The granular monoclinal wave: a dynamical systems survey

Abstract The theoretical existence of the granular monoclinal wave, based on the Saint-Venant equations for flowing granular matter, was reported recently by Razis et al. (J. Fluid Mech., vol. 843, 2018, pp. 810–846). The present paper focuses on the mathematical interpretation of its behaviour, treating the equation of motion that describes any granular waveform as a dynamical system, taking also into consideration the Froude number offset $\varGamma$ introduced by Forterre & Pouliquen (J. Fluid Mech., vol. 486, 2003, pp. 21–50). The critical value of the Froude number below which stable uniform flows are observed is determined directly from the stability analysis of the aforementioned dynamical system. It is shown that the granular monoclinal wave, represented as a heteroclinic orbit in phase space, can be categorized into two classes: (i) the mild class, for which the exact form of the waveform can be approximated by the non-viscous (first-order) adaptation of the granular Saint-Venant equations, and (ii) the steep class, for a description of which a second-order (viscous) term in the Saint-Venant equations is absolutely needed to capture the dynamics of the wave. The mathematical criterion that distinguishes the two classes is the changing sign of the trace of the Jacobian matrix evaluated at the fixed point corresponding to the waveform's lower plateau.


Introduction
The monoclinal flood wave is a travelling waveform that one can often encounter in shallow water flows (Moots & Mavis 1938;Whitham 1974;Le Méhauté 1976;Ferrick 2005;Shome & Steffler 2006;Fowler 2011). In recent years, many research groups have contributed to formulating a hydrodynamic-like framework for describing granular flows. In this context, our group established the theoretical existence of the monoclinal wave in granular chute flow (Razis, Kanellopoulos & van der Weele 2018 by employing the † Email address for correspondence: kanellop@math.upatras,gr  (Razis et al. 2018): a flowing sheet of uniform height h − and depth-averaged velocityū − overtakes a shallower and slower flow of height h + and velocityū + . The monoclinal wave, being the travelling shock structure connecting these two regions, is stable as long as the Froude number Fr − of the upper plateau does not exceed a critical value Fr cr given by (3.18). It propagates at a wave speed c exceeding the velocities of the granular materials in both plateaus, i.e. c >ū − >ū + . appropriately modified granular Saint-Venant equations (Savage & Hutter 1989;Forterre 2006;Viroulet et al. 2017).
A typical monoclinal wave is sketched in figure 1. It may be formed if we allow extra material to flow over a pre-existing thin, uniform sheet of height h + that propagates with mean velocityū + . Thus, a sheet of greater height h − and speedū − is created on the upstream flank. This waveform retains its stability as long as the Froude number in the upper plateau does not exceed a critical threshold, Fr cr . In the context of one-dimensional flow, the Froude number is defined as follows: h(x, t)g cos ζ , (1.1) whereū(x, t) is the depth-averaged velocity of the granular sheet, h(x, t) is its height, g is the gravitational acceleration and ζ is the inclination angle of the chute. The paper is organized as follows. In § 2 the governing equations, being the one-dimensional Saint-Venant equations adapted to the granular context, are introduced. We then give the fundamental solution representing the stable uniform flow. In § 3 the dynamical system that governs all possible wave solutions (travelling and standing) is derived, its fixed points are located and their local stability properties are determined. Then, in this context, the critical Froude number for the fundamental solution (stable uniform flow) is found. In § 4 the geometrical shape of the orbit of the granular monoclinal wave in phase space is determined. Based on the above findings, and on the form of the first-order approximation of the granular Saint-Venant equations, a classification of the monoclinal waveform is given. Finally, in § 5 the paper is concluded with a recapitulation of the main results and with a computational experiment confirming the stability of the granular monoclinal waves.

