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.
- BG model
- physical meaning
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  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 . van den Berg et al.  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. . Larson et al.  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  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. .
2. Derivation of the BG model
In the derivation of the sand transport equation of the BG model, we referred Bagnold  and the previous studies after Bagnold (Inman and Bagnold , Bowen , Bailard and Inman , and Bailard ). Bagnold  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  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 , Bailard and Inman , and Bailard  formulated the instantaneous sand transport flux on the basis of the sand transport equation for a unidirectional steady flow by Bagnold , 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  considers both bed load and suspended load, and this formula has been extensively used in the models for predicting beach changes. Kabiling and Sato  calculated the wave and nearshore current field using the Boussinesq equation and predicted 3D beach changes using the Bailard formula. Long and Kirby  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  and the energetics approach of Bagnold  is developed .
Figure 1 shows the definitions of the variables. Consider Cartesian coordinates (
The components of the sand transport vector are expressed as Eq. (4), the direction and magnitude of which give the direction of sand transport and a volumetric expression for the sand transport rate per unit width normal to the direction of sand transport and per unit time, respectively. In addition, can be expressed as the vector sum of the crossshore and longshore components in each direction of
When the gradient vector of
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 , and the sand transport equation for a unidirectional steady flow introduced by Bagnold  can be applied to each component.
Assuming that tan
Here, the subscript
The net sand transport flux due to waves, , is the sum of the components due to incoming and outgoing waves, as shown in Eq. (17), when the time-averaged sand transport rate in a period involving the action of incoming and outgoing waves is expressed by Eqs. (18) and (19).
Here, the subscripts + and – denote the values corresponding to incoming and outgoing waves, respectively, and and are the unit vectors in the directions of the shoreward and seaward flows of waves, respectively. Modifying Eq. (17) under the assumption that the directions of waves propagating shoreward and seaward are opposite, as given by Eq. (20), defining the slope satisfying zero net onshore or offshore sediment transport when waves are incident from the direction normal to the slope as the equilibrium slope, tan
Here, is the unit vector in the wave direction
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  and assuming that the coefficient
3. Physical meaning of the sand transport equation of the BG model
3.1 Statically stable condition
This equation demonstrates that the directions of the vectors on both sides of Eq. (27) and their absolute values are equivalent. When tan
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 . 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 (
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 .
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 .
3.4 Crossshore sand transport
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 . Shoreward transport is generated when the local slope is smaller than the equilibrium slope and vice versa (Figure 2). This represents the balance between the upslope flow asymmetry and the downslope component of gravity [15, 16, 17, 18].
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
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. . Offshore (shoreward) sand transport occurs when the local slope is larger (smaller) than the equilibrium slope.
When waves are obliquely incident to the shoreline, the equilibrium slope tan
3.5 Longshore sand transport
When the total sand transport