Stefania Ferrari 1 and Fausto Saleri 2

Transcription

1 ESAIM: M2AN Vol. 38, N o 2, 2004, pp DOI: 0.05/m2an: ESAIM: Mathematical Modelling and Numerical Analysis A NEW TWO-DIMENSIONAL SHALLOW WATER MODEL INCLUDING PRESSURE EFFECTS AND SLOW VARYING BOTTOM TOPOGRAPHY Stefania Ferrari and Fausto Saleri 2 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. Mathematics Subject Classification. 35L65, 65M60. Receied: January 2, 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. [2]) describes the flow of a Newtonian, iscous and incompressible fluid through the three-dimensional Naier Stokes equations (here called NS3d equations). 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. [ 4, 4]), in order to obtain a 2D Shallow Water approximation of NS3d model, atmospheric pressure and wind effects on the free surface, bottom topography and friction effects on the bottom through the Chézy law are taken into account; the iscosity is neglected in the deriation and a posteriori is added in the Shallow Water model by the so-called efficient-iscosity. The deriations of such models are heuristic and often unclear, een though they hae been used in hydraulics for oer twenty years. In particular, we will refer as classical Shallow Water model to the 2D model presented in []. Keywords and phrases. Naier-Stokes equations, Saint Venant equations, free surface flows. MOX, Dipartimento di Matematica F. Brioschi, Politecnico di Milano, ia Bonardi 9, 2033 Milano, Italy. Stefania.Ferrari@polimi.it 2 MOX, Dipartimento di Matematica F. Brioschi, Politecnico di Milano, ia Bonardi 9, 2033 Milano Italy. Fausto.Saleri@polimi.it c EDP Sciences, SMAI 2004

2 22 S. FERRARI AND F. SALERI In [7] Gerbeau and Perthame started from the free surface two-dimensional Newtonian Naier Stokes equations, taking friction effects on the (flat) bottom through a Naier condition into account and graity force as the only olume force. No stresses on the free-surface were taken into account and no turbulence model for the stress tensor was considered, although the Reynolds number associated to the physical phenomenon is ery large. The effect of the iscosity is recoered in the modelling of the friction term of zeroth order and replaces the classical Chézy term. They deried rigorously a D Shallow Water model which approximates the free surface two-dimensional newtonian Naier Stokes equations at the second order with respect to the ratio between the ertical and the longitudinal scales. A lot of articles are also present in the literature concerning the numerical discretization of systems of Shallow Water type, see e.g. Jin in [5] and Kurgano et al. in [6]. In the present paper we improe the classical 2D Shallow Water model by deriing a new 2D Shallow Water model, extending the rigorous deriation presented in [7]; we thus obtain a 2D hyperbolic system. We won t face to the difficulties of discretizing an hyperbolic system as in [5] or in [6]; in fact we will perturb our system through an artificial iscous term of second order and then deal (about numerical discretization) with a parabolic system as in [6]. We will start from the NS3d equations including a non-flat bottom topography with small slopes; a friction force is taken into account on the bottom and modeled through a Naier condition. Surface stresses are considered on the free surface, gien both by atmospheric pressure and wind. It is important to notice that considering atmospheric pressure effects allows the simulation of real phenomena such as 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. [8]), rigid lid models are considered. Graity force and Coriolis force both act on the fluid as olume forces. We also add a turbulence model for the stress tensor in NS3d, which we consider to present anisotropy with respect to the ertical and the longitudinal scale: therefore two different eddy iscosities (ertical and longitudinal) are introduced following [8]. The effect of the eddy (ertical) iscosity is recoered in the modelling of the friction term of zeroth order and replaces the classical Chézy term. The approximation between the physical NS3d model and our new model as well as the approximation between the physical stress tensor arising from NS3d equations and the approximated stress tensor arising from our model, is of the second order with respect to the ratio between the ertical and the longitudinal scales. We will refer to our new Shallow Water 2D model as the SW2d model. This paper is organized as follows. In Section 2 we proide the deriation of the conseratie model SW2d. In Section 3 the symmetrization of SW2d model, the SSW2d model in a not-conseratie form is presented; we also discuss there about second order iscous perturbations of SSW2d model and about the conergence of its solution to the one of the unperturbed problem, as the perturbation anishes. In Section 4 some preliminary numerical experiments on test cases are carried out in order to compare SSW2d model with respect to the classical Shallow Water model. 2. Deriation of the SW2d 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)andz = b(x): U := {(t, x, y, z) : t (0,T), (x, y) Ω, z (b(x, y),η(t, x, y))}, (2.) 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

