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| [[File:Southland Beach Almost Tropical.jpg|thumb|right|[[Breaking wave]]s on beaches induce variations in radiation stress, driving longshore currents. The resulting longshore [[sediment transport]] shapes the beaches, and may result in [[beach erosion]] or accretion.]]
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| In [[fluid dynamics]], the '''radiation stress''' is the depth-integrated – and thereafter [[phase (waves)|phase]]-[[average]]d – excess [[flux|momentum flux]] caused by the presence of the [[surface gravity wave]]s, which is exerted on the [[mean flow]]. The radiation stresses behave as a second-order [[tensor]].
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| The radiation stress tensor describes the additional [[force|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).
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| For the mean [[energy density]] in the [[oscillation|oscillatory part]] of the fluid motion, the radiation stress tensor is important for its [[dynamics (mechanics)|dynamics]], in case of an [[inhomogeneous]] mean-flow [[field (physics)|field]].
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| 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 [[Michael S. Longuet-Higgins|Longuet-Higgins]] and Stewart in 1960–1964.
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| Radiation stress derives its name from the analogous effect of [[radiation pressure]] for [[electromagnetic radiation]].
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| ==Physical significance==
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| 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:<ref>Longuet-Higgins & Stewart (1964,1962).</ref><ref>Phillips (1977), pp. 70–81.</ref><ref>{{Cite thesis | last = Battjes | first = J. A. | title = Computation of set-up, longshore currents, run-up and overtopping due to wind-generated waves | publisher = Delft University of Technology | accessdate = 2010-11-25 | year = 1974 | url = http://repository.tudelft.nl/view/ir/uuid%3Ae126e043-a858-4e58-b4c7-8a7bc5be1a44/ }}</ref>
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| * ''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);
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| * ''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 ''S''<sub>''xy''</sub> 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 [[geomorphology|morphology]];
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| * ''Bound long waves'' or ''forced long waves'' – for [[wave#modulated waves|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 (water waves)|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 (algebra)|square]] of the wave height, and is only significant in shallow water;
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| * ''[[Wave–current interaction]]'' – in varying [[mean flow|mean-flow]] [[field (physics)|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.
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| ==Definitions and values derived from linear wave theory==
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| ===One-dimensional wave propagation===
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| For uni-directional wave propagation – say in the ''x''-coordinate direction – the component of the radiation stress tensor of [[dynamics (mechanics)|dynamical]] importance is ''S''<sub>xx</sub>. It is defined as:<ref name="Mei_457">Mei (2003), p. 457.</ref>
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| :<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>
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| where ''p''(''x'',''z'',''t'') is the fluid [[pressure]], <math>\tilde{u}(x,z,t)</math> is the horizontal ''x''-component of the [[oscillation|oscillatory part]] of the [[flow velocity]] [[vector (mathematics and physics)|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 [[Earth's gravity|acceleration by gravity]], while an overbar denotes [[phase (waves)|phase]] [[average|averaging]]. The last term on the right-hand side, ½''ρg''(''h''+''{{overline|η}}'')<sup>2</sup>, is the [[integral]] of the [[hydrostatic pressure]] over the still-water depth.
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|
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| To lowest (second) order, the radiation stress ''S''<sub>xx</sub> for traveling [[periodic wave]]s can be determined from the properties of surface gravity waves according to [[Airy wave theory]]:<ref name="Mei_97">Mei (2003), p. 97.</ref><ref>Phillips (1977), p. 68.</ref>
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| :<math>S_{xx} = \left( 2 \frac{c_g}{c_p} - \frac12 \right) E,</math>
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| where ''c''<sub>p</sub> is the [[phase speed]] and ''c''<sub>g</sub> is the [[group speed]] of the waves. Further ''E'' is the mean depth-integrated wave energy density (the sum of the [[kinetic energy|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:<ref>Phillips (1977), p. 39.</ref>
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| :<math>E = \frac12 \rho g a^2 = \frac18 \rho g H^2,</math>
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| with ''a'' the wave [[amplitude]] and ''H'' = 2''a'' the [[wave height]]. Note this equation is for periodic waves: in [[random process|random waves]] the [[root-mean-square]] wave height ''H''<sub>rms</sub> should be used with ''H''<sub>rms</sub> = ''H''<sub>m0</sub> / √2, where ''H''<sub>m0</sub> is the [[significant wave height]]. Then ''E'' = {{frac|1|16}}''ρgH''<sub>m0</sub><sup>2</sup>.
