**Welcome to the official ADCIRCWiki site! The site is currently under construction, with limited information. ---**

# Difference between revisions of "Generalized Asymmetric Holland Model"

Luckygirl001 (talk | contribs) m (→Composite Wind Generation) |
Luckygirl001 (talk | contribs) m (→Calculation of the Radius to the Maximum Wind) |
||

Line 81: | Line 81: | ||

== Calculation of the Radius to the Maximum Wind == | == 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 | + | 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. |

− | First, the maximum sustained wind and the 34-, 50-, and/or 64-kt isotaches | + | As a standard procedure, for all data entries in the input file the <math> B_g </math> and <math> R_{max} </math> 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 | ||

<center> <math> V_{max} = \vert \frac{\overrightarrow{V_M} - \gamma\overrightarrow{V_T}}{W_{rf}} \vert = \frac{V_M - \gamma V_T}{W_{rf}} \quad </math> (14) </center> | <center> <math> V_{max} = \vert \frac{\overrightarrow{V_M} - \gamma\overrightarrow{V_T}}{W_{rf}} \vert = \frac{V_M - \gamma V_T}{W_{rf}} \quad </math> (14) </center> | ||

Line 93: | Line 95: | ||

which is the ratio of gradient wind speed to the maximum wind speed along a radial wind profile. Thus, <math> \gamma </math> is zero at storm center, and increases with <math> r </math> until reaches a maximum value of 1 at <math> R_{max} </math>, then gradually decreases outward to zero. | which is the ratio of gradient wind speed to the maximum wind speed along a radial wind profile. Thus, <math> \gamma </math> is zero at storm center, and increases with <math> r </math> until reaches a maximum value of 1 at <math> R_{max} </math>, 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 <math> \beta </math> according to the Queensland Government's Ocean Hazards Assessment (2001): | |

− | |||

− | |||

− | |||

− | |||

<center> <math> \beta = \begin{cases} | <center> <math> \beta = \begin{cases} | ||

Line 103: | Line 101: | ||

10^{\circ} + 75(r-R_{max})/R_{max}, & R_{max} \le r<1.2R_{max} \\ | 10^{\circ} + 75(r-R_{max})/R_{max}, & R_{max} \le r<1.2R_{max} \\ | ||

25^{\circ}, & r \ge 1.2R_{max} | 25^{\circ}, & r \ge 1.2R_{max} | ||

− | \end{cases} \quad (17) </math> </ | + | \end{cases} \quad (16) </math> </center> |

+ | |||

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

+ | |||

+ | <center> <math> \begin{align} | ||

+ | 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{align} \quad </math> (17) </center> | ||

+ | |||

+ | |||

+ | |||

+ | where <math> \overrightarrow{V_{isot}} </math> is the observed isotach wind speed with an unknown angle <math> \varepsilon </math>, and <math> \overrightarrow{V_{inflow}} </math> is the wind speed at radius to specified isotach before the inward rotation angle <math> \beta </math> is removed. | ||

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

<center> <math> V_r\cos(quad(i)+90+\beta)=V_{isot}\cos(\varepsilon)-\gamma {\mu}_T \quad </math> (18) </center> | <center> <math> V_r\cos(quad(i)+90+\beta)=V_{isot}\cos(\varepsilon)-\gamma {\mu}_T \quad </math> (18) </center> |

## Revision as of 07:04, 8 April 2020

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.

