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Effects of Tides on Maximum Tsunami Wave Heights: Probability Distributions*

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Effects of Tides on Maximum Tsunami Wave Heights: Probability Distributions* JANUARY 2007 M O F J E L D E T A L . 117 Effects of Tides on Maximum Tsunami Wave Heights: Probability Distributions* HAROLD O. MOFJELD,FRANK I. GONZÁLEZ,VASILY V. TITOV,ANGIE J. VENTURATO, AND JEAN C. NEWMAN NOAA/Pacific Marine Environmental Laboratory, Seattle, Washington (Manuscript received 29 November 2004, in final form 26 April 2006) ABSTRACT A theoretical study was carried out to understand how the probability distribution for maximum wave ␩ heights ( m) during tsunamis depends on the initial tsunami amplitude (A) and the tides. It was assumed that the total wave height is the linear sum of the tides and tsunami time series in which the latter is decaying exponentially in amplitude with an e-folding time of 2.0 days, based on the behavior of observed Pacific- wide tsunamis. Direct computations were made to determine the statistics of maximum height for a suite of different arrival times and initial tsunami amplitudes. Using predicted tides for 1992 when the lunar nodal f factors were near unity during the present National Tidal Datum Epoch 1983–2001, the results show that when A is small compared with the tidal range the probability density function (PDF) of the difference ␩ Ϫ ␩ Ϫ m A is closely confined in height near mean higher high water (MHHW). The m A PDF spreads in ␩ Ϫ height and its mean height o A decreases, approaching the PDF of the tides and MSL, respectively, when A becomes large compared with the tidal range. A Gaussian form is found to be a close approximation to ␩ Ϫ the m A PDF over much of the amplitude range; associated parameters for 30 coastal stations along the U.S. West Coast, Alaska, and Hawaii are given in the paper. The formula should prove useful in proba- bilistic mapping of coastal tsunami flooding. 1. Introduction tude tsunami wave is not the first one but occurs later in the wave train. When studying past tsunamis and mak- The tides can have a major effect on the maximum ing probabilistic height forecasts for future ones, it is wave heights experienced during a tsunami. By wave therefore essential to take the tides into account. height, we mean the total wave height (tsunami plus Observations have shown that Pacific-wide tsunamis tide) relative to a fixed reference level, such as mean form long wave trains that persist over several tidal lower low water (MLLW). Even when the first wave in cycles in which the envelopes encompassing the tsu- a tsunami wave train striking a coastal location has the nami energy and successive peak amplitudes decay ex- largest amplitude (height of the wave peak relative to ponentially in time (Miller et al. 1962; Van Dorn 1984, the background water level at the time of the peak), a 1987; Mofjeld et al. 2000). Mofjeld et al. (1997) used higher tide may combine with a smaller amplitude tsu- this fact to develop a short-term tsunami forecasting nami wave in the same wave train to produce a greater scheme for the total wave heights of later waves in net wave height. Whether this occurs depends on the tsunami wave trains. To better understand the influ- amplitudes of the successive waves in the tsunami wave ence of tides on the maximum tsunami wave heights train and the height of the tide at the time of each wave from a probabilistic point of view, the present theoret- peak. Also, it sometimes occurs that the largest ampli- ical study was carried out in which analytic tsunami wave trains that decay exponentially in time are com- * NOAA/Pacific Marine Environmental Laboratory Contribu- bined linearly with tidal time series. By varying the tion Number 2768. arrival time of the wave trains sequentially through a long, representative time series of the tides at a given location, it is then possible to generate probability den- Corresponding author address: Dr. Harold O. Mofjeld, NOAA/ Pacific Marine Environmental Laboratory, 7600 Sand Point Way sity functions (PDFs) that characterize the behavior of NE, Seattle, WA 98115-6349. the maximum wave height as a function of the initial E-mail: [email protected] tsunami amplitude and the local tides. This is very simi- DOI: 10.1175/JTECH1955.1 © 2007 American Meteorological Society Unauthenticated | Downloaded 09/27/21 05:53 AM UTC JTECH1955 118 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 24 lar to the method used by Houston and Garcia (1978) in their statistical study of tsunamis along the U.S. West Coast. Of particular interest in the present study are where the maximum wave heights tend to cluster in elevation above MLLW and how the spread in elevation of the PDFs depend on the initial tsunami amplitude relative to the tidal range. It is found