The Saint-Venant equations for granular chute flow
The dynamics of a flowing granular sheet, in analogy with the shallow water approximation in normal fluids, is described in terms of its height h(x, t) and depthaveraged velocityū(x, t). The latter is defined asū( where u (x, z, t) is the detailed velocity profile depending on the depth z. Flow variations in the crosswise direction are ignored; hence both h(x, t) andū(x, t) depend only on x and t (cf. figure 1). Furthermore, it is assumed that the granular sheet has a constant density, so it is treated as an incompressible fluid (Savage & Hutter 1989).
The height h(x, t) and the depth-averaged velocityū(x, t) are governed by a system of two coupled, nonlinear partial differential equations (PDEs) Razis et al. 2014;Edwards & Gray 2015;Razis et al. 2018;Edwards et al. 2019;Razis et al. 2019), namely the mass conservation (2. 2) The terms on the right-hand side of (2.2) represent the following.
(i) The gravity component along the x direction, with g = 9.81 m s −2 .
(ii) The force arising from variations in h(x, t), i.e. the negative gradient of the depth-averaged pressure. (iii) The frictional force exerted by the chute bed, modelled as a (height-and velocity-dependent) friction coefficient μ(h,ū) multiplied by the normal reaction force acting on the granular sheet.
For the friction coefficient, the modified expression for the fully dynamic regime introduced by Edwards et al. (2017) is adopted: (2.3) Here ζ 1 , ζ 2 , L, β and Γ are experimental parameters. The angles ζ 1 and ζ 2 denote the two inclination angles that delimit the interval in which uniform granular flows are possible. For ζ < ζ 1 the granular sheet remains at rest, whereas for ζ > ζ 2 it flows downwards in an accelerated fashion. The parameter L has its origins in the functional form of the experimental fit concerning the critical angle curves that determine the friction force (Pouliquen & Forterre 2002). It represents a characteristic thickness (usually ranging from 1 to 2 particle diameters) of flow over which a transition between the angles ζ 1 and ζ 2 occurs in the friction law (2.3). Its exact value depends both on the properties of the grains and on the bed roughness (Pouliquen & Forterre 2002;Edwards et al. 2017). The parameter β and the Froude number offset Γ are closely related (Forterre & Pouliquen 2003): where h stop (ζ ) denotes the thickness of the deposition layer that is left on the chute, at an inclination angle ζ , after a uniform flow has passed over it. A first version of (2.4) (without the Γ offset) was derived by Pouliquen (1999) by experimentally measuring the Froude number as a function of the quotient h/h stop for four different systems of glass beads, revealing their linear relation which is independent of the bead size, the inclination and the roughness condition. The Froude offset Γ was introduced by Forterre & Pouliquen (2003) in order to fit the experimental results concerning the mean velocity for different materials (sand and glass) into a single law, thus formulating (2.4). In this context we note that the factor 1/ √ cos ζ was absent from their definition of the Froude number Fr and the offset Γ . In Edwards et al. (2017) also the parameter β * was introduced, expressing the Froude number value at a height h = h * (ζ ) with h stop (ζ ) < h * (ζ ) < h start (ζ ). Here h * (ζ ) denotes the level of the thinnest (and therefore slowest) possible steady uniform flow which leaves a deposit of smaller thickness h stop (ζ ). In the same spirit, h start (ζ ) denotes the height for which a static layer is mobilized when the inclination is increased to an angle ζ . As such, it is guaranteed that the granular chute flow remains fully dynamic as long as Fr β * .
(iv) The viscous-like diffusive term arising from depth-averaging the in-plane stresses in the sheet. Here, the analytical expression derived by  is adopted: The granular monoclinal wave consists of two uniformly moving plateaus linked together via a shock front (Razis et al. 2018(Razis et al. , 2019. The steady uniform flow corresponds to the constant solution of the granular Saint-Venant equations (2.1)-(2.2) when all derivatives with respect to x and t vanish, leaving only tan ζ = μ (h,ū). This condition expresses the balance between the forces of gravity and friction (Razis et al. 2018). Now, we can express the velocity of the uniform plateaus as follows: (2.6) and the corresponding Froude number takes the form (2.7) The velocity of the shock front is given by the well-known formula (Whitham 1974) which reflects the fact that in the steady state, for an observer in the co-moving frame, the flux of material entering the shock structure must be equal to the flux leaving the shock, i.e.
