A Parametric Model of Tropical Cyclone Surface Winds for Sea and Land

a. Australian best track data

The Australian best track database (AustBT) is the Bureau of Meteorology’s official record of tropical cyclone tracks in the region and will be our source of climatological tropical cyclone information in the region. A limitation of the AustBT and other similar datasets is that they are quite nonuniform in quality, due to changes in observing systems and analysis techniques. The position data are likely complete after July 1979, coinciding with geostationary satellite coverage of the region. The Dvorak technique for tropical cyclone intensity estimation was developed and implemented from about 1972 to 1984. The advent of the QuikScat scatterometer in 1999 significantly improved data availability for estimating outer wind radii, although it took several years for these new data to be used systematically. Following the recommendations of Bureau of Meteorology (2018) regarding data reliability, we use intensity and structure data only after July 2003.

The AustBT is available from http://www.bom.gov.au/cyclone/tropical-cyclone-knowledge-centre/databases/ and a full description is given in Bureau of Meteorology (2011). For this work, we use the position, intensity (as maximum 10-min mean wind speed υ_max), and radii of maximum winds and gales. Data are at maximum 6-h intervals, but often more frequently near critical periods of the storm’s life such as landfall. Gale wind radii are given as either four quadrants or a single symmetric value; in the former case we average the quadrant values to give a single value when all four are present. Cyclone forward motion was calculated from the position data at the available frequency.

b. WDR parametric model

Our approach will be to represent the flow above the boundary layer in the absence of motion by an axisymmetric wind field in gradient balance, which is then modified for the effects of friction and motion. We use the axisymmetric profile of Willoughby et al. (2006, hereafter WDR) for this purpose.

c. Empirical surface wind model

We choose to represent the vector surface wind as a magnitude and direction (inflow angle), rather than as radial and azimuthal components, because many risk applications will require only the speed information, allowing these applications a simpler model. We work in Earth-relative coordinates because this is what most applications will require and to maintain simplicity. Had we used storm-relative coordinates, conversion to Earth-relative winds would require knowledge of the full wind vector.

We will develop statistical models for s(r) and m(r, θ) in section 3b, noting that these will be different for sea and land.

The surface r_max is typically less than that at gradient level. Within the boundary layer, this slope is due partly to friction and partly to the cyclone’s warm core (Powell et al. 2009). Although we develop a statistical model for the reduction in r_max due to friction, we will see that this difference is on average about 9% of r_max over sea (12% over land), too small to matter for risk applications.

d. Fitting the WDR profile

There will be multiple values of x and L₂ that satisfy (10). This choice determines whether the wind decrease from r_max to r_gale is relatively straight, or highly curved, such that the wind speeds in this annulus can differ by several meters per second, with potentially significant effect on the ocean response and other impacts of the storm. We now describe how to take the statistically optimum choice; that is, the maximum likelihood estimate.

With these constraints, the optimization procedure sometimes yields x = 0, and thus also addresses the problem of choosing between the single and double exponential forms of the profile.

For the remaining parameter, the blending length L_b, we adopt WDR’s recommendation L_b = 10 km for the single-exponential profile and 25 km for the double, using the test x < 0.1 to discriminate these cases. We further constrain L_b < 0.8r_max so that the blending zone’s inner boundary satisfies r > 0 even for tropical cyclones with very small r_max.

The constrained optimization procedure is not applicable to storms too weak to possess an r_gale, since there is no outer wind information to provide a constraint between x and L₂. In such cases, maximizing the conditional density function without r_gale simply gives the conditional mean of the outer structure of the storm, conditioned on υ_max, log(r_max) and λ. A discontinuity in storm structure then arises if r_gale is available at other times in the storm track. We avoid this discontinuity by blending the two methods (i.e., with and without r_gale) for storms that possess an r_gale, but where the diagnosed gradient wind speed at r_gale is relatively close to that at r_max. The conditional mean is used when those two gradient wind speeds are equal, and the constrained optimization when they differ by 4 m s⁻¹ or more, with a linear interpolation in between.

