Littoral drift and shoreline modelling

Littoral drift, the alongshore transport of sand, is a ubiquitous phenomenon along sandy coastlines. It occurs when incident waves approach the coast under an angle. This angle decreases by refraction when the waves propagate into shallower water. The incidence angle in the zone where waves are breaking is most relevant for the generation of littoral drift. Littoral drift is associated with the generation of longshore currents by gradients in radiation stress due to wave breaking.

Littoral drift plays a major role in shoreline dynamics because longshore gradients in littoral drift create accretion in zones of convergence and erosion in zones of divergence. Zones of erosion or accretion therefore occur where the orientation of the coastline changes relative to the direction of the incident waves. Modification of littoral drift by structures such as groynes, jetties and breakwaters, is another important cause of the generation of zones of accretion and coastal erosion (see Human causes of coastal erosion).

Measuring littoral drift in field situations is notoriously difficult and generally subject to great uncertainty. Indirect estimates can be obtained by tuning mathematical models to observed shoreline retreat or accretion. These models can then be used to explore the effect of interventions such as beach nourishments. A simple, popular model is the so-called one-line model, based on a simplified representation of the upper shoreface. In this article a generalized version of this model (called GENESIS ) is presented, together with conditions and prescriptions for practical application.

Longshore current

In the case of oblique wave incidence, a longshore current is generated in the surf zone by the cross-shore gradient in the average wave-induced momentum flow. An explanation is given in Shallow-water wave theory#Radiation Stress Components for Oblique Waves. The net momentum flow associated with propagating waves is equivalent to a stress, the so-called radiation stress. This stress decreases along the wave rays when wave energy is dissipated. The stress gradient then generates an acceleration of the water flow in the wave ray direction. By this process, obliquely incident waves generate a longshore current when breaking in the surfzone.

Wave dissipation in the surf zone, which in general is mainly due to wave breaking, is a complicated process for which a fundamental theory is still lacking. In Appendix B a strongly simplified model based on linear shallow-water wave theory is presented that yields an analytical expression of the longshore current [math]v_0[/math] generated by radiation stress,

[math]v_0 = K_v u_m \sin \alpha_b \; , \quad K_v=\dfrac{5 \pi}{8} \dfrac{\tan \beta}{C_D} \, . \qquad (1) [/math]

In this formula (and throughout this article), the subscript [math]_b[/math] denotes the cross-shore location of the wave breaker line (the wave breaking depth contour). Other symbols are: [math]\alpha_b=[/math] incidence angle to the breaker line, [math]u_m \approx \tfrac12 \sqrt{ g \gamma_b H_{b} } =[/math] amplitude of the cross-shore wave orbital velocity, [math]H_b=[/math] wave height at the breaker line, [math]\gamma_b = H_b/h_b =[/math] breaker index, [math]h_b =[/math] depth at the breaker line and [math]C_D=[/math] bottom friction coefficient. Based on field data of longshore currents, Komar and Inman (1970) suggested [math]K_v=2.7[/math]. This value is generally smaller than the estimate Eq. (1).

Littoral drift

Littoral drift results from the wave breaking process and the associated longshore current, and from the amount of sediment stirred up from the seabed. The lack of fundamental theories for these processes compels mathematical models to rely on (semi-)empirical descriptions.

Littoral drift is highly variable over many timescales. The strength varies strongly with wave height [math]H[/math]; field and laboratory data suggest a power relationship with an exponent of 2 to 3. Fluctuations in the incident wave angle [math]\alpha[/math] also have a great influence. Theoretical considerations suggest a dependence on [math]\sin 2 \alpha[/math], consistent with observations. The dependence of [math]Q[/math] on the wave incidence angle [math]\alpha[/math] is explained in Appendix A. See also Shallow-water wave theory#Longshore Currents for more detailed explanations.

Direct observations show that littoral drift is strongest in the wave breaking zone, but strong littoral drift has also been observed in the swash zone. The wave breaking and swash zones shift with wave height and with variations in mean water level due to tide and meteorology.