. Although in the shock region the inertial terms are not exactly zero, they are sufficiently small to justify this expression which is based on kinematic considerations alone, namely the mass conservation across the shock (Razis et al. 2018). Equation (2.8) with the help of equation (2.6) can be expressed in a form which does not contain the velocitiesū ± : From (2.9) one might have the impression that when Γ > 0, the shock speed decreases. This is not the case; in fact the opposite is true as may be seen by fixing the values of the Froude numbers in both plateaus. Equation (2.7) demonstrates that in the presence of the Γ offset, the corresponding heights of the plateaus increase and as a consequence, (2.9) returns a greater shock speed than for Γ = 0. This will become even more evident later, in § 3, when we introduce the non-dimensional equations.
Another aspect that is worth noting here is that the minimum height for the lower plateau, in the fully dynamic regime, can be found by setting Fr + = β * and solving equation (2.7) for h + . The resulting expression is (2.10) Finally, for completeness, we should note that the advective term ∂ x (hū 2 ) in (2.2) may be multiplied by a shape factor to obtain an optimal correspondence with experiments (Saingier, Deboeuf & Lagrée 2016;Lagrée et al. 2017). This factor, for a steady flow with a Bagnold velocity profile, is equal to α = 5/4 (GDR-MiDi 2004;Börzsönyi, Hasley & Ecke 2005;). However, for shallow granular flows with relatively small Froude number, as in the current paper, the solutions are quite insensitive to the shape factor's value (Saingier et al. 2016;Viroulet et al. 2017). In fact, in an experimental study by Forterre (2006) (see in particular figure 1 in that paper) it was found that the plug flow with shape factor value α = 1 gave a better correspondence with the experimental data than the Bagnold profile, for which α would be 5/4.

Travelling wave solution and stability analysis
The monoclinal wave is a travelling waveform propagating without change of shape and at a constant velocity c. Therefore its height and depth-averaged velocity are of the , and consequently the governing equations (2.1)-(2.2) become ordinary differential equations. The mass balance equation (2.1) reduces to where the integration constant K corresponds to the flux of the material per unit width of the channel in the co-moving frame. Using (2.6), this flux constant can be written as a function of height only: Solving (3.1) with respect toū, we find the first and the second derivativeū ξ ,ū ξξ in terms of h, and then we substitute into the momentum balance equation (2.2). This leads to the following second-order ordinary differential equation: where now μ(h) has the form (3.4) Equation (3.3) governs all the travelling and the standing, when c = 0, waveforms for granular chute flow that are sustained by the Saint-Venant equations (2.1)-(2.2).