We compute a central pressure by inwards integration of the gradient wind equation applied to the fitted WDR profile. The outer boundary condition, that the nominal environmental pressure is 1006 hPa, is applied at a nominal radius of 1200 km for consistency with WDR. Other values of environmental pressure may be appropriate in other regions, although, as discussed later, we also recommend adjusting the pressures as part of the process of blending with the environmental fields. This nominal radius is substantially larger than the typical cyclone circulation. The operational procedure in Australia is to estimate the environmental pressure as being 2 hPa greater than the pressure of the outermost cyclonic closed isobar (Courtney and Knaff 2009), whose radius is the r_oci. According to the AustBT, the r_oci has a skewed distribution with a mean of 298 km and standard deviation of 127 km. All the isobars of the synthetic vortices are cyclonically curved, so an alternative definition is needed here. Since the outer integration begins at radius 1200 km, a greater decrement than 2 hPa is appropriate, so we take the r_oci to be the radius at which the pressure drop is either 5 hPa, or half the overall pressure drop between the center and 1200 km, whichever is smaller. The latter bound prevents weak systems from having an absurdly small r_oci. The resulting values are reasonably consistent with the AustBT climatology (not shown).

The above procedure applies statistical relationships that WDR derived from aircraft reconnaissance data in the North Atlantic and east Pacific to storms in the Australian region. It is therefore reasonable to ask how similar the respective tropical cyclone climatologies are. The lack of aircraft reconnaissance in the Australian region precludes a full analysis but we can compare statistics from the historical record, shown in Table 1. We use the same period (post-2003) for HURDAT as for the Australian data. The HURDAT data are on average a little more intense, and about 18 km larger in both r_max and r_gale. These differences in size are largely due to a preponderance of relatively large systems in the North Atlantic and east Pacific, where over 10% of systems have r_max > 100 km or r_gale > 250 km. In Australia, these criteria are met by under 2% of storms. Storms with large r_max are underrepresented in the WDR dataset due to their QC procedure, making it more like the AustBT in this regard. Correlations between υ_max and the two radii are broadly similar in the two datasets.

Table 1.

Statistics of tropical cyclone intensity and size parameters for the Australian region and the North Atlantic and northeast Pacific combined, plus data where available from the QC flight level dataset used by WDR (their Table 1). The first three rows show mean values from the respective datasets, the next three show the frequency with which certain thresholds are exceeded, and the final three show correlation coefficients between variables. Note that the data from WDR involving r_max were calculated using log(r_max).

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Further similarity of observed storm structure between these regions can be inferred from the relationships between r_max and υ_max. WDR gives a regression relationship based on their flight-level data r_max = 46.4 exp(−0.0155υ_max + 0.0169|λ|) and Aijaz et al. (2019) give a similar relationship based on AustBT data, r_max = 40.4 exp(−0.0223υ_max + 0.030 495|λ|). A further relationship is given by Vickery and Wadhera (2008), r_max = 51.4 exp(−0.0223υ_max + 0.0281|λ|) also based on aircraft reconnaissance data. These are compared in Fig. 1, showing reasonably similar relationships between υ_max and r_max. Australian storms tend to be roughly 10 km smaller in r_max according to these relationships, except for the weaker systems at higher latitudes.

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Regression relationships between r_max, υ_max, and latitude, according to Willoughby et al. (2006) (green) and Vickery and Wadhera (2008) (red), and for the AustBT (blue). Latitudes of 10° are shown in solid and 25° are dashed.

Citation: Weather and Forecasting 38, 9; 10.1175/WAF-D-23-0028.1

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The distribution of r_gale, including interbasin differences, was studied using QuikSCAT data by Chan and Chan (2012). Their results are especially useful since the same methodology and data were applied globally. Their mean r_gale for the North Atlantic is 200 and 125 km for the northeast Pacific. Corresponding data for the South Indian zone are 178 km and for the South Pacific, 222 km, but these data are not representative of the Australian region. Johnny Chan (2015, personal communication) kindly provided the raw Southern Hemisphere data, from which we calculate that the mean r_gale for the Australian region is 142 km, smaller than the North Atlantic mean but larger than the northeast Pacific. Australian tropical cyclones thus fall within the envelope of basins from which WDR obtained their data, supporting the use of their statistics.

e. The boundary layer model

We use a dynamical diagnostic tropical cyclone boundary layer model (Kepert and Wang 2001; Kepert 2012, 2018) to train the statistical relationships in the surface wind parameterization. This model takes the pressure field at the top of the boundary layer (usually represented by the gradient wind) and the storm motion, and diagnoses the three-dimensional wind field by integrating the dry hydrostatic equations of motion to a near–steady state, with suitable parameterizations of surface fluxes and turbulent diffusion. The imposed gradient wind field, which is held fixed and represents the influence of the rest of the cyclone on the boundary layer, can be taken from parametric profiles (Kepert and Wang 2001; Kepert 2012, 2013, 2018), output from NWP models (Kepert and Nolan 2014; Zhang et al. 2017), or from composited aircraft observations (Yu et al. 2021). Here, the model is configured with a 3-km grid of size 900 km square, and 20 vertical levels on a stretched grid with the lowest at 10 m and the highest at 2.25 km.

