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Radiation stress
In fluid dynamics, the radiation stress is the depthintegrated – and thereafter phaseaveraged – 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 secondorder tensor.
The radiation stress tensor describes the additional forcing due to the presence of the waves, which changes the mean depthintegrated 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 (waveinduced 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 meanflow 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 LonguetHiggins and Stewart in 1960–1964.
Radiation stress derives its name from the analogous effect of radiation pressure for electromagnetic radiation.
Contents
Physical significance
The radiation stress – mean excess momentumflux due to the presence of the waves – plays an important role in the explanation and modeling of various coastal processes:^{[1]}^{[2]}^{[3]}
 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);
 Wavedriven 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 shearstress component S_{xy} of the radiation stress over the width of the surf zone. This provides the forcing of a wavedriven longshore current, which is of importance for sediment transport (longshore drift) and the resulting coastal morphology;
 Bound long waves or forced long waves – for wave groups the radiation stress varies along the group. As a result, a nonlinear 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 meanflow fields, the energy exchanges between the waves and the mean flow, as well as the meanflow forcing, can be modeled by means of the radiation stress.
Definitions and values derived from linear wave theory
Onedimensional wave propagation
For unidirectional wave propagation – say in the xcoordinate direction – the component of the radiation stress tensor of dynamical importance is S_{xx}. It is defined as:^{[4]}
 <math>S_{xx} = \overline{ \int_{h}^\eta \left( p + \rho \tilde{u}^2 \right)\; \text{d}z }  \frac12 \rho g \left( h + \overline{\eta} \right)^2,</math>
where p(x,z,t) is the fluid pressure, <math>\tilde{u}(x,z,t)</math> is the horizontal xcomponent 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 righthand side, ½ρg(h+η)^{2}, is the integral of the hydrostatic pressure over the stillwater depth.
To lowest (second) order, the radiation stress S_{xx} for traveling periodic waves can be determined from the properties of surface gravity waves according to Airy wave theory:^{[5]}^{[6]}
 <math>S_{xx} = \left( 2 \frac{c_g}{c_p}  \frac12 \right) E,</math>
where c_{p} is the phase speed and c_{g} is the group speed of the waves. Further E is the mean depthintegrated 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:^{[7]}
 <math>E = \frac12 \rho g a^2 = \frac18 \rho g H^2,</math>
with a the wave amplitude and H = 2a the wave height. Note this equation is for periodic waves: in random waves the rootmeansquare wave height H_{rms} should be used with H_{rms} = H_{m0} / √2, where H_{m0} is the significant wave height. Then E = ^{1}⁄_{16}ρgH_{m0}^{2}.
Twodimensional wave propagation
For wave propagation in two horizontal dimensions the radiation stress <math>\mathbf{S}</math> is a secondorder tensor^{[8]}^{[9]} with components:
 <math>\mathbf{S}= \begin{pmatrix} S_{xx} & S_{xy} \\ S_{yx} & S_{yy} \end{pmatrix}.</math>
With, in a cartesian coordinate system (x,y,z):^{[4]}
 <math>
\begin{align} S_{xx} &= \overline{ \int_{h}^\eta \left( p + \rho \tilde{u}^2 \right)\; \text{d}z }  \frac12 \rho g \left( h + \overline{\eta} \right)^2, \\ S_{xy} &= \overline{ \int_{h}^\eta \left( \rho \tilde{u} \tilde{v} \right)\; \text{d}z } = S_{yx}, \\ S_{yy} &= \overline{ \int_{h}^\eta \left( p + \rho \tilde{v}^2 \right)\; \text{d}z }  \frac12 \rho g \left( h + \overline{\eta} \right)^2, \end{align}
</math> where <math>\tilde{u}</math> and <math>\tilde{v}</math> are the horizontal x and ycomponents of the oscillatory part <math>\tilde{u}(x,y,z,t)</math> of the flow velocity vector.
