Snow avalanches striking water basins: behaviour of the avalanche’s centre of mass and front

Abstract

We study the behaviour of a low-density granular material entering a water basin by means of a simplified two-dimensional model, with the aim to understand the dynamics of a snow avalanche impacting a water basin like an alpine lake or a fjord. The low density of the impacting mass induces an uplift buoyancy force and, consequently, a complicated interaction between the solid and fluid phase. This paper provides an insight into the motion of the impacting mass, by presenting a simplified, two-dimensional model, where the snow is described by a low-density granular material. First, small-scale experiments, based on the Froude similarity with snow avalanches, are used to evaluate the motion of reference points of the impacting mass, i.e. the front (F), centre of mass (C) and deepest point (L). Then, applying the mass and momentum conservation principles to a fixed volume, we show that the mean motion of the impacting mass is similar to that of a damped oscillator. The stretch of the impacting mass motion is described through the motion of the reference points F and L.

Appendices

Appendix 1: Theoretical description of the mean motion of the impacting mass

We search for a simple equation for the mean motion of the solid phase by relaxing the hypothesis of diluted and submerged solid phase used in the theoretical model proposed by Zitti et al. (2016). After defining the local problem using the set of equations of a multi-phase mixture theory, they are integrated over a two-dimensional control volume representative of the problem. The integration leads to a differential equation in terms of mean value of the velocity of the impacting particles, which is hereafter referred to as the mean motion of the impacting mass.

First, conservation principles are formulated using the theory of multi-phase mixtures (Truesdell 1984; Meruane et al. 2010), i.e. assuming that each point is occupied by a volume fraction $\alpha_{i} \left( {\varvec{x},t} \right)$ of each ith phase. Three phases are present in the problem at hand: the air, the fluid water (subscript f) and the solid phase (subscript s) composed of particles with solid density slightly lower than the fluid density. Since the air density is negligible if compared to those of water and snow, the role of the air phase can be neglected in the conservation equations (see also Zitti et al. 2016). Assuming that both the fluid and the granular material are incompressible, the balance equations for the mass and momentum of each phase are (i = f,s):

$$\frac{{\partial \alpha_{i} \left( {\varvec{x},t} \right)}}{\partial t} + \nabla \cdot \left( {\alpha_{i} \left( {\varvec{x},t} \right)\varvec{u}_{i} \left( {\varvec{x},t} \right)} \right) = 0$$

(14)

$$\rho_{i} \left( {\frac{{\partial \left( {\alpha_{i} \left( {\varvec{x},t} \right)\varvec{u}_{i} \left( {\varvec{x},t} \right)} \right)}}{\partial t} + \nabla \cdot \left( {\alpha_{i} \left( {\varvec{x},t} \right)\varvec{u}_{i} \left( {\varvec{x},t} \right)\varvec{u}_{i}^{T} \left( {\varvec{x},t} \right)} \right)} \right) = \varvec{r}_{i} \left( {\varvec{x},t} \right)$$

(15)

where $\rho_{i}$ is the density of the ith phase, while $\varvec{u}_{i} \left( {\varvec{x},t} \right)$ and $\varvec{r}_{i} \left( {\varvec{x},t} \right)$ are, respectively, the velocity and the resultant of the external forces referring to the ith phase, at position $\varvec{x}$ and time $t$. Focusing on the solid phase, the resultant force is given by the divergence of the stress tensor, the gravitational body force and the interaction force with the fluid phase $\varvec{f}\left( {\varvec{x},t} \right)$:

$$\varvec{r}_{\text{s}} \left( {\varvec{x},t} \right) = \nabla \cdot \varvec{T}_{\text{s}} \left( {\varvec{x},t} \right) + \rho_{\text{s}} \alpha_{\text{s}} \left( {\varvec{x},t} \right)\varvec{g} + \varvec{f}\left( {\varvec{x},t} \right)$$

(16)

