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# Generalized Asymmetric Holland Model

The Generalized Asymmetric Holland Model (GAHM) is a parametric hurricane vortex model developed in ADCIRC for operational forecasting purpose. Based on the classic Holland Model, the GAHM removes the assumption of cyclostrophic balance at the radius of maximum wind, and allows for a better representation of a wide range of hurricanes. Another important feature of the GAHM is the introduction of a composite wind method, which when activated enables the usage of multiple storm isotaches in reconstructing the spatial pressure and wind fields, while only one isotach is used in the HM.

## The Classic Holland Model

The Holland Model (HM, 1980) is an analytic model that describes the radial pressure and wind profiles of a standard hurricane. To begin with, Holland found that the normalized pressure profiles of a number of hurricanes resemble a family of rectangular hyperbolas and may be approximated by a hyperbolic equation, which after antilogarithms and rearranging yields the radial pressure equation:

${\displaystyle P(r)=P_{c}+(P_{n}-P_{c})e^{-A/r^{B}}\quad }$ (1)

where ${\displaystyle P_{c}}$ is the central pressure, ${\displaystyle P_{c}}$ is the ambient pressure (theoretically at infinite radius), ${\displaystyle P(r)}$ is the pressure at radius ${\displaystyle r}$ from the center of the hurricane, and ${\displaystyle A}$ and ${\displaystyle B}$ are shape parameters that may be empirically estimated from observations in a hurricane.

Substituting (1) into the gradient wind equation, which describes a steady flow balanced by the horizontal pressure gradient force, the centripetal acceleration, and the Coriolis acceleration for a vortex above the influence of the planetary boundary layer where the atmospheric flow decouples from surface friction (Powell et al. 2009), gives the radial wind equation of a hurricane:

${\displaystyle V_{g}(r)={\sqrt {AB(P_{n}-P_{c})e^{-A/r^{B}}/\rho r^{B}+({\frac {rf}{2}})^{2}}}-{\frac {rf}{2}}\quad }$ (2)

where ${\displaystyle V_{g}(r)}$ is the gradient wind at radius ${\displaystyle r}$, ${\displaystyle \rho }$ is the density of air, ${\displaystyle f}$ is the Coriolis parameter. In the region of the maximum winds, if we assume that the Coriolis force is negligible in comparison to the pressure gradient and centripetal force, then the air is in cyclostrophic balance. By removing the Coriolis term in (2) we get the cyclostrophic wind

${\displaystyle V_{c}(r)={\sqrt {AB(P_{n}-P_{c})e^{-A/r^{B}}/\rho r^{B}}}\quad }$ (3)

By setting ${\displaystyle dV_{c}/dr=0}$ at radius to the maximum wind ${\displaystyle r=R_{max}}$, it is obtained that

${\displaystyle A=(R_{max})^{B}\quad }$ (4)

Thus the (${\displaystyle R_{max}}$) is irrelevant to the relative value of ambient and central pressures, and is solely defined by the shape parameters ${\displaystyle A}$ and ${\displaystyle B}$. Substituting (4) back into (3) to get rid of ${\displaystyle A}$, we get an estimate of ${\displaystyle B}$ as a function of the maximum wind speed

${\displaystyle B=(V_{max})^{2}\rho e/(P_{n}-P_{c})\quad }$ (5)

It was notable that the maximum wind speed is proportional to the square root of ${\displaystyle B}$ and irrespective of the (${\displaystyle R_{max}}$), given a constant pressure drop. It was also reasoned by Holland that a plausible range of ${\displaystyle B}$ would be between 1 and 2.5 for realistic hurricanes. Substituting (4) and (5) back into (1) and (2) yields the final radial pressure and wind profiles for the HM

${\displaystyle P(r)=P_{c}+(P_{n}-P_{c})e^{-(R_{max}/r)^{B}}\quad }$ (6)
${\displaystyle V_{g}(r)={\sqrt {(V_{max})^{2}e^{1-(R_{max}/r)^{B}}(R_{max}/r)^{B}+({\frac {rf}{2}})^{2}}}-{\frac {rf}{2}}\quad }$ (7)