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| ===Two-dimensional wave propagation===
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| For wave propagation in two horizontal dimensions the radiation stress <math>\mathbf{S}</math> is a second-order [[tensor]]<ref>Longuet-Higgins & Stewart (1961).</ref><ref>{{Citation | chapter=Wave setup | first1=R.G. | last1=Dean | first2=T.L. | last2=Walton | title=Handbook of Coastal and Ocean Engineering | editor=Young C. Kim | publisher=World Scientific | year=2009 | isbn=981-281-929-0 | pages=1–23 | postscript=. }}</ref> with components:
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| :<math>\mathbf{S}= \begin{pmatrix} S_{xx} & S_{xy} \\ S_{yx} & S_{yy} \end{pmatrix}.</math>
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| With, in a [[cartesian coordinate system]] (''x'',''y'',''z''):<ref name="Mei_457"/>
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| :<math>
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| \begin{align}
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| S_{xx} &= \overline{ \int_{-h}^\eta \left( p + \rho \tilde{u}^2 \right)\; \text{d}z }
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| - \frac12 \rho g \left( h + \overline{\eta} \right)^2, \\
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| S_{xy} &= \overline{ \int_{-h}^\eta \left( \rho \tilde{u} \tilde{v} \right)\; \text{d}z } = S_{yx}, \\
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| S_{yy} &= \overline{ \int_{-h}^\eta \left( p + \rho \tilde{v}^2 \right)\; \text{d}z }
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| - \frac12 \rho g \left( h + \overline{\eta} \right)^2,
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| \end{align}
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| </math>
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| where <math>\tilde{u}</math> and <math>\tilde{v}</math> are the horizontal ''x''- and ''y''-components of the oscillatory part <math>\tilde{u}(x,y,z,t)</math> of the flow velocity vector.
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| To second order – in wave amplitude ''a'' – the components of the radiation stress tensor for progressive periodic waves are:<ref name="Mei_97"/>
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| :<math>
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| \begin{align}
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| S_{xx} &= \left[ \frac{k_x^2}{k^2} \frac{c_g}{c_p} + \left( \frac{c_g}{c_p} - \frac12 \right) \right] E,
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| \\
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| S_{xy} &= \left( \frac{k_x k_y}{k^2} \frac{c_g}{c_p} \right) E = S_{yx},
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| \quad \text{and}
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| \\
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| S_{yy} &= \left[ \frac{k_y^2}{k^2} \frac{c_g}{c_p} + \left( \frac{c_g}{c_p} - \frac12 \right) \right] E,
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| \end{align}
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| </math>
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| where ''k''<sub>x</sub> and ''k''<sub>y</sub> are the ''x''- and ''y''-components of the [[wavenumber]] vector '''''k''''', with length ''k'' = |'''''k'''''| = {{radic|''k''<sub>x</sub><sup>2</sup>+''k''<sub>y</sub><sup>2</sup>}} and the vector '''''k''''' perpendicular to the wave [[crest (physics)|crests]]. The phase and group speeds, ''c''<sub>p</sub> and ''c''<sub>g</sub> respectively, are the lengths of the phase and group velocity vectors: ''c''<sub>p</sub> = |'''''c'''''<sub>p</sub>| and ''c''<sub>g</sub> = |'''''c'''''<sub>g</sub>|.
