On the hyperbolicity of two- and three-layer shallow water equations. 1 Introduction

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1 On the hyperbolicity of two- and three-layer shallow water equations M. Castro, J.T. Frings, S. Noelle, C. Parés, G. Puppo RWTH Aachen, Universidad de Málaga, Politecnico di Torino E-ail: Abstract The two-layer shallow water syste looses hyperbolicity if the agnitude of the shear velocity is above a certain threshold, essentially deterined by the density difference between the two layers. Introducing an additional third layer ight recover hyperbolicity in regions of strong shear. We deonstrate that this adaptive two/three-layer approach can cure soe of the shortcoings of the two-layer odel. Introduction In this paper we consider stratified flows. An eaple which has recently found attention in applied atheatics is the superiposition of the Mediterranean and the Atlantic in the Strait of Gibraltar, e.g. []. This layering persists as long as the velocity difference (shear velocity) is not too large. Under this condition the flow ay be odeled by two coupled shallow water odels, yielding a non-conservative hyperbolic syste in one or two space diensions which ay be solved efficiently and accurately by well-balanced finite volue schees, e.g. [3]. In regions of strong gradients in the botto topography, however, the lower layer Mediterranean water ay accelerate, causing a strong shear and hence a Kelvin-Helholtz instability. Matheatically, the hyperbolicity of the two-layer odel is lost, and usually the finite volue solvers break down or produce questionable results. The obvious, but epensive, solution would be to switch to the full D or 3D Navier-Stokes equations and apply the respective free surface incopressible solvers. A challenging question is whether one can odel such local instabilities without giving up the depth-averaging everywhere. Several authors have suggested such odels, e.g. LeVeque and Ki [6], Castro and Pares [5], Bouchut and Morales [] and Audusse []. Here we eplore an adaptive two/three layer odel. We introduce an interediate third layer to regain hyperbolicity in case it is lost.

2 Castro, Frings, Noelle, Pares, Puppo Multilayer flows The ultilayer shallow water syste The flow of i =,...,N layers of water, ordered fro top to botto, is governed by t h i + q i =, ( ( t q i + q i /h i + g/ h ) i i = ghi r jih j + ) N h j + b j= j=i+ (.) with densities ρ i < ρ i+ and r ij := ρi ρ j denoting the density ratios. The coordinate refers to the spatial position, t is the tie, and g is gravity. The unknowns q i (,t) and h i (,t) represent respectively the discharge and the height, i.e. thickness of the i-th layer at the section of coordinate at tie t. The botto topography is given by the function b. The syste (.) can be written in quasi-linear for w t + A(w)w = S(w)b, (.) cf. [3]. The eigenvalues of atri A(w) ay becoe cople even for physically relevant data, corresponding to the developent of shear instabilities. We will call a ultilayer state w hyperbolic, if the atri A(w) is diagonalizable and all its eigenvalues are real. Loss of hyperbolicity In the two-layer case, the atri A becoes a atri. An approiation forula for the eigenvalues in the liit of alost constant water level, alost equal densities and vanishing velocity difference is given in [7]. The authors distinguish two pairs of eigenvalues, the internal and the eternal ones: λ ± et = u h + u h ± (g(h + h )), h + h (.3) λ ± int = u [ h + u h ± g h h ( (u u ) )] h + h h + h g, (h + h ) (.) where g := ( r )g is a sall constant in this liit. The eternal eigenvalues λ ± et correspond to the eigenvalues of a single-layer shallow water syste with total water height H := h + h and an averaged total velocity u = uh+uh h +h. These eigenvalues are therefore distinct and real, thus they are not iportant regarding the hyperbolicity. If κ := (u u ) g >, (.5) (h + h ) the approiative internal eigenvalues becoe cople. In [5] it is shown that (.5) is a good approiative criterion for the loss of hyperbolicity when r is close to. The ratio κ can be interpreted as the

