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How to generate surface forcing for the SLOSH storm surge model

Riha, S.

1 Abstract

We review a numerical method to compute parametric hurricane wind profiles, first described by Jelesnianski and Taylor (1973). The profiles have been used for several decades to drive NOAA's operational SLOSH hurricane storm surge model. An ordinary differential equation for the inflow angle of surface winds is formulated, which contains a singularity at the origin. Jelesnianski and Taylor (1973) describe a numerical method that circumvents the complications arising from this singularity, allowing a stable, iterative computation of velocity, pressure and inflow angle for a given radius of maximum winds and pressure drop.

2 Introduction

The SLOSH model has been used for several decades by U.S. government agencies to estimate storm surge heights resulting from hurricanes. The model is of relatively low complexity compared to other numerical models (Jelesnianski et al., 1992). This makes it useful for probabilistic analysis, in which thousands of hypothetical storm events are simulated to obtain robust statistics. Each storm event is converted by SLOSH into a storm surge event, yielding a time-series of water heights in coastal areas. The simulation of a single storm surge event with SLOSH takes minutes or less on a single desktop computer. Lin et al. (2010) used the model for storm surge risk assessment on the city scale. It has also been used to generate hazard map data (Zachry et al., 2015), which we are using in one of our current projects for the insurance industry.

In this post we review a numerical method to compute hurricane wind fields for SLOSH, which was first described by Jelesnianski and Taylor (1973). The motivation arose from the necessity of validating our self-written SLOSH clone with the results published in the literature. A prerequisite for validation is the generation of identical forcing fields, i.e. wind stress and atmospheric pressure. The equations given in the SLOSH documentation ( Jelesnianski et al. 1992; Myers and Malkin 1961) are

$$V(r) = V_R\frac{2Rr}{R^2 + r^2}$$
$$ \frac{1}{\rho_a}\frac{dp}{dr} = \frac{k_sV^2}{\sin\phi} - V\frac{dV}{dr}$$
$$\frac{1}{\rho_a}\frac{dp}{dr} \cos\phi= fV +\frac{V^2}{r}\cos\phi-V^2\frac{d\phi}{dr}\sin\phi + k_n V^2$$

where \(V\) is the scalar surface wind speed as a function of the radial distance \(r\) from the storm center, \(V_R\) is the maximum surface wind speed, \(R\) is the radius of maximum winds, \(\rho_a\) is the air density\(f\) is the coriolis constant, and \(k_s\), \(k_n\) are tangential and radial friction coefficients. These equations are for a stationary, axisymmetric storm. Given any wind speed profile \(V\), the last two equations yield pressure \(p\) and the inflow angle \(\phi\). The pressure derivative can be eliminated to obtain an ordinary differential equation in \(\phi\). At first this looks quite simple and straightforward. However, when we actually tried to integrate these equations, we found that it's not quite so simple.

Conceptually, the appropriate initial conditions could be set at \(r=\infty\), where pressure is equal to the ambient pressure. Practically, this is difficult. Another possibility is setting initial conditions at \(r=0\), but note that the first equation above is undefined at \(r=0\). The mathematical model only describes two dimensional horizontal flow, and so it may not be surprising that there is a singularity in the center of the simplified model-hurricane, where in a real hurricane, the air moves mostly upward.

3 Approach of Jelesnianski and Taylor (1973)

After a day or two of headache, we reviewed the literature for any clues, and finally found that Jelesnianski and Taylor (1973) had already solved this problem. The solution comes in the form of a thorough discussion of the properties of the equations in their appendix C, focusing especially on numerical integration. They use a coordinate transformation \(w=Vru=Vr\cos\phi\) to obtain

$$G(w,r):=\frac{dw}{dr} = \frac{k_swVr}{\sqrt{V^2r^2-w^2}}-fr-k_nVr.$$

Figure 1 shows solutions \(w(r)\) for \(V_R\approx 160\, \rm km/h\) and \(R\approx 48.3\,\rm km\) at a latitude of Atlantic City, New Jersey. The black dashed curve shows \(Vr=Vr\cos 0\). All solutions (except the blue one) start close to zero and stay close together for some interval, until they rapidly diverge, either approaching \(Vr\) or zero.

