On the linear stability of one- and two-layer Boussinesq-type Equations for wave propagation over uneven beds

Transcription

1 On the linear stability of one- and two-layer Boussinesq-type Equations for wave propagation over uneven beds Gonzalo Simarro Marine Sciences Institute (ICM, CSIC), 83 Barcelona, Spain Alejandro Orfila IMEDEA, 719 Esporles, Spain Carlos M. Mozos, Rosa E. Pruneda School of Civil Engineering (UCLM), 13 Ciudad Real, Spain Abstract Bousssinesq-type Equations are a powerful tool to model the wave propagation from intermediate waters to the shore. By construction, these equations have a good performance in weakly dispersive conditions, and a great effort has been done during the last years to increase their range of application to deeper waters; the improved equations introduce free coefficients that are chosen for this purpose. Some of the improved sets of equations show instabilities when numerically solved over uneven beds. In this work we show how these instabilities can be due to the equations (including the values of the involved coefficients) and not to the numerical scheme. We further introduce new sets of coefficients that optimize the linear performance while improving corresponding author address: simarro@icm.csic.es (Gonzalo Simarro) Preprint submitted to Ocean Engineering July 7, 1

2 the linear stability of the equations. Keywords: Boussinesq-type Equations, wave propagation, linear dispersion, linear shoaling, linear stability Introduction As water waves travel to the coast, the water depth, h, is reduced from hundreds or thousands of meters in the open sea to a few meters, and eventually zero, at the shore. In the open sea the wave height, H, is much smaller than the water depth, so that H/h 1. Conversely, in the nearshore region the wavelength, λ, is usually much larger than the water depth, i.e. λ/h 1 (or kh 1, with k π/λ being the wavenumber). Now the horizontal velocity profile is nearly uniform in the water column and the wave celerity is independent of the wave period, so that there is no frequency dispersion. The very well-known Shallow Water Equations (SWEs) apply in this region. There is an intermediate zone where H/h 1 and kh 1. The Boussinesq Equations (BEs) were developed to represent water wave propagation in this region. BEs can be seen as an extension of the SWEs that includes dispersion in a perturbative way. BEs by Peregrine (1967) were obtained for weakly dispersive and weakly non-linear conditions. The extension to weakly dispersive but arbitrary (or fully ) non-linear conditions are very popular nowadays (Green and Naghdi, 1976; Wei and Kirby, 199; Madsen and Schaffer, 1998; Carmo, 13), and are usually referred to as Serre s Equations, after Serre (193), or also as Boussinesq type Equations (BTEs hereafter). BEs and BTEs have ensured a good performance under weakly dispersive conditions (BEs for weakly non-linear conditions and BTEs for arbitrary non-

3 linear conditions), by construction. In order to assess the performance under stronger dispersive conditions, BEs and BTEs are linearized and compared to linear and fully dispersive theories such as Airy and Mild Slope Equations (Dean and Dalrymple, 1984). The comparison is made in terms of wave celerity (linear dispersion) and wave shoaling over mild slopes (linear shoaling). The weakly non-linear performance is, also, usually compared to the second order Stokes theory for flat beds (Schaffer, 1996). Since BEs and their corresponding BTEs are identical in their linear weakly dispersive terms, the comparisons give the same results using the BEs or their corresponding fully non-linear extensions (BTEs). Much of the research in this area during the last years has been devoted to improve the linear properties of the equations. Leaving aside higher order (in dispersive terms) equations (Gobbi et al., ), which include spatial derivatives of order five, three main different approaches can be distinguished to this end: i) Madsen and Sorensen (199) proposed an enhancement technique so as to introduce new terms that improve the dispersive performance, and Beji and Nadaoka (1996) proposed an alternative set of enhanced equations; ii) Nwogu (1993) introduced a new set of BEs written for the velocity at z α = αh (instead of the depth averaged velocity), and chose α =.396 to improve the linear dispersion upto kh 3; the corresponding BTEs (fully non-linear extension) were introduced by Wei et al. (199); and iii) Lynett and Liu (4) proposed a multilayer approach (previous equations were onelayer). The above three techniques have been further combined: Madsen and Schaffer (1998) combined the first two approaches, and Simarro et al. (13) all three approaches. Actually, Lynett and Liu (4) already used the ap- 3

