A new two-dimensional Shallow Water model including pressure effects and slow varying bottom topography

Transcription

1 A new two-dimensional Shallow Water model including pressure effects and slow arying bottom topography Stefania Ferrari Fausto Saleri Abstract The motion of an incompressible fluid confined to a shallow basin with a slightly arying bottom topography is considered. Coriolis force, surface wind and pressure stresses, together with bottom and lateral friction stresses are taken into account. We introduce appropriate scalings into a three-dimensional anisotropic eddy iscosity model; after aeraging on the ertical direction and considering some asymptotic assumptions, we obtain a two-dimensional model, which approximates the three-dimensional model at the second order with respect to the ratio between the ertical scale and the longitudinal scale. The deried model is shown to be symmetrizable through a suitable change of ariables. Finally, we propose some numerical tests with the aim to alidate the proposed model. 1 Introduction Simulation of free surface flows plays an important role in many engineering applications such as in the coastal exploitation and naigation. Classical mechanics (see e.g. [14]) describes the flow of a Newtonian, iscous and incompressible fluid through the three-dimensional Naier-Stokes equations (here called NS3d equations). In particular, we will consider the NS3d equations with an anisotropic stress tensor, where different eddy iscosities appear, in the horizontal and in the longitudinal scales. We further suppose the stresses on the free surface to be gien both by atmospheric pressure and wind; on the bottom a friction force is taken into account. Graity force and Coriolis force both act on the fluid as olume forces. It is well known that the numerical solution of NS3d equations for a free surface flow is generally ery onerous. For this reason, in order to make the simulation of the water leel as efficient as possible, when the ratio between the ertical and the horizontal scales is small, it is usual to consider the so called Shallow Water approximation. In the literature (see e.g. [5]), in order to obtain a Shallow Water approximation (D or 3D) of NS3d model, the iscosity is neglected in the deriation and a posteriori is added in the Shallow Water model by the so-called efficient-iscosity (see e.g. [15]). In this paper, following the idea of Gerbeau and Perthame in [9], we want to include the effect of the iscosity directly in the deriation of a two-dimensional Shallow Water system. In this way, we obtain a (real) iscous Shallow Water (or Saint Venant) system in which the effect of the physical iscosity can Dipartimento di Matematica F.Enriques, Uniersità degli Studi di Milano, ia Saldini 50, 0100 Milano, ferrari@mat.unimi.it Dipartimento di Matematica F.Brioschi, Politecnico di Milano, ia Bonardi 9, 0100 Milano, Fausto.Saleri@polimi.it 1

2 z Free surface η( t,x,y) x H(t,x,y) b(x,y) Rier bottom Figure 1: Vertical cross-section of the domain be recoered. This model, referred to in the sequel as SWd, represents an improement oer the classical model for the description of a rier flow in which the longitudinal scale is larger then the transersal one. Our model can be iewed as an extension of the one-dimensional model proposed in [9] and the differences between the physical NS3d model and this one will be pointed out. Finally, the difference between the physical stress tensor arising from NS3d equations and the approximated stress tensor arising from our model is estimated. This paper is organized as follows. In Section we proide the deriation of the conseratie model SWd. In Section 3 the symmetrization of SWd model, the SSWd model in a notconseratie form is presented; we also discuss there about second order iscous perturbations of SSWd model and about the conergence of its solution to the one of the unperturbed problem, as the perturbation anishes. In Section 4 we present some preliminary numerical experiments are carried out in order to alidate the SSWd model with respect to the classical shallow water system. Deriation of the SWd model Let us consider an incompressible fluid with constant density ρ > 0 in a three dimensional domain U which is normal with respect to the ertical direction z and ertically bounded by the surfaces z = η(x, t) and z = b(x): U := {(t, x, y, z) : t (0, T ), (x, y) Ω, z (b(x, y), η(t, x, y))}, (.1) where T > 0, η : [0, T ] Ω R (T > 0) denotes the eleation, b : Ω R the bottom depth with respect to the same reference leel and H = η b the total height of the fluid from the bottom to the free surface (see Fig.1). We denote by Ω the projection of U on the xy-plane and, for the sake of simplicity, we assume Ω be the following rectangular domain: Ω = {(x, y) R : y ( L /, L /), x (x in, x out )}, (.) where L > 0 and x in < x out (for the sequel, we set L 1 = x out x in ). The goerning equations for the motion of an incompressible fluid in U (0, T ], T > 0, are

3 the Naier-Stokes equations (in the sequel, NS3D) that can be written as: u t + u + (u) + (uw) + 1 p z ϱ = l + 1 ( σ11 ϱ + σ 1 + σ ) 13, z t + (u) + + (w) + 1 p z ϱ = lu + 1 ( σ1 ϱ + σ + σ ) 3, z w t + (uw) + (w) + w z + 1 p ϱ z = g + 1 ( σ31 ϱ + σ 3 + σ ) 33, z = 0, (.3) where = (u,, w) T and p are the elocity and the pressure fields, respectiely, l is the Coriolis coefficient and g is the graity acceleration (oriented downward). σ is the following symmetric tensor (with null trace): µ h D 11 µ h D 1 µ D 13 σ = (σ ij ) = µ h D 1 µ h D µ D 3, (.4) µ D 31 µ D 3 µ h D 33 where D = + ( ) T, µ h, µ are two positie constants, representing the horizontal and ertical eddy iscosity coefficients with µ h << µ (see, for instance, [19]). As usual, the physical stress tensor σ (symmetric), is obtained adding pressure to σ: σ = pi + σ. (.5) Remark.1. For an isotropic Newtonian fluid a unique iscosity coefficient µ usually appears. In this paper we will assume the Shallow Water approximation, then the stress tensor (.4) has to presere isotropy only with respect to the ertical axis, so that the introduction of two different (eddy) iscosities in the tensor is physically justified. Obsere further that tensor (.4) has the same form as the tensor proposed by Leermore and Sammartino in [18], proided one considers the horizontal eddy coefficient µ h to coincide with the bulk iscosity coefficient, in their work µ e. The authors made the physical assumption that µ e should be much smaller than µ h with the rigid lid approximation; in our case, we take µ h and µ e of the same order of magnitude. System (.3) must be completed by the initial condition for the elocity field, for the eleation and by suitable boundary conditions. In particular, the boundary of U can be splitted into 5 different sides: the free surface S, the bottom surface B, the inflow surface L in, the outflow surface L out and the closed surface L c± (see Fig.). On S and on B both dynamical (inoling the stress tensor) and kinematical conditions are set, while on L in, L out and L c± only kinematical conditions will be set. In particular, wind and atmospheric stresses are gien on the free-surface. Therefore, s xy = C W W, s z = p a, (.6) where s is the total three-dimensional stress acting on S, W is the wind elocity, C ia suitable coefficient and p a is the (positie) atmospheric pressure. 3

