On Some Background Flows for Tsunami Waves

With the aim to describe the state of the sea in a coastal region prior to the arrival of a tsunami, we show the existence of background flow fields with a flat free surface which model isolated regions of vorticity outside of which the water is at rest.


Introduction
Tsunami waves are generated by a sudden vertical displacement of a body of water on a massive scale, caused by landslides, volcanic eruptions or, most commonly, by undersea earthquakes [2]. Tectonic collisions in the form of thrust (or normal) faults sometimes make the ocean floor rise (or drop) by a few meters, causing the column of water directly above to rise (or fall) as well and thereby creating an initial wave profile of elevation (or of depression), as it was the case with the December 2004 tsunami cf. [4,5,9,10,16,23]. Tsunami waves are a special type of gravity water waves, with typical wavelength of hundreds of kilometers. They can travel over thousands of kilometers at very high speed with little loss of energy, a spectacular example being the May 1960 tsunami that originated near the Chilean coast (due to the largest earthquake ever recorded) and propagated across the Pacific Ocean devastating coastal areas in Hawaii and Japan, 10,000 km respectively 17,000 km far from the Chilean coast [8,24]. Away from the shore, where the ocean can be assumed to have uniform depth over large distances (e.g. the ocean floor of the Central Pacific Basin is relatively uniform, with a mean water depth of about 4,300 m cf. [8]), the evolution of the wave is governed essentially by linear water wave theory, the typical wave speed being √ gh with g the gravitational constant of acceleration and h the average depth of the sea [11,23]. The amplitude of a tsunami wave out in the open sea is typically very small (roughly about 0.5 m cf. [23]), but when it approaches a gently sloping beach, the front of the wave slows down causing the water to pile up vertically since the back of the wave is still hundreds of kilometers out in the sea, travelling at much higher speed. The enormous amounts of water involved in this process, account for much of the devastating effects tsunami waves have in coastal areas. Before the arrival of the tsunami waves at the shore, the water in that region is unlikely to be still: even in the presence of surface waves of small amplitude or for a flat free surface, beneath the surface there could be considerable motion due to the presence of currents (already for irrotational flows with a free surface, an underlying uniform current complicates considerably the dynamics of the flow since without a current all particle paths describe a non-closed loop [3] whereas certain currents can produce closed particle paths [14]). Taking into account currents, it seems essential in a reasonable model for tsunami waves to allow for some kind of background flow field, which models the motion of water in the absence of waves. While most investigations are restricted to irrotational flows which model background states

Physical Assumptions and the Formulation of the Problem
We can reasonably model the evolution of tsunami waves in a two dimensional setting, a simplifying assumption which is justified for the December 2004 tsunami off the coast of Indonesia [23] and the 1960 Chile tsunami [4]. The direction of propagation of tsunami waves was mainly perpendicular to the fault line, with the length of the rupture zone exceeding the wavelength, and the ocean depth over which the tsunami waves travelled was relatively uniform. Furthermore, we assume the water to be inviscid and consider its density to be constant. As we are concerned with gravity water waves, we neglect surface tension. We want the model to admit a shoreline and assume that at the bottom we have a fixed impermeable bed. In Cartesian coordinates (x, y), let the origin be the intersection of the flat free surface and the seabed at the shoreline x = 0. Let the horizontal x-axis be in the direction of the incoming right-running waves and the vertical y-axis pointing upwards. We assume the fluid to extend to −∞ in the negative horizontal direction and let the bed's topography for a gently sloping beach be given by In the open sea we assume uniform depth h 0 such that b(x) = h 0 for x far away from the shoreline x = 0. We will denote the fluid domain by D = {(x, y) ∈ R 2 : x < 0, b(x) < y < 0}. In the two-dimensional setting we can introduce a stream function ψ, such that the fluid's velocity field is given by (ψ y , −ψ x ). We consider the vorticity ω to be a function of ψ, ω = γ(ψ), where γ is called vorticity function. Clearly ω = γ(ψ) specifies a vorticity distribution throughout the flow and notice that in the absence of stagnation points (that is, points where ∇ψ = (0, 0)), one can prove that the vorticity distribution is specified by means of a vorticity function, cf. the discussion in [7,13]. The equations governing a background state with flat free surface can be reformulated in terms of ψ as given a vorticity distribution γ and the bottom profile b of the fluid domain D. For a detailed discussion of how these equations governing the fluid motion can be derived from the principle of mass conservation and the Euler equations we refer to [10,13]. Our aim is to show existence of an isolated region of non-zero vorticity in the fluid domain, outside of which the water is at rest (see Fig. 1). That is, we have to find a suitable vorticity distribution γ and prove that (2.1) has a non-trivial radially symmetric solution with compact support in D. Radial solutions are obtained via the Ansatz turning the system (2.1) into the semi-linear second order differential equation