To second order – in wave amplitude a – the components of the radiation stress tensor for progressive periodic waves are:^{[5]}
 <math>
\begin{align}
S_{xx} &= \left[ \frac{k_x^2}{k^2} \frac{c_g}{c_p} + \left( \frac{c_g}{c_p}  \frac12 \right) \right] E, \\ S_{xy} &= \left( \frac{k_x k_y}{k^2} \frac{c_g}{c_p} \right) E = S_{yx}, \quad \text{and} \\ S_{yy} &= \left[ \frac{k_y^2}{k^2} \frac{c_g}{c_p} + \left( \frac{c_g}{c_p}  \frac12 \right) \right] E,
\end{align} </math>
where k_{x} and k_{y} are the x and ycomponents of the wavenumber vector k, with length k = k = √k_{x}^{2}+k_{y}^{2} and the vector k perpendicular to the wave crests. The phase and group speeds, c_{p} and c_{g} respectively, are the lengths of the phase and group velocity vectors: c_{p} = c_{p} and c_{g} = c_{g}.
Dynamical significance
The radiation stress tensor is an important quantity in the description of the phaseaveraged dynamical interaction between waves and mean flows. Here, the depthintegrated dynamical conservation equations are given, but – in order to model threedimensional mean flows forced, or interacting with, surface waves – a threedimensional description of the radiation stress over the fluid layer is needed.^{[10]}
Mass transport velocity
Propagating waves induce a – relatively small – mean mass transport in the wave propagation direction, also called the wave (pseudo) momentum.^{[11]} To lowest order, the wave momentum M_{w} is, per unit of horizontal area:^{[12]}
 <math>\boldsymbol{M}_w = \frac{\boldsymbol{k}}{k} \frac{E}{\rho\, c_p},</math>
which is exact for progressive waves of permanent form in irrotational flow. Above, c_{p} is the phase speed relative to the mean flow:
 <math>c_p = \frac{\sigma}{k} \qquad \text{with} \qquad \sigma=\omega  \boldsymbol{k}\cdot\overline{\boldsymbol{v}},</math>
with σ the intrinsic angular frequency, as seen by an observer moving with the mean horizontal flowvelocity v while ω is the apparent angular frequency of an observer at rest (with respect to 'Earth'). The difference k⋅v is the Doppler shift.^{[13]}
The mean horizontal momentum M, also per unit of horizontal area, is the mean value of the integral of momentum over depth:
 <math>\boldsymbol{M} = \overline{\int_{h}^\eta \rho\, \boldsymbol{v}\; \text{d}z}
= \rho\, \left( h + \overline{\eta} \right) \overline{\boldsymbol{v}} + \boldsymbol{M}_w,</math>
with v(x,y,z,t) the total flow velocity at any point below the free surface z = η(x,y,t). The mean horizontal momentum M is also the mean of the depthintegrated horizontal mass flux, and consists of two contributions: one by the mean current and the other (M_{w}) is due to the waves.
Now the mass transport velocity u is defined as:^{[14]}^{[15]}
 <math>\overline{\boldsymbol{u}} = \frac{\boldsymbol{M}}{\rho\, \left( h + \overline{\eta} \right)}
= \overline{\boldsymbol{v}} + \frac{\boldsymbol{M}_w}{\rho\, \left( h + \overline{\eta} \right)}.</math>
Observe that first the depthintegrated horizontal momentum is averaged, before the division by the mean water depth (h+η) is made.
Mass and momentum conservation
Vector notation
The equation of mean mass conservation is, in vector notation:^{[14]}
 <math>\frac{\partial}{\partial t}\left[ \rho \left( h + \overline{\eta} \right) \right] + \nabla \cdot \left[ \rho \left( h + \overline{\eta} \right) \overline{\boldsymbol{u}} \right] = 0,</math>
with u including the contribution of the wave momentum M_{w}.
The equation for the conservation of horizontal mean momentum is:^{[14]}
 <math>\frac{\partial}{\partial t}\left[ \rho \left( h + \overline{\eta} \right) \overline{\boldsymbol{u}} \right] + \nabla \cdot \left[ \rho \left( h + \overline{\eta} \right) \overline{\boldsymbol{u}} \otimes \overline{\boldsymbol{u}} + \mathbf{S} + \frac12 \rho g (h+\overline{\eta})^2\, \mathbf{I} \right] = \rho g \left( h + \overline{\eta} \right) \nabla h + \boldsymbol{\tau}_w  \boldsymbol{\tau}_b,</math>
where u ⊗ u denotes the tensor product of u with itself, and τ_{w} is the mean wind shear stress at the free surface, while τ_{b} is the bed shear stress. Further I is the identity tensor, with components given by the Kronecker delta δ_{ij}. Note that the right hand side of the momentum equation provides the nonconservative contributions of the bed slope ∇h,^{[16]} as well the forcing by the wind and the bed friction.
