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Difference between revisions of "Generalized Asymmetric Holland Model"

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<math> B = (V_{max})^2\rho e/(P_n - P_c) \quad </math> (5)
 
<math> B = (V_{max})^2\rho e/(P_n - P_c) \quad </math> (5)
  
It was notable that the maximum wind speed is proportional to the square root of <math> B </math> and irrespective of the (<math> RMW </math>), given a constant pressure drop. It was also reasoned by Holland that a plausible range of <math> B </math> 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
+
It was notable that the maximum wind speed is proportional to the square root of <math> B </math> and irrespective of the (<math> R_{max} </math>), given a constant pressure drop. It was also reasoned by Holland that a plausible range of <math> B </math> 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
  
 
<math> P(r) = P_c + (P_n - P_c)e^{-(R_{max}/r)^B} \quad </math> (6)
 
<math> P(r) = P_c + (P_n - P_c)e^{-(R_{max}/r)^B} \quad </math> (6)
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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 <math> R_{max} </math> and shape parameter <math> B </math> may be estimated by fitting data into the radial wind equation, which in turn allow us to compute <math> P(r) </math> and <math> V_g(r) </math> along the radius <math> r </math> of the hurricane. However, discrepancies between wind observations and computed winds were sometimes found, and were negatively correlated to the Rossby number at <math> r = R_{max} </math>, defined as
 
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 <math> R_{max} </math> and shape parameter <math> B </math> may be estimated by fitting data into the radial wind equation, which in turn allow us to compute <math> P(r) </math> and <math> V_g(r) </math> along the radius <math> r </math> of the hurricane. However, discrepancies between wind observations and computed winds were sometimes found, and were negatively correlated to the Rossby number at <math> r = R_{max} </math>, defined as
  
<math> R_0 = \frac{Nonlinear Acceleration}{Coriolis force} ~ \frac{V_{max}^2/R_{max}}{V_{max}f} = \frac{V_{max}}{R_{max}f} \quad </math> (8)
+
<math> R_o = \frac{Nonlinear Acceleration}{Coriolis force} ~ \frac{V_{max}^2/R_{max}}{V_{max}f} = \frac{V_{max}}{R_{max}f} \quad </math> (8)
  
By definition, a large <math> R_0 (\approx 10^3) </math> 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 <math> (\approx 10^{-2} \sim 10^2) </math> 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 <math> R_{max} </math> made in HM is mostly valid for describing an intense and narrow (small <math> R_{max} </math>) hurricane with a large <math> R_0 </math>, but not applicable for a weak and broad hurricane with a small <math> R_0 </math>. 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).   
+
By definition, a large <math> R_o (\approx 10^3) </math> 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 <math> (\approx 10^{-2} \sim 10^2) </math> 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 <math> R_{max} </math> made in HM is mostly valid for describing an intense and narrow (small <math> R_{max} </math>) hurricane with a large <math> R_o </math>, but not applicable for a weak and broad hurricane with a small <math> R_o </math>. 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).   
  
 
<!--  *********************  chapter  2 ******************* -->
 
<!--  *********************  chapter  2 ******************* -->
 
== Derivation of the GAHM ==
 
== Derivation of the GAHM ==
  
The GAHM also starts with the same radial pressure and wind equations (1)&(2) with shape parameters <math> A </math> and <math> B </math> as in the HM. Without assuming cyclostrophic balance at <math> R_{max} </math>), we take <math> dV_g/dr = 0 </math> at <math> r = R_{max} </math> to get the adjusted shape parameter <math> B_g </math> as
+
The GAHM also starts with the same radial pressure and wind equations (1)&(2) with shape parameters <math> A </math> and <math> B </math> as in the HM. Without assuming cyclostrophic balance at <math> R_{max} </math>, we take <math> dV_g/dr = 0 </math> at <math> r = R_{max} </math> to get the adjusted shape parameter <math> B_g </math> as
  
<math> B_g = \frac{(V_{max}^2 + V_{max}R_{max}f)\rho e^\phi}{\phi(P_n - P_c)} = B \frac{(1+1/R_0)e^{\phi - 1}}{\phi} \quad </math> (9)   
+
<math> 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 </math> (9)   
  
where <math> {\phi} </math> is a scaling parameter introduced to simplify the derivation process, defined as
+
where <math> {\varphi} </math> is a scaling parameter introduced to simplify the derivation process, defined as
  
