniform circulations, the hurricane's structure is determined by different mechanisms in different regions. Emanuel  developed a hurricane model that regards the wind profile in the outer region of the storm as being determined by the balance between radiatively controlled subsidence in the free troposphere and Ekman suction in the boundary layer, and the outer part of the storm's eyewall being controlled by thermal wind balance, convective neutrality in the vortex, and entropy and momentum balance in the boundary layer. He then derived asymptotic solutions of the model for each region and, by patching the solutions together and extending the asymptotic limit to the storm center (based on numerical simulations from a dynamical model), he obtained a wind profile for the entire storm structure. Validated against flight level observations, this wind profile (hereafter E04) is described, for r ≤ Ro, as
where b, m, and n are empirical parameters governing the shape of the wind profile: b = 0.25, m = 1.6, and n = 0.9.
 Emanuel and Rotunno  improved Emanuel's  model for the outer part of the storm eyewall by assuming a constant (critical) Richardson Number in the storm outflow, which determines the variation of the outflow temperature and thereby the radial structure of the storm outside its radius of maximum wind. The model was shown (by numerical simulations) to produce physically realistic results. An asymptotic solution of the model gives an analytical gradient wind profile (hereafter E11) for V (V ≥ 0),
where the ratio of the exchange coefficients for enthalpy and momentum in the original model disappear as we have assumed the two coefficients to be approximately equal in magnitude based on observations [Powell et al., 2003] and sensitivity analysis [Emanuel and Rotunno, 2011]. This wind profile is most accurate near the radius of maximum wind but is less accurate for the outer region of the storm, where the critical Richardson Number assumption may be violated. Also, similar to other steady state models, this wind profile is not accurate near the center of the storm, as the effect of radial diffusion in the eye is not accounted for. Curiously, if the terms associated with the Coriolis parameter, f, are neglected, equation (5) becomes S92 in equation (3).
 For illustration, Figure 4 compares these parametric wind profiles applied to two storms: the most and least intense at landfall among the selected storms for Tampa. Moving radially outward from the center to the radius of maximum wind, Rm, azimuthal wind speed increases most rapidly in E11 and S92, followed by E04, and finally both Holland profiles. Given that E04 is the only one among these profiles that is made asymptotically consistent with dynamic solutions (that account for the effect of radial diffusion in the storm center; Emanuel  showed a comparison), it may more accurately reproduce storm inner structure. Meanwhile, the storm structure outside Rm varies greatly among the wind profiles. In the outer region of the storm's eyewall, wind speed decreases more rapidly in the H80 and E04 profiles relative to E11 for the intense storm (Figure 4a). For the relatively weak storm (Figure 4b), the H80 profile decays more slowly with radius, approaching the E11 profile. In the outer region of the storm, E04 makes use of the information of the storm's outer radius, Ro, to define outer profiles; E11 has short profile tails and underestimates Ro; and H80 has long profile tails and lacks a well-defined Ro. S92 decays relatively slowly for the whole region outside Rm, and its Ro is ill-defined. Note that H80c gives much higher outer profile winds compared to H80, especially for the weak storm (see Figure 4b). This approximate profile is shown here for comparison; it is not used in the following sensitivity investigation as it is not recommended for wind or surge analysis in practice, but in Section 4 we will demonstrate the magnitude of errors this cyclostrophic approximation may induce if it is used.
Figure 4. Comparison of parametric wind profiles (a) for a relatively intense storm and (b) for a relatively weak storm, making landfall near Tampa. The storm for Figure 4a has the following parameters: φ = 28.0°, ΔP = 88.3 mb, Vm = 80.2 m/s, Rm = 20.5 km, and Ro = 400 km. The storm for Figure 4b has φ = 28.1°, ΔP = 30.3 mb, Vm = 39.5 m/s, Rm = 31.6 km, and Ro = 400 km.