## Contents

## 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:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(r) = P_c + (P_n - P_c)e^{-A/r^B} \quad }**(1)

where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_c }**
is the central pressure, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_c }**
is the ambient pressure (theoretically at infinite radius), **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(r) }**
is the pressure at radius **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r }**
from the center of the hurricane, and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A }**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_g(r) }**
is the gradient wind at radius **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r }**
, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho }**
is the density of air, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_c(r) = \sqrt{AB(P_n - P_c)e^{-A/r^B}/\rho r^B} \quad }**(3)

By setting **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dV_c/dr = 0 }**
at radius to the maximum wind **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = R_{max} }**
, it is obtained that

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = (R_{max})^B \quad }**(4)

Thus the (**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{max} }**
) is irrelevant to the relative value of ambient and central pressures, and is solely defined by the shape parameters **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A }**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B }**
. Substituting (4) back into (3) to get rid of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A }**
, we get an estimate of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B }**
as a function of the maximum wind speed

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{max} }**
), given a constant pressure drop. It was also reasoned by Holland that a plausible range of

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(r) = P_c + (P_n - P_c)e^{-(R_{max}/r)^B} \quad }**(6)

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_g(r) = \sqrt{(V_{max})^2e^{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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{max} }**
and shape parameter **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(r) }**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_g(r) }**
along the radius **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = R_{max} }**
, defined as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_o = \frac{Nonlinear Acceleration}{Coriolis force} ~ \frac{V_{max}^2/R_{max}}{V_{max}f} = \frac{V_{max}}{R_{max}f} \quad }**(8)

By definition, a large **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_o }**
, but not applicable for a weak and broad hurricane with a small **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dV_g/dr = 0 }**
at **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = R_{max} }**
to get the adjusted shape parameter **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_g }**
as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\varphi} }**
is a scaling parameter introduced to simplify the derivation process, defined as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi = \frac{A}{R_{max}^B} \quad }**or

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \quad A = \varphi R_{max}^B \quad }**(10)

and later derived as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\varphi} }**
, as Equation (11) indicates that **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\varphi} \ge 1 }**
. Numerical solutions for **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_g }**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\varphi} }**
can be solved iteratively in the model using Equation (9)&(11). Figure 1 illustrates how **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_g/B }**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi }**
vary with **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_{10}R_o }**
given different **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_g/B }**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi }**
remain close to 1 when **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_{10}R_o }**
is within the range of [1,2], but increase noticeably as **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_{10}R_o }**
decreases below 1, and the smaller the value of

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_g/B }**(left panel) and

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi }**(right panel) with respect to

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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(r) = P_c + (P_n - P_c)e^{-\varphi(R_{max}/r)^{B_g}} \quad }**(12)

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_g(r) = \sqrt{V_{max}^2(1+1/R_o)e^{1-(R_{max}/r)^{B_g}}(R_{max}/r)^B_g + (\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_o }**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=0 }**
, which corresponds to an infinitely large

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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r/R_{max} }**
, the Holland **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_g = V_{max} }**
at **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r/R_{max} = 1 }**
on the plane of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_g/V_{max} = 1 }**
, no matter what values of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_g/V_{max} =1 }**
as

To have a dissective look of the surface plots in Figure 2, we draw slices perpendicular to the axis of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_g = V_{max} }**
at **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_{10}R_o =0, 1, 2 }**(or correspondingly

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_g }**
and

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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overrightarrow{V_M} }**
is the reported maximum sustained wind at 10 meter height assuming in the same direction as **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overrightarrow{V_T} }**
, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overrightarrow{V_T} }**
is the storm translational speed calculated from successive storm center locations, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma }**
is the damp factor for **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_T }**
. The following formula of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma }**
is employ in the ASWIP program:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma }**
is zero at storm center, and increases with

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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta }**
according to the Queensland Government's Ocean Hazards Assessment (2001):

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta = \begin{cases} 10^{\circ}, & r<R_{max} \\ 10^{\circ} + 75(r-R_{max})/R_{max}, & R_{max} \le r<1.2R_{max} \\ 25^{\circ}, & r \ge 1.2R_{max} \end{cases} \quad (16) }**

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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} 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{align} \quad }**(17)

where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overrightarrow{V_{isot}} }**
is the observed isotach wind speed with an unknown angle **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon }**
, and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overrightarrow{V_{inflow}} }**
is the wind speed at radius to specified isotach before the inward rotation angle **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta }**
is removed.

