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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 414060, 26 pages
-Advanced Models for Tsunami and Rogue Waves
Department of Mathematics, East Carolina University, Greenville, NC 27858, USA
Received 9 March 2012; Revised 19 May 2012; Accepted 21 May 2012
Academic Editor: Ferhan M. Atici
Copyright © 2012 D. W. Pravica et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A wavelet , that satisfies the q-advanced differential equation for , is used to model N-wave oscillations observed in tsunamis. Although q-advanced ODEs may seem nonphysical, we present an application that model tsunamis, in particular the Japanese tsunami of March 11, 2011, by utilizing a one-dimensional wave equation that is forced by . The profile is similar to tsunami models in present use. The function is a wavelet that satisfies a q-advanced harmonic oscillator equation. It is also shown that another wavelet, , matches a rogue-wave profile. This is explained in terms of a resonance wherein two small amplitude forcing waves eventually lead to a large amplitude rogue. Since wavelets are used in the detection of tsunamis and rogues, the signal-analysis performance of and is examined on actual data.
Tsunami or maremoto waves occur in response to earthquakes or landslides on the seafloor of large bodies of water, as discussed in [1–4]. The consequential runup to the shore is such that the tide goes out, then returns as a large surge, only to be followed by several diminishing cycles of similar events . An understanding of this behavior involves a consideration of the effects from the seafloor near the shore where the wave velocity decreases [6, 7]. We study a short-lived forcing that predominantly generates a traveling-wave profile where solves a multiplicatively advanced differential equation (MADE) whose profile resembles a typical tsunami (see Figure 1). In contrast to the -wave profile proposed in , the -MADE profiles are asymmetric wavelets that are flat on a half-line.
In an apparently unrelated phenomena, rogue, freak, or monster waves are caused by small ripples or currents in layers near the water's surface [9, 10]. Various methods have been used to model these rare events, as in [11, 12]. Here, we demonstrate why rogue waves may be a type of resonance wherein an arbitrarily low amplitude forcing, for a sufficiently long period of time, can produce any size of localized wave. Our models use square-integrable versions of the sine and cosine functions that we call and (see Figure 2).
Tsunami and rogue waves are perturbations of the water-surface elevation function . The surface velocity components are small and the time-averaged mean-flow velocity is assumed to remain 0 everywhere. Subject to assumptions, the functions , and will be used in the construction of the forcing terms in the Matsuno equations  as given by where is the depth of the water, is the vertical velocity of the wave surface, and are variable external forcings, and is a mass source term. The acceleration due to gravity accounts for buoyancy, and the Coriolis parameter addresses the rotation of the earth (only required for very long waves over the earth ). The friction coefficient acts as a generic sink that reduces the amplitude over time .
This work considers localized plane waves, or wavelets, on a flat sea propagating only in the -direction. Thus, we set In deep water is a fixed constant and is a constant surface-wave speed (called the celerity). Then, for small amplitudes, the elevation function satisfies These equations are found in [8, Equation ], [15, Equation ], and [16, Equation ], respectively. The new coefficients are defined as and . Equation (1.4) is referred to as the KdV-top model in . Inclusion of an additional boundary-turbulence term in (1.5) would result in a loss of conservation properties for (considered in  but not here).
The forcings in (1.3) will be constructed from bounded wavelets that satisfy MADEs. As such they satisfy the following conditions: Physically, if represents the displacement of the water-level from equilibrium, conservation of mass necessitates that the conditions in (1.6) hold. Note that, as discussed in , solitons are not expected to occur in (1.4) since the total spatial integral of vanishes. We find that if is a wavelet in the variable (for fixed ), then so is away from the shore (using -integration of either (1.3) or (1.4)). During the runup, however, looses its wavelet properties due to the variability of the coefficients in (1.5).
It seems natural that wavelets should appear in the study of surface water waves. The wavelets presented here are particularly well suited for surface waves. In particular, we show that when the forcing is expressed in terms of , and in (2.2)-(2.3), the solution can be reexpressed in terms of these functions. An objective of this paper is to demonstrate that the modeling as well as the detection and analysis of an observed wave profile can be achieved efficiently in terms of the wavelets (2.2)-(2.3), see [18–20]. Hence, a brief discussion on the topics of signal analysis and recovery on real data is presented using and . We apply these techniques to the Japanese tsunami of March 11, 2011. The paper is completed with the details of a perturbation analysis in that is needed to establish the existence of a resonance for rogue waves.
