CHAPTER 4 ANALYSIS OF THE GOVERNING EQUATIONS

4.1 INTRODUCTION

The mathematical nature of the systems of governing equations deduced in Chapters 2 and 3 is investigated in this chapter. The systems of PDEs that express the quasi-equilibrium approximation are studied in greater depth. The analysis is especially important for the systems whose solutions are likely to feature discontinuities, as a result of strong gradients growing steeper, or because the initial data is already discontinuous. The geomorphic shallow-water flows, such as the dam-break flow considered in Chapter 3, are the paradigmatic example of flows for which discontinuities arise fundamentally because of the initial conditions.

Laboratory experiments show that, in the first instants of a sudden collapse of a dam, vertical accelerations are strong and a bore is formed, either through the breaking of a wave (Stansby et al. 1998) or due to the incorporation of bed material (Capart 2000, Leal et al. 2002). Intense erosion occurs in the vicinity of the dam and a highly saturated wave front is likely to form at ttt≡≈0 4 , thg00= , where h0 is the initial water depth in the reservoir. The saturated wave front can be seen forming in figure 3.1(a). Unlike the debris flow resulting from avalanches or lahars, the saturated front is followed by a sheet-flow similar to that

303 encountered in surf or swash zones (Asano 1995), as seen in figures 3.2(b) and (c). The intensity of the sediment transport decreases in the upstream direction as the flow velocities approach fluvial values. While the flow is highly erosive in the wave front region, sediment debulking may result into generalised deposition as the flow velocity decreases. Thus, the solution of the competent system of equations comprises continuous reaches eventually separated by discontinuities.

If the collapse of a dam should be idealised as an instantaneous removal of a vertical barrier, initially separating two constant states that extend indefinitely on both up- and downstream directions, as seen in figure 4.1, the dam-break problem is a Riemann problem. Riemann problems admit self-similar solutions relatively to the variable x tgh0 if the hyperbolic equations are homogeneous, i.e., if G = 0. For special cases of the flux vector, F, explicit expressions for the dependent variables, functions of time and spatial co-ordinates, are attainable, as it is the case of the flat- fixed-bed solution for the shallow water equations presented by Ritter in 1887. The latter is generalized and thoroughly described by Stoker (1958), pp. 311-326, 333-341 and 513-522.

The importance of explicit theoretical solutions is threefold: i) they are computationally simpler than numerical solutions, ii) they provide an order of magnitude and important phenomenological insights on the behaviour of the system under more general conditions and iii) they provide a way to access the quality of numerical discretization techniques. Stoker’s or Ritter’s solutions have been used to verify the quality of, virtually, all numerical models build ever since Stoker’s reference book was published. They also provided an elemental proof of existence of a weak solution for the shallow water equations and sparkled important theoretical advances on the hydrodynamics of unsteady open-channel flow (e.g. Su & Barnes 1970, Hunt 1983, 1984). The practical use of Stoker’s solution was extended to play the role of a reference situation for the interpretation of experimental results on dam-break flows. As an example, Ritter’s value for the velocity of the dam-break wave front, 2 gh0 , is the reference order of magnitude of the dam-break flood wave propagation, to which every experimental result is compared and at whose light is discussed.

Pure hydrodynamic models, Stoker’s solution included, fail to reproduce the characteristic time and length scales of the dam-break flow when morphological impacts are important. Because of this fundamental inadequacy, research in geomorphic dam-break flows has been conducted through a combination of fieldwork, laboratory physical modelling, theoretical analysis and numerical simulation. Research projects like CADAM and IMPACT provided the framework for a number of studies, notably Capart & Young (1998), Fraccarollo & Capart (2002) or Leal et al. (2005), that resulted in major advances, comparable to those proportioned by the landmark works of Dressler (1952, 1954), Whitham (1955) and Stoker, op. cit., in the conceptualisation of the phenomena involved and in the development of simulation capabilities.

Especially relevant is the study of Fraccarollo & Capart (2002) whom, in the wake of works by Capart & Young (1998) and Fraccarollo & Armanini (1999), have built a solution for the Riemann problem posed by the homogeneous geomorphic shallow-water equations subjected to initial conditions comprising a jump in the water level. The solution is not explicit because of the strong non-linearity of the closure equations. Numerical computations are unavoidable because the invariants of the simple centred waves can not be explicitly found.

304 y/h0 RL

YL

RR

YR YbL

YbR

0 x0/h0 x/h0

FIGURE 4.1. Graphic depiction of the initial conditions for the Riemann problem posed to the geomorphic shallow water equations. The variables, Y, R, and Yb, are, respectively, the water level, the unit mass discharge and the bed elevation. The subscripts L and R stand, respectively, for the left and right states, respectively.

The main objectives of this chapter are, in the wake of Fraccarollo & Capart (2002), the development of a weak solution of the Riemann problem for the geomorphic shallow water equations and the description of the main features of the wave structure. Special attention will be devoted to the condition of existence of alternative wave structures, depending on the initial data. The initial values for the Cauchy problem are the left and right states, characterised by the water elevation, Y, bed elevation, Yb, and total mass discharge per unit width, R. Adding to the discontinuity in the water level, the initial discontinuity in the bed elevation is, thus, explicitly addressed.

Most of the chapter is dedicated to the study of the characteristic fields for which discontinuities are likely to develop. It is necessary to investigate the fundamental properties of the characteristic fields, notably signal, monotonicity and non-linearity. In addition, because the debate on the role of non-linear algebraic equations, describing important physical phenomena such as sediment transport capacity or bulk flow resistance, is yet to be closed, questions concerning existence and uniqueness of the solution must, thus, be posed.

The existence of the solutions for the Riemann problem was proved by Lax (1957) for a finite set of strictly hyperbolic, genuinely non-linear, conservation equations. However, the proof is valid for the case of small discontinuities in the initial data, which is generally not the case in the dam-break problem. Further works by Glimm (1965), the first proof for arbitrary initial values, Smoller (1969), di Perna (1973), Dafermos (1973) or Liu (1974) helped building a library of theoretical results that may be used as a guide to establish the conditions of existence and uniqueness of the Riemann solution of the geomorphic shallow water equations. Considerations on the existence and unicity of the solution of the Riemann problem for the geomorphic shallow water equations are, thus, legitimate and will be addressed.

The text is structured so as to highlight the main objectives mentioned above. Wave-like description of wave forms and hyperbolicity are discussed in §4.2.1. The governing equations

305 subjected to analysis are described in §4.2.2, namely in what concerns the embedded hypotheses. The following sections, §4.2.3 to §4.2.5 are dedicated to the mathematical analysis of the system of equations. Special attention is given to the type of hyperbolicity and non-linearity of the system of equations. The properties of each of the characteristic fields are investigated, notably signal, monotonicity and non-linearity.

Two possible Riemann wave structures are identified in §4.3. The conditions for the existence of each of the types of solution is discussed in §4.3.2. Entropy-compatible solutions are calculated in §4.4, with shocks determined by the Rankine-Hugoniot jump equations. The existence of Riemann invariants for the simple waves encountered is also discussed.

4.2 MATHEMATICAL ANALYSIS OF THE CHARACTERISTIC FIELDS

4.2.1 Notes on hyperbolicity and non-linear propagation of non-linear hyperbolic waves

4.2.1.1 Source terms and hyperbolicity

The generic quasi-linear, autonomous, non-conservative form of the governing equations is

AB∂ VVG+∂ = (4.1) tix( ) i ( ) where A e Bi, i = 1…m, are real bounded matrix-valued functions of the dependent variables, m is the dimension of the number of space-like variables, V is the n-dimensional vector of dependent variables and G, the source term, is a n-dimensional vector valued bounded function of the dependent variables.

The source terms are of paramount importance in what concerns the quality of the solutions, understood as its physical plausibility and agreement with observations. They are less important for the study of the mathematical properties of the system, a claim that is substantiated next. It should be made clear that the study of the mathematical properties are, in this chapter, restricted to the qualitative discussion of the solutions, namely existence and unicity, and to the study of the nature of the propagation of information, in particular type and number of conditions at the contour of the solution domain.

Regarding the existence and unicity of the solutions, if the initial conditions are smooth bounded functions and the components of the matrixes A to Bm are smooth functions of the dependent variables, there is a region around the initial conditions in which the solution exists and is unique provided that the source term, G, is integrable. If the initial conditions are less well behaved, existence and unicity are difficult to establish (Dafermos 2000, p. 50) but the necessary condition concerning G is still its integrability. As seen below, it is easily shown that the source terms devised in Chapters 2 and 3 do posses the regularity requirements that warrant its integrability.

As for the nature of the propagation of the information inside the solution domain, the fundamental propagation typologies are the hyperbolic, the elliptic, the parabolic and the respective hybrids. The source terms are not fundamental for the filiation of (4.1) in the later categories. Support for this claim can be found in the elegant account of non-linear wave-like propagations of Jeffrey & Taniuty (1964), p. 3-9. In particular, the nature of the source terms is irrelevant for the definition of the type and number of boundary and initial conditions. Thus,

306 in the remainder of this chapter, the source terms G of (4.1) will be discarded and the homogeneous system

AB∂ VV0+∂ = (4.2) tix( ) i ( ) will be investigated.

4.2.1.2 Wave-like description of wave forms

The solution of (4.2) is sought as a combination of wave forms. A wave form is imagined as a bounded piece-wise continuous vector valued function of the space and time co-ordinates that is superimposed to an equilibrium state. These regularity properties are enough to allow for a Fourier description of the wave form and hence, without loss of generality, it is assumed that the wave form is obtained by linear superposition of an infinite number of harmonic waves. The nature of the propagation of the wave form is discussed next.

The idea underlying the search for solutions with a wave-like behaviour for the quasi-linear system (4.2) is to take advantage of some of the well known properties of the linear propagation of periodic waves. For instance, it is known that the Cauchy problem for the simple advection equation, ∂+λ∂=tx()vv( ) 0 , where λ is a real constant, admits the solution vxt(,)=−λ gx ( t ) when the initial condition is vx(,0)== gx () Re{ Ax ()eiκx } , where κ is real and A(x) is piecewise continuous. Figure 4.2 shows an example where g(x) is defined as above, where A(x) is a real smooth function with compact support. It is not important that the initial condition is not periodic as long as it can be extended to display periodicity. As seen in figure 4.2, A(x) was chosen to be zero outside the interval [0,b], b =πκ2 . A periodic function can be obtained by repeating A(x) in the intervals ⎣⎦⎡⎤( jj−πκπκ12) , 2 , for all integer j.

v(x,t)

t

a dt(X) = λ b x FIGURE 4.2. Solution of the scalar linear advection equation for the Cauchy problem iκx ∞ vx(,0)Re= { Ax ()e } where Ax()∈ C0 .

307 As shown in figure 4.2, the solution of the simple advection problem in the space-time domain is simply the displacement, over a distance equal to λt, of the profile exhibited at t = 0. The amplitude v(x,0) corresponding to each point in the line t = 0 is conveyed, unaltered, along a line whose slope in the space-time domain is dx/dt = λ.

This result is easy to obtain if it is noticed that each value of vx( 0 ,0) is associated to a value of kx0 , x0 ∈π[0, 2 k]. Similarly, at a given value of t, to each value of vx()1 −λ tt, corresponds a value of kx()1 −λ t, x1 ∈[λπ+λtkt,2 ] . Equation d()t( Xt) =λ (figure 4.2) can also be written kx()−λ t = K, which implies kx01= k() x−λ t = K if x10=+λxt.

Finally, if kx01=−λ k() x t then vx( 01,0) =−λ vx( tt , ) for all x0 ∈[0, 2π k], t > 0 and x10=+λxt. Thus, the solution at all times is easily obtained if it is kept in mind that v is constant along the line, called the phase of the wave, Σ( x,()tkxtK) =−λ=.

It would be important to understand to what extent the procedures valid for the linear solution can be of use in more complex situations. In this text, while looking for the solution for the initial-boundary value problem for quasi-linear systems of physically meaningful PDEs, it is of considerable interest to find variables and coordinates for which initial profiles are purely advected, by which it is meant unaffected by diffusion or attenuation. This quest leads to the notion of hyperbolicity. It will be seen that these variables and coordinates are possible to be found only if the system (4.2) is hyperbolic.

Further discussion of hyperbolicity requires the formal identification of wave form sought as a solution for (4.2). It is a n-dimensional vector, n being the number of dependent variables, that can be written as

+∞ ()p Vx()p (,)t = Vk ()ei(ω+ (kkx )t i ) d k (4.3) ∫−∞ where the components of x are the space co-ordinates, t is the time, the components of k are the wave numbers of the elemental harmonics in each of the space directions, ω()k is the angular frequency of each of the elemental harmonics, i is the imaginary unit and Vk() are the weighting factor of each harmonic.

It should be made clear that in a m-dimensional space, there are n vectors kj, j = 1…n, which are m-dimensional. Each of these corresponds to each of the n entries of the wave form vector. They are all linearly dependent, i.e., kej = ς jk where ς j are constants and ek is the direction of propagation of the wave form. The phase of an harmonic corresponding to the wave number kj, j = 1…n, is ω+()kkxj j i , where ω()k j is the angular frequency. It should also be stressed that the harmonics are weighted in phase and amplitude by V j ()k j . For the sake of simplicity, (4.3) is written for the case where the wave number k is the same for all of the n components of the vector of the dependent variables, i.e., ς j =ςi for i and j. This restriction does not pose limitations to the following developments and can be lifted whenever necessary.

308 If the problem was linear, each of the harmonics would be propagated at constant speed and the solution would be retrieved by superimposing the displaced harmonics at the desired time. In quasi-linear problems this procedure is not feasible. Yet, it is possible to write the wave form in a way that resembles the solutions of the linear advection equation. As seen in Annex 4.1, equation (4.3) can be written as

()p ()pp () i(,)Σ x t Vx(,)tt= Vx0 (,)e (4.4) provided that the concept of phase, represented by Σ, is generalised. As is the case for linear problems, it may be imagined that each Vx()p (,0) in the surface t = 0 propagates along lines of constant phase. In an autonomous system such as (4.2), each value of the wave form at the origin of the time is propagated with a unique speed. More precisely, at small times near the origin, there is an injective continuous application that maps a propagation speed to each Vx(,)t . Unlike linear waves, though, the wave form may endure deformation because its ()p points do not necessarily propagate with the same speed. In this case, the function V0 in (4.4) must be a function of the space and time coordinates. It is now assumed that the application that maps propagation speeds and Vx(,)t exists for larger times. The method of constant phase is based on this assumption. The propagation of the wave form is observed as the evolution of the locus of the points with the same phase, which can be written as Σ=ϖ+=()pp(,)x tt () κix K (4.5)

In (4.5) K is a constant, κ is a wave number and ϖ is the angular frequency corresponding to κ. The wave number should be close to that corresponding to the main harmonic contribution in (4.3) and can be computed as shown in Annex 4.1.

4.2.1.3 Constant phase

In a m-dimensional metric space, the locus of the points whose phase is K is a (m−1)- dimensional manifold. For instance, in 3 , the locus of the points with the same phase would be represented by two-dimensional manifold as seen in figure 4.3. The progression of a three dimensional wave in the direction of κ is represented by the position of the two-dimensional manifold Σ(x,t) − K = 0 in two distinct instants, dt apart. If the most distinctive feature of the wave form is a sharp gradient following the crest (see figure 3.1, p. 182). It is generally called a wave front. Without loss of generality, and with considerable gain of visual suggestion, a surface such as that represented in figure 4.3 may be thought to represent the propagation of a wave front.

In a two-dimensional space, the locus of a given constant phase is a line. The progression of a two-dimensional wave front can be observed in figure 4.4. The two-dimensional manifold shown in figure 4.4 represents the locus, in the space-time domain, of the wave front, i.e., its successive positions over time.

A simpler case, though less easy to depict graphically, is the propagation of a one- dimensional wave. In a one-dimensional domain the position of the wave front is represented by a point and its direction is the x direction, the only spatial co-ordinate. The wave progression, in the space time domain, is represented by a line. Figure 4.5 shows such a line;

309 the wave front is, following the constant phase method, represented by the phase isoline Σ−=(,)xt K 0.

x2

κ

x1

x3

FIGURE 4.3. Propagation, in the direction of the wave number κ, of a three-dimensional wave form in 3 .

t

t = t0 x2 t = t0 + dt

t = t0 + 2dt κ

x1

FIGURE 4.4. Propagation of a wave form in a two-dimensional metric space. The locus of the successive points of the wave front is a two-dimensional manifold in 2 × + .

310 For a one-dimensional wave, (4.5) is simplified to Σ(,)x ttxK=ϖ +κ = (4.6) where the symbols maintain their previous definitions. Again, the phase can be interpreted as a potential and, along one of its isolines, one has

ddd0Σ=∂tx( Σ) tx +∂( Σ) = (4.7)

Rearranging the terms of (4.7), it becomes

∂Σ( ) ϖ dx t = =− (4.8) ∂Σx() κ dt

Let (4.6) be written as function of a parameter s, so that x = Xs(), X : + → and tTs= (), T : ++→ are continuously differentiable mappings. A vector c()s can be defined as c = [ X ()sTs ()] .

t

n* C ∂Σt( ) −κ

∂x(Σ) ϖ

x FIGURE 4.5. Propagation of a wave form in a one-dimensional domain.

The derivative of c with respect to s is the tangent of the line of constant phase and is defined as

ddsxsts()cc=∂( ) ( X ) +∂( c) d(T ) ⇔

ddsxsts()ce=+( X ) e d(T ) (4.9)

Without loss of generality, let st= . In that case, attending to (4.8) and to the fact that dddt()X = xt, one has

Ccee≡−κ()dsxt( ) = ϖ+( −κ) (4.10) The vector C, depicted in figure 4.5, is, following its definition, tangent to Σ−=(,)xt K 0. The direction of C is, from (4.8) and (4.10), normal to a vector defined as neeee* =∂x() Σxt +∂ () Σ t =κ x +ϖ t.

311 Thus, bearing (4.10) in mind, equation (4.8) expresses the result that the gradient of a potential is perpendicular to its isolines. The physical meaning of the construction, seen in 4.5, is that any disturbance associated with a particular phase, Σ, is carried, in the space-time domain, along the respective isoline with a velocity, called phase velocity, equal to ds()c . By disturbance it is meant any superimposition to a state of equilibrium, in accordance to the notion of wave form (see also Whitham 1974, p. 127).

Re-writing (4.9), the phase velocity is written ϖ d ()ceeee= −+=λ+ (4.11) s κ xt xt where λ is the slope of the direction of Σ, since dx ϖ = d()()Xt ≡λ=− (4.12) dt t κ The role of λ is fundamental in the study of the qualitative behaviour of the solution of PDEs and also for its quantification. Given a point P in the solution domain, it is important to know how many independent lines of constant phase cross that point (the value of p in (4.4)), how fast will the information propagate along these lines, i.e., how large is λ corresponding to the pth wave, and what is the nature of the information carried along such lines.

4.2.1.4 Classification of systems of PDEs

The number of independent propagation directions that exists for a system of PDEs describing physical phenomena is investigated next. The analysis is restricted to one dimensional systems of more than one dependent variable (m = 1, n ≥ 2), i.e., the governing equations are in the form of

AB∂tx(VV0) +∂( ) = (4.13) and the wave form solutions are

()p ()pp () i(,)Σ x t (4.14) VV= 0 (,)xte

Equation (4.13) is promptly obtained from (4.2) by setting B1 ≡ B. Introducing (4.6) in the wave-form solution (4.14) and the latter in (4.13), one has

AB∂+∂+∂+∂=VV()ppi(ϖ+κtx ) () i( ϖ+κ tx ) VV () pp i( ϖ+κ tx ) () i( ϖ+κ tx ) 0 { ttxx()00ee( )} { () 00 ee( )}

i(ϖ+κtx )()pp () i( ϖ+κ tx ) () pp () ABee{∂+ϖ+tx()VV00ii} { ∂( VV0 00) +κ=} (4.15) the term ei(ϖ+κtx ) can be factored out. It does not constitute a solution because it is different from zero for all finite real values of the phase. Equation (4.15) becomes

()pp () () p AB∂+∂+κ+ϖ=tx(VV) ( ) i{} BA V0 (4.16) {00} 0 =0

312 As explained above, solutions of (4.13) are sought as wave-forms, written as (4.14). If (4.14) is a solution of (4.13), it is so for all values of the phase. Thus, if the constant K in (4.6) is zero, (4.14) reduces to ()pp () and (4.13) becomes ()pp () . VV(,)x txt= 0 (,) AB∂tx(VV000)()+∂ =

This justifies the elimination of the first two terms in (4.16). The remainder of equation (4.16) can be written as

()pp () (BAκ +ϖ)V00 = and, since λ=−ϖκ()pp () (equation (4.12))

()pp () (4.17) (BA− λ=)V00

()p Equation (4.17) states that non-trivial solutions V0 can be found provided that the matrix BA−λ is singular. Thus, the computation of the direction of the phase is a eigenvalue problem. The condition of singularity of BA− λ is expressed by the condition of zero determinant det(BA− λ=) 0 (4.18)

Equation (4.18) is the characteristic polynomial of A−1B , admitting that A is non-singular1. The order of the polynomial is equal to the rank of A−1B or, equivalently, to the number, n, of dependent variables. The eigenvalues of A−1B are the p ≤ n distinct roots of the characteristic polynomial, simply called characteristics. From (4.11) and (4.12) and from figure 4.5 it is clear that they are the directions of the lines of constant phase. The phase velocities of system (4.13) are fully determined by the characteristics of AB−1 . As a result, the term characteristic, denoted λ, is taken as a substitute of phase velocity when referring to the direction of a given line of constant phase.

The number and the properties of the characteristics, i.e., the roots of (4.18), determine the mathematical properties of (4.13) in what concerns information transfer and well-posedness of initial and boundary conditions, i.e., type and number of boundary and initial conditions.