Here, it is convenient to introduce non-dimensional variables, denoted by a tilde, as follows: h = h −h and ξ = h −ξ (and also h + = h −h+ ). That is, all length scales are measured in terms of the thickness h − of the incoming flow. With this rescaling, the differential equation (3.3) takes the non-dimensional form d 2h and the granular Reynolds number defined by Equation (3.5) first appeared in a study by  of granular roll waves and also featured in Razis et al. (2019) in the context of the transition from the granular monoclinal waves to (granular) roll waves. Here, despite the fact that (3.5) is identical, the constitutive relations (3.6), (3.7) and (3.8) are different, due to the inclusion of the Γ offset. Of course, for Γ = 0 one recovers the exact same expressions as in Razis et al. (2019). We also note that the shock speed given by (3.7), expressed only as a function ofh + , is always greater than unity forh + > 0 and thatc|h + =0 = 1. Equation (3.7) also implies that in the presence of the Γ offset, the dimensionless shock speed depends not only onh + but also on the Froude number. In general, given a fixed value of Fr − , a non-zero Γ offset causes an increase of the (dimensionless) shock speed. Now, we can write (3.5) as two coupled first-order ordinary differential equations, forming a dynamical system: The fixed points of the above dynamical system lie at the intersections, in the (h,s) phase space, of the two nullcline curves: Substituting (3.10a) into (3.10b), the necessary and sufficient condition for uniform flow, tan ζ = μ(h), appears yielding two fixed points: This means that both fixed points correspond to uniform flow conditions. To determine their local stability we consider the Jacobian matrix: and we determine the corresponding eigenvalues on each fixed point. The analytical expressions of the above eigenvalues are four functions (two for each fixed point) depending onh + , Fr, Γ, ζ 1 , ζ and ζ 2 . Their formulas are rather long, so they are not presented here for economy of space. For the granular monoclinal waveform, in the fully dynamic regime, where β * Fr < Fr cr ,c > 1 andh + < 1 (and of course ζ 1 < ζ < ζ 2 ), it is found that (h + , 0) has always two real eigenvalues with opposite sign, meaning that it is a saddle, while (1, 0) is always an unstable node with two real positive eigenvalues. This result is in full agreement with Razis et al. (2019) where the Γ offset was taken to be zero. At this point, let us write down the explicit expression of the trace of the Jacobian matrix, ats = 0, since it will play a key role in our study: .

Critical Froude number for stable uniform flow
In the situation of uniform flow along the whole chute (μ(h) = tan ζ ) it applies thath − = h + (=1), and so the dynamical system (3.9) has a unique fixed point at (1, 0). Now, the Jacobian matrix (3.11) at this fixed point takes the form (3.14) The uniform threshold flow, i.e. the marginal value ofh above which the uniform flow cannot be sustained, is neither stable nor unstable (in fact, mathematically it is a cusp point; see Razis et al. (2019)), so both eigenvalues of the matrix J c are zero. One eigenvalue is already zero by default and thus we have only to consider the second eigenvalue.
( 3.16) Equation (3.7) for the (dimensionless) shock speed in this case gives and by substituting into (3.16), we conclude that Here we should also note that (3.17) and (3.18) coincide with the results of Forterre & Pouliquen (2003). They obtain the corresponding equations from the depth-averaged PDEs, by performing a stability analysis around the uniform flow and deriving the associated dispersion relation.
In figure 2, a numerical experiment is conducted in order to validate the result found in (3.18). The granular Saint-Venant equations (2.1) and (2.2) are solved numerically, with a randomly perturbed initial condition, of maximum amplitude 0.3 mm around h + = h − (denoted by the blue curve) and cyclic boundary conditions. The Froude number of the incoming flow remains always fixed at Fr exp = 0.6, while the value of the Γ offset is gradually increased. For the rest of the parameters, we use the values measured by Edwards et al. (2017) (carborundum particles on a bed of glass beads). The outcome is denoted by the red curve and is depicted after 60 time steps (representing seconds). Starting from Γ = 0, in figure 2(a), where the critical Froude number Fr cr = 2/3 is well beyond Fr exp , we witness the restoration of a stable uniform flow. In figure 2(b) we set Γ = 0.07 which gives Fr cr = 0.62, still larger than Fr exp . As expected the uniform flow is still stable. The case when Fr cr = 0.6 = Fr exp (Γ = 0.1) is shown in figure 2(c). The perturbed initial condition can no longer evolve into a uniform flow, and thus it forms and maintains small-amplitude undulations of relatively long wavelength. Finally, in figure 2(d) we choose Γ = 0.2, which corresponds to Fr cr = 0.533 < Fr exp . Now, stable roll waves are formed as foreseen for flows that exceed the critical Froude number Razis et al. 2014Razis et al. , 2019. To be more specific, as we can witness, the coarsening of roll waves has not finished after 60 s. In due time the first roll wave will merge with the second forming a single wave, while the third will merge with the fourth. The detailed physical mechanism that allows the larger (and faster) granular roll wave to merge with a smaller (and slower) one, forming a single travelling wave afterwards, was studied in Razis et al. (2014). For this computational experiment the numerical scheme that was used was the method of lines (Schiesser 1991;Razis et al. 2018), with a space step of x = 0.003 m over a total length of x max = 2 m. The grid independence of the result has been checked by using smaller space steps as well.