<math> \phi = \frac{A}{R_{max}^B} \quad </math> or   <math> A = \phi R_{max}^B \quad </math> (10)
+
<math> \varphi = \frac{A}{R_{max}^B} \quad </math> or <math> \quad A = \varphi R_{max}^B \quad </math> (10)
  
 
and later derived as
 
and later derived as
  
<math> \phi = 1 + \frac{V_{max}R_{max}f}{B_g(V_{max}^2+V_{max}R_{max}f)} = 1 + \frac{1/R_0}{B_g(1+1/R_0)}  \quad </math> (11)
+
<math> \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 </math> (11)
  
Thus, the <math> R_{max} </math> in the GAHM is not entirely defined by the shape parameters <math> A </math> and <math> B </math> as in the HM, but also by the scaling factor <math> {\phi} </math>, as Equation (11) indicates that <math> {\phi} \ge 1 </math>. Numerical solutions for <math> B_g </math> and <math> {\phi} </math> can be solved iteratively in the model using Equation (9)&(11). Figure 1 illustrates how <math> B_g/B </math> and <math> \phi </math> vary with <math> \log_{10}R_0 </math> given different <math> B </math> values. It is evident that values of both <math> B_g/B </math> and <math> \phi </math> remain close to 1 when <math> \log_{10}R_0 </math> is within the range of [1,2], but increase noticeably as <math> \log_{10}R_0 </math> decreases below 1, and the smaller the value of <math> B </math>, the bigger the changes.
+
Thus, the <math> R_{max} </math> in the GAHM is not entirely defined by the shape parameters <math> A </math> and <math> B </math> as in the HM, but also by the scaling factor <math> {\varphi} </math>, as Equation (11) indicates that <math> {\varphi} \ge 1 </math>. Numerical solutions for <math> B_g </math> and <math> {\varphi} </math> can be solved iteratively in the model using Equation (9)&(11). Figure 1 illustrates how <math> B_g/B </math> and <math> \varphi </math> vary with <math> \log_{10}R_o </math> given different <math> B </math> values. It is evident that values of both <math> B_g/B </math> and <math> \varphi </math> remain close to 1 when <math> \log_{10}R_o </math> is within the range of [1,2], but increase noticeably as <math> \log_{10}R_o </math> decreases below 1, and the smaller the value of <math> B </math>, the bigger the changes.
  
 
Substituting (9)&(11) back into (1)&(2) yields the final radial pressure and wind equations for the GAHM
 
Substituting (9)&(11) back into (1)&(2) yields the final radial pressure and wind equations for the GAHM
  
<math> P(r) = P_c + (P_n - P_c)e^{-\phi(R_{max}/r)^B_g} \quad </math> (12)
+
<math> P(r) = P_c + (P_n - P_c)e^{-\varphi(R_{max}/r)^{B_g}} \quad </math> (12)
  
<math> V_g(r) = \sqrt{V_{max}^2(1+1/R_0)e^{1-(R_{max}/r)^B_g}(R_{max}/r)^B_g + (\frac{rf}{2})^2} - \frac{rf}{2} \quad </math> (13)
+
<math> 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 </math> (13)
  