4.2.1.5 Domains of dependence and influence and conditions at the contour

If all the roots of (4.18) are complex, system (4.13) expresses a diffusive phenomenon. In that case, system (4.13) is said to be elliptic. If some of the eigenvalues are complex and some real, the system is said to be hybrid. If the rank of BA− λ is odd, and there are complex roots, the system is necessarily hybrid because the complex roots are conjugate pairs.

Most notably, if all the eigenvalues of A−1B are complex, then there are no real characteristic lines. If there are no characteristic lines in the space time domain, the information cannot be propagated from the contour to the solution domain along lines of constant phase such as that shown in figure 4.5. The value of V(x,t) at a given point in the

1 If A is singular, the roots of the characteristic polynomial are still propagation paths. Because the number of roots is less than the rank of A, the solution exhibits a parabolic behaviour (see the discussion in the next page). Ponce & Simmons (1977) discuss, in the context of the shallow water equations, the physical consequence of the absence of the time derivatives that, in some coordinate frame, make A a singular matrix. In the following text it will be assumed that A is non-singular.

313 solution domain H (see figure 4.6) cannot be tracked to any specific region in H or any specific point in ∂H (figure 4.6). Thus, solutions in the form of (4.14) are not possible for elliptical problems.

Equation (4.14) can, nevertheless, be used to intuit the type of solution obtainable for elliptic problems. If the eigenvalues of AB−1 are complex, then, by (4.12), so are the angular frequencies. Consequently, from (4.6) and (4.14) the wave form would be written

()pp () −ϖIm( )t iRe()(κ+x ϖt) VV(,)xt= 0 (,) xtee (4.19)

The effect of e−ϖIm( )t in (4.19) is that of attenuation of the wave amplitude as the time increases, hence the diffusive character of the elliptic solutions.

Having no definite directions of propagation, system (4.13) admits, in each point P = (xi,ti) belonging to H (figure 4.6), a solution V(P) that depends potentially on all other values of V(x,t), (x,t) ∈ H. Thus, the domain of dependence of P is the entire domain H. Reciprocally, the value of V in P may affect the solution on all other points of H, i.e., the domain of influence of P is also the whole solution domain. This feature of elliptic systems is depicted in figure 4.6.

t Vj(x,t1), j ≤ 4 ∂Η

P Η

Vj(x,t0), j ≤ 4

domain of influence and x of dependence

FIGURE 4.6. Elliptic systems; domains of influence and of dependence of a given point P. The solution domain is enclosed by the dashed line .

Obviously, the variable t is not time-like. Inexistence of definite paths for information transfer implies physical reversibility, incompatible with a time-like behaviour. Thus, in the contour ∂Η (figure 4.6) of the solution domain, it is necessary to provide information where it is physically relevant. Because there are no time-like variables, the Cauchy problem does not make sense for elliptic problems.

The solution of the homogeneous problem (4.13) is non-diffusive iff the eigenvalues of (4.18) are real numbers. Non-diffusive phenomena are related to propagation problems in the broad sense. If the number of roots of (4.18) is p < n, where n is the rank of the matrix BA−λ and all the roots are real, the system is said to be parabolic. In that case, the algebraic multiplicity of some of the eigenvalues λ()p is larger than one. Figure 4.7 shows an idealised situation, with n = 4, where the algebraic multiplicity of the two roots is equal to two. There are less independent directions of propagation than dependent variables. In general, the information carried by each characteristic line concerns all n dependent variables. Let it

314 be assumed that there are p coordinate transformations such that the information conveyed by each of the p characteristic lines concerns only one dependent variable2. In the case depicted in figure 4.7 p = 2 and the characteristics would be associated to V1 and V3. Thus, the initial conditions would not be sufficient to specify the solution at P as no information regarding V2 and V4 would have travelled from earlier times.

Boundary information, i.e., information placed on a time-like line such as x = x0 in figure 4.7 would have to be called to complete the solution. An imprecise generalisation of the notion of characteristic is often performed. Since the boundary conditions imposed on t ≤ ti, where ti is the time coordinate of P, affect the solution of P, the horizontal line t = ti is conceived as a generalised characteristic line. Physically, a horizontal characteristic line means the information is propagated with infinite velocity. Mathematically, such a horizontal line would be a space-like line and, as seen in next section, well posed problems do not admit the specification of boundary conditions on characteristic lines. Thus, the initial condition cannot specify all the dependent variables.

t ∂Η

+ Η P Η Η −

(3) (4) λ(1) ≡ λ(2) λ ≡ λ

Vj(x,t0) j = 1,3 x domain of influence domain of dependence

FIGURE 4.7. Parabolic system with n = 4; domains of influence and of dependence of a given point P. The solution domain is enclosed by the dashed line .

From the above considerations it is easy to verify that parabolic systems are often associated to i) systems where not all the time derivatives exist or, in general, where the matrix A in (4.13) is singular and ii) systems where the initial state is not disturbed by wave-like phenomena, but whose evolution occurs in time.

The domain of dependence of P is the half-plane H(,):iH− = {}xt t≤∩ t , as seen in figure 4.7. This is a direct result of the existence of phenomena that propagates with infinite velocity. The domain of influence of P is HH+ = − , the complement of H. For the problem idealised in figure 4.7, with n = 4, the initial conditions can specify the only two of the dependent variables because the x axis is a characteristic line for the remaining two. Boundary conditions must be introduced, distributed so to match the mathematical and physical requirements of the system. In general, for the wave-like solutions, the boundary conditions are placed in time-like lines associated to the b characteristics that satisfy Cn()b i < 0 (4.20)

2 The circumstances for which this transformation is possible will be discussed below.

315 where n is the outfacing normal to ∂H and C is given by (4.10) (see also Hirsch 1988, p. 99). For the phenomena that propagate with infinite speed, the boundary information should be placed in accordance to physical or numeric criteria. In the case shown in figure 4.7, at x = x0 (3) there is one characteristic for which (4.20) holds. At x = x1 it is the characteristic λ that verifies (4.20). The remaining information was arbitrarily placed at x = x0. A well-known example of these type of systems can be drawn from river hydraulics. The shallow water equations without the local inertia in the momentum equation are

∂+∂=huh0 and ∂+=β−τρ1 ughg2 sin( ) ()w h tx() ( ) xb( 2 ) () where the variables assume their usual definition. This is imprecisely called the diffusive model (Ponce & Simmons 1977) despite the fact that there is no diffusion but propagation without wave-like character. Only one variable can be specified at t = 0 while the other must be computed from one of the equations. Usually, the initial condition specifies q(x,0) = u(x,0) h(x,0), the unit discharge. In that case, the water depth, h(x,0), is computed from backwater equation

23()w 2 dxb()hqxgh{ 1−=β−τρ− ( ,0)()} sin( )() ghqxqxgh '( ,0) ( ,0) () where qx'( , 0)= d q is the derivative of the unit discharge, in case it is not constant. It x()t=0 is easily seen that the absence of time derivatives in the momentum equation is equivalent to the assumption of infinite propagation velocities in the channel. In fact, the characteristic polynomial would yield the roots d0t = and ddxt≡λ(1) = u( Fr1Fr22 − ) . The root dt = 0 represents the physical requirement that, at each time level, the flow instantaneously adapts to any constraint. The finite root depends on the Froude number; the direction of propagation is positive if Fr > 1 (supercritical regime, upstream hydraulic control) and negative if Fr < 1 (subcritical regime, downstream hydraulic control). The boundary conditions specify both variables at the upstream reach if 10−qgh23() there will be boundary information at both downstream and upstream reaches. The problem is ill-posed if qgh23( ) = 1 .

The remaining propagation phenomena are of dispersive or of hyperbolic type (c.f., Whitham 1974, pp. 4-10). In either case, the number of roots of (4.18) is p = m, i.e., there are as many eigenvalues of AB−1 as dependent variables. Equivalently, the number of independent propagation directions, or lines of constant phase, is equal to the rank of BA−λ . In the general case the angular frequency is a function of the wave number. If the second derivatives of the angular frequency are null, as was the case in the derivation of (4.4) performed in the Annex 4.1, the system is hyperbolic. In this case, the characteristics (phase velocities) are independent of the wave number since λ()pp=−ϖ () κ and ϖ∝κ()p . On the contrary, in a dispersive system, the second derivatives of the angular frequency are not zero. It can be shown (Whitham 1974, p. 99) that the individual waves segregate and that the phase velocity is the propagation velocity of the wave train.

316 Figure 4.8 shows a trivially simple solution domain of a hyperbolic system with n = 4. At each point P in H, four characteristic directions can be identified. These directions can be tracked back in time, to the beginning of the times, to form a closed envelop. Such envelope, represented in light green in figure 4.8, is limited by the “fastest” positive and negative characteristic lines. It represents the set of points (x,t) whose values of V(x,t) influence the solution at P = (xi,ti). Because all the propagation speeds are finite, no information that can affect the solution at P is coming from outside its domain of dependence. Similarly, there are four characteristic lines issuing from P to future times. The envelope formed by the fasted positive and negative characteristics, represented in light blue in figure 4.8, is the domain of influence of P. Again as a consequence of the finite propagation velocities, the solution at P cannot influence any region of H outside this domain.

t Η +

P Η ∂Η Η −

(1) (2) (3) λ λ λ λ(4)

domain of influence x domain of dependence

FIGURE 4.8. Hyperbolic system with n = 4; domains of influence and of dependence of a given point P. The solution domain is enclosed by the dashed line .

Initial and boundary conditions must be made available in ∂H. The number of initial conditions is simply the number of intersections between a space-like direction and the characteristic lines. In fact, this is the basis for another definition of hyperbolicity (c.f., Jeffrey & Taniuty 1964, p. 15). System (4.13), with n dependent variables, is hyperbolic iff any space-like direction intersects n characteristic lines while satisfying Cn()k i < 0 , k = 1...n (4.21) Figure 4.8 shows the intersection of the four characteristic lines with the space-like boundary. Equation (4.21) is necessary to ensure that the initial conditions are prescribed in the correct space-like boundary, i.e., the one relative to the past times. A more explicit depiction of the necessary and sufficient conditions to be prescribed at the proper space-like boundary is shown in figure 4.9.

The number of boundary conditions in the time-like boundaries is prescribed in accordance to (4.20). For the simple situation idealised in 4.8 there are three positive characteristics and one negative. Thus, at a point U in x = x0 there must be 3 independent equations which, complemented with the information travelling along λ(4), allow for the computation of the solution V(U). Boundary information at x = x1 is specified in accordance to the same principles. One equation is required, corresponding to the negative characteristic line. Both situations are depicted in figure 4.9.

317 U D H

3 independent 1 boundary boundary condition conditions (4) (3) λ λ(1) λ(2) λ I space-like ∂H 4 independent initial conditions FIGURE 4.9. Summary of boundary and initial conditions for a hyperbolic system whose solution is sought in a simple path-connected set, H, with two time-like and two space- like boundaries and n = 4.

4.2.1.6 Ill-posedness and characteristics

Consider the PDE AB∂+∂=tx()VV0( ) . Let V be prescribed at some curve Β≡{(,)xt ∈ : x = X (), η t = T () η} . Let the implicit functional representation of that curve be Φηη=()XT(),() 0.

Then, if both VVΦ = ( XT(),()ηη) and Φηη( XT(),()) are known, the tangential derivative is known to exist if

η

VVΦξ()η =∂() d ξ ∫0 is a Lipschitz continuous bounded function. The derivative is thus defined except at a countable number of points. Along the curve, one also knows that the directional derivative is

∂=ηη()VVVd+d( XT) ∂xt( ) η( ) ∂( ) (4.22) since the unit vector in the tangential direction is

T eeeηηη=η+η=lim{}() dxt dxt() d d⎡ d() XT d ()⎤ d0η→ ⎣ ⎦

In the direction normal to the curve, the derivative of VΦ is unspecified and unknown. Yet, T since the direction normal to eη is, eνη=−⎣⎡ dd()TX η ( )⎦⎤ , the directional derivative can be written

∂=−∂νη()VVVd+d(TX) xt( ) η( ) ∂( ) (4.23)

It is now searched on which lines does the specification of V allow for the computation of the normal derivatives and, hence, the partial derivatives w.r.t. x and t. The system of equations to be solved is, in matricial form

318 ⎡⎤ AB0⎡⎤∂=t()V0⎡ ⎤ ⎢⎥⎢⎥⎢ ⎥ ⎢⎥ddηη()TXII0 ( ) ⎢⎥∂x()VV⎢ d η()⎥ (4.24) ⎢⎥⎢⎥d V ⎢ ⎥ ⎣⎦ddηη()XTIII−− () ⎣⎦ν() ⎣ 0 ⎦ It is clear that the system has a unique solution iff

det()AB()− dηη()XT−−( d()) ≠ 0 (4.25)

If d0η()T ≠ , which means simply that η ≠ x, then

d ( X ) detBA− η ≠ 0 (4.26) ( dη()T )

Assuming that the function are injective in certain intervals, then, at least locally,

dη( X ) dt()X = dη()T and

det(BA− dt( X )) ≠ 0 (4.27)

It was seen that there is one class of curves for which det(BA− λ=) 0

These are the characteristic curves, whose directions are λ. Thus, in order to make sure that system (4.24) has a solution, it is necessary that dt( X ) ≠ λ . It is thus concluded that if Φηη=Σηη()()XT(),() XT (),() is a characteristic curve, the specification of V renders the problem ill-posed, in the sense that (4.24) does not have a unique solution.

As a corollary, boundary or initial conditions cannot be specified over a characteristic line (in the context of the shallow water/ Exner equations, cf. Ferreira 1998, p. 45).

4.2.1.7 Characteristic variables and compatibility equation

Further attention is now given to the actual computation of the solution of hyperbolic problems. In a n-dimensional hyperbolic problem, the solution of (4.13) at a point P in the interior of H can, at least formally, be written as a superposition of n wave forms. It could be written as

nn ()p VV(,)x txtxt==()pp (,)ei(κ−λxt ) V () (,) (4.28) ∑∑0 pp==11

It was seen that n independent characteristic directions cross at P (figure 4.8) each carrying independent and complementary information. The recombination of that information provides the necessary and sufficient conditions to compute the solution at P. Obviously, it is implied that, in general, each characteristic line carries information concerning all the dependent variable with physical meaning, i.e., the primitive and the conservative variables.

319 Thus, it is legitimate to ask if there is a transformation of variables (alluded to while discussing parabolic systems) for which each characteristic line conveys information related to one variable only.

(1) (1) T ()nn () One looks for solutions in the form of W = ⎣⎡w ... 0⎦⎤ … W = ⎣⎦⎡⎤0 ... w that satisfy (4.28). Equivalently, it can be stated that a transformation, not necessarily linear, and

(1) (n ) T a new set of variables W = ⎣⎦⎡⎤ww... are sought, such that (4.28) becomes

(1) (1)iμ−λ1 ( xt) (1) Wxt1(,)== w0 (,) xte w (,) xt ... iμ−λxt()n ()nnn () () (4.29) Wxtwn (,)==0 (,) xte w (,) xt

(p) If W exists, along each characteristic line p there travels information regarding Wp ≡ w only. The quasi-linear system of PDEs would then be amenable to a decomposition such that its solution would be the superimposition of solutions of scalar quasi-linear equations. To solve these equations, analogies drawn from the linear scalar equation (see p. 307) are useful. It will be discussed next how and when the intended decomposition is possible. It will be seen that while the set of quasi-linear scalar differential equations may be derived for all hyperbolic systems, not all will allow for explicit determination of W. In order to find the transformation between V and W, one might use the concept underlying (4.28) and look for linear combinations of the equations that compose (4.13). Then, a combination of derivatives is searched such that it is equivalent to the derivative of the desired new set of variables. The linear combinations of the equations that compose (4.13) may be written as

()p lVV0()AB∂+∂=tx( ) ( ) ⇔

()pp () ⇔∂+∂= aVbV0tx() () (4.30) provided that la()p A = ()p and lb()p B= ()p . System (4.30) is a set of p = n PDEs, each composed of n derivatives of the n dependent variables. Each equation (4.30) represents also a directional derivative in the space-time domain. The coefficients a(p) and b(p) may be written in such a way that the direction of the derivative is made explicit. Without loss of generality, let the direction of the derivatives be taken as the tangent to the path Γ(x,t) = cte. As usual, let this path be parameterized for s such that x = Xs() and tTs= (). System (4.30) becomes

()pp () ddstsx()TXξ ∂ (V) +∂=( )ξ (V) 0 (4.31) whenever

()pp () ()pp () l A = ξ ds(T ) ; l B = ξ ds( X ) (4.32)(a)(b)

At this point it is convenient to show that the path Γ(x,t) = cte, whose tangents are the directions for which the directional derivatives (4.30) are taken, is a characteristic line. Solving (4.32)(a) and (b) for ξ()p and equalling the result, it is obtained

320 ()p l0(ddss(TX)BA− ( ) ) = (4.33) Assuming that both x = Xs() and tTs= ()are injective Lipschitz continuous mappings in some neighbourhood of (xi,ti), (4.33) becomes, from the implicit function theorem

()p l0(BA− dt( X ) ) = (4.34)

Equation (4.34) expresses that the vectors l(p) are the left eigenvectors of AB−1 . Non-trivial (p) solutions for l are possible if the matrix is singular, i.e., if det(BA− dt()X ) = 0 . Two conclusions can be drawn from (4.34): i) its non-trivial solutions lead to the same eigenvalue problem, i.e., to the same characteristic polynomial, that was early obtained and expressed in (4.18); ii) the coefficients, organised in the vector l(p), of the linear combinations of PDEs, system (4.30), are the entries of the right eigenvectors of A−1B . Thus, the only directions that enable rewriting system (4.13) as a combination of derivatives, ()p all taken in the same direction, are the directions dt( X ) = λ of the characteristic lines. This proposition actually represents another, broader, definition of hyperbolicity. System −1 (4.13) is totally hyperbolic if the eigenspace of A B is of the same dimension of the space of the dependent variables, i.e., the system admits as many linearly independent eigenvectors as dependent variables. According to this definition, a nxn system of PDEs that have less than n eigenvalues is still hyperbolic if it has n independent eigenvectors (for an example cf. Whitham 1974, p. 76).

Let (4.31) be written so as to highlight the fact that the derivatives of the dependent variables are being taken along the characteristic lines. Without loss of generality let t ≡ s as in p. 311. The following notation for the combination of derivatives can be used

()pp () ξ∂+∂++ξ∂+∂=1 {dtt()TV (11 ) d t ( X ) x( V)} ... nttnt{ d( TV) ( ) d( X) xn() V} 0

()pp () (4.35) ξ++ξ=1 Dtntn()VV1 ... D() 0

()p The derivative Dt(Vk) is the time derivative taken along dt( X ) = λ and it can be interpreted as a Lagrangian derivative. It should be noted that (4.35) is the differential analogous to (4.28). The meaning of both formulations is that the solution at a point P in H is achieved through the composition of n independent sources of information, each, in general, carrying information related to the n dependent variables. Equation (4.35) is called the compatibility equation of (4.13). The coefficients ()p are easily ξk obtained from (4.33) up to an arbitrary scale factor. What is left to know is whether or not there is a combination of derivatives of and respective ()p such that new variables can V ξk W be obtained. For that purpose, the compatibility equation (4.35) can be written as

()pp () () p () p dtt()TV()ξ∂11( 11) + ... +ξ∂ ntntx( VXV) +d( )( ξ∂( ) + ... +ξ∂ nxn() V) = 0 (4.36) and, then

d+d0ttptxp()TW∂ ( ) () X∂=( W)

()p ∂λ∂=tp()WW+0 xp () (4.37)

321 provided that

()pp () () pp () (4.38)(a)(b) ξ∂11x()VVWVVW11 +... +ξ∂nxn() =∂ x() p ; ξ∂ t() + ... +ξ∂ ntn() =∂ t() p for each p-characteristic. Equation (4.37) can be further simplified for

()p D=0tp()W along dt( X ) = λ (4.39)(a)(b)

The meaning of (4.39) is clear: each variable Wp, herein called characteristic variable, does not change along the corresponding characteristic line. The analogy of the linear scalar advection equation is now pertinent. Along each family of lines of constant phase (characteristic lines) there is one and only one variable whose value remains constant along that direction. Furthermore, the new system is well-posed because the system is hyperbolic and there are as many variables Wp as characteristic directions. 4.2.1.8 Examples of computation using characteristic variables

Knowing that there are n independent eigenvectors and that the combination of derivatives taken along characteristic lines may allow for the definition of a new set of variables - characteristic variables -, let the procedures that led to (4.39) be condensed and rewritten in a simplified way. Assuming that A is non singular, let M = A−1B. Then, (4.13) becomes

∂tx(VV0) +∂M ( ) = (4.40)

−1 −1 let S ∂=∂tt()VW ( ) and S ∂=∂xx()VW ( ). Then (4.40) can be written

SMS∂+∂=tx(WW0) ( ) (4.41) which leads to SS−−11∂+()WW0 SMS ∂ () = tx ∂+∂=tx()WW0Λ () (4.42) where Λ = SMS−1 . If A is non-singular and x(s) and t(s) are indeed injective Lipschitz continuous mappings in some neighbourhood of (xi,ti), without loss of generality, it can be taken ξ()pp= l (). In that case the transformation matrix S is, from equation (4.38), defined as

S−1T= ⎡l(1) ⎤ (4.43) ⎢ ⎥ ⎢ ... ⎥ ⎢ ()n T ⎥ ⎣l ⎦ From elemental algebra, it is known that there are vectors r such that ⎧rl()kli ()= 1 if kl =

⎨ ()kl () (4.44) ⎩rli = 0 if kl ≠

−1 ()p These are the right eigenvectors of A B , directly obtained from (BA−=dt()X )r0. (1) (n ) The transformation matrix can more easily be written as S = ⎣⎡rr... ⎦⎤ .