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 The differences in the parametric wind profiles are reflected in the differences in the peak wind and surge estimates at the sites using these profiles, as shown in Figure 5. The peak winds at the sites (Figures 5a and 5c) are often generated when the storms are relatively strong and their radius of maximum wind pass near or directly over the sites. Compared to E11, H80 and E04 often give lower peak wind estimates due to their weaker profile winds in the storm's inner and outer eyewall regions (when the storms are relatively strong). Although H80 and E04 have higher profile winds than E11 in the outer region of the storm, the peak winds at the sites usually do not come from these far wind field regions (generated when the storms are relatively far away from the sites). S92 often gives higher wind estimates, compared to E11, due to its stronger profile winds outside the radius of maximum wind. The deviations of the wind estimates using H80, E04, and S92 from those using E11 are on average small to moderate, with the mean deviations being −1%, −2.4%, and 0.6%, respectively, for NYC and −2.6%, −4%, and 0.45%, respectively, for Tampa. Over all profiles the largest mean variation of the wind estimates is 3.3% for NYC and 5% for Tampa, due to the deviation of the estimates using S92 from those using E04.
Figure 5. Comparison of the wind and surge estimates using various gradient-wind profiles with those of the control case using E11, for (a and b) NYC (295 storms) and (c and d) Tampa (135 storms).
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 The surge estimates (Figures 5b and 5d) are much more sensitive than the wind estimates to different wind profiles. Usually the stronger the profile winds, the larger the storm surge; and the surge is mainly determined by the inner and outer eyewall regions of the profile winds. Therefore, in most cases, the surge estimates using H80 and E04 are lower than the surge estimates using E11, with the mean deviations being −6.1% and −13%, respectively, for NYC and −6.1 and −11%, respectively, for Tampa; the surge estimates using S92 are higher than those using E11, with the mean deviations being 5.4% for NYC and 5.3% for Tampa. There exist cases, however, in which the surge estimates using H80 are higher than those using E11, with deviations greater than 4.7% for NYC and 11% for Tampa. These higher surges under H80 may be generated mainly from the moments when the storms are relatively weak so that the H80 winds are very close to the E11 winds in the outer eyewall region while the H80 winds are much stronger than the E11 winds in the farther out region of the storm (see Figure 4b). This effect does not show up in the wind estimates (Figures 5a and 5c) as the peak winds are often generated when the storms are relatively strong. Over all profiles the largest mean variation of the surge estimates is 21.7% for NYC and 18.8% for Tampa, again due to the deviation of the estimates using S92 from those using E04.
 To further examine the impact of applying different wind profiles on the wind and surge estimates, we compare these estimates using each of the four wind profiles, H80, E04, S92, and E11, with the averages of these estimates over the four profiles, as shown in Figure 6. The deviations from the averages are relatively small for wind estimates (Figures 6a and 6c). Wind estimates using H80 and E11 are closest to the averages; a small number of the H80 estimates are lower than the averages and a small number of the E11 estimates are higher than the averages. Most E04 and S92 wind estimates are also very close to the averages, with some E04 estimates lower and some S92 estimates higher than the averages. Compared to the wind estimates, the deviations from the averages are much larger for the surge estimates (Figures 6b and 6d). Most E04 surge estimates are lower than the averages by about 10%, and most S92 surge estimates are higher than the averages by about 10%. H80 surge estimates have wide-spread deviations, especially for Tampa, from −30% to 10%. E11 surge estimates are most likely to be around the averages, but some E11 surge estimates are higher than the averages by about 10%. In summary, these results are consistent with those in Figures 4 and 5. Local peak wind estimates are not very sensitive to the wind profiles used, but peak surge estimates can vary greatly with the wind profiles. Since surge is mainly determined by the wind fields in the vicinity of the eyewall region, E11 is most likely to give accurate estimates as its outer eyewall structure is developed based on physical arguments and is numerically validated. Although it may overestimate the surge in some cases, as it may overestimate the winds toward the storm center, over all it gives more precise estimates than the other wind profiles.
