A wind-wave interaction explanation for Jelesnianski's open-ocean storm surge estimation using Hurricane Georges' (1998) measurements.

Abstract

On 28 September 1998, Hurricane Georges made its final landfall near Biloxi, Mississippi with maximum one-minute sustained surface winds of 46 m[s.sup.-1] (90 kt) and a minimum central pressure of 960 hPa. The measured peak storm surge was approximately 3 m (10 ft) at Pascagoula, Mississippi. Using Jelesnianski's nomograph for open-ocean surge along with adjustments for shoaling factor and storm motion, the peak surge on the coast is estimated to be 10 ft, in excellent agreement with the measurement. It is shown that Jelesnianski's open-ocean peak-surge nomograph can be further substantiated by the recent advance in wind-wave-surge interaction studies using the data from a buoy located near Hurricane Georges' track. For rapid estimation of the surge before shoaling, an analytical formula incorporating both the wind-stress tide and the barometric tide is also provided for operational use.

1. Introduction

According to Dean and Dalrymple (2002), the NOAA/National Weather Service uses the SLOSH (Sea, Lake, and Overland Surges from Hurricanes) model (Jelesnianski et al. 1992) operationally to predict storm surges. The numerical SLOSH model runs on a grid system. Convective accelerations are neglected, but some nonlinearities are included (particularly in shallow water). Comparisons with historical storms show that the model is accurate to within [+ or -] 20% of observed values.

On the other hand, according to the USACE (1977), Jelesnianski (1972) combined empirical data with his theoretical calculations to produce a set of nomographs that permit the rapid estimation of peak surge for any geographical location when a few storm parameters are known.

In September 1998, Hurricane Georges made eight landfalls in its 17-day journey, from islands in the northeastern and northern Caribbean Sea to the Florida Keys and finally to Biloxi, Mississippi (Pasch et al. 2001). Figure 1a shows a portion of the storm track plotted on a visible satellite image from the NOAA-14 polar-orbiting environmental satellite. A low-level wind circulation analysis of Hurricane Georges near the satellite overpass time is provided in Fig. 1b. Storm Surges related to Hurricane Georges were measured at locations along the Gulf Coast from Louisiana to Florida (Fig. 2). A NOAA/NWS/National Data Buoy Center buoy (42040; see Fig. 1a) was located near Georges' track and recorded wave characteristics necessary for the wind-wave interaction study related to this storm.

The purpose of this investigation is to estimate the peak open-ocean surge using Jelesnianski's nomograph and to further substantiate this nomograph by recent wind-wave interaction formulations.

2. Application of Jelesnianski's Nomograph

According to Jelesnianski (1972), the corrected peak surge on the coast, Sp, can be estimated by

[S.sub.P] = [S.sub.I][F.sub.S][F.sub.M] (1)

where [S.sub.I] is the peak open ocean surge (i.e., before shoaling), [F.sub.s] is a shoaling factor, and [F.sub.M] is a correction factor for storm motion. Nomographs for [S.sub.I], [F.sub.S], and [F.sub.M] are available (e.g., USACE 1977; see Figs. A1 - A4 in Appendix).

Around 1130 UTC 28 September 1998, Georges made its final landfall near Biloxi, Mississippi, with maximum sustained surface winds of 46 m [s.sup.-1] (90 kt), and a minimum central pressure ([P.sub.0]) of 960 hPa measured by an Air Force Reserve Command (AFRC) "Hurricane Hunter" reconnaissance aircraft at 0503 UTC (Pasch et al. 2001). The radius of maximum wind (R) was approximately 50 km (30 miles; Hsu et al. 2000). Since [P.sub.0] = 960 hPa, [DELTA]P = (1010 hPa - [P.sub.0]) = 50 hPa. Using [DELTA]P = 50 hPa and R = 30 miles, one finds [S.sub.I] = 11.2 ft from Jelesnianski's nomograph (see Fig. A1). Furthermore, [F.sub.S] is approximately 1.1 (see Fig. A2). The speed of the storm was about 3.6 m [s.sup.-1] (7 kt or 8 mph). If one approximates its track to be nearly perpendicular to the shore, [F.sub.M] = 0.8 (from Fig. A4). Therefore, from Eq. (1) [S.sub.P] = 9.9 ft. From Fig. 2, the surge is about 2.9 m (9.6 ft) at Pascagoula, Mississippi. Hence, we can say that Eq. (1) is a useful analytical formula.

