Abstract
The BG model (a model for predicting 3D beach changes based on the Bagnold’s concept) was introduced, and the fundamental aspects of the model were explained. The BG model is based on the concepts such as (1) the contour line becomes orthogonal to the wave direction at any point at the final stage, (2) similarly, the local beach slope coincides with the equilibrium slope at any point, and (3) a restoring force is generated in response to the deviation from the statically stable condition, and sand transport occurs owing to this restoring force. The same concept has been employed in the contour-line-change model and N-line model. In these studies, the movement of certain contour lines was traced, but in the BG model, 3D beach changes were directly calculated.
Keywords
- BG model
- derivation
- physical meaning
1. Introduction
The BG model is based on the concepts such as (1) the contour line is orthogonal to the wave direction at any point at the final stage, (2) similarly, the local beach slope coincides with the equilibrium slope at any point, and (3) a restoring force is generated in response to the deviation from the statically stable condition, and sand transport occurs owing to this restoring force. The same concept has been employed not only in the contour-line-change model [1] but also in the N-line model [2, 3, 4, 5, 6, 7]. In these studies, the movement of certain contour lines was traced, but in the BG model, the depth change on the 2D horizontal grids was directly calculated. Falqués et al. [8, 9] developed a medium- to long-term model for beach morphodynamics named Q2D-morfo. In their model, similar expressions regarding crossshore and longshore sand transport equations as the BG model were employed. In particular, they used the beach slope measured on a real coast as the equilibrium slope, similar to the BG model, and the prediction of beach changes bounded by two groynes was carried out, but the prediction period was as short as 35 days [8]. van den Berg et al. [10] predicted the development of a sand wave associated with large-scale beach nourishment due to shoreline instability under the oblique wave incidence at a large angle using the model proposed by Falqués et al. [8]. Larson et al. [11] proposed the crossshore sand transport equation in the swash zone using the concept of the equilibrium slope and predicted the foreshore evolution. Similarly, Larson and Wamsley [12] proposed crossshore and longshore sand transport formulae in the swash zone, and they used the similar equations as the BG model. Their equations were also employed in the 3D beach change model in the swash zone by Nam et al. [13].
2. Derivation of the BG model
In the derivation of the sand transport equation of the BG model, we referred Bagnold [14] and the previous studies after Bagnold (Inman and Bagnold [15], Bowen [16], Bailard and Inman [17], and Bailard [18]). Bagnold [14] derived the sand transport equation for a unidirectional steady flow with an explicit expression of the seabed slope by applying the energetics approach. Inman and Bagnold [15] assumed that sand transport in a wave field is the sum of the components caused by shoreward flow during the motion of incoming waves and those caused by seaward flow during the motion of outgoing waves and defined the slope satisfying zero net onshore or offshore sediment transport as the equilibrium slope. Their equilibrium slope is the slope when upslope effect due to the asymmetry in action of incoming and outgoing waves and downslope effect due to the gravity balance each other.
Regarding the sand transport equation under waves, Bowen [16], Bailard and Inman [17], and Bailard [18] formulated the instantaneous sand transport flux on the basis of the sand transport equation for a unidirectional steady flow by Bagnold [14], assuming that the wave dissipation rate is proportional to the third power of the instantaneous velocity. Then, the net sand transport formula was derived by integrating the instantaneous sand transport flux over one wave period. Furthermore, they derived the equilibrium slope equation using the wave velocity parameters. Out of these studies, the sand transport flux formula by Bailard and Inman [17] considers both bed load and suspended load, and this formula has been extensively used in the models for predicting beach changes. Kabiling and Sato [19] calculated the wave and nearshore current field using the Boussinesq equation and predicted 3D beach changes using the Bailard formula. Long and Kirby [20] also carried out the numerical simulation of beach changes using the Bailard formula and Boussinesq equation. However, the application of their model to the long-term prediction of the topographic changes in an extensive calculation domain is limited because the recurrent calculations in solving the time-dependent equation of the wave field are time consuming. On a real coast, a longitudinal profile maintains its stable form as a whole, as a result of the wave action for a long period of time, apart from the short-period seasonal variation of the beach, suggesting the existence of an equilibrium slope on a real coast. In contrast, in their studies, the beach slope does not necessarily agree with the equilibrium slope after long-term prediction, even though an equilibrium slope exists, and it is difficult to explain the phenomena really observed on a coast.