Plot showing solutions of the variable w as function of the radial coordiante.
Figure 1: Various solutions \(w(r)\) for \(V_R\approx 160\, \rm km/h\) and \(R\approx 48.3\,\rm km\). The black dashed curve shows \(Vr=Vr\cos 0\), where \(G(w,r)\) is undefined. The pink, purple and green solutions are obtained by picking an initial value slightly smaller than \(Vr\), the brown, red and orange curves have initial values slightly larger than zero. All solutions are integrated backwards from the intial value towards zero. The initial value of the blue curve lies outside the figure domain. Integrating backwards, all solutions quickly converge towards the physically relevant solution (see text). Compare this figure with figure C3 of Jelesnianski and Taylor (1973).

Jelesnianski and Taylor (1973) argue that the only physically relevant solution is the one that neither terminates at \(Vr\), nor zero. Within the limits of figure Fig. 1, this corresponds to the blue curve. Since it is hard or impossible to find this solution (the blue curve shown in the figure does in fact approach \(Vr\) slightly to the right of figure bounds, which is not visible), the strategy is to pick an arbitrary curve that is reasonably close to the true solution up until a sufficient large value of \(r\). In practice, this may be the same value that is chosen to define the ambient pressure of the storm.

Plot showing wind speed, pressure difference and inflow angle as a function of the radial coordinate.
Figure 2: Wind speed, pressure difference to the ambient pressure \(\Delta p\), and inflow angle \(\phi\) as a function of radial distance from the stationary storm center. \(V_R\) and \(R\) are chosen approximately as in figure 2 of Jelesnianski et al. (1992). We use a latitude of about \(39.4^{\circ}N\).

4 Caveats

The study of Jelesnianski and Taylor (1973) is somewhat dated, even Jelesnianski et al. (1992) still uses the Imperial system of measurement (e.g. statute miles). One problem we encountered was the magnitude of the friction coefficients \(k_n\) and \(k_s\). These are defined by Jelesnianski et al. (1992) as

$$k_s = \alpha \sqrt{\frac{10^{-4}R}{0.3V_R+60}}$$
$$k_n = 0.87k_s,$$

where \(R\) is the radius of maximum winds in statute miles and \(V_R\) is the maximum wind speed in miles per hour for a stationary storm, and \(\alpha=1\) over the ocean (as opposed to lakes). Despite the formula for the magnitude of \(k_s\), it is essential to note that both \(k_s\) and \(k_n\) have dimensions of \(1/L\), where \(L\) stands for length. This is imposed by the dynamical equations further above, and this yields a meaningful conversion of these parameters to the metric system.

5 Summary and discussion

We reviewed a numerical method proposed by Jelesnianski and Taylor (1973) to obtain the surface inflow angle and pressure distributions for a simplified, parametric model of surface winds in a stationary cyclone. The wind can be used as forcing function for models similar to SLOSH, and in other numerical models such as ADCIRC or ROMS, to compare their performance to SLOSH. Recently, Lin et al. (2010) used SLOSH in a model-based storm surge risk assessment study for New York City. In this context, SLOSH is the "deterministic link" between storm surge event distributions on the one hand, and hypothetical storm event distributions on the other hand. Interestingly, Lin et al. (2010) state on the one hand that

In order to make a consistent comparison between the SLOSH and ADCIRC surge simulations, the SLOSH wind field model Jelesnianski et al. (1992) is used to generate the wind fields in the ADCIRC simulations.

On the other hand, they also state that

For simplicity, the pressure field is generated using the Holland pressure distribution model Holland (1980) ...

We are currently unaware whether these pressure distributions differ, but assuming they do, one reason why they use the Holland (1980) pressure could be that they encountered similar difficulties as we did.

Some sources report that the SLOSH wind model may be updated in the near future. Alaka (2017) notes that the current model does not accurately capture surge responses for disorganized, asymmetric, and transitional storms and presents various examples. If not for anything else, the present note will continue to be relevant for validation studies of storm surge models. There exists a significant amount of published SLOSH validations using the model of Jelesnianski and Taylor (1973). Given that more comprehensive (computationally expensive) models like ADCIRC outperform SLOSH already in terms of accuracy (not efficiency), these published SLOSH results might best be viewed as a minimal performance benchmark for newly developed models.

References