Figure 6. Comparison of the wind and surge estimates using each of the four gradient-wind profiles with the averages of the estimates over these profiles, for (a and b) NYC (295 storms) and (c and d) Tampa (135 storms). The lower panel of each subplot shows the distribution of the deviation (%) of the (wind or surge) estimates using each profile from the average of the estimates over the four profiles.
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3.3. Surface Wind Adjustment
 In this study, the wind fields at gradient height are adjusted to the surface (at 10 m) through the use of the empirical surface wind reduction factor, SWRF, and storm inflow angle. Various values of SWRF, ranging from 0.65 to 0.95 over the ocean, have been used in the literature (a review is given by Vickery et al. [2009a]). As the value of SWRF is uncertain and it may also vary with the wind speed [Powell et al., 2005], uncertainties in wind and surge estimates may be induced when using a SWRF. To estimate how large the uncertainties can be, we analyze the sensitivity of wind and surge estimates to this parameter in the range of 0.7–0.9, as shown in Figure 7. Both wind and surge estimates change greatly with SWRF, because it directly affects the magnitude of the surface wind associated with the storm, which dominates the wind field for extreme events. Every incremental increase (decrease) of the value of SWRF by 0.05 from 0.85 increases (decreases) the magnitude of the wind speeds associated with the storm by about 5.9%, and thus it increases (decreases) the peak wind estimates by a slightly different amount, on average about 5.6% for NYC and 5.5% for Tampa (Figures 7a and 7c); this difference and the variation among storms reflect the effect of the (unchanged) background wind component. Curiously, every incremental increase (decrease) of the value of SWRF by 0.05 from 0.85 increases (decreases) the surge estimates on average by about 7% for NYC and 6% for Tampa (Figures 7b and 7d); the change of the surge is only slightly larger than the change of the wind when it is applied uniformly over the entire wind field. This nearly linear response of the surge to the wind is not observed in the analysis of surge sensitivity to different gradient wind profiles, where the wind field is varied differently over space and the change of the surge is much larger than the change of the wind (see Figures 5 and 6). The largest mean variation of the wind estimates is 27% for NYC and 26.2% for Tampa and the largest mean variation of the surge estimates is 36% for NYC and 29.6% for Tampa, due to the deviations of the estimates using SWRF = 0.9 from those using SWRF = 0.7.
Figure 7. Comparison of the wind and surge estimates using various values of the surface wind reduction factor, SWRF, with those of the control case using SWRF = 0.85, for (a and b) NYC (295 storms) and (c and d) Tampa (135 storms).
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 We identify two common empirical forms of storm inflow angle, measured inward from the azimuthal direction, as functions of the distance to the center of the storm. The NWS recommends approximating the inflow angle to increase linearly from 10° at the storm center to 20° at Rm and then to 25° at 1.2Rm, and to remain at 25° beyond 1.2Rm [Bretschneider, 1972]. Queensland Government  uses a somewhat different expression for the inflow angle, which increases linearly from 0° at the storm center to 10° at Rm and then to 25° at 1.2Rm, and remains at 25° beyond 1.2Rm. The difference between these two forms is that Queensland' inflow angle is smaller than that of NWS in the storm inner region. The sensitivities of both the wind and surge estimates to these two inflow angle representations are relatively small, as shown in Figure 8. The wind estimates using the Queensland inflow angle are slightly higher than those using the NWS inflow angle, with a mean deviation of 1.8% for NYC and 0.34% for Tampa. The surge estimates using the Queensland inflow angle, however, may be lower or higher than those using the NWS inflow angle, with a mean deviation of −1.1% for NYC and 0.13% for Tampa. This difference exists because the inflow angle of storms moving along the Atlantic coast usually enhances the transport of water into New York Harbor, while the opposite is true in Tampa Bay, which is consistent with the different effects of the background wind rotation parameter β on the two sites (see Figure 3).
Figure 8. Comparison of the wind and surge estimates using Queensland's inflow angle with those of the control case using NWS's inflow angle, for (a and b) NYC (295 storms) and (c and d) Tampa (135 storms).
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