2. Preliminaries on Special Functions
The first quantity that we introduce is the function that satisfies the MADE We set for , and define Note that satisfies the wavelet conditions in (1.6), see . Furthermore, as in , the reproducing kernel of gives the functions which are displayed in Figure 2 for . The normalization constant is chosen so that . It is shown in  that the -advanced harmonic oscillator equations, hold, which are second-order MADEs. Each of (2.2)-(2.3) lies in . We show as a consequence of Theorem 2.1 that and pointwise for as . Note, we also have uniform convergence on compact sets, which was obtained in . Thus, and can be viewed as approximations of and , respectively, which are solutions of the limit of equations in (2.4) as .
2.1. Theta Functions
As part of our study, we employ the Jacobi theta function, defined for , This function will only be used in regions where and . Clearly the function is for all and grows faster than any rational polynomial at and 0. As such, for each one has that is a Schwartz function that is flat at . It also satisfies the algebraic identity which plays an important role in our analysis, .
Theorem 2.1. For , let Then is a delta sequence in , at and has the following properties:
Proof. From , for and for one has Multiplication of (2.9) and (2.10) by , followed by reciprocation, gives that for Now, follows from (2.11) and the fact that for , . For , also follows from (2.11). To obtain , we handle some preliminaries. First, it was shown in  that The functional identity , established in , allows for the replacement of by in (2.12) to obtain the following: This in turn gives that where the last equality follows from the -Wallis limit, , from . This proves and finishes the proof of the theorem.
2.2. -Advanced Wavelets That Solve MADEs
The appearance of theta functions is a consequence of the Laplace transform of (2.2), In , we use the inverse Fourier transform to compute the expressions and these expressions will be used in the proof of a resonance for rogue waves. The representation in (2.15) verifies the MADE in (2.1), and (2.16) implies the identities which verify the MADEs in (2.4). Now Theorem 2.1 gives that both and are delta sequences in at , respectively. These applied to (2.16), in combination with the -Wallis limit, give that pointwise For , the functions in (2.15) and (2.16) are in the Schwartz space. In particular, their rates of decay for large are typically slower than exponential but faster than any reciprocal polynomial. It can be shown that so that for and ,
Remark 2.2. The use of theta functions in frequency space provides decaying versions of cosine and sine while preserving many differential properties of these functions.
3. Tsunami Modeling Using MADEs
A tsunami wave is the consequence of a spontaneous change in elevation on the seafloor, which creates a variable pressure field throughout the volume of water. This sets up forces that extend to the surface of the ocean causing it to be moved up and down, locally. The perturbed wave height then propagates away from this disturbance. In still water, the surface wave speed mainly depends on the average depth of the ocean. For tsunamis, the wavelength is much longer than and so the shallow-water wave equation applies (see [22, page 195]). Near the shore nonlinear effects need to be introduced to account for the sloping of the shoreline [4, 12, 23, 24].
Suppose that the faultline on the seafloor is parallel to the coordinate. Then the -dependence can be ignored even after the tsunami-causing event has taken place. In this situation, the wave front will travel in the direction only.
3.1. -Advanced Tsunami Model
Consider the forced one-dimensional wave equation for the water-level function [14, 22], where the condition on the right in (3.1) constitutes the boundary and initial conditions. The forcing is expected to satisfy the conditions, for fixed , , and is related to the depth of the ocean floor by . The models in [4, 8] start with ground-motion profiles or where respectively. In , it is noted that using the first model gives a force consistent with a landslide that continues for all time. In , the second model suggests an earthquake that continues for all time. In these settings, (3.2) does not hold. However, these models are still used as part of the initial forcings for the wave equation since they lead to integrable solutions whose evolutions resemble that of actual tsunamis. As a comparison, we propose the following -advanced model: where now satisfies the conditions in (3.2). When (3.4) is substituted into (3.1), solved and simplified, one obtains a unique solution to the forced wave equation. To express the solution, define the two phase functions, corresponding to right and left propagation, where the parameters are related by . Define, for any and , Then, for and , the reader can verify that where a particular solution to (3.1), with the forcing in (3.4), is A smooth solution that satisfies , for , is obtained by choosing One can see from (3.5)–(3.9) that the solution has the basic wavelet properties for each . To analyze the long-term behavior of the solution , we use (2.1), (2.17), and integration by parts, twice, to obtain Proposition 3.1.
Proposition 3.1. For any and the following identity holds:
The expression in (3.11) demonstrates that , as used in (3.8), can be written as a series of localized (bound) states and traveling (free) states. These different terms appear as translated and scaled versions of the wavelets , and . Thus, such functions provide a good match for a wave profile that was generated by the family of forcings in (3.4). Furthermore, for , and (reasonable for tsunamis), Finally, observe that in (3.4) is -advanced in time compared to the ground-motion profile . We also find that the forcing precedes the response for the -advanced models. Figure 4(a) illustrates a case where a forcing profile generates a similarly shaped tsunami in Figure 4(b).
Remark 3.2. Similar results hold for height functions .