Differently from Zitti et al. (2016), here we will not neglect the stress tensor, since the solid phase is not always loosely packed. Furthermore, the interaction force is composed of two contributions: the pressure force and the drag force on the solid phase, i.e.

$$\varvec{f}\left( {\varvec{x},t} \right) = - \rho_{\text{f}} \sigma \left( {\varvec{x},t} \right)\alpha_{\text{s}} \left( {\varvec{x},t} \right)\varvec{g} + \varvec{ }\sigma \left( {\varvec{x},t} \right)\delta \left( {\varvec{x},t} \right)\alpha_{\text{s}} \left( {\varvec{x},t} \right)\left( {\varvec{u}_{\text{f}} \left( {\varvec{x},t} \right) - \varvec{u}_{\text{s}} \left( {\varvec{x},t} \right)} \right)$$

(17)

where $\sigma \left( {\varvec{x},t} \right)$ is the fraction of the solid phase interacting with the fluid phase (i.e. submerged) and $\delta \left( {\varvec{x},t} \right)$ is a positive function, representing the drag distribution, defined as:

$$\delta \left( {\varvec{x},t} \right) = \frac{3}{4}c_{\text{D}} \left( {\varvec{x},t} \right)\frac{{\rho_{\text{s}} }}{{d_{\text{s}} }}\left| {\varvec{u}_{\text{f}} \left( {\varvec{x},t} \right) - \varvec{u}_{\text{s}} \left( {\varvec{x},t} \right)} \right|f_{\text{v}} \left( {\varvec{x},t} \right)$$

(18)

where $c_{\text{D}} \left( {\varvec{x},t} \right)$ is the drag coefficient, $d_{\text{s}}$ is the grain diameter and $f_{\text{v}} \left( {\varvec{x},t} \right)$ is the voidage function (Di Felice 1994). Function (18) provides the amount of drag force associated with the fraction $\sigma \left( {\varvec{x},t} \right)$ of the solid volume $\alpha_{\text{s}} \left( {\varvec{x},t} \right)$ interacting with the fluid, related to the velocity gradient $\left| {\varvec{u}_{\text{f}} \left( {\varvec{x},t} \right) - \varvec{u}_{\text{s}} \left( {\varvec{x},t} \right)} \right|$. Equations (14) to (18) provide the theoretical foundations for developing the equation that describes the mean motion of the impacting mass.

We define a control volume representative of the experimental model, i.e. a two-dimensional volume V, illustrated in Fig. 6, that includes the fluid and gas phases, while the solid phase, composed of grains with mean diameter $d_{\text{s}}$, enters across a section $s\left( t \right)$ of the edge AB, with average velocity $\varvec{u}_{0} = u_{0} \left[ {\begin{array}{l} {\cos \theta } \\ {\sin \theta } \\ \end{array} } \right]$. The entering mass is characterized by the bulk density $\rho_{0} = \phi_{0} \rho_{\text{s}}$, where $\phi_{0}$ is the solid fraction of the granular bulk entering the control volume. Volume Eqs. (14) and (15) are integrated over the control volume V.

Integration of the mass conservation Eq. (14) for the solid phase gives:

$$\frac{{{\text{d}}V_{\text{s}} \left( t \right)}}{{{\text{d}}t}} = \phi_{0} u_{0} \cos \theta s\left( t \right)b$$

(19)

where $V_{\text{s}} \left( t \right)$ is the volume of the solid phase and $b$ is the width of the control volume. Integration of the momentum Eq. (15) leads to:

$$\beta_{1} \rho_{s} \frac{{{\text{d}}\left( {V_{\text{s}} \left( t \right)\bar{\varvec{u}}_{\text{s}} \left( t \right)} \right)}}{{{\text{d}}t}} - \rho_{\text{s}} \phi_{0} u_{0} \cos \theta \varvec{u}_{0} s\left( t \right)b = \varvec{R}_{\text{s}} \left( t \right)$$