322 From (4.37) and (4.38) it follows that the transformation Λ = SMS−1 renders necessarily a diagonal matrix whose main diagonal is composed by the eigenvalues of the system. Expanding the last equation of (4.42), one verifies that the set of characteristic variables allows for a formally decoupled system of PDEs, equivalent to (4.13), that reads

(1) ∂+λ∂=tx(WW11) ( ) 0 ... (4.45)

()n ∂+λ∂=tn()WW xn ()0

()p As seen above, along the direction dt( X ) = λ (4.45) can be written as

D0t(W1 ) = ...

D0tn()W = (4.46)

It was showed that the problem of finding a transformation of variables and of coordinates such that each characteristic line conveys information regarding one variable only may have a solution. At a point P in H, the solution may be found through (4.46), a set of equations that is valid along the n characteristic lines that intersect at P. Formally, this is equivalent to what is expressed in (4.29): the information regarding each characteristic variables is transported by the respective characteristic line alone. Formally, this is an improvement relatively to what is expressed in (4.28) or in (4.35). These express that the solution is obtained at the cross of n lines of constant phase, each of which, in general variables, conveys information related to the n dependent variables. Summing up the above discussion, it can be said that the problem of finding a transformation of variables and of coordinates has a solution if i) the system is totally hyperbolic and ii) if (4.38) has a solution, not necessarily unique. Assuming that the characteristic lines have the regularity properties assumed above, (4.38) becomes

−11− ∂=∂⇔∂=ss()WVSS () V () W (4.47)

Not all nxn equations in (4.47) are independent. Nevertheless, a system of n equations to n unknowns can be derived. The usual choice is

∂+∂++∂=+++WW... WSS−11−− ... S 1 VV12()11 () Vn ( 111121) n ... ∂+∂++∂=+++WW... WSS−−11 ... S − 1 (4.48) Vn12() Vn () Vn () n n12 n nn

If system (4.48) has a solution, the problem of reducing (4.13) to a quasi-decoupled system of scalar quasi-linear equations has a simple solution. Unfortunately, there is no guarantee that (4.48) does have a solution for n > 2. In fact, it is possible to find solutions for (4.48) only for n ≤ 2 (cf. Hirsch 1988, p. 566). It will be seen later that, in the absence of characteristic variables, it is always possible to write the compatibility equation (4.35). Its analytical- numerical solution is also generally well defined, namely at the boundaries. The analytical- numerical procedures that make use of (4.35) are broadly called method of characteristics. An

323 application of the method of characteristics is shown in Chapter 5, while solving the governing equations derived in Chapter 2.

The following example reports one case where the solution is easily found. It is the case of the shallow water equations for the flow of an incompressible fluid over a horizontal, fixed, perfectly smooth bed.

Example 4.1

Written in the primitive variable non-conservative form, the system of conservation laws obtained from the depth-integration of the incompressible Navier-Stokes equations, given appropriate cinematic boundary conditions and in accordance to the shallow-water approximation, is

∂+∂+∂=txx()huhhu( ) ( ) 0

∂+∂+∂=txx()ughuu( ) ( ) 0

The eigenvalues are λ=+(1) ugh and λ=−(2) ugh. The corresponding left eigenvectors are (1) ⎡⎤ and (2) ⎡⎤. Thus, the inverse and the transformation l = ⎣⎦gh 1 l =−⎣⎦gh 1 matrixes are

S−1 = ⎡⎤gh 1 ; S = 1 h ⎡ 11⎤ ⎢⎥ 2 g ⎢ ⎥ ⎣⎦⎢⎥gh −1 ⎣⎢ gh− gh⎦⎥

Equations (4.48) are, in this case

∂+∂=+hu()WW11( ) 1 hg

∂+∂=−hu()WW22 ()1 hg (4.49)

Admitting that the derivatives can be separated, W1 can be integrated first in h. It is obtained

(1) ∂=h()WghW11 ⇔=+ 2 ghfu( )

Deriving the expression for W1, obtained above, with respect to u and integrating the result, one obtains the first characteristic variable

(1) ∂=uu(Wf1 ) d( ) =⇔ 1

(1) ⇔≡=+ Ww1 u 2 gh

The second characteristic variable can be derived using the same procedures. The result is

(2) Ww2 ≡=− u2 gh

It is thus retrieved the well-known result that the shallow water equations can be written as

ughK+=2 (1) along λ=+(1) ugh and

ughK−=2 (2) along λ=−(2) ugh

324 (1) (2) where K and K are constants that can be quantified from the initial conditions. The following example provides further insights on the solution procedure, using the concept of wave form. The solution procedure for the scalar linear advection equation is used as an analogy.

Example 4.2

Consider the following system of PDEs, written in conservative form

∂tx(uuu112) +∂( ) =0

2 ∂−−∂−uu1 uu =0 (4.50)(a)(b) tx()122 (() 12)

The corresponding characteristic polynomial is (λ −+(uu12))( λ+−( uu 12)) =0 from which (1) (2) the eigenvalues λ=+uu12 and λ=−−()uu12 are obtained. The diagonal matrix in (4.42) is

(1) Λ ==⎡⎤λ 0 ⎡uu12+ 0 ⎤

⎢⎥(2) ⎢ ⎥ ⎣⎦0 λ ⎣ 0 −−()uu12⎦ T T and the right eigenvectors are r(1) = []11 and r(2) = []11− . The transformation matrix and its inverse are

−1 1 S = ⎡⎤11 ; S = 2 ⎡11⎤ ⎢⎥ ⎢ ⎥ ⎣⎦11− ⎣11− ⎦

In order to find the characteristic variables, equations (4.47) and (4.48) are invoked. The characteristic variables are obtained from the following steps ∂=⇔=+ϕWWuu11 (1) ( ) u1 ()111222 ∂=ϕ=⇔ϕ=Wuud(1)11 (1) ( ) uu22()122()22 (1) 1 (4.51) Ww112≡=2 () uu + ∂=⇔=+ϕWWuu11 (2) ( ) u1 ()221222 ∂=ϕ=−⇔ϕ=−WUUd(2)11 (2) ( ) uu22()222() 22 (2) 1 (4.52) Ww212≡=2 () uu −

System (4.50) can now be written in the characteristic variable formulation. Attending to the formulation of the characteristics of the system, the following representations are equivalent

(1) (1) (1) (1) (1) (1) ∂+λ∂=⇔∂+∂=tx()ww ()0 t( www) 2 x( ) 0

(2) (2) (2) (2) (2) (2) ∂+λ∂=⇔∂−∂=tx()ww ()0 t () www 2 x () 0 (4.53)(a)(b)

In order to find the solution of (4.53) one can profit from the fact that the system is truly decoupled, i.e., each of the equations is a scalar quasi-linear PDE in the characteristic variable formulation.

325 Consider the following Cauchy problem. The governing equations are (4.50) and the initial conditions are u1(x,0) and u2(x,0), x ∈ Ω ≡ [0,1]. The latter functions are depicted in figure 4.10. From the data in figure 4.10, the initial values of the characteristic variables are computed, attending to (4.51) and (4.52).

3.0 3.0

2.0 2.0 (-) (-) 1.0 2 1.0 1 u u

0.0 0.0

-1.0 -1.0 00.5100.51 x (L) x (L)

FIGURE 4.10. Initial conditions for the system (4.50) in example 4.2.

The equivalent Cauchy problem, in terms of the characteristic variables, is composed of (4.53) and the initial data depicted in figures 4.11(a) and (b). A Fourier decomposition of the initial conditions for w(1) and w(2) is performed, in order to choose an appropriate wave number for the wave form description. In a one-dimensional domain, the choice of the wave number cannot influence the space-like direction of propagation, as this is obviously the x direction. The wave-number is chosen to simplify the expressions of ()p and of the phase in V0 ()p (4.14) or of w0 and respective phase in (4.29).

Figures 4.11(b) and (c) show the Fourier decomposition of the functions shown in figure 4.11. It is clear that the dominant modes for w(1) are around k = 3 and k = 6. As for w(2) , the dominant modes are around k = 3. According to the procedures explained in Annex 4.1, the wave number for the wave form associated to w(1) is κ = 3. For simplicity, this same value is chosen for the wave form associated to w(2) . The solution of (4.53), can be obtained graphically with the help of the scalar linear advection analogy. The initial conditions are interpreted as wave forms and the value of the phase is computed in the domain, Ω, of the initial condition. The values of the initial conditions, wx()k (,0), k = 1,2, and the respective phases, computed from (4.6), are plotted in figure 4.11. In the same ()k figure, the associated values of the w0 , computed from (4.29), are also plotted.

(1) (2) The characteristics, λ=+uu12 and λ=−−()uu12, are computed at each point of Ω. From each point x ∈ Ω, two lines, corresponding to the two characteristics fields, are issued. Along each of these lines, the corresponding characteristic variables remain constant, i.e., ()kkk () () wxtwx(,)=−λ− ( d, ttt d), k = 1,2, a result that follows (4.46). Thus, if at t = 0 and x = x0, (1) (1) , then, at , one has (1)1 (1) (1) because the characteristics wx(,0)0 ≡ w0 t = t +dt wx(d,d)0 +=2 wtt00 w of the first characteristic field are, in this example, λ(1)= 2w (1) . Similarly, for the second characteristic field, (2) (2) and (2)1 (2) (2) , because, for the second wx(,0)0 ≡ w0 wx(d,d)0 −=2 wtt00 w characteristic field, and λ=−(2)2w (2) .

326 2.0 2.0 (a) (b) 1.0 1.0 (-) (-)

1 0.0 2 0.0 w w

-1.0 -1.0

-2.0 -2.0 00.5100.51 x (L) x (L)

3 3 (c) (d) 2.5 2.5 2 2 (-) (-)

0 1.5 0 1.5 w w 1 1

0.5 0.5 0 0 0102030 0 102030 k (1/L) k (1/L)

(1) FIGURE 4.11. Initial conditions for the system (4.53) in example 4.2. Initial values of (a) w and (b) w(2). Fourier decomposition, in terms of wave numbers and amplitudes, of the modes that make up the initial conditions of (c) w(1) and (d) w(2).

So far, the solution procedure is analogous to that of the scalar linear advection equation. The result of this similar procedure is, nonetheless, quite different, as the resulting profiles, after a given elapsed time, are seen to deform. This feature is clearly seen in figure 4.12. After a given elapsed time, the profile of the solution of w(1) is sharpened in the positive direction whereas the profile of w(2) is sharpened in the negative direction. This is a consequence of the application of the method of constant phase to a system whose characteristics are proportional to the values of the variables. The constructions on figure 4.12 show that the value of the phase at a given point x0 will be at x0 + λt and that the larger the value of λ the farther away that particular value will be found. The value of w associated with that phase will remain constant along the line of constant phase, thus, higher values of w will travel faster than lower values of w. Thus, the profiles of w()k will sharpen their gradients in the direction of the respective k-characteristic, as is clear from figure 4.12.

Finally, the solution in terms of the original variables u1 and u2 can be obtained from the profiles of the characteristic variables. The required transformation is the inverse of that represented by equations (4.51) and (4.52). In this case, the inversion poses no problems because it is linear. The results are seen in figure 4.13.

327 3.0 3.0 (a) 2.5 2.5 2.0 2.0

1.5 1.5 S (-) S (-) 1.0 1.0

0.5 0.5 0.0 0.0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 3.0 3.0 (b)

2.0 2.0

1.0 1.0 w1(-) w1(-)

0.0 0.0

-1.0 -1.0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 3.0 3.0 (c) 2.0 2.0 (-) (-) 1.0 1.0 10 10 w w

0.0 0.0

-1.0 -1.0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x (L) x (L)

FIGURE 4.12. Construction of the solution of (4.53), example 2. Left: first characteristic variable; right: second characteristic variable. From top to bottom: (a) phase, (b) wave form and (c) ()k . The thin lines ( ) stand for the initial conditions and the thick w0 lines ( ) stand for the final solutions. Constructions show how the method of constant phase allows for the construction of the solution.

It is a matter of some interest to observe that there is strong deformation of the wave forms with apparent attenuation of the wave maximum. Earlier, it was stated that hyperbolicity is associated to wave propagation without attenuation. This is true for each of the elemental wave forms that compose (4.28) or (4.29). This can be verified by looking at figure 4.12. Both wave forms of w(1) and w(2) are indeed propagating without attenuation. It is the superposition of the wave forms that causes the change in the maximum amplitudes. This is easy to understand observing the transformation of the characteristic variables into primitives ones, equations (4.51) and (4.52), as these operations involve explicit summations and subtractions.

328 3.0 t = 0.0 t = 0.12 2.5 (a) t = 0.24 2.0 1.5 (-)

1 1.0 u 0.5 0.0 -0.5 -1.0 0 0.2 0.4 0.6 0.8 1 3.0 2.5 (b)

2.0 t = 0.24 1.5 t = 0.13 t = 0.0 (-)

2 1.0 u 0.5 0.0 -0.5 -1.0 0 0.2 0.4 0.6 0.8 1 x (L)

FIGURE 4.13. Solution of the system (4.50) of example 2. Results at three distinct times for (a) u1 and (b) u2.

()p It should also be kept in mind that the functions V0 represent also phase averages. Thus, the superposition of non-attenuating elements in equation (4.28) will generally result on deformation and attenuation.

4.2.1.9 Non-linear wave propagation and shock formation; the scalar case

One last remark concerning non-linear hyperbolic wave propagation should be made at this stage. It was seen that the waves of example 2 deform and grow steeper in the direction of its characteristics. This is true because the characteristics are proportional to the value of the solution at each point of the domain. Later, this condition will be generalised. For now, it its sufficient to bear in mind that the behaviour shown in example 2 occurs necessarily when 2 the modulus of the flux function is convex, i.e., d()0w ()fw > . This is the case of both 2 2 fw()()(1)= w (1) and fw()()(2)=− w (2) .

It was seen in figure 4.12(a) that the values of the phase along the x-axis, initially a straight line, deform as time grows. Eventually, there is point, to the right of the x-axis for w(1) and to the left in the case of w(2), for which there will be more than one value of the phase, i.e., the thick lines in figure 4.12(a) will become vertical.

329 In fact, because the fluxes in equation (4.53) are convex, the characteristics are increasing functions of w(1) and w(2) . In the space time domain, this is represented by the convergence of the characteristic lines until eventually intercepting at a finite time. This is shown in figures 4.14(a) and (b), for the case of λ(1)-characteristics of example 2. Figure (a) shows the formation of the shock whereas figure (b) shows two w(1) profiles before and after the formation of the shock.

0.50

0.40

0.30 ( T ) t 0.20

0.10

0.00 2.5

2.0 t = 0.10 t = 0.40

1.5 (-) 1 u 1.0

0.5

0.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 x (L)

FIGURE 4.14. Formation of a shock in non-linear hyperbolic wave propagation. (a) characteristic and shock paths in the xt domain; (b) profiles of w(1) before and after the formation of the shock.

In the case of the quasi-linear equations of example 2, the time for which the shock is formed is easy to compute. The time for which the shock is initiated corresponds to the point, in the space-time domain, where the first two characteristic lines intercept. Let the path of the shock be represented by the set of points, Γ≡{(,)x txst ∈ : = ()} . Behind the shock the value of w(1) is designated by w− . Similarly w+ is the value of w(1) immediately after the shock. With the help of a Taylor expansion around (ww− + + ) it can be proved that the 2 1 +−⎛⎞ −+ direction of the shock, Sst≡ d()t(), is such that Sww=λ+λ+O⎜⎟ − , where 2 ( ) ⎝⎠ λ=λ±±()w (cf., Dafermos 2000, p. 148).

330 Assuming that the shock is sufficiently weak, the above result states that its velocity lies between the values of the characteristics on each side of the discontinuity. Since the shock occurs as a consequence of the convergence of the characteristics, i.e., in a region where the (1) initial condition is monotone decreasing, d(,0)dxx(wx ) = (λ≤0 ) 0, then, at the shock −+ −+ formation, λ>λ00 and λ>00S >λ. Let xA be chosen such that λA > S and xB chosen such that λB < S. Since the initial condition is monotone decreasing in this region, the set of all xA is disjoint of the set of all xB and there is xm simultaneously supremum of the set of all xA and infimum of the set of all xB. Thus, the shock is originated over a characteristic λm issued from xm.

Let (xI,tI) be the coordinates of a point of interception of λ A and λB <λA over the shock path. Then, xIAAI=+λxt and xIBBI= xt+λ , where, at the origin of the time, xA is the point from which λA was issued and xB is the point from which λB was issued. Simultaneously, (,)xIItxtxt∈Γ≡{ (,) ∈ : = γ ()} , i.e., (xI,tI) belongs to the path of the shock. The time can be eliminated to obtain

xB − xA tI =− λB −λA

The critical time for the formation of the shock is the minimum of tI and belongs to the set of all tm such that

− 11 + 11 tm =−lim =− , tm =−lim =− (4.54) xx→ − λ−λmA − xx→ + λBm−λ + Am d ()λ Bm d ()λ xxmA− x 0 m xxBm− x 0 m Because the initial distribution of the characteristics is continuous differentiable, the left and right derivatives given by (4.54) are equal and

−1 t =−d λ (4.55) mx{ ()0 m}

In order to locate xm, the following argument can be pursued. Any point in the region of monotone decreasing initial conditions is a candidate and must be tested. The shock is initiated when the characteristics first cross. This means that one is looking for the minimum tm, computed by (4.55), that is allowed by the initial data. Since, in this region (1) d(,0)dxx()wx ∝λ≤()0 0, this corresponds to the maximum absolute value of dx()λ0 or, equivalently, its necessarily negative minimum. Thus, the time for which a local shock is formed is

−1 tcx=−min{} d () λ0 (4.56)

A more general deduction of tc is given in Rhee et al. (1989), p. 45-47. 4.2.1.10 Weak solutions, shocks and simple rarefaction waves

Once formed, the shock will endure permanently in the solution because there is no mechanism to dampen it away. The solution of (4.13) is called regular while is smooth. Once a shock is developed, or if the initial conditions are discontinuous, the derivatives in (4.13) are

331 ill-defined. An extended definition of solution is required in order to accommodate its non- smoothness. A weak solution V(x,t) of the conservation law

∂+∂=tx(UV()) ( FV ()) 0 (4.57) is a bounded measurable vector valued function that satisfies

{}UF∂φ+∂tx() () φddx tx = V0 φ d (4.58) ∫∫H ∞

∞ for all smooth test functions φ with compact support, i.e., φ∈CH0 ()∞ , with

H,:∞ ≡−∞<<∞∧≥{xt x t 0} . Because φ is zero outside and at the boundary of the solution domain, H (see figure 4.8), equation (4.58) becomes

{}UF∂tx()φ+ ∂ () φ = 0 (4.59) ∫H The solution domain can be divided in two regions separated by the path of the shock. Let these two disjoint regions be H(,)H:()L ≡∈<{ x yxst} and H(,)H:()R ≡∈>{ x yxst}. Equation (4.59) can be written

{}UF∂tx()φ+∂φ ()ddxt +{} UF ∂φ+∂φ tx() () dd xt = 0 (4.60) ∫∫HH LR

Each of the above integrals obeys to

{}UF∂φ+∂tx() () φ dxt d = ∫H m

{}∂φtx()UF +∂φ () dxt d −{} ∂ tx() UF +∂ ()φddx t (4.61) ∫H ∫H m m

The second integral in the right hand side of (4.61) is zero because its integrand is (4.57), the 1 differential conservation law, and V is at least C in Hm, m ≡ L or m ≡ R. The first integral on the right hand side can be expanded with the help of Green’s theorem3 in the plane. For instance, for the region left of the path of the shock, one has

(4.62) {}∂φtx()UF +∂φ ()ddx txt = φ()()∂∂HHLL{} U d − F () ∂ H L d ∫∫HH ∂ LL

3 For the purposes of this text, Green’s theorem for two variables states that (Apostol 1980, p. 380)

∂−∂uvxxuxvxdd = d + d {}xx12() () 12{} 1 2 ∫HH∫ ∂

332 The circulation integral in (4.62) is evaluated considering that φ is necessarily zero in the “exterior” boundary of HL but might be different from zero in the boundary between HL and HR. Thus

φ−=()()∂∂HHLL{}UF ddxt () ∂ H L ∫ ∂H L ++ + ∂∂HH φ−+φ{}UF (sttx ( ), )d ( sttt ( ), )d 0 {}UF()LLddx − ()t ∫Γ↑ ∫∂ΓH\ L and

++ + {}∂φtx()UF +∂φ ()ddx t = φ{} U ((),)d s tt x − F ((),)d s tt t (4.63) ∫∫H Γ↑ L

A change of variables yields

++ + ++ + φ−=φ−{}UF((),)dsttx ((),)d sttt{} U ((),)d sttt() s F ((),)d stt t (4.64) ∫∫Γ↑ Δt Likewise, in the region to the right of the path of the shock one has

−− − −− − φ−=−φ−{}UF((),)dsttx ((),)d sttt{} U ((),)d sttt() s F ((),)d stt t (4.65) ∫∫Γ↓ Δt ∞ since φ is continuous across the shock path (it should not be forgotten that φ∈CH0 ()) then φ+ = φ− . Equalling (4.64) and (4.65) one obtains ()UU+−− S −−=( FF +−) 0 (4.66) where UU±±= ()stt(), , FF±±= ()stt(), and the velocity of the shock is, as seen before,

Ss= dt(). The system of equations (4.66) is known as the Rankine-Hugoniot shock conditions. It governs the solution of (4.57) in the points in which its derivatives do not make sense.

In the case of the quasi-linear scalar equation the shock velocity is obtained from Δ( f ()w ) SwΔ=Δ⇔=() ( fw() ) S (4.67) Δ()w

For the case of the particular equation whose solution is depicted in figure 4.14, equation (4.53)(a), the velocity of the shock is

22 ff− ()wwLR− LR 1 (4.68) Sww== =+=λ+λ()LR2 () LR wwLR−− ww LR

This result confirms the assumptions made in the derivation of the time corresponding to the initiating of the shock.