4 y Inflow boundary n in L in (1,0,0) x in Closed boundary n (0, 1,0) c L c n out ( 1,0,0) Outflow boundary x out x L out (0,1,0) n c+ Closed boundary L c+ Figure : Cut of domain U at leel z = z (b(x, y), η(t, x, y)) At the bottom B we assume that s = k(t, x, y, b), where k > 0 is the friction coefficient. Finally, we assume that the boundary Γ of the two-dimensional domain Ω (.) may be partitioned as follows: Γ = Ω = Γ Γ Γ, (.7) where Γ = Γ + Γ, Γ = {(x, y) Γ : x = x in, y ( L/, L/)}, (.8) Γ = {(x, y) Γ : x = x out, y ( L/, L/)}, Γ = {(x, y) R : x (x in, x out ), y = L/}. (.9) The suffixes in, out and c stay respectiely for inflow, outflow and closed, and refer to a possible physical characterization of the boundary edges. Correspondingly, we define: L = {(t, x, y, z) [0, T ] R 3 : (t, x, y) (0, T ) Γ, z (b(x, y), η(t, x, y))} (.10) and analogously L in, L out and L c±. Normals to L are depicted in Fig.. Following the preious notations, for eery t (0, T ], we complete the Naier-Stokes system with the following set of boundary conditions: on S : on B : on L : 1 ϱ σ n + s = 0, w = η t + η; (.11) 1 ϱ σ n + s = 0, w = b; (.1) n = 0 on L c± n > 0 on L in (.13) n < 0 on L out where n k with k equal to b, s, ±, in and out denotes the outward normals to the bottom, the free-surface and the close boundaries. 4

5 .1 Adimensionalization of the NS3d system In this section, we introduce an adimensional form of the NS3d equations, assuming that the ratio ε between the ertical and the longitudinal scale is small. In particular, let us consider the following absolute scales: for the total length: L 1, for the width: L, for the heigth: A, for the x-component of the elocity: U. (.14) As we are in the Shallow Water assumptions, we consider A L 1 and A L. Moreoer, since we are interested in rier flow simulation, we assume that L L 1. On theses assumptions, we set: ε := A/L 1, γ := L /L 1, ν h := µ h ρul 1, ν := µ ρul 1. (.15) Note that ν h and ν represent the inerse Reynolds numbers with respect to the eddy iscosities µ h and µ, while γ is considered to be a finite ratio such that ε γ. The corresponding deried scales are then: for the time: L 1 /U, for the pressure/density: U, for the y-component of the elocity: V = γu, for the z-component of the elocity: W = εu. (.16) For the sake of simplicity we indicate again by η, b, H, u,, w, p/ϱ, respectiely, eleation, bottom leel, total distance from the bottom leel to the eleation leel, elocity components and specific pressure, after rescaling. The graity acceleration g, the Coriolis coefficient l, the surface stress s and the friction coefficient k, are respectiely rescaled as G, λ, S and α. Moreoer, let us denote again by p a the rescaling of the atmospheric pressure. It is worth mentioning the rescaling criteria for these last quantities. Denoting with square brackets the term whose dimension we need to the rescaling, we hae: [[ ]] 1 p g G = G U ϱ z A, l λ U, L s k αu, s ( S ( = = k x S ( αu [[ p η ϱ x εu, S ]], S [[ ]] p, αγ ϱ y [[ p η ϱ ), εu y γ, p au = ( αuu, αγu, αwεu ). [[ ]] p, αwε ϱ. Asymptotic assumptions and integral aerages ]] [[ ]]) p, p a = ϱ [[ ]]) p = ϱ (.17) On the basis of physical assumptions, we assume that the ertical eddy iscosity is much greater than the horizontal one and that friction on the bottom depends linearly on the (relatie) depth of the rier, taken into account by the parameter ε. We can suppose that: ν = εν 0, ν h = O(ε ), α = εα 0. (.18) 5

6 Rescaling (.3) together with boundary conditions (.11)-(.13) and using (.14) (.17), we obtain the following system: u t + u + (u) + (uw) + 1 p z ϱ λ + ( ) u ν h + ( ) νh u γ + ν h + + ( ) ν u z ε z + ν w, ( γ t + (u) + + (w) ) + 1 p z ϱ λu + ( u ν h + ν hγ ) + + ( ) ν h + ( ν γ ) z ε z + ν w, (.19) ( w ε t + (uw) + (w) ) + w + 1 p z ϱ z G + ( u ν z + ν ε w ) + + ( ν z + ν ε ) w γ + ( ) w ν h, z z = 0. The moduli of the rescaled normals are: N(ε, η) = [ 1 + ε ( ( η ) + 1 γ ( ) )] 1 [ ( η ( ) b, N(ε, b) = 1 + ε + 1 γ ( ) )] 1 b, together with boundary conditions on S and on B respectiely. Therefore, indicating by [σ 11], [σ 1], [σ ], [σ 3], [σ 33] the rescaling of the stress tensor s components, we hae on S: ( ) η p ϱ ν u h + η ( ν ) h u γ ν h + ν u ε z + ν w = = S xn(ε, η), ( η u ν h ν hγ ) + η ( ) p ϱ ν γ h + ν ε z + ν w = = S yn(ε, η), [ ( η p ) ϱ + ν w h z = ε [σ 11] + η η γ [σ 1] + 1 ( ) (.0) η γ [σ ]+ ] η S x N(ε, η) 1 η γ S yn(ε, η) p a N(ε, η), w = η t + u η + η. Note that the third equation aboe is the result of a linear combination of the three equations representing the dynamical condition on the free surface (see (.11)). 6

