Models the gradient wind speed at a certain radius from
a storm's center. To do this, it uses different equations and subfunctions
depending on how large the radius is (see details). This function requires,
as inputs, Willoughby wind model parameters calculated using the
add_wind_radii function.
will1(cdist, Rmax, R1, R2, vmax_gl, n, A, X1, X2 = 25)Distance (in km) from center of tropical cyclone to grid point.
Numeric vector of the radius at which the maximum wind occurs, in kilometers
A numeric vector of one of the parameters of the Willoughby model.
A numeric vector of one of the parameters of the Willoughby model.
Numeric vector of the tangential wind component of the maximum gradient wind speed, in meters per second
A numeric vector of one of the parameters of the Willoughby model.
A numeric vector of one of the parameters of the Willoughby model.
A numeric vector of one of the parameters of the Willoughby model.
A numeric vector of one of the parameters of the Willoughby model.
Returns a numeric vector with gradient wind speed at a radius of \(r\) from the storm's center, in meters per second.
If \(r \le R_1\), this function is calculating the equation:
$$V(r) = V_i = V_{max} \left( \frac{r}{R_{max}} \right)^n$$
where:
\(V(r)\): Maximum sustained gradient wind speed at a radius of \(r\) from the storm's center
\(r\): Radius from the storm center, in kilometers
\(V_{max,G}\): Maximum sustained gradient wind speed of the storm, in meters per second
\(R_1\): A parameter for the Willoughby wind model (radius to start of transition region)
\(R_{max}\): Radius (in kilometers) to highest winds
\(n\): A parameter for the Willoughby wind model
If \(R_2 < r\), this function is calculating the equation:
$$V(r) = V_o = V_{max}\left[(1 - A) e^\frac{R_{max} - r}{X_1} + A e^\frac{R_{max} - r}{X_2}\right]$$
where:
\(V(r)\): Maximum sustained gradient wind speed at a radius of \(r\) kilometers from the storm's center
\(r\): Radius from the storm center, in kilometers
\(V_{max,G}\): Maximum sustained gradient wind speed of the storm, in meters per second
\(R_{max}\): Radius (in kilometers) to highest winds
\(A\), \(X_1\), \(X_2\): Parameters for the Willoughby wind model
If \(R_1 < r \le R_2\), this function uses the equations:
$$\xi = \frac{r - R_1}{R_2 - R_1}$$
and, if \(0 \le \xi < \le 1\) (otherwise, \(w = 0\)):
$$w = 126 \xi^5 - 420 \xi^6 + 540 \xi^7- 315 \xi^8 + 70 \xi^9$$
and then:
$$V(r) = V_i (1 - w) + V_o w, (R_1 \le r \le R_2)$$
where, for this series of equations:
\(V(r)\): Maximum sustained gradient wind speed at a radius of \(r\) kilometers from the storm's center
\(r\): Radius from the storm center, in kilometers
\(w\): Weighting variable
\(R_1\), \(R_2\): Parameters for the Willoughby wind model
Willoughby HE, Darling RWR, and Rahn ME. 2006. Parametric representation of the primary hurricane vortex. Part II: A new family of sectionally continuous profiles. Monthly Weather Review 134(4):1102-1120.