333 One last remark concerns the unicity of the solution. Once discontinuities are developed, weak solutions are potentially non-unique. The following classical result (cf. Prasad 2001, p.

12) highlights one such case. Consider equation ∂tx(ww) +∂( w) =0 , with initial conditions ⎧−1, x ≤ 0 wx(,0)= ⎨ (4.69) ⎩1, x > 0 An admissible solution is

⎧−≤−1, x t ⎪ x (4.70)(a) wxt(,)= ⎨ t , −< t x ≤ t ⎪ ⎩1, xt> as it verifies the differential equation for all times larger than zero. But it is also easily verifiable that ⎧−≤−1, xt ⎪ x ⎪ t , − tx<≤− at ⎪ ⎪−aatx, −<≤0 wxt(,)= ⎨ (4.70)(b) ⎪axat, 0<≤ ⎪ x , at<≤ x t ⎪ t ⎩⎪1, xt> with 0 ≤ a ≤ 1. The graphic depiction of (4.70) is shown in figure 4.15. Solution (4.70)(b) also verifies the governing differential equation in the continuous regions. Across the stationary shock at x = 0, the solution verifies the Rankine-Hugoniot conditions (4.68). The solution is thus not unique.

Uniqueness can, in some circumstances, be attained by limiting the set of solutions to those physically admissible. The work of Olga Oleinik (cf. Magenes 1996) is among the first to provide a sound basis for the choice of physically admissible solutions. In the scalar case, the solution of the equation ∂+∂=tx()wfw (() ) 0 is physically admissible if

f ()()ξ−fw−−++ fw () − fw () fw () − f () ξ ≥≥ (4.71) ξ−wwww−−++ − −ξ for every ξ between w+ and w− . If it is convex, which is the case for most applications in this text, (4.71) is equivalent to

λ− ≥≥λS + (4.72)

This is easily shown by introducing (4.67) in (4.71) and taking the limits ξ→w− and ξ→w+ . The role of ξ in (4.71) is to check the regularity of f(w) between f ()w+ and f ()w− . The convexity of f(w) is required in (4.72) to ensure the regularity of the flux. For general fluxes, it can be imagined that the small variations in the strength of the shock could render (4.72) false. This is shown in figure 4.16.

334 1.5 1.5 1.5 1 a = 0 1 a = 0.5 1 a = 1 0.5 0.5 0.5

w 0 w 0 w 0 -0.5 -0.5 -0.5 -1 -1 -1 -1.5 -1.5 -1.5 -5 0 5 -5 0 5 -5 0 5 x /t x /t x /t FIGURE 4.15. Family of solutions of the Burgers equation for the initial data (4.69) parameterised by a.

+ If, for instance, due to a change in the initial conditions, the right state changed from w1 to + − + w2 , then λ < λ2 and (4.72) could become not valid. This would question the well-posedness of the solution given small variations of the initial conditions.

(b) − + (a) − + + − λ > λ1 λ > λ , ∀w

− d (f(w−)) ≡λ− λ f(w−) w f(w−) + λ2 + f(w2 ) f(w+) + + + + f(w1 ) λ1 dw(f(w )) ≡λ + − + + − w w w w1 w2 w w

FIGURE 4.16. Implication of fluxes of ∂tx(wfw) +∂( ()) = 0 on the admissibility condition (4.72). (a) convex flux and (b) non-convex, non-concave flux.

The inequalities (4.72) are known as Lax E-condition. It loosely states that expansive shocks are not admissible, only compressive ones. The first are depicted in figure 4.17(b) as a set of rays issued from the shock path. The second are shown in 4.17(a) and were already identified in figure 4.14; the characteristics converge into the shock path.

A suggestive interpretation recurs again to the information transfer metaphor. In compressive shocks, the information carried by the incoming characteristics is lost. A different way to transfer information across the shock is required; this is represented by the Rankine- Hugoniot conditions (4.66). In the case of expansive shocks, new information is continuously generated at the shock as if the initial conditions were folded into a line, the shock path, in a given singular point. It is shown that this production of information would correspond to a decrease of the entropy of the system, hence its inadmissibility. The details are outside the scope of this text and can be consulted in, e.g., Dafermos 2000, p. 160-163. It should be noticed that, in most of the references, the (mathematical) entropy considered is the negative of the physical entropy. Thus, admissible shocks are, in those references, those for which the (mathematical) entropy decreases.

335 (a) (b) t λ− > S > λ+ t λ− > S > λ+

λ−

λ− + λ λ+ x x FIGURE 4.17. Admissible and non-admissible shocks according to the Lax entropy condition (4.72). (a) admissible (compressible) shock; (b) non-admissible (expansive) shock.

Clearly, the only admissible solution corresponding to the initial conditions (4.69) is (4.70)(a). The shock appearing in (4.70)(b) is clearly not compressive as seen in figure 4.18(b). The blue lines are the characteristics that are issued from the points along the shock path. They correspond to the constant state around the shock observed in figure 4.15, a = 0.5.

1 (a) 1 (b) 0.75 0.75 ( T) ( T) 0.5 0.5 t t 0.25 0.25 a = 0.5 0 0 -2 0 2 -2 0 2 x /t x /t FIGURE 4.18. Characteristics and shock paths corresponding to (a) solution (4.70)(a) and (b) solution (4.70)(b), a = 0.5. Blue lines correspond to the characteristics issued from the shock.

The admissible solution (4.70) could not present any shock, a result that follows from the fact that ww−+< in the initial condition. No shock would be able to verify (4.72) in the vicinity of the initial conditions. The admissible solution, (4.70), is, in terms of the characteristic lines, represented by the fan seen in figure 4.18(a). This configuration is designated an expansive rarefaction wave. It should be noticed that the solution shown in figure 4.15, a = 0.5, features a rarefaction wave split by a constant shock at the place where a vertical (zero) characteristic should be. It is frequent to find numerical discretisations featuring this erroneous solution when facing the problem of computing a rarefaction wave through a critical flow point (cf. Ferreira & Leal 1998). In this case, entropy corrections are enforced to keep physically relevant solutions only (see Hirsch 1988, p. 434 and Chapter 5, §5.3.1 and §5.6.5.5).

The properties of the weak solutions will be discussed at length in §4.3.3 of the present chapter. For further details on the properties of discontinuous solutions see, e.g., LeVeque (1990), p. 8, Prasad (2001), p. 23, Toro 1998, p. 64, Dafermos (2000) §8, pp. 147-175 or LeFloch (1988).

336 4.2.1.11 Summary of the notes on hyperbolicity

Hyperbolicity and some properties of non-linear hyperbolic wave propagation were discussed in the previous paragraphs. The main results will be frequently used in the remainder of this text. Thus, it is appropriate to summarize the main topics addressed. These can be organized as follows: i) hyperbolic propagation represents propagation without attenuation; for a scalar problem the general solution is wxt(,)=+ w0() x dw( f ( w )) t ; ii) for systems of PDEs, it is noted that the information supplied at the beginning of the times is conveyed along lines of constant phase (equation (4.6)); for this purpose, it is admitted that the wave forms are describable by a wave-like description (equation (4.4)); iii) at a given point in the solution domain, the solution is attained by superimposing necessary and sufficient information; such information travels along the lines of constant phase; iv) the computation of the direction of the lines of constant phase is amenable to a eigenvalue problem (expressed in equation (4.18)); the roots of the characteristic polynomial corresponding to a given system (characteristics of the system) are the directions of the lines of constant phase; v) strict hyperbolicity requires that the number of characteristics must be equal to the number of dependent variables of the system; vi) a weaker condition for hyperbolicity requires that there are as many independent eigenvectors as dependent variables; vii) in some simple 2x2 systems, there are variables, called characteristic variables, such that its values are propagated along the respective characteristics; for all other variables, each of the characteristics convey information regarding all the dependent variables; viii) if a characteristic variable formulation is not feasible, the compatibility equation (4.35) can always be written; this is at the root of the method of characteristics; ix) imposing information on a characteristic line renders the problem ill-posed; x) the initial conditions must specify all dependent variables; xi) at the boundaries, there are as many equations as negative characteristics relatively to the exterior normal at that boundary. xii) compressive shocks will occur if the initial conditions are monotone decreasing and the flux is convex; xiii) expansive shocks are not admissible as they represent non-physical sources of information; xiv) across a shock, the derivatives of (4.13) or (4.57) are not defined; the solution is specified by the Rankine-Hugoniot conditions (4.66)

Having addressed the general features of non-linear hyperbolic wave propagation, the following sections are dedicated to presentation of a summary of the conservation equations, whose particular properties are to be studied with greater depth with the techniques so far discussed.

337 4.2.2 Governing equations

4.2.2.1 Equations of conservation

The mathematical model applicable to geomorphic dam-break flows, was developed in Chapter 3, namely in §3.5, §3.6 and §3.7. It features unsteady flow hydro- and sediment dynamics and channel morphology and it resorts to semi-empirical formulations to account for flow resistance and depth-averaged velocity. In what concerns sediment dynamics, the dense limit approximation of granular flow theory of Jenkins & Richman (1985) and (1988), rooted in Enskog’s kinetic theory of dense gases (Chapman & Cowling 1970, §16, pp. 297- 322), provided the theoretical background for the core of the model.

Two-dimensional governing equations are integrated in an idealised layered domain shown in figure 3.32, p. 266. The lowermost layer is the bed, composed of grains with no appreciable vertical or horizontal mean motion. While the total flow depth and the depth-averaged velocity are, respectively, h and u, the flow is subdivide into two regions: i) the contact load layer, whose thickness, depth-averaged velocity and concentration are hc, uc and Cc, respectively, and ii) the suspended sediment layer, whose thickness, velocity and concentration are hs = h – hc and us = (uh – uchc)/hs respectively. The bed is defined as the surface that connects the centres of gravity of the uppermost layer of the pack of immobile grains. Above, the contact load layer is characterized by high concentrations and the stresses are generated during collision events among particles. Linear and angular momentum are transferred mostly by inelastic collisions. The submerged weight of the sediment grains is equilibrated by a reaction force in the bed, solicited by the quasi- permanent contacts in the boundary between the transport layer and the bed.

Above the contact load layer, the uppermost flow layer transports a comparatively small amount of wash load or no sediment at all. Hydrodynamic lift and drag are expected to play some role in this more diluted region. The grains are not sustained by collisional interactions but by lift forces originated from turbulent fluctuations (Sumer & Deigaard 1981, Nezu & Nakagawa 1993, §12, p. 251-258). Turbulent stresses are expected to be dominant in this region.

This layered structure is expected to retain the essential mechanisms of the two dimensional conceptual model while being mathematically simpler. In the process of integration of the governing two-dimensional equations, the shallow flow hypotheses are employed. It is thus assumed that: i) flow depth is small in comparison to the representative longitudinal length scale of the flow, ii) local and convective accelerations normal to the flow direction are negligibly small and iii) slope in the longitudinal direction is small. Conservation equations are thus found for each of the solid and fluid constituents in each of the identified layers. A reasonable compromise between computational simplicity and phenomenological complexity can be achieved if the equations are combined into the following system of five equations to five unknowns, equations (3.221) to (3.225), shown in p. 278.

Equations (3.221) to (3.225) can be solved for the total flow depth, h = hs + hc, for the layer- averaged flow velocity uuhuhhh=+()s sccsc( +) , for the flux-averaged concentrations Cs and Cc and for the bed elevation Yb. Because of relevance is immediately perceived, the latter variables form the set of the so-called primitive variables. Closure equations are required for

338 the layer thickness hc and average velocity uc in the contact load layer, for the bed shear stress, τcb, and for the mass density fluxes between the bed and the contact load layer and the contact load layer and the suspended sediment layer.

The remaining variables are the average flow density ρ=ρmsssccc( uh +ρ uh) uh or, simply ()w ρ=ρm ()1( +sC − 1) , and the average concentration CCuhCuhuhuh=+( s ss ccc)( ss + cc ). No special formulations are advanced to account for the saturated flow that may occur in the wave front, as seen in figures 3.1(a) and (b). However, it is suggested that an earth pressure coefficient k’c > 1 should be used when the hc = h. Because of the source terms, equations (3.221) to (3.225) are not amenable to theoretical treatment. Fraccarollo & Capart (2002) went beyond simply discarding the source terms. They attempted to show that there is a time window where homogeneous equations drawn from equations similar to (3.221) to (3.225) describe the phenomena with sufficient accuracy. They proposed characteristic time scales for the geomorphic and frictional phenomena that occur in erosional dam-break flows. Before a given characteristic time, the transport capacity is different from the actual sediment load. That time defines the geomorphic time scale, tg. Beyond this moment, it is expected that local equilibrium is a valid hypothesis. The frictional time scale, tf, is the characteristic time beyond which frictional effects become dominant in the momentum conservation equation.

Under the assumed hypotheses and for the specific set of formulations used, Fraccarollo &

Capart (2002) showed that there is indeed a time window such that tttg < < f . Furthermore, they also retrieved theoretically what has been observed in laboratory experiments (cf. Leal et al. 2003), that, for heavy granular materials like sand, local equilibrium conditions appear to be always valid, whereas, for light material, there seems to occur non-equilibrium sediment transport. This is explained by the fact that the characteristic hydrodynamic time scale, thg00= , is of the order of magnitude of tg for heavy material, while, for light material, tt0 g . In the first case, the shallow-water and the equilibrium hypotheses become valid at the same time. In the second case, the solution is valid, i.e., the shallow-water assumptions are valid, before the local equilibrium hypothesis becomes admissible.

Although Fraccarollo & Capart’s (2002) results are strictly valid for erosional flows and for a particular conception of the frictional time scale, it will be assumed that there is indeed a time window for which the homogeneous equations are a sufficiently good account of the geomorphic dam-break flow. Furthermore, given the fact that the concentrations of suspended sediment are expected to be one order of magnitude smaller than the concentration of the contact load (cf. Bagnold 1966), Cs will be discarded. Under these assumptions, system (3.221) to (3.225) can be simplified and written in quasi-conservative form

∂tx(Yuh) +∂( ) =0 (4.73)

∂+∂ρ+ρ+∂ρ+ρ+ρ=Ruhuhghhhh22()www1 () 22 () 2 txcccssxs() ()2 ( ( sccc)) ()w −gh()ρcc+ ρ h s∂−τ x( Y b) bc (4.74)

∂tx(ZCuh) +∂( ) =0 (4.75)

339 where YhY=+b is the water elevation, R = ρmuh is the mass discharge per unit width and Z =−(1pY ) bcc + Ch is an equivalent bed elevation that takes into account the sediment stored in the contact load layer. Equations (4.73) to (4.75) are solved for h, u and Yb. No closures are required for the fluxes normal to the flow but the concentration in the contact load layer, Cc, must be prescribed. The earth pressure coefficient k’c will not be considered in this analysis since it is a constant that would not change the nature of the pressure terms.

For the application envisaged in Chapter 2, it was shown, in §2.2.2.6, the momentum associated to the near-bed sediment transport is negligible if compared to the momentum of the water flow. In that case, the above equations can be simplified to

∂+∂+∂+∂=ttbx()hYuhhu ( )( ) x( ) 0 (4.76) u ∂−+tstb()uApCACCY() 1() (1 −−+ )() − ∂ ( ) h ⎛⎞u2 +∂ACu()2 +1 ghA − ∂ () ChCC + − + g ∂ () h ⎜⎟hs()2 () h() s x ⎝⎠h +∂ACu()2 +1 ghuACuCC − ∂ () + − +∂ u () ugY +∂ () = ( us()2 () u() s) x xb ()w −τbs( ρ(1 + (sCh − 1) ) ) (4.77)

(1−∂pY )tb ( ) +() ChC s +∂ h ( s ) ∂ t ( hhC ) +∂ u ( s ) ∂ t ( u )

(4.78) ++∂()()CChuhCCuhuhx ( ) ∂ ( ) ++∂ ux ( ) ∂ ( ) = 0 in which A =−()ssC11(1)( +− s ) allows for the inertia of the suspended sediment to be considered in the momentum equation. This system admits the unknowns h, u and Yb. Closure equations are required for Cs, C and τb. The correct use of this model requires that the time scales of the morphological processes associated to sediment transport are sufficiently low to admit that the quasi-equilibrium model is sufficiently good.

4.2.2.2 Closure and constitutive equations

Equations for hc, uc and τbc were derived in §3.7.2, pp. 278-287, from the granular flow theoretical framework that produced the conservation equations. The contact load layer is divided into a frictional sub-layer, a region were stresses are mainly rate-independent, and a collisional sub-layer, were stresses are born from momentum exchange due to inter-particle collisions (figure 3.32). Considering that the dam-break flow is a shear flow in a gravitic field, it is argued that the normal stresses increase with increasing depth, thus increasing the sediment concentration and restricting the fluctuating movement. As a result, enduring contacts among particles become frequent and frictional stresses become predominant in the lower areas of the contact load layer. Thus, the frictional sub-layer corresponds to the bottom region of the idealised contact load layer. Two mechanisms govern the dynamics of the frictional sub-layer: i) increasing the shear rate results in the increase of the sediment load and, thus, the contact load thickness would increase and, ultimately, so would the normal stresses at the bottom; as a result, the thickness of frictional sub-layer would increase; ii) increasing shear rate would also increase the flux of granular temperature towards the bottom; the kinetic energy per unit time supplied to the fluctuating motion would tend to repel

340 particles and reduce the sediment concentration and the normal stresses; enduring frictional bonds would be broken and the thickness of the frictional sub-layer would diminish.

Both mechanisms act on the triad: concentration, contact load thickness and granular temperature. If the equation of conservation of the fluctuating energy is invoked, the granular temperature can be expressed as a function of the former. Thus, it should be possible to draw a relation between concentration and contact load thickness from the dynamics of the frictional sub-layer. A control volume analysis of the momentum exchange in the frictional sub-layer provided, in §3.7.2, the means to derive the relation between hc and Cc. Assuming that: i) the superposition of the above described mechanisms render the local inertia negligible (not the convective inertia), ii) the frictional sub-layer acts as a buffer, transforming collisional stresses above into frictional ones below and iii) equilibrium occurs when frictional and collisional stresses cancel on each side of the sub-layer, the searched relation can be expressed as

θds Cc = (4.79) tan(ϕbc )h

()w where θ=τbc ρgs(1) − d s is the Shields parameter. The thickness of the contact load layer was determined from the equation of conservation of the fluctuating kinetic energy. The resulting expression was compared with the experimental data of Sumer et al. (1996). It was found that thickness of the contact load layer could be well approximated by h c = 1.7+θ 5.5 (4.80) ds

As for the velocity in the contact load layer, it was found that the velocity profile in the contact load layer would be well approximated by a power law whose leading term would be 3 4 ()y ds . Its depth integration renders

3 hc 4 1 − 1 ⎛⎞h ()g 10 4 c (4.81) uucx=ξξ=−θ()d7 gsd ( 1) s⎜⎟ hdcs∫0 ⎝⎠

()g where uytx (,) is the longitudinal velocity at a height y above the bed.

From dimensional arguments, namely that the shear stress scales with the square of the shear rate in the collisional granular flow regime (cf. Lun et al. 1984, Jenkins & Richman 1988 or Campbell 1989), and re-plotting the data of Sumer et al. (1996), the shear stress may be written as

()w 2 τ=ρbcCu f (4.82) where the resistance coefficient is given by CCdhuwf = fs0 ( )( s) , in an initial stage and, 2 in a latter stage, CuCdhuwf =−ϒ()01 + fs()() s. The involved coefficients must be determined empirically.

Physical restrictions must be added to equations (4.79) to (4.81) in order to specify Cc, hc and uc. The depth-averaged sediment concentration cannot exceed 1 – p, where p is the bed porosity. The thickness and the depth-averaged velocity of the contact load layer are limited

341 by the total flow depth, h, and by the depth-averaged flow velocity, u, respectively. Figure 4.19 shows a virtual sequence of uniform flows, performed with the plastic granular material of the experiments of Sumer et al. (1996), illustrating the need for restrictions. At a given flume inclination, the sheet flow occupies most of the flow depth and velocity of the contact load layer becomes the depth-averaged velocity. From this point on, the flow is saturated and may be treated as fully mature debris flow. It should be noticed that since the expressions for hc and uc were developed independently, the saturation point does not occur at the same Shields parameter in both sequences. This represents but a minor physical imprecision if it is guaranteed that uuc → faster than hhc → , in order to ensure that u is continuous. In this context, saturation means that the whole of the flow depth is laden with sediment. Thus, when + uu− c 0 , one must have Cpc ≤−1 .

10 100 u

u c h h/ds hc/ds h/ds u/u* uc/u* u/u*

h c 1 10 110 Shields parameter

FIGURE 4.19. Left axis: non-dimensional flow velocity depth-averaged over the contact layer

(uc, ) and over the totality of the flow depth (u, ); velocities are made non- dimensional by the friction velocity u*. Right axis: non-dimensional thickness of the contact load layer (hc, ) and total flow depth (h, ); depths made non-dimensional by the representative grain diameter. Each value of the Shields parameter corresponds to a virtual uniform sheet flow achieved in a 0.3 m wide with ds = 0.003 m, ws = 0.119 m/s, s = 1.27, and Cf0 = 0.05. The sequence represents the gradual tilting of the flume from ic = 0.01 to 0.107.

The main shortcoming posed by the closure equations (4.79) to (4.81) is that they are expressed by continuous but non-differentiable functions. This implies that the functions that 1 express for the characteristic speeds are not C ()× + , which, as seen later, represents a serious mathematical problem. Thus, throughout the remainder of the text, for computational purposes, it will be imposed that uuc = , ∂hc(u ) = 0 , ∂uc(u ) = 1, hhc = , ∂=hc()h 1 and

∂=uc()h 0 .

2 2 The errors committed are Ehh=εOrd( ) and Euu= Ord(ε ) where ε=hc()hh − h and

ε=uc()uu − u. After the solution is found, the thickness of the transport layer can be retrieved by (4.80).