4 proach by Nwogu (1993) within each layer. All the above BTEs include free coefficients which are chosen so as to mimic the linear and fully dispersive theory. In general, the more coefficients the better performance. When applied to certain bathymetries (particularly when they have steep slopes), the numerical solution of the above BTEs may show instability problems. These instabilities may arise from the equations themselves (depending on choice for the free coefficients), and not from the numerical scheme employed to solve them. For instance, for the simplest flat bed case, the coefficient α of Nwogu must be (this is shown in Section 3) 1.8 α.4, (1) for the equations to be linearly stable. Nwogu (1993) proposed α =.396, which falls within the above range. Similar conditions are already known for other sets of equations, for the flat bed case. These conditions do not ensure, however, the stability of the equations for uneven bathymetries and, to the authors knowledge, no further research has been done in this regard. The goal of this work is to obtain the sets of coefficients that optimize the linear properties of a wide range of BTEs while improving the linear stability over uneven bathymetries. The work is structured as follows: the following Section introduces the sets of BTEs considered and shows how to assess the linear dispersion and linear shoaling errors in a new simple way; Section 3 introduces the problem of the linear stability for these equations and presents the strategy followed in Section 4 to obtain the convenient coefficients. The final Section shows the applications in the numerical solution of the equations. 4

5 Governing equations We consider one- and two-layer BTEs in this study. The equations under consideration are fully described in Appendix A. In this Section we describe the equations only in terms of the free coefficients that they introduce as well as in their linear properties. One layer equations The one-layer BTEs analysed here are those presented by Beji and Nadaoka (1996) and by Simarro et al. (13), shown in Appendix A.1. Other systems of weakly dispersive and fully non-linear equations are not analysed here, but the same treatment presented below is applicable. The equations by Beji and Nadaoka (1996) were introduced as a simpler alternative to those by Madsen and Sorensen (199). They include one free parameter, β, and have shown to be particularly well conditioned to represent linear dispersion and shoaling (Simarro, 13). Beji and Nadaoka (1996) first derived their equations for weakly non-linear conditions (BEs); in Appendix A.1 we introduce their fully non-linear extension (BTEs). The BTEs by Simarro et al. (13) depart from the sets by Madsen and Schaffer (1998) and Galan et al. (1) to further improve weakly non-linear and weakly dispersive performance, and include eight free coefficients: α, α ɛ, δ, δ ɛ, δ h, γ, γ ɛ and γ h. The equations by Wei et al. (199) and hence Nwogu (1993), Madsen and Schaffer (1998), Kennedy et al. (1) or Galan et al. (1) are recovered as particular cases setting some of these coefficients null (Table 1). The coefficients α ɛ, δ ɛ and γ ɛ affect exclusively the non-linear performance

6 Authors β α α ɛ δ δ h δ ɛ γ γ h γ ɛ B96-1 S13-1 G1-1 M98-1 K1-1 W9-1, N93-1 Table 1: Free coefficients for the one-layer BTEs (marked with ; means does not apply ). B96-1: Beji and Nadaoka (1996); S13-1: one-layer Simarro et al. (13); G1-1: Galan et al. (1); M98-1: Madsen and Schaffer (1998); K1-1: Kennedy et al. (1); W9-1: Wei et al. (199); N93-1: Nwogu (1993) of the above BTEs. This work focuses on linear aspects, which are independent of these three coefficients; for completeness, however, the values of α ɛ, δ ɛ and γ ɛ will be provided following the strategy by Schaffer (1996). Also, in the linear case, the equations K1-1 (Kennedy et al., 1), W9-1 (Wei et al., 199) and N93-1 (Nwogu, 1993) reduce to the same equations, and here we will refer to them as N93-1 when dealing with linear aspects. Similarly, S13-1 (Simarro et al., 13), G1-1 (Galan et al., 1) and M98-1 (Madsen and Schaffer, 1998) coincide in the linear case, and we will refer to them as M98-1. For later use, we recall that an alternative to ξ kh to express the range of validity of BTEs is κ ω h/g, with ω the wave angular frequency and g the gravitational acceleration. For linear waves (Airy theory) κ = ξ tanh ξ so that κ ξ for ξ 1 and κ ξ for kh 3. This number, κ, is proportional 6

7 to the parameter h/λ used by other authors (Madsen et al., 1991; Nwogu, 1993), where λ πg/ω is the wavelength in deep waters. Linear properties Being c = ω/k the wave celerity, the dispersion equation of the one-layer BTEs is { } c κ gh = ξ = 1 + ρ 1ξ 1 + ρ 3 ξ { D (ξ) }, () 1 + ρ ξ 1 + ρ 4 ξ where the coefficients ρ j depend on β for B96-1, on α for N93-1 and on α, δ and γ for M98-1. Some of the coefficients ρ j can be null. For N93-1, e.g. 114 ρ 1 = ρ =, ρ 3 = 3α + 6α + 6, ρ 4 = α + α. (3) The error in the linear wave celerity (linear dispersion) is defined as ɛ c (κ) c c A 1, (4) where c A is the celerity obtained from the linear fully dispersive theory (Airy). The error is expressed as a function of κ for the same reasons argued by Galan et al. (1). It is known that ω/ k obtained from the above dispersion equation does not satisfy A c g = constant, with A the wave amplitude (Beji and Nadaoka, 1996; Schaffer and Madsen, 1998, e.g.). The proper procedure to assess the linear shoaling of BTEs originally considered the so-called shoaling gradient (Madsen and Sorensen, 199), and, in order to obtain the error for the wave amplitude, a weighted integral of the errors in the shoaling gradient (Chen and Liu, 199). 7