2)
Vol. 14 (2012) Tsunami Background Flows 143  where denotes the derivative with respect to r. Note that for solutions with compact support the boundary conditions in (2.1) will be trivially satisfied, as ψ ≡ 0 outside some compact region. To be able to uniquely determine a solution to (2.2), we have to specify initial values for ψ and ψ at r = 0, say (ψ(0), ψ (0)) = (a, 0).
We require ψ (0) = 0 to produce classical solutions. The boundary value problem (2.1) is over-determined and it is expected that a non-trivial solution will only exist for certain classes of functions γ. The fact that our model admits a shoreline and we require the water to be still outside the region of vorticity imposes restrictions on the regularity of γ. For linear vorticity functions γ(ψ) = aψ + b it can be shown (see [10]) that system (2.1) admits only trivial solutions. The argument relies mainly upon maximum principles and the fact that the streamlines of the flat free surface and the seabed intersect at the shoreline and are equal to zero. We can therefore not hope to find non-trivial solutions with compact support inside a circular boundary for a linear vorticity distribution as suggested for example in [1], since these arise in the context of an unbounded fluid which is at rest at infinity. As we are interested in classical solutions, γ has to be at least continuous. However, requiring γ ∈ C 1 precludes radially symmetric solutions with compact support in the fluid domain, since we could find a value T > 0 sufficiently large, such that ψ(T ) = ψ (T ) = 0. Then, by the backward uniqueness property [15] for (2.2) with γ ∈ C 1 , we would have ψ(r) ≡ 0 for all values of r > 0. The discussions in [6,12] lead us to consider the vorticity function ( Fig. 2)

Proof of the Main Result
Instead of solving the second order initial value problem (2.2)-(2.3), consider the equivalent planar system of first order ordinary differential equations with initial values The proof of Theorem 2.1, using a dynamical system approach which relies upon basic theory of ordinary differential equations, follows essentially from the results of the following two propositions, which we will prove in Sects. 3.1 and 3.2, respectively.

Proposition 3.2.
There exists a > M α > a α such that for the corresponding unique solution (ψ, β) of Proof of Theorem 2.1. By virtue of Propositions 3.1 and 3.2 there exists a value of a > M α > a α such that for the corresponding uniquely defined C 2 -solution to (3.1)-(3.2), we can find T > 0 such that ψ(T ) = β(T ) = 0. Then by setting ψ(r) = 0 for r ≥ T we obtain a compactly supported solution of (2.2) defined for all r ≥ 0. Furthermore, recall from Proposition 3.1 that ψ(r) > 1 for r ∈ [0, 1], which in view of (2.4) yields ω = γ(ψ) > 0. Since ψ has compact support, we obtain an isolated region of non-zero vorticity ω which contains a ball of unit radius where ω > 0, outside of which the water is at rest.