3 /3-layer Shallow Water 3 balance between the stabilizing influence of the difference in density and the destabilizing one of the velocity shear. We quantify this loss of hyperbolicity by the indicator δ as follows: Definition. The hyperbolicity indicator δ for a state U is given by δ(u) := R(λ + int (U)) R(λ int (U)) I(λ+ int (U)) I(λ int (U)), (.6) where R denotes the real part and I the iaginary part. This indicator is nonnegative if the eigenvalues are real and negative if they are cople conjugate. It is etended to the ultilayer case by applying it to each pair of internal eigenvalues, then taking the iniu. 3 The three-layer adaptation The breakdown of hyperbolicity corresponds to a Kelvin-Helholtz instability and the onset of iing [7]. Our interest is to odel this situation in the contet of shallow water odels. We introduce an interediate layer around the original interface with constant values for velocity and density attached to it, assuing that all the possible turbulence and iing is contained within this layer and does not affect the horizontal velocities of the other layers in a substantial way. A strategy for choosing the interediate layer Given a twolayer state [(ρ,h,u ),(ρ,h,u )] we are now looking at a way to find a corresponding three layer state [( ρ, h,ũ ),(ρ,h,u ),( ρ, h,ũ )] such that the discharge (Q), ass (M) and total water height (H) are conserved: ρ ih i u i = ρ i h i ũ i + ρ h u (Q) i i ρ ih i = ρ i hi + ρ h (M) i = h i + h (H) i i h i In the upper and lower layer the densities and the velocities are set to ũ = u, ũ = u, ρ = ρ, ρ = ρ. (C.) Net, aong several possible choices we pick ρ = (ρ + ρ ) in the interediate layer, thus enforcing constant density throughout the layer, and consider h a free paraeter. This gives h = h h, h = h h, u = ρu+ρu ρ +ρ. (C.) The density ρ is independent of and t. Thus even for ultiple adaption steps the three-layer contet persists, siplifying the construction of an adaptive solver.

4 Castro, Frings, Noelle, Pares, Puppo Hyperbolic doains We test the hyperbolicity of. randoly chosen two-layer test states and copare with those of the corresponding three-layer states gained by the transforation forula (C). The ratio r =.98 is kept fi. The hyperbolicity is tested by inserting the test states into the atri A in (.) and then nuerically calculating the eigenvalues. In case of all real, distinct eigenvalues we plot a blue dot in the left graph in Fig. 3., otherwise we plot a red dot in the right graph. The aes are (u u ) and h + h. With this choice, the hyperbolic and the non-hyperbolic region are approiately separated by the straight line κ =. NON hyperbolic h +h h + h hyperbolic u u u u Figure 3.: Hyperbolic doain for -layer test cases, r =.98 NON hyperbolic a.5. h /H h + h h + h hyperbolic in in a heu heuristic..5. u u u u.95.5 κ..5 Figure 3.: Left: Hyperbolic doain for -3-layer adaptation, Θ =.; and heuristic h heu Right: h, H To check whether we gain hyperbolicity with our adaptation strategy, we try to find any interediate layer height for the given test states Note that there are indeed soe hyperbolic states with κ > close to the approiative boundary. Also, there is a second region of hyperbolicity for very large shear velocities, but this is not in the focus of our work.

5 /3-layer Shallow Water NON hyperbolic h/h h + h h + h hyperbolic u u 5 in a heu.5 u u κ..5.3 Figure 3.3: Left: Hyperbolic doain for -3-layer adaptation, Θ =.; and heuristic h heu Right: h, H NON hyperbolic h /H h + h h + h hyperbolic u u in a heu. u u.. κ.6.8 Figure 3.: Left: Hyperbolic doain for -3-layer adaptation, Θ =.; Right: h and heuristic h heu, H that gives a hyperbolic state with three positive layer heights. The interediate layer height h is chosen to aiize the indicator δ aong all states with positive layer heights. Again we plot dots indicating the hyperbolicity, this tie of the three-layer syste, but at the location given by the values of the associated original two-layer syste. Thus we can observe fro the plots the change in hyperbolicity triggered by the two/three-layer adaptation. We can indeed notice a gain in hyperbolicity but a clear picture only eerges if the data are ordered by the ratio Θ := hh with H := h + h the total water height. The plots on the left in Figs. 3., 3.3 and 3. show the gain for different values of Θ. Note that there is a sall area in Fig. 3. where the hyperbolicity is lost when transferring fro two to three layers. This does not pose a proble as those states are hyperbolic for the two-layer odel.