8 Departing from the energy balance equation, Simarro (13) has recently shown that the celerity v g so that A v g = constant for BTEs is not ω/ k but v g = ω k a r, a r (1 + ρ 1ξ ) σ 1 (1 + ρ ξ ) σ (1 + ρ 3 ξ ) σ 3 (1 + ρ 4 ξ ) σ 4, () where σ j depend on β for B96-1, on α for N93-1 and on α, δ, γ, δ h and γ h for M98-1. It is noteworthy that a r = 1 for B96-1, and therefore v g = ω/ k as already claimed in the original work Beji and Nadaoka (1996). For N σ 3 = 9α + 6α 3α + 6α +, σ 4 = 3α + α α + α, (recall that in this case ρ 1 = ρ =, so that σ 1 and σ are irrelevant according to Equation ()). The error in the linear wave amplitude (linear shoaling) is now ɛ s (κ) = vg c g,a 1, (6) with c g,a the group celerity correspoding to Airy theory. We emphasize that the above error is the same as the obtained by Chen and Liu (199) through the integration of the errors in the shoaling gradient. Two-layer equations Two-layer BTEs analysed herein are the ones presented by Simarro et al. (13), and shown in Appendix A.. These equations are an enhancement of those by Lynett and Liu (4) in the same way that M98-1 enhance W9-1. The equations by Simarro et al. (13) include the 1 free parameters shown 8

9 in Table ; the original equations by Lynett and Liu (4) are recovered as a particular case (Table ). Again, the parameters α 1,ɛ α,ɛ, β 1,ɛ, δ ɛ and γ ɛ affect exclusively the nonlinear performance, and will be introduced only in the final part of the work. Authors α 1 α ɛ,1 β 1 β ɛ,1 α α ɛ, δ δ h δ ɛ γ γ h γ ɛ S13- L4- Table : Free coefficients for the two-layer BTEs (martked with ). L4- Lynett and Liu (4), S13- two-layer Simarro et al. (13). 146 The dispersion equation is now { } c ω gh = gk h = 1 + ρ 1ξ 1 + ρ 3 ξ 1 + ρ ξ { D (ξ) }, (7) 1 + ρ ξ 1 + ρ 4 ξ 1 + ρ 6 ξ where ρ j (different that those in the one-layer case) depend on α 1, β 1 and α for L4- and on α 1, β 1, α, δ and γ for S13-. Following the same procedure that Simarro (13) for the one-layer case, now v g = ω k a r, a r (1 + ρ 1ξ ) σ 1 (1 + ρ ξ ) σ (1 + ρ 3 ξ ) σ 3 (1 + ρ 4 ξ ) σ 4 (1 + ρ ξ ) σ (1 + ρ 6 ξ ) σ 6. (8) The errors ɛ c and ɛ s are computed using equations (4) and (6) respectively. It is remarkable the simplicity of v g and the evaluation of the linear shoaling error as compared to the weighted integral of the shoaling grandient error by Chen and Liu (199), specially if we take into account that the expressions for the shoaling gradient are complicated (they are usually not written out). 9

10 The coefficients ρ j and σ j, for both one- and two-layer sets of equations are at Coefficients choice The free coefficients are usually chosen so as to optimize the performance of the equations in a given range of values of κ ω h/g, i.e., κ κ max. Simarro et al. (13) reported that the coefficients are often chosen so as to reduce ɛ c while they can give large errors ɛ s, and proposed new coefficients for different BTEs. For example, for κ max = 3 the coefficient α =.396 that was proposed by Nwogu (1993) gives ɛ c 1% but ɛ s 13%, while setting α =. one gets { ɛ c, ɛ s } 4.%. Hereafter, ɛ l (κ max ) max { ɛ c (κ), ɛ s (κ) }. (9) κ κ max Simarro et al. (13) provided their sets of coefficients minimizing ɛ l for given κ max. Here the coefficient will further be required to safisty the stability conditions presented in the next section Linear stability Flat bed case For the flat bed case, linearizing the BTEs and assuming h = constant it is found the condition for the solutions not to grow indefinitely in time (i.e., stable). For the one-layer case this condition is (Madsen and Schaffer, 1998, e.g.) 1