 +
Influence of the Coriolis force on the radial pressure and wind profiles are evidenced by the presence of <math> R_o </math> and <math> \varphi </math> in (12)&(13). A special case scenario is when we set <math> f=0 </math>, which corresponds to an infinitely large <math> R_o </math>, then (12)&(13) in the GAHM reduce to (6)&(7) in the HM. However,for a hurricane with a relatively small <math> R_o </math>, 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 <math> r/R_{max} </math>, the Holland <math> B </math> parameter, and <math> R_o </math>. In both panels, each colored surface represents the normalized gradient winds corresponding to a unique Holland B value. By definition, we get <math> V_g = V_{max} </math> at <math> r = R_{max} </math>, which means all the surfaces in each panel should intersect with the plane of <math> r/R_{max} = 1 </math> on the plane of <math> V_g/V_{max} = 1 </math>, no matter what values of <math> R_o </math>. However, the line of intersection (shown by the black line) shown in the left panel deviates from the plane of <math> V_g/V_{max} =1 </math> as <math> \log_{10}R_o </math> decreases from 2 to close to 0 (<math> R_o </math> decreases from 100 to 1), while remains on the plane regardless of how <math> R_o </math> changes in the right panel, demonstrating that the GAHM is mathematically more coherent than the HM. 
 +
 +
To have a dissective look of the surface plots in Figure 2, we draw slices perpendicular to the axis of <math> \log_{10}R_o </math> 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 <math> V_g = V_{max} </math> at <math> r = R_{max} </math> consistently in the right panel for the GAHM regardless of the value of <math> R_o </math>. The HM in the left panel, however, generates distorted wind profiles with underestimated maximum winds skewed inward towards the storm center, espeically when <math> \log_{10}R_o < 1 </math>. As a results, when both models being applied to real hurricane cases, the GAHM will perform more consistently than the HM.
  
  
  
 
<!--  *********************  chapter  3 ******************* -->
 
<!--  *********************  chapter  3 ******************* -->
== Azimuthally-Varying RMW ==
+
== Calculation of the <math> R_{max} </math> ==
 +
 
 +
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) as input files, which contain storm information 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. As a standard procedure, the quadrant-varying  <math> B_g </math> and <math> R_{max} </math> are pre-computed in the ASWIP program (an external FORTRAN program developed by Flemming et al. and further developed in this study to accommodate the GAHM) prior to running an ADCIRC simulation forced with the GAHM wind model.
 +
 
 +
First, the maximum sustained wind and the 34-, 50-, and/or 64-kt isotaches in NHC’s forecast or “best track” advisories, normally reported at 10 meter height, must be scaled to the gradient wind level to remove the influence of the boundary layer effect. Practically, the maximum gradient wind can be directly calculated as
 +
<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)
 +
 
 +
where <math> \overrightarrow{V_M} </math> is the reported maximum sustained wind at 10 meter height assuming in the same direction as <math> \overrightarrow{V_T} </math>, <math> \overrightarrow{V_T} </math> is the storm translational speed calculated from successive storm center locations, <math> W_{rf} = 0.9 </math> 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 <math> \gamma </math> is the damp factor for <math> V_T </math>. The following formula of <math> \gamma </math> is employ in the ASWIP program:
 +
 
 +
<math> \gamma = \frac{V_g}{V_{max}} \quad </math> (15)
 +
 
 +
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. 
 +
 
 +
The gradient wind speed at the radii to specified storm isotaches in different storm quadrants can be calculated from the observed isotaches similarly as
 +
 
 +
<math> V_r = \vert \overrightarrow{V_r}\vert = \frac{\vert\overrightarrow{V_{isot}} - \gamma\overrightarrow{V_T}  \vert} \quad </math> (16)
 +
 
 +
where <math> \overrightarrow{V_{isot}} </math> is the observed isotach wind speed with an unknown angle <math> \varepsilon </math>, and <math> \overrightarrow{V_r} </math> is the gradient wind speed with an inward rotation angle  <math> \beta </math>, defined as Equation (19) according to the Queensland Government's Ocean Hazards Assessment (2001):
 +
 
 +
<math> \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  </math>    (17)
 +
 
 +
 
 +
Rewriting (16) in x- and y-components yields:
 +
 
 +
<math> V_r\cos(angle(i)+90+\beta)=V_{isot}\cos(\varepsilon)-\gamma {\mu}_T \quad </math>  (18)
 +
 
 +
<math> V_r\sin(angle(i)+90+\beta)=V_{isot}\sin(\varepsilon)-\gamma {\nu}_T \quad </math>  (19)
  
 +
where <math> angle(i) </math> is the <math> i-th </math> of the NE, SE, SW, NW storm quadrants at <math>  45^\circ, 135^\circ, 225^\circ, 315^\circ </math>, <math> V_{isot}\cos(\varepsilon) </math> and <math> V_{isot}\sin(\varepsilon) </math> are the zonal and meridional components of <math> \overrightarrow{V_{isot}} </math>, <math> {\mu}_T </math> and <math> {\nu}_T </math> are the zonal and meridional components of <math> \overrightarrow{V_T} </math>.
  