3.2. Numerical Solution of a -Advanced Tsunami Wave Event
Here, we model the Japan tsunami of March 11, 2011, using (1.3), (1.4), and (1.5) with a forcing term in (3.4) with environmental parameters chosen to be, in units, which are based on past experience . The sea-depth function is chosen to be which models the sea-floor near Wake Island. This is a required modification of the model used in . For stability, we employ a Lax-Wendroff correction term in the numerical method. A good match with typical tsunami profiles and run-up profiles was obtained by using . In this case, , which justifies the approximation in (3.12).
Figure 3 represents data from the Japan tsunami of March 11, 2011, at the first oceanic observation site DART 21418. This data includes a preliminary forcing profile from the (local) times 5 min to 20 min, along with the actual tsunami profile from the (local) times 42 min to 72 min.
Figure 4(a) again shows the Dart 21418 forcing profile from the times 5 min to 20 min, along with a -advanced forcing profile as in (3.4), with and chosen to be and to effect comparable forcing profiles.
Figure 4(b) shows the Dart 21418 actual tsunami profile from the times 42 min to 72 min along with the numerically propagated theoretical tsunami for the same time interval that was generated by (1.3) and (1.4) for the forcing in (3.4) as above (with parameters and ).
Figure 5 shows the actual tsunami profile at a later time and greater distance at oceanic observation DART 21413 from the (local) times 5 min to 65 min. Also shown for comparison is the numerically propagated theoretical tsunami for the same time interval that was generated by (1.3) and (1.4) for the forcing in (3.4) with parameters and .
Figure 6 shows the actual run-up profile taken from the tide gauge at Wake Island at an even later time and greater distance. This is compared with our predicted run-up profile generated by (1.3) and (1.4) together with (1.5) for the forcing in (3.4) with parameters and . The run-up data and theoretical profile have similar profiles initially, for the first few oscillations.
4. Rogue-Wave Modeling Using Solutions of MADEs
A debate continues on the physical cause of rogue waves, . One possible mechanism is a natural outcome of a constructive interference between rippling surface waves that propagate in different directions [22, page 191]. We construct a localized plane wave using a -generated solution of the wave equation, for , Substituting into (3.1) results in a small but persistent forcing. The model in (4.1) ignores possible translational behavior, which will be discussed at the end of this section. Here, and are parameters that satisfy for a constant celerity . The forcing required to obtain from a calm distant past consists of two terms and it is easily seen that . We show that as while from (4.1) it follows that remains constant for all . This is a type of triad resonance with peak at . To proceed with our analysis, estimates on the differences between -advanced functions, which are solutions of MADEs, are needed.
4.1. Analysis of Forcing Terms for and -Type Rogue Waves
In this section, we show that small amplitude forces, over long periods of time, can naturally produce large rogue waves. This demonstrates the existence of a resonance for the system externally forced by (4.2).
Proposition 4.1. Let be given. Then there exists such that for all with one has for all ; also there exists such that for all with one has for all .
The proof of this result is given in the last section. It is used here to show that small amplitude forces, over long periods of time, can naturally produce rogue waves. That is, for sufficiently near , we now apply Proposition 4.1 to show that for arbitrarily small forcing terms there are large rogue solutions of the forced wave equation (4.5) (and of (4.10)).
Proof. Without loss of generality, set and . Then one has, for all , that giving (4.5). Choose . Now let , and let be chosen so that for one has by Proposition 4.1. Then, for all , where (4.8) follows from the fact that . Now, since , we have .
Theorem 4.3. Let . Let be given, and let be sufficiently close to . Define Then satisfies the forced wave equation where , but for all .
Proof. From (4.9), one has for all that giving (4.10). Choose . Now let , and let be chosen so that for one has | by Proposition 4.1. Then, for all , where (4.14) follows from the fact that , as in . Now, since for sufficiently close to , we have that .
4.2. Slowly Moving Rogue Waves
When a slight drift in the rogue-generating current is present, there may be a speed to the wave-height profile. A model for such a rogue can be given by where . The peak of this wave still occurs at but moves to the right at speed . The techniques used in the previous section show that by choice of parameters, a small amplitude forcing, over a long period, will create a moving rogue of arbitrary size.
Theorem 4.4. Let and . Define . Let be given, and let be sufficiently close to . Define the surface-height function and let the forcing be given by Then satisfies the forced wave equation where is fixed, but for all .
Proof. Applying the operator to (4.17) yields (4.19) for in (4.18). The magnitude of the first term in (4.18) is handled as follows: where the last inequality follows from the fact that . The remaining expressions in (4.18) are controlled by noticing that can be brought below for sufficiently close to by paralleling steps (4.7) through (4.8) in Theorem 4.2 and applying Proposition 4.1. The result is now proven.