7 Following the same step, we rescale the conditions on B as follows: ( b p ) ϱ + ν u h + b ( ) νh u γ + ν h ν u ε z ν w = = α 0 un(ε, b), ( b u ν h + ν hγ ) + b ( p ) ϱ + ν γ h ν ε z ν w = = α 0 γ N(ε, [ b), ( p ) ϱ + ν w b h z = ε [σ 11] + b b γ [σ 1] + 1 ( ) b γ [σ ]+ ] + b α 0uN(ε, b) + b α 0γN(ε, b) + α 0 wn(ε, b), w = u b + b. (.1) Now, the idea is to neglect quantities which are O(ε ) in (.19)-(.1). In iew of such approximations, we may then rewrite (.19) as follows: u t + u + (u) + (uw) + 1 p z ϱ = λ + ( ) ν u z ε z + ν w, ( γ t + (u) + + (w) ) + 1 p z ϱ λu + ( ν γ ) z ε z + ν w, 1 p ϱ z = G + ( ) u ν + ( ) (.) ν, z z = 0. Moreoer, as soon as the gradient of the free surface remains bounded and recalling that the bathymetry surface is assumed to be regular on Ω, we hae that: N(ε, η) = 1 + O(ε ), N(ε, b) = 1 + O(ε ). We then complete (.) with upper and lower boundary conditions: on S : η p ϱ + ν u ε z + ν w = S x, η p ϱ + ν γ ε z + ν w = S y, p ϱ = p a, w = η t + u η + η, on B : b p ϱ + ν u ε z + ν w = α 0u, b p ϱ + ν γ ε z + ν w = α 0γ, p ϱ = 0, w = u b + b. (.3) (.4) 7

8 The initial condition and the boundary conditions on (0, T ) Γ are not affected by the rescaling. As we suppose the ertical scale to be small enough with respect to the longitudinal scale, we don t loose much in precision (in the following we will specify exactly how much) if in (.) we substitute to the elocity components and to the pressure, the corresponding integral aerages on the ertical axis from z = b(x, y), (x, y) Ω to z = η(t, x, y), (t, x, y) [0, T ] Ω. Recalling that H = η b and using the notation 1 η(t,x,y) f(t, x, y) = f(t, x, y, z) dz, (.5) H(t, x, y) b(x,y) if in (.) we integrate the first two equations from the bottom to the free surface and the third equation from the bottom to a gien ertical leel z, we obtain on (0, T ] Ω (using (.3) and (.4)): t (Hū) + (Hu ) + (Hu) + (H 1ϱ ) p = λh S x α 0u(t, x, y, b), [ γ t (H ) + (Hu) + ] (H ) + (H 1ϱ ) p = λhū S y γ α 0 (t, x, y, b), 1 ϱ p(t, x, y, z) = p u a + G(η(t, x, y) z) + ν (t, x, y, η) ν u (t, x, y, z)+ +ν (t, x, y, η) ν (t, x, y, z), H t + (Hū) + (H ) = 0. (.6) Recalling that normals to Ω (see Fig.) don t depend on z, and assuming = (ū, ) T, the boundary conditions are now set on (0, T ) Γ by: n = ± (t, x, y) = 0 on (0, T ) Γ, n (t, x, y) = ū(t, x, y) > 0 on (0, T ) Γ, (.7) n (t, x, y) = ū(t, x, y) < 0 on (0, T ) Γ. Initial conditions on {t = 0} Ω are gien by: = 0, H = H 0. (.8) Let us now consider terms with an approximation of O(ε). From (.), (.3), (.4) and (.18), one obtains: 1 ϱ p(t, x, y, z) = S z + G(η(t, x, y) z) + O(ε), (.9) Therefore: u u (t, x, y, z) = O(ε), z (t, x, y, z) = O(ε), z u (t, x, y, b) = O(ε), z (t, x, y, b) = O(ε), z (t, x, y, η) = O(ε), z (t, x, y, η) = O(ε). z (.30) u(t, x, y, z) = ū(t, x, y) + O(ε), (t, x, y, z) = (t, x, y) + O(ε), (.31) 8

9 and straightforward we conclude that u (t, x, y) = ū (t, x, y) + O(ε ), (t, x, y) = (t, x, y) + O(ε ), u = ū + O(ε ). (.3) In order to derie our model, some further restrictions on gien data hae to be taken into account. Firstly, we assume that b = O(ε), p a = O(ε), (.33) that is, we are supposing a slow arying bathymetry and a small atmospheric pressure gradient. Moreoer, as in [5], we suppose the horizontal components of the elocity to admit a linear asymptotic expansion to the second order with respect to ε as: u = ū + εu 1 + O(ε ), = + ε 1 + O(ε ), (.34) where the zero order term is gien by the mean alue of quantities u and respectiely, computed as in (.5). This allows us to deduce from (.31) that: u ϕ = ū ϕ + O(ε), ϕ = + O(ε) where ϕ {t, x, y} (.35) ϕ As in [9], Proposition 4.1, we obtain for the elocity and for the pressure the following approximations to the second order with respect to ε: u(t, x, y, b) = ū(t, x, y) 1 + α 0εH 3ν 0 (H 1ϱ ) p = (Hp a) + G + O(ε ), (t, x, y, b) = H + O(ε ), (H 1ϱ ) p = (Hp a) + G (t, x, y) 1 + α + O(ε 0εH ),, 3ν 0 γ H + O(ε ). (.36) Substituting (.31) and (.3) in (.6), one obtains on (0, T ) Ω, with a precision of O(ε ): (Hū) t + (Hū ) λh (Hp a) γ ( (H ) t + (Hū ) λhū (Hp a) + (Hū ) S + G + (H ) S H t + (Hū) + (H ) = 0. H = x α α ū, 0εH ) 3ν 0 + G H = y α 0γ 1 + α, 0εH 3ν 0 γ (.37) 9