Equation (3.18) implies that in order to witness granular monoclinal waves in experiments, it is preferable to use a material that minimizes the Γ offset parameter. In fact, most known experimental set-ups that include a non-zero Γ offset are not In the direction of applying our analysis in a real set of measurements, even though in all the analytic expressions we will include the Γ offset, for the numerical calculations in § 4, the parameter values measured by Russell et al. (2019), concerning glass ballotini on a bed of the same material, will be used: ζ 1 = 21.27 • , ζ 2 = 33.89 • , L = 0.2351 mm, Γ = 0.0, β = 0.143, β * = 0.19. Moreover, in § 4.3, a hypothetical material with non-zero Γ offset value, based on the measurements of Edwards et al. (2017), will be assumed in order to illustrate that the parameter Γ alters the results only quantitatively and not qualitatively.

Classification of the granular monoclinal waves
4.1. The heteroclinic orbit and its properties In phase space, the monoclinal wave is represented as a heteroclinic orbit connecting the saddle's stable manifold with the nearest (always repelling) manifold of the unstable node (see figure 3) (Razis et al. 2018(Razis et al. , 2019. This means that the reason for the granular monoclinal wave to exist (from the dynamical systems perspective) is the presence of the stable manifold of the saddle point. The exact position, i.e. the specific negative slope with respect to the horizontal axis, of that manifold as the value ofh + varies determines the entire shape of the granular monoclinal wave.
The contribution of the unstable node at (h,s) = (1, 0) is rather simple by comparison. Its trace and its determinant are always positive, repelling any initial condition in its neighbourhood. The saddle has a more interesting contribution. In a two-dimensional phase space, the two manifolds of a saddle are rotated as the trace of the Jacobian matrix (evaluated at the saddle point) varies, while the always negative determinant regulates the relative angle between the two manifolds. This general behaviour is depicted in figure 4 for a minimal dynamical system. The saddle point lies at the origin while the manifolds are the straight lines that intersect with it. In figure 4(a,b), the trace of the corresponding Jacobian matrix is kept constant at Tr = 0 while the determinant becomes smaller, from Det = −1 in figure 4(a) to Det = −2 in figure 4(b). Evidently, the two manifolds move away from each other with respect the x axis and the angle α increases without any rotation. On the other hand, in figure 4(c,d) where the determinant is kept constant at Det = −1 and the trace changes values from Tr = 1 in figure 4(c) to Tr = −1 in figure 4(d), the relative distance of the manifolds is kept intact but they are rotated. More specifically, the angle α increases as the trace becomes smaller. Back to our system, the analytical expression of the determinant, evaluated at the saddle point (h + , 0), as a function ofh + is where the prefactor C(ζ ) is given by 27 (tan(ζ ) − tan(ζ 1 )) (tan(ζ ) − tan(ζ 2 )) 4 tan(ζ ) (tan(ζ 2 ) − tan(ζ 2 )) .
(4.2) Equation (4.1) expresses a negative, monotonically increasing function for 0 <h + < 1 and β * Fr < Fr cr (see figure 5). This means that as the two plateaus of the granular monoclinal wave come closer to each other (h + approaches 1) the determinant of the saddle approaches zero. This implies that the slope of the stable manifold has the tendency to decrease as the fixed points come closer to each other. The trace of the Jacobian matrix evaluated at the saddle point is given by .

