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The One-Line Concept

Figure 1. Shoreline change and associated bottom profiles.

Figure 2. Definition sketch for shoreline change calculation.
A second geometrical-type assumption is that sand is transported alongshore between two well-defined limiting elevations on the profile. The shoreward limit is located at the top of the active berm (DB in Fig. 1), and the seaward limit is located where no significant depth changes occur, the so-called depth of closure (DC in Fig. 1). Restriction of profile movement between these two limits provides the simplest way to specify the perimeter of a beach cross-sectional area by which changes in volume, leading to shoreline change, can be computed. Thus, it is assumed that the beach profile translates seaward or shoreward along a section of shore without changing shape when a net amount of sand enters or leaves the section during a time interval Δt. The change in shoreline position is Δy (Fig. 2) , the length of the shoreline segment is Δx, and the profile moves within a vertical extent defined by the height of the berm DB and the depth of closure DC. The change in volume of the section is ΔV = ΔxΔy(DB + DC), and it is determined by the net rate of sand entering and leaving the section from its two sides. The volume change results if there is a difference in the longshore sand transport rates Q at the lateral sides of the cells. This net volume change is ΔV = ΔQΔt. Rearrangement of terms yields the governing equation for the rate of change of shoreline position:

 \frac{\Delta y}{\Delta t} + \frac{1}{(D_B + D_C)} \frac{\Delta Q}{\Delta x} = 0  \qquad (1)


 Q = (H^2 C_g)_b (a_1 \sin {2 \alpha_{bs}} + a_2 \cos {\alpha_{bs}} + \frac {\part H}{\part x})    \qquad (2)

in which H = wave height, Cg = wave group speed given by linear wave theory, b = subscript denoting wave breaking condition, and αbs = angle of breaking waves to the local shoreline.

The nondimensional parameters a1 and a1 are given by

 a_1 = \frac {K_1}{16 (s - 1)(1 - p)W}     \qquad (3a)


 a_1 = \frac {K_2}{8 (s - 1)(1 - p)W \tan \beta}     \qquad (3b)

Figure 3. Coupled calculation cells alongshore.
in which K1 and K2 = empirical coefficients, treated as a calibration parameters, s = ρs/ρ, ρs = density of sand (taken to be 2.65 103 kg/m3 for quartz sand), ρ = density of water (1.03 103 kg/m3 for sea water), p = porosity of sand on the bed (taken to be 0.4), tanß = average bottom slope, and W = a numerical factor (W = 1.4165/2) used to convert from significant wave height, the statistical wave height required by GENESIS, to root-mean-square (rms) wave height. Figure 3 shows the representation of several cells along a coastal stretch downdrift of a short groyne.

Representation of Currents

By writing the transport rate in terms of immersed weight, Il (N/s), Eqs. (1) and (2) may be expressed as,

 I_l = (E C_g)_b (K_1 \sin { \alpha_{b}} \cos {\alpha_{b}} - K_2 \frac { \cos {\alpha_{b}}}{ \tan {\beta}} \frac {\part H}{\part x})     \qquad (4)

where E is the wave energy density (J/m2). The classical ‘CERC’-equation for calculating the longshore sediment transport rate is formulated as,


which is identical to the first term in the GENESIS transport relation as given in Eq. (2). Following the formula proposed by Bagnold (1963)[2], Eq. (5) can be modified to explicitly represent a longshore current as,

 I_l = K_3 (E C_g)_b \cos {\alpha_{b}} \frac{v_l}{u_m}    \qquad (6)


 u_m = ( \frac {2 E_b} {\rho h_b} )^{1/2} = ( \frac {\gamma g H_b} {4 * 1.416} )^{1/2}             \qquad (7)

where hb is the breaker depth, γ is the breaker index, and g is the acceleration due to gravity. By using Eq. (6), the longshore sediment transport rate is no longer restricted to being generated exclusively by breaking waves; the longshore current may originate from other mechanisms as discussed further by Komar and Inman (1970)[3] and Kraus et al. (1982)[4], for example, as generated by the tide or the wind.