3. Application of Wind-Wave Interaction Formulas

In order to further substantiate Eq. (1) physically, formulas related to wind-wave interaction are employed. According to the Shore Protection Manual (USACE 1977), the total water level rise at the coast during a hurricane is

[S.sub.T] = [S.sub.x] + [S.sub.y] + [S.sub.[DELTA]P] + [S.sub.e] + [S.sub.A] + [S.sub.W] + [S.sub.L] (2)

where [S.sub.T] = total setup,

[S.sub.x] = x-component setup (i.e., the direct transport by the wind stress) or the wind stress tide;

[S.sub.y] = y-component setup (i.e., the Ekman transport by the wind stress) or the Coriolis tide;

[S.sub.[DELTA]P] = atmospheric pressure setup or the barometric tide;

[S.sub.e] = initial water level;

[S.sub.A] = astronomical tide;

[S.sub.W] = wave setup; and

[S.sub.L] = local conditions, such as freshwater runoff from land into bays or rivers.

An example of the relative magnitude for the above terms is provided for Hurricane Camille of 1969 as follows:

[S.sub.T] = 7.6 m (25 ft);

[S.sub.x] = 6.1 m (20 ft), or 80% of [S.sub.T];

[S.sub.y] = 0.6 m (2 ft), or 8% of [S.sub.T];

[S.sub.[DELTA]P] = 0.3 m (1 ft), or 4% of [S.sub.T];

[S.sub.e] = 0.4 m (1.2 ft), or 4.8% of [S.sub.T]; and

[S.sub.A] = 0.2 m (0.8 ft), or 3.2% of [S.sub.T].

It is clear that 80% of the total surge was contributed by the x-component setup during this hurricane.

The x-component setup along the hurricane track before shoaling has been parameterized by Hsu (1999):

[S.sub.x] = ([[rho].sub.a]/[[[rho].sub.w][K.sup.2]])([u.sub.*]/[U.sub.10])[.sup.2]([D.sub.s]/[H.sub.z])[.sup.-1][H.sub.s] (3)

where [[rho].sub.a] and [[rho].sub.w] are the air and water densities, respectively. K (= 0.0016) is the wind-wave interaction coefficient, [u.sub.*] is the friction velocity, [U.sub.10] is the wind speed at a height of 10 m, [H.sub.s] is the significant wave height, and [D.sub.s] is the shoaling depth. Note that the parameters ([u.sub.*]/[U.sub.10]) and ([D.sub.s]/[H.sub.s]) are normalized friction velocity and shoaling depth, respectively. It follows from Eq. (3) that from the viewpoint of dimensional analysis, [S.sub.x] has the same units as [H.sub.s] (meters or feet).

In order to estimate [S.sub.x] from [H.sub.s], we must further parameterize ([u.sub.*]/[U.sub.10]) and ([D.sub.s]/[H.sub.s]). This parameterization is accomplished as follows. In the atmospheric surface boundary layer under hurricane conditions, the stability is near neutral (see Simpson and Riehl 1981, p. 201) so that

[U.sub.Z] = [[u.sub.*]/[kappa]] ln [Z/[Z.sub.0]] (4)

where [U.sub.Z] is the wind speed at height Z, [kappa] (0.4) is the von Karman constant, and [Z.sub.0] is the roughness length.

According to Taylor and Yelland (2001),

[Z.sub.0]/[H.sub.s] = 1200 ([H.sub.s]/[L.sub.p])[.sup.4.5] (5)

[L.sub.P] = [g[T.sub.P.sup.2]]/[2[pi]] (6)

where [L.sub.p] is the deepwater dominant wave length, g is the gravitational acceleration, and [T.sub.p] is the dominant wave period at the spectral peak. From Eq. (4) and setting Z = 10 m,

[u.sub.*]/[U.sub.10] = [0.4/[ln[10/[Z.sub.0]]]] (7)

and from Eqs. (5) and (6)

[Z.sub.0] = 1200 [H.sub.s] ([2[pi][H.sub.s]]/[g[T.sub.p.sup.2]])[.sup.4.5] (8)

Simultaneous measurements of [H.sub.s] and [T.sub.p] during Hurricane Georges in 1998 are provided in Table 1. The mean and standard deviation during this 30-h period for [u.sub.*]/[U.sub.10] are 0.054 [+ or -] 0.0057. Therefore, the coefficient of variation (ratio of s.d. / mean) is 10.6%. Note that this mean value is not significantly different from that of 0.0504 for [U.sub.10] > 20 m [s.sup.-1] as suggested by Amorocho and DeVries (1980), since the difference between the two means is less than 7%.