10 Recoering dimensions in (.37) we hae on (0, T ) Ω: (Hū) + (Hū ) + (Hū ) t lh (Hp a) s (H ) t + (Hū ) lhū (Hp a) + (H ) s + g H = x k 1 + kh + g H t + (Hū) + (H ) = 0, ū, 3µ H = y k 1 + 3µ kh, which in compact form reads as: (H ) + (H ) + gh H = t = lh ˆ (Hp a ) s xy k 1 + 3µ kh, (.38) (.39) H t + (H ) = 0. x,y rep- Gradient and diergence operators are meant here as two-dimensional operators, s resents the ector of the first two components of the surface stress, is the two-dimensional ector of the two horizontal elocity components ( = (u, )), while through ector ˆ = (, ū) we represent the Coriolis term. We refer to (.39) as our SWd model, completed by boundary and initial conditions (.7)- (.8). System (.39) has now become two-dimensional, meaning that spatial ariables inoled are only two: x and y; the same way, only two aeraged elocity components are inoled: ū and. Note further, that momentum conseration equations are reduced from three (in (.3)) to only two. Remark.. Let us recall the classical uniscid two-dimensional Shallow Water system (see, for instance, [1]) on (0, T ) Ω: (H ) + (H ) + gh H = t = lhˆ H p a + s sup xy g, (.40) H t + (H ) = 0, C 1 where we are considering notation (.6). The friction term is modeled through the so-called Chézy formula which inoles a suitable proportion constant C 1. Notice, in comparison with our model (.39), that classical model presents a different treatment of the friction term and of the surface stresses, while conectie and pressure terms are left the same. 10

11 .3 Recoering 3d-like informations. Thanks to the continuity equation in (.), recalling the last relation in (.4), we recoer the ertical component of the elocity by the relation: w(t, x, y, z) = u(t, x, y, b(x, y)) b b (x, y) + (t, x, y, b(x, y)) (x, y)+ z u z (t, x, y, r) dr (t, x, y, r) dr. b(x,y) b(x,y) (.41) Moreoer, by (.31), (.41) takes the form w(t, x, y, z) = ū(t, x, y) b (x, y) + (t, x, y) b Then, we calculate the aerage ertical elocity as ū (x, y) (z b(x, y)) (t, x, y)+ w(t, x, y) = ū(t, x, y) b b (x, y) + (t, x, y) (x, y) H H (t, x, y) (t, x, y) + O(ε). (z b(x, y)) (t, x, y) + O(ε). (.4) (t, x, y) ū (t, x, y)+ It is important to notice that we can obtain also the aerage alue of the total stress tensor (.5) with an approximation of O(ε ). In fact, if we consider the rescaling of the total stress tensor (.5), using the notation in (.14) (.17), we hae: [[ σ ϱ p ϱ + ν u h ]] = U ν h u γ + ν hγ ν u ε z + ν ε w ν h γ u + ν hγ p ϱ + ν h z + ν ε γ ν γ ε w ν u ε z + ν ε w γ ν ε z + ν ε w. γ p ϱ + ν w h z Then, approximating to O(ε ), using (.4), (.31) and (.9) we conclude that [[ ]] σ U (p a + G(η(t, x, y) z)) I 3 3. (.43) ϱ Aeraging and recoering dimensions, we obtain, modulus an O(ε ): ( σ ϱ = p a + g H ) I 3 3. (.44) Remark that only pressure terms don t drop through the O(ε ) approximation. 11

12 .4 Further remarks Let us obsere that if we aerage model (.37) along y-axis (where y ( L/, L/)) then it is not possible anymore to extend the asymptotic reasoning described in the last paragraph to obtain a one-dimensional model from the two-dimensional SWd model (.39). It is important to note that the whole asymptotic analysis preiously described starts from the alidity of (.30), but a relation similar to (.30) like: u (t, x, y, z) = O(ε), (t, x, y, z) = O(ε), does not hold as one can erify from (.). In fact, in sight of (.18), it is eident that terms u z and z appear, respectiely, in the first and in the second equation of (.) with an ε factor at the denominator, which permits (.30) to be alid. On the other side, it can be seen that this fact doesn t hold for terms like u and. Remark that an analogous deriation of our model (.39) together with boundary and initial conditions (.7)-(.8) can be obtained with a general bounded domain with a piecewise smooth boundary, proided boundary conditions are of the form n 0, where n doesn t depend on z, or of Dirichlet type. 3 The symmetrized model (SSWd) Let us consider again the SWd system (.38) in conseratie form, denoting, for the sake of simplicity, ū, and µ, respectiely with u, and µ on (0, T ) Ω. This system is completed with initial and boundary conditions (.7)-(.8). In quasi-linear form the system (.38) can be rewritten in the form: V t + A 1(t, x, y, V ) V + A (t, x, y, V ) V = G(t, x, y, V ), (3.1) where V = (Hu, H, H) T, u 0 gh + p a u A 1 (t, x, y, V ) = u u, (3.) and G(t, x, y, V ) = u u A (t, x, y, V ) = 0 gh + p a, lh H p a s x ku 1 + kh 3µ lhu H p a s y 1 + kh 3µ 0 1.

13 Remark that the dependence of A 1, A and G on t, x and y, is due to the explicit dependence of terms p a and s xy on those ariables. The following proposition holds: Proposition 3.1. Consider: A(ξ, t, x, y, V ) = ξ 1 A 1 (t, x, y, V ) + ξ A (t, x, y, V ), ξ R \ {0}, where A 1 and A are gien by (3.). The three eigenalues of the SWd system are then gien by: λ (ξ, t, x, y, V ) = ξ 1 u + ξ, λ 1,3 (ξ, t, x, y, V ) = ξ 1 u + ξ gh + p a (ξ1 + ξ ). (3.3) Therefore, the SWd system is a (first order-quasilinear) strictly hyperbolic system. In fact, since p a is a positie constant (as atmospheric pressure is always positie), we hae that: λ 1 (ξ, t, x, y, V ) < λ (ξ, t, x, y, V ) < λ 3 (ξ, t, x, y, V ). Then, following the definition cited in Serre [3] ol.i, Chapter 3, the model is (quasilinear) hyperbolic. A wide analytical theory on hyperbolic problems has been deeloped in the case they are symmetrizable. Therefore, we will first consider a symmetrization of (first order hyperbolic) SWd system, gien by (3.1), in order to obtain the symmetric system, denoted by SSWd, which we show to be hyperbolic as well. We will then exploit if SSWd model may be considered as the iniscid limit of a sequence of artificial iscous perturbed (second order) parabolic problems. We concentrate our attention in the case when p a is a non-null function of x, y and t. In this case, we postulate that a Shallow Water flow on which an atmospheric pressure is acting, behaes equialently (from a physical point of iew) to a Shallow Water flow with eleation fictitiously increased by the constant alue p a /g with no atmospheric pressure acting on it (see Fig.3). Formalizing this fact, let us define η = η + p a /g, and derie newly the system as in the preious sections (when p a = 0). In practice, we are considering that the free surface is now gien by: We set S = {(t, x, y, z) [0, T ] R 3 s.t. z = η (t, x, y), (t, x, y) [0, T ] Ω}. (3.4) Ĥ = H + p a g, (3.5) obsering that Ĥ is always strictly positie (as p a g ). Taking these considerations into account, if we reduce the NS3d model, we obtain on (0, T ) Ω the system: (Ĥu) t (Ĥ) t + (Ĥu ) + (Ĥu) + (Ĥu) + (Ĥ ) + g + g Ĥ t + (Ĥu) + (Ĥ) = 0. Ĥ lĥ s x k 1 + kĥ 3µ Ĥ lĥu s y k u, 1 + kĥ 3µ, (3.6) 13