 +
Given an initial guess of <math> R_{max} </math>, values of <math> B_g </math> and <math> \varphi </math> can be solved iteratively from (9) and (11) until both converge, and <math> V_r </math> can be estimated by combining (15), (17), (18), and (19). Plugging <math> V_{max} </math> from (14), the above calculated <math> B_g, \varphi, V_{max}, V_r </math> and the radius <math> r </math> at <math> V_r </math> back into (13), a new <math> R_{max} </math> can be inversely solved by a root-finding algorithm. Since the above calculations are carried out based on an initial guess of <math> R_{max} </math>, wWe need to repeat the entire process until <math> R_{max} </math> converges.
  
 +
In case where multiple isotaches are given in the forecast/best track advisories, the <math> R_{max} </math> for the highest isotach will be calculated using the above procedure, and used as the pseudo <math> R_{max} </math> for the entire storm (physically, there is only one <math> R_{max} </math> found along a radial wind profile ). For each lower isotach, <math> R_{max} </math> will be calculated with the pseudo <math> R_{max} </math> set as its initial value to determine the inward rotation angle <math> \beta </math> following the above process only once. The use of the pseudo <math> R_{max} </math> 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 <math> V_{max} < V_T </math>, which violate the (13) so <math> R_{max} </math> couldn't be calculated   
  
  
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=== Single-Isotach Approach ===
 
=== Single-Isotach Approach ===
 
=== Multiple-Isotach Approach ===
 
=== Multiple-Isotach Approach ===
 +
 +
 +
 +
<math>  </math>
 +
 +
<math> \frac{\nu^2}{r} + f\nu - \frac{1}{\rho_0}\dfrac{\partial p}{\partial r} = 0 \quad</math> ()
 +
 +
<math> P(r) = P_c + (P_n - P_c)e^{-A/r^B} \quad </math> (1)
 +
 +
<math> V_g(r) = \sqrt{AB(P_n - P_c)e^{-A/r^B}/\rho r^B + (\frac{rf}{2})^2} - \frac{rf}{2} \quad </math> (2)
 +
 +
<math> V_c(r) = \sqrt{AB(P_n - P_c)e^{-A/r^B}/\rho r^B} \quad </math> (3)
 +
 +
<math> A = (R_{max})^B \quad </math> (4)
 +
 +
<math> B = (V_{max})^2\rho e/(P_n - P_c) \quad </math> (5)
 +
 +
<math> P(r) = P_c + (P_n - P_c)e^{-(R_{max}/r)^B} \quad </math> (6)
 +
 +
<math> V_g(r) = \sqrt{(V_{max})^2e^{1-(R_{max}/r)^B}(R_{max}/r)^B + (\frac{rf}{2})^2} - \frac{rf}{2} \quad </math> (7)
 +
 +
<math> R_o = \frac{V_{max}}{R_{max}f} \quad </math> (8)
 +
 +
<math> 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 </math> (9)
 +
 +
<math> \varphi = \frac{A}{R_{max}^B} \quad </math> or  <math> A = \varphi R_{max}^B \quad </math> (10)
 +
 +
<math> \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 </math> (11)
 +
 +
<math> P(r) = P_c + (P_n - P_c)e^{-\varphi(R_{max}/r)^B_g} \quad </math> (12)
 +
 +
<math> 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 </math> (13)
 +
 +
<math> V_{max} = \frac{V_M - \gammaV_T}{W_{rf}} \quad </math> (14)
 +
 +
<math> \gamma = \frac{V_g}{V_{max}} \quad </math> (15)
 +
 +
<math> V_r = \vert \overrightarrow{V_{r\_inflow}\vert = \frac{\vert\overrightarrow{V_{isot}} - \gamma\overrightarrow{V_T}  \vert} \quad </math> (16)
 +
 +
<math> \quad </math>
 +
 +
<math> \quad </math>
 +
 +
<math> </math>
 +
 +
<math> \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}
 +
\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}=0,f_{34}=1
 +
\end{array}\quad </math>
 +
 +
 +
<math> \beta = \left\{\begin{matrix}
 +
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{matrix}\right \quad </math>

Revision as of 01:16, 6 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.