According to Taylor and Yelland (2001), a large increase in the sea surface roughness is predicted for shoaling waves if the depth is less than about 0.2 [L.sub.p] (where [L.sub.p] is the peak wavelength for the combined sea and swell spectrum). If one approximates that

[D.sub.s] = 0.2 [L.sub.p] = 0.2 [[g[T.sub.p.sup.2]]/[2[pi]]] (9)

then

[D.sub.s]/[H.sub.s] = [0.2g[T.sub.p.sup.2]]/[2[pi][H.sub.s]] = [0.2/2[pi]]([H.sub.s]/[g[T.sub.p.sup.2]])[.sup.-1] (10)

According to USACE (1984, p. 3-85, Eq. 3-64),

[T.sub.p] = 12.1 [square root of [[H.sub.s]/g]] (11)

[therefore][H.sub.s]/g[T.sub.p.sup.2] = 0.0068 (12)

Eq. (12) is verified in Table 1 since the mean [H.sub.s]/g[T.sub.p.sup.2] is 0.0068.

Substituting Eq. (12) into Eq. (10) yields

[D.sub.s] = 4.68 [H.sub.s] (13)

Now, substituting the following values into Eq. (3)

[[rho].sub.a] = 1.2 kg [m.sup.-3]

[[rho].sub.w] = 1025 kg [m.sup.-3]

K = 1.6* [10.sup.-3]

[u.sub.*]/[U.sub.10] = 0.054

and Eq. (13), we have

[S.sub.x] = 0.285 [H.sub.s] (14a)

From Table 1, the maximum [H.sub.s] was 10.88 m, therefore Eq. (14a) becomes

[S.sub.x max] = 0.285 * 10.88 m = 3.1 m (14b)

According to Dean and Dalrymple (2002; p. 84), the total storm surge is the sum of four components: the barometric tide, the wind-stress tide, the Coriolis tide, and the wave setup. Since both Coriolis tide and wave setup have greatest effect in the nearshore region, we postulate from Eq. (2) that, before shoaling,

[S.sub.I] = [S.sub.x] + [S.sub.[DELTA]P] (15)

Also, from Dean and Dalrymple (2002; p. 81)

[S.sub.[DELTA]P] = 0.0104 [DELTA] P (16)

Since [DELTA]P = 50 mb, [S.sub.[DELTA]P] = 0.5 m. [S.sub.x] = 3.1, hence the maximum [S.sub.I] is 3.6 m (11.8 ft) from Eq. (15). The difference between this value and that of 3.4 m (11.2 ft; see Section 2) is about 5%, therefore we can say that Jelesnianski's nomograph is further substantiated by the physics of wind-wave interaction.

4. Further Simplification of Equation 14a

Because measurements of [H.sub.s] are often not as available as those of [DELTA]P, we need to use the pressure difference in Eq. (14a). This is done as follows:

According to Hsu (1991, 1994) and Hsu et al. (2000),

[H.sub.s] = 0.2 [DELTA] P (17a)

[DELTA] P = (1013 hPa - [P.sub.0]) (17b)

Equation (17b) is used traditionally in wind-wave interaction (see, e.g., USACE 1977). However, under hurricane conditions, [DELTA]P = (1010 hPa - [P.sub.0]) is normally applied (J. Chen, personal communication). To be consistent for the tropical environment, Eq. (17b) is adjusted so that

[H.sub.s] = 0.2(1013 hPa - [P.sub.0]) = A(1010 hPa - [P.sub.0]) (17c)

where

A = 0.2(1013 hPa - [P.sub.0])/(1010 hPa - [P.sub.0]) (17d)

Since [P.sub.0] = 960 hPa for Georges,

A = 0.21

[therefore] [H.sub.s] = 0.21(1010 hPa - [P.sub.0]) (17e)

Now substituting Eq. (17e) into Eq. (14a), we have

[S.sub.x] = 0.060(1010 hPa - [P.sub.0]) (18)

From equations (15), (16), and (18), we get

[S.sub.I] = 0.070(1010 hPa - [P.sub.0] (19)

Using [DELTA]P = 50 hPa as before, Eq. (19) estimates that [S.sub.I] = 3.5 m (11.5 ft). This value is in excellent agreement with that of 3.4 m (11.2 ft) as estimated from Jelesnianski's nomograph as discussed in Section 2. Therefore, the following equation is useful, that from Eq. (1) and Eq. (19)