11 D (ξ) >, (1) where D (ξ) is defined in Equation (). From that equation, the function D (ξ) depends on coefficients ρ j, and for the one-layer case it is well known that the above condition (1) holds if (and only if) all the coefficients ρ j are positive. In fact, for N93-1, the range 1.8 α.4 in Equation (1) is obtained from the Equation (3) and imposing {ρ 3, ρ 4 }. For the two-layer case, as it is shown below, the stability condition is also the one in Equation (1), but now with D (ξ) defined in Equation (7). Again, the stability condition will impose conditions on the free coefficients. Condition in Equation (1) ensures linear stability for the flat bed case, but not for arbitrary uneven bathymetries. While these authors do not know of a way to analyze the linear stability of BTEs for arbitrary bathymetries, from the flat bed case we learn that the stability can depend on the values given to the coefficients: some coefficient sets (and presumably BTEs formulations) should be more stable than others. We will use Fourier series in space to gain a better understanding about this question. Spatial Fourier analysis Let us first consider the SWEs for illustrative purposes. SWEs correspond to the non-dispersive part of BEs and BTEs. The 1D linearized SWEs read (see Equation (A.1) in Appendix A.1) η t + (hu) =, (11a) x u t + g η x =. (11b) 11

12 with η the free surface elevation over the mean sea level, h the water depth at rest and u the depth averaged velocity. Given a spatial domain [, L = π/k], assuming periodic boundary conditions we can write h (x) = ĥ n exp (inkx), η (x, t) = u (x, t) = n= n= n= ˆη n (t) exp (inkx), û n (t) exp (inkx). 198 In this way, the Equations (11) read ˆη n t + ink l ĥ l û n l =, (1a) û n t + inkgˆη n =, (1b) 199 for all n. The above can be written in matrix form as L 11 L t ˆη û = R 1 R 1 ˆη û, (13) where ˆη and û contain the (infinite) Fourier components, L 11 = L are infinite sized identity matrices, R 1 is an infinite diagonal matrix with inkg in the diagonal terms, and R 1 is, in general, an infinite full matrix including the weighted Fourier components of h, ĥn. At this point, we recall that the ODEs system a/ t = M a is stable if the real part of the eigenvalues of the matrix A is (this is the A-stability 1

13 6 7 8 condition). If h is set constant (i.e., flat bed) the matrix R 1 in the Equation (13) becomes diagonal, so that the system is decoupled and for each Fourier component n we get ˆη n t û n = inkh inkg ˆη n û n, (14) i.e., a set of two ODEs with eigenvalues λ ± = ±ink gh, i.e., purely imaginary indicating stability. If the same procedure (Fourier analysis) is applied to the one-layer BTEs we get the same system in Equation (13), but with more complicated matrices. Again, for constant h the submatrices become diagonal, and imposing the eigenvalues of the ODEs system to be purely imaginary one recovers the condition in Equation (1). As seen, for constant h (flat bed case) the Fourier analysis recovers known results. Unfortunately, this approach does not allow to study the stability for arbitrary bathymetries. However, because from a practical point of view it is of interest to improve the stability of the BTEs over uneven bathymetries, we consider in the following to study the simplest uneven bathymetry, i.e. h = h c + h w cos (kx), (1) with h w < h c (to ensure h > ). Similar to condition in Equation (1) for flat beds, the stability of the equations for this simple sinusoidal bathymetry will be a necessary condition, introducing the bed slope, for the coefficients. We remark, however, that it will not be a sufficient condition. The Fourier components ĥn for h in Equation (1) are all null except for 13

14 ĥ = h c and ĥ+1 = ĥ 1 = h w /. In this case, the submatrices in Equation (13), which are full in the general case and diagonal for flat beds, turn out to be banded (tridiagonal for SWEs above, but wider for BTEs due to the inclusion of h and h 3 ). We will follow here the often used method of truncating the system in Equation (13) to a finite number of harmonics, n max, in order to analyse the stability (Shivakumar et al., 1987) Influence of n max To use the truncation method, it is convenient to have some understanding of the influence of the trunction size n max on the analysis. For this purpose, Figure 1 shows the eigenvalues of the system in Equation (13) obtained for n max = 8 and for n max = 1 for SWEs (left) and for L4- (right). The original coefficients for L4- are used, i.e., α 1 =.18, α =.618 and β 1 =.6. The bathymetry is, in both cases, obtained from Equation (1) with k = π/1 m 1, h c = m and h w = 4 m (the bold line in Figure ). We use g = 9.81 m/s. Naturally, the number of eigenvalues increases with n max, since the size of the matrices is proportional to n max. Also, as n max increases the imaginary parts reach larger values, i.e., higher frequencies. This is because the values of the reachable wavelengths, i.e. nk, grow with n max, and larger wavenumbers correspond to larger frequencies from the embedded dispersion relationship. More interestingly, the eigenvalues with smaller imaginary parts, which are the ones showing instability for L4- (positive real parts), are unaffected as the truncation n max increases. Furthermore, the eigenfunctions associated to these eigenvalues also converge as n max grows. For example, Figure shows the eigenfunctions η (free surface elevation), u α,1 (characteristic velocity of 14