Proof of Proposition 3.1
We claim that for any a > a α there exists a unique C 2 -solution (ψ, β) to (3.1)-(3.2) which depends continuously on the initial data (a, 0) on any compact interval on which ψ 2 (r) + β 2 (r) > 0 and for which ψ > 1 for r ∈ [0, 1]. This is not immediately clear for two reasons: • the right hand side of (3.1) displays a discontinuity at r = 0, so the system is not a classical initial value problem. • since the vorticity function γ(ψ) fails to be locally Lipschitz when ψ = 0 the right hand side of (3.1) is not locally Lipschitz and we cannot apriori expect uniqueness of solutions or continuous dependence on initial data from the standard theory of ordinary differential equations. In the first part of the proof, summed up in Lemma 3.3, we consider the system in the vicinity of the discontinuity, for r ∈ [0, 1]. By a simple change of variables (3.6) we overcome the problem of the discontinuity and solve the equivalent system (3.4) using an integral Ansatz and Banach's fixed point theorem. We ensure continuous dependence of solutions on the initial data (a, 0) and find that the solutions of Vol. 14 (2012) Tsunami Background Flows 145 the integral equation (3.7) are always greater than one. In Lemma 3.4 we introduce an important functional (3.14) which decreases along solutions and is essential in deriving results throughout the proofs of both Propositions 3.1 and 3.2 as it ensures global existence of solutions. In Lemma 3.5 we tackle the second part of the proof by analyzing the system away from the discontinuity. The difficulty in this case lies in the fact that the right hand side of (3.1) fails to be locally Lipschitz continuous whenever ψ = 0. By rewriting the system in polar coordinates we obtain another equivalent formulation (3.16), for which existence and uniqueness of solutions as well as continuous dependence on initial data follows from standard results whenever the right hand side is C 1 . In the vicinity of points where ψ = 0 an application of the inverse function theorem yields yet another local reformulation (3.20), shifting the lack of Lipschitz continuity in the dependent variable for (3.16) to the independent variable for the new system and thereby gaining C 1 -regularity of the dependent variable for (3.20). We thus obtain local uniqueness and continuous dependence also at points where the right hand side of (3.16) fails to be Lipschitz.
where the initial values (3. Proof. We perform the change of variables and find that ( This follows from the fact that (3.1) is equivalent to (2.2) which in view of The restriction 0 ≤ r ≤ 1 is equivalent to s ≥ 0 in the new variable. Let an arbitrary a > a α be fixed. We can deal with local existence and uniqueness issues of a solution to (3.1)-(3.2) by considering the integral equation The corresponding asymptotic behavior (3. where we used the rule of de l'Hospital in the second equality. As long as v(s) ≥ 1 we have that v is non-decreasing, since γ(v) ≥ 0 which in view of (3.8) gives v (s) ≥ 0. We even have v(s) > a 1−α > 1 for s ≥ 0. (3.9) Indeed, if this were not so, define s 1 := sup{s ≥ 0 : v(s) = a 1−α }. Then for all s ≥ s 1 we have 1 < a 1−α ≤ v(s) ≤ a, which in view of (3.7) yields a contradiction as These considerations allow us to view the solution of the integral equation (3.7) as the unique fixed point of the contraction T a defined by The upper bound follows from the fact that the integral is positive, since for v ∈ X a , v ≥ 1 and thus γ(v) ≥ 0. For the lower bound, we use the same reasoning as in the proof of (3.9). Next we show that T a as defined above is a contraction. Since the vorticity function γ defined in (2.4) is C 1 on [1, ∞), by the mean value theorem (cf. [17]) there exists ξ ∈ (v, w) for v, w ≥ 1 such that γ(v) − γ(w) = γ (ξ)(v − w). This yields

11)
Vol. 14 (2012) Tsunami Background Flows 147 since γ (ξ) ≤ 1 for ξ ≥ 1. Then for s ≥ 0 we have which shows that T a is a contraction on X a with contraction constant K ≤ 1 4 . Therefore, according to Banach's contraction principle, T a has a unique fixed point, i.e. the integral equation (3.7) has a unique solution v ∈ X a which is of class To show continuous dependence of the solution on the parameter a, let v 1 ∈ X a1 , v 2 ∈ X a2 . Then the integral equation (3.7) yields, in view of (3.11), that for s ≥ 0 and therefore so we actually even obtain that the solution is stable, cf. [15].
Before we proceed to the case where r ≥ 1, we prove the following useful  Proof. As long as a solution to (3.1)-(3.2) exists, we have E (r) = − 1 r β 2 , since the derivative with respect to r of the function E(r) given by (3.14) in view of (3.1) can be computed as

Lemma 3.5.
For r ≥ 1 system (3.1) can be equivalently reformulated as As long as R > 0 this system of first order differential equations has a unique C 2 -solution which depends continuously on the initial data (θ(1), R(1)), which in turn depends continuously on the parameter a.
Notice that this time we choose cos(θ(r)) = τ instead of −τ to preserves monotonicity of the respective independent variables. As before, we transfer (3.16) into a system which differs from (3.20) only by a change of sign in the second equation. Thus by the same reasoning as above we deduce that uniqueness and continuous dependence on a of the solution to (3.16) also holds in neighborhoods of points where the solution intersects the vertical axis in the upper half plane. This procedure can be repeated for all values of r where the right hand side of (3.16) fails to be locally Lipschitz, as long as R > 0. Summing up, we can say that for values of r where cos(θ(r)) = 0, that is, where θ(r) = − π 2 + 2kπ or θ(r) = π 2 + 2kπ for k ∈ Z, the above local transformations guarantee uniqueness and continuous dependence on a of the solution to (3.16) also in neighborhoods of such values as long as R > 0. In between these values of r, the right hand side of (3.16) is C 1 and everything follows from standard results.
This concludes the proof of Proposition 3.1, as we have seen that for any a > a α there exists a unique C 2 -solution (ψ, β) to (3.1)-(3.2) for which ψ > 1 on [0, 1] by virtue of (3.9) and which depends continuously on the initial data (a, 0) on any compact interval on which ψ 2 (r) + β 2 (r) > 0.