The HM was implemented in the ADICRC as a wind module with NWS = 19. When sparse observations of a hurricane are given, estimates of the ${\displaystyle R_{max}}$ and shape parameter ${\displaystyle B}$ may be estimated by fitting data into the radial wind equation, which in turn allow us to compute ${\displaystyle P(r)}$ and ${\displaystyle V_{g}(r)}$ along the radius ${\displaystyle r}$ of the hurricane. However, discrepancies between wind observations and computed winds were sometimes found, and were negatively correlated to the Rossby number at ${\displaystyle r=R_{max}}$, defined as

${\displaystyle R_{o}={\frac {NonlinearAcceleration}{Coriolisforce}}~{\frac {V_{max}^{2}/R_{max}}{V_{max}f}}={\frac {V_{max}}{R_{max}f}}\quad }$ (8)

By definition, a large ${\displaystyle R_{o}(\approx 10^{3})}$ describes a system in cyclostrophic balance that is dominated by the inertial and centrifugal force with negligible Coriolis force, such as a tornado or the inner core of an intense hurricane, whereas a small value ${\displaystyle (\approx 10^{-2}\sim 10^{2})}$ signifies a system in geostrophic balance where the Coriolis force plays an important role, such as the outer region of a hurricane. As a result, the assumption of cyclostrophic balance at ${\displaystyle R_{max}}$ made in HM is mostly valid for describing an intense and narrow (small ${\displaystyle R_{max}}$) hurricane with a large ${\displaystyle R_{o}}$, but not applicable for a weak and broad hurricane with a small ${\displaystyle R_{o}}$. This intrinsic problem with the HM calls our intention to develop a generalized model that will work consistently for a wide range of hurricanes, which theoretically can be accomplished by removing the above cyclostrophic balance assumption and re-derive the radial pressure and wind equations (6)&(7).

## Derivation of the GAHM

The GAHM also starts with the same radial pressure and wind equations (1)&(2) with shape parameters ${\displaystyle A}$ and ${\displaystyle B}$ as in the HM. Without assuming cyclostrophic balance at ${\displaystyle R_{max}}$, we take ${\displaystyle dV_{g}/dr=0}$ at ${\displaystyle r=R_{max}}$ to get the adjusted shape parameter ${\displaystyle B_{g}}$ as

${\displaystyle B_{g}={\frac {(V_{max}^{2}+V_{max}R_{max}f)\rho e^{\varphi }}{\varphi (P_{n}-P_{c})}}=B{\frac {(1+1/R_{o})e^{\varphi -1}}{\varphi }}\quad }$ (9)

where ${\displaystyle {\varphi }}$ is a scaling parameter introduced to simplify the derivation process, defined as

${\displaystyle \varphi ={\frac {A}{R_{max}^{B}}}\quad }$ or ${\displaystyle \quad A=\varphi R_{max}^{B}\quad }$ (10)

and later derived as

${\displaystyle \varphi =1+{\frac {V_{max}R_{max}f}{B_{g}(V_{max}^{2}+V_{max}R_{max}f)}}=1+{\frac {1/R_{o}}{B_{g}(1+1/R_{o})}}\quad }$ (11)

Thus, the ${\displaystyle R_{max}}$ in the GAHM is not entirely defined by the shape parameters ${\displaystyle A}$ and ${\displaystyle B}$ as in the HM, but also by the scaling factor ${\displaystyle {\varphi }}$, as Equation (11) indicates that ${\displaystyle {\varphi }\geq 1}$. Numerical solutions for ${\displaystyle B_{g}}$ and ${\displaystyle {\varphi }}$ can be solved iteratively in the model using Equation (9)&(11). Figure 1 illustrates how ${\displaystyle B_{g}/B}$ and ${\displaystyle \varphi }$ vary with ${\displaystyle \log _{10}R_{o}}$ given different ${\displaystyle B}$ values. It is evident that values of both ${\displaystyle B_{g}/B}$ and ${\displaystyle \varphi }$ remain close to 1 when ${\displaystyle \log _{10}R_{o}}$ is within the range of [1,2], but increase noticeably as ${\displaystyle \log _{10}R_{o}}$ decreases below 1, and the smaller the value of ${\displaystyle B}$, the bigger the changes.