3 A NEW 2D SHALLOW WATER MODEL INCLUDING PRESSURE EFFECTS 23 z Free surface η( t,x,y) x H(t,x,y) b(x,y) Rier bottom Figure. Vertical cross-section of the domain. (see Fig. ). 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 2 : y ( L 2 /2,L 2 /2),x (x in,x out ) }, (2.2) where L 2 > 0andx in <x out are the inflow and outflow abscissae of the portion of fluid flow, we are focusing on. In fact we consider our fluid flow to coer a total lenght L much larger than the segment (x in,x out ). The goerning equations for the motion of an incompressible fluid in U (0,T], T>0, are the Naier Stokes equations (in the sequel, NS3D) that can be written as: u t + u2 + (u) t + (u) w t + (uw) =0, + (uw) + p z ϱ = l + ϱ + p ϱ = lu + ϱ + w2 z + p ϱ z = g + ϱ (w) z + (w) ( σ + σ 2 + σ ) 3, z + σ 22 + σ ) 23, z ( σ3 + σ 32 + σ ) 33, z ( σ2 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). To obtain an appropriate form for the stress tensor σ appearing in (2.3), we introduce a turbulence model which is gien trough a linear constitutie relationship between σ and the strain-rate tensor D, which is the traceless, symmetric tensor gien by: (2.3) D = +( ) T. (2.4) Following Leermore and Sammartino in [8], we obsere that this relationship cannot be assumed to be isotropic because the horizontal and the ertical length scales will be of different orders in a shallow water approximation. Because the scale of the horizontal eddies will be much larger than that associated with ertical eddies, the eddy iscosity associated with horizontal eddies (µ h ) will be much larger than that associated with ertical eddies (µ ), i.e. µ h >> µ (see also [9]). The relationship should be symmetric with respect to the ertical direction; that is to say, it should be inariant under rotations about a ertical axis. In [8] authors introduced a further eddy iscosity, say the bulk iscosity µ e and they made the physical assumption that µ e should be much smaller than µ h ; but they were dealing with the rigid lid approximation; in our case, without any rigid lid approximation, we take µ h and µ e of the same order of magnitude.

4 24 S. FERRARI AND F. SALERI The null-trace symmetric stress tensor σ is therefore gien by: µ h D µ h D 2 µ D 3 σ =(σ ij )= µ h D 2 µ h D 22 µ D 23. (2.5) µ D 3 µ D 32 µ h D 33 As usual, the physical total stress tensor σ T (symmetric), is obtained adding pressure to σ as: σ T = pi + σ. (2.6) System (2.3) 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 split 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. 2). 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 sup xy = C W W, s sup z = p a, (2.7) where s sup is the total three-dimensional stress acting on S, W is the wind elocity, C is a suitable coefficient and p a is the (positie) atmospheric pressure. At the bottom B we assume that s bfr = k(t, x, y, b), where k>0isthe friction coefficient. Finally, we assume that the boundary Γ of the two-dimensional domain Ω (2.2) may be partitioned as follows: where Γ c = Γ c+ Γ c, Γ = Ω =Γ in Γ out Γ c, (2.8) Γ in = {(x, y) Γ : x = x in, y ( L/2,L/2)}, (2.9) Γ out = {(x, y) Γ : x = x out, y ( L/2,L/2)}, Γ c± = { (x, y) R 2 : x (x in,x out ), y = L/2 }. (2.0) 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)) } (2.) and analogously L in, L out and L c±. Normals to L are depicted in Figure 2. Following the preious notations, for eery t (0,T], we complete the Naier Stokes system with the following set of boundary conditions: on S : ϱ σ T n s + s sup = 0, w = η (2.2) t + η; on B : on L : ϱ σ T n b + s bfr = 0, w = b; n ± =0 onl c±, n in > 0 on L in, n out < 0 on L out, (2.3) (2.4) where n k with k equal to b, s, ±, in and out denotes the outward normals to the bottom, the free-surface and the closed boundaries.

5 A NEW 2D SHALLOW WATER MODEL INCLUDING PRESSURE EFFECTS 25 y Inflow boundary Closed boundary n (0,,0) c n in L in (,0,0) x in L c n out (,0,0) Outflow boundary x out x L out (0,,0) nc+ Closed boundary L c+ Figure 2. Cut of domain U at leel z = z (b(x, y),η(t, x, y)). 2.. 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, for the width: L 2, for the height: A, for the x-component of the elocity: U. (2.5) As we are in the Shallow Water assumptions, we consider A L and A L 2. On these assumptions, we set: ε := A/L, γ := L 2 /L, ν h := µ h ρul, ν := µ ρul (2.6) 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 /U, for the y-component of the elocity: V = γu, for the pressure/density: U 2, for the z-component of the elocity: W = εu. (2.7) 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, thesurfacestresss sup and the friction coefficient k, are respectiely rescaled as G, λ, S sup 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: [[ ]] p g G = G U 2 ϱ z A, l λ U, L 2 ( [[ ]] [[ ]] [[ ]]) p η p η p s sup S sup x, S sup y,p a = (S sup x εu 2 εu 2 ), S sup y ϱ ϱ ϱ γ,p au 2, (2.8) k αu,