High variability is not the only difficulty for measuring littoral drift. Direct measurements of littoral drift are especially difficult because most of the sand transport occurs very close to the seabed and under high-wave conditions. Estimates of littoral drift therefore also use indirect methods: tuning the parameters of an empirical transport formula to observed trends in shoreline position and beach level. One such indirect method consists of comparing field data of erosional or accretional trends with simulations using the one-line model discussed below. Formulations of the total net littoral drift [math]Q[/math] (usually expressed in m³/year) can be tested in this way. Such indirect methods have several drawbacks, however: parameters are site-specific and gradients in littoral drift are not the only possible cause of accretion or erosion.

Many empirical formulas for the bulk littoral drift [math]Q[/math] have been proposed in the literature. The uncertainty margins of these empirical formulas are considerable. Predictions from different empirical transport formulas can differ by a factor of two or more, depending on the wave climate, sediment characteristics and coastal morphology. An analysis of factors inducing uncertainty suggested that, for open coasts, 40% to 50% of the total uncertainty was due to the choice of models that extrapolate offshore wave conditions to wave conditions in the surf zone (wave transformation models). Another 30% to 50% of the uncertainty was due to the choice of longshore sediment transport formulas and the interaction of these models with wave transformation methods.

A few formulas of the longshore drift [math]Q \; [m^3/s][/math] that are often used in practice are the empirical formula of Kamphuis (1991)

[math]Q = K_{Kamphuis} \, \Large\frac{H_{sb}^3}{T}\normalsize \Big( \Large\frac{H_{sb}}{L_0}\normalsize \Big)^{-1.25} \Big(\Large\frac{H_{sb}}{d_{50}}\normalsize \Big)^{0.25} \big( \tan \beta \big)^{0.75 } \big(\sin (2 \alpha_b) \big)^{0.6} \, , \quad K_{Kamphuis} = 8.5 \times 10^{-4} \qquad (2) [/math]

and the empirical formula of van Rijn (2014)

[math]Q = K_{vanRijn} \, g^{0.5} H_{sb}^{3.1} d_{50}^{-0.6} \big( \tan \beta \big)^{0.4} \sin(2 \alpha_b) ,\qquad K_{vanRijn} = 1.8 \times 10^{-4} \, max \Bigg( 1, \; min \Big( 1.5 , \; 1.5 - 10 \big( \Large\frac{H_{sb}}{L_0}\normalsize - 0.01 \big) \Big) \Bigg) \, .\qquad (3) [/math]

In these formulas, [math]H_{sb} =[/math] significant wave height at the wave breaking depth contour, [math]g=[/math] gravitational acceleration, [math]\tan \beta=[/math] mean bed slope of the breaker zone, [math]d_{50}=[/math] median sand grainsize, [math]T=[/math] peak wave period and [math]L_0=[/math] offshore wave length. The formulas (2) and (3) are largely based on parameter fitting to data from field measurements and laboratory experiments. The empirical CERC formula is based on the assumption that littoral drift is proportional to the wave power (or wave energy flux), according to linear wave theory:

[math] Q \propto c_{gb} \, H_{sb}^2 \sin (2\alpha_b) \, , \qquad (4) [/math]

where [math]c_{gb}[/math] is the group velocity at the breaker line, which approximately equals the shallow-water wave celerity [math]c_b \approx \sqrt{g h_b}[/math]. The depth [math]h_b[/math] and the significant wave height are related through the breaker index [math]\gamma_{ib} = H_{sb} / h_b[/math] for an irregular wave field at the breaker line (see Breaker index).

The empirical transport formulas differ in the importance assigned to wave height, wave period, beach slope and sediment grain size. Their predictive performance depends on the characteristics of the coast and the wave climate. Calibration against field observations is therefore desirable whenever reliable data are available. None of the proposed transport formulas performs best for all of the available data from field observations and laboratory experiments.

The One-Line Concept

Figure 1. Shoreline change and associated bottom profiles.

Gradients in littoral drift are in general the primary cause of structural shoreline change. Shoreline change also arises from cross-shore transport, which can move the shoreline landward or seaward in response to alternating episodes of calm and stormy wave conditions. However, these incidental changes in shoreline position usually tend to recover after some time, as discussed in Shoreline retreat and recovery.

The one-line concept rests on a common observation that the beach profile maintains an average shape that is characteristic of the particular coast, apart from times of extreme change as produced by storms. For example, steep beaches remain steep and gently sloping beaches remain gentle in a comparative sense and in the long term; the deviation from an average beach slope over the total active profile is relatively small.