The model has been compared to observations by Kepert (2006a,b), Schwendike and Kepert (2008) and Yu et al. (2021), and to output from NWP systems by Kepert and Nolan (2014) and Zhang et al. (2017). When compared to NWP systems, the model tends to underestimate the surface inflow by up to about 2 m s⁻¹, because it omits that part of the secondary circulation forced by latent heat release in the eyewall. Kepert and Nolan (2014) compared simulations of an eyewall replacement cycle using WRF with simulations of the boundary layer flow using this model forced by the azimuthal-mean pressure field from WRF, and similar parameterizations of surface fluxes and turbulent diffusion, and found that the 10-m inflow was always within about 2 m s⁻¹, but that WRF was usually stronger. A similar analysis by Zhang et al. (2017), but using an ensemble of WRF simulations, similarly found a slight underestimate of the inflow by the diagnostic model, which they attributed to the omission of that component of the inflow due to latent heating. Stern et al. (2015) used 3DVPAS to separate the secondary circulation from a WRF simulation into heating-induced and frictional components. They obtained good agreement on the total, and that heating contributes about 2 m s⁻¹ or less within the boundary layer.

We will use this diagnostic model to train regression relationships between the gradient and surface winds over sea and land. While other analyses have used observations for this task (e.g., Franklin et al. 2003; Powell et al. 2009; Giammanco et al. 2013; Zhang and Uhlhorn 2012), we prefer a modeling approach since we also require data over land, where instruments such as the GPS dropsonde and SFMR are unavailable, and surface heterogeneity complicates interpretation of point measurements (Powell et al. 1996).

The hierarchy of available tropical cyclone boundary layer diagnostic models was reviewed by Kepert (2010a). Most such models are axisymmetric and therefore lack information on the motion-induced asymmetry. Slab (vertically averaged) models are computationally cheaper and can provide asymmetry information but have significant limitations. They simulate the boundary layer mean wind, not the 10-m wind, can produce spurious oscillations and singularities (Shapiro 1983; Vogl and Smith 2009; Kepert 2010a; Williams 2015), and have inaccurate representations of both the surface drag and the nonlinear terms (Kepert 2010b). Vickery et al. (2000) needed to make further adjustments to the slab model winds to calculate the 10-m wind speed from the boundary layer mean value. Given the limitations of slab models, we prefer to use a model that explicitly represents the vertical structure.

We selected all systems with position, intensity, r_max, and r_gale data in the AustBT from July 2003 to June 2018. Where data were reported more frequently than every 6 h, for example for storms close to landfall, they were thinned to 6-hourly, leaving 1203 fixes in total. An earlier version of the WDR parameter computation process in section 2d was applied to derive a WDR profile for each storm. This earlier version differed from the final version in that the wind reduction factors at r_max and r_gale were mean values from a smaller set of simulations.

The set of test storms thus reflects the tropical cyclone climatology of the region, except that those systems too weak to possess an r_gale are omitted. As such storms make a small contribution to wind risk, this sampling bias was deemed acceptable.

The diagnostic model was run out to 24 h for each of these 1203 cases over three different surfaces: sea, land, and landfall. Sea is defined by a capped drag coefficient 10³C_D = min(0.7 + 0.065υ₁₀, 2.3), and land by a constant roughness length of 0.03 m representing open terrain. The turbulence closure uses the first-order closure of Louis (1979) as recommended by Kepert (2012). Landfall cases have both surface types. The coastline moves across the storm-following coordinate system, oriented perpendicularly to the direction of motion, and reaches the cyclone center at 24 h of simulation. The sea and land cases were used for development of the statistical models by the cross-validated process described in section 3b, while all three sets of simulations were used for verification in sections 4b and 4c.

The 10-m wind data from the boundary layer model simulations were postprocessed to compute the azimuthal mean and the phase and magnitude of the azimuthal wavenumber-1 asymmetry, as functions of radius, for the wind speed and inflow angle.

a. Comparison of KW model results to other studies

The 1203 runs of the KW model represent a large, statistically representative set of tropical cyclone boundary layer data. It is therefore of interest to compare the surface wind statistics computed from this dataset (Table A3) with other estimates, particularly those based on observations.