Remark 4.5. For sufficiently small (as well as sufficiently near ), the term in Theorem 4.4 is small, and one then has that can be made arbitrarily small compared with the rogue amplitude , independently of . For smaller values of , the moving rogue wave maintains a large amplitude near for a longer period of time.
Remark 4.6. There is a corresponding theorem for the moving rogue wave given by .
Remark 4.7. Note that, as is demonstrated in Figure 7, classic rogue wave profiles emerge from smaller forcings even for relatively far from .
5. Wavelet Signal Analysis, Inversion, and the Frame Operator
We now have a collection of solutions to differential equations that give the qualitative behavior of a physical phenomena. Next, to detect, analyze, store, and recover a tsunami waveform, it is common to use a wavelet analysis . The process begins by identifying a discrete set of functions, called an affine frame, With some effort, it can be shown that , for appropriate and sufficiently small . This leads a wavelet transform where The range of is a subspace of and has an adjoint defined to be where are the elements of the dual frame to . The frame operator is invertible for sufficiently small. If is an orthonormal basis for , then one can use . The wavelets , , and generate frames whose inner product structures are nearly orthogonal . Consequently, as will be shown below, a reasonable analysis of the different waveforms is obtained without computing the dual to .
The results of a signal analysis and synthesis, briefly presented here, consist of choosing 256 equallyspaced points from the data, computing the inner products with elements of , sorting and identifying inner products with the largest magnitudes, reconstructing the wavelet-based waveform, normalizing in , and computing the normalized error.
By , the sizes chosen for the parameters are and , with slight adjustments being made to improve the result. The reconstructed signal has an amplitude about of the size of the data profile. The need for this scaling was explained in  and is due to the fact that the operator norm of is estimated to be .
5.1. Tsunami Wavelet Analysis
On March 11, 2011, an earthquake of magnitude 9.0 occurred off the coast of Japan causing a tsunami no less than high. Surface wave levels were detected by buoys operated by DART. To detect, analyze, store, and recover a tsunami waveform, we choose the wavelet frame with and . Then the application of on tsunami data gives coefficients which can be plotted in a versus diagram. The largest values of indicate position (due to ) and narrowness (due to ). Only the largest 20 coefficients in magnitude, for the ranges and , were used to obtain the fit on the bottom right of Figure 8. The normalized- error was computed to be . The bubble plots of the relative sizes of the coefficients are shown with the discrete time translation variable displayed horizontally, and the discrete frequency exponent displayed vertically.
5.2. Rogue Wavelet Analysis
On January 1, 1995, a rogue wave was detected on the Draupner platform in the North Sea. Surface wave heights were recorded using a laser-detection method . Here we use the wavelet frame with and . Here, we used 30 coefficients, for the ranges and , to obtain the fit on the bottom right. The normalized RMS error is . See Figure 9 for the results of this analysis.
6. Estimates for and
In this section, the estimates for the differences of -advanced trigonometric functions are proven. First we record useful Fourier transform expressions. From (2.3) we have the Fourier transforms of , and , respectively, are given by where the algebraic identity , that follows from (2.6), was used to obtain (6.2) and (6.4). Further details are presented in .
Proof of Proposition 4.1. We first prove the estimate for the differences involving. From (6.1) and (6.2), one has
where the triangle inequality gives the inequality in (6.6), and the evenness of allows for reduction to integration over in (6.6).
The change of variables is made on the first integral in (6.7), and the algebraic identity is used to obtain Now (6.8) is used to reexpress the bound (6.7) as
From (2.9), we have that for . Deploying the bound (6.10) within the integral in the bound (6.9) gives where It follows from  that
We now show that can be made arbitrarily small, independently of , for all sufficiently close to . In light of (6.13), this is accomplished by first showing the corresponding statement holds for the bracketed expression in (6.11).
Let be arbitrary, with to be specified later. The integral in over the interval in (6.11) is now subdivided into two integrals, the first over and the second over . First, considering restricted to the interval , one has that Now the function assumes its maximum value on at the right endpoint . In addition, for all sufficiently close to , the function assumes its maximum value on at the right endpoint . The latter statement follows since is increasing, and one needs only to compare endpoints and to determine the larger value. An application of L'Hopital's rule gives that . Thus, for near , on the interval , one has Thus, we bound the integral in (6.14) by the length of the interval times the bound (6.15) on the numerator. This gives An application of L'Hopital's rule gives that which implies that in (6.16) we have Combining (6.18) with (6.16) and (6.14) gives that
We next estimate the portion of the integral in (6.11) over the interval . by bounding the integrals of the summands in (6.20). For each exponent one has where a completion of squares gives (6.21). We use the following estimate from  which is applied to (6.21) with , and to obtain