The closure equations corresponding to the model based on equations (4.76) to (4.78) are the following. The thickness of the bedload layer is

342 hdguusdbs=Ψυ−min{ ( , ,** crs , ), 4 } (4.83)

The velocity and the flux-averaged concentration of sediment in the bedload layer are, respectively,

uubwbcb=−α−(11 C( * )) (4.84) and hu2 Csgd=−3.1x 10−8 ( 1) * θθ−θ (4.85) cb s2 () c υα* where uuwb = 7.1 * is the velocity of the water in that layer and α* = uuwb cb where ucb is the velocity of the particles travelling as bedload.

The resistance law is equation (4.82) but the friction coefficient is 1 C = (4.86) f 2 ()2.5ln()hds + 5.22

This closure sub-model does not require additional physical restrictions because the thickness of the bedload layer is always smaller than the flow depth. Some care in the implementation of the equations is required since, in incipient transport conditions, ucb ≈ 0 and α* 1 . However, in the same conditions, Ccb 1 ; hence, the product Ccbα* in (4.84) is indeterminate. This problem is tackled by computing the flux-averaged concentration in the first place and only then the average velocity. If the concentration, given by (4.85), is below an arbitrarily small number, the velocity is set uubwb= = 7.1 u* .

Since both systems of equations, (4.73) to (4.75) and (4.76) to (4.78), express a layered physical system, its structure is fundamentally the same. The following analysis will be based on system composed by equations (4.73) to (4.75) because it is this system that is subjected to the occurrence of discontinuities and, hence, requires conservative formulations.

4.2.3 Conservative and quasi-linear formulations

Written in vector notation, the first order, non-homogeneous, system of PDEs represented by equations (4.73), (4.74) and (4.75) becomes

∂+∂=tx()UV( ) ( FV( )) GV( ) (4.87) where V :0,×+∞→] [ 3 is the vector of dependent primitive variables, U : 33→ is T defined by U = []YRZ, where the entries are defined in §2.1, equations (4.73) to (4.75), F : 33→ is defined as

F = ⎡⎤uh ⎢⎥ uh22()www uh1 g () h 22 () hh h 2 (4.88) ⎢⎥ρ+ρ+ρ+ρ+ρcc c s s2 () s sc cc ⎢⎥ ⎣⎦⎢⎥Cuh and G : 33→ is the vector of source terms, defined as

343 GV()= ⎡ 0 ⎤ ⎢−⋅∂fuh,; Y⎥ ⎢ bxb()()⎥ (4.89) ⎣⎢ 0 ⎦⎥

()w where fbccs()uh,;⋅= g() ρ h +ρ h .

Variable t is meant to stand for time, thus physically featuring non-negativity and irreversibility. It does not necessarily follows that t is mathematically a time-like variable. For that purpose, it is necessary and sufficient that there is a smooth transformation φ : VU and a linear operator, δ , such that δ=δ(UV) A ( ) . The latter condition is true iff A ∈ 33× is non-singular (details at Hirsch 1988, p. 148, although the author requires that the matrix is positive definite). It follows that x = constant is a time-like surface and that U and F are of different nature. To highlight this claim, it is noted that, if G = 0, and if the solution V vanishes outside a closed bounded interval of , one has −∞ UM(,)dξ t ξ= (4.90) ∫−∞ where M is independent of t, while, on the contrary, it is generally not true that +∞ F(,)dx ξξis independent of x. Physically, (4.90) means that quantities U are conserved ∫0 in time and, hence, called the vector of conservative variables, and A is the Jacobian matrix of the transformation between primitive and conservative variables. Obviously, equation (4.90) is verified if V :0,×+∞→] [ 3 is zero outside some arbitrarily large interval Ω⊂ ×]0, +∞[ .

In order to show that A is indeed non-singular, let its coefficients be computed accordingly to the definition A =∂V (U) . One obtains

A = ⎡ 10 1⎤ ⎢ ⎥ (4.91) ⎢KK120 ⎥ ⎣⎢NN121− p⎦⎥ where

Ku1 =ρmhm+∂ hu(ρ )

Kh2 = ρ+mum hu ∂ρ( )

Nh1 =∂ch() C c +∂ h sh( C s) + C ch ∂( h c) + C sh ∂( h s)

NhC2 =∂cu() c +∂ hC su( s) + C cu ∂( hC c) + su ∂( h s)

Thus, it is clear that A is singular only for the trivial case h = u = 0.

344 The concept of hyperbolicity, in the context of the classification of PDEs, is associated to the − + shape of the domains of dependence, ΩP , and of influence, ΩP , of a given point P0,∈× ] +∞[ . In a hyperbolic problem, the region of influence of P is delimited by lines that are neither space-like nor time-like. It should be recalled that given any time-like line, Τ, − + there is at least one point P such that T ⊃ ΩΩPP∪ . Hence, information travels from the boundaries (time-like or space-like) is with finite velocity. Also, any problem posed in a bounded domain for a hyperbolic system is amenable to an initial value (or Cauchy) problem.

A system of conservation laws ∂+∂=tx(UF0) ( ) is hyperbolic iff the system of PDEs

AJ∂+∂=tx()VV0 () is hyperbolic, where A =∂V (U) is non-singular and J =∂V ()F . As for the non-homogeneous system, it is necessary to make explicit some of its properties. While the grouping of the variables in equations (4.73) and (4.75) has no special physical meaning, equation (4.74) bears visible traces of the control volume analysis from which it was derived. The conserved variable is R, the unit mass discharge, i.e., the depth-averaged momentum per unit width and unit channel length. The flux of momentum is ρ⋅uSund ∫∫S 2 from which ρmuh is obtained after time and space integration. The material derivative of

22()w the momentum is thus represented by ∂+∂ρ+ρtxcccss()R ( uh uh) and, from Newton’s first law, it is equal to the sum of external forces. These are divided into surface and mass forces. The former are only represented by the pressure-related terms, 1 gh∂ρ()ww22 +2 ρ () hhh +ρ , since the dissipative forces per unit flow area, τ , were 2 xs( () sccc) bc discarded. The latter is the force of gravity per unit horizontal area ()w −ρgh()cc +ρ h s ∂ x() Y b . It is assumed that ∂≈x(Yggbyx) throughout the flow depth.

From the above discussion, it is clear that the vector F, designated flux vector, incorporates the flux of the conservative variables and all the terms related to the external forces that can be written as a gradient. Similarly, the source term incorporates all the forces that cannot be written as a simple derivative. Therefore, there is no physical argument underlying the grouping represented by (4.87), but merely a formal one. On the contrary, there is an important argument to keep fbx∂ ()Y b in the system: the fact that it represents a strong coupling mechanism between geomorphic and hydrodynamic phenomena. It will now be seen that it is possible to keep it in the search for the solution.

Because of the peculiar form of the source vector, namely its dependence on the gradient of a primitive variable, Yb, system (4.87), can be written in a standard quasi-linear, autonomous, non-conservative form

AB∂tx(VV0) +∂( ) = (4.92) where * BA=∂V()F + (4.93)

345 * * * and A is such that A23 = fb and Aij = 0 for all other values of i and j. The coefficients of the matrices B and A are shown in Annex 4.2. In order to keep the equations comparable to those of previous works (eg. de Vries 1965, Lyn 1987, Sloff 1993, Morris & Williams 1996), the second line of A is pivoted to eliminate the coefficient of ∂t(h) .

Thus, matrix B is not a Jacobian matrix. Furthermore, it follows from (4.93) and (4.91) that it * is not possible to write fbx∂ ()Y b in conservative form. If that was possible, A would be a Jacobian matrix of the transformation between and . But this is impossible δ(U) δ(V) because, in what concerns (4.92), the relation between and is already specified δ(U) δ(V) by A. The fact that (4.92) has no true conservative form is of little importance for the purpose of this section. In fact, it can be shown that the system (4.92) has the essential property of a conservative system; equation (4.90) is valid.

Proposition 3.1

Let (,):hu ×+∞→] 0, [ 2 be piecewise continuous with compact support,

fb(h(x,t),u(x,t)) smooth and of bounded variation and Yb(x,t) Lipschitz continuous a.e. with compact support. If Yb is independent of (u,h), then +∞ * fbb∂ξ=ξ()YMd 2 ∫−∞ * where M 2 is independent of t.

The proof of the above proposition is direct. The space integration of fbx∂ ()Y b will be first carried out and integration by parts applied

+∞ m

fYbb∂ξ=ξξ()dlim fY bb ∂ξ= () d m→+∞ ∫∫−∞ −m ⎧⎫m ⎪⎪ lim⎨⎬Ymtfhmtumtbb()( , (,),(,) )−− Y b ( mtfh , )( b ( − mtu ,),( − mt ,) ) − Y bb ∂ξ() f d ξ m→+∞ ⎩⎭⎪⎪∫−m

Since Yb is of compact support, it can be assumed that it vanishes outside arbitrarily large [ab, ]∈ and

+∞ b mb

fYbb∂ξ=∂ξ=ξξ()dd fY bb () − limYfbb∂ξ=−∂ξξξ() d Yf bb () d m→+∞ ∫∫−∞ a ∫−ma∫ From Riez representation theorem (Giles 2000, p. 107), for which the premises of this proposition are necessary, the latter integral exists. Because the primitive variables are independent among themselves, or, at least, Yb is independent of (u,h), fbb≠ Y and

∂≠∂x()fbxb ()Y , wherever the derivatives exist. Under these conditions,

346 bb

fYbb∂ξ=−∂ξξξ()dd Y bb () f ∫∫aa only if the integrand does not depend on h, u and Yb. It is easy to see that this is equivalent to time invariance, since the integrand varies with t only through h, u and Yb. It follows from Proposition 3.1 that (4.90) is true for system (4.92) provided that the premises are true. The verification that fb is of bounded variation is elemental; this function can be expressed as a difference of two monotone functions (see §3.1, equation (4.89)). The

Lipschitz continuity of Yb will be verified by computing the solution. It can be advanced that 1 Yb is piecewise C in its domain and that the derivative is indeed bounded. As a final remark, it is intuitive that G affects the shape of the domains of dependence and influence of a given point in the domain of V, as G depends on the gradient of a primitive variable. Thus, the computation of the characteristics of the system should include fb . Thus, it is utterly desirable to keep fbx∂ ()Y b in the system of conservation laws. The shortcomings of this approach will be explained later.

4.2.4 Eigenstructure and monotonicity

The nature of (4.92) is investigated through the characteristic polynomial BA−λ =0 , where the operator ⋅ stands for the determinant operator. The characteristic polynomial can be written as 32 λ+aaa123 λ+ λ+ =0 (4.94) where the coefficients a1, a2 and a3 depend on u and h and on its derivatives and are required to be C1 () . The expressions for these coefficients are shown in Annex 4.2. It is now clear why it was necessary to introduce the approximations referred to in §4.3.2.2, p. 343: since a1, a2 and a3 depend on the derivatives of the closures (4.79), (4.80) and (4.81) the latter must be C1 () with respect to h and u so that the former are C1 () .

The characteristic polynomial (4.94) possesses three real and different roots, such that λ(1)>λ(2)>λ(3), called the characteristics of the system. Consequently, there are three independent eigenvalues associated to each of the λ(k) eigenvalues which form the base of eigenspace whose dimension is equal to the rank of BA− λ . It is thus concluded that the system is strictly hyperbolic (cf. Witham 1974, p. 116). The characteristics are functions of u and h and can be plotted as a function of the Froude number. Figure 4.20(a) shows the characteristics corresponding to system (4.76) to (4.78). It is shown that the influence of the sediment inertia does not change the structure of the characteristic fields, namely that λ(1)>λ(2)>0 and that λ(3) is always negative. Figure 4.20(a), relative to the system (4.73), (4.74) and (4.75), shows that the characteristic fields are also λ(1)>λ(2)>0 and that λ(3)<0. The magnitude of the characteristics depends on the mobility of the sediment. Increasing the mobility of the sediment by changing the internal friction angle and the density of the sediment, it is observed that all the characteristic fields most affected are affected those corresponding to λ(2) and λ(3). However, the structure of the fields and the order of magnitude remains the same.

347 3.0 3.0

2.0 2.0 (-) (-) 0.5 0.5 ) ) 0 1.0 0 1.0 gh gh /( /(

0.0 0.0

-1.0 -1.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Fr (−) Fr (−)

FIGURE 4.20. Eigenvalues of (4.92) as a function of the Froude number. a) Influence of the sediment density; red line ( ) stands for the characteristics of system (4.76) to (4.78) with (s − 1) = 0 (A = 0) and blue line ( ) stands for the characteristics of the same system with s = 2.65. b) Influence of sediment mobility; blue line ( ) stands

for tan(ϕb) = 0.7, s = 2.65, red line ( ) and stands for tan(ϕb) = 0.45, s = 1.65.

The remaining analysis is carried out with the characteristics of system (4.73), (4.74) and (4.75). The expressions for each of the λ(k) are too long to be meaningful. It is preferable to plot the values of λ(k) within the physical domain of V. The results are shown in figures 4.21, 4.22, 4.23 and 4.24. The physical domain of V (the space of the dependent variables) can be reduced to a square (k) of ++× , since the entries of B and A, and thus of λ , are functions of h and u, and it is assumed that the solution will feature non-negative velocities and water depths. The upper bounds of the square are determined from the initial conditions. The maximum water depth is the initial water depth in the reservoir h0 = YL – YbL (figure 4.1). It is expected that the maximum flow velocity for a geomorphic dam-break flow does not exceed the velocity of the wave-front obtained under the ideal conditions of the Ritter solution, namely, clear water and fixed smooth bed (Stoker 1958, p. 340). It can be shown that this is the case for the horizontal bed problem, but it is not necessarily under the more general initial conditions shown in figure 4.1, namely YbL > YbR. Lacking a better estimate, it will thus be admitted that the upper bound for the velocity is ugh00= 2 .

Thus, the space of the dependent variables is the bounded interval

Δ=hu {()hu,:002 ∈++ × <≤∧<≤ h h00 u gh } .

Any solution will be represented by a line in Δhu . The remaining dependent variable, Yb, is completely determined in that line because, as it will be seen, the solution is piecewise monotone and self-similar. Thus, the transformations ξ h()ξ and ξ u()ξ , ξ=x / t are −1 injective and it is always possible to find transformations ξ =ξuu() Yb() or −1 ξ=hh() Yb() ξ .

348 u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ gh0 26

è!!!!!!! 1.675 4 1.334 2 l 1 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 13 0 gh0 H L 0.2 0.672 62-2 h0.4 è!!!!!!!!! ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅh 0.6 41.33 0.331 h h0 0.8 20.67u u 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ hh gh 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅ 1 gh0 0 h 0.2 0.4 0.6 0.8 1 0h0 è!!!!! a) è!!!!!!!!! u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!! gh0 26

è!!!!!!! 1.675 4 1.334 2 l 2 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 13 0 gh0 H L 0.2 0.672 62-2 h0.4 è!!!!!!!!! ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅh 0.6 41.33 0.331 h0 0.8 20.67u u h0 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ h ÅÅÅÅÅÅÅÅÅÅÅÅÅh 1 gh0 gh0 0 0.2 0.4 0.6 0.8 1 h0h0 è!!!!! b) è!!!!!!!!! u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!! gh0 26

è!!!!!!! 1.675 4 1.334 2 l 3 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 13 0 gh0 H L 0.2 -2 0.672 62 h0.4 è!!!!!!!!! ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅh 0.6 41.33 0.331 h0 0.8 20.67u u h0 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ hh 1 gh gh 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅ 0 0 h0 0.2 0.4 0.6 0.8 1 h0 è!!!!! c) ()k FIGURE 4.21. Surface and è!!!!!!!!!density plots of λ gh . a) k = 1; b) k = 2; c) k = 3. 0 è!!!!! Computations performed with Cf0 = 0.1, tan(ϕb) = 0.5, s = 2.65 and ds = 0.003 m. Scale: 4 3 2 1 0 1 2 - -

349

Scale 4

3

4 2

2 l 1 1 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 0 gh0 0 H L 0 -2 0.25 62 h è!!!!!!!!! ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅh0.5 41.33 -1 h0 0.75 20.67u u h0 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 1 0 gh0 gh0 è!!!!! -2 ()k FIGURE 4.22. Surface plot of λ gh0 . è!!!!!!!!! It is shown that the system is always stricktly (1) (2) (3) hyperbolic since λ >λ >λ in all of Δhu . Computations performed with Cf0 = 0.1,

tan(ϕb) = 0.5, s = 2.65 and ds = 0.003 m.

Figures 4.21, surface and density plots, and 4.22, a comparative surface plot, show the behaviour of each λ(k)(u,h) regarding signal and monotonicity. In particular, it is shown in figure 4.22 that the characteristics are strictly λ(1)>λ(2)>λ(3), throughout the domain. Figures 4.23, 4.24 and 4.25 are solely concerned with monotonicity, as they show the gradients of λ(k) regarding h and u.

The first characteristic field is always positive in Δhu and it is simultaneously monotone in both u and h (figures 4.21(a) and 4.23) except in a narrow strip near h = 0. This is easily seen in figures 4.23(a) and 4.23(b). The u-derivative is greater than zero in Δhu but, near h = 0, (1) ∂λu () appear to be smaller than zero. Varying Cf0, and thus the intensity of the sediment transport, it is observed that this narrow region does not widen. Except for the strip near h = 0, whose importance is neglected because it does not vary with the sediment transport parameters, the first characteristic field is essentially monotone, according to the following definition.

Definition 3.1: F : 2 → is essentially monotone iff, for every 2 AhuBhu=∈(,AA ),(,) BB , there is at least one bijection φ→:0,] uh00] ] 0, ] such that AB−>⇒0 FB( ) ≥ FA( )

or AB−>⇒0 FB( ) ≤ FA( )

for a given norm, and uhA =φ()A and uhB =φ( B ) .

350 u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ gh0 26

è!!!!!!! 1.675 2 1 1.334

0 h0 1 1 ÅÅÅÅÅÅÅ ∑h l 3 -1 g 0.20.2 0.67 62-2 H L 2 h0.4 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅh 0.6 1.334 $%%%%%%%% H L 0.33 hh00 0.8 u 1 20.67ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu 1 gh gh hh 0 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅ 0 h0 è!!!!! 0.2 0.4 0.6 0.8 1 h0 a) è!!!!!!!!! u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!! gh0 26

è!!!!!!! 1.675 2 1 1.334

0 1 ∑u l 13 -1 H L 0.2 -2 0.672 26 H L hh0.4 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 0.6 41.33 0.331 h 0h0 0.8 20.67u u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ hh 1 gh 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅ gh0 0 h0 0.2 0.4 0.6 0.8 1 h0 è!!!!! b)

è!!!!!!!!! (1) h0 (1) (1) FIGURE 4.23. Surface and density plots of: (a) ∂λ and (b) ∂λ .è !λ!!!! is a h ()g u () essentially monotone characteristic field except in a narrow strip near h = 0. Computations performed with Cf0 = 0.1, tan(ϕb) = 0.5, s = 2.65 and ds = 0.003 m. Scale: 2 1 0 1 2 - -

From the above definition it is clear that a function defined in a sub-interval of 2 is essentially monotone if it does not possess any crest-like or valley-like lines, i.e., lines of zero gradient in some direction. Without loss of generality, the norm may be the Euclidean b2– norm. This is a necessary condition for non-linearity. Thus, it can be anticipated that the λ(1)-field is genuinely non-linear. Genuinely non-linear fields assume as solutions only shocks or rarefaction waves, contact discontinuities are precluded. Hence, it can also be anticipated that, as in the clear water shallow water equations, the solution for λ(1)-field is expected to be a shock.

351 u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ gh0 26

è!!!!!!! 1.675 2 1 1.334

0 h0 3 ÅÅÅÅÅÅÅ ∑h l 13 -1 g 0.20.2 62-2 H L0.672 h0.4 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅh 0.6 41.33 $%%%%%%%% H L 0.33 hh00 0.8 0.67u u 1 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 1 h gh0 gh0 h 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅ h0 è!!!!! 0.2 0.4 0.6 0.8 1 h0 a) è!!!!!!!!! u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!! gh0 26

è!!!!!!! 1.675 2 1 1.334

0 3 ∑u l 13 -1 H L 0.2 -2 0.672 0.4 26 H L h ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 0.6 1.334 0.331 hh 00 0.8 20.67u u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ hh 1 gh0 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅ gh0 h0 0.2 0.4 0.6 0.8 1 h0 è!!!!! b) è!!!!!!!!! (2) h0 (2) è!!!!! (2) FIGURE 4.24. Surface and density plots of: (a) ∂λ and (b) ∂λ . λ is a h ()g u ()

essentially monotone characteristic field in Δhu . Computations performed with Cf0 =

0.1, tan(ϕb) = 0.5, s = 2.65 and ds = 0.003 m. Scale: 2 1 0 1 2

- - The second characteristic field is such that the values of λ(2) are always positive and smaller (1) (2) (2) than the values of λ inΔhu (figure 4.22). The derivatives of λ are such that ∂λu () >0 (2) and ∂λh () <0 , as seen in figure 4.21(b) and, explicitly, in figure 4.24. Therefore, it is clear that λ(2)-field is also essentially monotone. It is anticipated that there is a lack of symmetry between the wave structure of the solution of the Riemann problem for the one- dimensional Euler equations and the geomorphic shallow-water equations. It is expected that the λ(2)-field is genuinely non-linear while the middle characteristic field for the Euler equations is linearly degenerate (cf., eg, LeVeque 1990, pp. 89-93)

352 u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ gh0 26

è!!!!!!! 1.675 2 1 1.334

0 h0 3 1 ÅÅÅÅÅÅÅ ∑h l 3 -1 g 0.20.2 0.67 62-2 H L 2 hh0.4 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 0.6 41.33 $%%%%%%%% H L 0.33 hh00 0.8 0.67u u 1 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 1 gh gh hh 0 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅ 0 h0 è!!!!! 0.2 0.4 0.6 0.8 1 h0 a) è!!!!!!!!! u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!! gh0 26

è!!!!!!! 1.675 2 1 1.334

0 3 ∑u l 13 -1 H L 0.2 -2 0.672 62 H L h0.4 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅh 0.6 41.33 0.331 hh00 0.8 2 u u 0.67ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ hh 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅ 1 gh0gh0 h0 0.2 0.4 0.6 0.8 1 h0 è!!!!! b) è!!!!!!!!! (3) h0 (3) è!!!!! FIGURE 4.25. Surface and density plots of: (a) ∂λ and (b) ∂λ , thick line h ()g u () (3) (3) stands for ∂λu () =0 , hence λ is not a strictly monotone characteristic field in

Δhu (see figure 10). Computations performed with Cf0 = 0.1, tan(ϕb) = 0.5, s = 2.65 and

ds = 0.003 m. Scale: 2 1 0 1 2 - -

(3) In the Δhu domain, the values of λ are always negative, as seen in figures 4.21(c) and 4.22. (3) There is strict monotony with respect to h, i.e., ∂h (λ<) 0 as easily seen in figures 4.21(c) and 4.25(a), but not with respect to u. This is an important feature that can be observed neither in figure 4.21(c) nor in figure 4.25(b). Detailed contour plot of figure 4.21(c) is revealed in figure 4.26. Figure 4.26 was computed with a much lighter sediment, s = 1.5, and with unreasonably high values of the friction coefficient. In this case it used Cf0 = 0.3 in order to magnify the non- monotonicity regarding u. Observing this figure one detects a crest line which separates a

353 (3) (3) zone where ∂h(λ ) < 0 and ∂u(λ ) > 0 (lower part of the plot) from a zone where the field is (3) (3) concave relatively to u, i.e., ∂h(λ )h < 0 and ∂u(λ ) < 0.

u 0.61

0.8

0.6

0.3

0.4

0.2

0.00 0.0 0.01 0.02 0.0 0.03 0.04 0.05 0.1 h (3) FIGURE 4.26. Detailed contour plot of λ gh0 . It is shown that characteristic field is not (3) essentially monotone in all of Δhu . Thick line ( ) represents ∂u(λ ) = 0. Dotted

line ( ) represents Fr = 0.7. Computations performed with Cf0 = 0.3, tan(ϕb) = 0.5, s = 1.5 and ds = 0.003 m.