We find a mean surface wind factor at the RMW, s(r_max) = 0.802. Franklin et al. (2003) give a range of 0.75 to 0.9 (depending on flight altitude) based on the analysis of 630 GPS dropsondes. Of their data, we consider the flight altitudes of 850 and 925 hPa the most comparable, since these are likely above the boundary layer but have less thermal shear than the 700-hPa level, reducing the range to 0.75–0.80. Powell et al. (2009) analyzed flight level in situ and SFMR surface winds and found a mean of 0.84 with a standard deviation of 0.09. It should be noted that the definition of the quantity is different in these two studies; Franklin et al. (2003) compare the flight level wind to that at the dropsonde splash location, including slant in both azimuth and radius, while Powell et al. (2009) compare the maximum flight level wind to the maximum remotely sensed surface wind and explicitly note that theirs is a slant reduction.

Powell et al. (2009) give a regression equation for the eyewall wind reduction factor against $\sqrt{M}$, where M is the relative angular momentum, of $s(r_{\text{max}})=1.028-1.27\times {10}^{-4}\sqrt{M}$ and accounting for 32% of the variance. A similar univariate regression on our data yields $s(r_{\text{max}})=0.903-9.05\times {10}^{-5}\sqrt{M}$ accounting for 41% of the variance. They also analyzed the azimuthal variation, reporting a wavenumber one structure with an amplitude of 0.05 and the maximum in the right rear quadrant. The mean normalized asymmetry amplitude here of 0.76, multiplied by our mean motion speed of 3.56 m s⁻¹ and divided by the mean intensity 23.2 m s⁻¹, yields 0.12, suggesting a substantially stronger dependency, likely because motion is the sole source of asymmetry in our simulations, whereas their data would have included other factors such as vertical wind shear that obscured the motion signal.

At larger radius, Franklin et al. (2003) give surface wind factor values for the outer vortex, both in and out of convection, of 0.75–0.80 and 0.75, respectively, where we have again taken their middle two flight altitudes. These are somewhat higher than our value of 0.69. The difference may be partly because they include all dropsondes outside of the eye region, many of which would have been inside of r_gale, leading to higher values. The HURDAT mean r_gale of 155 km (Table 1), is close to the typical flight leg length of 150 km (Willoughby and Rahn 2004), so most dropsondes are sampling within the radius of gales and would therefore report a higher surface wind factor. In support of this hypothesis, we find mean values of 0.72 and 0.75 at the radii of storm-force and hurricane winds, respectively.

Powell et al. (2009) also reported values for the ratio of the surface r_max to that at flight level, with a mean of 0.875 that compares closely to our mean value of 0.911. Their data exhibit substantial spread, with values ranging from 0.2 to over 2 (their Fig. 7), due to the difficulty in defining this quantity in storms undergoing eyewall replacement cycles.

Surface inflow angle has been less studied than wind speed, with Zhang and Uhlhorn (2012) being the most comprehensive observational study. They found a mean inflow angle of −22.6°, steeper than our values at r_max of −16.8° but close to our r_gale value of −21.6°. They note that the inflow angle decreases with radius by −0.53° for every unit of radius normalized by r_max, although their Fig. 6 suggests a steeper increase near the r_max and a mean inflow angle of about −20° there. While they discuss the azimuthal variation, it is only in terms of the inflow angle of the storm-relative data, which are difficult to compare to Earth-relative data since that conversion requires knowledge of the wind speed also.

Further information on the surface inflow asymmetry is given by Powell (1982), who analyzed Hurricane Frederick and found that the inflow angle was greatest in the right rear in this Northern Hemisphere case, with near-eyewall values in the range of −30° to −50°, and least to the left and left front where the inflow angle was in the range of 0° to −5°. Given storm motion of 5 m s⁻¹, these suggest a normalized inflow asymmetry in Frederic of roughly 4° s m⁻¹, more than double our mean value of 1.71° s m⁻¹.

b. Surface wind model and statistics

In this section, we construct statistical models for the four functions s(r), m(r, θ), ϕ_s(r), and ψ(r, θ) that relate the surface wind and direction to the cyclone’s symmetric gradient wind and motion in Eqs. (5) and (6).

Examination of the output from the boundary layer model revealed that the asymmetric structure is strongly dominated by azimuthal wavenumber one. It further revealed that it would suffice to develop models for the four functions s, m, ϕ_s and ψ at the surface r_max and surface r_gale, and to interpolate in radius between these values. Outside of this radius range, the values are held fixed. We similarly define values of the symmetric inflow angle ϕ_s and amplitude and phase of the scaled inflow angle asymmetry ψ at r_max and r_gale, interpolate for intermediate radii, and hold fixed for radii outside this range.