14 Fictitiously increased free surface z s supz /g ^ H(t,x,y) η( t,x,y) b(x,y) x Physical free surface Rier bottom Figure 3: Vertical cross-section with eleation fictitiously increased Remark that assumption (.33) on the smallness of p a becomes not restrictie for model (3.6), as we are supposing that no atmospheric pressure is acting. In quasi-linear form system (3.6) reads as: ˆV t + Â1( ˆV ) ˆV + Â( ˆV ) ˆV = Ĝ(t, x, y, ˆV ), (3.7) where we posed ˆV = (Ĥu, Ĥ, Ĥ)T,  1 ( ˆV u 0 gĥ u u u ) = u u,  ( ˆV ) = 0 gĥ, (3.8) and Ĝ(t, x, y, ˆV ) = lĥ s x ku 1 + kĥ 3µ lĥu s y k kĥ 3µ Now we can symmetrize system (3.7) i.e., performing a suitable change of ariables we transform matrices Â1 and  into symmetric ones (the right hand side Ĝ will be consequently changed). Let us introduce the new ector ariable: U = (u,, H g ) T, H g = gĥ = gh + p a, (3.9) where H g, known as the celerity ariable, is always positie een if the effectie eleation H is anishing. In fact we hae: H g p a > 0. (3.10) Using the new set of ariables, we can rewrite system (3.7) in the following form ˆV U U t + Â1( ˆV ) ˆV U U + Â( ˆV ) ˆV U U = Ĝ(t, x, y, ˆV ). (3.11) 14.

15 We debote  0 ( ˆV ) ˆV U Ĥ 0 0 Ĥ 0 0 Ĥu gĥ Ĥ gĥ Ĥ gĥ 1 Ĥ 0 ṷ H, with  0 ( ˆV ) Ĥ Ĥ. (3.1) gĥ 0 0 Ĥ Matrix  0 and their inerse are is well posed as Ĥ is always strictly positie. Multiplying to the left all terms defined in (3.11) by  1 0 ( ˆV ), we finally obtain the symmetric system: U t + S 1(U) U + S (U) U = F (t, x, y, U), (3.13) where: and H g u S 1 (U) = 0 u 0 H, S H g (U) = 0 g H, (3.14) g 0 u 0 F (t, x, y, U ) = l 4gs x H g lu 4gs y H g 0 H g H g 4gku ( 1 + kh g 1gµ 4gk ( ) 1 + kh g 1gµ ). (3.15) We will refer the symmetric system (3.13) as two-dimensional Symmetric Shallow Water system, briefly denoted by SSWd in the sequel. By components, in nonconseratie form on (0, T ] Ω it looks like: u t + u u + u + H g t + u + + H g H g t where = (u, ) T. + (H g ) H g H g l + 4gs H g H g + lu + 4gs = 0, H g x y + + H g H g 4gku ( ) = 0, 1 + kh g 1gµ 4gk ( ) = 0, 1 + kh g 1gµ (3.16) 15

16 In the following, we will always deal with the SSWd system together with initial and boundary conditions (.7)-(.8). Let us now show some analytical properties. Of course, as soon as p a = O(ε), the solutions of (3.16) may be interpreted as the solutions of the preiously SWd model. Proposition 3.. The eigenalues of system SSWd are gien by: λ (ξ, U) = ξ 1 u + ξ, λ 1,3 (ξ, U)ξ 1 u + ξ H g and the SSWd system is a quasilinear strictly hyperbolic system. In fact, as H g > 0 (see (3.10)), we hae that (ξ 1 + ξ ) (3.17) λ 1 (ξ, U) < λ (ξ, U) < λ 3 (ξ, U) (3.18) Then the model is (quasilinear) hyperbolic. It is also important to note that the eigenalues of the SSWd system are the same as the eigenalues of the SWd system. Remark 3.1 (boundary conditions). If 0 < u < H g /, then Γ in and Γ out defined in (.8) are not-characteristic boundaries for SSWd model. Further, on Γ in the number of positie eigenalues is p =, while on Γ out we hae p = 1. For all x 0 Γ in, take ξ = n in (x 0 ) = (1, 0); from (3.17) we hae: while for all x 0 λ 1 (ξ, U) = u H g, λ (ξ, U) = u, λ 3 (ξ, U) = u + H g, Γ out, take ξ = n out (x 0 ) = ( 1, 0): λ 1 (ξ, U) = u H g, λ (ξ, U) = u, λ 3 (ξ, U) = u + H g. On the other side, from boundary conditions (.7), we hae that Γ c defined in (.9) is a characteristic boundary for SSWd model on our bounded domain (.). Further on, we hae on Γ c exactly a null, a positie and a negatie eigenalue. For all x 0 Γ c, take ξ = (0, ±1); from (3.17) we hae: λ 1 (ξ, U) = ± H g λ (ξ, U) = ±, λ 3 (ξ, U) = ± + H g. In the following, we will always consider for the sake of simplicity the following so-called subsonic hypothesis: ( u 0, H ) [ g, 0, H ) g. (3.19) 3.1 Viscous perturbation of the SSWd model Let us consider a iscous perturbations of the SSWd system (3.13), defined on the bounded domain Ω (.) in the eolution time interal [0, T ]. To each equation of the SSWd system we add a Laplacian term multiplied by the artificial iscosity coefficient ν and we complete the system with initial conditions and the following Dirichlet boundary conditions: U ν + S 1 (U ν ) U ν t + S (U ν ) U ν ν U ν = F (t, x, y, U ν ) on (0, T ] Ω U ν Γ = a on Γ [0, T ] (3.0) U ν t=0 = U 0 on Ω {t = 0} 16