Pelnard-Considère (1956) originated a mathematical theory of shoreline response to wave action under the assumption that the beach profile moves parallel to itself, i.e., that it translates shoreward and seaward without changing shape in the course of eroding and accreting. If the profile shape does not change, any point on it is sufficient to specify the location of the entire profile with respect to a baseline (Fig. 1). Thus, one contour line can be used to describe change in the beach plan shape and volume as the beach erodes and accretes. This contour line is conveniently taken as the readily observed shoreline, and the model is therefore called the "shoreline change" or "shoreline response" model. Sometimes the terminology "one-line" model, a shortening of the phrase "one-contour line" model is used with reference to the single contour line.

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 ([math]D_B[/math] in Fig. 1). The seaward limit is represented by the closure depth ([math]D_C[/math] in Fig. 1), conceived as the offshore limit beyond which cross-shore sediment transport, or its net contribution to the coastal sediment balance, is negligible for the timescale considered. 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 [math]\Delta t[/math]. The change in shoreline position is [math]\Delta y[/math] (Fig. 2) , the length of the shoreline segment is [math]\Delta x[/math], and the profile moves within a vertical extent defined by the height of the berm [math]D_B[/math] and the closure depth [math]D_C[/math]. The change in volume of the section is [math]\Delta V=\Delta x \Delta y (D_B+D_C)[/math], 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 [math]Q[/math] at the lateral sides of the cells. This net volume change is [math](1-p)\Delta V=\Delta Q \Delta t[/math], where [math]p \sim 0.4[/math] is the seabed porosity. Rearrangement of terms yields the governing equation for the rate of change of shoreline position:

[math] \large\frac{\Delta y}{\Delta t}\normalsize + \large\frac{1}{(D_B + D_C)(1-p)} \frac{\Delta Q}{\Delta x}\normalsize = 0 \; . \qquad (5) [/math]

In this formula, [math]Q[/math] represents the volume of sediment grains transported per unit time. The shoreline changes where longshore sediment transport changes along the coast: transport convergence causes accretion and transport divergence causes erosion. A large littoral drift by itself does not imply rapid shoreline change provided the same amount of sediment enters and leaves a coastal section.

Figure 3. Coupled calculation cells alongshore.

Figure 3 shows the representation of several cells along a coastal stretch downdrift of a short groyne. The model requires predictive expressions for the total longshore sand transport rate. For open-coast beaches, the transport rate is a function of the breaking wave height and direction alongshore. The predictive formula for the longshore sand transport rate [math]Q \; [m^3/s][/math] (volume transport of sediment grains) used in GENESIS is the semi-empirical bulk transport relation (often referred to as the CERC formula),

[math] Q = {H_{sb}}^2 c_{gb} \left(a_1 \sin 2 \alpha_b - a_2 \dfrac{\cos \alpha_b }{\tan \beta} \dfrac {\partial H_{sb}}{\partial x} \right) \; , \qquad (6) [/math]

where [math]a_1[/math] and [math]a_2[/math] are dimensionless parameters. This formula follows from the assumption that the littoral drift is related to the mean longshore current and the wave energy received per unit contour length. The first term represents the contribution of radiation stress to the mean longshore current. The second term represents the contribution of alongshore gradients in the wave set-up, which depends linearly on the wave height [math]H_s[/math] at the breaker contour, see wave set-up. If the wave height is variable along the breaker depth contour (the [math]x[/math]-direction), the variable wave set-up will induce a mean longshore current proportional to [math]\partial H_{sb} / \partial x[/math]. Using [math]c_{gb} \approx c_b \approx \sqrt{g h_b} = \sqrt{g H_{sb} / \gamma_{ib}}[/math], formula (6) can be written

[math] Q = \sqrt{\dfrac{g}{\gamma_{ib}}} H_{sb}^{2.5} \left(a_1 \sin 2 \alpha_b - a_2 \dfrac{\cos \alpha_b }{\tan \beta} \dfrac {\partial H_{sb}}{\partial x} \right) \; . \qquad (7) [/math]

Instead of the nondimensional parameters [math]a_1[/math] and [math]a_2[/math], often the empirical coefficients [math]K_1[/math] and [math]K_2[/math] are used. They are related by