15 the upper layer, Appendix A.) and u α, (bottom layer) corresponding to the eigenvalue with real part for L4- (Figure 1, right). Figure shows the results for n max = 8 and for n max = 1, but they are indistinguishable. While the above is not a rigorous demostration, it shows that the truncation method can be used as a tool to analyse the stability of the BTEs over sinusoidal bathymetries. Hereafter we consider n max = 1. Influence of the bathymetric shape Once the influence of n max can be neglected, the eigenvalues will depend on h c, h w, k (bathymetry), g and the BTEs. Figure 3 shows the eigenvalues for four different bathymetries and, for each bathymetry, for five (linearized) equations: N93-1 (with the original α =.396), B96-1 (original β =.), M98-1 (original α =.41, δ =.3917, δ h =.1443, γ =.1, γ h =.13), L4- (original α 1 =.18, α =.618 and β 1 =.6) and S13- (original values for κ max = ). From Figure 3, the equations S13- (for the given coefficients) are unstable in all cases. Actually, their coefficients do not satisfy the flat bed stability condition, and this is why there is instabilty even in the milder cases. Conversely, B96-1 and N93-1 are always stable (of course, SWEs are also always stable, not shown). Finally, M98-1 and L4- are stable for some bathymetries and unstable for other. Because the eigenvalues depend on h c, h w, k and g (and on the equations and corresponding coefficients), applying Buckinghams Π theorem (Buckingham, 1914) one gets 1

16 ν gk = f (Π h h w /h c, Π w kh w, equations (coefficients) ), (16) where ν are the eigenvalues and < Π h < 1 because < h w < h c. Note that Π w is proportional to the maximum bed slope. We define s as the maximum of the real part of all the normalized eigenvalues ν/ gk, so that s > indicates instability. According to the Equation (16) above s = f (Π h, Π w, equations (coefficients) ). (17) Figure 4 shows the influence of Π h and Π w on s for M98-1 for the original coefficients. The area below the line s = 1 1 is the stability region, and the instability function s grows with Π h and Π w. For L4-, the function s also grows with the slope Π w, which is intuitive, but there is not a clear trend for Π h (not shown). For the equations SWEs, B96-1 (with β =.) and N93-1 (with α =.396), s = for the whole range explored in Figure New sets of coefficients For each set of equations (B96-1, N93-1, M98-1, L4-, S13-1), new sets of coefficients are found to optimize the linear performance, i.e., to minimize the error ɛ l, and so that they satisfy stability over flat beds: ρ j ; stability over sinusoidal beds: s = for any Π h and Π w. Since the second condition can not be proved exhaustively, we restrict to check that s = for all combinations of Π h and Π w where Π h {.1,.,.9} 16

17 and Π w {1, }. Note that we only consider large values of Π w since they correspond to the most demanding situations. Also, we emphasize here that the above conditions are only necessary conditions, and do not ensure linear stability for non sinusoidal bathymetries. Table 3 and 4 show the results obtained for the different BTEs considered and for κ max = 3. Further results, including other values of κ max and the nonlinear coefficients, are shown in Appendix B. For the one-layer models (see Table 3) we find that, for B96-1 and N93-1, the values are those obtained by Simarro et al. (13) and Simarro (13) disregarding stability considerations. In these two cases, the global minimum (not requiring stability) is actually stable. We recall that the values improve the original: B96-1 proposed β =. (yielding ɛ l = 9.1% for the same range) and N93-1 proposed α =.396 (ɛ l = 13%). Similar arguments hold for M98-1, but the coefficients for M98-1 in Table 3 are new, since Simarro et al. (13) provided coefficients for κ max =. β α δ δ h γ γ h ɛ l (%) B M N Table 3: Free coefficients for the one-layer BTEs considered and for κ max = Table 4 shows the new coefficients and errors for the two-layer models. For L4-, the errors are clearly larger than those obtained imposing only the stability condition over flat beds. Actually, using the original coefficients for L4- (α 1 =.18, α =.618 and β 1 =.6, which satisfy the flat bed stability condition) the error is ɛ l =.6% for the same range κ 3 17

18 313 (this error is.8% for the proposed, more stable, coefficients). α 1 α β 1 δ δ h γ γ h ɛ l (%) S L Table 4: Free coefficients for the two-layer BTEs considered and for κ max = The interest of the proposed coefficients is shown in Figure. This figure shows, for the same bathymetries in Figure 3, the eigenvalues of the different sets of equations now using the coefficients in Tables 3 and 4. From the figure it is clear that the new coefficients fix the instability problems that showed up for M98-1, L4- and S13- in Figure Numerical stability To further show the usefulness and limitations of the new sets of coefficients, we present the results of the numerical linear stability for the BTEs. We emphasize that the above results are independent of the numerical scheme used to solve the equations. They are also independent of the fact that the equations are written in conservative or non-conservative form. Usual numerical schemes employed to solve BTEs include finite differences (Nwogu, 1993; Wei et al., 199), finite elements (Sorensen et al., 3) and, more recently, finite volume schemes (Fang et al., 1; Lannes and Marche, 1, e.g.). Most often, when using finite volume schemes, the non-dispersive part of the BTEs (that corresponds to SWEs) is written in conservative form and then the numerical scheme applies finite volume techniques for the nondispersive part and finite differences for the dispersive terms. 18