17 where a(t, x, y) is a function to be prescribed on (0, T ] Γ and taking its alues in R 3. We used here as in (.13) the same function-name a, to indicate Dirichlet conditions for a iscous problem (system NS3d (.3) or the aboe iscous perturbed SSWd system (3.0)); of course, we are dealing with two different functions, een if they show some logical relationship. At present, an existence and uniqueness result global in time for smooth solutions of parabolic problems like (3.0) is aailable (see e.g. Ladyzenskaia, Solonniko, Ural cea [16], Theorem 7.1, chap. VII, pgg ) only for a smooth and bounded boundary, with initial and boundary conditions sufficiently smooth, satisfying suitable compatibility conditions. Howeer, as far as we know, existence and uniqueness results for initial boundary alue hyperbolic problems like: U t + S 1(U) U + S (U) U = F (t, x, y, U) on Ω (0, T ] U Γ C(a) on Γ [0, T ] (3.1) U t=0 = U 0 on Ω {t = 0} and results on conergence of (3.0) to (3.1) in a suitable space-time norm, are not yet aailable on a bounded domain like (.), een if one smooths its edges. When the domain is a half-space in R n, some literature is aailable on the conergence as ν 0 of perturbed symmetric problems like (3.0) to hyperbolic IVB problems like (3.1). Let us obsere that we may consider our bounded domain Ω (.) as the intersection of four half-spaces: Ω = Ω in Ω out Ω c+ Ω c, where, recalling Fig., Ω in = {(x, y) R s.t. x > x in }, Ω out = {(x, y) R s.t. x < x out }, Ω c+ = {(x, y) R s.t. y > L/}, Ω c = {(x, y) R s.t. y < L/}. (3.) If conditions (3.19) are satisfied and, in iew of boundary conditions (.7), Ω in and Ω out hae not-characteristic boundaries, while Ω c+ and Ω c may hae characteristic boundaries. We will consider separately first the not-characteristic case, then the characteristic case (if = 0). 1. The not-characteristic case: if we set our second order iscous perturbed SSWd model (3.0) into a half-space of the form Ω in (or Ω out ), then by [16], we hae that (3.0) has a unique smooth solution U ν globally in time, while by [0] the hyperbolic problem (3.1) has a unique smooth solution U 0 locally in time. Thanks to Grenier and Guès in [1], we may also conclude that there exists a time T 0 such that the following conergence result is alid: U ν U L ((0,T 0 ) Ω 0 as ν 0. A deeper analysis of the compatibility between the theory exposed by Grenier and Guès in [1] and our case is shown in [7].. The characteristic case: for this case, literature is not so extensie as in the preious one. We hae a result by Guès in [13] for the semi linear case. In this case, we don t hae much information on the admissible set of boundary conditions for the hyperbolic iniscid limit, as we had in the not-characteristic case. Another result, by Grenier in [11], deals with the quasilinear case, but only in the case of a scalar problem. 17

18 Let us end this section by remarking that the classical Shallow Water model (.40) is an hyperbolic model as well, and that it may be symmetrized the same way as our SWd model; we shall refer to this one as to the CSSWd model. In terms of ariable U = (u,, H g) T, where we posed: H g = gh. (3.3) The CSSWd model on (0, T ) Ω looks like: U t + S 1 (U ) U + S (U ) U = G(t, x, y, U ), (3.4) where matrices S 1 and S are the same as in (3.14), while the right hand side G is now gien by: l p a + 4gs x H 4g ku g H g G(t, x, y, U ) = lu p a + 4gs y H 4g k, (3.5) g H g 0 assuming that H g > 0 iff H > 0. Remark 3.. About the introduction of a iscous perturbation, we want to mention the paper by Sammartino and Caflisch in []. They consider the conergence of the three-dimensional incompressible Naier-Stokes equations to the iniscid incompressible Euler equations; the physical second order iscous term which appears in the Naier-Stokes equation is considered to be anishing, thus perturbing the momentum equation in the Euler system, but the continuity equation is left unaltered. In our paper, instead, following Grenier and Guès in [1], we consider an artificial second order iscous perturbation of all the equations of a hyperbolic system (with suitable properties). Finally, it is important to notice that the presence of the atmospheric pressure term p a in (3.13) allows the simulation of real phenomena s.t. storms, where the arying of that quantity is compulsory. In our work it is worth noting the originality of treating atmospheric pressure, where we plugged it into a free surface (reduced) model, while in the oceanographic literature (see e.g. [10]), rigid lid models are considered. 4 Numerical experiments In this section we present an academic test case in order to compare the results of the proposed model with the results obtained with the iscous perturbed CSSWd model, when the atmospheric pressure and the Coriolis effects are neglected (s x = s y = 0 and l = 0). Numerical approximation is based on the space-time approximation introduced in [8]: in particular, we consider piecewise-quadratic finite element in space and a first order semi-implicit finite difference scheme in time. Notice that on the test cases, we will refer to the MKS system of measurement; for the sake of simplicity, we will often omit in the following the unit of measurement. All the numerical results are obtained using the C++ library FreeFem++ [17]. 18