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| ==Dynamical significance==
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| The radiation stress tensor is an important quantity in the description of the phase-averaged dynamical interaction between waves and mean flows. Here, the depth-integrated dynamical conservation equations are given, but – in order to model three-dimensional mean flows forced, or interacting with, surface waves – a three-dimensional description of the radiation stress over the fluid layer is needed.<ref>{{Citation | first1=D. J. R. | last1=Walstra | first2=J. A. | last2=Roelvink | first3=J. | last3=Groeneweg | contribution=Calculation of wave-driven currents in a 3D mean flow model | title=Proceedings of the 27th International Conference on Coastal Engineering | location=Sydney | pages=1050–1063 | publisher=[[American Society of Civil Engineers|ASCE]] | year=2000 | doi=10.1061/40549(276)81 }}</ref>
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| ===Mass transport velocity===
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| Propagating waves induce a – relatively small – mean [[mass flux|mass transport]] in the wave propagation direction, also called the wave (pseudo) [[momentum]].<ref>{{Citation | doi = 10.1017/S0022112081001626 | volume = 106 | pages = 331–347 | last = Mcintyre | first = M. E. | title = On the 'wave momentum' myth | journal = Journal of Fluid Mechanics | year = 1981 |bibcode = 1981JFM...106..331M }}</ref> To lowest order, the wave momentum '''M'''<sub>w</sub> is, per unit of horizontal area:<ref>Phillips (1977), p. 40.</ref>
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| :<math>\boldsymbol{M}_w = \frac{\boldsymbol{k}}{k} \frac{E}{\rho\, c_p},</math>
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| which is exact for progressive waves of permanent form in [[irrotational flow]]. Above, ''c''<sub>p</sub> is the phase speed relative to the mean flow:
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| :<math>c_p = \frac{\sigma}{k} \qquad \text{with} \qquad \sigma=\omega - \boldsymbol{k}\cdot\overline{\boldsymbol{v}},</math>
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| with ''σ'' the ''intrinsic angular frequency'', as seen by an observer moving with the mean horizontal flow-velocity {{overline|'''''v'''''}} while ''ω'' is the ''apparent angular frequency'' of an observer at rest (with respect to 'Earth'). The difference '''''k'''''⋅{{overline|'''''v'''''}} is the [[Doppler shift]].<ref>Phillips (1977), pp. 23–24.</ref>
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| The mean horizontal momentum '''''M''''', also per unit of horizontal area, is the mean value of the integral of momentum over depth:
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| :<math>\boldsymbol{M} = \overline{\int_{-h}^\eta \rho\, \boldsymbol{v}\; \text{d}z}
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| = \rho\, \left( h + \overline{\eta} \right) \overline{\boldsymbol{v}} + \boldsymbol{M}_w,</math>
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| 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 depth-integrated horizontal mass flux, and consists of two contributions: one by the mean current and the other ('''''M'''''<sub>w</sub>) is due to the waves.
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| Now the mass transport velocity {{overline|'''''u'''''}} is defined as:<ref name="Phillips_61_63"/><ref>Mei (2003), p. 453.</ref>
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| :<math>\overline{\boldsymbol{u}} = \frac{\boldsymbol{M}}{\rho\, \left( h + \overline{\eta} \right)}
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| = \overline{\boldsymbol{v}}
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| + \frac{\boldsymbol{M}_w}{\rho\, \left( h + \overline{\eta} \right)}.</math>
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| Observe that first the depth-integrated horizontal momentum is averaged, before the division by the mean water depth (''h''+''{{overline|η}}'') is made.
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| ===Mass and momentum conservation===
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| ====Vector notation====
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| The equation of mean mass conservation is, in [[vector notation]]:<ref name="Phillips_61_63">Phillips (1977), pp. 61–63.</ref>
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| :<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> | |
|
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| with {{overline|'''''u'''''}} including the contribution of the wave momentum '''''M'''''<sub>w</sub>.