Fig 1. Profiles of ${\displaystyle B_{g}/B}$ (left panel) and ${\displaystyle \varphi }$ (right panel) with respect to ${\displaystyle \log _{10}R_{o}}$, given different ${\displaystyle B}$ values as shown in different colors.

Substituting (9)&(11) back into (1)&(2) yields the final radial pressure and wind equations for the GAHM

${\displaystyle P(r)=P_{c}+(P_{n}-P_{c})e^{-\varphi (R_{max}/r)^{B_{g}}}\quad }$ (12)

${\displaystyle V_{g}(r)={\sqrt {V_{max}^{2}(1+1/R_{o})e^{1-(R_{max}/r)^{B_{g}}}(R_{max}/r)_{g}^{B}+({\frac {rf}{2}})^{2}}}-{\frac {rf}{2}}\quad }$ (13)

Influence of the Coriolis force on the radial pressure and wind profiles are evidenced by the presence of ${\displaystyle R_{o}}$ and ${\displaystyle \varphi }$ in (12)&(13). A special case scenario is when we set ${\displaystyle f=0}$, which corresponds to an infinitely large ${\displaystyle R_{o}}$, then (12)&(13) in the GAHM reduce to (6)&(7) in the HM. However，for a hurricane with a relatively small ${\displaystyle R_{o}}$, the influence of the Coriolis force can only be addressed by the GAHM. It meets our expectation that the GAHM’s solution approaches to that of the HM’s when the influence of Coriolis force is small, but departs from it when the Coriolis force plays an important role in the wind system.

The above reasoning can be demonstrated by the 3D plots in Figure 2, which show the normalized gradient winds of the HM (left panel) and the GAHM (right panel) as functions of the normalized radial distances ${\displaystyle r/R_{max}}$, the Holland ${\displaystyle B}$ parameter, and ${\displaystyle R_{o}}$. In both panels, each colored surface represents the normalized gradient winds corresponding to a unique Holland B value. By definition, we get ${\displaystyle V_{g}=V_{max}}$ at ${\displaystyle r=R_{max}}$, which means all the surfaces in each panel should intersect with the plane of ${\displaystyle r/R_{max}=1}$ on the plane of ${\displaystyle V_{g}/V_{max}=1}$, no matter what values of ${\displaystyle R_{o}}$. However, the line of intersection (shown by the black line) shown in the left panel deviates from the plane of ${\displaystyle V_{g}/V_{max}=1}$ as ${\displaystyle \log _{10}R_{o}}$ decreases from 2 to close to 0 (${\displaystyle R_{o}}$ decreases from 100 to 1), while remains on the plane regardless of how ${\displaystyle R_{o}}$ changes in the right panel, demonstrating that the GAHM is mathematically more coherent than the HM.

Fig 2. The normalized gradient wind profiles of the HM (left panel) and the GAHM (right panel) as functions of the normalized radial distances and ${\displaystyle \log _{10}R_{o}}$, given different Holland ${\displaystyle B}$ values.

To have a dissective look of the surface plots in Figure 2, we draw slices perpendicular to the axis of ${\displaystyle \log _{10}R_{o}}$ at three different values 0, 1, 2, and plot the lines of intersection with each surface in Figure 3. It is evident that we get ${\displaystyle V_{g}=V_{max}}$ at ${\displaystyle r=R_{max}}$ consistently in the right panel for the GAHM regardless of the value of ${\displaystyle R_{o}}$. The HM in the left panel, however, generates distorted wind profiles with underestimated maximum winds skewed inward towards the storm center, espeically when ${\displaystyle \log _{10}R_{o}<1}$. As a results, when both models being applied to real hurricane cases, the GAHM will perform more consistently than the HM.

Fig 3. Slices of the normalized gradient wind profiles (as shown in Figure 2) at ${\displaystyle \log _{10}R_{o}=0,1,2}$ (or correspondingly ${\displaystyle R_{o}=1,10,100}$).