[math] a_1 = \dfrac {K_1}{16(s-1)} \approx \dfrac{K_1}{26} \, , \quad a_2 = \dfrac {K_2}{8 \sqrt{2}(s-1)} \approx \dfrac{K_2}{19} \; , \qquad (8) [/math]

where [math]s=\rho_s/\rho \approx 2.65[/math] is the ratio of sediment density to seawater density. The factor [math](s-1)[/math] arises because the coefficients [math]K_1, K_2[/math] relate the immersed weight of the transported sediment to the wave power. Here [math]K_1, K_2[/math] are defined with respect to the significant wave height (different from the GENESIS convention). The Coastal Engineering Manual recommends [math]K = 0.39[/math] for the CERC formula written in terms of significant wave height. A more recent update by Mil-Homens et al. (2013) suggests [math]K_1 = \left[4.5+2233 \, \big(H_{sb}/L_0 \big)^{1.45} \right]^{-1}[/math]. Hanson and Kraus recommend [math]K_2 = (0.5-1) \times K_1[/math]. In practice, [math]K_1[/math] and [math]K_2[/math] are calibration parameters that need to be determined empirically for each site.

An empirical formula that relates the CERC formula to a longshore current [math]v_0[/math] is

[math]Q = \dfrac{K_3}{8 (s-1)} \sqrt{\dfrac{g}{\gamma_{ib}}} H_{sb}^{2.5} \cos \alpha_b \dfrac{v_0}{u_m} \, , \qquad (9)[/math]

where [math]K_3 = K_1/K_v \;[/math], if [math]v_0[/math] is given by Eq. (1).

The CERC formula represents wave-driven longshore transport through breaking-wave energy and angle. Where substantial tidal, wind-driven or other longshore currents occur, a transport formulation based explicitly on the longshore current can be more appropriate. Such currents can be obtained from hydrodynamic models and incorporated in the sediment-transport calculation.

Representation of wave refraction

The wave height and wave incidence angle at the wave breaking depth contour in Eqs. (6,7) have to be specified. Existing wave transformation models including refraction and diffraction can be used for this purpose.

For sufficiently smooth bathymetries, and no significant wave dissipation seaward of the wave breaking depth contour, an analytical expression can be derived relating the wave height [math]H_{sb}[/math] and wave incidence angle [math]\alpha_b[/math] at breaking with the wave height [math]H_s[/math] and incidence angle [math]\alpha[/math] in deep water, where the wave group velocity [math]c_g[/math] is approximately half the wave celerity [math]c[/math]. The conservation of the cross-shore wave energy flux is used,

[math] c_b \, H_{sb}^2 \cos \alpha_b = \frac{1}{2} c \, H_s^2 \cos \alpha \quad[/math] and Snell's refraction law, [math]\quad \Large\frac{\sin \alpha_b}{c_b}\normalsize = \Large\frac{\sin \alpha}{c}\normalsize .[/math]

Here [math]c \approx \Large\frac{gT}{2 \pi}\normalsize [/math] and [math]c_b \approx \sqrt{g h_b} = \sqrt{g H_{sb} / \gamma_{ib} }[/math].

[math]H_{sb}[/math] and [math]\alpha_b[/math] can be eliminated from these equations if it is assumed that the wave incidence angle at breaking is sufficiently small for approximating [math]\cos^{0.2} \alpha_b \approx 1[/math]. Substituting the result in [math]Q = a_1 H_{sb}^2 c_{gb} \sin 2\alpha_b[/math] gives

[math]Q = Q_1 \, (\cos \alpha)^{1.2} \, \sin \alpha \, , \quad Q_1 = a_1 2^{-0.4} \pi^{-0.2} \gamma_{ib}^{-0.4} g^{0.6} T^{0.2} H_s^{2.4} . \qquad (10)[/math]

The deep-water wave incidence angle is defined relative to the shoreline orientation, thus changes as the shoreline orientation changes.

Input wave data

Small deviations of the wave input data from the actual wave climate can make a large difference and compromise the quality of model results. Input data directly from observations are rarely available due to the remoteness of the nearest wave buoy. Wave climate data can be retrieved from models such as WAVEWATCH, which provide hindcasted deep-water wave records with global coverage (significant wave height, wave direction and wave period at 3-hour intervals). The deep-water data can be extrapolated to the nearshore zone with wave models such as SWAN. Long series of wave data are generally needed to reliably represent the wave climate, including rare extreme conditions that can significantly affect the long-term evolution of the coast. The need for lengthy model runs can be overcome by synthesizing the wave climate in a small number of representative wave conditions and associated durations.