The surface r_max is typically around 90% of that at gradient level. Within the boundary layer, this slope is partly due to the interaction between frictional inflow and inertial stability, and partly to the cyclone’s warm core (Kepert 2017; Powell et al. 2009). For interest, we develop a statistical model, with the same predictors, for this slope also.

Candidate predictors were chosen based on their availability in AustBT data and their connection to tropical cyclone boundary layer dynamics. The intensity υ_max, inner and outer size r_max and r_gale, and motion speed s_TC are obvious choices. Powell et al. (2009) found that $\sqrt{M}$ was a useful predictor, so we include an approximated version of that. Theory (Kepert and Wang 2001) and observations (Kepert 2006a,b) have shown that the “steepness” of the wind profile (i.e., how rapidly the wind speed decreases with radius outside of r_max) strongly impacts the tropical cyclone boundary layer. We measure steepness by an approximation to the exponent α in a modified Rankine vortex.

Prior to developing statistical models, scatterplots were made of all predictands against all potential predictors. In nearly all cases, visual inspection indicated that the relationship was either approximately linear, or relatively weak. There were two notable exceptions to this situation, the phases of the motion-induced speed and inflow angle asymmetries at r_max.

The distribution of motion-induced speed asymmetry, normalized by the translation speed, is shown in Fig. 2a. In this figure, if the motion-induced asymmetry equals the translation speed, the datum falls on the unit circle, indicated in red. Most data fall in the left-forward quadrant as expected for Southern Hemisphere tropical cyclones, with relatively few to the right front and almost none in the rear half. Nearly all are inside the unit circle, indicating that the magnitude of the motion-induced asymmetry is smaller than the motion speed. The mean of the joint distribution is at radius 0.76, rotated 65° anticyclonically from the front of the storm. The distribution of the magnitude is reasonably symmetric, but those of the azimuth (Fig. 2c), and front–back and left–right components (not shown), are skewed. We therefore transform the azimuth data,

$m_{T,p}(r_{\text{max}})=\log [90°+m_p(r_{\text{max}})],$(18)

to reduce the skewness (Fig. 2d) and improve the linearity of the relationships with the potential predictors. Here, the angle is defined such that −90° < m_p(r_max) ≤ 270° and is measured clockwise from the front of the storm.

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(a) Joint frequency distribution of wind speed asymmetry at the RMW, normalized by motion speed. The cyclone motion is toward the top of the figure. The unit circle (red) corresponds to cyclones where the motion-induced asymmetry equals the motion speed; most cases fall within that circle. The mean asymmetry, indicated by the red dot, is at normalized magnitude 0.76 and azimuth 65° anticyclonically from the front of the storm (on the left in this Southern Hemisphere diagram). Most of the data are in the left-forward quadrant, with a tail of low density extending into the right front. (b) As in (a), but for the scaled inflow direction asymmetry. Inflow angles are defined as negative inward, and this diagram shows the location of the most positive inflow, which is the weakest in magnitude. (c) Marginal distribution of the azimuth of the maximum wind. (d) Marginal distribution of azimuth of maximum wind speed transformed by Eq. (18).

Citation: Weather and Forecasting 38, 9; 10.1175/WAF-D-23-0028.1

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Table A1 in the appendix, which presents linear correlation coefficients between the (transformed) predictands and predictors, encapsulates some physically interesting relationships. The columns for υ_max and α stand out as having relatively large-magnitude correlation coefficients in nearly all rows, whether they relate to predictands at r_max or at r_gale. We agree with Franklin et al. (2003) that more intense storms tend to have larger wind reduction factors s(r_max), and add that this ratio is also strongly negatively correlated with r_max. The greater slope of the r_max with height in intense storms is apparently contrary to the results of Stern and Nolan (2009), who found little variation with intensity, but we note that they considered only slope above the boundary layer, while we are examining slope within the boundary layer, for which the cause is different. The relatively strong motion-induced asymmetry, and the tendency for it to be closer to the front of the storm than to the left, for more intense and/or more peaked storms, appears new, although Knaff et al. (2018) found that the wind maximum rotated toward the front of low-latitude storms. In contrast, the motion speed has generally low correlations, likely because those asymmetry parameters one would expect for physical reasons to depend strongly on this parameter have already been normalized by this parameter.