## Calculation of the Radius to the Maximum Wind

Both the HM and the GAHM use processed forecast advisories (during active hurricanes) or best track advisories (post-hurricanes) from the National Hurricane Center (NHC) in ATCF format as input files, which contain a time series of storm parameters (usually at 6-hour intervals) such as storm location, storm movement, central pressure, 1 minute averaged maximum wind, radii to the 34-, 50-, and/or 64-kt storm isotaches in 4 storm quadrants (NE, SE, SW, NW), etc. See meteorological input file with NWS = 20 for more details.

As a standard procedure, for all data entries in the input file the ${\displaystyle B_{g}}$ and ${\displaystyle R_{max}}$ are pre-computed in 4 storm quadrants of all available isotaches in the ASWIP program (an external FORTRAN program developed by Flemming et al. and further developed here to accommodate the GAHM) and appended to the input file prior to running an ADCIRC simulation forced with the GAHM wind model.

First, the influence of the boundary layer effect must be removed to bring the maximum sustained wind and the 34-, 50-, and/or 64-kt isotaches from 10 meter height to the gradient wind level. Practically, the maximum gradient wind can be directly calculated as

${\displaystyle V_{max}=\vert {\frac {{\overrightarrow {V_{M}}}-\gamma {\overrightarrow {V_{T}}}}{W_{rf}}}\vert ={\frac {V_{M}-\gamma V_{T}}{W_{rf}}}\quad }$ (14)

where ${\displaystyle {\overrightarrow {V_{M}}}}$ is the reported maximum sustained wind at 10 meter height assuming in the same direction as ${\displaystyle {\overrightarrow {V_{T}}}}$, ${\displaystyle {\overrightarrow {V_{T}}}}$ is the storm translational speed calculated from successive storm center locations, ${\displaystyle W_{rf}=0.9}$ is the wind reduction factor for reducing wind speed from the gradient wind level to the surface at 10 meter height (Powell et al., 2003), and ${\displaystyle \gamma }$ is the damp factor for ${\displaystyle V_{T}}$. The following formula of ${\displaystyle \gamma }$ is employ in the ASWIP program:

${\displaystyle \gamma ={\frac {V_{g}}{V_{max}}}\quad }$ (15)

which is the ratio of gradient wind speed to the maximum wind speed along a radial wind profile. Thus, ${\displaystyle \gamma }$ is zero at storm center, and increases with ${\displaystyle r}$ until reaches a maximum value of 1 at ${\displaystyle R_{max}}$, then gradually decreases outward to zero.

In addition to the scalar reduction in wind speed, surface friction and continuity also cause the vortex wind to flow inward across isobars, with an inward rotation angle ${\displaystyle \beta }$ according to the Queensland Government's Ocean Hazards Assessment (2001):

${\displaystyle \beta ={\begin{cases}10^{\circ },&r

Thus, the gradient wind at the radii to specified storm isotaches in 4 storm quadrants can be obtained from the observed isotaches as

{\displaystyle {\begin{aligned}V_{r}&=\vert {\overrightarrow {V_{r}}}\vert =\vert {\overrightarrow {V_{inflow}}}\vert \\&={\frac {\vert {\overrightarrow {V_{isot}}}-\gamma {\overrightarrow {V_{T}}}\vert }{W_{rf}}}\end{aligned}}\quad } (17)

where ${\displaystyle {\overrightarrow {V_{isot}}}}$ is the observed isotach wind speed with an unknown angle ${\displaystyle \varepsilon }$, and ${\displaystyle {\overrightarrow {V_{inflow}}}}$ is the wind speed at radius to specified isotach before the inward rotation angle ${\displaystyle \beta }$ is removed.