Benedet et al. (2016) compared several wave climate schematization methods and found good results using wave classes with equal time-integrated deep-water energy flux. In their applications, a limited number of wave classes (of the order of ten) provided a satisfactory representation of the wave climate.

Other numerical one-line models

GENESIS was the first successful and widely applied numerical one-line model. Other numerical applications of the one-line concept for shoreline evolution by longshore transport processes have been developed since. These models also assume an invariant beach profile that translates as the shoreline position evolves cross-shore, and they are best suited to coasts with relatively uniform alongshore profiles. The assumption of an invariant profile should be interpreted as an approximation for the average active beach profile over the timescale considered. Short-term profile changes caused by storms and subsequent recovery are not explicitly represented.

The LX-Shore model (Robinet et al., 2018) combines a one-line longshore-transport model with an equilibrium-based cross-shore shoreline model and couples these with the spectral wave model SWAN to account for refraction and diffraction over more complex bathymetries (e.g., nearshore bars), but it does not resolve cross-shore profile evolution (e.g. bar migration). The ShorelineS model (Roelvink et al., 2020) removes the fixed-baseline restriction of classical one-line models by representing the coast with a freely moving chain of shoreline points. This allows large rotations, spits, islands, tombolos and merging or splitting coastlines to be simulated. Cross-shore sediment transport processes are not resolved explicitly and must be represented by additional parameterizations when relevant.

Classical one-line models describe shoreline change caused by alongshore sediment-transport gradients but neglect direct cross-shore shoreline response. This is often adequate for longshore-dominated open coasts, but not where storm erosion and recovery, sea-level rise or other cross-shore processes make a substantial contribution. Modern reduced-complexity models therefore increasingly combine one-line longshore transport with simplified cross-shore response models.

Analytical one-line model

An analytical solution of the one-line model Eq. (5) for a uniform coast with small wave incidence angles is presented in the article Beach nourishment. In the Appendix of this article an explicit analytical expression is given of the evolution of a Gaussian-shaped nourishment on a longshore-uniform coast.

Table of symbols

symbol	meaning	symbol	meaning	symbol	meaning
subscript [math]\; _b[/math]	subscript denoting wave breaking location	[math]K_1, K_2, K_3, K_v[/math]	empirical coefficients (calibration parameters)	[math]x \quad [m][/math]	longshore coordinate
[math]c \quad [m/s][/math]	wave celerity	[math]L_0 = g T^2 / (2 \pi) \; [m][/math]	offshore wave length	[math]y (x,t) \quad [m][/math]	shoreline distance from [math]x-[/math]axis
[math]c_g \quad [m/s][/math]	wave group speed	[math]p \approx 0.4[/math]	porosity of sand on the bed	[math]\alpha \quad [rad][/math]	deep-water wave incidence angle relative to the shoreline
[math]d_{50} \quad [m][/math]	median sediment grainsize	[math]Q \quad [m^3/s][/math]	longshore sediment transport	[math]\alpha_b \quad [rad][/math]	wave incidence angle at the breaker line
[math]D=D_B+D_C \quad [m][/math]	sum of berm height and closure depth	[math]s=\rho_s/\rho[/math]	relative density	[math]\tan \beta[/math]	average slope of the surf zone
[math]E=\frac{1}{16} \rho g H_s^2 \quad [J/m^2][/math]	wave energy density	[math]t \quad [s][/math]	time	[math]\gamma_b = H_b / h_b[/math]	breaker index regular waves
[math]g \quad [m/s^2][/math]	acceleration due to gravity	[math]T \quad [s][/math]	(peak) wave period	[math]\gamma_{ib} = H_{sb} / h_b[/math]	breaker index irregular waves
[math]h_b \quad [m][/math]	depth at the breaker line	[math]u_m \quad [m/s][/math]	maximum wave-induced near-bottom horizontal velocity	[math]\rho_s \quad [kg/m^3][/math]	sediment density
[math]H_s \quad [m][/math]	significant wave height	[math]v_0 \quad [m/s][/math]	average longshore current in the surf zone	[math]\rho \quad [kg/m^3][/math]	density of water