(4.3)
In figure 5 the value of the trace is depicted for fixed Froude number value Fr = 0.6 and parameter values taken from Russell et al. (2019). As expected, the trace of the saddle can be positive, negative or even zero as the value ofh + varies. The slope of the stable manifold becomes milder as the trace increases its value and vice versa, in full correspondence with the effects of the determinant. In fact, when Tr (h + ,0) J = 0, the two manifolds of the saddle are fully symmetric as far as the slopes of its manifolds are concerned (locally near the saddle point), as the eigenvalues have the same absolute value (but opposite sign).

Mild and steep granular monoclinal waves
The vanishing of the saddle's trace constitutes a criterion that can be used to classify the granular monoclinal waves, in the sense that it determines the interval of validity of the first-order approximation of the equation of motion (3.5). If we assume the inviscid limit (ν(ζ ) = 0), then the non-dimensional equation (3.5) takes the form which, with (3.13), can also be written as .
(4.5) Equation (4.5) constitutes a dimensionless and more general form (due to the inclusion of the Γ offset) of (4.1) from Razis et al. (2018), where the authors investigate the region of validity of the inviscid description for the granular monoclinal wave. Now, the trace of the unstable node is always positive, Tr (1,0) J > 0, and thus when also Tr (h + ,0) J > 0 holds, it means that the approximate trajectory of the granular monoclinal wave in phase space, given by (4.5), does not encounter any singularity in the interval h + h 1 (c is always larger than 1 in the same interval). Indeed, in this case, and especially if Tr (h + ,0) J is well above zero, one can safely use the first-order approximation rather than (3.5) (see figure 6a). In fact, one can find here, to a very good approximation, the orbit of the granular monoclinal wave in phase space algebraically, from the right-hand side of (4.5), without solving the dynamical system (3.9). On the other hand, if Tr (h + ,0) J < 0, (4.5) will produce a singularity inside the intervalh + h 1, rendering the inviscid approximation invalid, as can be seen in figure 6(c). The critical state is witnessed in figure 6(b), where we seth =h + + 0.001 (to achieve visualization for this case we must deviate slightly fromh =h + and Tr (h + ,0) J = 0). Here, the use of the inviscid approximation is still marginally possible but, as can be seen in the inset, is inadvisable. The above analysis prompts us to classify the monoclinal waves into two major categories, the mild (M) and the steep (S), depending on whether the sign of the trace of the Jacobian matrix evaluated at the saddle (h + , 0) is positive or negative, respectively.
The algebraic expression of the threshold height of the lower plateauh +,thres with respect only to incoming Froude number Fr − can be found analytically by solving the equation Tr (h + ,0) J = 0 given in (4.3) with the help of (3.7). Discarding the two complex solutions and the one real solution that givesh +,thres > 1, we concludẽ   Figure 6. The classification of the granular monoclinal waves for fixed Froude number Fr = 0.6 < Fr cr = 2/3. The threshold value ofh + , taken from (4.6), ish +,thres = 0.6771243445. (a) Mild regime. Forh + = 0.75 >h +,thres we witness that the heteroclinic orbit, representing the monoclinal wave in phase space (red thick curve), is in very good agreement with the inviscid approximation given by (4.5) (dark blue thin curve). Theh value for which the first-order approximation becomes singular is well beforeh + (vertical line), validating its use. The good agreement can be seen also in the corresponding profile depicted in the right-hand panel.