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 B } and irrespective 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} } ), 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 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

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 B } may be estimated by fitting data into the radial wind equation, which in turn allow us to compute 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_{max} } made in HM is mostly valid for describing an intense and narrow (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_{max} } ) hurricane with 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 } , 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 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 } as in the HM. Without assuming 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_{max} } , we take 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 R_{max} } in the GAHM is not entirely 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 } as in the HM, but also by the scaling 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 {\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 } values. It is evident that values of both 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 } , the bigger the changes.

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 \varphi } in (12)&(13). A special case scenario is when we set 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 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 } , then (12)&(13) in the GAHM reduce to (6)&(7) in the HM. However,for a hurricane with a relatively 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 } , 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 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 B } parameter, 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 R_o } . In both panels, each colored surface represents the normalized gradient winds corresponding to a unique Holland B value. By definition, we get 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} } , which means all the surfaces in each panel should intersect with 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 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 R_o } . However, the line of intersection (shown by the black line) shown in the left panel deviates from 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 } 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 from 2 to close to 0 (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 } decreases from 100 to 1), while remains on the plane regardless of 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 R_o } changes in the right panel, demonstrating that the GAHM is mathematically more coherent than the HM.

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 \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 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} } consistently in the right panel for the GAHM regardless of 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 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 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.


Calculation 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} }

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) as input files, which contain storm information 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. As a standard procedure, the quadrant-varying 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 R_{max} } are pre-computed in the ASWIP program (an external FORTRAN program developed by Flemming et al. and further developed in this study to accommodate the GAHM) prior to running an ADCIRC simulation forced with the GAHM wind model.

First, the maximum sustained wind and the 34-, 50-, and/or 64-kt isotaches in NHC’s forecast or “best track” advisories, normally reported at 10 meter height, must be scaled to the gradient wind level to remove the influence of the boundary layer effect. 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 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 } until reaches a maximum value of 1 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_{max} } , then gradually decreases outward to zero.

The gradient wind speed at the radii to specified storm isotaches in different storm quadrants can be calculated from the observed isotaches similarly 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_r = \vert \overrightarrow{V_r}\vert = \frac{\vert\overrightarrow{V_{isot}} - \gamma\overrightarrow{V_T} \vert} \quad } (16)

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_r} } is the gradient wind speed 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 } , defined as Equation (19) 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 } (17)


Rewriting (16) 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(angle(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(angle(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 angle(i) } is 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 } of the NE, SE, SW, NW storm quadrants 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 } , 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 R_{max} } , 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 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 from (9) and (11) until both converge, 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_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 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 } 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 V_r } back into (13), a new 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} } can be inversely solved by a root-finding algorithm. Since the above calculations are carried out based on 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 R_{max} } , wWe need to repeat the entire process until 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} } converges.

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 R_{max} } for the highest isotach will be calculated using the above procedure, and used as the pseudo 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} } for the entire storm (physically, there is only one 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} } found along a radial wind profile ). For each lower 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 R_{max} } will be calculated with the pseudo 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} } set as its initial value to determine 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 } following the above process only once. The use of the pseudo 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} } 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 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_T } , which violate the (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 R_{max} } couldn't be calculated


A Linearly-weighted Composite Wind Method

Case Studies

Single-Isotach Approach

Multiple-Isotach Approach


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 \frac{\nu^2}{r} + f\nu - \frac{1}{\rho_0}\dfrac{\partial p}{\partial r} = 0 \quad} ()

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)

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)

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)

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)

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)

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)

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{V_{max}}{R_{max}f} \quad } (8)

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)

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 A = \varphi R_{max}^B \quad } (10)

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)

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)

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} = \frac{V_M - \gammaV_T}{W_{rf}} \quad } (14)

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)

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 = \vert \overrightarrow{V_{r\_inflow}\vert = \frac{\vert\overrightarrow{V_{isot}} - \gamma\overrightarrow{V_T} \vert} \quad } (16)

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 }

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 }


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} \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}=0,f_{34}=1 \end{array}\quad }


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 = \left\{\begin{matrix} 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{matrix}\right \quad }