Related articles

Modelling coastal hydrodynamics

Stability Models: Linear and nonlinear

Shallow-water wave theory

Beach nourishment

Dealing with coastal erosion

Rhythmic shoreline features

Human causes of coastal erosion

Typical examples of structural erosion

Groynes as shore protection

Accretion and erosion for different coastal types

Port breakwaters and coastal erosion

Sand spit

Appendix

A: Dependence of littoral drift on wave incidence angle

Fig. A1. The longshore component of the wave energy flux through the breaker depth contour per unit contour length is given by [math]F \cos\alpha_b \sin\alpha_b[/math].

The amount of transmitted wave energy along a wave ray is given by [math]F=c_g E[/math], where [math]c_g[/math] is the wave group speed and [math]E=\frac{1}{16} \rho g H_s^2[/math] the wave energy. Since depth contours are assumed parallel to the shoreline, the wave energy flux through a depth contour of unit length is given by [math]F \cos \alpha_b[/math] (see Fig. A1). The alongshore component of the transmitted wave energy is obtained by multiplying by [math]\sin \alpha_b[/math]. The product [math]2 \cos \alpha_b \sin \alpha_b = \sin 2 \alpha_b[/math].

For a small amplitude shoreline bulge, the gradient of the littoral drift [math]\partial Q / \partial x [/math] is proportional to [math]\cos 2 \alpha_b \times d \alpha_b / dx[/math]. When the wave angle [math]\alpha_b[/math] is smaller than 45°, littoral drift increases with increasing wave angle, thus increases along the transport direction over the bulge. A shoreline bulge therefore generates transport divergence that tends to spread the bulge alongshore. For wave angles greater than 45°, littoral drift decreases with increasing wave angle. The response to shoreline perturbations is then reversed: shoreline irregularities can grow instead of being smoothed.

The critical angle of 45° follows from the simplified dependence of littoral drift on wave incidence at breaking. When wave refraction and the dependence of wave energy received per unit shoreline length on coastal orientation are included, the critical deep-water incidence angle [math]\alpha[/math] is approximately 42° (Ashton and Murray, 2006), according to Eq. (10). Larger incidence angles may give rise to features such as sand waves and flying spits, see also Rhythmic shoreline features and Sand spit.

B: Balance equation longshore current

This appendix reproduces the equations from which the longshore current [math]v_0 [/math] can be determined using numerical methods. An approximate analytical expression is also derived based on simplifying assumptions. This appendix uses a representative regular wave with height [math]H = H_{rms}[/math], rather than irregular-wave statistics.

We consider a longshore current driven by a surface wave field [math]\eta_w (x,t)[/math] incident obliquely to the shore of a longshore uniform coast.

Symbols are (attention: [math]x[/math] and [math]y[/math] swapped here compared to the previous sections!)

[math]x=[/math] shore-perpendicular onshore coordinate, [math]y=[/math] alongshore coordinate , [math]z=[/math] vertical upward coordinate, [math]h=[/math] still water depth, [math]g=[/math] gravitational acceleration, [math]\Big\langle … \Big\rangle \, =[/math] wave-averaged value (averaged over one or more wave cycles, encompassing the turbulence time scale), [math]\; u(x,z,t), \, v(x,z,t), \, w(x,z,t) \,=[/math] shore-normal, longshore, vertical velocity; [math]\; u_0 = \langle u\rangle, \, v_0=\langle v\rangle, \, w_0=\langle w\rangle [/math], [math]\; u_w, \, v_w, \, w_w \,=[/math] shore-normal, longshore, vertical wave orbital velocities, [math]\; u', \, v', \, w' \, =[/math] turbulent velocity fluctuations.