Proof of Proposition 3.2
We show that there exists a value of a > a α such that for the corresponding solution to (3.1)-(3.2) we can find some 0 < T < ∞ with ψ(T ) = β(T ) = 0. The idea is to perform a detailed qualitative analysis for the system (3.1)-(3.2), similar to the phase-plane analysis of autonomous systems. We introduce two sets Ω ± defined by the solution sets of the equation E(ψ, β) = 0, where E is the functional defined in Sect. 3.1. In Lemma 3.4 we show that for initial data a large enough the solution can enter the region Ω ± only for values of r > 2 α . We find that the value of r at which the solution can enter Ω ± tends to infinity as a → ∞. After that, Lemma 3.5 ensures that there exists an initial value a + such that the corresponding solution stays outside Ω − ∪ Ω + for all r ≥ 0. Finally, in Lemma 3.6, we prove that for Vol. 14 (2012) Tsunami Background Flows 151 Fig. 3. The solution set of E = 0 in the plane (ψ, β) with arrows indicating the dynamics of the system solutions corresponding to such initial data a + there exists a finite value T > 0 such that E(T ) = 0, and therefore also ψ(T ) = β(T ) = 0. Let us start with defining the sets Ω ± . From (3.14) in Lemma 3.4 we have that In the plane (ψ, β) the set where E < 0 consists of the interiors Ω ± of the closed curves representing the solution set of the above equation. These curves are symmetrical with respect to the vertical and the horizontal axis and are tangential to one another and to the vertical axis at the origin. Note from At β = 0 we have ψ = β = 0 and β = − 1 r β − ψ + ψ|ψ| −α > 0 when ψ|ψ| −α > ψ which is true for 0 < ψ < 1, whereas β < 0 for ψ > 1. In the left half plane, we have exactly the opposite situation. Therefore, solutions intersect the horizontal axis perpendicularly from the upper to the lower half plane for ψ > 1 and for −1 < ψ < 0. On the complement of these sets, they intersect the axis in the opposite direction. For ψ = 0 and β > 0 we have that ψ > 0 and β < 0, which means that solutions intersect the vertical axis from left to right in the upper half plane. In the lower half plane, the opposite is true (see Fig. 3). By Lemma 3.4, E is strictly decreasing as long as (ψ, β) / ∈ {(0, 0), (±1, 0)}. Therefore, once a solution reaches the boundary of Ω ± at a point other than (0, 0) it will enter Ω ± . Once inside, a solution will stay in either Ω + or Ω − for all subsequent times, as E is strictly decreasing. Recall from (3.3) that we defined cos(θ(r)) > a 1−α > a α . If a solution with initial data a enters the region Ω + ∪ Ω − , then for some value of Vol. 14 (2012) Tsunami Background Flows 157

Limiting Cases of the Parameter α
In the case where α = 1, the vorticity function γ simplifies to γ(ψ) = ψ − ψ |ψ| , (3.35) which has a point of discontinuity at ψ = 0. As we are only interested in classical solutions, we will not consider this case. When α = 0 we simply have γ(ψ) ≡ 0. (3.36) Thus, system (2.3)-(2.2), for which we seek compactly supported C 2 -solutions, reads ψ + 1 r ψ = 0, r > 0, ψ(0) = a, ψ (0) = 0, (3.37) which we can solve easily, obtaining ψ(r) = C 1 ln(r) + C 2 , for some constants C 1 , C 2 ∈ R. In view of the boundary conditions, C 1 = 0 and C 2 = a, and we conclude that ψ(r) ≡ a is constant for all r ≥ 0. In the setting of ψ being the stream function on the fluid domain D, this means that ψ ≡ a is constant throughout the flow field. The boundary conditions ψ = ψ y = 0 on the flat free surface require this constant to be zero. So ψ ≡ 0 and the water is still throughout the fluid domain, which is why we do not consider this case.