Rewriting (17) in x- and y-components yields:

${\displaystyle V_{r}\cos(quad(i)+90+\beta )=V_{isot}\cos(\varepsilon )-\gamma {\mu }_{T}\quad }$ (18)
${\displaystyle V_{r}\sin(quad(i)+90+\beta )=V_{isot}\sin(\varepsilon )-\gamma {\nu }_{T}\quad }$ (19)

where ${\displaystyle quad(i)}$ is the azimuth angle of the ${\displaystyle i-th}$ storm quadrant (NE, SE, SW, NW at ${\displaystyle 45^{\circ },135^{\circ },225^{\circ },315^{\circ }}$, respectively), ${\displaystyle V_{isot}\cos(\varepsilon )}$ and ${\displaystyle V_{isot}\sin(\varepsilon )}$ are the zonal and meridional components of ${\displaystyle {\overrightarrow {V_{isot}}}}$, ${\displaystyle {\mu }_{T}}$ and ${\displaystyle {\nu }_{T}}$ are the zonal and meridional components of ${\displaystyle {\overrightarrow {V_{T}}}}$.

Given an initial guess of ${\displaystyle R_{max}}$, values of ${\displaystyle B_{g}}$ and ${\displaystyle \varphi }$ can be solved iteratively from (9) and (11) until both converge, and ${\displaystyle V_{r}}$ can be estimated by combining (15), (17), (18), and (19). Plugging ${\displaystyle V_{max}}$ from (14), the above calculated ${\displaystyle B_{g},\varphi ,V_{max},V_{r}}$ and the observed radius ${\displaystyle R_{r}}$ to ${\displaystyle V_{r}}$ back into (13), a new ${\displaystyle R_{max}}$ can be inversely solved by a root-finding algorithm. Since the above calculations are carried out based on an initial guess of ${\displaystyle R_{max}}$, wWe need to repeat the entire process until ${\displaystyle R_{max}}$ converges.

In case where multiple isotaches are given in the forecast/best track advisories, the ${\displaystyle R_{max}}$ for the highest isotach will be calculated using the above procedure, and used as the pseudo ${\displaystyle R_{max}}$ for the entire storm (physically, there is only one ${\displaystyle R_{max}}$ found along a radial wind profile ). For each lower isotach, ${\displaystyle R_{max}}$ will be calculated with the pseudo ${\displaystyle R_{max}}$ set as its initial value to determine the inward rotation angle ${\displaystyle \beta }$ following the above process only once. The use of the pseudo ${\displaystyle R_{max}}$ across all storm isotaches ensures that the cross-isobar frictional inflow angle changes smoothly along the radius according to (17).

Occasionally, we have to deal with situations where ${\displaystyle V_{max}, which violate (13) so ${\displaystyle R_{max}}$ couldn't be calculated. These situations mostly happen in the right hand quadrants (in the Northern Atmosphere) of a weak storm with a relatively high translational speed. For cases like this, we assign ${\displaystyle V_{max}=V_{r}}$, which is equivalent to assigning ${\displaystyle R_{max}=R_{r}}$.

After the ASWIP program finishes processing the input file, it can be readily used by the GAHM to construct spatial pressure and wind fields in ADCIRC for storm surge forecast.

## Composite Wind Generation

Since storm parameters are only given in 4 storm quadrants (assuming at ${\displaystyle 45^{\circ },135^{\circ },225^{\circ },315^{\circ }}$ azimuthal angles, respectively) at 3 available isotaches in the input file, spatial interpolation of storm parameters must take place first at each ADCIRC grid node. Traditionally, the single-isotach approach is used, in which storm parameters will be interpolated azimuthally from the highest isotach only. To take advantage of the availability of multiple isotaches, a new composite wind method is introduced in the GAHM, the multiple-isotach approach, in which storm parameters will be interpolated both azimuthally and radially from all available isotaches.

To begin, the relative location of a node to the storm center at given time ${\displaystyle t}$ is calculated, specified by the azimuth angle ${\displaystyle \theta }$ and distance ${\displaystyle d}$. The angle ${\displaystyle \theta }$ places the node between two adjacent quadrants ${\displaystyle i}$ and ${\displaystyle i+1}$, where ${\displaystyle quad(i)<\theta \leq quad(i+1)}$. For each storm parameter ${\displaystyle P}$ to be interpolated, its value at ${\displaystyle (\theta ,d)}$ are weighted between its values at two pseudo nodes ${\displaystyle (quad(i),d)}$ and ${\displaystyle (quad(i+1),d)}$:

${\displaystyle P(\theta ,d)={\frac {P(quad(i),d)(90-\theta )^{2}+P(quad(i+1),d)\theta ^{2}}{(90-\theta )^{2}+\theta ^{2}}}\quad }$ (20)

The distance ${\displaystyle d}$ then places each pseudo node between the radii of two adjacent isotaches in its quadrant, and the value at the pseudo node is interpolated using the inverse distance weighting (IDW) method:

${\displaystyle P(quad,d)=f_{34}P_{34}+f_{50}P_{50}+f_{64}P_{64}\quad }$ (21)

where ${\displaystyle P_{34},P_{50},P_{64}}$ are parameter values computed from the 34-, 50-, and 64-isotach, ${\displaystyle f_{34},f_{50},f_{64}}$ are distance weighting factors for each isotach, calculated as

${\displaystyle {\begin{array}{lll}\mathrm {I} .&r (22)

and ${\displaystyle f_{34}+f_{50}+f_{64}=1}$.

The above procedure is performed at each node of an ADCIRC grid. After all storm parameters are interpolated, the pressure and gradient winds can be calculated using (12)&(13). To bring the gradient wind down to the standard 10 meter reference level, the same wind reduction factor ${\displaystyle W_{rf}}$ is applied, and the tangential winds are rotated by an inward flow angle β according to (16). Then, the storm translational speed is added back to the vortex winds. Last but not least, a wind averaging factor is applied to convert resulted wind field from 1-min to 10-min averaged winds in order to be used by ADCIRC. This new composite wind method is simple and efficient, and more importantly, it assures that the constructed surface winds match all observed storm isotaches provided in NHC’s forecast or “best track” advisories.

## Case Studies

Preliminary evaluation of the GAHM was carried out based on seven hurricanes that struck the Gulf of Mexico and the Eastern United States: Katrina (2005), Rita (2005), Gustav (2008), Ike (2008), Irene (2011), Isaac (2012), and Sandy (2012), see Table 1. Ranging from category 1 to 5 on the Saffir-Simpson Hurricane Wind Scale, these storms vary in storm track, forward motion, size, intensity, and duration, but all caused severe damages to coastal states due to destructive winds, wind-induced storm surges, and ocean waves. Their “best track” advisories were retrieved from NHC’s ftp site (ftp://ftp.nhc.noaa.gov/atcf; previous years’ data are located in the archive directory) and pre-processed using the ASWIP program. The “best track” file contains an estimate of the radius to the maximum wind for each data entry, but will solely be used for model validation purpose as both the GAHM and HM calculate their own spatially-varying ${\displaystyle R_{max}}$.

Table 1. Seven selected hurricanes used for preliminary evaluation of the GAHM
Hurricane Saffir-Simpson Wind Scale Maximum Sustained Wind (knot) Minimum Central Pressure (mbar) Period from Formation to Dissipation
Katrina 5 150 902 08/23-08/30, 2005
Rita 5 150 902 09/18-09/26, 2005
Gustav 4 135 941 08/23-09/04, 2008
Ike 4 125 935 09/01-09/14, 2008
Irene 3 105 942 08/21-08/30, 2011
Isaac 1 70 965 08/21-09/03, 2012
Sandy 3 95 940 10/22-10/01, 2012

Besides the maximum wind speed, both Holland ${\displaystyle B}$ and ${\displaystyle R_{o}}$ can be used as key parameters to characterize the development of the storm. Figure 4 depicts the change of ${\displaystyle V_{M}}$, ${\displaystyle B}$, and ${\displaystyle \log _{10}R_{o}}$ during different stages of the hurricanes along their best tracks. Typically, both ${\displaystyle B}$ and ${\displaystyle R_{o}}$ increase as hurricane strengthens, and decrease as hurricane dissipates, within range of (0, 2.5). Previously via analytical evaluation we have demonstrated that the GAHM behaves consistently better than the HM, especially under situations where ${\displaystyle \log _{10}R_{o}<1}$. Here, evaluation of model performance will be carried out by comparing the modeled winds with the observed winds in the "best track" data, as well as the SLOSH (Sea, Lake, and Overland Surges from Hurricanes) winds, re-analysis H*Wind and hindcast OWI winds.