(20)

where $\bar{\varvec{u}}_{\text{s}} \left( t \right) = \frac{1}{{V_{\text{s}} \left( t \right)}}\int\limits_{{V_{\text{s}} \left( t \right)}} {\varvec{u}_{\text{s}} \left( {\varvec{x},t} \right){\text{d}}V}$ is the mean velocity of the solid phase (hereafter the overbar stands for the volume average); the coefficient $\beta_{1}$, taken as constant for simplicity sake, is such that $\overline{{\alpha_{\text{s}} \left( {\varvec{x},t} \right)\varvec{u}_{\text{s}} \left( {\varvec{x},t} \right)}} = \beta_{1} \bar{\alpha }_{\text{s}} \left( {\varvec{x},t} \right)\bar{\varvec{u}}_{\text{s}} \left( {\varvec{x},t} \right)$. $\varvec{R}_{\text{s}} \left( t \right)$ is the resultant of the external forces on the whole solid phase, obtained integrating Eq. (16).

The resultant of the external forces $\varvec{R}_{\text{s}} \left( t \right)$ must be carefully analysed because it is made of three different contributions:

• (1) The divergence of the solid stress tensor

  $$\varvec{t}_{\text{s}} \left( t \right) = \int\limits_{V} \nabla \cdot \left( {\varvec{T}_{\text{s}} \left( {x,t} \right)} \right){\text{d}}V$$

  (21)

  which represents the particle–particle interaction force;

• (2) The weight of the solid phase

  $$\int\limits_{V} {\rho_{\text{s}} \alpha_{\text{s}} \left( {x,t} \right)g {\text{d}}V} = \rho_{\text{s}} V_{\text{s}} \left( t \right)g;$$

  (22)

• (3) The particle–water interaction force, composed of the pressure and drag terms (see Eq. 17). The pressure term corresponds to the buoyancy force integrated over the volume

  $$- \int\limits_{V} {\rho_{\text{f}} \sigma \left( {\varvec{x},t} \right)\alpha_{\text{s}} \left( {\varvec{x},t} \right)\varvec{g}{\text{d}}V} = - \rho_{\text{f}} \sigma \left( t \right)V_{\text{s}} \left( t \right)\varvec{g}$$

  (23)

  where $\sigma \left( t \right)$ identifies the submerged fraction of the entire solid volume. The drag term becomes

  $$\int\limits_{V} {\sigma \left( {\varvec{x},t} \right)\delta \left( {\varvec{x},t} \right)\alpha_{\text{s}} \left( {\varvec{x},t} \right)\left( {\varvec{u}_{\text{f}} \left( {\varvec{x},t} \right) - \varvec{u}_{\text{s}} \left( {\varvec{x},t} \right)} \right){\text{d}}V} = \beta_{2} \sigma \left( t \right)V_{\text{s}} \left( t \right)\delta \left( t \right)\left( {\bar{\varvec{u}}_{\text{f}} \left( t \right) - \bar{\varvec{u}}_{\text{s}} \left( t \right)} \right)$$

  (24)

  where $\bar{\varvec{u}}_{f} \left( t \right)$ is the average velocity of the fluid phase; $\delta \left( t \right)$ is a volume-averaged drag function, and the coefficient $\beta_{2}$, taken as constant for simplicity sake, is such that $\overline{{\sigma \left( {\varvec{x},t} \right)\delta \left( {\varvec{x},t} \right)\alpha_{\text{s}} \left( {\varvec{x},t} \right)\left( {\varvec{u}_{\text{f}} \left( {\varvec{x},t} \right) - \varvec{u}_{\text{s}} \left( {\varvec{x},t} \right)} \right)}} = \beta_{2} \sigma \left( t \right)\bar{\alpha }_{\text{s}} \left( {\varvec{x},t} \right)\delta \left( t \right)\left( {\bar{\varvec{u}}_{\text{f}} \left( t \right) - \bar{\varvec{u}}_{\text{s}} \left( t \right)} \right)$. Note that the functions $\sigma \left( t \right)$ and $\delta \left( t \right)$ have a physical meaning different from, respectively, $\sigma \left( {\varvec{x},t} \right)$ and $\delta \left( {\varvec{x},t} \right)$. The latter ones represent the local forces of particle–fluid interaction expressed in terms of pressure and drag, while $\sigma \left( t \right)$ and $\delta \left( t \right)$ are global functions representing, respectively, the fraction of submerged mass and an averaged drag coefficient.