that a simple Gaussian form fits the PDFs reasonably well. Furthermore, modi- fied exponential laws are found to closely approximate the dependences of the PDF mean heights and standard deviations on the initial tsunami amplitude; these two parameters are sufficient to quantitatively determine the Gaussian PDF. The coefficients in the exponential laws provide a concise characterization of the tidal ef- fects on the maximum tsunami heights for a given lo- cation. In the next section, theoretical tsunamis are super- imposed on predicted tides on the open coast at Sea- side, Oregon. These mixed semidiurnal tides are typical FIG. 1. Example of a water level time series consisting of a of those along the U.S. West Coast. This serves to il- theoretical tsunami adding to predicted tides at Seaside, OR. The lustrate the influence of the tides on the maximum of tsunami arrives at 0000 UTC 2 Jan 2004 and then decays expo- the total wave height (tsunami wave plus tide) for wave nentially in time after the first wave. At the time resolution trains of various initial amplitudes. This height is la- shown, the rapidly oscillating (20-min period) time series appears as a solid distribution between decaying envelopes. beled by the arrival time of the first tsunami wave peak so that time series (loci) of maximum wave heights can be plotted as functions of the arrival time. In section 3, (NOAA), and the U.S. Geological Survey (USGS). De- it is convenient to use the differences between these caying exponentially in amplitude, the maximum tsu- heights and the initial tsunami amplitude in order to nami height (4.91 m above MLLW) for this event then compare PDFs when the initial amplitude is varied. occurs at 1604 UTC, the next higher high water. The Subsequent sections justify the modified exponential use of an exponentially decaying envelope to charac- laws for the coefficients in the Gaussian approximation terize the maximum wave heights along sections of tsu- to the tsunami PDFs and present values of these coef- nami time series is justified by the case studies and ficients for 29 U.S. Pacific tide stations plus the Seaside stochastic modeling of Mofjeld et al. (1997, 2000). In location. The discussion addresses the influence of the Fig. 1 as well as elsewhere in the study, the decay co- 18.6-yr nodal cycle of the tides, the insensitivity to using efficient (␶ ϭ 2.0 days) is set by observations of Pacific- observed or predicted tides for fitting the coefficients, wide tsunamis (e.g., Mofjeld et al. 2000). The predicted and limiting the application of the theory to situations tides are based on 37 harmonic constants, where those where nonlinear interactions between tsunamis and the for O1, K1, N2, M2, and S2 are from the Eastern North tides can be neglected. Pacific 2003 (ENPAC 2003) tide model (for details, see Spargo 2003; Mofjeld et al. 2004a); the others are in- 2. Tsunami time series ferred from observed relationships at South Beach, Or- Figure 1 shows a typical tsunami time series used in egon (44°37.5ЈN, 124°02.6ЈW). this study. The wave period of 20 min is within Moving the tsunami arrival time to progressively the middle range (10–40 min) of major transpacific later times then generates time series for the maximum tsunamis striking the U.S. West Coast. The theoretical wave height as functions of tsunami amplitude and ar- tsunami arrives at Seaside, Oregon (46Љ00.1°N, rival time. Figure 2 shows the maximum wave height for 123Љ55.7°W), at 0000 UTC 2 January 2004. The tides at successive time series when arrival time is moved for- Seaside are typical of those along the northwestern sec- ward every 15 min over 1-min sampled predicted tides tion of the West Coast; this is also the site for a proba- and the tsunamis have initial amplitudes ranging from bilistic tsunami pilot study recently carried out by the 0.5 to 9.0 m. The result is a set of serrated patterns that Federal Emergency Management Agency (FEMA), the grow in amplitude as the tsunami amplitude increases. National Oceanic and Atmospheric Administration For small amplitude tsunamis, the maximum heights Unauthenticated | Downloaded 09/27/21 05:53 AM UTC JANUARY 2007 M O F J E L D E T A L . 119 FIG. 3. Probability distribution functions (PDFs) of the maxi- FIG. 2. Locus of the maximum tsunami wave heights (tsunami mum tsunami wave heights at Seaside, OR, based on exponen- plus tide) as a function of the arrival time of the first wave peak tially decaying tsunamis with the indicated amplitudes. The tsu- in each tsunami wave train. For each initial amplitude A (height of nami amplitudes have been subtracted from the heights of the the first wave peak relative to the background water level), the corresponding PDF. Also shown is the PDF for the predicted time series are generated by moving the arrival time sequentially tides.
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