19 4.1 The gaussian hill test case We deal with the spreading of a body of water initially haing a Gaussian-hill distribution. The closed water basin is Ω = [0, 10] [0, 10] with a flat bathymetry b(x, y) = 1. The final time is T = Initial conditions are: C(x, y, 0) = 5 + 5exp(( (x 5) (y 5) + 1)), u 0 = 0, 0 = 0, (4.1) while for any time t (0, T ] the boundary conditions are gien by: u = 0, = 0, C = 5. The ariable C is H g (see (3.9)) when the SSWd model is considered, Hg (see (3.3)) for the CSSWd model. Remark that in the iscous perturbed SSWd model, the atmospheric pressure term is comprised into the celerity term H g, while in the iscous perturbed CSSWd model, pressure is considered apart from the celerity ariable Hg. On a uniform triangular grid (with space-step equal to 0.5) we compare the different behaiors of our perturbed SSWd model with the iscous perturbed CSSWd model, considering different alues of the artificial iscosity parameter ν; we first point out the difference in treating the friction term (letting the atmospheric pressure be a constant) and the atmospheric-pressure term, considering an atmospheric pressure wae acting on the Gaussian hill. In both cases we neglect for simplicity the wind effect and the Coriolis force effect; we set the time-step t = 0.001, the final eolution time T = 0.81 and the physical iscosity parameter µ = 0.. In the following simulations we will compare the isoalues of the eleation η (see the beginning of Section ) at the final time t = T with the ertical cuts of the eleation profiles in y = 5 at t = T obtained by the iscous perturbed SSWd model and the iscous perturbed CSSWd model with artificial iscosity alues ν = 0.1 and ν = respectiely. Friction effects. Let us choose k = 50 and p a = 1. The initial plotting of eleation in the case of our model is ery similar to that for the classical model, as the atmospheric pressure is supposed to be a constant. Remark that our model eoles more rapidly than the classical one; in fact at the final time (Fig. 4(a), Fig. 4(c) and Fig. 5(a), Fig. 5(c)) we notice that the wae depicted by our model is already propagating toward the boundary, loosing much of its initial eleation, while the wae depicted by the classical model (Fig. 4(b), Fig. 4(d) and Fig. 5(b), Fig. 5(d)) is still going down from its initial eleation. It is clear that the classical model is quasi-insensitie with respect to ariations of the artificial iscosity, while our model is ery sensitie to ariations on the artificial iscosity. Physically, water, which is an almost iniscid flow, behaes near to simulations with an artificial iscosity ν = , which is a alue of the same magnitude order of water physical iscosity. This fact is in accordance with respect to the conergence theory recalled in Section 3.1 (een if in particular cases) and with respect to our accurate model deriation exposed in Section 1. Atmospheric pressure effects. In order to point out atmospheric pressure effects, let us consider an experiment where k = 0 and atmospheric pressure is gien by: p a = x + t (4.) 19

20 Since atmospheric pressure is not constant, the initial plotting of eleation for the iscous perturbed CSSWd model is just the same as if pressure is constant (as the classical model is pressure independent). But in the case of our SSWd model, where eleation takes the atmospheric pressure into account, then the plotting of initial eleation is quite different from the constant-pressure case (compare Fig.6(a) and Fig.6(b)). We obsere that eleation η might take negatie alues, as it is in the case of our model (see Fig. 7(a), Fig. 7(c) and Fig. 8(a), Fig. 8(c)). In this situation, both the perturbed CSSWd (see Fig. 7(b), Fig. 7(d) and Fig. 8(b), Fig. 8(d)) and the perturbed SSWd model are sensitie to the change of artificial iscosity. In particular for ν = both the perturbed SSWd model and the perturbed CSSWd model behae much less regularly as for ν = 0.1. Notice that for a low alue of the artificial iscosity parameter ν, some peaks are generated. Obsering the wae profiles, one may conclude that results obtained by our model look closer to physical expectation than the classical one. As a pressure wae moing horizontally toward the positie orientation of x-axis is acting, one expects the wae to be lower on that side; that is what happens by our new model, while the classical model behaes thoroughly unphysically. 5 Conclusions and open problems We newly deried a two-dimensional Shallow Water model (SWd model) on a bounded domain and set physical boundary conditions on elocity. A specification of the nature (characteristic, not-characteristic) of boundary edges is set. An open problem is to demonstrate the conergence of the artificial iscous-perturbed model to our model on a two-dimensional domain, with inhomogeneous initial and boundary data and with different combinations of characteristic and not-characteristic edges. It would be also interesting to find out what is the time-interal while our system eoles smoothly, in relation with initial and boundary data and supposing that we are in the subsonic assumptions (3.19). Another task which could be performed, is to find out some relations between initial and boundary conditions such that the subsonic hypothesis, which are supposed a priori, are effectiely point-wise fulfilled. Acknowledgments We thank warmly Dr. Debora Amadori (Uniersità Statale degli Studi dell Aquila, Dipartimento di Matematica Pura e Applicata) for the substantial help she gae us in structuring the paper and in punctualizing the analytical setting. We are also grateful to Prof. Marco Sammartino (Uniersità Statale degli Studi di Palermo, Dipartimento di Matematica e Applicazioni): his contribution in erifying our model has been of greatest help. We finally owe much gratitude to Prof. Benoît Perthame (École Normale Supérieure - DMA, INRIA Rocquencourt) for the many suggestions he gae us in order to improe the present work. 0

21 Eleazione al tempo t= 0.81sec. IsoValue Eleazione al tempo t= 0.81sec. IsoValue (a) (b) (c) (d) Figure 4: (a) Final total eleation for the gaussian hill case: the iscous perturbed SSWd model, ν = , friction k = 50, atmospheric pressure p a = 1. (b) Final total eleation for the Gaussian hill case: the iscous perturbed CSSWd model, ν = , friction k = 50, atmospheric pressure p a = 1 (c) Eleation profile of the final total eleation for the Gaussian hill case: the iscous perturbed SSWd model, ν = , friction k = 50, atmospheric pressure p a = 1 (d) Eleation profile of the final total eleation for the Gaussian hill case: the iscous perturbed CSSWd model, ν = , friction k = 50, atmospheric pressure p a = 1 1

22 Eleazione al tempo t= 0.81sec. IsoValue Eleazione al tempo t= 0.81sec. IsoValue (a) (b) (c) (d) Figure 5: (a) Final total eleation for the Gaussian hill case: the iscous perturbed SSWd model, ν = 0.1, friction k = 50, atmospheric pressure p a = 1 (b) Final total eleation for the Gaussian hill case: the iscous perturbed CSSWd model, ν = 0.1, friction k = 50, atmospheric pressure p a = 1 (c) Eleation profile of the final total eleation for the Gaussian hill case: the iscous perturbed SSWd model, ν = 0.1, friction k = 50, atmospheric pressure p a = 1 (d) Eleation profile of the final total eleation for the Gaussian hill case: the iscous perturbed CSSWd model, ν = 0.1, friction k = 50, atmospheric pressure p a = 1