Using Eqs. (21) to (24), we obtain:

$$\varvec{R}_{\text{s}} \left( t \right) = \varvec{t}_{\text{s}} \left( t \right) + \left( {\rho_{\text{s}} - \rho_{\text{f}} \sigma \left( t \right)} \right)V_{\text{s}} \left( t \right)\varvec{g} + \beta_{2} \sigma \left( t \right)V_{\text{s}} \left( t \right)\delta \left( t \right)\left( {\bar{\varvec{u}}_{\text{f}} \left( t \right) - \bar{\varvec{u}}_{\text{s}} \left( t \right)} \right)$$

(25)

In this manner the resultant that appears in Eq. (20) is formulated, as much as possible, as function of the variables that already appear in Eq. (20).

The integrated conservation principles (19) and (20) together with expression (25) give a rather complicated form of the equation for the mean motion of the solid mass $\bar{\varvec{u}}_{\text{s}} \left( t \right)$:

$$\begin{aligned} \beta_{1} \rho_{\text{s}} \frac{{{\text{d}}V_{\text{s}} \left( t \right)}}{{{\text{d}}t}}\bar{\varvec{u}}_{\text{s}} \left( t \right) + \beta_{1} \rho_{\text{s}} V_{\text{s}} \left( t \right)\frac{{{\text{d}}\bar{\varvec{u}}_{\text{s}} \left( t \right)}}{{{\text{d}}t}} - \rho_{\text{s}} \frac{{{\text{d}}V_{\text{s}} \left( t \right)}}{{{\text{d}}t}}\varvec{u}_{0} \hfill \\ \quad = \varvec{t}_{\text{s}} \left( t \right) + \left( {\rho_{\text{s}} - \rho_{\text{f}} \sigma \left( t \right)} \right)V_{\text{s}} \left( t \right)\varvec{g} + \beta_{2} \sigma \left( t \right)V_{\text{s}} \left( t \right)\delta \left( t \right)\left( {\bar{\varvec{u}}_{\text{f}} \left( t \right) - \bar{\varvec{u}}_{\text{s}} \left( t \right)} \right) \hfill \\ \end{aligned}$$

(26)

In order to simplify it, we exploit the fact that the first and third terms on the right-hand side are both dissipative terms; hence, we collect them and assume that the resultant is in opposition to the solid average velocity:

$$\varvec{t}_{\text{s}} \left( t \right) + \beta_{2} \sigma \left( t \right)V_{\text{s}} \left( t \right)\delta \left( t \right)\left( {\bar{\varvec{u}}_{\text{f}} \left( t \right) - \bar{\varvec{u}}_{\text{s}} \left( t \right)} \right) = - \varvec{D}\left( t \right)\bar{\varvec{u}}_{\text{s}} \left( t \right)$$

(27)

$\varvec{D}\left( t \right)$ being a diagonal positive tensor. Substituting Eq. (27) into Eq. (26), after some algebra we get:

$$\begin{aligned} \beta_{1} \rho_{\text{s}} V_{\text{s}} \left( t \right)\frac{{{\text{d}}\bar{\varvec{u}}_{\text{s}} \left( t \right)}}{{{\text{d}}t}} + \left[ {\beta_{1} \rho_{\text{s}} \frac{{{\text{d}}V_{\text{s}} \left( t \right)}}{{{\text{d}}t}}\varvec{I} + \varvec{D}\left( t \right)} \right]\bar{\varvec{u}}_{\text{s}} \left( t \right) \hfill \\ \quad = \rho_{\text{s}} \frac{{{\text{d}}V_{\text{s}} \left( t \right)}}{{{\text{d}}t}}\varvec{u}_{0} + \left( {\rho_{\text{s}} - \rho_{\text{f}} \sigma \left( t \right)} \right)V_{\text{s}} \left( t \right)\varvec{g} \hfill \\ \end{aligned}$$