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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_r\cos(quad(i)+90+\beta)=V_{isot}\cos(\varepsilon)-\gamma {\mu}_T \quad }**(18)

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_r\sin(quad(i)+90+\beta)=V_{isot}\sin(\varepsilon)-\gamma {\nu}_T \quad }**(19)

where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle quad(i) }**
is the azimuth angle of the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i-th }**
storm quadrant (NE, SE, SW, NW at **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 45^\circ, 135^\circ, 225^\circ, 315^\circ }**
, respectively), **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{isot}\cos(\varepsilon) }**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{isot}\sin(\varepsilon) }**
are the zonal and meridional components of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overrightarrow{V_{isot}} }**
, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mu}_T }**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\nu}_T }**
are the zonal and meridional components of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overrightarrow{V_T} }**
.

Given an initial guess of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_r }**
can be estimated by combining (15), (17), (18), and (19). Plugging **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{max} }**
from (14), the above calculated **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_g, \varphi, V_{max}, V_r }**
and the observed radius **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_r }**
to **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_r }**
back into (13), a new

In case where multiple isotaches are given in the forecast/best track advisories, the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta }**
following the above process only once. The use of the pseudo

Occasionally, we have to deal with situations where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{max} < V_r }**
, which violate (13) so **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{max} = V_r }**
, which is equivalent to assigning **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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

The GAHM uses radially and azimuthally varying storm parameters to construct its spatial pressure and wind fields for a hurricane vortex in ADICRC. Since storm parameters are only given in 4 storm quadrants (assuming at **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 45^\circ, 135^\circ, 225^\circ, 315^\circ }**
azimuthal angles, respectively) at 3 available isotaches in the input file, spatial interpolation must take place first at each ADCIRC grid node. Traditionally, only the highest isotach is used for storm parameter interpolation (the single-isotach approach). To take advantage of the availability of multiple isotaches, a new composite wind method is introduced in the GAHM (the multiple-isotach approach).

To begin, the relative location of a node to the storm center at given time **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t }**
is calculated, specified by the azimuth angle **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta }**
and distance **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d }**
. The angle **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta }**
places the node between two adjacent quadrants **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i }**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i+1 }**
, where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle quad(i) < \theta \le quad(i+1) }**
. For each storm parameter **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P }**
to be interpolated, its value at **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\theta,d) }**
are weighted between its values at two pseudo nodes **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (quad(i),d) }**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (quad(i+1),d) }**
:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(quad,d) = f_{34}P_{34}+f_{50}P_{50}+f_{64}P_{64} \quad }**(21)

where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{34}, P_{50}, P_{64} }**
are parameter values computed from the 34-, 50-, and 64-isotach, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{34}, f_{50}, f_{64} }**
are distance weighting factors for each isotach, calculated as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{lll} \mathrm{I}. & r<R_{64}& f_{64}=1,f_{50}=0,f_{34}=0 \\ \mathrm{II}. & R_{64}\le r<R_{50} & f_{64}=(r-R_{64})/(R_{50}-R_{64}),f_{50}=(R_{50}-r)/(R_{50}-R_{64}), f_{34}=0 \\ \mathrm{III}. & R_{50}\le r<R_{34} & f_{64}=0,f_{50}=(r-R_{50}/(R_{34}-R_{50})),f_{34}=(R_{34}-r/(R_{34}-R_{50})) \\ \mathrm{IV}. & r \le R_{34} & f_{64}=0,f_{50}=0,f_{34}=1 \end{array} \quad }**(22)

and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{34}+f_{50}+f_{64}=1 }**
.

The above procedure is performed at each node of an ADCIRC grid, and after all storm parameters are interpolated the pressure and gradient wind can be calculated using (12)&(13).

Once the pressure and gradient wind are calculated at the gradient wind level, they are reduced back down to the standard 10 meter reference level, using the same wind reduction factor **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_{rf} }**
. Then the tangential winds are rotated by an inward flow angle β according to (16); and last but not least, the storm translational speed is added back to the vortex winds to obtain the final wind speed at 10 meter level. 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.