[S.sub.P] = (0.070 [DELTA] P)[F.sub.S][F.sub.M] (20a)

where

[DELTA] P = (1010 hPa - [P.sub.0]) (20b)

5. Some Comments on the [S.sub.I] Nomograph

According to Jelesnianski (1972), the nomograph for [S.sub.I] was constructed for a hurricane moving perpendicular to the coastline at a speed of ~6.7 m [s.sup.-1] (15 mph). Under these conditions the highest surge elevations occur with large values of [DELTA]P for R = ~48 km (30 statute miles). This curve may be digitized so that [S.sub.I] varies approximately linearly with [DELTA]P as

[S.sub.I] = 0.069 [DELTA] P (21)

where [S.sub.I] is in meters and [DELTA]P in hPa.

It is interesting to note that this R value (48 km) is also the composite mean (with a standard deviation of only 3 km) between categories 2 and 4 based on 59 hurricanes affecting the U.S. coastline from 1893 through 1979 (Hsu and Yan 1998). The composite mean R is 47 km for all hurricanes as compiled with central pressures between 909 and 993 hPa. Since Eq. (21) and Eq. (19) are nearly the same, and since 90% of the hurricanes studied were within categories 2 and 4, Eq. (20a) is an excellent approximation for operational use.

Furthermore, some information about the general movement or speed of a typical hurricane may be useful. According to Simpson and Riehl (1981, Table 36), there were 48 hurricanes between 1893 and 1979 with major open-coast storm surges affecting the U.S. for which definitive speeds were obtained. The mean speed is 5.5 m [s.sup.-1], with a standard deviation of 1.7 m [s.sup.-1]. The coefficient of variation is approximately 0.3; consequently about 70% of the hurricanes moved at a speed of 5.5 m [s.sup.-1] (12 mph). Since this mean speed is not very much different from that of 15 mph as used in construction of the [S.sub.I] nomograph, this further reinforces our recommendation that Eq. (20a) and Eq. (20b) are useful operationally. This rapid estimation method may be applied as a supplement to the numerical simulation such as SLOSH model as mentioned in Section 1.

6. Conclusions

Several conclusions can be drawn from this study:

a. The measured peak storm surge along the northeast Gulf coast induced by Hurricane Georges in 1998 was approximately 3 m (10 ft), and was located within the radius of maximum wind as expected;

b. Estimation of the peak onshore surge is also about 3 m (10 ft) based on Jelesnianski's nomographs;

c. Jelesnianski's open ocean surge nomograph is further substantiated physically by recent advances in wind-wave-surge interaction studies;

d. For operational use, an analytical formula Eq. (20a) is provided. It is recommended that this equation be subjected to further verification and improvement using more pertinent tropical cyclone datasets.

Acknowledgments

This study is supported in part by an endowed professorship from Chevron-Texaco to the Coastal Studies Institute, Louisiana State University. Comments from the two reviewers to improve this paper are also appreciated.

Author

Dr. S. A. Hsu has been a Professor of Meteorology at LSU since 1969, after he earned his Ph.D. in Meteorology from the University of Texas at Austin. He is the author of Coastal Meteorology (Academic Press, 1988) and numerous papers on coastal and marine meteorology and air-sea interaction. Dr. Hsu is also an AMS Certified Consulting Meteorologist. Dr. Hsu can be contacted at the LSU Coastal Studies Institute, 308 Howe-Russell Geoscience Building, Baton Rouge, Louisiana 70803-7527; e-mail: [e-mail].

References

Amorocho, J., and J. J. DeVries, 1980: A new evaluation of the wind stress coefficient over water surfaces. J. Geophys. Res., 85(C1), 433-442.

Dean, R. G., and R. A. Dalrymple, 2002: Coastal Processes With Engineering Applications. Cambridge University Press, 475 pp.

Hsu, S. A., 1991: Forecasting hurricane waves. Mariners Weather Log, 35(2), 57-58.

______, 1994: Estimating hurricane waves. Mariners Weather Log, 38(1), 68.

______, 1999: Effects of wave dynamics on the wind setup in deep water. The Wind-Driven Air-Sea Interface, M. L. Banner, Ed. School of Mathematics, The University of New South Wales, Sydney, Australia, 57-64. [Available from CEMAP, School of Mathematics, University of New South Wales, Sydney, NSW, 2052, Australia.]

