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# Difference between revisions of "Generalized Asymmetric Holland Model"

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− | The Generalized Asymmetric Holland Model | + | <!-- markdown file for GAHM on ADCIRCWIKI--> |

+ | 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. | ||

− | + | <!-- ********************* chapter 1 ******************* --> | |

− | |||

− | |||

== The Classic Holland Model == | == 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: | ||

+ | |||

+ | <math> P(r) = P_c + (P_n - P_c)e^{-A/r^B} \quad </math> (1) | ||

+ | |||

+ | where <math> P_c </math> is the central pressure, <math> P_c </math> is the ambient pressure (theoretically at infinite radius), <math> P(r) </math> is the pressure at radius <math> r </math> from the center of the hurricane, and <math> A </math> and <math> B </math> 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: | ||

+ | |||

+ | <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) | ||

+ | |||

+ | where <math> V_g(r) </math> is the gradient wind at radius <math> r </math>, <math> \rho </math> is the density of air, <math> f </math> 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 | ||

+ | |||

+ | <math> V_c(r) = \sqrt{AB(P_n - P_c)e^{-A/r^B}/\rho r^B} \quad </math> (3) | ||

+ | |||

+ | By setting <math> dV_c/dr = 0 </math> at radius to the maximum wind <math> r = R_{max} </math>, it is obtained that | ||

+ | |||

+ | <math> A = (R_{max})^B \quad </math> (4) | ||

+ | |||

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

+ | |||

+ | <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 | ||

+ | |||

+ | <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) | ||

+ | |||

+ | 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) | ||

+ | |||

+ | 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). | ||

+ | |||

+ | <!-- ********************* chapter 2 ******************* --> | ||

== Derivation of the GAHM == | == Derivation of the GAHM == | ||

− | |||

− | |||

− | |||

− | |||

− | |||

− | The | + | 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) | ||

+ | |||

+ | where <math> {\phi} </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) | ||

− | + | and later derived as | |

− | <math> | + | <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> | + | 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. |

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

− | <math> | + | <math> P(r) = P_c + (P_n - P_c)e^{-\phi(R_{max}/r)^B_g} \quad </math> (12) |

− | <math> | + | <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) |

− | |||

− | |||

− | |||

− | < | + | <!-- ********************* chapter 3 ******************* --> |

+ | == Azimuthally-Varying RMW == | ||

− | |||

− | |||

− | |||

− | |||

− | < | + | <!-- ********************* chapter 4 ******************* --> |

− | + | == A Linearly-weighted Composite Wind Method == | |

− | |||

− | |||

− | |||

− | < | + | <!-- ********************* chapter 5 ******************* --> |

− | + | == Case Studies == | |

+ | === Single-Isotach Approach === | ||

+ | === Multiple-Isotach Approach === |

## Revision as of 02:58, 3 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 RMW }**
), 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_0 = \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_0 (\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_0 }**
, 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_0 }**
. 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^\phi}{\phi(P_n - P_c)} = B \frac{(1+1/R_0)e^{\phi - 1}}{\phi} \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 {\phi} }**
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 \phi = \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 = \phi 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 \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 }**
(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 {\phi} }**
, 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 {\phi} \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 {\phi} }**
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 \phi }**
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_0 }**
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 \phi }**
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_0 }**
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_0 }**
decreases below 1, and the smaller the value of

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^{-\phi(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_0)e^{1-(R_{max}/r)^B_g}(R_{max}/r)^B_g + (\frac{rf}{2})^2} - \frac{rf}{2} \quad }**
(13)