6 6 Castro, Frings, Noelle, Pares, Puppo Choice of interediate layer height We fi Θ and randoly generate a set with two-layer test states with h /H = Θ. For each test state, using the transforation rules (C), we search for a three-layer state with all positive layer heights that aiizes δ. This gives an interediate height h. Now plotting (κ,h /H) we found interesting structures as shown by the eeplary results in the plots on the right of Figs. 3., 3.3 and 3.. Θ is fied for each plot, but varies fro plot to plot. Considering several values of Θ, the points (κ,h /H) see to for a surface h (κ,θ). Note that for each Θ, there is a value κ a such that for κ > κ a the third layer cannot cure the hyperbolicity loss, at least not without deteriorating to a two-layer state again. There are three additional alleged curves in these plots. The two sets of pairs (κ,h in/a ) are defined as follows: define H for a given κ as the largest interval of hyperbolic points containing h. This interval eists as the eigenvalues depend soothly on the state (and thus on h ). Then H =: [h in,h a ]. The reaining curve is given by a heuristic approach for the interediate layer height. Heuristic interediate layer height As a result of the structure revealed by the Figs. 3., 3.3 and 3. we can define a heuristic ethod to deterine h. We fit curves to the point values of h given for different fied values of Θ and then use interpolation between the discrete curves h heu to get a function depending on Θ and κ. The resulting function h heu is defined only for < κ < κ a, i.e. for those values of κ for which hyperbolicity could be regained. h heu is shown in the figures entioned above as solid lines lying within the ial intervals H found for the tested states. In the interval [(κ a +)/,κ a ) it is defined as a leastsquares linear fit to the ial heights. In the interval [,(κ a +)/) the definition is h heu (κ,θ) := c Θ ( κ) αθ, where c Θ,α Θ > are chosen in such a way that the linear fit in κ a / is atched with C soothness. For κ = it connects to the corresponding two-layer hyperbolic state. Loss of hyperbolicity for a two-layer tidal flow over a hup We present the details of a tidal flow over a hup as shown in Fig The initial data are set to: η := h + h + b =, h = +.5 atan(.5( +.75)) b, q =., b = e. q =., with r =.98 and g = 9.8. The spatial doain is I = ( 3,3) subdivided into cells and CFL-Nuber is cfl =.8. Absorbing boundary conditions are applied.

7 /3-layer Shallow Water 7 surface, interface(s) and botto at tie surface, interface(s) and botto at tie 3.95 height height 3 3 velocities of the layers velocities of the layers.5 velocity velocity δ indicator δ δ indicator δ Figure 3.5: Left: Initial state; Right: Situation close to loss of hyperbolicity surface, interface(s) and botto at tie 3.95 surface, interface(s) and botto at tie.69 height height 3 3 velocities of the layers velocities of the layers.5 velocity velocity δ indicator δ δ indicator δ Figure 3.6: Left: Refined state, hyperbolicity regained; Right: Situation close to second loss of hyperbolicity This test proble is an idealized version of a natural two-layer flow being found e.g. at the Strait of Gibraltar, where the denser Mediterranean Sea fors the lower layer and spills over a hup while being oved by tidal forces. Our goal is to design an efficient nuerical solver that retains ost of the physical features of the real flow. The siulation runs up until tie t = 3.95, where the indicator

8 8 Castro, Frings, Noelle, Pares, Puppo shows that hyperbolicity is close to being lost, see Fig At this tie an adaptation step is carried out using our heuristic approach to deterine the interediate height. The siulation then continues to run for soe tie steps until tie t =.689 when the three-layer syste loses hyperbolicity in the sae cell that was arked critical before the adaptation step, see Fig This is caused ainly by the transport of the interediate layer. Mass is flowing away fro the forerly critical cell and the interediate layer height drops below h in for that state. The adaptive strategy we have designed can cure hyperbolicity at a given fied tie. However, our results suggest that the cure ay not be stable under tie evolution. Several iproveents are possible, and they will be the subject of future research. References [] E. Audusse, A ultilayer Saint-Venant odel: derivation and nuerical validation., Discrete Contin. Dyn. Syst. Ser. B, 5():89, 5. [] F. Bouchut and T. Morales, An entropy satisfying schee for twolayer shallow water equations with uncoupled treatent., MAN Math. Model. Nuer. Anal., (): , 8. [3] M. Castro, J. M. Gallardo, and C. Parés, High order finite volue schees based on reconstruction of states for solving hyperbolic systes with nonconservative products. Applications to shallow-water systes., Math. Cop. 75(55):3 3, 6. [] M. Castro, J. Macías, and C. Parés. A Q-schee for a class of systes of coupled conservation laws with source ter. Application to a two-layer -D shallow water syste., MAN Math. Model. Nuer. Anal, 35():7 7,. [5] M. J. Castro Daz, E. D. Fernndez-Nieto, J. M. Gonzlez-Vida and C. Pars. Nuerical treatent of the loss of hyperbolicity of the two-layer shallow-water syste., Subitted to Journal of Scientific Coputing. [6] J. Ki and R. J. LeVeque, Two-layer shallow water syste and its applications, Proceedings of the Twelth International Conference on Hyperbolic Probles, Maryland 8. [7] J. B. Schijf and J. C. Schonfeld. Theoretical considerations on the otion of salt and fresh water., in: Proceedings of the Minn. Int. Hydraulic Conv., Joint Meeting IAHR Hydro. Div. ASCE. (Septeber 953), pp , 953.