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| The equation for the conservation of horizontal mean momentum is:<ref name="Phillips_61_63"/>
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| :<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>
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| where {{overline|'''''u'''''}} ⊗ {{overline|'''''u'''''}} denotes the [[tensor product]] of {{overline|'''''u'''''}} with itself, and '''τ'''<sub>w</sub> is the mean wind [[shear stress]] at the free surface, while '''τ'''<sub>b</sub> is the bed shear stress. Further '''I''' is the identity tensor, with components given by the [[Kronecker delta]] δ<sub>ij</sub>. Note that the [[right hand side]] of the momentum equation provides the non-conservative contributions of the bed slope ∇''h'',<ref>By [[Noether's theorem]], an inhomogeneous medium – in this case a non-horizontal bed, ''h''(''x'',''y'') not a constant – results in non-conservation of the depth-integrated horizontal momentum.</ref> as well the forcing by the wind and the bed friction.
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| In terms of the horizontal momentum '''''M''''' the above equations become:<ref name="Phillips_61_63"/>
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| :<math>
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| \begin{align}
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| &\frac{\partial}{\partial t}\left[ \rho \left( h + \overline{\eta} \right) \right]
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| + \nabla \cdot \boldsymbol{M} = 0,
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| \\
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| &\frac{\partial \boldsymbol{M}}{\partial t}
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| + \nabla \cdot \left[ \overline{\boldsymbol{u}} \otimes \boldsymbol{M} + \mathbf{S}
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| + \frac12 \rho g (h+\overline{\eta})^2\, \mathbf{I} \right]
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| = \rho g \left( h + \overline{\eta} \right) \nabla h
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| + \boldsymbol{\tau}_w - \boldsymbol{\tau}_b.
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| \end{align}
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| </math>
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| ====Component form in Cartesian coordinates====
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| In a [[Cartesian coordinate system]], the mass conservation equation becomes:
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| :<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>
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| with {{overline|''u''}}<sub>x</sub> and {{overline|''u''}}<sub>y</sub> respectively the ''x'' and ''y'' components of the mass transport velocity {{overline|'''''u'''''}}.
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| The horizontal momentum equations are:
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| :<math>
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| \begin{align}
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| \frac{\partial}{\partial t}\left[ \rho \left( h + \overline{\eta} \right) \overline{u}_x \right]
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| &+ \frac{\partial}{\partial x}
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| \left[ \rho \left( h + \overline{\eta} \right) \overline{u}_x \overline{u}_x + S_{xx} + \frac12 \rho g (h+\overline{\eta})^2 \right]
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| + \frac{\partial}{\partial y}
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| \left[ \rho \left( h + \overline{\eta} \right) \overline{u}_x \overline{u}_y + S_{xy} \right]
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| \\
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| &= \rho g \left( h + \overline{\eta} \right) \frac{\partial}{\partial x} h
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| + \tau_{w,x} - \tau_{b,x},
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| \\
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| \frac{\partial}{\partial t}\left[ \rho \left( h + \overline{\eta} \right) \overline{u}_y \right]
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| &+ \frac{\partial}{\partial x}
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| \left[ \rho \left( h + \overline{\eta} \right) \overline{u}_y \overline{u}_x + S_{yx} \right]
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| + \frac{\partial}{\partial y}
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| \left[ \rho \left( h + \overline{\eta} \right) \overline{u}_y \overline{u}_y + S_{yy} + \frac12 \rho g (h+\overline{\eta})^2 \right]
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| \\
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| &= \rho g \left( h + \overline{\eta} \right) \frac{\partial}{\partial y} h
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| + \tau_{w,y} - \tau_{b,y}.