19 We will consider here a simple finite difference scheme, splitting the space discretization and the time-domain solution, as usual when using finite differ- ences or finite elements schemes. For simplicity, we consider periodic bound- ary conditions. We illustrate the numerical scheme for the linearized SWEs in Equation (11). Let here η, u and h be the vectors containing the nodal values for η, u (functions of time) and h respectively, and let M a be a diagonal matrix so that the elements in the diagonal are the components of the vector a. The finite differences discretization of the Equations (11) reads η t u = A η, u A = D M h gd, (18) where is the null matrix and D is the cyclic matrix for the first derivative. If we use second order approximations of the first derivative, D is tridiagonal (with zeros in the diagonal). The problem is therefore reduced to a system of ODEs for the nodal values in Equation (18). Again, to analyse the A-stability we look at the eigenvalues of the matrix A. The stability of the numerical solution is greatly dependent on the spatial discretization. For example, if the SWEs are rewritten in an equivalent form as η t + h u x + h x u =, u t + g η x =, (19a) (19b) 349 the corresponding spatial discretization reads 19

20 η t u = A η u, A = M h D + M D h gd, () For a given bathymetry, h, and given differenciation matrix, D, the matri- ces A and A have different eigenvalues. Further, A is prone to instabilities, while A has shown to be stable for all the tested bathymetries. For example, Figure 6 shows the eigenvalues of A and A for D corresponding to the fourth order approximation of the first derivative and h shown in the figure, with nodes. All the eigenvalues of A are purely imaginary (indicating stability) while the maximum real part of the eigenvalues of A is.3 > (i.e., unstable). If we use the second order approximation of the spatial derivative to build the derivation matrix D, the eigenvalues of A remain purely imag- inary while the real part of the eigenvalues of A reach larger values (not shown). The dispersive terms of the BTEs introduce expressions such as ( h u ) x x = h u x x + u h x, and, therefore, the stability of the scheme can change depending on the way these terms are manipulated. Keeping in mind the above considerations, we will consider the straight- forward discretization of the equations as they are written in Appendix A. In doing so, the spatial discretization of the non-dispersive part of the equations reduces to Equation (18), which is stable. Figure 7 shows the eigenvalues obtained for four different bathymetries (flat, sinusoidal, biharmonic and bar-like) using the same sets of BTEs and

21 coefficients as in Figure 3 (original coefficients). In all cases we use 1 nodes. Similar to Figure 3, S13- is unstable (except for the flat bed and, surprisingly, for the bar-like bathymetry), and M98-1 and L4- are more unstable than simpler models such as B96-1 and N93-1. The results for the coefficients in Tables 3 and 4 are shown in the Figure 8. Compared to the results in Figure 7, the new coefficients are more stable also in the numerical scheme. There are though some unstable cases: N93-1 and M98-1 for the bar-like bathymetry. There are two possible sources for these instabilities: first, we recall that the coefficients are found so as to be stable for sinusoidal bathymetries, but not for any bathymetry, so that the instabilities can arise from the equations (or coefficients) themselves. Second, the spatial discretization can introduce instabilities, particularly in the dispersive terms. Small instabilities such as those in Figure 8 can be balanced introducing some difussion (filters) in the time solution (Wei and Kirby, 199, e.g.) Conclusions In this work we analysed the linear stability of one- and two-layer Boussinesq type equations (BTEs) over uneven bathymetries. Given the complexity of the problem, we have limited the study to sinusoidal bathymetries, so as to include bed slope. This represents an improvement, since the analysis of the linear stability of BTEs was restricted to the flat bed case to date. New sets of coefficients are stable for sinusoidal bathymetries. Even though the stability over sinusoidal bathymetries does not guarantee stability over arbitrary beds, the new coefficients have shown to give more stability in practical problems. 1

22 One main result of the present analysis is that one-layer BTEs seem to be more stable than two-layer ones. Among the one-layer models, the equations by Beji and Nadaoka (1996) seem to be particularly well conditioned, both for stability and linear errors (disperion and shoaling) Acknowledgement The first author thanks support from the Spanish government through the Ramon y Cajal program. Thanks to Ricardo Álvarez.