6 26 S. FERRARI AND F. SALERI s bfr = k ( αu [[ ]] p,αγ ϱ [[ ]] p,αwε ϱ [[ ]]) p ϱ = ( αuu 2,αγU 2,αwεU 2) Asymptotic assumptions and integral aerages On the basis of physical assumptions, we assume that the ertical eddy iscosity is much larger 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 2 ),α= εα 0. (2.9) Rescaling (2.3) together with boundary conditions (2.2) (2.4) and using (2.5) (2.8), we obtain the following system: u t + u2 + (u) + (uw) + p z ϱ λ + ( ) u 2ν h + ( ) νh u γ 2 + ν h + ( ) ν u z ε 2 z + ν w, ( γ 2 t + (u) (w) ) + p z ϱ λu + ( u ν h + ν hγ 2 ) + ( ) 2ν h + ( ν γ 2 ) z ε 2 z + ν w, (2.20) ( w ε 2 t + (uw) + (w) ) + w2 + p z ϱ z G + ( u ν z + ν ε 2 w ) + ( ν z + ν ε 2 ) w γ 2 + ( ) w 2ν h, z z =0. The moduli of the rescaled normals are: N(ε, η) = [ +ε 2 ( ( η ) 2 + γ 2 ( ) )] [ ( 2 2 ( ) 2 η b, N(ε, b) = +ε 2 + γ 2 ( ) )] 2 2 b, together with boundary conditions on S and on B respectiely. Therefore, indicating by [σ T ], [σ T 2 ], [σ T 22 ], [σ T 23 ], [σ T 33 ] the rescaling of the stress tensor s components, we hae on S: ( ) η p ϱ 2ν u h + η ( ν ) h u γ 2 ν h + ν u ε 2 z + ν w = S sup x N(ε, η), ( η u ν h ν hγ 2 ) + η ( ) p ϱ 2ν γ 2 h + ν ε 2 z + ν w = S sup y N(ε, η), [ ( p ) 2 ϱ +2ν w η h z = ε2 [σ T ]+ 2 η η γ [σ T 2]+ ( ) 2 η γ 2 [σ T 22]+ (2.2) ] η S sup x N(ε, η) η γ 2 S sup y N(ε, η) p a N(ε, η), w = η t + u η + η

7 A NEW 2D SHALLOW WATER MODEL INCLUDING PRESSURE EFFECTS 27 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 (2.2)). Following the same step, we rescale the conditions on B as follows: ( b p ) ϱ +2ν u h + b ( ) νh u γ 2 + ν h ν u ε 2 z ν w = α 0uN(ε, b), ( b u ν h + ν hγ 2 ) + b ( p ) ϱ +2ν γ 2 h ν ε 2 z ν w = α 0γ 2 N(ε, b), [ ( p ) 2 ϱ +2ν w b h z = ε2 [σ T ]+ 2 b b γ [σ T 2]+ ( ) 2 b γ 2 [σ T 22] (2.22) ] + b α 0uN(ε, b)+ b α 0γN(ε, b)+α 0 wn(ε, b), w = u b + b Now, the idea is to neglect quantities which are O(ε 2 ) in (2.20) (2.22). In iew of such approximations, we may then rewrite (2.20) as follows: ( ν u t + u2 + (u) + (uw) + p z ϱ = λ + u z ε 2 z + ν w ( γ 2 t + (u) (w) ) + p z ϱ λu + ( ν γ 2 z ε 2 p ϱ z = G + ( ) u ν + ( ) ν, z z =0. ), z + ν ) w, (2.23) 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) =+O(ε 2 ), N(ε, b) =+O(ε 2 ). We then complete (2.23) with upper and lower boundary conditions: on S : η p ϱ + ν u ε 2 z + ν w = S sup x, η p ϱ + ν γ 2 ε 2 z + ν w = S sup y, p ϱ = p a, w = η t + u η + η, on B : b p ϱ + ν u ε 2 z + ν w = α 0u, b p ϱ + ν γ 2 ε 2 z + ν w = α 0γ 2, p ϱ =0, w = u b + b The initial condition and the boundary conditions on (0,T) Γ are not affected by the rescaling. (2.24) (2.25)

8 28 S. FERRARI AND F. SALERI 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 (2.23) 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) Ωtoz = η(t, x, y), (t, x, y) [0,T] Ω. Recalling that H = η b and using the notation f(t, x, y) = H(t, x, y) η(t,x,y) b(x,y) f(t, x, y, z)dz, (2.26) if in (2.23) 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, weobtainon(0,t] Ω (using (2.24) and (2.25)): t (Hū)+ (Hu2 )+ (Hu)+ (H ϱ ) p = λh S sup x α 0 u(t, x, y, b), [ γ 2 t (H )+ (Hu)+ ] (H2 ) + (H ϱ ) p = λhū S sup y γ 2 α 0 (t, x, y, b), 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) (2.27) +ν (t, x, y, η) ν (t, x, y, z), H t + (Hū)+ (H ) =0. Recalling that normals to Ω (see Fig. 2) 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 c, n in (t, x, y) =ū(t, x, y) > 0 on (0,T) Γ in, (2.28) n out (t, x, y) = ū(t, x, y) < 0 on (0,T) Γ out. Initial conditions on {t = 0} Ω are gien by: = 0, H = H 0. (2.29) Let us now consider terms with an approximation of O(ε). From (2.23) (2.25) and (2.9), one obtains: ϱ p(t, x, y, z) =S sup z + G(η(t, x, y) z)+o(ε), (2.30) 2 u (t, x, y, z) =O(ε), z2 u (t, x, y, b) =O(ε), z u (t, x, y, η) =O(ε), z (2.3) 2 (t, x, y, z) =O(ε), (t, x, y, b) =O(ε), (t, x, y, η) =O(ε). z2 z z Therefore: u(t, x, y, z) =ū(t, x, y)+o(ε), (t, x, y, z) = (t, x, y)+o(ε), (2.32) and straightforward we conclude that u 2 (t, x, y) =ū 2 (t, x, y)+o(ε 2 ), 2 (t, x, y) = 2 (t, x, y)+o(ε 2 ), u =ū + O(ε 2 ). (2.33) 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), (2.34)