In this study, we return to the starting point of Bagnold’s basic study, and simple sand transport equations are derived. Then, a model for predicting 3D beach changes by applying the concept of the equilibrium slope introduced by Inman and Bagnold [15] and the energetics approach of Bagnold [14] is developed [21].
Figure 1 shows the definitions of the variables. Consider Cartesian coordinates (x, y) and the seabed elevation Z (x, y, t) with reference to the still water level as a variable to be solved, where t is the time. Assume that waves are obliquely incident on a coast with a slope of tan β. n and s are the local coordinates taken along the directions normal (shoreward) and parallel to the contour lines, respectively. The n-axis makes an angle of θn measured counterclockwise from the x-axis,

Figure 1.
Setup of the coordinate system and definition of variables.
The components of the sand transport vector
When the gradient vector of Z is defined as Eq. (8),
The fluid motion due to waves near the sea bottom becomes oscillatory, and a sand particle moves back and forth in the crossshore direction. Sand transport in a wave field is assumed to be the sum of the components caused by shoreward flow during the motion of incoming waves and those caused by seaward flow during the motion of outgoing waves, as suggested in [15], and the sand transport equation for a unidirectional steady flow introduced by Bagnold [14] can be applied to each component.
Assuming that tan β is infinitesimal, the flow makes a sand particle move in the direction of the flow, and gravity causes downslope action; the sand transport flux of a unidirectional flow is expressed by Eq. (16) as a linear approximation in terms of tan β [15, 16, 17, 18].
Here, the subscript u denotes the unidirectional flow,
The net sand transport flux due to waves,
Here, the subscripts + and – denote the values corresponding to incoming and outgoing waves, respectively, and
Here,
In this study, we used the seabed slope measured on real coasts as the equilibrium slope instead of using the formulated results of the equilibrium slope. The measured slope is assumed to be given a priori because the real seabed topography includes every effect of past events, and it has a stable form, except for seasonal short-period variations, in the long term.
Applying the energetics approach [14] and assuming that the coefficient A in Eq. (23) is proportional to the wave energy dissipation rate Φ, the total longshore sand transport rate is obtained by integrating Eq. (23) over the depth. This has the same form as the CERC-type formula given by Komar and Inman [27], and the coefficient A is determined as Eq. (24) from the equivalence of both results as mentioned later. Finally, the fundamental equation of sand transport flux due to waves is given by Eq. (26).
Here, K1 is the coefficient of longshore sand transport, Φ is the wave energy dissipation rate per unit time and unit seabed area, C0 is the coefficient through which the sand transport rate expressed in terms of the immersed weight is related to the volumetric sand transport rate, ρs and ρ are the sand and water densities, respectively, g is the acceleration of gravity, and p is the porosity of the sediment.
3. Physical meaning of the sand transport equation of the BG model
3.1 Statically stable condition
When we set
This equation demonstrates that the directions of the vectors on both sides of Eq. (27) and their absolute values are equivalent. When tan β and
Finally, the conditions required for the formation of a statically stable beach are (1) the contour line is orthogonal to the wave direction at any point and (2) the local beach slope coincides with the equilibrium slope at any point. This concept was also employed in the model for predicting a statically stable beach [28]. According to Eq. (26), a restoring force is generated in response to the deviation from the statically stable condition, and sand transport occurs owing to this restoring force.
3.2 Topographic changes
Topographic changes can be determined from the mass conservation equation.