(28)

This is a differential equation for the mean motion of the solid phase $\bar{\varvec{u}}_{\text{s}} \left( t \right)$. We recognize an inertial term, involving the average acceleration (first term on the left-hand side), a dissipative term, involving the average velocity (second term on the left-hand side) and a non-homogeneous contribution on the right-hand side. The dissipative term is reduced to a unique term defining the dissipative function $\varvec{\varphi }\left( t \right)$:

$$\varvec{\varphi }\left( t \right) = \beta_{1} \rho_{\text{s}} \frac{{{\text{d}}V_{\text{s}} \left( t \right)}}{{{\text{d}}t}}\varvec{I} + \varvec{D}\left( t \right)$$

(29)

Thus, we achieve the simple form:

$$\beta_{1} \rho_{\text{s}} V_{\text{s}} \left( t \right)\frac{{{\text{d}}\bar{\varvec{u}}_{\text{s}} \left( t \right)}}{{{\text{d}}t}} + \varvec{\varphi }\left( t \right)\bar{\varvec{u}}_{\text{s}} \left( t \right) = \rho_{\text{s}} \frac{{{\text{d}}V_{\text{s}} \left( t \right)}}{{{\text{d}}t}}\varvec{u}_{0} + \left( {\rho_{\text{s}} - \rho_{\text{f}} \sigma \left( t \right)} \right)V_{\text{s}} \left( t \right)\varvec{g}$$

(30)

Its dimensionless form can be obtained by dividing it by $\rho_{s} V_{s} \left( t \right)g$, after substituting the terms involving $V_{s} \left( t \right)$ using Eq. (19) and its solution:

$$\beta_{1} \frac{{{\text{d}}\bar{\varvec{u}}_{\text{s}}^{*} \left( {t^{*} } \right)}}{{{\text{d}}t^{*} }} + \varvec{\varphi }^{\varvec{*}} \left( {t^{*} } \right)\bar{\varvec{u}}_{\text{s}}^{*} \left( {t^{*} } \right) = \frac{{s^{*} \left( {t^{*} } \right)}}{{\mathop \smallint \nolimits_{0}^{{t^{*} }} s^{*} \left( \tau \right){\text{d}}\tau }}\varvec{u}_{0}^{*} + \left( {1 - \frac{{\rho_{\text{f}} }}{{\rho_{\text{s}} }}\sigma \left( {t^{*} } \right)} \right)\varvec{n}_{\varvec{y}}$$

(31)

where $t^{*} = t\sqrt {g/h}$ is the dimensionless time, $s^{*} \left( {t^{*} } \right) = s\left( t \right)/h$ is the dimensionless avalanche thickness, $\bar{\varvec{u}}_{\text{s}}^{*} \left( {t^{*} } \right) = \bar{\varvec{u}}_{\text{s}} \left( t \right)/\sqrt {gh}$ and $\varvec{u}_{0}^{*} = \varvec{u}_{0} /\sqrt {gh}$ are dimensionless velocities, $\varvec{\varphi }^{\varvec{*}} \left( {t^{*} } \right) = \varvec{\varphi }\left( t \right)/\rho_{\text{s}} V_{\text{s}} \left( t \right)\sqrt {g/h}$ is the dimensionless dissipative function, and $\varvec{n}_{\varvec{y}} = \left[ {\begin{array}{l} 0 \\ 1 \\ \end{array} } \right]$ is the direction of the gravitational field. Equation (31) has now the form of the law of motion of a single body with varying coefficients. In fact, the mean velocity $\bar{\varvec{u}}_{s}^{*} \left( {t^{*} } \right)$ multiplies the dissipative function $\varvec{\varphi }^{\varvec{*}} \left( t \right)$, the non-homogeneous term (right-hand side) is function of both the relative avalanche thickness $s^{*} \left( {t^{*} } \right)$ and the dimensionless fraction of submerged mass $\sigma \left( {t^{*} } \right)$.