9 A NEW 2D SHALLOW WATER MODEL INCLUDING PRESSURE EFFECTS 29 that is, we are supposing a slow arying bathymetry and a small atmospheric pressure gradient. Moreoer, as in [23], 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 + O(ε 2 ), = + ε + O(ε 2 ), (2.35) where the zero order term is gien by the mean alue of quantities u and respectiely, computed as in (2.26). This allows us to deduce from (2.32) that: u ϕ = ū ϕ + O(ε), ϕ = + O(ε) where ϕ {t, x, y} (2.36) ϕ As in [7], Proposition 4., 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) + α 0εH 3ν 0 (H ϱ ) p = (Hp a) + G H O(ε2 ), (t, x, y) + α + O(ε 0εH 2 ), 3ν 0 γ 2 (H ϱ ) p = (Hp a) + G H O(ε2 ). + O(ε 2 ), (t, x, y, b) = (2.37) Substituting (2.32) and (2.33) in (2.27), one obtains on (0,T) Ω, with a precision of O(ε 2 ): (Hū) t + (Hū2 ) λh (Hp a) γ 2 ( (H ) t + (Hū ) λhū (Hp a) + (Hū ) H t + (Hū)+ (H ) =0. Recoering dimensions in (2.38) we hae on (0,T) Ω: (Hū) + (Hū2 ) t (H ) t lh (Hp a) + (Hū ) lhū (Hp a) + G H 2 2 = α 0 S sup x + α ū, 0εH ) 3ν 0 + (H 2 ) + G 2 S sup y α 0γ 2 + (Hū ) + (H 2 ) H t + (Hū)+ (H ) =0, H 2 = + α, 0εH 3ν 0 γ 2 + g H 2 2 = k s sup x + 3µ kh ū, + g H 2 2 = k s sup y + 3µ kh, (2.38) (2.39)

10 220 S. FERRARI AND F. SALERI which in compact form reads as: t (H )+ (H )+gh H = lhˆ k (Hp a ) s sup xy + 3µ kh, H t + (H ) =0. (2.40) Gradient and diergence operators are meant here as two-dimensional operators, s sup x,y represents 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 (2.40) as our SW2d model, completed by boundary and initial conditions (2.28), (2.29). System (2.40) 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. Notice further, that the number of momentum conseration equations has been reduced from three (in (2.3)) to only two. Remark that although we are supposing the bottom surface is slow arying (see (2.34)), in our model (2.39) we don t hae any term like b. In fact all the terms connected to the bottom topography arying are canceled through aeraging NS3d equations, taking boundary conditions on the bottom surface into account. Remark 2.. Let us recall the classical iniscid two-dimensional Shallow Water system (see, for instance, []) on Ω (0,T]: t (H )+ (H )+gh H = lhˆ H p a + s sup xy g C 2, (2.4) H + (H ) =0, t where we are considering notation (2.7). The friction term is modeled through the so-called Chézy formula which inoles a suitable proportion constant C. Notice, in comparison with our model (2.40), 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 Recoering 3d-like informations Thanks to the continuity equation in (2.23), recalling the last relation in (2.25), 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 b(x,y) u (t, x, y, r) dr z (t, x, y, r) dr. (2.42) b(x,y) Moreoer, by (2.32), (2.42) takes the form w(t, x, y, z) =ū(t, x, y) b (x, y)+ (t, x, y) b (x, y) (z b(x, y)) ū (t, x, y) (z b(x, y)) (t, x, y)+o(ε). (2.43)

11 A NEW 2D SHALLOW WATER MODEL INCLUDING PRESSURE EFFECTS 22 Then, we calculate the aerage ertical elocity as w(t, x, y) =ū(t, x, y) b b (x, y)+ (t, x, y) (x, y) H (t, x, y) ū 2 (t, x, y) H 2 (t, x, y) (t, x, y)+o(ε). (2.44) It is important to notice that we can obtain also the aerage alue of the total stress tensor (2.6) with an approximation of O(ε 2 ). In fact, if we consider the rescaling of the total stress tensor (2.6), using the notation in (2.5) (2.8), we hae: [[ σt ϱ p ϱ +2ν u h ν h γ u + ν hγ ]] = U 2 ν h u γ + ν hγ p ϱ +2ν h ν u ε z + ν ε w ν γ ε z + ν ε γ w z + ν ε w z + ν ε w. γ p ϱ +2ν w h z ν u ε γ ν ε Then, approximating to O(ε 2 ), using (2.43), (2.32) and (2.30) we conclude that [[ σt ϱ Aeraging and recoering dimensions, we obtain, modulus an O(ε 2 ): ]] U 2 (p a + G(η(t, x, y) z)) I 3 3. (2.45) σ T ϱ ( = p a + g H ) I 3 3. (2.46) 2 Remark that only pressure terms don t drop through the O(ε 2 ) approximation Further remarks Let us obsere that if we aerage model (2.38) along y-axis (where y ( L/2,L/2)) 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 SW2d model (2.40). It is important to notice that the whole asymptotic analysis preiously described starts from the alidity of (2.3), but a relation similar to (2.3) like: 2 u 2 (t, x, y, z) =O(ε), 2 (t, x, y, z) =O(ε), 2 does not hold as one can erify from (2.23). In fact, in sight of (2.9), it is eident that terms 2 u z and 2 2 z 2 appear, respectiely, in the first and in the second equation of (2.23) with an ε factor at the denominator, which permits (2.3) to be alid. On the other side, it can be seen that this fact doesn t hold for terms like 2 u 2 and 2 2 Remark that an analogous deriation of our model (2.40) together with boundary and initial conditions (2.28), (2.29) 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.