In terms of the horizontal momentum M the above equations become:^{[14]}
 <math>
\begin{align}
&\frac{\partial}{\partial t}\left[ \rho \left( h + \overline{\eta} \right) \right] + \nabla \cdot \boldsymbol{M} = 0, \\ &\frac{\partial \boldsymbol{M}}{\partial t} + \nabla \cdot \left[ \overline{\boldsymbol{u}} \otimes \boldsymbol{M} + \mathbf{S} + \frac12 \rho g (h+\overline{\eta})^2\, \mathbf{I} \right] = \rho g \left( h + \overline{\eta} \right) \nabla h + \boldsymbol{\tau}_w  \boldsymbol{\tau}_b.
\end{align} </math>
Component form in Cartesian coordinates
In a Cartesian coordinate system, the mass conservation equation becomes:
 <math>\frac{\partial}{\partial t} \left[ \rho \left( h + \overline{\eta} \right) \right] + \frac{\partial}{\partial x} \left[ \rho \left( h + \overline{\eta} \right) \overline{u}_x \right] + \frac{\partial}{\partial y} \left[ \rho \left( h + \overline{\eta} \right) \overline{u}_y \right] = 0,</math>
with u_{x} and u_{y} respectively the x and y components of the mass transport velocity u.
The horizontal momentum equations are:
 <math>
\begin{align}
\frac{\partial}{\partial t}\left[ \rho \left( h + \overline{\eta} \right) \overline{u}_x \right] &+ \frac{\partial}{\partial x} \left[ \rho \left( h + \overline{\eta} \right) \overline{u}_x \overline{u}_x + S_{xx} + \frac12 \rho g (h+\overline{\eta})^2 \right] + \frac{\partial}{\partial y} \left[ \rho \left( h + \overline{\eta} \right) \overline{u}_x \overline{u}_y + S_{xy} \right] \\ &= \rho g \left( h + \overline{\eta} \right) \frac{\partial}{\partial x} h + \tau_{w,x}  \tau_{b,x}, \\ \frac{\partial}{\partial t}\left[ \rho \left( h + \overline{\eta} \right) \overline{u}_y \right] &+ \frac{\partial}{\partial x} \left[ \rho \left( h + \overline{\eta} \right) \overline{u}_y \overline{u}_x + S_{yx} \right] + \frac{\partial}{\partial y} \left[ \rho \left( h + \overline{\eta} \right) \overline{u}_y \overline{u}_y + S_{yy} + \frac12 \rho g (h+\overline{\eta})^2 \right] \\ &= \rho g \left( h + \overline{\eta} \right) \frac{\partial}{\partial y} h + \tau_{w,y}  \tau_{b,y}.
\end{align} </math>
Energy conservation
For an inviscid flow the mean mechanical energy of the total flow – that is the sum of the energy of the mean flow and the fluctuating motion – is conserved.^{[17]} However, the mean energy of the fluctuating motion itself is not conserved, nor is the energy of the mean flow. The mean energy E of the fluctuating motion (the sum of the kinetic and potential energies satisfies:^{[18]}
 <math>\frac{\partial E}{\partial t} + \nabla \cdot \left[ \left( \overline{\boldsymbol{u}} + \boldsymbol{c}_g \right) E \right] + \mathbf{S}:\left( \nabla \otimes \overline{\boldsymbol{u}} \right) = \boldsymbol{\tau}_w \cdot \overline{\boldsymbol{u}}  \boldsymbol{\tau}_b \cdot \overline{\boldsymbol{u}}  \varepsilon,</math>
where ":" denotes the doubledot product, and ε denotes the dissipation of mean mechanical energy (for instance by wave breaking). The term <math>\mathbf{S}:\left( \nabla \otimes \overline{\boldsymbol{u}} \right)</math> is the exchange of energy with the mean motion, due to wave–current interaction. The mean horizontal waveenergy transport (u + c_{g}) E consists of two contributions:
 u E : the transport of wave energy by the mean flow, and
 c_{g} E : the mean energy transport by the waves themselves, with the group velocity c_{g} as the waveenergy transport velocity.