Unlike the non-essentially monotone region in the λ(1)-field, the magnitude of the concave region beyond the crest line revealed in figure 4.26 grows visibly with the intensity of sediment transport. This is the reason why this apparently insignificant feature deserves this amount of attention. Concave characteristic fields are not common in water flow problems and arise, for instance, in oil-recovery problems (Isaacson 1986, Rhee et al. 1989, p. 63-72). Formally, the existence of the crest line may signify that the field is not genuinely non-linear in Δhu . In this case, both a shock and a rarefaction wave can, simultaneously, constitute the Riemann solution for the λ(3)– field, as in the Buckley-Leverett flux equation. This is an unlikely, although theoretically possible solution for a dam-break flow.

Being the only negative field, the third characteristic field must be responsible for the connection between the undisturbed left state (figure 4.1) and a constant state in the vicinity of the dam. A solution involving both a rarefaction wave and an upstream propagating shock, that leaves a smaller water depth on its downstream side, would have no correspondence to any observation done on dam-break flows with or without movable bed. It is in this sense that this solution is unlikely.

In order to evaluate the degree of dependence of each of the characteristic fields with the magnitude of the sediment transport, it is important to note that, as C → 0, the eigenvalues of matrix B become the solutions of λ(λ − u + gh)(λ − u − gh ) = 0 . This is easily shown by taking the limit C→0 of the expressions of a1, a2 and a3 shown in the Annex 4.2. The characteristics of the resulting shallow-water system are the well known λ(1) ≡ λ+ = u + gh , λ(2) = 0 and λ(3) ≡ λ− = u − gh . The relative magnitude of the characteristics of the geomorphic shallow water equations are seen in figure 4.27. It is clear from this figure that the first characteristic field is almost unaltered by the sediment phase. On the contrary, the geomorphic variables deeply affect (3) and (2) , especially for Froude λ λ numbers between 0.7 and 1.5. The concave zone in figure 4.26 is likely to be sensitive to the

354 parameters that control the transport of sediment because it occurs near the lines that represent that range of Froude numbers, as seen in figure 4.27.

4.2.5 Non-linearity

The study of the non-linearity of the characteristic fields requires the computation of the right eigenvectors of matrix BA−λ . These are obtained from (BA− λ=)r0 (4.95)

Simple algebraic manipulations lead to the following expressions, as function of λ(k)

()k ()kk () (4.96) rphNN1 =−()(1 ) −52 +λλ

()k ()kk () (4.97) rpuNpN2 =−()(1 − ) +41 +() (1 − ) − λ λ

rNhNuNNhNu()k =−+−+ −λλ()kk () (4.98) 3 45( 512( )) where the coefficients are those listed in Annex 4.2. The compatibility with the clear water case must be ensured. It is easy to show that if the sediment concentrations and their derivatives are zero, equations (4.96) to (4.98) become (1) , (1) , (1) , r1 = 1 rghh2 = r3 = 0 (2) (2) (2) (3) (3) (3) r1 = 0 , r2 = 0 , r3 = 0 , r1 = 1, rghh2 =− and r3 = 0 . It is clear that (1) (1) (1) and (3) (3) (3) are the eigenvectors of the 2 2 Jacobian matrix of the r = ()rr12, r = ()rr12, x shallow water equations. The second eigenvector simply states that the second characteristic field degenerates into a linear field for which the solutions are stationary waves of zero amplitude.

The non-linearity of the characteristic fields should be verified, at least for the second and third characteristic fields, since the first field appears to be similar to the one of the shallow- water equations for fixed bed. A characteristic field is genuinely non-linear if

∂λ()kkir () ≠0 ∀ (4.99) VV() ∈Δ∪Δhu Y b where Δ is the domain for Yb, as generated by any of the remaining variables. The left-hand Yb side of (4.99) is the directional derivative of the characteristic in the direction of r. dV It should be recalled that r()k = , where W ()k is the characteristic variable dW ()k ()k ()kk () () k corresponding to the λ – field, such that ∂tt(WW) +λ ∂( ) =0 . Hence, condition (4.99) ensures that across a k-wave, the path of the solution will not be overtaken by a line of local maxima or minima of λ()k .

355 u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ gh0 26

è!!!!!!! 1.675 5 1.334 λ+ 0 l 1 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅλ(1) 31 gh0 -5 H L 0.2 0.672 62 h0.4 è!!!!!!!!! ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅh 0.6 41.33 0.331 hh0 0.8 20.67u u 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ hh 1 gh 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅ gh0 0 h 0.2 0.4 0.6 0.8 1 0h0 è!!!!! a) è!!!!!!!!! u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!! gh0 62

è!!!!!!! 1.675 4 1.334 2 λ− l 2 λÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ(2) 31 0 gh0 H L 0.67 0.2 -2 2 62 h0.4 è!!!!!!!!! ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅh 0.6 41.33 0.331 h h00 0.8 20.67u u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ h ÅÅÅÅÅÅÅÅÅÅÅÅÅh 1 gh0 gh0 0 0.2 0.4 0.6 0.8 1 h0h0 è!!!!! b) è!!!!!!!!! u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!! gh0 26

è!!!!!!! 1.675 4 1.334 2 − λl 3 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ(3) 31 0 λ gh0 H L 0.2 0.672 26-2 h0.4 è!!!!!!!!! ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅh 0.6 41.33 0.331 hh0 0.8 20.67u u 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ hh gh 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅ 1 gh0 0 h0 0.2 0.4 0.6 0.8 1 h0 è!!!!! c) + (1) − (2) − (3) FIGURE 4.27. Surface and densityè!!!!!!!!! plots of: (a) λ λ , (b) λ λ and (c) λλ . è!!!!! Computations performed with Cf0 = 0.1, tan(ϕb) = 0.5, s = 2.65 and ds = 0.003 m.

Scale: 5 0 5 10 10 -

-

356 A pulse in a simple wave corresponding to a genuinely non-linear characteristic field necessarily deforms. If the path of the solution is such that the characteristic is monotone increasing, the pulse is self-sharpening and gives raise to a shock, even if the initial conditions are smooth. If the characteristic is monotone decreasing, the simple wave is a rarefaction wave. Thus, the only admissible waves in a non-linear characteristic field are rarefaction waves or shocks; contact discontinuities are not admissible (cf. Rhee et al. 1989, pp. 59-60).

It is well known that inequality (4.99) is verified for the fixed-bed shallow-water equations. In order to verify that the same happens with the geomorphic shallow-water equations, a plot of ∂λ()kkir ()is shown in figure 4.28. V ()

It is observed that condition (4.99) holds for both the first and the second characteristic fields, as seen in figures 4.28(a) and (b). In fact, except for a limited region near h = 0 outside (1) (1) (1) (1) figure 4.28(a), ∂λV ()ir >0 and ∂V (λ<)ir 0 .

As for the third characteristic field, it can be shown that essential monotonicity is a necessary condition for non-linearity. Thus, the crest line seen in figure 4.25(b) and 4.26 indicates that this field might have points where (4.99) does not hold. It is clear from figure 4.28(c) that (3) (3) there is indeed a line in Δhu where ∂λV ( )ir =0 .

(3) Furthermore, a detailed analysis of λ reveals that it changes its sign only once in Δhu . Rhee et al. (1989), pp. 57-59, show that the λ(3) -characteristic field admits, as solutions, rarefaction waves combined with shocks and name these solutions as semi-shocks. It is easy to understand how a semi-shock can occur in the present λ(3) − field. Given that λ(3) is always negative, let it be imagined that the 3-wave connects a left constant state, with high h (3) (3) and low u, and a right constant state with low h and high u. If the line ∂λV ()ir =0 stands between the left and the right states, the 3-wave would span until that line is reached and then fold back until the values of u and h on the right side are reached. This is described in figure 4.29. Such a solution is not a good description of any observed laboratorial dam- break flow (cf. Chapter 5, §5.2.5).

If ∂λ()kkir () is taken as a measure of the degree of the non-linearity of the field, it can V () be concluded from figure 4.28 that the strength of the 1-waves is expected to be larger than that of the 2-and 3- waves. For instance, if a shock develops as a solution of the second characteristic field, its strength is expected to be lower than the strength of a shock in the first characteristic field.

357 u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ gh0 62

è!!!!!!! 1.675 4 1.334 2 1 0 h ∑ 1 l ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ0 r ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 31 -2 H ghL 0 H L 0.20.2 -4 0.672 62 H L h0.4 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 0.6 41.33 è!!!!!!!!! 0.331 hh00 0.8 u 0.672ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu 1 gh0 gh h 0 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅh è!!!!! 0.2 0.4 0.6 0.8 1 h0 h0 a) è!!!!!!!!! u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!! gh0 26

è!!!!!!! 1.675 4 1.334 2 2 0 h ∑ 2 l ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ0 r ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 31 -2 H ghL 0 H L 0.20.2 -4 0.672 62 H L h0.4 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 0.6 41.33 è!!!!!!!!! 0.331 hh00 0.8 u 20.67ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu 1 gh0 hh gh0 ÅÅÅÅÅÅÅÅÅÅÅÅÅ 0 h è!!!!! 0.2 0.4 0.6 0.8 1 0h0 b) è!!!!!!!!! u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!! gh0 26

è!!!!!!! 1.675 4 1.334 2 3 0 h ∑ 3 l ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ0 r ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 31 -2 H ghL 0 H L 0.20.2 -4 0.672 62 H L h0.4 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 0.6 41.33 è!!!!!!!!! 0.331 hh00 0.8 u 20.67ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu 1 gh gh h 0 0 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅh è!!!!! 0.2 0.4 0.6 0.8 1 h h0 0 c) è!!!!!!!!! ()kkk () () FIGURE 4.28. Surface and density plots of Ε=∂λhghir : (a) k = 1, (b) k = 2, 00V () è!!!!! (c) k = 3, thick line in the density plot of stands for Ε(3) = 0 . Computations performed with Cf0 = 0.1, tan(ϕb) = 0.5, s = 2.65 and ds = 0.003 m. Scale: 4 2 0 2 4

- -

358

maximum (a) λ(3) t shock

x (b) solution of h solution of u

hL uR

h R uL

x/t

FIGURE 4.29. Formation of a semi-shock as a solution for the third characteristic field. (a) behaviour of the characteristic lines in the x-t domain. (b) qualitative behaviour of the (3) (3) solution. Left and right states are on opposite sides of the line ∂V ()λ=ir 0 .

As final conclusion, it can surely be stated that none of the characteristic fields will develop contact discontinuities, hence breaking the symmetry between the Euler equations and the geomorphic shallow-water equations.

4.3 DESCRIPTION OF WAVES THAT CHARACTERIZE TO THE POSSIBLE RIEMANN SOLUTIONS

4.3.1 Existence of solutions

A Riemann problem is the Cauchy problem

∂+∂=tx(UFU0) ( ( )) (4.100)

⎧UL0, x ≤ x UU0 ≡=(,0)x ⎨ (4.101) ⎩UR0, x > x where (4.100) is hyperbolic. Lax (1957) showed that the solution of the Riemann problem exists, for a generic system of n equations, provided that: i) system (4.100) is strictly

359 hyperbolic, ii) F ∈ C1 (n ) (is Lipschitz continuous, see, e.g. Ambrosio et al. (2000), p. 12), iii) all the characteristic fields are genuinely non-linear and iv) UULR− is small. In that case, the solution is composed of n+1 constant states separated by shocks or centred simple waves (or rarefaction waves). All waves are, thus, centred at the origin, as seen in figure 4.30. Following Lax’s (1957) work, a large body of research on the existence of solutions for (4.100) was built over the time. Less restrictive conditions were studied by Glimm (1965), Smoller (1969) or Liu (1974), but most research was performed for systems of two equations.

t wave associated to λ(2)

wave associated to λ(3) constant state (2) constant state (1)

wave associated to λ(1)

undisturbed L-state undisturbed R-state

x FIGURE 4.30. General wave structure of the Riemann solution for the geomorphic dam-break problem.

System (4.92) cannot, as seen in §4.2.3, be written in the form of (4.100). Also, it cannot be ensured that dam-break problems feature small UULR− . Thus, it makes sense to discuss the existence of solutions of (4.92), subjected to (4.101). For that purpose, let (4.92) be written as

* (4.102) ∂+∂+∂=tx()UV() ( FV()) A x() V 0 Dafermos (1973), working on a 2x2 system, proved the existence of solutions of (4.100) and 2 (4.101) by taking the limit ε→0 of ∂+∂=ε∂tx()UF ()t x () U. This viscosity method can be used to prove the existence of solutions of (4.102) and (4.101), where the initial data, U0, is that of figure 4.1. First, it is noted that (4.102) is invariant under the transformation x,,, t ax at ∀ , which is the same as asserting that the Riemann problem admits () ( )a∈ + \0 self-similar solutions. Therefore, if ω = x t , the system

* 2 ∂+∂+∂=ε∂tx()UV()(,)x txtxttxt() FV() (,)A xx()() V (,) V (,) is equivalent to

* 2 −ω∂ωω()UV()() ω + ∂( FV( () ω)) +A ∂ ωω() V() ω = ε ∂( V () ω ) (4.103) subjected to the initial data

UU()−∞ = L , and UU()+∞= R (4.104)

In order to prove the existence of solutions for the geomorphic shallow water equations, the following theorems will be invoked.

360

Theorem 4.1 (Dafermos 1973) Assume that there is M, independent of m, such that every possible solution U()ω of

(4.103) subjected to UU()−=m L and UU()m = R m ≥ 1, satisfies sup UM(ω) < −<ω

Theorem 4.2 (Dafermos 1973)

For a fixed ε>0 , let Uε()ω denote the solution of (4.103), subjected to (4.104). Suppose

that the set {Uε :0<ε< 1} is of uniformly bounded variation. Then, there exists a function U()ω of bounded variation such that U( x/t) is a weak solution of (4.102) subjected to (4.101).

If the conditions of Theorem 4.1 are met, then Uε(ω) exists. It follows that the existence of solutions for (4.102), (4.101) is dependent on the boundedness of the solution. In particular, it + is necessary to prove that the there is a sequence {ε j } , such that ε→j 0 as j →∞, and a function U()ω of bounded variation, such that UUεεj(ω) →ω( ) , a.e. on ]−∞, +∞[ as j →∞. The results of Dafermos are applicable in this study provided that the unicity of the solutions of (4.103) is proved. The following proposition must be proved true.

* Proposition 4.1. Let the ∂V()F and A be continuous in Δuh . For given ε>0 the Cauchy problem for (4.103) has a unique solution for a class of initial conditions that comprises a null-measured set of discontinuities.

* If ∂=V()U I and if A and ∂V()F are Lipschitz continuous, proposition 4.1 would follow from the general theory of ODEs (cf. Braun 1993, p. 129). Unfortunately, A* is not amenable to the identity matrix and, since the derivatives of some of the closure equations are unbounded, a proof must be constructed. For that purpose, let VA and VB be two solutions of (4.103) AB defined in some bounded interval centred in ω0 such that VV()ω=00 () ω and ii AB VV()ω=00 () ω, where the overdot stands for derivative to ω. Defining ii D()()VVω≡AB () ω− V () ω, it is noted that, from the mean value theorem,

FVAB−=∂− FV FV V AB V (4.105) ()( ) V ( ( ))( )

Because the derivative at the mean point is a constant, it is true that

ω AB FV()()−=∂ FVV() FV()D()d() V ξξ ∫ω 0

361 On the other hand

UVAB−=∂− UV UV V AB V () ( ) V ( ( ))( )

∂−∂=∂UVAB UV UVD() V (4.106) ωω( ()) ( ( )) V ( ())

If (4.102) is integrated

ωω ω * 2 −ω∂ωωωω()UV()() ωω+∂d+dd{}() FV()() ωA ∂() V ω=ε∂() V() ωω ∫∫ωω ∫ ω 00 0

FV()()ω− FV( () ω00) =ε∂()ωω( V() ω−∂) ( V() ω)

ω * (4.107) −ω∂ω−∂{}ωω()UV()() A () V d ω ∫ω 0

From (4.105), (4.106) and (4.107)

ω ∂ξξ=ε∂−∂FVD()d V VAB V V()() () ( ωω() ()) ∫ω 0 ω −ω∂⎡⎤UVAB −∂ UV −∂A* ⎡⎤ V AB −∂ V d ω {}⎢⎥⎣⎦ωω( ()) ( ()) ⎣⎦ ωω() () ∫ω 0

ωω ∂ξξ=εω−ω∂−ωFVD()dD()() V ( V ) ⎡⎤ UVA* D() V d V()() ⎣⎦V ()() ∫∫ωω 0 0

ω ⎡⎤∂−ξ∂+FV UVA* D()dD()() V ξξ=εω ( V ) (4.108) ⎣⎦V()() V ()() ∫ω 0 since ∂−ξ∂+FV UV A* is continuous, it follows from (4.108) and from V()( ) V ()( ) Gronwall’s inequality (Hirsch et al. 2004, p. 393), with K = 0, that D()(V0ω=) and, hence, VVAB= , which completes the proof. Proposition 4.1 and Theorem 4.1 allow for the application of Theorem 4.2. Hence, it is proved that there are indeed weak solutions of (4.102) subjected to (4.101).

The unicity of the weak solution must be addressed now. It should be recalled that a function U(,)x t is a weak solution of (4.100) iff there exists a function φ×→: , 1 φ∈C0 () ×[[0, +∞ , i.e., smooth and with compact support, such that

+∞ +∞ +∞

()UFU∂φ+tx()() ∂ () φ ddx tx = U00 φ d (4.109) ∫∫0 −∞ ∫ −∞

362 The derivatives of the potentially discontinuous V(,)x t are transferred to the smooth φ and conservation, thus allowing for the generalised definition of the solution. Hence, a weak solution admits discontinuities in its domain provided the set of points where they occur is of null measure. In other words, a weak solution is smooth almost everywhere.

Given that the characteristic fields are genuinely non-linear, the only discontinuities in the profiles of the dependent variables are shock waves. Across the shock waves, equations (4.100) or (4.102) are not valid because the derivatives do not exist. Only the integral form of the conservation laws is valid. Hence, shock conditions, or Rankine-Hugoniot conditions, must be derived from (4.109). A system of conservation laws written in conservative form admits the shock relations ΔU S = ΔF (4.110) where ΔU = Udownstream – Uupstream , ΔF = Fdownstream – Fupstream and S is the velocity of the shock.

Back to the problem of unicity, it is recalled that, in general, there can be more that one weak solution for a given set of initial conditions. The works of Lax (1957) and Olga Oleinik, 1959 (cited by Magenes 1996) set the fundamental of the theory of the unicity of PDEs. In this paper, the fundamental result is that unicity is ensured if Liu’s (1974) extended entropy condition

()kkk () () λLR>>λS (4.111)

(1)kkk− () (1)+ λ>RLS >λ (4.112) is met. In (4.111) and (4.112) S(k) stands for the shock velocity. Equation (4.111) states that, in an entropy satisfying shock, the characteristic lines corresponding to two adjacent constant states must converge into the shock path. Equation (4.112) ensures that a shock in a given characteristic field is not overtaken by characteristics of a different type.

It was seen before, in §4.2.3, that (4.102) has the properties of a conservation law. Yet, since the derivation of the Rankine-Hugoniot conditions from (4.109) involves the use of Green’s theorem, the conservation laws must be written in divergence form. As seen in §3.1, this is not, in general, possible for system (4.102). Hence, a mathematical trick must be employed: matrix A* must be linearised across the discontinuities, i.e., it should be a combination of the values of V at the left and right states adjacent to the discontinuity. The shortcoming envisaged at the end of §3.1 is clear now: the solution of (4.102) is unique but only for the chosen linearization. Furthermore, not all linearisations are admissible. If the shock strength tends to zero, the chosen combination of left and right states must converge to the expression of A* valid in smooth regions. It was shown that the generic unique Riemann solution of (4.102) subjected to (4.101), has the generic structure shown in figure 4.30. The particular structure corresponding to the expressions embedded in (4.102) will be discussed next.