23 (a) (b) Figure 6: (a) Initial total eleation for the Gaussian hill case: the iscous perturbed SSWd model, atmospheric pressure p a = x+t (b) Initial total eleation profile for the Gaussian hill case: the iscous perturbed CSSWd model, atmospheric pressure p a = x + t 3

24 Eleazione al tempo t= 0.81sec. IsoValue Eleazione al tempo t= 0.81sec. IsoValue (a) (b) (c) (d) Figure 7: (a) Final total eleation for the Gaussian hill case: the iscous perturbed SSWd model, ν = , friction k = 0, atmospheric pressure p a = x + t (b) Final total eleation for the Gaussian hill case: the iscous perturbed CSSWd model, ν = , friction k = 0, atmospheric pressure p a = x + t (c) Eleation profile of the final total eleation for the Gaussian hill case: the iscous perturbed SSWd model, ν = , friction k = 0, atmospheric pressure p a = x + t (d) Eleation profile of the final total eleation for the Gaussian hill case: the iscous perturbed CSSWd model, ν = , friction k = 0, atmospheric pressure p a = x + t 4

25 Eleazione al tempo t= 0.81sec. IsoValue Eleazione al tempo t= 0.81sec. IsoValue (a) (b) (c) (d) Figure 8: (a) Final total eleation for the Gaussian hill case: the iscous perturbed SSWd model, ν = 0.1, friction k = 0, atmospheric pressure p a = x + t (b) Final total eleation for the Gaussian hill case: the iscous perturbed CSSWd model, ν = 0.1, friction k = 0, atmospheric pressure p a = x+t (c) Eleation profile of the final total eleation for the Gaussian hill case: the iscous perturbed SSWd model, ν = 0.1, friction k = 0, atmospheric pressure p a = x + t (d) Eleation profile of the final total eleation for the Gaussian hill case: the iscous perturbed CSSWd model, ν = 0.1, friction k = 0, atmospheric pressure p a = x + t 5

26 References [1] V.I.Agoshko, D.Ambrosi, V.Pennati, A.Quarteroni, F.Saleri, Mathematical and numerical modelling of shallow water flow, Comput. Mech. 11 (1993), [] V.I.Agoshko, A.Quarteroni, F.Saleri, Recent deelopments in the numerical simulation of shallow water equations. Boundary conditions, Appl. Numer. Math. 15 (1994), [3] I.Babuska, The Finite Element Method with Penalty, Math. Comp. 7 (1973), 1 8 [4] S.C.Brenner, L.R. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, 15 (1994), Springer-Verlag, New-York [5] C.N.Dawson, M.L.Martinez-Canales, A characteristic-galerkin approximation to a system of shallow water equations, Numer.Math. 86 (000), [6] R.E.Ewing, T.F.Russell, M.F.Wheeler, Conergence analysis of an approximation of miscible displacement in porous media by mixed finite element methods and a modified method of characteristics, Comput. Methods Appl. Mech. Engrg. 47 (1984), 73 9 [7] S.Ferrari, A new two-dimensional Shallow Water model: physical, mathematical and numerical aspects, PhD Thesis, a.a. 00/003, Dottorato M.A.C.R.O., Uniersità degli Studi di Milano. [8] S.Ferrari, Conergence analysis of a space-time approximation to a two-dimensional system of Shallow Water equations, in preparation. [9] J.F.Gerbeau, B.Perthame, Deriation of iscous Saint-Venant system for laminar shallow water; numerical alidation, Discrete Contin. Dyn. Syst. Ser. B 1 (001), [10] R.H. Goodman, A.J. Majda, D.W. Mclaughlin, Modulations in the leading edges of midlatitude storm tracks, SIAM, J.Appl.Math. 6 (00), [11] E.Grenier, Boundary layers for parabolic regularizations of totally characteristic quasilinear parabolic equations, J.Math.Pures Appl. 76 (1997), [1] E.Grenier, O.Guès, Boundary layers for iscous perturbations of noncharacteristic quasilinear hyperbolic problems, J. Differential Equations 143 (1998), [13] O.Guès, Perturbations isqueuses de problèmes mixtes hyperboliques et couches limites, Ann. Inst. Fourier (Grenoble) 45 (1995), [14] M.E.Gurtin, An introduction to continuum mechanics, 1981, Academic Press, New York [15] J.M.Herouet, A.Watrin, Code TELEMAC (système ULYSSE): Résolution et mise en oeure des équations de Saint-Venant bidimensionnelles, Théorie et mise en oeure informatique, Rapport EDF HE43/87.37 (1987). [16] O.A.Ladyzenskaja, V.A.Solonniko, N.N.Ural cea, Linear and quasilinear equations of parabolic type, American Mathematical Society, 3, Proidence, Rhode Island. [17] F.Hecht, O.Pironneau, FreeFem++:Manual ersion 1.3, FreeFem++ is a free software aailable at: [18] D.Leermore, M.Sammartino, A shallow water model with eddy iscosity for basins with arying bottom topography, Nonlinearity 14 (001), [19] E.Miglio, A.Quarteroni, F.Saleri, Finite element approximation of a quasi 3D shallow water equation, Comput. Methods Appl. Mech. Engrg 174 (1999), [0] J.Rauch, F.Massey, Differentiability of solutions to hyperbolic initial-boundary alue problems, Trans. Amer. Math.Soc., 189 (1974),

27 [1] T.F.Russell, Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media, SIAM J. Numer. Anal. (1985), [] M.Sammartino, R.E.Caflisch, Zero iscosity limit for analytic solutions of the Naier Stokes equations on a half-space. I.Existence for Euler and Prandtl Equations; II.Construction of the Naier Stokes solution., Comm. Math. Physics 19 (1998), and [3] D.Serre, Sytems of conseration laws, I and II, 1996, Cambridge Uniersity Press, Cambridge [4] A.Quarteroni, A.Valli, Numerical approximation of partial differential equations, 1991, Springer Verlag, Berlin [5] G.B. Whitham, Linear and nonlinear waes, 1974, John Wiley & Sons, New York. 7