12 222 S. FERRARI AND F. SALERI 3. The symmetrized model (SSW2d) Let us consider again the SW2d system (2.39) 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 (2.28), (2.29). In quasi-linear form the system (2.39) can be rewritten in the form: V t + A (t, x, y, V ) V + A 2(t, x, y, V ) V = G(t, x, y, V ), (3.) where V =(Hu,H,H) T, and A (t, x, y, V )= 2u 0 gh + p a u 2 u u, (3.2) 0 0 A 2 (t, x, y, V )= u u 02gH+ p a 2, 0 0 G(t, x, y, V )= lh H p a s sup x lhu H p a s sup y 0 ku + kh 3µ + kh 3µ Remark that the dependence of A, A 2 and G on t, x and y, is due to the explicit dependence of terms p a and s sup xy on those ariables. The following proposition holds: Proposition 3.. Consider:. A(ξ,t,x,y,V )=ξ A (t, x, y, V )+ξ 2 A 2 (t, x, y, V ), ξ R 2 \{0}, where A and A 2 are gien by (3.2). The three eigenalues of the SW2d system are then gien by: λ 2 (ξ,t,x,y,v )=ξ u + ξ 2, λ,3 (ξ,t,x,y,v )=ξ u + ξ 2 gh + p a ξ 2 + ξ2 2. (3.3) Therefore, the SW2d 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 (ξ,t,x,y,v ) <λ 2 (ξ,t,x,y,v ) <λ 3 (ξ,t,x,y,v ). Then, following the definition cited in Serre [22] (Vol. I, Chap. 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) SW2d system, gien by (3.), in order to obtain the symmetric system, denoted by SSW2d, which we show to be hyperbolic as well. We will then exploit if SSW2d model may be considered as the iniscid limit of a sequence of artificial iscous perturbed (second order) parabolic problems. We concentrate our attentiononthecasewhenp a is a non-null function of x, y and t.

13 A NEW 2D SHALLOW WATER MODEL INCLUDING PRESSURE EFFECTS 223 Fictitiously increased free surface z p a /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. Physical Postulate. A Shallow Water flow on which atmospheric pressure is acting, behaes equialently (from a physical point of iew) to a Shallow Water flow without atmospheric pressure, in which the eleation is fictitiously increased by the constant alue p a /g. Let us explain in detail what we mean as a Shallow Water flow without atmospheric pressure, in which the eleation is fictitiously increased by the constant alue p a /g, going through SW2d model deriation exposed in Section 2. We consider our incompressible fluid confined into a three dimensional domain U which is ertically bounded by the surfaces z = η (x,t), where η = η + p a /g and z = b(x), so that (2.) becomes: U := {(t, x, y, z) : t (0,T), (x, y) Ω, z (b(x, y),η (t, x, y))}, (3.4) denoting again by Ω the domain introduced in (2.2), where T>0, we hae that η :[0,T] Ω R (T >0) is the fictitious eleation (see Fig. 3), while b : Ω R is as before the bottom depth with respect to the same reference leel; H,gienby: H = η b = H + p a g (3.5) is always strictly positie (as pa g ) and represents the total height of the fluid from the bottom to the fictitious free surface. Let S be the free surface connected to the domain U,letB be the bottom surface and L be the lateral surface gien by a formula analogous to (2.). Let us consider again NS3d equations (2.3) with stress tensor assigned by (2.5) and total stress tensor assigned by (2.6). Now, the surface stress tensor is gien by s sup so that (2.7) becomes: s sup xy = C W W, s sup z = 0; (3.6) in fact the horizontal components of the surface stress tensor (wind effect) are left unaltered, while we are neglecting the effect of the atmospheric pressure as the ertical component of the surface stress (we are recoering this effect adding the fictitious increment pa g to the eleation H). We then consider boundary conditions analogous to (2.2, 2.3) and (2.4) and perform adimensionalizations, asymptotic assumptions and integral aerages as in Sections 2. and 2.4. Assumption (2.34) on the smallness of p a becomes now not restrictie as we are supposing that no atmospheric pressure is acting as the ertical component of the surface stress tensor.