The velocities [math]u, \, v, \, w[/math] and surface elevation [math]\eta[/math] are decomposed as

[math]u = u_0 + u_w + u' \, , \; v = v_0 + v_w + v' \, , \; w = w_0 + w_w + w' \, , \; \eta = \langle\eta \rangle + \eta_w \, , \; d= h + \langle \eta \rangle . \qquad (B1)[/math]

The momentum balance equation in the longshore direction is (instantaneous momentum equation with molecular viscous stresses neglected):

[math]\dfrac{\partial v}{\partial t} + \dfrac{\partial uv}{\partial x} + \dfrac{\partial v^2}{\partial y} + \dfrac{\partial w v}{\partial z} + \dfrac{1}{\rho}\dfrac{\partial p}{\partial y} = 0 \, , \qquad (B2)[/math]

where [math]p=p(x,y,z,t)=[/math] pressure. Integration over the depth and averaging over one or more wave cycles gives

[math]\Big\langle \dfrac{\partial}{\partial x} \int_{-h}^{\eta} uv dz \Big\rangle + \Big\langle \dfrac{\partial}{\partial y} \int_{-h}^{\eta} v^2 dz \Big\rangle + \dfrac{1}{\rho} \Big\langle \int_{-h}^{\eta} \dfrac{\partial p}{\partial y} dz \Big\rangle - \dfrac{1}{\rho } \Big\langle \tau_s - \tau_b \Big\rangle = 0 \, , \qquad (B3)[/math]

where [math]\tau_b \approx - \rho \Big\langle v'_b w'_b \Big\rangle \, , \; \tau_s \approx - \rho \Big\langle v'_s w'_s \Big\rangle [/math] and subscripts [math]_b, \, _s[/math] denote values at the seabed [math]z=-h[/math] and the surface [math]z=\eta[/math] respectively. For the permutation of integration and derivation in Eq. (B3) we have used the surface and bottom boundary conditions which state that the surface and the seabed are streamlines: [math]\quad W_s=\dfrac{\partial \eta}{\partial t} + U_s \dfrac{\partial \eta}{\partial x} + V_s \dfrac{\partial \eta}{\partial y} \; , \; W_b = - U_b\dfrac{dh}{dx} - V_b \dfrac{dh}{dy} \, , [/math] where [math]U = u_0+u_w \, , \, V = v_0+v_w \, , \, W = w_0+w_w[/math].

Equation (B3) is greatly simplified if it is assumed that the coast is uniform in longshore direction, as this eliminates all [math]y-[/math]derivatives. With this assumption, the longshore current is driven by the cross-shore gradient of the wave-induced radiation stress [math]S_{xy} = \Big\langle \int_{-h}^{\eta} \rho u_w v_w dz \Big\rangle \quad[/math], see also Shallow-water wave theory#Radiation Stress Components for Oblique Waves. In linear shallow-water wave theory [math]S_{xy}[/math] is given by

[math]S_{xy} = \dfrac{c_g}{c} E \cos \alpha \sin \alpha \, , \quad c_g = \dfrac{c}{2} \Big( 1 + \dfrac{2kh}{\sinh 2kh} \Big) \approx c \approx \sqrt{gh} \, , \quad E = \frac{1}{8} \rho g H^2 \, , \qquad (B4)[/math]

where [math]H[/math] is the root-mean-square wave height and shallow water is assumed.

The first term in Eq. (B3) can be written [math]\Big\langle \int_{-h}^{\eta} u v dz \Big\rangle = \dfrac{1}{\rho} S_{xy} + \Big\langle \int_{-h}^{\eta} u_0 v_0 dz \Big\rangle +\Big\langle \int_{-h}^{\eta} u' v' dz \Big\rangle \, .[/math]

Evaluation of [math]\Big\langle \int_{-h}^{\eta} u_0 v_0 dz \Big\rangle[/math] requires solving the undertow equation, as explained in the article Undertow. The term [math]\Big\langle \int_{-h}^{\eta} u' v' dz \Big\rangle[/math] represents cross-shore diffusion of the longshore current. In a gradient-type diffusion model with diffusion coefficient [math]K_H[/math] this term can be represented by [math]- d \, K_H \partial v_0 / \partial x \, .[/math] The bed stress [math]\tau_b = - \rho \Big\langle v' w' \Big\rangle[/math] can be related through a drag coefficient [math]C_D[/math] to the cross-shore and longshore instantaneous velocities [math]u[/math] and [math]v[/math] in the wave boundary layer, [math]\tau_b = - \rho \Big\langle v' w' \Big\rangle = \rho C_D \Big\langle v \, \sqrt{u^2+v^2} \Big\rangle \, .[/math]

The longshore current for a longshore uniform coast can then be determined numerically from the momentum balance equation