Solvability of Eq. (31) depends on the proper definition of the time-dependent functions, which is verified in the following. The physical meaning of the relative avalanche thickness $s^{*} \left( {t^{*} } \right)$ and of the dimensionless fraction of submerged mass $\sigma \left( {t^{*} } \right)$ ensure they are positive definite functions, while the dissipative function $\varvec{\varphi }^{\varvec{*}} \left( t \right)$ requires more attention, since it includes several components. The explicit form of $\varvec{\varphi }^{\varvec{*}} \left( t \right)$ is:

$$\varvec{\varphi }^{\varvec{*}} \left( t \right) = \beta_{1} \frac{{s^{*} \left( {t^{*} } \right)}}{{\mathop \smallint \nolimits_{0}^{{t^{*} }} s^{*} \left( \tau \right){\text{d}}\tau }}\varvec{I} + \frac{1}{{\rho_{\text{s}} \phi_{0} u_{0} \cos \theta \mathop \smallint \nolimits_{0}^{t} s\left( \tau \right){\text{d}}\tau b}}\varvec{D}\left( t \right)$$

(32)

$\varvec{\varphi }^{\varvec{*}} \left( t \right)$ is well defined, being $\varvec{I}$ and $\varvec{D}\left( t \right)$ diagonal positive tensors, if the denominator of its components is not zero. Since they include $\mathop \int \limits_{0}^{{t^{*} }} s^{*} \left( \tau \right){\text{d}}\tau$, the divergence of the function to infinity occurs when the solid phase is not in the control volume yet. Such condition is easily avoided by assuming the initial time ($t = 0$) to correspond with the impact of the first particle on the water surface, as assumed in Sect. 3, thus ensuring the convergence of Eq. (31) almost everywhere.

Appendix 2: Graphical approach for the analysis of the vertical mean motion of the impacting mass

The analysis of the vertical mean motion of the solid phase is based on use of the $\left( {\sigma \left( {t^{*} } \right),\psi \left( {t^{*} } \right)} \right)$-plane (see Sect. 3), whose reading is here explained in detail. In the $\left( {\sigma \left( {t^{*} } \right),\psi \left( {t^{*} } \right)} \right)$-plane the motion is represented in terms of both the submerged fraction of the solid phase and dissipation. By the definition of $\sigma \left( {t^{*} } \right)$, the representation of the motion of a floating mass is bounded vertically between 0 and 1 (shaded area in Fig. 8a). Further, being $\varphi_{y}^{*} \left( {t^{*} } \right)$ positive, $\psi \left( {t^{*} } \right)$ has the same sign of the vertical velocity $\bar{u}_{{{\text{s}}y}}^{*} \left( {t^{*} } \right)$. Therefore, the right-hand side of the plane corresponds to positive (downward) velocities, the left-hand side corresponds to negative (upward) velocities, and the condition of zero velocity coincides with the vertical line $\psi \left( {t^{*} } \right) = 0$. The condition of zero acceleration is given by the line $\sigma \left( {t^{*} } \right) = \frac{{\rho_{\text{s}} }}{{\rho_{\text{f}} }}\left( {1 - \psi \left( {t^{*} } \right)} \right)$, which divides the plane into two parts: the upper part of positive (downward) acceleration and the lower part of negative (upward) acceleration. In the specific case of snow avalanches, the ratio is $\frac{{\rho_{\text{s}} }}{{\rho_{\text{f}} }}\sim0.9$ and the sector of the plane with upward acceleration and upward velocity is small.