14 224 S. FERRARI AND F. SALERI We therefore obtain on (0,T) Ω the system: (H u) + (H u 2 ) + (H u) t (H ) t H t + (H u) + (H 2 ) + (H u)+ (H )=0. + g H 2 2 lh s sup x + g H 2 2 lh u s sup y k u, + kh 3µ k, + kh 3µ (3.7) In quasi-linear form system (3.7) reads as: whereweposed ˆV =(H u, H, H ) T, ˆV t + Â( ˆV ) ˆV + Â2( ˆV ) ˆV = Ĝ(t, x, y, ˆV ), (3.8)  ( ˆV )= 2u 0 gh u 2 u u,  2 ( ˆV )= u u 02gH 2, (3.9) and Ĝ(t, x, y, ˆV )= lh ku s sup x + kh 3µ lh k u s sup y + kh 3µ 0. Remark 3.. We underline that system (3.) may not be recoered by taking relation (3.5) into account and substituting into (3.7). Now we can symmetrize system (3.8) i.e., performing a suitable change of ariables we transform matrices  and Â2 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 =2 gh =2 gh + p a, (3.0) where H g,knownasthecelerity ariable, is always positie een if the effectie eleation H is anishing. In fact we hae: H g 2 p a > 0. (3.) Using the new set of ariables, we can rewrite system (3.8) in the following form ˆV U U t + Â( ˆV ) ˆV U U + Â2( ˆV ) ˆV U U = Ĝ(t, x, y, ˆV ). (3.2)

15 A NEW 2D SHALLOW WATER MODEL INCLUDING PRESSURE EFFECTS 225 We denote  0 ( ˆV ) ˆV U H H u gh 0 H H gh, with  0 ( ˆV H 0 u H ) 0 H H H gh. (3.3) 0 0 gh Matrix Â0 and their inerse are well defined as H is always strictly positie. Multiplying to the left all terms defined in (3.2) by  0 ( ˆV ), we finally obtain the symmetric system: U t + S (U) U + S 2(U) U H = F (t, x, y, U), (3.4) where: u 0 H g S (U) = 0 u 0, S H g 2(U) = 0 2 H g 2 0 u 0 H, (3.5) g 2 and l 4gs sup x 4gku H 2 ( ) g Hg 2 + kh2 g 2gµ F (t, x, y, U) = lu 4gs sup y 4gk Hg 2 ( ). (3.6) Hg 2 + kh2 g 2gµ 0 We will refer to the symmetric system (3.4) as two-dimensional Symmetric Shallow Water system, briefly denoted by SSW2d in the sequel. By components, in nonconseratie form on (0,T] Ωitreads: u t + u u + u + H g H g 2 l + 4gs sup x 4gku Hg 2 + ( ) =0, Hg 2 + kh2 g 2gµ t + u + + H g 2 H g t + (H g ) H g 2 H g + lu + 4gs sup y Hg 2 + =0, H 2 g 4gk ( ) =0, + kh2 g 2gµ (3.7) where =(u, ) T. In the following, we will always deal with the SSW2d system together with initial and boundary conditions (2.28), (2.29). Let us now show some analytical properties. Of course, as soon as p a = O(ε), the solutions of (3.7) may be interpreted as the solutions of the preiously SW2d model. Proposition 3.2. The eigenalues of system SSW2d are gien by: λ 2 (ξ, U) =ξ u + ξ 2, λ,3 (ξ, U) =ξ u + ξ 2 H g ξ ξ2 2 (3.8) and the SSW2d system is a quasilinear strictly hyperbolic system.

16 226 S. FERRARI AND F. SALERI In fact, as H g > 0 (see (3.)), we hae that λ (ξ, U) <λ 2 (ξ, U) <λ 3.(ξ, U). (3.9) Then the model is (quasilinear) hyperbolic. It is also important to note that the eigenalues of the SSW2d system are the same as the eigenalues of the SW2d system. Remark 3.2 (boundary conditions). If 0 <u<h g /2, then Γ in and Γ out defined in (2.9) are non-characteristic boundaries for SSW2d model. Further, on Γ in the number of positie eigenalues is p = 2, while on Γ out we hae p =. For all x 0 Γ in,takeξ = n in (x 0 )=(, 0); from (3.8) we hae: while for all x 0 λ (ξ, U) =u H g 2, λ 2(ξ, U) =u, λ 3 (ξ, U) =u + H g 2, Γ out,takeξ = n out (x 0 )=(, 0): λ (ξ, U) = u H g 2, λ 2(ξ, U) = u, λ 3 (ξ, U) = u + H g 2. On the other side, from boundary conditions (2.28), we hae that Γ c defined in (2.0) is a characteristic boundary for SSW2d model on our bounded domain (2.2). Further on, we hae on Γ c exactly a null, a positie and a negatie eigenalue. For all x 0 Γ c,takeξ =(0, ±); from (3.8) we hae: λ (ξ, U) =± H g 2 λ 2 (ξ, U) =±, λ 3 (ξ, U) =± + H g 2 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. (3.20) Viscous perturbation of the SSW2d model Let us consider a iscous perturbations of the SSW2d system (3.4), defined on the bounded domain Ω (2.2) in the eolution time interal [0,T]. To each equation of the SSW2d 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 (U ν ) U ν t + S 2(U ν ) U ν ν U ν = F (t, x, y, U ν ) on Ω (0,T], U ν Γ = a on Γ [0,T], (3.2) U ν t=0 = U 0 on Ω {t =0}, where a(t, x, y) is a function to be prescribed on Γ (0,T] and taking its alues in R 3. We used here as in (2.4) the same function-name a, to indicate Dirichlet conditions for a iscous problem (system NS3d (2.3) or the aboe iscous perturbed SSW2d system (3.2)); of course, we are dealing with two different functions, een if they show some logical relationship. the terms subsonic and supersonic are used in hydraulics, see e.g. [23].