4.3.2 The structure of the Riemann solution

Given that the λ(k)-fields are such that λ(1) > λ(2) > λ(3), it is expected that the fastest wave, separating the undisturbed right state (R-state) from the constant state (1), would be

363 associated to the λ(1)-field, as seen in figure 4.28. The wave associated to the λ(2)-field ought to be the middle wave, connecting constant states (1) and (2), and should travel downstream, since λ(1) > λ(2) > 0 > λ(3). Associated to the only negative characteristic, λ(3) is an upstream moving wave, separating the undisturbed left state and the constant state (2).

Physically admissible shocks verify the condition (4.111). Rarefaction waves are simple centred waves that connect two constant states through a smooth transition. A rarefaction wave is graphically identified by a fan of characteristic lines whose bounding lines obey

()kk () λLR<λ (4.113) Judging upon the monotonicity of the characteristic fields and considering the inequalities

(4.111) and (4.113), one can determine the zones in Δuh for which the solution is a rarefaction wave or a shock wave.

Let the point (h0,u0), a generic point in an essentially monotone region of Δuh, represent the constant state upstream from a given wave. It should be recalled that all of Δuh is essentially monotone for the first and the second characteristic fields (figures 4.21(a) and (b) and figures 4.23 and 4.24). On the contrary, the third characteristic field presents a crest line (figures (3) (3) (3) 4.25(b) and 4.26). Also, there is a line such that ∂V (λ=−κ=)ir uh() 0. The region (3) (3) CR = {u,h : u > κ (h)} will herein be called the concave sub region of Δuh . From the density plots of figures 4.21, 4.23, 4.24 and 4.25, one can determine the sub- regions of existence of shock and rarefaction waves for each of the characteristic fields as a function of the location of the downstream point (h1,u1). Indeed, any sub-region for which (k) (k) λ (u0,h0) > λ (u1,h1) is a region for which a shock is the admissible solution. It should be noticed that this is a necessary condition and not a sufficient one since it is nowhere proved that, in between those characteristics, there is a shock speed obeying inequality (4.111) and (k) (k) (4.112). Contrarily, the set of all points for which λ (u0,h0) < λ (u1,h1) is a sub-region for which the solution is, in accordance to (4.113), a rarefaction wave.

The results are shown in figures 4.31(a), (b) and (c). The origin of these plots is point (h0 , u0) = (0.20, 1.2), but the results are qualitatively valid for any other point. The thick line in figures 4.29(a), (b) and (c) is the isoline that separates the shock from the rarefaction wave region and it has the following equation

(k) (k) (k) Ψ (u,h) = λ (h,u) – λ (h0,u0) = 0 or, if it can be made explicit for u, u = ψ(k)(h) (4.114) (1) (1) The solution for the λ -field can be a rarefaction wave if (h1,u1) belongs to RW = {u,h : u (1) > ψ (h)}. This region, shown in figure 4.31(a), includes the first quadrant of that plot, (u>u0 ∧ h>h0), the upper part of the second quadrant, (u>u0 ∧ hh0). A shock wave would be expected if (h1,u1) lies in SW = {u,h : u < ψ(1)(h)}. As explained above, the λ(1)-field is such that λ(1) > λ(2) > 0 > λ(3). It is, thus, expected that the constant state downstream of this wave is the undisturbed initial right state or R-state (see figures 4.1 and 4.30), for which the velocity is typically zero (or very small) and the

364 water depth is typically small. The most likely situation in a dam-break flow is, thus, the downstream point (h1,u1) lying on the third quadrant (u

u > u 0 0.5 u > u 0

h < h 0 h > h 0 0.25Rarefaction wave 0.5 ) L h

g 0 )/( 0

u -0.5 -0.25 0 0.25 0.5 - u ( Shock-wave-0.25

u < u 0 u < u 0

h < h 0 h > h 0 -0.5

(h - h 0)/h L a)

u > u 0 0.5 u > u 0

h < h 0 h > h 0 Rarefaction0.25 wave 0.5 ) L h

g 0 )/( 0

u -0.5 -0.25 0 0.25 0.5 - u ( -0.25Shock-wave

u < u 0 u < u 0

h < h 0 h > h 0 -0.5

(h - h 0)/h L b)

u > u 0 0.5 u > u 0

h < h 0 h > h 0 Concave 0.25 region Rarefaction 0.5 ) L h

wave g 0 )/( 0

u -0.5 -0.25 0 0.25 0.5 - u ( -0.25Shock-wave u < u 0 u < u 0

h < h 0 h > h 0 -0.5

(h - h 0)/h L c) (k) FIGURE 4.31. Regions of possibility for shock- or rarefaction waves for each of the λ -fields. a) k = 1; b) k = 2; c) k = 3. Computations performed with Cf0 = 0.3, tan(ϕb) = 0.5, s = 1.5 and ds = 0.003 m.

365 (2) (2) The solution for the λ -field can be a rarefaction wave, if (h1,u1) belongs to RW = {u,h : u (2) (2) (2) > ψ (h)}, or a shock wave, if (h1,u1) belongs to SW = {u,h : u < ψ (h)} (see figure 4.31(b)). These results were determined with the aid of the discussion on the monotony of the field, based on figures 4.21(c) and 4.24. Being the middle wave (figure 4.30), it is hard to infer the type of wave if one possesses only information about the initial state and the form of the closure equations.

This issue will be addressed in detail later, in the certainty that the particular shape of equation (4.114), k=2, is determined by the initial conditions and by parameters of the closure equations. For the purposes of determining the structure of the Riemann solution, one should expect either a shock or a rarefaction wave as a solution for the λ(2)-field. (3) A concave sub-region appears in Δuh for the λ -field (figures 4.21(c), 4.25 and 4.26). Outside this region, the solution can be either a rarefaction wave or a shock wave. In the first (3) (3) case, (h1,u1) would belong to RW = {u,h : u > ψ (h)} whereas, in the second case, (h1,u1) would belong to SW(3) = {u,h : u > ψ(3)(h)}. The depth averaged velocity in a typical reservoir is zero. The water depth measured at the reservoir is necessarily the larger in the system (figure 4.1). The wave solution for the λ(3)- field will, thus, separate the constant L-state (upstream, see figures 4.1 and 4.30) from the constant state (2). Relatively to the former, the latter state is characterised by a larger flow velocity and a smaller water depth, i.e., quadrant 2 in figure 4.31(c). As seen in that figure, and under the hypothesis that (h1,u1) lies in the essentially monotone region of Δuh, the solution for the λ(3)-field will be a rarefaction wave. + If, on the contrary, (h1,u1) belongs to the concave region of uh , the solution comprises an expansion wave and a shock wave. The velocity of shock wave is determined from the considerations explained in §4.3.1 and illustrated in figure 4.29. Further attention will be given to this matter in the next chapter.

After ruling out some of the wave combinations, according to the reasoning explained above, figure 4.30 can be updated. The structures of the possible Riemann solutions for the geomorphic dam-break problem are depicted in figure 4.32. The conditions for which each of the wave-structures occur will be discussed in the next section.

4.3.3 Mathematical treatment of shock and rarefaction waves

Once the structure of the solution is known, there remains the problem of computing the values of u, h and Yb in the constant states and across the rarefaction waves and the values of the shock velocities.

Let u1, h1, Yb1 and u2, h2, Yb2 be the values of the primitive variables in the constant states (1) (1) and (2), respectively. Let S1 be the velocity of the shock-wave corresponding to the λ – (2) field and, if that is the case, let S2 be the velocity of the λ – field. The problem admits 8 unknowns if the structure of the solution is of type A (see figures 4.32a or 4.23). If it is pf type B (figure 4.30b) the problem admits 7 unknowns. If the solution for the λ(3)-field is both a shock and a rarefaction wave, the velocity of that shock constitutes an extra unknown.

366

rarefaction wave t shock associated to λ(2) associated to λ(3)

constant constant state (1) state (2)

shock associated to λ(1)

undisturbed L-state undisturbed R-state

x

rarefaction wave t rarefaction wave (3) associated to λ associated to λ(2) constant state (2) constant state (1)

shock associated to λ(1)

undisturbed L-state undisturbed R-state

x FIGURE 4.32. Possible wave structures of the Riemann solution for the geomorphic dam-break problem. Solution for λ(2)-field: (a) shock wave, type A; (b) rarefaction wave, type B.

In order to determine the solution for a particular Riemann problem, it is sufficient to use the relations that involve the unknown quantities of the constant states upstream and downstream which, according to Lax’s (1957) theorem, bound a given wave. Across a rarefaction wave, the Riemann invariants (Prasad 2001, pp. 84-86) are computed from the differential equations ddVVdV 12==3 (4.115) ()kkk () () rrr123 Equation (4.115) can be cast in a 2x2 system of ODE’s

()k du r = 2 =⋅f ()k ()uh,; (4.116) dh ()k u r1

dY r()k b = 3 =⋅f ()k ()uh,; (4.117) dh ()k z r1 subjected to (ui , hi) and (Yb i , hi), and where −−(1pu ) + N +() (1 − p ) − N λ()k f ()k = 41 (4.118) u ()k (1−−+λph ) N52 N

367 ()kk () Nh45−+−+ Nu( N 512 Nh N( u −λλ)) f ()k = (4.119) z ()kk () ()(1−−+λλph ) N52 N

The coefficients in (4.118) and (4.119) are identified in Annex 4.2. From (4.116) to (4.119) it (k) (k) is clear that u and Yb change across the rarefaction wave whenever f u and f z are different from zero.

As for the shock waves, jump discontinuities in the profiles of the dependent variables, the Rankine-Hugoniot conditions (4.110) apply. It should be reminded that the governing equations could not be cast in divergence form. As stated before, a local, across the discontinuity, linearization is required for the term related to the gravity force in the momentum conservation equation.

A control volume analysis of the discontinuity renders the following integral momentum equation

xR +− + − ∂ξ+−t ()R d SRx()() Rx () −() F22() x − F () x ∫x L

+− ++ −Φ(VV,0)(Yxbb( ) − Yx( )) = (4.120) where Φ→: 2 is a continuous function of the flow depth and velocity at the left and the + – right states. Taking the limit as (x2 – x1)→ 0 and x2 → x ; x1 → x one obtains

+− ++ SRΔ−Δ F2 −Φ(VV,0)(Yxbb( ) − Yx( )) = (4.121) which is the generalised form of (4.110) for the linearization Φ . The function Φ is chosen to ensure the compatibility between the integral and the differential formulae in smooth regions. In smooth regions it is required that

x2 +− 1 limΦ∂ξ=⋅∂VV ,ξ()YfhuYbb d() , ; xb() ()xx−→0 ()xx− 21 21∫x 1

+− If the limit exists then limΦ=⋅(VV ,) fb()hu , ; , which is the only mathematical ()xx21−→0 requirement that Φ must obey. There is a physical constrain to the choice of Φ: the mechanical energy of flow must decrease across a discontinuity. Possible choices for this function are

⎧ f hu++, ;⋅ if xtS / > ⎪ b ( ) Φ=⎨ ⎪ f hu−−, ;⋅ if xtS / < ⎩ b () or the arithmetic mean

Φ=1 fhu++,; ⋅ + fhu −− ,; ⋅ 2 { bb( ) ( )}

The latter formulation is used throughout this work.

368 It is now possible to build the system of equations necessary to find the Riemann solutions under investigation (figure 4.32). The system corresponding to the 8x8 problem depicted in figure 4.32(a), i.e., two shocks and one rarefaction wave, is composed of

YYSuhuh (4.122) ()R11−=( )R1 −( )

221 22 ()RRSR11−=ρ−ρ+ρ−ρmm uh uhg mm h h ()R1( ) 2 (( ) R1( ) ) +ρgh1 +ρ hYY − (4.123) 2 ()()()mmbbR1()R1

YYSCuhCuh (4.124) ()bbR11−=( )R1 −( )

YYS uhuh (4.125) ()122−=−( )12( )

221 22 ()RRS122−=ρ−ρ+ρ−ρmm uh uhg mm h h ()12( ) 2 (( ) 12( ) ) +ρ+ρgh1 hYY − (4.126) 2 ()()()mmbb12()12

YYS CuhCuh (4.127) ()bb122−=−( )12( ) du = fuh(3)(),;⋅ (4.128) dh u 22 dY b = fuh(3)(),;⋅ (4.129) dh z 22 Equations (4.122) to (4.124) are the RH conditions across the shock associated to λ(1) while equations (4.125) to (4.127) represent the jump conditions for the shock associated to λ(2). Equations (4.128) and (4.129) are obtained from the Riemann invariants across the rarefaction (3) (k) (k) wave associated to λ . Since the general form of both f u and f z, equations (4.118) and (4.119), in (4.128) and (4.129) is highly non-linear, those equations are not, in general, amenable to an analytic solution. Thus, the solution of (4.128) and (4.129) implies the numerical computation of these ODEs from the boundary condition u0 = uL, h0 = hL and Yb0 = YbL, h0 = hL to the point h = h2. The system of equations necessary to determine the 7x7 problem of figure 4.32b), i.e., two rarefaction waves and one shock wave, is composed of equations (4.122) to (4.124), (4.128), (4.129) and du = f (2)()uh,;⋅ (4.130) dh u 11 dY b = f (2)()uh,;⋅ (4.131) dh z 11 The conditions across the second rarefaction wave, associated to the λ(2)-field, are expressed in (4.130) and (4.131). These require the boundary conditions u0 = u2, h0 = h2 and Yb0 = Yb2, h0 = h2 to the point h = h1. The proprieties of equations (4.118) and (4.119) will be discussed next. Across the rarefaction waves of figure 4.32, the variation of the primitive variables h, u and Yb is

369 necessarily smooth and monotone. Thus, two questions arise: i) do (4.118) and (4.119) ensure the monotone variation of h, u and Yb in all of Δuh and ii) what are the implications of the singularity achieved by zeroing the denominator of (4.118) and (4.119), whose equation is

()k ()k β=−(1ph ) − N52 +λ N (4.132)

Both questions are answered by looking at figures 4.33, 4.34, 4.35 and 4.36. Figures 4.33 and 4.34 show surface and density plots of the variation of (4.118) and (4.119) for each of the λ(2)- and λ(3)-fields. Figure 4.35(a) show that β(2) is always positive, which means that (4.118) and (4.119) have no discontinuity for k = 2. On the contrary, it is clear from figure 4.35(b) that there is a line for which β(3) = 0. u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ gh0 62

è!!!!!!! 1.675 4 1.334 2 0 h r 2 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ0 2 31 -2 gh0 -4 H L 0.2 0.672 0.4 62 hh è!!!!!!!!! ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 0.6 41.33 0.331 h h00 0.8 20.67u u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ h ÅÅÅÅÅÅÅÅÅÅÅÅÅh 1 gh0 gh0 0 0.2 0.4 0.6 0.8 1 h0h0 è!!!!! a) è!!!!!!!!! u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!! gh0 26 è!!!!!!! 1.675 4 2 1.334 0 2 r3 13 -2 -4 H L0.672 0.2 62 0.4 hh ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 0.6 41.33 0.331 h0h 0 0.8 20.67u u h ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅh 1 ghgh0 0 0.2 0.4 0.6 0.8 1 h0 h0 è!!!!! b) (2) (2) FIGURE 4.33. Surface and density plots of non-dimensional (a) f , (b) f .Computations è!!!!!!!!! u z è!!!!! performed with Cf0 = 0.1, tan(ϕb) = 0.5, s = 2.65 and ds = 0.003 m. Scale: 4 2 0 2 4 - -

(3) (3) Figure 4.34(a) shows that f u is negative in a sub-region of Δuh bounded by the line β = 0. (3) (3) At that line the denominator of (4.118) is zero and f u is infinitely large. Approaching β = 0

370 (3) from the right, f u tends to infinity over negative values whereas approaching it from the (3) left, f u tends to infinity over positive values. u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ gh0 62 è!!!!!!! 1.675 4 4 2 1.33 0 h r 3 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ0 2 31 -2 gh0 -4 H L 0.2 0.672 62 h0.4 è!!!!!!!!! ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅh 0.6 41.33 0.331 h h00 0.8 0.67u u 2 h ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ h 1 gh0 gh0 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅ 0.2 0.4 0.6 0.8 1 h h0 è!!!!! 0 a) è!!!!!!!!! u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!! gh0 62

è!!!!!!! 1.675 4 2 1.334 0 3 r3 31 -2 -4 H L 0.672 0.2 26 0.4 hh ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 0.6 1.334 0.331 hh 00 0.8 0.672 u u h ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅh 1 ghgh0 0 0.2 0.4 0.6 0.8 1 h0h0 è!!!!! b) (3) (3) FIGURE 4.34. Surface and density plots of non-dimensional (a) f , (b) f .Computations è!!!!!!!!! u z è!!!!! performed with Cf0 = 0.1, tan(ϕb) = 0.5, s = 2.65 and ds = 0.003 m. Scale: 4 2 0 2 4

- -

(3) (3) Let D be the sub-region of Δuh on the “right-hand side” of the line defined by β = 0 (3) (figure 4.36). Upstream from the expansion wave associated to λ one has the point h/h0 = H 0.5 (3) = 1, u/(g h0) = U = 0 (see figure 4.32), clearly lying on D (see figure 4.36). The downstream point cannot be found outside D(3) because the solution of (4.128) or (4.129), with boundary conditions defined on D(3), is valid only in that sub-domain. D(3) is thus the domain of occurrence of the wave associated to λ(3). (3) (3) Given that f u is negative in D , the velocity increases as the water depth decreases, the flow accelerates from the undisturbed left state to the second constant state, as expected (cf. (3) (3) Stoker 1958, p. 333). As for f z, its values are positive in D (figure 4.34(b)), which means that, along the λ(3) expansion wave, the bed elevation decreases with decreasing water depth.

371 This result is in agreement to the most elementary physical intuition and also with the results of Fraccarollo & Capart (2002).

1 1 0.5 0.5 0 b 2 0 b 3 ÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ -0.5h0 -0.5h0 H L H L 0.2 -1 0.2 -1 62 62 0.4 0.4 h h ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 0.6 1.334 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 0.6 1.334 h h h00 0.8 2 u u h00 0.8 20.67u u 0.67ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 1 ghg0h0 1 gh0gh0 è!!!!! a) è!!!!! b) (k) FIGURE 4.35. Surface plot è!!!!!!!!!of β . (a) k = 2; (b) k = 3. Computations performedè!!!!!!!!! with Cf0 = 0.1, tan(ϕb) = 0.5, s = 2.65 and ds = 0.003 m.

u ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ gh0 62

è!!!!!!! 1.675 (3) 1.334 β = 0

31

0.672

0.331

h 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅh 0.2 0.4 0.6 0.8 1 h0h0

(3) (3) (3) (3) (3) FIGURE 4.36. Density plot of ∂λ +∂λff +∂λ =0 . The discontinuity huuYz() ( ) b ( è!!!!!) (3) (3) (3) corresponds to β=0 . Thick line ( ) corresponds to ∂λV ()r =0 .

Computations performed with Cf0 = 0.1, tan(ϕb) = 0.5, s = 2.65 and ds = 0.003 m.

It was earlier identified that the solution for the λ(3)-field could be composed of an expansion wave and a shock wave if the downstream point would lie in the concave region of Δuh (see figures 4.25(c), 4.29 and 4.31). It is now clear that such a solution is possible only if D(3) (3) contains the essentially monotone region. Otherwise, the crest line defined by ∂u(λ ) = 0 lies outside D(3) and λ(3) becomes essentially monotone in this domain. Hence, the only admissible solutions are rarefaction waves. Observing figure 4.35 it is clear that the crest line defined by (3) (3) (3) ∂u(λ ) = 0 does indeed lie outside D . The λ -field is, thus, essentially monotone in its domain and the only admissible solution is a simple rarefaction wave.

372 (2) (2) (2) As for the expansion wave associated to λ , figure 4.33 shows that both f u and f z are (2) negative throughout Δuh. Since it is strictly β > 0, as seen in figure 4.34(a), the solution of a (2) centred simple rarefaction wave is smooth throughout Δuh. In other words, the domain D of (2) occurrence of the expansion wave associated to λ is the totality of Δuh. Before trying to solve (4.122) to (4.129), it should be recalled that the waves associated to (1) (2) (k) (k) the λ - and the λ -fields can be shock waves because λ upstream > λ downstream (k=1,2, figure 4.32). This last condition is a necessary one; the necessary and sufficient conditions would be the fulfilment of the entropy conditions (4.111) and (4.112) for shock obeying the RH conditions (4.110). It will be assumed that the necessary and sufficient conditions are met and, following the computation of the solution, it will be shown that (4.111) and (4.112) were indeed true.

4.4 SOLUTIONS OF THE GEOMORPHIC DAM-BREAK PROBLEM

The computational procedure depends on whether the solution is composed of one expansion wave and two shocks (type A solution, figure 4.32(a)) or two expansion waves and one shock (type B solution, figure 4.32b)). Latter it will be discussed what are the domains of validity of each solution and what are the appropriate parameters to express it.

Assuming that the solution comprises an expansion wave in the 3rd characteristic field and shock waves in the 1st and 2nd ones (solution of type A), the solution procedure can be described as follows. (3) The variation of u and Yb across the maximum possible expansion wave associated to λ are computed first, as a function of h. The undisturbed upstream left state provides the boundary conditions for the ODE’s (4.128) and (4.129). These equations are simultaneously solved until (3) the water depth, and corresponding velocity, approach a point (hβ,uβ) along the singularity β = 0. The numerical solution of the ODE’s was achieved through a 4th order Runge-Kutta scheme. The values of h, u and Yb along the expansion wave are stored in a database.