[math] \dfrac{\partial S_{xy}}{\partial x} + \rho C_D \Big\langle v \, \sqrt{u^2+v^2} \Big\rangle = \tau_s - \rho \dfrac{\partial}{\partial x} \Big\langle \int_{-h}^{\eta} u_0 v_0 dz \Big\rangle + \rho \dfrac{\partial}{\partial x} \Big( d\, K_H \dfrac{\partial v_0}{\partial x} \Big) \, . \qquad (B5) [/math]

The terms in the right-hand-side of this equation are discussed in the article Wave set-up. These terms are much smaller than the terms on the left-hand side if the surface roller stress [math]\tau_s[/math] is small, the mean cross-shore velocity [math]u_0[/math] ('undertow') is small and the cross-shore diffusion of longshore momentum is small. Neglecting these terms gives

[math] \dfrac{\partial S_{xy}}{\partial x} \approx - \rho C_D \Big\langle v \, \sqrt{u^2+v^2} \Big\rangle \, . \qquad (B6)[/math]

This equation can be further simplified by assuming that linear wave theory is applicable in the zone around the breaker line and that cross-shore wave orbital velocities [math]u_w[/math] are most of the time much larger than the longshore velocities [math]v[/math] and the cross-shore mean velocity [math]u_0[/math]. In this case,

[math]\dfrac{\partial S_{xy}}{\partial x} \approx - \rho C_D \Big\langle v \, \sqrt{u^2+v^2} \Big\rangle \approx - \dfrac{2}{\pi}\rho \, C_D \, u_m \, v_0 \, . \qquad (B7)[/math]

In shallow water, the amplitude of the cross-shore wave orbital velocity, [math]u_m \approx \dfrac{H}{2h} \sqrt{gh} = \frac{1}{2} \sqrt{\gamma_b g H} [/math], where we have assumed a constant breaker index near the breaker line, [math]\gamma_b = H/h[/math].

The group velocity is approximately equal to the phase velocity, [math]c_g=c=\sqrt{gh}=\gamma_b^{-1/2} g^{1/2} H^{1/2}[/math]. The surf zone slope is written [math]\tan \beta = - dh/dx = - \gamma_b^{-1} \partial H / \partial x [/math].

If the wave incidence angle at the breaker line, [math]\alpha_b[/math], is small, we may simplify Eq. (B4)

[math]S_{xy} \Big|_b = E_b \cos \alpha_b \sin \alpha_b \approx E_b \sin \alpha_b \approx \frac{1}{8} \rho g H_b^2 \sin \alpha_b \, . \quad (B8)[/math]

Snell's law, [math]\sin \alpha / c =constant[/math], is used for evaluating the derivative of the radiation stress,

[math]\dfrac{\partial S_{xy}}{\partial x} = \dfrac{\partial E \cos \alpha \sin \alpha}{\partial x} = \dfrac{\sin \alpha}{c} \dfrac{\partial c E \cos \alpha}{\partial x} \approx \dfrac{\sin \alpha}{c} \dfrac{\partial cE}{\partial x} = - \frac{5}{16} \rho \, g \, \gamma_b \, \tan \beta \, H \, \sin \alpha = - \frac{5}{4} \rho \, u_m^2 \, \tan \beta \, \sin \alpha \, . \qquad (B9)[/math]

Substitution in Eq. (B7) yields Eq. (1), with [math]K_v=\dfrac{5 \pi}{8} \dfrac{\tan \beta}{C_D} [/math].

Typical values of the drag coefficient [math]C_D[/math] are of the order 0.01. The theoretical value of [math]K_v[/math] is in most cases larger than the empirical value 2.7, indicating that the use of linear wave theory likely leads to an overestimation of the longshore current.

References

The main authors of this article are Hanson, Hans and Job Dronkers Please note that others may also have edited the contents of this article. Citation: Hanson, Hans; Job Dronkers; (2026): Littoral drift and shoreline modelling. Available from http://www.coastalwiki.org/wiki/Littoral_drift_and_shoreline_modelling [accessed on 2-11-2026] For other articles by this author see Category:Articles by Hanson, Hans For other articles by this author see Category:Articles by Job Dronkers For an overview of contributions by this author see Special:Contributions/Tvrl-hha For an overview of contributions by this author see Special:Contributions/Dronkers J