In a Cartesian coordinate system, the above equation for the mean energy E of the flow fluctuations becomes:
 <math>
\begin{align}
\frac{\partial E}{\partial t} &+ \frac{\partial}{\partial x} \left[ \left( \overline{u}_x + c_{g,x} \right) E \right] + \frac{\partial}{\partial y} \left[ \left( \overline{u}_y + c_{g,y} \right) E \right] \\ &+ S_{xx} \frac{\partial \overline{u}_x}{\partial x} + S_{xy} \left( \frac{\partial \overline{u}_y}{\partial x} + \frac{\partial \overline{u}_x}{\partial y} \right) + S_{yy} \frac{\partial \overline{u}_y}{\partial y} \\ &= \left( \tau_{w,x}  \tau_{b,x} \right) \overline{u}_x + \left( \tau_{w,y}  \tau_{b,y} \right) \overline{u}_y  \varepsilon.
\end{align} </math>
So the radiation stress changes the wave energy E only in case of a spatialinhomogeneous current field (u_{x},u_{y}).
Notes
 ^ LonguetHiggins & Stewart (1964,1962).
 ^ Phillips (1977), pp. 70–81.
 ^ Battjes, J. A. (1974). Computation of setup, longshore currents, runup and overtopping due to windgenerated waves (Thesis). Delft University of Technology. Retrieved 20101125.
 ^ ^{a} ^{b} Mei (2003), p. 457.
 ^ ^{a} ^{b} Mei (2003), p. 97.
 ^ Phillips (1977), p. 68.
 ^ Phillips (1977), p. 39.
 ^ LonguetHiggins & Stewart (1961).
 ^ Dean, R.G.; Walton, T.L. (2009), "Wave setup", in Young C. Kim, Handbook of Coastal and Ocean Engineering, World Scientific, pp. 1–23, ISBN 9812819290.
 ^ Walstra, D. J. R.; Roelvink, J. A.; Groeneweg, J. (2000), "Calculation of wavedriven currents in a 3D mean flow model", Proceedings of the 27th International Conference on Coastal Engineering, Sydney: ASCE, pp. 1050–1063, doi:10.1061/40549(276)81
 ^ Mcintyre, M. E. (1981), "On the 'wave momentum' myth", Journal of Fluid Mechanics 106: 331–347, Bibcode:1981JFM...106..331M, doi:10.1017/S0022112081001626
 ^ Phillips (1977), p. 40.
 ^ Phillips (1977), pp. 23–24.
 ^ ^{a} ^{b} ^{c} ^{d} Phillips (1977), pp. 61–63.
 ^ Mei (2003), p. 453.
 ^ By Noether's theorem, an inhomogeneous medium – in this case a nonhorizontal bed, h(x,y) not a constant – results in nonconservation of the depthintegrated horizontal momentum.
 ^ Phillips (1977), pp. 63–65.
 ^ Phillips (1977), pp. 65–66.
References
 Primary sources
 LonguetHiggins, M. S.; Stewart, R. W. (1960), "Changes in the form of short gravity waves on long waves and tidal currents", Journal of Fluid Mechanics 8 (4): 565–583, Bibcode:1960JFM.....8..565L, doi:10.1017/S0022112060000803
 LonguetHiggins, M. S.; Stewart, R. W. (1961), "The changes in amplitude of short gravity waves on steady nonuniform currents", Journal of Fluid Mechanics 10 (4): 529–549, Bibcode:1961JFM....10..529L, doi:10.1017/S0022112061000342
 LonguetHiggins, M. S.; Stewart, R. W. (1962), "Radiation stress and mass transport in gravity waves, with application to ‘surf beats’", Journal of Fluid Mechanics 13 (4): 481–504, Bibcode:1962JFM....13..481L, doi:10.1017/S0022112062000877
 LonguetHiggins, M. S.; Stewart, R. W. (1964), "Radiation stresses in water waves; a physical discussion, with applications", Deep Sea Research 11 (4): 529–562, Bibcode:1964DSROA..11..529L, doi:10.1016/00117471(64)900014
 Further reading
 Mei, Chiang C. (2003), The applied dynamics of ocean surface waves, Advanced series on ocean engineering 1, World Scientific, ISBN 9971507897
 Phillips, O. M. (1977), The dynamics of the upper ocean (2nd ed.), Cambridge University Press, ISBN 0521298016