A trial value for h2, to which corresponds a value of u2 and Yb2, is chosen from the database. Equations (4.122) to (4.127) are then used to compute h1, u1, Yb1 (variables in the constant state (1)), S2 and S1 (velocities of the shocks). It should be noticed that at this stage there are 6 equations but only 5 quantities to be computed, since the choice of h2 implies choosing also u2 and Yb2. One of the equations should be used to compute the error associated to a particular h2. The error should be brought to zero by choosing a different h2. This shooting method can be rationalised and refined to save computational time.

The solution composed of expansion waves in the 3rd and 2nd characteristic fields and a shock wave associated to the λ(1)– field (solution of type B) was determined from the following procedure.

As it is the case with the solution of type A, the variation of u, Yb and h across the expansion wave associated to λ(3) are computed first, through a procedure identical, in every step, to the one described above. The resulting values are stored in a database.

Trial values for h2, u2 and Yb2, are chosen from the database. These values provide the initial conditions for the ODE’s (4.130) and (4.131). These ODE’s are numerically discretized and th solved by the same 4 order Runge-Kutta scheme until a given trial value of h1 is reached.

373 Associated to this value of h1 one obtains also the values of u1 and Yb1. The remaining unknown is the shock velocity S1. There are three independent equations, (4.125), (4.126) or (4.127), to compute this parameter. One of these can be used to compute S1 while the remaining must be kept to evaluate the error of the iterative process.

New trial values of h2 and h1 should then be tested until the values of S1 are identical irrespectively of the equation used. Again, this shooting method should be refined to minimise the time involved in the computations.

To illustrate the procedures explained above, two computation examples are given. The equations are solved for two different sets of initial conditions and types of sediment. Solutions of types A and B where computed with pumice and sand as bed material, respectively. The parameters for the closure equations and the values of the initial states are displayed in table 4.1.

TABLE 4.1. Parameters of the examples of computation

Solution hL hR YbL Cf0 ds s (m) (m) (m) (-) (m) (-) Type A 0.4 0.1 0.08 0.1 0.003 2.65 Type B 0.4 0.00004 0.08 0.1 0.003 1.5

Figure 4.37 shows the structure of the solution of type A. The corresponding profiles of the primitive variables are shown in figure 4.38. As stated before, the solution for the λ(1)-field is a shock wave. Characteristic lines converge into the shock path, as seen in figure 4.37(a), confirming that this shock satisfies the entropy condition (4.111). Across the rarefaction wave the λ(1)-characteristics bend (1st derivative discontinuous) as their value increases as a result of the increasing velocity. Across the shock associated to the 2nd characteristic field there is a small change of direction due to the decrease of the velocity field in the constant state (1). Entropy condition (4.112) is also verified. The solution for the λ(2)-field is, as seen in figure 4.37(b), a shock wave for the initial conditions chosen. It is an entropy-satisfying shock as the λ(2) characteristic lines of constant states (1) and (2) converge to the shock as required by (4.111). The value of the λ(2) characteristics is zero in the upstream constant state (vertical characteristic lines) and increases as the lines cross the rarefaction wave. The values of λ(2) for this type of solution are very much dependent on the sediment transport rate. It is through this characteristic field that the coupling between water and sediment is most visibly expressed. In fact, the main difference between the clear water dam-break problem and the movable bed problem lies in the existence of a second shock wave, a positive jump in the flow velocity (like an hydraulic jump in the water phase) associated to an agradational “sediment bore”. The 3rd characteristic field admits a rarefaction wave as a solution as seen in figure 4.37(c). Characteristic lines are deflected as they pass the shock paths as a result of the abrupt changes in the water depth and flow velocity.

Figure 4.38(a) shows the geometric configuration of the solution. The two downstream- moving discontinuities and an upstream moving depression wave are clearly visible. The

374 dotted line represents the sediment transport layer of thickness hc. This layer was computed considering (4.79) and (4.80).

7.00 6.00 5.00 4.00 ' (-)

T 3.00 2.00 1.00 0.00 -10 -5 0 5 10 15 X '(-) a)

7.00 6.00 5.00 4.00 ' (-)

T 3.00 2.00 1.00 0.00 -10 -5 0 5 10 15 X ' (-) b)

7.00 6.00 5.00 4.00 ' (-)

T 3.00 2.00 1.00 0.00 -10 -5 0 5 10 15 X ' (-) c) (1) FIGURE 4.37. Characteristic lines and Riemann wave structure for solution of type A. a) λ - field; b) λ(2)-field; and c) λ(3)-field. Initial conditions and parameters of the closure equation as shown in table 4.1.

375 Figure 4.38(b) shows the depth averaged flow velocity, u, the total discharge per unit width, q, and the total sediment concentration, C. It is clear from figures 4.38(a) and 4.38 (b) that, when hR is sufficiently large, the strongest sediment transport rate occurs at the constant state (2) and not at the tip of the shock associated to the λ(1)-field.

1.50 1.25 1.00 0.75 ' (-)

Z 0.50 0.25 0.00 -0.25 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 X ' (-) a)

1.00

0.75 (-) C ', 0.50 Q ', U 0.25

0.00 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 X ' (-) b) FIGURE 4.38. Example of solution of type A. The profiles are shown in a self-similar

referential, X '/= ()xt ghL , and are: a) the non-dimensional water elevation, Y/hL ( ), the non-dimensional elevation of the upper interface of the contact load layer

h/hL + hc/hL ( ) and the non-dimensional bed elevation, Yb/hL ( ); b) the

non-dimensional flow velocity, Uugh' = L ( ), the non-dimensional unit

3 discharge, Qqgh' = L ( ) and the flow-averaged sediment concentration, C’ ( ). Initial conditions and parameters of the closure equation as shown in table 4.1.

This result is in agreement with the experimental results shown in Leal et al. (2003). A more complete discussion will be conducted latter. It should be noticed that the total discharge increases across the shock associated to λ(2), which is not strange since the flux of momentum is larger in constant state (2).

Solutions of type B are similar to those presented by Fraccarollo & Capart (2002). Figure 4.39 shows the structure of the solution of type B while the profiles of the primitive variables are shown in figure 4.40. As is the case for type A solutions, the 1st characteristic field admits

376 a shock wave as a solution whereas the 3rd field admits an expansion wave. Contrarily to type A, the solution for the λ(2)-field is an expansion wave (figures 4.37(b) and 4.39(b)). The shock wave in the λ(1)-field is entropy satisfying, as it could be verified from figure 4.39(a) where it is visible that the characteristic lines converge into the shock path. Across the rarefaction wave associated to the λ(3)– field, it is visible that the values of λ(1) increase. 5 t (s)

4

3

2

1

0 -2 -1 0 1 234 Characteristics path x (m) 5 t (s)

4

3

2

1

0 -2 -1 0 1 234 x (m) 5 t (s)

4

3

2

1

0 -2 -1 0 1 234 x (m)

(1) FIGURE 4.39. Characteristic lines and Riemann wave structure for solution of type B. (a) λ - field; (b) λ(2)-field; and (c) λ(3)-field. Initial conditions and parameters of the closure equation as shown in table 4.1. Adapted from Ferreira & Leal (1998).

377 The characteristic lines bend as it was the case for the solution of type A. Across the rarefaction wave associated to the λ(2)– field (figure 4.39b) the values of λ(1) increase, as a result of the increase of the flow velocity and the decrease of the water depth. Here resides the main difference between solutions of types A and B in of what concerns the behaviour of the 1st characteristic field.

1.25

1.00

0.75

' (-) 0.50 Z 0.25

0.00

-0.25 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 X ' (-)

1.20

1.00

0.80 ' (-) ' C

' , 0.60 Q

' , 0.40 U 0.20

0.00 -1.5 -1 -0.5 0 0.5 1 1.5 X ' (-)

FIGURE 4.40. Example of solution of type B. The profiles are shown in a self-similar

referential, X '/= ()xt ghL , and are: a) the non-dimensional water elevation, Y/hL ( ), the non-dimensional elevation of the upper interface of the contact load layer

h/hL + hc/hL ( ) and the non-dimensional bed elevation, Yb/hL ( ); b) the

non-dimensional flow velocity, Uugh' = L ( ), the non-dimensional unit

3 discharge, Qqgh' = L ( ) and the flow-averaged sediment concentration, C’ ( ). Initial conditions and parameters of the closure equation as shown in table 4.1.

As shown by Fracarollo & Armanini (1999) for the horizontal bed case, as the sediment transport rate vanishes, the expansion waves of the 3rd and 2nd characteristic fields merge, giving rise to the expansion wave of the Stoker solution.

378 4.5 CONCLUSIONS

A mathematical analysis of the systems of governing equations deduced in Chapters 2 and 3 was performed in this chapter. The emphasis was placed on the systems that express quasi- equilibrium (or capacity) sediment transport, in which case the mass and momentum fluxes among layers are zero. Mathematically, the absence of vertical fluxes means that, except for the friction slope in the total momentum conservation equation, the systems are homogeneous. Since the fundamental mathematical properties of a system of equations are imposed by its homogeneous part (see §4.2.1.1, p. 306), it was decided to study the quasi- equilibrium approximations only.

The applications of the model developed in Chapter 2 are not susceptible to feature discontinuous flows, since the model was designed for low Froude numbers (see §2.1, p. 14). On the contrary, the geomorphic shallow-water model developed in Chapter 3 was meant to be applied in highly unsteady flow problems, where discontinuities, hydraulic jumps or bores, are likely to be in the solution. Hence, most of the analysis is performed on the geomorphic shallow-water equations, written as in §4.2.2.1.

Before analysing the full set of equations, some fundamental results on hyperbolicity, wave propagation, ill-posedness and prescription of information at the computational boundaries is reviewed. It is underlined that the computation of the direction of lines of constant phase, associated to a given wave form, is amenable to an eigenvalue problem, expressed in the characteristic polynomial (4.18) (see §4.2.1.3 and §4.2.1.4). The roots of that the characteristic polynomial (the characteristics of the system) are the directions of the lines of constant phase and express propagation of information in the space-time domain. Hence, a well-posed hyperbolic problem possesses as many characteristic lines as dependent variables (§4.2.1.4). It was also seen that, in the computational boundaries, this principle applies and, as a consequence, independent information must be prescribed for each characteristic line that “enters” the computational domain (§4.2.1.5 and §4.2.1.6). Compatibility equations are also derived, as they are easily discretized.

Shock formation and weak solutions are discussed in §4.2.1.9 and §4.2.1.10. The conditions for obtaining entropy admissible solutions are reviewed and the Rankine-Hugoniot equations are derived for a general system of hyperbolic equations.

The governing conservation and closure equations, object of the analysis, are shown in §4.2.2. Additional conditions for obtaining a solution are identified, addressing mostly the problem of the growth of the contact-load layer. Indeed, if the thickness of the contact load layer is a monotone increasing function of the mean flow velocity, a condition must be imposed limiting that thickness to the local flow depth. The mathematical consequences, namely on the continuity and the differentiability of the fluxes, are discussed.

Since discontinuities are expected, the system of conservation equations must be written in conservative form. It is claimed in §4.2.3 that a strict conservative formulation is impossible because the term that expresses the force of gravity in the momentum equation, fbxb()hu,;⋅∂() Y , is, physically, not a flux and, mathematically, cannot be written in conservative form. However, it is proved that, under the conditions of proposition 3.1, there is indeed, a vector or conservative variables and, hence, solutions are possible.

379 The eingenstructure of the system of conservation equations is presented in §4.2.4. The properties of the characteristic fields are discussed, namely signal and monotonicity. It is shown in §4.2.5 that all fields are genuinely non-linear, which excludes the possibility of featuring contact discontinuities as solutions.

The solutions of the Riemann problem for the geomorphic shallow water equations, which, in some conditions can be used to describe the dam-beak problem (see §4.1), are identified in §4.3. It is shown that solutions exist but are unique only in relation to the specific linearization performed to the term fb()hu,;⋅ in fbxb(hu,;⋅∂) ( Y ) . This is a consequence of the non- purely conservative form of geomorphic shallow water equations.

Two types of solutions are encountered in §4.3.2. Type B solution is similar to that found by Fraccarollo & Capart (2002) and it is composed of two rarefaction waves and a downstream propagating shock. Type A solution features two shock waves, a faster bore-like shock and a slower hydraulic jump-like discontinuity. It is shown that composite shock-rarefaction wave solutions are not possible for the 3rd characteristic filed. The threshold conditions for the occurrence of each of the solutions are investigated. In §4.3.3, the sets of equations necessary to compute each of the solutions are presented and discussed.

Examples of computations of solutions of types A and B are shown in §4.4. Some properties of the solution of type A are briefly discussed. A thorough analysis of the properties of the Riemann solution is left to be performed in Chapter 5. It is envisaged that these theoretical solutions may substitute the classic Stoker solution as the preferred reference situation for geomorphic flows.

4.6 REFERENCES

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FRACCAROLLO, L. & ARMANINI, A. (1999) A semi-analytical solution for the dam-break problem over a movable bed. Proc. 1st IAHR Symposium on River, Coastal and Estuarine Morphodynamics (RCEM), Genoa, Vol. 1, pp. 361-369.

FRACCAROLLO, L. & CAPART, H. (2002) Riemann wave description of erosional dam-break flows. J. Fluid Mech. 461, 183-228.

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HIRSCH, C. (1988) Numerical computation of internal and external flows. Vol. 1: Fundamentals of numerical discretization. 1994 Reprint, Willey-Interscience Publication.

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381 HIRSCH, M., SMALE, S. & DEVANEY, R. L. (2004) Differential equations, dynamical systems & an introduction to chaos. 2nd Ed., Elsevier Academic Press.

HUNT, B. (1983) Asymptotic solution for dam-break on sloping channel. J. Hydraul. Res., 109 (12), pp. 1098-1104.

HUNT, B. (1984) Perturbation solution for dam-break floods. J. Hydraul. Res., 110 (8), pp. 1058-1071.

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JENKINS, J. T. & RICHMAN, M. W. (1988) Plane simple shear of smooth inelastic circular disks: the anisotropy of the second moment in the dilute and dense limits. J. Fluid Mech. 192, 313- 328.

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LEAL, J. G. A. B.; FERREIRA, R. M. L & CARDOSO A. H. (2005). Dam-break flood wave over mobile beds. J. Hydraul. Eng., (Accepted for publication).

LEAL, J. G. A. B.; FERREIRA, R. M. L; CARDOSO, A. H. & ALMEIDA, A. B. (2003). Comparison between Numerical and Experimental Results on Dam-Break Waves over Dry Mobile Beds. Proceedings of the 3rd IMPACT Workshop, Louvain-La-Neuve. http://www.samui.co.uk/ impact-project/cd3/default.htm.

LEAL, J.G.A.B.; FERREIRA, R.M.L. & CARDOSO, A.H. (2002) Dam-break waves on movable bed. In River Flow 2002 D. Bousmar & Y. Zech (ed.). Proc. of the International Conference on Fluvial Hydraulics, RIVER FLOW, Balkema. Louvain-la-Neuve, Belgium, pp. 981-990.

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382 MAGENES, E. (1996) On the scientific work of Olga Oleinik. Rendiconti di Matematica, Serie VII, Vol. 16 , Roma. pp. 347-373

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383

384

ANNEX 4.1 – Calculation of the wave number of a wave form.

Consider the wave form (4.3), build as superposition of harmonic waves, weighted in phase and amplitude, by Vk(), where k is the wave number

+∞ it(())kxi +ω k Vx(,)t = Vk ()e d k (A4.1.1) ≡ (4.3) ∫−∞ It is shown that (A4.1.1) can be transformed into (4.4). Formally, (4.4) is advantageous for the manipulations performed in §4.2. Let the wave number be expressed by an asymptotic expansion around κ, truncated to first order k = κη+ε (A4.1.2)

The wave number κ should be chosen as that of the most representative harmonic or, simply as the average wave number

+∞ κ = p()kk d k (A4.1.3) ∫−∞ where p()κ is an appropriate function of density of probability. For the sake of simplicity it can be chosen as that of the uniform distribution. I any case, the chosen κ must allow for the expansion of the angular frequency in a Taylor series around ω()κ ≡ϖ. The angular frequency becomes

3 ω=ω+∇ω+()k ()κ ikk1 T H k + O k (A4.1.4) κκ2 ( ) where H is the Hessian matrix of the angular frequency (the matrix of second derivatives). The case of non-dispersive waves is of particular interest in this text. In this case H0= . Introducing (A4.1.2) and (A4.1.4) in (A4.1.1) and writing V()()κη+ ε=f η ε2 , the following manipulations are possible

+∞ Vx(,)t = V (κη+ε )eei(κiiiix+ϖttt ) i( εη x +∇ ( ω )κη +ε∇ ( ω ) ) ε 2 d η = ∫−∞ +∞ f ()η eei(κiiiix+ϖttt ) i( εη x +∇ ( ω )κη +ε∇ ( ω ) )ε= 2dη ε2 ∫−∞ +∞ eei(κiix+ϖtt ) i ∇ ( ω )κηf ()ηη e i ε i (x +∇ ( ω ) t ) d = ∫−∞

385 +∞ eei(κiix+ϖt ) − i τ f ()ηη e iηχ d =

∫−∞ i(κix+ϖt ) V0 (,)χ τ e (A4.1.5)

In (A4.1.5) a change of variables is performed. The following identities are used τ =−∇ω()iκ t (A4.1.6)

χ = ε−∇ω(x ()t) (A4.1.7)

The new phase of the wave form is, after these manipulations, written as Σ(,)x tt=−ϖκix (A4.1.8) where ϖ≡ω()κ . Formally, the wave can thus be written as

i(,)Σ x t Vx(,)tt= V0 (,) x e (A4.1.9) ≡ (4.4)

The specification of Vx0 (,)t requires initial conditions and this function is sought sufficiently regular to admit Fourier transforms. From (A4.1.9) it is clear that the propagation of the wave form is formally the propagation of Vx0 (,)t along the surfaces of constant phase, the latter expressed by the exponential term.

386

ANNEX 4.2 – Coefficients of the matrixes A and B of equation (4.91)

The coefficients of the matrixes A and B are presented. These read A = ⎡⎤10 1 B = ⎡ uh0 ⎤ ⎢⎥ ⎢ ⎥ (A4.2.1)(a)(b) ⎢⎥0 KK23 ⎢KKK456⎥ ⎣⎦⎢⎥NN121− p ⎣⎢NN450 ⎦⎥ It should be noticed that matrix A as expressed by equation (4.91) was subjected to a pivoting operation in order to eliminate the coefficient of ∂t(h) in the momentum equation.

Obviously, the operation implies not only the indroduction of K3 but also of a redefinition of the second line of B. Traditionally, the full non-conservative momentum equation has been written with a term proportional to ∂tb(Y ) . In fluvial regimes, i.e., for low Froude numbers, this term is neglected on physical basis. In a dam-break flow, ∂tb(Y ) may be of the order of

∂t()u and either of the formulations can be retained. For the purpose of comparison with previous works, the term proportional to ∂tb(Y ) was preferred.

The coefficients are

Kh2 =+−+∂(1( s 1)( Cu () Cu)) (A4.2.2)

KusC3 =−(1( + − 1)( +∂h () Ch)) (A4.2.3)

123456 KKKKKKK4444444=+++++ (A4.2.4) where

1 Kus4 =chcccchccchcc()( −∂ 1)() Chu + 21(1) ( +− sC ) ∂( uh) ++−( 1( sC 1) )() ∂ hu

2 Kus4 =s (( −∂ 1)h( Chu sss) + 21(1)( +− sC shss) ∂( uh) ++−( 1( sC 1) shss) ∂( hu) )

KgsCh32=−∂++−∂k ' (1) 21(1) sChh 4 2 ( hcc() ( chcc )())

KgsCh42=−∂++−∂1 (1) 21(1) sChh 4 2 ( hss() ( shss )())

5 KkgsChhsChhsChh4 ='( ( −∂ 1)hssc( ) ++−( 1 ( 1) shsc) ∂( ) ++−( 1 ( 1) shcs) ∂( ) )

62 Ku4 =−(1( + sC − 1)( +∂h () Ch))

123456 KKKKKKK5555555=+++++ (A4.2.5) where

1 Kus5 =cuccccucccucc()( −∂ 1)() Chu + 21(1) ( +− sC ) ∂( uh) ++−( 1( sC 1) )() ∂ hu

387 2 Kus5 =s ()( −∂ 1)u() Chu sss + 21(1) ( +− sC suss ) ∂( uh) ++−( 1( sC 1) suss)() ∂ hu

32k ' KgsCh5 =−∂++−∂2 ((1)ucc() 21(1) ( sChh cucc )())

KgsCh42=−∂++−∂1 (1) 21(1) sChh 5 2 ( uss() ( suss )())

5 KkgsChhsChhsChh5 ='() ( −∂ 1)ussc( )( ++− 1 ( 1) susc ) ∂( ) ++−( 1 ( 1) sucs)() ∂

6 KuhsC5 =−(1( + − 1)( +∂h () Ch))

K6 =+−gsChhghgsCh()()1( 1)cc +=+− s ( 1) cc (A4.2.6)

NChCh1 =∂( hcchss( ) +∂( )) (A4.2.7)

NChCh2 =∂( uccuss( ) +∂( )) (A4.2.8)

NuCCh4 =+∂( h ( ) ) (A4.2.9)

NhCCu5 =+∂( u ( ) ) (A4.2.10)

The coefficients of the characteristic polynomial, equation (4.94), are 1 aKNKpKpuKNKNKN16252=−−−−+−+[](1 ) (1 ) 514224 Dλ 1 ++−⎣⎡KN35() NuNh 2 1⎦⎤ (A4.2.11) Dλ

1 ahKNhKpKpuKNNu26145=−−+−−+⎣⎦⎡⎤(1 ) (1 ) 652() Dλ 1 +−+−⎣⎡()()KN45 NK 45 hKN 34 KNu 35 ⎦⎤ (A4.2.12) Dλ 1 a3654=− K() uN hN (A4.2.13) Dλ where

DKλ =−−−22132(1 pKNKN ) (A4.2.14)

388