17 A NEW 2D SHALLOW WATER MODEL INCLUDING PRESSURE EFFECTS 227 At present, an existence and uniqueness result global in time for smooth solutions of parabolic problems like (3.2) is aailable (see e.g. Ladyzenskaia, Solonniko, Ural cea [7], Th. 7., Chap. VII, pp ) 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 (U) U + S 2(U ) U = F (t, x, y, U) on Ω (0,T] U Γ C(a) on Γ (0,T] (3.22) U t=0 = U 0 on Ω {t =0} and results on conergence of (3.2) to (3.22) in a suitable space-time norm, are not yet aailable on a bounded domain like (2.2), 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 (3.2) to hyperbolic IVB problems like (3.22). Let us obsere that we may consider our bounded domain Ω (2.2) as the intersection of four half-spaces: where, recalling Figure 2, Ω=Ω in Ω out Ω c+ Ω c, Ω in = {(x, y) R 2 s.t. x > x in }, Ω out = {(x, y) R 2 s.t. x < x out }, Ω c+ = {(x, y) R 2 s.t. y > L/2}, Ω c = {(x, y) R 2 s.t. y < L/2} (3.23) If conditions (3.20) are satisfied and, in iew of boundary conditions (2.28), Ω in and Ω out hae non-characteristic boundaries, while Ω c+ and Ω c may hae characteristic boundaries. We will consider separately first the noncharacteristic case, then the characteristic case (if =0). () The non-characteristic case: if we set our second order iscous perturbed SSW2d model (3.2) into a half-space of the form Ω in (or Ω out ), then by [7], we hae that (3.2) has a unique smooth solution U ν globally in time, while by [20] the hyperbolic problem (3.22) has a unique smooth solution U 0 locally in time. Thanks to Grenier and Guès in [0], we may also conclude that there exists a time T 0 such that the following conergence result is alid: U ν U L 2 ((0,T 0) Ω 0 as ν 0. A deeper analysis of the compatibility between the theory exposed by Grenier and Guès in [0] and our case is shown in [5]. (2) The characteristic case: for this case, literature is not so extensie as in the preious one. We hae a result by Guès in [] 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 noncharacteristic case. Another result, by Grenier in [9], deals with the quasilinear case, but only in the case of a scalar problem. Let us end this section by remarking that the classical Shallow Water model (2.4) is an hyperbolic model as well, and that it may be symmetrized the same way as our SW2d model; we shall refer to this one as to the CSSW2d model. In terms of ariable U =(u,, H g) T, where we posed: The CSSW2d model on (0,T) Ωreads: H g =2 gh. (3.24) U t + S (U ) U + S 2(U ) U = G(t, x, y, U ), (3.25)

18 228 S. FERRARI AND F. SALERI where matrices S and S 2 are the same as in (3.5), while the right hand side G is now gien by: assuming that H g > 0iffH>0. G(t, x, y, U )= l p a + 4gs sup x H 2 4g2 ku g H 2 g lu p a + 4gs sup y H 2 4g2 k g H 2 g 0, (3.26) Remark 3.3. About the introduction of a iscous perturbation, we want to mention the paper by Sammartino and Caflisch in [2]. 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 [0], we consider an artificial second order iscous perturbation of all the equations of a hyperbolic system (with suitable properties). 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 CSSW2d model, when the wind and the Coriolis effects are neglected (s sup x = s sup y =0andl = 0). Numerical approximation is based on the space-time approximation introduced in [6]: in particular, we consider piecewise-quadratic finite element in space and a first order semiimplicit 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++ [3]. 4.. 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, 0] [0, 0] with a flat bathymetry b(x, y) =. The final time is T =0.8. Initial conditions are: C(x, y, 0) = 5 + 5exp(( (x 5) 2 (y 5) 2 +)),u 0 =0, 0 =0, (4.) 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 (3.0)) when the SSW2d model is considered, H g (see (3.24)) for the CSSW2d model. Remark that in the iscous perturbed SSW2d model, the atmospheric pressure term is comprised into the celerity term H g, while in the iscous perturbed CSSW2d model, pressure is considered apart from the celerity ariable H g. On a uniform triangular grid (with space-step equal to 0.5) we compare the different behaiors of our perturbed SSW2d model with the iscous perturbed CSSW2d model, considering different alues of the artificial