There, the area of interest is magnified in the inset in order to visualize the small differences. (b) Critical regime. Ath + =h +,thres the first-order approximation is marginally valid, as the singular point lies ath =h + (the plots are generated by settingh + = 0.678 =h +,thres + 0.001 in order to enable the visualization of this case). (c) Steep regime. Here, whenh + = 0.6 <h +,thres , the singularity lies inside the intervalh + h 1, rendering the first value approximation invalid. The system parameters are taken from Russell et al. At the same time, however, one must take into account the lower boundary for the fully dynamic regime, β * Fr + , which dictates the minimum height of the lower plateauh + . Equation (2.10) can be written in dimensionless variables as h +,min = β * + Γ Fr − + Γ . (4.7) A combined plot of (4.6) together with (4.7) is depicted in figure 7 for parameter values taken from Russell et al. (2019). The dark shaded area belowh +,min (blue curve) denotes the values of Froude number andh + for which the regime is not fully dynamic. The light shaded region, which lies aboveh +,min and to the right ofh +,thres (red curve), is labelled with the letter 'S' and represents the area where the steep granular monoclinal waves will be formed. There, the use of the second-order differential equation (3.5) is the only valid choice. On the other hand, the region labelled with the letter 'M', above thẽ h +,min curve and to the left of theh +,thres curve, is the mild regime where the granular monoclinal wave can be approximately described by the inviscid limit (4.4). The gradient grey shading near theh +,thres curve reflects the fact that the inviscid approximation (4.4) becomes increasingly inaccurate. The intersection of the two curves is found by setting h +,thres =h +,min . For Froude number values smaller than Fr ≈ 0.5 here, the granular moniclinal waves are exclusively of the mild type. The above classification of the granular monoclinal waves into two regimes, the mild and the steep, must be seen mainly from a mathematical point of view. Indeed, the viscous equation of motion, (3.5), is capable of fully describing dynamic granular monoclinal waves (as well as all the other waveforms, given the appropriate parameter values) regardless of to which regime they belong. The element that makes this classification important, combined with the dynamical systems view, is that in the mild regime we can approximate, quite accurately, the granular monoclinal wave in phase space, without solving the full system but by directly plotting (4.5) as a function ofh. This constitutes an insightful and direct way to reveal the dynamics of any granular monoclinal wave in the mild regime.

A case study
It is conceivable that an experimental set-up can be constructed, in which fully dynamic granular monoclinal waves can be detected despite the fact that the chosen material displays a non-zero Γ offset. For that reason, a case study of a hypothetical material that includes a Γ offset and is nevertheless capable of establishing this waveform is studied in this subsection. The parameter values of Edwards et al. (2017) are adopted with the exception of the Γ offset value. Here we take Γ = 0.1 (instead of Γ = 0.4), and so the full set of the parameter values becomes: ζ 1 = 31.1 • , ζ 2 = 47.5 • , L = 0.44 mm, β = 0.63, β * = 0.466, Γ = 0.1. In addition, the value of ζ = 33.0 • is used. The critical Froude number in this case, see (3.18), is Fr cr = 0.6, and thus the fully dynamic interval is restricted to 0.466 Fr < 0.6. Using equation (4.7), the corresponding interval of validity forh + is found to be 0.83235 h + < 1.
Taking these values into consideration, the incoming Froude number is chosen to be Fr − = 0.58, while the corresponding lower plateau threshold that separates the mild from the steep regime, see (4.6), ish +,thres = 0.876359. In figure 8 the classification of the granular monoclinal waves for this hypothetical material is depicted. In figure 8(a) where the valueh + = 0.9 >h +,thres is chosen, one can see a mild granular monoclinal wave, as expected. In the right-hand panel, the orbit's excellent agreement with the inviscid approximation is witnessed. In figure 8(b) the near-critical case is presented:h + = 0.8765 =h +,thres + 0.0002. Finally, the steep case can be seen in figure 8(c) whereh + = 0.85 <h +,thres . Comparing these results with the findings of figure 6, the two systems are qualitatively identical; the differences are only quantitative. This is evident also in figure 9 where the correspondingh +,thres curve is plotted along with the correspondingh +,min curve, as a function of the incoming Froude number, for this hypothetical set-up.
The phase diagrams of figures 7 and 9 provide a guideline for further study of granular monoclinal waves. Equations (4.6) and (4.7) have a simple algebraic form and they depend only on the parameters Γ and β * . This fact makes them a very useful tool for theorists and experimentalists alike.

