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# Generalized Asymmetric Holland Model: Difference between revisions

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

## The Classic Holland Model

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

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

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

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

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

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

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

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

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

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

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

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

${\displaystyle P(r)=P_{c}+(P_{n}-P_{c})e^{-(R_{max}/r)^{B}}\quad }$ (6)

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

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

${\displaystyle R_{0}={\frac {NonlinearAcceleration}{Coriolisforce}}~{\frac {V_{max}^{2}/R_{max}}{V_{max}f}}={\frac {V_{max}}{R_{max}f}}\quad }$ (8)

By definition, a large ${\displaystyle R_{0}(\approx 10^{3})}$ 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 ${\displaystyle (\approx 10^{-2}\sim 10^{2})}$ signifies a system in geostrophic balance where the Coriolis force plays an important role, such as the outer region of a hurricane. As a result, the assumption of cyclostrophic balance at ${\displaystyle R_{max}}$ made in HM is mostly valid for describing an intense and narrow (small ${\displaystyle R_{max}}$) hurricane with a large ${\displaystyle R_{0}}$, but not applicable for a weak and broad hurricane with a small ${\displaystyle R_{0}}$. 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 ${\displaystyle A}$ and ${\displaystyle B}$ as in the HM. Without assuming cyclostrophic balance at ${\displaystyle R_{max}}$), we take ${\displaystyle dV_{g}/dr=0}$ at ${\displaystyle r=R_{max}}$ to get the adjusted shape parameter ${\displaystyle B_{g}}$ as

${\displaystyle B_{g}={\frac {(V_{max}^{2}+V_{max}R_{max}f)\rho e^{\phi }}{\phi (P_{n}-P_{c})}}=B{\frac {(1+1/R_{0})e^{\phi -1}}{\phi }}\quad }$ (9)

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

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

and later derived as

${\displaystyle \phi =1+{\frac {V_{max}R_{max}f}{B_{g}(V_{max}^{2}+V_{max}R_{max}f)}}=1+{\frac {1/R_{0}}{B_{g}(1+1/R_{0})}}\quad }$ (11)

Thus, the ${\displaystyle R_{max}}$ in the GAHM is not entirely defined by the shape parameters ${\displaystyle A}$ and ${\displaystyle B}$ as in the HM, but also by the scaling factor ${\displaystyle {\phi }}$, as Equation (11) indicates that ${\displaystyle {\phi }\geq 1}$. Numerical solutions for ${\displaystyle B_{g}}$ and ${\displaystyle {\phi }}$ can be solved iteratively in the model using Equation (9)&(11). Figure 1 illustrates how ${\displaystyle B_{g}/B}$ and ${\displaystyle \phi }$ vary with ${\displaystyle \log _{10}R_{0}}$ given different ${\displaystyle B}$ values. It is evident that values of both ${\displaystyle B_{g}/B}$ and ${\displaystyle \phi }$ remain close to 1 when ${\displaystyle \log _{10}R_{0}}$ is within the range of [1,2], but increase noticeably as ${\displaystyle \log _{10}R_{0}}$ decreases below 1, and the smaller the value of ${\displaystyle B}$, the bigger the changes.

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

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

${\displaystyle V_{g}(r)={\sqrt {V_{max}^{2}(1+1/R_{0})e^{1-(R_{max}/r)_{g}^{B}}(R_{max}/r)_{g}^{B}+({\frac {rf}{2}})^{2}}}-{\frac {rf}{2}}\quad }$ (13)