19 A NEW 2D SHALLOW WATER MODEL INCLUDING PRESSURE EFFECTS 229 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 =0.00, the final eolution time T =0.8 and the physical iscosity parameter µ =0.2. In the following simulations we will compare the isoalues of the eleation η (see the beginning of Section 2) at the final time t = T with the ertical cuts of the eleation profiles in y =5att = T obtained by the iscous perturbed SSW2d model and the iscous perturbed CSSW2d model with artificial iscosity alues ν = 0. and ν = respectiely. Friction effects. Let us choose k =50andp a =. 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. 4a, Fig. 4c and Fig. 5a, Fig. 5c) 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. 4b, Fig. 4d and Fig. 5b, Fig. 5d) 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 =0.0000, 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. (eenifinparticularcases)andwith respect to our accurate model deriation exposed in Section. 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. (4.2) Since atmospheric pressure is not constant, the initial plotting of eleation for the iscous perturbed CSSW2d model is just the same as if pressure is constant (as the classical model is pressure independent). But in the case of our SSW2d 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. 6a with Fig. 6b). We obsere that eleation η might take negatie alues, as it is in the case of our model (see Fig. 7a, Fig. 7c and Fig. 8a, Fig. 8c). In this situation, both the perturbed CSSW2d (see Fig. 7b, Fig. 7d and Fig. 8b, Fig. 8d) and the perturbed SSW2d model are sensitie to the change of artificial iscosity. In particular for ν = both the perturbed SSW2d model and the perturbed CSSW2d model behae much less regularly as for ν = 0.. 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 (SW2d model) on a bounded domain and set physical boundary conditions on elocity. A specification of the nature (characteristic, non-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 non-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 (3.20). Another task which could be performed, is to find out some relations between

20 230 S. FERRARI AND F. SALERI Eleazione al tempo t= 0.8sec. IsoValue Eleazione al tempo t= 0.8sec. IsoValue (a) (b) (c) (d) Figure 4. (a) Final total eleation for the Gaussian-hill case: the iscous perturbed SSW2d model, ν =0.0000, friction k = 50, atmospheric pressure p a =. (b) Final total eleation for the Gaussian-hill case: the iscous perturbed CSSW2d model, ν =0.0000, friction k = 50, atmospheric pressure p a =. (c) Eleation profile of the final total eleation for the Gaussianhill case: the iscous perturbed SSW2d model, ν = , friction k = 50, atmospheric pressure p a =. (d) Eleation profile of the final total eleation for the Gaussian-hill case: the iscous perturbed CSSW2d model, ν =0.0000, friction k = 50, atmospheric pressure p a =. initial and boundary conditions such that the subsonic hypothesis, which are supposed apriori, are effectiely point-wise fulfilled. Acknowledgements. 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.

21 A NEW 2D SHALLOW WATER MODEL INCLUDING PRESSURE EFFECTS 23 Eleazione al tempo t= 0.8sec. IsoValue Eleazione al tempo t= 0.8sec. IsoValue (a) (b) (c) (d) Figure 5. (a) Final total eleation for the Gaussian-hill case: the iscous perturbed SSW2d model, ν =0., friction k = 50, atmospheric pressure p a =. (b) Final total eleation for the Gaussian-hill case: the iscous perturbed CSSW2d model, ν = 0., friction k = 50, atmospheric pressure p a =. (c) Eleation profile of the final total eleation for the Gaussian-hill case: the iscous perturbed SSW2d model, ν =0., friction k = 50, atmospheric pressure p a =. (d) Eleation profile of the final total eleation for the Gaussian-hill case: the iscous perturbed CSSW2d model, ν =0., friction k = 50, atmospheric pressure p a = (a) (b) Figure 6. (a) Initial total eleation for the Gaussian-hill case: the iscous perturbed SSW2d model, atmospheric pressure p a = x + t. (b) Initial total eleation profile for the Gaussian-hill case: the iscous perturbed CSSW2d model, atmospheric pressure p a = x + t.

22 232 S. FERRARI AND F. SALERI Eleazione al tempo t= 0.8sec. IsoValue Eleazione al tempo t= 0.8sec. IsoValue (a) (b) (c) (d) Figure 7. (a) Final total eleation for the Gaussian-hill case: the iscous perturbed SSW2d model, ν =0.0000, friction k = 0, atmospheric pressure p a = x + t. (b) Final total eleation for the Gaussian-hill case: the iscous perturbed CSSW2d model, ν = , friction k = 0, atmospheric pressure p a = x+t. (c) Eleation profile of the final total eleation for the Gaussianhill case: the iscous perturbed SSW2d 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 CSSW2d model, ν =0.0000, friction k = 0, atmospheric pressure p a = x + t.

23 A NEW 2D SHALLOW WATER MODEL INCLUDING PRESSURE EFFECTS 233 Eleazione al tempo t= 0.8sec. IsoValue Eleazione al tempo t= 0.8sec. IsoValue (a) (b) (c) (d) Figure 8. (a) Final total eleation for the Gaussian-hill case: the iscous perturbed SSW2d model, ν =0., friction k = 0, atmospheric pressure p a = x + t. (b) Final total eleation for the Gaussian-hill case: the iscous perturbed CSSW2d model, ν = 0., 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 SSW2d model, ν =0., 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 CSSW2d model, ν =0., friction k = 0, atmospheric pressure p a = x + t.