Concluding remarks
In this paper, the mathematical properties of the granular monoclinal wave are clarified. As a travelling waveform, it can be well described by casting the granular Saint-Venant equations into a two-dimensional dynamical system. In order to make the formulation as general as possible, the modified expression of the friction coefficient introduced by Edwards et al. (2017), which includes the Γ offset, has been adopted.
Exploiting the stability analysis findings, the critical Froude number above which the uniform flow becomes unstable was determined as a function of the Γ offset (equation (3.18)). From that relation it is clear that -given the fact that the granular monoclinal wave is a travelling shock structure that connects two plateaus of steady and uniform flow -a material with Γ = 0 is more likely to develop such waves.  Figure 9. Phase diagram showingh +,thres (red curve) together withh +,min (blue curve) as a function of the incoming Froude number for the hypothetical set-up of § 4.3. As in figure 7, the dark shaded region denotes the area where the dynamic regime is impossible. The letter 'M' above theh +,min curve and to the left of theh +,thres curve denotes the area where the mild granular monoclinal waves appear, while the light shaded area with the letter 'S' to the right of theh +,thres curve and above theh +,min curve denotes the steep regime. The gradient shading inside the mild regime represents the increasing lack of accuracy of the inviscid approximation close to theh +,thres curve. 3) with Fr = 0.6 and a height h + corresponding toh + = 0.92 (mild regime): an initial perturbation (black thin curve) is positioned on the wave's upper plateau and its evolution is computed from the granular Saint-Venant equations (2.1)-(2.2). The perturbation is seen to diminish in time, and after 180 s we witness that the monoclinal wave has recovered its initial shape. The red dashed curve denotes the unperturbed initial monoclinal wave, and is presented here for direct comparison with the solution taken from the PDEs (solid lines). The evolution of the system has been evaluated using the method of lines (Schiesser 1991;Razis et al. 2018), with a computational space step of x = 0.01 m over a total length of x max = 50 m. The system parameters are taken from Russell et al. Based on our understanding of the properties of the saddle's stable manifold, a classification of the granular monoclinal waves was possible. It was shown that when the trace of the saddle is positive, the actual profile can be approximated by the inviscid limit (4.4) of the full, viscous equation of motion (3.5) (mild regime). On the other side of the spectrum, when the saddle's trace is negative, this approximation becomes invalid (steep regime). In the same context, an analytical expression relating the height of the lower plateau with the value of the incoming Froude number that separates these two regimes is given (4.6). Along with the curve that determines the minimum level ofh + in the fully dynamic regime (4.7), they constitute a map of the dynamical properties of the granular monoclinal waves.
A study of the possible formation of granular monoclinal waves in a hypothetical material with non-zero Γ offset is included in this paper. The results were qualitatively very similar to those obtained with the experimental set-up of Russell et al. (2019), which lacks the Γ offset parameter, illustrating the relevance of the case study in question.
In order to address the important issue of the stability of the solutions obtained from the dynamical system, one must return to the original time-dependent PDEs, i.e. the granular Saint-Venant equations (2.1)-(2.2). The stability of the monoclinal profiles found earlier can be demonstrated by inserting them, in perturbed form, as initial conditions into these PDEs. In figure 10, as an example, we show the time evolution of the thus perturbed 'mild' monoclinal wave. The perturbation (a small surplus of material in the shape of a Gaussian-like bump) is seen to disappear in the course of time, yielding again the unperturbed profile which remains unchanged afterwards.
As a final remark, it may be noted that this analysis applies to normal fluids too. The modified Saint-Venant equations introduced by Kranenburg (1992) in the context of the evolution of roll waves in normal fluids (see also Yu & Kevorkian 1992;Yu, Kevorkian & Haberman 2000;Balmforth & Mandre 2004) give rise to a similar dynamical system with many analogies. This will be reported in a forthcoming publication.