In fluid dynamics, the radiation stress is the depth-integrated – and thereafter phase-averaged – excess momentum flux caused by the presence of the surface gravity waves, which is exerted on the mean flow. The radiation stresses behave as a second-order tensor. The radiation stress tensor describes the additional forcing due to the presence of the waves, which changes the mean depth-integrated horizontal momentum in the fluid layer. As a result, varying radiation stresses induce changes in the mean surface elevation (wave setup) and the mean flow (wave-induced currents). For the mean energy density in the oscillatory part of the fluid motion, the radiation stress tensor is important for its dynamics, in case of an inhomogeneous mean-flow field. The radiation stress tensor, as well as several of its implications on the physics of surface gravity waves and mean flows, were formulated in a series of papers by Longuet-Higgins and Stewart in 1960–1964. Radiation stress derives its name from the analogous effect of radiation pressure for electromagnetic radiation.
Physical significance The radiation stress – mean excess momentum-flux due to the presence of the waves – plays an important role in the explanation and modeling of various coastal processes:
Wave setup and setdown – the radiation stress consists in part of a radiation pressure, exerted at the free surface elevation of the mean flow. If the radiation stress varies spatially, as it does in the surf zone where the wave height reduces by wave breaking, this results in changes of the mean surface elevation called wave setup (in case of an increased level) and setdown (for a decreased water level); Wave-driven current, especially a longshore current in the surf zone – for oblique incidence of waves on a beach, the reduction in wave height inside the surf zone (by breaking) introduces a variation of the shear-stress component Sxy of the radiation stress over the width of the surf zone. This provides the forcing of a wave-driven longshore current, which is of importance for sediment transport (longshore drift) and the resulting coastal morphology; Bound long waves or forced long waves, part of the infragravity waves – for wave groups the radiation stress varies along the group. As a result, a non-linear long wave propagates together with the group, at the group velocity of the modulated short waves within the group. While, according to the dispersion relation, a long wave of this length should propagate at its own – higher – phase velocity. The amplitude of this bound long wave varies with the square of the wave height, and is only significant in shallow water; Wave–current interaction – in varying mean-flow fields, the energy exchanges between the waves and the mean flow, as well as the mean-flow forcing, can be modeled by means of the radiation stress.
Definitions and values derived from linear wave theory
One-dimensional wave propagation For uni-directional wave propagation – say in the x-coordinate direction – the component of the radiation stress tensor of dynamical importance is Sxx. It is defined as:
S x x = ∫ − h η ( p + ρ u ~ 2 ) d z ¯ − 1 2 ρ g ( h + η ¯ ) 2 , {\displaystyle S_{xx}={\overline {\int _{-h}^{\eta }\left(p+\rho {\tilde {u}}^{2}\right)\;{\text{d}}z}}-{\frac {1}{2}}\rho g\left(h+{\overline {\eta }}\right)^{2},}
where p(x,z,t) is the fluid pressure, u ~ ( x , z , t ) {\displaystyle {\tilde {u}}(x,z,t)} is the horizontal x-component of the oscillatory part of the flow velocity vector, z is the vertical coordinate, t is time, z = −h(x) is the bed elevation of the fluid layer, and z = η(x,t) is the surface elevation. Further ρ is the fluid density and g is the acceleration by gravity, while an overbar denotes phase averaging. The last term on the right-hand side, 1/2ρg(h+η)2, is the integral of the hydrostatic pressure over the still-water depth. To lowest (second) order, the radiation stress Sxx for traveling periodic waves can be determined from the properties of surface gravity waves according to Airy wave theory:
S x x = ( 2 c g c p − 1 2 ) E , {\displaystyle S_{xx}=\left(2{\frac {c_{g}}{c_{p}}}-{\frac {1}{2}}\right)E,}
where cp is the phase speed and cg is the group speed of the waves. Further E is the mean depth-integrated wave energy density (the sum of the kinetic and potential energy) per unit of horizontal area. From the results of Airy wave theory, to second order, the mean energy density E equals:
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