The description in the $\left( {\sigma \left( {t^{*} } \right),\psi \left( {t^{*} } \right)} \right)$-plane of the mean vertical motion of the avalanche is subdivided into several steps:

(1) At impact ($t^{*} = 0$), the fraction of the submerged mass in the control volume $V$ is zero ($\sigma (0) = 0$) and $\varphi_{y}^{*} \left( {t^{*} } \right)$ does not converge yet (see Appendix 1); hence, the representation of the initial stages of motion is close to the asymptotic condition. Since the initial velocity is downward (i.e. positive), $\psi \left( 0 \right)$ is positive, and the asymptotic condition that represents the initial condition of the motion is $\psi \left( 0 \right) \to + \infty$.

(2) Such initial state of motion, named O in Fig. 8, is located in a region of the plane where the velocity is downward, this increasing the value of $\sigma \left( {t^{*} } \right)$, and where the acceleration is upward, this reducing the downward velocity itself and, subsequently, $\psi \left( {t^{*} } \right)$. Hence, the trajectory moves left-upward in the plane (curve OA in Fig. 8a).

(3) In the specific case of snow avalanches the condition of zero velocity is reached before the condition of zero acceleration (point A in Fig. 8a), as proven by experiments. In fact, this condition represents a maximum for the mean vertical position, which occurs in all experiments reported in the bottom panel of Fig. 3. In some cases, the mass could be totally submerged for a while and point A becomes a segment lying on the upper bound.

(4) Since point A is in a region of upward acceleration, the velocity changes its sign and the motion becomes upwards, this gradually reducing the fraction of the submerged mass. Then, the trajectory moves left-downward in the plane, until it reaches zero acceleration (point B in Fig. 8a).

(5) Being B in a region of the plane where the velocity is upward, $\sigma \left( {t^{*} } \right)$ decreases and the point on the trajectory is pushed down in a region where the acceleration is downward. Thus, the value of the upward velocity decreases and the point representing the motion moves towards the condition of zero velocity (point C).

(6) Point C is placed in a region of downward acceleration, then the downward velocity increases, together with $\sigma \left( {t^{*} } \right)$, until the trajectory reaches the condition of zero acceleration (Point D).

(7) Point D is characterized by downward velocity, then the trajectory continues upwards in a region where the acceleration is upwards, this decreasing the downward velocity until it vanishes, like in point A. Then, the motion is periodical with smaller amplitudes characterizing each loop, i.e. it is a damped motion.

(8) The oscillations approach the quiescent condition, represented in Fig. 8a through point S, which satisfies both conditions of zero velocity and zero acceleration. For the density ratio typical of the case at hand, the distance between B and S is small and we assume that only one oscillation is needed to attain the quiescent condition.

The representation in the $\left( {\sigma \left( {t^{*} } \right),\psi \left( {t^{*} } \right)} \right)$-plane supports the qualitative description of the dimensionless mean depth in time $y_{\text{C}}^{*} \left( {t^{*} } \right)$: the depth function derived from the representation in Fig. 8a is reported in Fig. 8b. In fact, the position of each point representing the motion illustrates if the depth is increasing or decreasing (downward or upward velocity regions) and if the function is convex or concave (downward or upward acceleration regions). In particular, the points with zero velocity (A and C) correspond to maximum and minimum depths, respectively, and the points with zero acceleration (B and D) are inflection points for the function. In more details: lines OA and DS of Fig. 8a are in a region of upward velocity, that implies an increasing function in Fig. 8b, while ABC is decreasing in Fig. 8b because in Fig. 8a the curve ABC is placed in a region of downward velocity. Further, curve OAB of Fig. 8a is in a region of upward acceleration, this implying the related function of motion to be concave in Fig. 8b and A to be a maximum. Instead, line BCD is in a region of downward acceleration, then the correspondent function is convex in Fig. 8b and B is a local minimum. Since only one oscillation is assumed to occur, the remaining concave curve CS leads to the quiescent condition of zero motion (constant depth in Fig. 8b). In conclusion, the information given by the representation of the motion in the $\left( {\sigma \left( {t^{*} } \right),\psi \left( {t^{*} } \right)} \right)$-plane has produced a function for the mean depth of the solid phase that has the typical behaviour of a solution of a damped harmonic oscillator.