Our set of six candidate predictors are correlated, because of tropical cyclone behavior (e.g., υ_max and r_max), and because some parameters are defined in terms of others. Principal component analysis of the sample correlation matrix shows that first four components explain 98.4% of the variance. These mutual correlations must be accounted for in developing the statistical models. We use a standard cross-validation process to select predictors (Wilks 2011, section 7.4.4). Within individual cyclone tracks, we expect serially correlated random errors in the storm parameters, and therefore, when dividing the data into development and test subsets, we ensure that individual storms are entirely within one group or the other. We divide our set of 1203 cases into 5 groups of nearly equal size such that each track is within only one group. We then take each such group in turn as the verification data and use the remaining four for model fitting. For each possible number of predictors n_pred, we fit all possible models with n_pred predictors using the training set and compute the RMSE against the verification set. We choose the best RMSE for each number of predictors, average these RMSEs across the five different sets of training data and choose the number of predictors as that beyond which overfitting becomes apparent, as described in Wilks (2011, p. 252). Finally, we refit the model to all data and find the best model using that number of predictors.

The model coefficients are in Tables A3 and A4 for the sea and land surfaces, respectively. The most frequently used predictors are r_gale and the steepness parameter α. Storm motion is selected only twice, because both the speed and inflow angle asymmetry have already been scaled by this parameter. Predictors that depend only on conditions at r_max (υ_max, r_max, and $\sqrt{M}$) are never used at r_gale over sea and only seldom over land, but predictors involving r_gale (r_gale and α) are often used at r_max, reflecting that storm outer structure can affect the boundary layer flow in the core (Kepert and Wang 2001; Kepert 2006a,b; Chavas and Lin 2016; Chavas and Knaff 2022).

The radius to gales is an important part of the method, for r_gale is both a predictor and serves as the outer radius at which the surface wind parameters are defined. This raises the practical difficulty that r_gale is (by definition) unavailable in weaker systems. For such storms, we replace r_gale by 3r_max as the outer radius for calculation of the surface wind parameters and perform a similar statistical analysis using only weak storms (those with υ_max < 25 m s⁻¹) in the training set, to define surface wind parameters at r_max and 3r_max. The predictors r_gale and α are omitted since r_gale is unavailable in weaker systems. On this reduced dataset, r_max and $\sqrt{M}$ are correlated at 0.98, so using both would provide very little additional information and carry the risk that the regressions would choose one or the other essentially at random. We therefore omit $\sqrt{M}$ also. The coefficients suitable for systems too weak to possess an r_gale are in Table A5 (sea) and Table A6 (land).

c. Systems partly over land

Two effects may apply here. The first is the development of internal boundary layers. We begin by computing two wind fields, one for the entire domain having a sea surface, and the other assuming land. Then, for onshore flow, we estimate the length of overland trajectory of the gradient wind,¹ and transition from the “sea” winds immediately at the coast to “land” winds further inland by taking a weighted mean, with the weight decaying exponentially with a length scale of 4 km from the coast. For the opposite transition, a similar weighting applies, only over the sea. The distance scale of 4 km was chosen based on examination of the landfall simulations and is consistent with the observational analysis of Alford et al. (2020). These transitions are deemed to be complete after 12 km. Away from the transition region, the land or sea surface wind models are applied as appropriate. Note that, although the simulations with the dynamical model all cross the coast perpendicularly, this approach allows for the cyclone to cross at any angle.

The second potential effect is the larger-scale response to asymmetric friction examined by Kepert (2002a,b) and Hlywiak and Nolan (2022). The available evidence is that this leads to an offshore acceleration of the surface wind speed, but no substantial increase in risk over land. At the present stage of development, we choose to not account for it, noting that there is typically less exposed infrastructure over the sea than over land. Nevertheless, we caution that the model likely underestimates the hazard in the offshore flow in the inner core.

a. Comparison to observations

Here, we compare simulated winds to anemometer observations at selected coastal sites. The selected observation sites, shown in Fig. 3, were chosen as being near-coastal or offshore stations with wind observations at half-hourly intervals (or better) and minimal topography inland. All wind observations were 10-min means, and where available more frequently than half-hourly, were thinned to that interval. We then selected from the AustBT all cyclones since July 2003 that passed within 200 km of at least one station with an intensity of at least 20 m s⁻¹, and that possessed r_gale data to enable reliable estimation of the outer structure. These storms are also shown in Fig. 3.

