Runup and Reverse Shoaling

This Javascript app calculates swash runup elevation on the foreshore. Input parameters include wave height H, wave period T, water depth h, beach slope β_f, and γ, the ratio of H_b/h_b (wave height to water depth at the break point).

Runup height estimates are based on beach slope in the foreshore region (landward of the break point) and deepwater parameters height H_o and wavelength L_o. Deepwater wavelength depends only on wave period as L_o = gT²/(2π) where g is gravitational acceleration (9.81 m/s). Runup height is a combination of setup <η> (elevation of the sea surface by shoaling) and variance of the swash associated with incident waves S_inc and low-frequency infragravity waves S_IG (sometimes called surf beat). Total runup R₂ (based on the highest 2%) is calculated as R₂ = 1.1 (<η>+S/2) where S = sqrt(S_inc² + S_IG²; Stockdon et al., 2006, eqns. 9 and 7). S_inc and S_IG are calculated from Stockdon et al. (2006), eqns. 11 and 12. For very dissipative conditions (low-angle beaches with Iribarren number ξ = β_f sqrt(L_o/H_o)< 0.3), R₂ = 0.043 sqrt(H_o L_o) (Stockdon et al., 2006, eqn. 18).

Runup distance x = R₂/sin(arctanβ_f) is the distance that swash extends across the sloping shoreface from the mean waterline.

There are three different ways to use these calculations, depending on where the waves are measured. The radio-button selection allows you to specify this:

1. Wave heights are measured at the break point h_b – In this case, the associated water depth is the depth at the break point h_b = H_b/γ, and the offshore conditions are determined from reverse shoaling (see below).
2. Wave heights are measured offshore in deep water – In this case, H_o is the input wave height H, and the break height is computed from an empirical formula: H_b = γ/g^1/5 (C_go H_o²)^2/5 (Plant et al., 1999) where C_go = 1/2 gT/(2π) is group velocity for deepwater waves and break depth is computed as h_b = H_b/γ. Wavenumber k and shoaling coefficient K_s are computed for the input depth h.
3. Wave heights measured at an intermediate depth h – In this case, deepwater wave height H_o is determined from reverse shoaling, and break height and break depth are computed as in (2) above.

Reverse shoaling involves estimating deepwater conditions from measurements at shallow or intermediate depths, using linear wave theory as follows. The wavenumber k at measurement depth h (or h_b) is calculated from the dispersion relationship, shoaling coefficient K_s is calculated according to Neilsen (2009, eqn. 1.7.5), and deepwater wave height H_o is determined as H_o = H_b/K_s. Because wave period T remains constant as waves shoal, deepwater wavelength can be calculated as above.

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