23 Appendix A. One- and two-layer BTEs We briefly presente the one- and two-layer BTEs equations analyzed in this work. Herein (and also throughout the paper), h is the local water depth, η is the free surface elevation around the mean water level, t is time, g is the gravitational acceleration, u is the depth averaged horizontal velocity, u α is the horizontal velocity at z = z α and = ( / x, / y) is the horizontal gradient Appendix A.1. One-layer equations Non-linear Shallow Water Equations. SWEs for η and u are η + ( (η + h) u) =, t (A.1a) u t + g η + 1 (u u) =. (A.1b) Equations by Beji and Nadaoka (1996). the continuity equation The BTEs for η and u are 41 η + ( (η + h) u) =, t (A.a) and the momentum equation u t + g η + 1 (u u) + t ( f 1 X + f ( η X ) Y ) + t + η Y t [ ( + u (f 1 η) X + f ) ] η (X + ηy) Y + =, (A.b) 3

24 413 where X (hu), Y u and f 1 η h, and f η ηh + h. 3 The equations by Beji and Nadaoka (1996) are for the weakly non-linear case (BEs), because they are an enhancement of the BEs by Peregrine (1967). Applying the same technique to Equations (A.), in order to get the equiva- lent enhanced BTEs we must add the term β ( f 1 (hz ) + f ) Z =, to the left hand side of Equation (A.b), being β a dimensionless free coeffi- cient and Z u t + g η + 1 (u u) Equations by Simarro et al. (13). (13) are for η and u α. Continuity is The equations by Simarro et al. η t + M + (δ δ h ) (h X ) + δ h (h X ) + δ ɛ (hη X ) =, (A.3a) 4 where δ, δ ɛ and δ h are free dimensionless coefficients and ) ( ) M du α + (z α d η h z X α + α d η3 + h 3 Y α, 6 X η t + (du α), 4

25 43 44 with d η+h, z α = αh+α ɛ η (α and α ɛ are free coefficients), X α (hu α ) and Y α u α. Momentum equation is u α t + g η + 1 (u α u α ) + ( ) z α X α + z α t Y α ( η X ) α + η Y α t t [ ( ) ] + u α (z α η) X α + z α η Y α + (X α + ηy α ) + (γ γ h ) h Z + γ h h (hz ) γ ɛ (η (hz ) ) =, (A.3b) 4 where γ, γ ɛ and γ h are free dimensionless coefficients and, here, Z u α t + g η + 1 (u α u α ). The above equations (A.3) reduce to: Galan et al. (1): if α ɛ = ; Madsen and Schaffer (1998): if α ɛ = δ ɛ = γ ɛ = ; Kennedy et al. (1): if δ = δ ɛ = δ h = γ = γ ɛ = γ h = ; Wei et al. (199): if α ɛ = δ = δ ɛ = δ h = γ = γ ɛ = γ h = ; Nwogu (1993): if α ɛ = δ = δ ɛ = δ h = γ = γ ɛ = γ h = and the Boussi- nesq hypothesis is used, i.e., non-linear dispersive terms are ignored. Appendix A.. Two-layer equations Two-layer equations divide the vertical domain into an upper layer (z > ζ 1 ) and a bottom layer (z ζ 1 ). The interface is at ζ 1 = β 1 h + β ɛ,1 η, where

26 β 1 and β ɛ,1 are free coefficients. The enhanced equations by Simarro et al. (13) are for η, u α,1 and u α, and depart form the equations by Lynett and Liu (4). Continuity is η t + M + (δ δ h ) (h X ) + δ h (h X ) + δ ɛ (hη X ) =, (A.4a) 439 with δ, δ ɛ and δ h are free coefficients, and ( M = d 1 u α,1 + z α,1 d 1 η ζ1 + d u α, + (z α, d ζ 1 h X = η t + (d 1u α,1 + d u α, ), ) ( z α,1 d 1 X α,1 + ) X α, + ( z α, d η3 ζ ζ3 1 + h 3 6 ) Y α,1 ) Y α,, with d 1 η ζ 1 and d ζ 1 + h the depths of the layers; u α,1 and u α, the velocities at z α,1 = α 1 h + α ɛ,1 η (in the upper layer) and z α, = α h + α ɛ, η (in the bottom layer), with α 1, α ɛ,1, α, α ɛ, free coefficients; and Y α, u α,, X α, (hu α, ), Y α,1 u α,1 and X α,1 X α, + ζ 1 (Y α, Y α,1 ). The momentum equation is 6

27 u α,1 t + g η + 1 (u α,1 u α,1 ) + t ( η X α,1 t + [ u α,1 ( z α,1 X α,1 + z α,1 + η ( (z α,1 η) X α,1 + z α,1 η ) Y α,1 + t ) Y α,1 Y α,1 ) ] + (X α,1 + ηy α,1 ) + (γ γ h ) h Z + γ h h (hz ) γ ɛ (η (hz ) ) =, (A.4b) 44 with γ, γ ɛ and γ h free coefficients and Z = u α,1 + g η + 1 t (u α,1 u α,1 ). Finally, the matching condition that ensures continuity of the velocity at the interface is u α,1 + (z α,1 ζ 1 ) X α,1 + z α,1 ζ 1 Y α,1 = u α, + (z α, ζ 1 ) X α, + z α, ζ 1 Y α,. (A.4c) The above Equations (A.4) reduce to the two-layer equations by Lynett and Liu (4) if δ = δ ɛ = δ h = γ = γ ɛ = γ h = Appendix B. Full sets of coefficients Tables B. and B.6 the coefficients proposed for the different sets of equations. The linear coefficients ar found to minimize the error ɛ l for the range κ 3 (i.e., κ max = 3) and the non-linear coefficients (α ɛ, δ ɛ, γ ɛ,... ) so as to minimize the non-linear error, here ɛ ɛ, defined by Schaffer (1996) for κ 1. The linear coefficients correspond to those in Tables 3 and 4. 7