______, M. F. Martin, Jr., and B. W. Blanchard, 2000: An evaluation of the USACE's deepwater wave prediction techniques under hurricane conditions during Georges in 1998. J. Coastal Res., 16, 823-829.

______, and Z. Yan, 1998: A note on the radius of maximum wind for hurricanes. J. Coastal Res., 14, 667-668.

Jelesnianski, C. P., 1972: SPLASH (Special Program to List Amplitudes of Surges for Hurricanes): 1. Landfall storms. NOAA Tech. Memo. NWS TDL-46, 52 pp. [NTIS COM-72-10807.]

______, J. Chen, and W. A. Shaffer, 1992: SLOSH: Sea, lake, and overland surges from hurricanes. NOAA Tech. Rep. NWS 48, 71 pp. [Available from NOAA/AOML Library, 4301 Rickenbacker Causeway, Miami, FL 33149.]

Pasch, R. J., L. A. Avila, and J. L. Guiney, 2001: Atlantic hurricane season of 1998. Mon. Wea. Rev., 129, 3085-3123.

Simpson, R. H., and H. Riehl, 1981: The Hurricane and Its Impact. Louisiana State University Press, Baton Rouge, LA, 398 pp.

Taylor, P. K., and M. J. Yelland, 2001: The dependence of sea surface roughness on the height and steepness of the waves. J. Phys. Oceanogr., 31, 572-590.

U.S. Army Corps of Engineers (USACE), 1977: Shore Protection Manual. 3rd ed. US Government Printing Office, 3-101-3-145.

_____, 1984: Shore Protection Manual. 4th ed. US Government Printing Office, 1088 pp.

Appendix

Figures from Jelesnianski (1972)

S. A. Hsu

Coastal Studies Institute Louisiana State University Baton Rouge, Louisiana

Table 1. Measured wave parameters from NDBC buoy 42040 in the Gulf of
Mexico during Hurricane Georges, 27-28 September 1998. (Data source:
http://seaboard.ndbc.noaa.gov)

Date  Time (UTC)  [H.sub.s] (m)  [T.sub.P] (sec)

27    11           8.21          12.5
      12           9.36          14.29
      13           9.97          12.5
      14           8.56          12.5
      15           8.84          12.5
      16           9.48          12.5
      17           8.98          12.5
      18           9.87          11.11
      19          10.88          12.5
      20           9.83          12.5
      21           8.99          11.11
      22           7.86          11.11
      23           7.13          10.0
28     0           7.43           9.09
       1           7.02           9.09
       2           6.09           9.09
       3           7.16          10.0
       4           6.51           9.09
       5           6.40           9.09
       6           5.88           9.09
       7           6.13           9.09
       8           5.78           9.09
       9           5.76           9.09
      10           5.37          11.11
      11           5.19           9.09
      12           5.21           9.09
      13           5.40           9.09
      14           5.02          10.0
      15           5.17           9.09
      16           5.06           7.69
Mean
Standard Deviation
Coefficient of Variation

                                                        [u.sub.*]/
Date  Time (UTC)          [H.sub.s]/[g[T.sub.p.sup.2]]  [U.sub.10]

27    11                   0.0054                        0.048
      12                   0.0047                        0.045
      13                   0.0065                        0.055
      14                   0.0056                        0.049
      15                   0.0058                        0.050
      16                   0.0062                        0.053
      17                   0.0059                        0.051
      18                   0.0082                        0.064
      19                   0.0071                        0.059
      20                   0.0064                        0.054
      21                   0.0074                        0.059
      22                   0.0065                        0.053
      23                   0.0073                        0.056
28     0                   0.0092                        0.066
       1                   0.0087                        0.063
       2                   0.0075                        0.056
       3                   0.0073                        0.056
       4                   0.0080                        0.059
       5                   0.0079                        0.058
       6                   0.0073                        0.055
       7                   0.0076                        0.056
       8                   0.0071                        0.054
       9                   0.0071                        0.054
      10                   0.0044                        0.042
      11                   0.0064                        0.050
      12                   0.0064                        0.050
      13                   0.0067                        0.051
      14                   0.0051                        0.044
      15                   0.0064                        0.050
      16                   0.0087                        0.060
Mean                       0.0068                        0.054
Standard Deviation         0.0012                        0.0057
Coefficient of Variation  17.6%                         10.6%