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Verification observation sites (black) and tracks of all tropical cyclones since July 2003 to come within 200 km while possessing r_gale data (gray), according to the AustBT. Cyclones mentioned in the text are shown in color: green is TC Damien and red is TC Monica.

Citation: Weather and Forecasting 38, 9; 10.1175/WAF-D-23-0028.1

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The wind model was then used to simulate wind fields on a 0.05° grid at half-hourly intervals for each storm, based on the AustBT data. The AustBT was linearly interpolated in time for this purpose, and winds were only simulated when r_gale data were present. These wind grids were then bilinearly interpolated to the observation locations.

Figure 4 shows time series of observation against model for TC Damien, chosen because it was an intense storm that impacted a relatively dense part of the observing network, including an eyewall passage over Karratha. Wind speed and direction are both simulated reasonably well at stations at a range of distances from the landfall point. The observed double peak in wind speed at Roebourne and a secondary maximum in wind speed at Barrow Island, consistent with the formation of a local maximum in the offshore flow, were not simulated by the model.

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(a) Observed (blue) and modeled (green) wind speed at Karratha, using the statistical wind model, during TC Damien of 2020. (b) As in (a), but for wind direction. (c),(d) As in (a) and (b), but at Roebourne. (e),(f) As in (a) and (b), but at Mardie. (g),(h) As in (a) and (b), but at Barrow Island.

Citation: Weather and Forecasting 38, 9; 10.1175/WAF-D-23-0028.1

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Examination of similar plots for all storms revealed a systematic tendency for the model winds at Maningrida, Milingimbi, and Gove to be higher than observed during offshore flow. These sites have extensive areas of savannah woodland inland, with a substantially higher roughness length than the 3 cm assumed by the model formulation. The largest error in the comparison arose during TC Monica’s landfall immediately to the west of Maningrida, produced an observed wind peak there of 54 m s⁻¹, well in excess of the modeled peak of 26 m s⁻¹, although the observed and modeled minimum pressures were in good agreement at 986 and 983 hPa, respectively. Stations between Learmonth and Broome, which border extensive desert regions, do not exhibit this bias.

Figure 5 shows a joint density plot of observed and modeled wind speed and direction for all cases where the storm center was within 200 km of the observation. Summary statistics for the differences are as follows: wind speed bias = 2.5 m s⁻¹, mean absolute error = 3.9 m s⁻¹, root-mean-square error = 5.1 m s⁻¹; wind direction bias = 0.1°, mean absolute error = 16.7°, root-mean-square error = 21.6°; MSLP errors bias = −0.3 hPa, mean absolute error = 3.0 hPa, root-mean-square error = 3.8 hPa. The wind speed positive bias of 2.5 m s⁻¹ reduces to 1.0 m s⁻¹ if the analysis is restricted to the stations between Learmonth and Broome, i.e., that border desert.

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Joint density plots of statistical wind model against observations for tropical cyclones in the Australian region from July 2003: (a) wind speed, (b) wind direction, and (c) mean sea level pressure.

Citation: Weather and Forecasting 38, 9; 10.1175/WAF-D-23-0028.1

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Reasons for model–observation differences include limitations of the input AustBT data, phenomena excluded from the model such as wind accelerations near rainbands, observation errors including the effects of turbulence not removed by the 10-min average, topographic modification of the flow, and land roughness lengths other than the 3 cm assumed, as well as errors in the model itself. Operational use is expected to be comparable to these statistics, except that operational analyses are less accurate than the best track.

b. Comparison to dynamical boundary layer model

As a further verification step in the landfall wind model, we compare the simulated wind fields from this model to those from the dynamical boundary layer model. For each storm in the training set, we compute the RMS difference of the wind speed and direction between the two datasets, over storm-centered circles of radius 50, 100, 150, and 200 km. These scores are then averaged over all storms, and the results shown in Table 2.

Table 2.

Verification scores for the statistical wind model. Each entry is the root-mean-square difference between the dynamical and statistical surface wind models, calculated over the circle indicated, and averaged over all storms, for the surface type indicated.

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For wind speed, results over sea are generally better than those over land, perhaps because the rougher land surface leads to greater frictional inflow, and therefore greater influence of the nonlinear terms in the equations of motion. These nonlinear terms are explicitly included in the dynamical model, but their influence is only discerned statistically in the model developed here. Scores for landfalling storms are worse than either of the others at the smaller radii, but fall between those for land and sea at larger radii. The deterioration at small radii may reflect the limitations of the “cut-and-paste” approach to generating a landfall wind field, which omits the inner core dynamics during landfall explored by Kepert (2002a,b), J. D. Kepert (2023, unpublished manuscript), and Hlywiak and Nolan (2022).