When the sand transport fluxes in Eq. (26) are expressed by the components in (x, y) coordinates and are substituted into Eq. (29), the following two-dimensional diffusion equation is obtained, assuming that the coefficient A, equilibrium slope, and wave direction are constant:
The left term and the terms in the parenthesis on the right represent the rate of topographic changes and the spatial curvature of the topography, respectively. In other words, beach changes cause the smoothing of an uneven topography, and in a closed system of sand transport, a statically stable beach is obtained such that the direction of the contours at any point becomes orthogonal to the wave direction, and the local slope is equivalent to the equilibrium slope. These characteristics are the same as those of the contour-line change model [1].
3.3 Dynamically stable beach
In addition to the formation of a statically stable beach, a stable beach can also be dynamically stable, which occurs when the divergence of the sand transport flux in Eq. (30) becomes 0. The dynamically stable beach topography satisfies the Laplace equation, and the relationship between the dynamically stable topography and the sand transport flux has an analogy with the two-dimensional potential flow in fluid dynamics [29].
3.4 Crossshore sand transport
Using Eqs. (6) and (26), the crossshore component of sand transport qn is obtained as
Here, n is the coordinate in the crossshore direction and
Eq. (32) has the characteristics that crossshore sand transport diminishes when waves are incident from the direction normal to the shoreline, and the local slope is equal to the equilibrium slope

Figure 2.
Stabilization mechanism of the beach profile based on the sand movement during the one wave period.
Figure 2(a) shows the stabilization mechanism of a beach profile based on sand movement during one wave period, a sand particle moves from point 1 to point 2 during incoming waves and from point 2 to point 3 during outgoing waves, and it returns to the same position after one wave period. The net movement of the sand particle is zero, resulting in the formation of a stable beach profile. The seabed slope tan β under this condition is equivalent to the equilibrium slope tan βc. Figure 2(b) shows the movement of a sand particle in the case that the local slope is larger than the equilibrium slope tan βc. Because of the increase in the effect of gravity, the sand particle moves seaward as from point 1 to point 2 and then to point 3, resulting in net seaward sand transport and the beach attains a stable slope. To the contrary, when the local slope is gentler than the equilibrium slope, the sand particle is transported landward because of the decrease in the effect of gravity, as schematically shown in Figure 2(c). In Figure 2, an extremely simple condition that a sand particle sets on the slope is assumed as an imaginary case to enhance the understanding of the physical meaning of sand movement. In fact, a sand body to be transported due to waves was conceptually regarded as “a sand particle,” instead of the method of tracking one sand particle.
Figure 3 shows a summary of the movement of a sand particle. Crossshore sand transport is zero when the local seabed slope is equivalent to the equilibrium slope, similar to the stabilization mechanism of the longitudinal profile described by Serizawa et al. [1]. Offshore (shoreward) sand transport occurs when the local slope is larger (smaller) than the equilibrium slope.

Figure 3.
Equilibrium between the gravity effect and the wave action.
When waves are obliquely incident to the shoreline, the equilibrium slope tan βc′ can be obtained as Eq. (33) after setting qn = 0 in Eq. (31). Here, the breaker angle αb is substituted into α as an approximation.
Although tan βc′ is smaller than tan βc by a factor of cos αb, the approximation of
3.5 Longshore sand transport
Using Eqs. (7) and (26), the longshore component of sand transport q can be expressed as
Here, s and

Figure 4.
Formation of stable contour lines.
When the total sand transport Qs is calculated by integrating Eq. (34) in the crossshore direction, it coincides with the CERC-type formula [1], as in Eq. (35), under the assumptions that the integral of Φ in the crossshore direction is equal to the energy flux per unit length of the coastline at the breaking point and that α is approximately given by the breaker angle αb.
Here, (ECg)b and αb are the wave energy flux at the breaking point and the breaker angle, respectively. This CERC-type formula [30] has been employed in the one-line model in practical engineering [31, 32]. Thus, the integral of the longshore component of sand transport in the BG model in the crossshore direction is equivalent to the total longshore sand transport, so both bedload and suspended sediment transport are automatically included in the total sand transport. The fundamental equations of the BG model are compatible with the CERC total sand transport formula, which has often been employed in the prediction of beach changes in the practical applications.