Coastal Erosion and Longshore Drift

What is modelled

A cliff coast seen from the sea. Waves arrive at an angle, refract and shoal as the water gets shallower, break, and push sand along the beach (longshore drift). A one-line model moves the shoreline according to the gradient of that drift. At the back of the beach a cliff retreats whenever waves reach its foot at high tide, and the retreat feeds new sand to the beach. You can add a groyne, a sea wall or rock armour, raise the sea level and trigger storms.

Equations

• Linear wave theory: ω2 = g k tanh(kh), phase speed c = ω/k, group speed cg = n c. Refraction (Snell): sinα / c = const. Shoaling with refraction: H = H0 √(cg0/cg) √(cosα0/cosα). Waves break where H/h = γ = 0.78 (found by bisection).
• Longshore sediment transport (CERC formula): Q = K ρ √g Hb5/2 sin(2αb) / [16 √γ (ρs - ρ)(1 - p)] with K = 0.39, ρ_s = 2650, ρ = 1025 kg/m³, porosity p = 0.4.
• One-line shoreline model (Pelnard-Considere): ∂y/∂t = -(1/D) ∂Q/∂x, with Q evaluated for the local angle between the wave crests and the shoreline, so a curved beach turns the drift. D (berm plus closure depth) is the active profile height. A groyne sets Q = 0 at its face until the fillet reaches its tip, then sand bypasses.
• Sea-level rise moves the whole profile landward by the Bruun rule Δy = S / tanβ with the shoreface slope 0.03.
• Cliff retreat (Sunamura-type): dX/dt = Rmax exp(-Fc/Fw), with Fw = ρ g Htoe, wave height at the cliff foot Htoe = min(Hb, γ dtoe) and water depth at the foot dtoe = tide + surge + sea-level rise - tanβ · (beach width), averaged over six tidal phases. R_max = 6 m/yr, F_c is the rock strength you choose. A wide beach lifts the foot above the water and protects the cliff.
• Sand supply: a share f of the retreated cliff volume stays on the beach, Δy = f Hc ΔX / D.

Checks run in the page

• At the breaking point: Snell invariant, energy-flux conservation and Hb/hb = 0.78.
• A Gaussian shoreline bump diffuses with σ2 = σ02 + 2Gt, G = Q'(α)/D (small test waves, no cliff).
• Groyne fillet against the analytical Pelnard-Considere solution y(0, t) = 2 tanα √(Gt/π).
• Live sediment budget: beach volume change = drift in - drift out + cliff feed - Bruun loss.

Honest simplifications

• This is a plan-view coastline model, not a 3D hydrodynamic and sediment model. The beach profile keeps a fixed shape (slopes 0.06 and 0.03), there is no cross-shore transport, no rip currents, no wave diffraction behind the groyne and no tidal currents. The CERC constant and the cliff-erosion law are empirical; the numbers (R_max, rock strengths in kPa) are illustrative orders of magnitude, not a calibration for any real cliff.
• The waves are one monochromatic train; real seas are a spectrum with varying direction. Storms are 2.5-day events with doubled wave height and a 1.5 m surge. The model beach cannot narrow below zero width, and that small correction is counted in the budget.
• Time is strongly accelerated (years per second), the cliff keeps its height, and rock falls are smoothed into a continuous retreat. Real cliffs retreat in landslides after weeks or decades of undercutting. The sea surface and foam are drawn for illustration from the breaking depth and are not a wave simulation.

• Switch on the groyne with oblique waves: the beach grows on the updrift side and the downdrift cliff retreats faster.
• Compare Chalk coast with Granite headland, then try a Sea wall: the cliff stops, but it no longer feeds the beach, which narrows.
• Raise the storm rate and the sea-level rise and watch the beach width fall and the lighthouse distance shrink.