Thus, by assuming shallow water conditions at the location of wave breaking, the wave group velocity Cgb may be approximated,

 C_{gb} ~ C_b ~ \sqrt {g h_b} \sqrt { g H_b / \gamma}   \qquad (8)

where Cb is the wave breaking celerity. Inserting Eq. (7) into Eq. (6) yields

  I_l = \frac {K_3}{4 * 1.416^2} ( g / \gamma )^{1/2} \rho H_b^{1.5} C_g v_l \alpha_{bs} \qquad (9)

If the longshore current vl comprise of a wave generated current vb alone, it may be calculated as (Lonquet-Higgins 1970[5]; Komar and Inman 1970)


where K4 is an empirical coefficient for which Komar and Inman (1970) suggested K4 = 2.7. By combining Eqs. (5), (6), and (10) we obtain K3 = K1/K4 = 0.28. Thus, with only wave generated currents taken into account, Eqs (6) and (10) together are compatible with the ‘CERC’ equation but are still able to account for currents of different origin.

Along the same lines, Hanson et al. (2001)[6] proposed a generalized version of Eq. (9) which includes the effect of longshore gradients as,

 v_b = K_4 u_m \sin \alpha_b - K_5 \frac {4}{ \gamma g} u_m \frac {\part u_m^2}{\part x} \frac {1}{ \tan \beta}  \qquad (11)


Representation of Offshore Contours

Figure 4. Impact of predefined offshore contour line.

Representation of Variable Transmission Coefficient for Submerged Breakwaters

Figure 5. Parameters in wave transmission predictive formulas.
The transmission coefficient is a leading parameter in controlling beach response to detached breakwaters (Hanson and Kraus 1990)[9]. Previously the GENESIS model only represented a constant transmission coefficient (Kt = Ht / Ho in Fig. 5) for detached breakwaters. To improve the predictive capability of the GENESIS model, several published empirical formulae for the wave transmission coefficient were evaluated. In the revised GENESIS, the user may choose either a constant value of Kt for each structure or allow the model to calculate appropriate values based on time-varying water level and wave height, and structure characteristics. If the variable-Kt option is selected, water level is read from an input file at a specified input time interval. For each structure, the user specifies geometric properties (crest height and width, slopes on seaward and landward sides, and median rock size) and can select between the calculation methods of Ahrens (2001)[10], Seabrook and Hall (1998)[11], and d'Angremond et al. (1996)[12]. The method selected should be based upon structure type and configuration.

Based on the input values describing the structure, water level, and calculated wave properties, a corresponding Kt is calculated for each structure at each time step. The calculated Kt will have a strong influence on the wave field behind and adjacent to the structure as it influences wave transmission and diffraction. Through an iterative procedure for calculating wave breaking, Kt also influences the breaking wave height and direction alongshore, thereby determining the associated shoreline response to the structure (Hanson and Kraus 1989[13], 1990, 1991a[14], 1991b[15]).

Representation of Tombolos


The procedure is illustrated in Fig. 6, a plan view of an idealized beach protected by a DBW at an arbitrary moment in time. A tombolo has developed in Cell i+1 in a previous time step. In the adjacent Cell i, allowing the calculated influx Qi from Cell i-1 to enter, the shoreline would advance beyond the DBW (Fig. 6a), which is not allowed.

Thus, the initially calculated transport rate Q6 must be adjusted to Q6* (Fig. 6b) to cause the shoreline to advance up to the detached breakwater but no further, giving y6* = yt. With the new transport now going out of Cell 5, the shoreline location in this cell will be adjusted from y5 to y5*. In this particular case, only two cells were recalculated. In the general case, the correction may be carried through any number of cells until the criterion that the shoreline may not advance beyond the DBW is not violated.

See also

Models (Theme 6)

Modelling Review (all generic types) V.Magar, J.Sutherland, H.Hanson

How to apply models I. Broker

Process-based Morphological Models – Applications to longer Time Frame I. Broker

Stability Models: Linear and nonlinear A. Falques Serra

Estuary models D. Prandle

Geomorphological time scales and processes K. Pavlopoulos


The main author of this article is Hanson, Hans