Models
The main functions of the
StormR package, spatialBehaviour() and
temporalBehaviour(), allow to compute characteristics of
the storm surface wind field, as re-constructed from storm track data
and a parametric cyclone model. Three parametric models are implemented
in this package: Holland (1980),
Willoughby et al. (2006), and
Boose et al. (2004). The use of
one model or the other is defined using the method argument
in spatialBehaviour() and temporalBehaviour()
functions.
The original Holland (1980) and
Willoughby et al. (2006) models
provide a symmetrical wind field around the cyclone centre. However,
cyclonic winds are not symmetric, and an order zero asymmetry is caused
by the storm translation (forward motion). We therefore suggest using an
asymmetric version of the parametric wind fields that takes into account
storm motion. In the StormR package the methods developed
by Miyazaki et al. (1962) and
Chen (1994) that allow to take this
asymmetry into account can be used to adjust the outputs of the
symmetrical models accordingly. These can be activated by using the
asymmetry argument of the spatialBehaviour()
and temporalBehaviour() functions. The model of Boose
et al. (2004) is already an
asymmetrical version of the Holland (1980)
model. Contrary to the Holland (1980) and
Willoughby et al. (2006), this
model considers different parameter settings over water or over
lands.
By default the spatialBehaviour() and
temporalBehaviour() functions use the Willoughby et
al. (2006) model adjusted using the
Chen (1994) method.
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 × 7.29 × 10−5sin (ϕ)
where,
vr is the
tangential wind speed (in m.s−1),
b is the shape parameter,
ρ is the air density
set to 1.15kg.m−3,
e is the base of
natural logarithms (~2.718282),
vm the maximum
sustained wind speed (in m.s−1),
poci
is the pressure at outermost closed isobar of the storm (in Pa),
pc is the
pressure at the centre of the storm (in Pa),
r is the distance to the eye of the
storm (in km),
Rm is the
radius of maximum sustained wind speed (in km),
f is the Coriolis force (in N.kg−1),
and
ϕ 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 × vm − 0.4522 × ln (Rm) − 0.0038 × |ϕ| X1 = 287.6 − 1.942 × vm + 7.799 × ln (Rm) + 1.819 × |ϕ| A = 0.5913 + 0.0029 × vm − 0.1361 × ln (Rm) − 0.0042 × |ϕ| and A ≥ 0
where,
vr is the
tangential wind speed (in m.s−1),
vm is the
maximum sustained wind speed (in m.s−1),
r is the distance to the eye
of the storm (in km),
Rm
is the radius of maximum sustained wind speed (in km),
ϕ 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,
V⃗ 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.
Miyazaki et al. (1962)
$C = e^{(-\frac{r}{500} \times
\pi)}$
Chen (1994)
$C = \frac{3 \times R_m^{\frac{3}{2}} \times r^{\frac{3}{2}}}{R_m^3 + r^3 +R_m^{\frac{3}{2}} \times r^{\frac{3}{2}}}$
where,
Rm is the radius
of maximum sustained wind speed (in km).
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 speed
Wind speed is computed as follows,
$$ v_r = F\left(v_m - S \times (1 - \sin(T)) \times \frac{v_h}{2} \right) \times \sqrt{\left(\frac{R_m}{r}\right)^b \times e^{1 - \left(\frac{R_m}{r}\right)^b}} $$
with, $$ b = \frac{\rho \times e \times v_m^2}{p_{oci} - p_c} $$
where,
vr is the
tangential wind speed (in m.s−1),
F is a scaling parameter for
friction (1.0 in water, 0.8 in land),
vm is the
maximum sustained wind speed (in m.s−1),
S is a scaling parameter for
asymmetry (usually set to 1),
T is the oriented angle
(clockwise/counter clockwise in Northern/Southern Hemisphere) between
the forward trajectory of the storm and a radial line from the eye of
the storm to point r,
vh is the
storm velocity (in m.s−1),
Rm is the
radius of maximum sustained wind speed (in km),
r is the distance to the eye of the
storm (in km),
b is the shape parameter,
ρ = 1.15 is the air
density (in kg.m−3),
poci
is the pressure at outermost closed isobar of the storm (in Pa), and
pc is the
pressure at the centre of the storm (pressure
in Pa).
Wind direction
Wind direction is computed as follows,
$$
\left\{
\begin{aligned}
D = A_z - 90 - I \quad if \quad \phi > 0 \quad(Northern \quad
Hemisphere) \\
D = A_z + 90 + I \quad if \quad \phi \leq 0 \quad(Southern \quad
Hemisphere) \\
\end{aligned}
\right.
$$ where,
D is
the direction of the radial wind,
Az is the
azimuth from point r to the eye of the storm,
I is the cross isobar inflow angle
(20 in water, 40 in land), and
ϕ is the latitude.
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)
## 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... /tmp/RtmpyOK8nW/Rinstf75607dc40b/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)