For wind direction, the sea surface outperforms the land at all radii, and the landfall fields are worse than either.

Difference plots of the two wind fields were examined for a large sample of the dataset. No systematic differences were observed over either land or sea surfaces. Figure 6 plots a typical (near-median RMS differences) oversea case, and Fig. 7 for a relatively poor oversea case (selected because wind speed and direction RMS differences for the 100-km annulus were both near the 98th percentile). In the second case, the maximum winds are located to the left of the storm, rather than to the left front. The inflow angle is generally too weak, and the inner core of this field lacks the cyclonic rotation apparent in the dynamical model. Nevertheless, even this relatively poor performer is probably sufficiently accurate for many purposes.

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(a) Wind speed from the KW model for Cyclone Narelle at 1200 UTC 10 Jan 2013, a median performer. (b) Wind speed from the statistical wind model over sea for the same case. (c) Difference between the wind speed from the two models. (d)–(f) As in (a)–(c), but for inflow angle. The dashed black circles in each panel indicate r_max and r_gale, and the black arrow shows the storm motion of magnitude 3.3 m s⁻¹. Note that the contour values in (c) are 0, ±0.5, ±1, ±2, ±4, ±8 m s⁻¹ and in (f) they are 0°, ±1°, ±2°, ±4°, ±8°, ± 16°.

Citation: Weather and Forecasting 38, 9; 10.1175/WAF-D-23-0028.1

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As in Fig. 6, but for Cyclone Durga at 0000 UTC 24 Apr 2008, a relatively poor performer. The motion speed is 6.1 m s⁻¹.

Citation: Weather and Forecasting 38, 9; 10.1175/WAF-D-23-0028.1

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Difference plots for the same case as in Fig. 6, except as if it was making landfall and the center had just crossed the coast, are shown in Fig. 8. While the blending technique clearly captures the differences reasonable well, the wind is too weak and has too small an inflow angle in most of the semicircle to the right of the storm motion where the flow is offshore, while the opposite applies in the semicircle with onshore flow. A similar wind speed bias was present in the majority of the cases examined, with the statistical model weakly underpredicting the wind speed in the offshore flow and overpredicting in the onshore flow.

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As in Fig. 6, but as if the cyclone was at the point of landfall. The effects of the coastal discontinuity are apparent in each panel. The sea surface is to the right, and the land to the left, of this boundary.

Citation: Weather and Forecasting 38, 9; 10.1175/WAF-D-23-0028.1

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We chose above to perform the radial interpolation of the surface wind parameters in log(r) for wind speed, but in r for wind direction, based on the examination of the simulations with the KW model. We have repeated this verification process with all interpolation in either log(r) or r and found that in both cases the scores deteriorate.

c. Wind–pressure relationship

One advantage of our decision to use the parametric wind field to provide a gradient wind, and then apply a statistical boundary layer, is that a dynamically consistent pressure field is also available. We use these pressure data to construct a wind–pressure diagram, shown for sea and land surfaces in Figs. 9a and 9c. Comparing this with a similar figure taken directly from the AustBT data (i.e., using the AustBT central pressures, Fig. 9b) shows good agreement. The reason for the greater breadth of the modeled distribution is unclear, but may reflect that the WDR profile allows a greater variety of shape of the wind profile than was implicit in Courtney and Knaff (2009), and also that the central pressure and intensity data in the AustBT are not independent, but in most cases reflect the application of a wind–pressure relationship.

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(a) Density plot for the wind–pressure relationship from the wind model over sea. The black curves are the typical relationships from Courtney and Knaff (2009, see their Fig. 2) as being typical for small, fast-moving, and low-latitude tropical cyclones (continuous) and large, slow-moving and high-latitude tropical cyclones (dashed). (b) As in (a), but for the AustBT data post–July 2003. (c) As in (a), but over land. The black curves are repeated from the at-sea panels for reference, but should not be used over land.

Citation: Weather and Forecasting 38, 9; 10.1175/WAF-D-23-0028.1

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Wind–pressure relationships are also potentially useful over the land surface but have not to our knowledge been previously presented. Figure 9c shows the wind–pressure relationship for land points, computed from the overland simulations with the wind model. These data are displaced to the lower right of the sea data, albeit with some overlap of the two distributions.