28 Imaginary Part 1 1 n max = 8 n max = x 1 3 Imaginary Part 1 1 n max = 8 n max = Real Part x 1 3 Real Part Figure 1: Eigenvalues for SWEs (left) and L4- (right). The small plots are details around the origin. 1 η u α, 1 u α, x (m) Figure : Eigenfunctions corresponding to the eigenvalue with real part in Figure 1 (right). The bold line stands for the bathymetry. 8

29 h(m) 4 bathymetry 6 1 x(m) B h(m) bathymetry 1 x(m) B h(m) 4 bathymetry 6 1 x(m) B96 1 h(m) bathymetry 1 1 x(m) B x 1 3 N x 1 3 N x 1 3 N x 1 3 N x 1 3 M x 1 3 M x 1 3 M x 1 3 M x 1 3 ML4 1.. ML4 1 1 x 1 3 ML4 1 ML4 1 1 x 1 3 S x 1 3 S S13 S13 1 Figure 3: Eigenvalues (horizontal = real part; vertical = imaginary part) for four different bathymetries and for different systems of equations (details in the main text). In all cases n max = 1. 9

30 1e Π w e 1 1e Π h Figure 4: Instability function s = f (Π h, Π w ) for M98-1 with the original coefficients. 3

31 h(m) 4 bathymetry Π h.1 Π w x(m) B h(m) bathymetry Π h.8 Π w x(m) B h(m) 4 bathymetry Π h.1 Π w x(m) B96 1 h(m) bathymetry Π h.8 Π w x(m) B x 1 3 N x 1 3 N x 1 3 N x 1 3 N x 1 3 M x 1 3 M x 1 3 M x 1 3 M x 1 3 ML x 1 3 ML x 1 3 ML4 1 1 x 1 3 ML4 1 1 x 1 3 S x 1 3 S x 1 3 S x 1 3 S x x x x 1 3 Figure : Eigenvalues (horizontal = real part; vertical = imaginary part) for four different bathymetries and for different systems of equations using the coefficients in Tables 3 and 4. 31

32 Imaginary Part A A* Real Part x 1 3 Figure 6: Illustration of the influence of the finite differences discretization on the numerical stability for SWEs. 3

33 h(m) bathymetry 1 1 x(m) B96 1 h(m) 4 6 bathymetry 8 1 x(m) B h(m) 4 bathymetry 6 1 x(m) B h(m) 4 bathymetry 6 1 x(m) B x 1 3 N x 1 3 N x 1 3 N x 1 3 N x 1 3 M x 1 3 M x 1 3 M98 1 x 1 4 M x 1 3 L4.. L4 1 1 L4 x 1 4 L4 1 1 x 1 3 S x 1 3 S13 1 x 1 4 S x 1 3 S x x 1 3 Figure 7: Finite different discretization: eigenvalues (horizontal = real part; vertical = imaginary part) for four different bathymetries and for different systems of equations using the original coefficients. 33

34 h(m) bathymetry 1 1 x(m) B96 1 h(m) 4 6 bathymetry 8 1 x(m) B h(m) 4 bathymetry 6 1 x(m) B h(m) 4 bathymetry 6 1 x(m) B x 1 3 N x 1 3 N x 1 3 N x 1 3 N x 1 3 M x 1 3 M x 1 3 M x 1 4 M x 1 3 L x 1 3 L x 1 3 L4 x 1 4 L4 1 1 x 1 3 S x 1 3 S x 1 3 S x 1 3 S x x x x 1 3 Figure 8: Finite different discretization: eigenvalues (horizontal = real part; vertical = imaginary part) for four different bathymetries and for different systems of equations using the coefficients in Tables 3 and 4. 34

35 Authors β α αɛ δ δh δɛ γ γh γɛ B S G M K W9-1, N93-1. Table B.: Proposed coefficients for the one-layer BTEs. B96-1: Beji and Nadaoka (1996); S13-1: one-layer Simarro et al. (13); G1-1: Galan et al. (1); M98-1: Madsen and Schaffer (1998); K1-1: Kennedy et al. (1); W9-1: Wei et al. (199); N93-1: Nwogu (1993). Authors α1 αɛ,1 α αɛ, β1 βɛ,1 δ δh δɛ γ γh γɛ S L Table B.6: Proposed coefficients for the two-layer BTEs. L4- Lynett and Liu (4), S13- two-layer Simarro et al. (13). 3

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