Models

Holland (1980) symmetric wind field

The Holland model (1980), widely used in the literature, is based on the gradient wind balance in mature tropical cyclones. The wind speed distribution is computed from the circular air pressure field, which can be derived from the central and environmental pressure and the radius of maximum winds.

\[ v_r = \sqrt{\frac{b}{\rho} \times \left(\frac{R_m}{r}\right)^b \times (p_{oci} - p_c) \times e^{-\left(\frac{R_m}{r}\right)^b} + \left(\frac{r \times f}{2}\right)^2} - \left(\frac{r \times f}{2}\right) \]

with, \[ b = \frac{\rho \times e \times v_m^2}{p_{oci} - p_c} \] \[ f = 2 \times 7.29 \times 10^{-5} \sin(\phi) \]

where,
\(v_r\) is the tangential wind speed (in \(m.s^{-1}\)),
\(b\) is the shape parameter,
\(\rho\) is the air density set to \(1.15 kg.m^{-3}\),
\(e\) is the base of natural logarithms (~2.718282),
\(v_m\) the maximum sustained wind speed (in \(m.s^{-1}\)),
\(p_{oci}\) is the pressure at outermost closed isobar of the storm (in \(Pa\)),
\(p_c\) is the pressure at the centre of the storm (in \(Pa\)),
\(r\) is the distance to the eye of the storm (in \(km\)),
\(R_m\) is the radius of maximum sustained wind speed (in \(km\)),
\(f\) is the Coriolis force (in \(N.kg^{-1}\)), and
\(\phi\) is the latitude.

Willoughby et al. (2006) symmetric wind field

The Willoughby et al. (2006) model is an empirical model fitted to aircraft observations. The model considers two regions: inside the eye and at external radii, for which the wind formulations use different exponents to better match observations. In this model, the wind speed increases as a power function of the radius inside the eye and decays exponentially outside the eye after a smooth polynomial transition across the eyewall (see also Willoughby (1995), Willoughby et al. (2004)).

\[ \left\{ \begin{aligned} v_r &= v_m \times \left(\frac{r}{R_m}\right)^{n} \quad if \quad r < R_m \\ v_r &= v_m \times \left((1-A) \times e^{-\frac{|r-R_m|}{X1}} + A \times e^{-\frac{|r-R_m|}{X2}}\right) \quad if \quad r \geq R_m \\ \end{aligned} \right. \]

with, \[ n = 2.1340 + 0.0077 \times v_m - 0.4522 \times \ln(R_m) - 0.0038 \times |\phi| \] \[ X1 = 287.6 - 1.942 \times v_m + 7.799 \times \ln(R_m) + 1.819 \times |\phi| \] \[ A = 0.5913 + 0.0029 \times v_m - 0.1361 \times \ln(R_m) - 0.0042 \times |\phi| \quad and \quad A\ge0 \]

where,
\(v_r\) is the tangential wind speed (in \(m.s^{-1}\)),
\(v_m\) is the maximum sustained wind speed (in \(m.s^{-1}\)),
\(r\) is the distance to the eye of the storm (in \(km\)),
\(R_m\) is the radius of maximum sustained wind speed (in \(km\)),
\(\phi\) is the latitude of the centre of the storm, and
\(X2 = 25\).

Adding asymmetry to Holland (1980) and Willoughby et al. (2006) wind fields

The asymmetry caused by the translation of the storm can be added as follows,

\(\vec{V} = \vec{V_c} + C \times \vec{V_t}\)

where,
\(\vec{V}\) is the combined, asymmetric wind field,
\(\vec{V_c}\) is symmetric wind field,
\(\vec{V_t}\) is the translation speed of the storm, and
\(C\) is function of \(r\), the distance to the eye of the storm (in \(km\)).

Two formulations of C proposed by Miyazaki et al. (1962) and Chen (1994) are implemented.

Boose et al. (2004) asymmetric model

The Boose et al. (2004) model, or “HURRECON” model, is a modification of the Holland (1980) model (see also Boose et al. (2001)). In addition to adding asymmetry, this model treats of water and land differently, using different surface friction coefficient for each.

Wind fields comparison

Here, we compare wind fields generated by different models that can be used in StormR for the same time and location (tropical cyclone Pam near Vanuatu)

sds <- defStormsDataset()

## Warning in checkInputsdefStormsDataset(filename, sep, fields, basin, seasons, : No basin argument specified. StormR will work as expected
##              but cannot use basin filtering for speed-up when collecting data

## === Loading data  ===
## Open database... /private/var/folders/v1/6y176c8d2jz06qr8rd3jq_480000gn/T/RtmpSAC0zx/Rinst11d381788fbb5/StormR/extdata/test_dataset.nc opened
## Collecting data ...
## === DONE ===

st <- defStormsList(sds = sds, loi = c(168.33, -17.73), names = "PAM", verbose = 0)
PAM <- getObs(st, name = "PAM")

pf <- spatialBehaviour(st, product = "Profiles", method = "Holland", asymmetry = "None", verbose = 0)
terra::plot(pf$PAM_Speed_41, main = "Holland (1980)", cex.main = 0.8, range = c(0, 90))
terra::plot(countriesHigh, add = TRUE)
lines(PAM$lon, PAM$lat, lty = 3)

pf <- spatialBehaviour(st, product = "Profiles", method = "Willoughby", asymmetry = "None", verbose = 0)
terra::plot(pf$PAM_Speed_41, main = "Willoughby et al. (2006)", cex.main = 0.8, range = c(0, 90))
terra::plot(countriesHigh, add = TRUE)
lines(PAM$lon, PAM$lat, lty = 3)

pf <- spatialBehaviour(st, product = "Profiles", method = "Holland", asymmetry = "Miyazaki", verbose = 0)
terra::plot(pf$PAM_Speed_41, main = "Holland (1980) + Miyazaki et al. (1962)", cex.main = 0.8, range = c(0, 90))
terra::plot(countriesHigh, add = TRUE)
lines(PAM$lon, PAM$lat, lty = 3)

pf <- spatialBehaviour(st, product = "Profiles", method = "Willoughby", asymmetry = "Miyazaki", verbose = 0)
terra::plot(pf$PAM_Speed_41, main = "Willoughby et al. (2006) + Miyazaki et al. (1962)", cex.main = 0.8, range = c(0, 90))
terra::plot(countriesHigh, add = TRUE)
lines(PAM$lon, PAM$lat, lty = 3)

pf <- spatialBehaviour(st, product = "Profiles", method = "Holland", asymmetry = "Chen", verbose = 0)
terra::plot(pf$PAM_Speed_41, main = "Holland (1980) + Chen (1994)", cex.main = 0.8, range = c(0, 90))
terra::plot(countriesHigh, add = TRUE)
lines(PAM$lon, PAM$lat, lty = 3)

pf <- spatialBehaviour(st, product = "Profiles", method = "Willoughby", asymmetry = "Chen", verbose = 0)
terra::plot(pf$PAM_Speed_41, main = "Willoughby et al. (2006) + Chen (1994)", cex.main = 0.8, range = c(0, 90))
terra::plot(countriesHigh, add = TRUE)
lines(PAM$lon, PAM$lat, lty = 3)

pf <- spatialBehaviour(st, product = "Profiles", method = "Boose", verbose = 0)
terra::plot(pf$PAM_Speed_41, main = "Boose et al. (2004)", cex.main = 0.8, range = c(0, 90))
terra::plot(countriesHigh, add = TRUE)
lines(PAM$lon, PAM$lat, lty = 3)

par(oldpar)