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| \end{align}
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| </math>
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| ===Energy conservation===
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| 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.<ref>Phillips (1977), pp. 63–65.</ref> 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 energy|kinetic]] and [[potential energy|potential energies]] satisfies:<ref>Phillips (1977), pp. 65–66.</ref>
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| :<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>
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| where ":" denotes the [[dyadics|double-dot 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 wave-energy transport ({{overline|'''''u'''''}} + '''c'''<sub>g</sub>) ''E'' consists of two contributions:
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| * {{overline|'''''u'''''}} ''E'' : the transport of wave energy by the mean flow, and
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| * '''''c'''''<sub>g</sub> ''E'' : the mean energy transport by the waves themselves, with the [[group velocity]] '''''c'''''<sub>g</sub> as the wave-energy transport velocity.
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| In a Cartesian coordinate system, the above equation for the mean energy ''E'' of the flow fluctuations becomes:
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| :<math>
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| \begin{align}
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| \frac{\partial E}{\partial t}
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| &+ \frac{\partial}{\partial x} \left[ \left( \overline{u}_x + c_{g,x} \right) E \right]
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| + \frac{\partial}{\partial y} \left[ \left( \overline{u}_y + c_{g,y} \right) E \right]
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| \\
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| &+ S_{xx} \frac{\partial \overline{u}_x}{\partial x}
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| + S_{xy} \left( \frac{\partial \overline{u}_y}{\partial x} + \frac{\partial \overline{u}_x}{\partial y} \right)
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| + S_{yy} \frac{\partial \overline{u}_y}{\partial y}
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| \\
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| &= \left( \tau_{w,x} - \tau_{b,x} \right) \overline{u}_x
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| + \left( \tau_{w,y} - \tau_{b,y} \right) \overline{u}_y
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| - \varepsilon.
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| \end{align}
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| </math>
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| So the radiation stress changes the wave energy ''E'' only in case of a spatial-[[homogeneity and heterogeneity|inhomogeneous]] current field ({{overline|''u''}}<sub>x</sub>,{{overline|''u''}}<sub>y</sub>).
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| {{refbegin}}
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| ;Primary sources
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| * {{Citation | first1=M. S. | last1=Longuet-Higgins | first2=R. W. | last2=Stewart | title=Changes in the form of short gravity waves on long waves and tidal currents | journal=Journal of Fluid Mechanics | year=1960 | volume=8 | issue=4 | pages=565–583 | doi=10.1017/S0022112060000803 |bibcode = 1960JFM.....8..565L }}
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| * {{Citation | first1=M. S. | last1=Longuet-Higgins | first2=R. W. | last2=Stewart | title=The changes in amplitude of short gravity waves on steady non-uniform currents | journal=Journal of Fluid Mechanics | year=1961 | volume=10 | issue=4 | pages=529–549 | doi=10.1017/S0022112061000342 |bibcode = 1961JFM....10..529L }}
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| * {{Citation | first1=M. S. | last1=Longuet-Higgins | first2=R. W. | last2=Stewart | title=Radiation stress and mass transport in gravity waves, with application to ‘surf beats’ | journal= Journal of Fluid Mechanics | year=1962 | volume=13 | issue=4 | pages=481–504 | doi=10.1017/S0022112062000877 |bibcode = 1962JFM....13..481L }}
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| * {{Citation | first1=M. S. | last1=Longuet-Higgins | first2=R. W. | last2=Stewart | title=Radiation stresses in water waves; a physical discussion, with applications | journal= Deep Sea Research | year=1964 | volume=11 | issue=4| pages=529–562 | doi=10.1016/0011-7471(64)90001-4 }}
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| | |
| ;Further reading
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| * {{Citation | title=The applied dynamics of ocean surface waves | volume=1 | series=Advanced series on ocean engineering | first=Chiang C. | last=Mei | author-link=Chiang C. Mei | publisher=World Scientific | year=2003 | isbn=9971-5-0789-7 }}
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| * {{Citation | first=O. M. | last=Phillips | title=The dynamics of the upper ocean |publisher=Cambridge University Press | year=1977 | edition=2nd | isbn=0-521-29801-6 }}
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| {{refend}}
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| {{Physical oceanography}}
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| [[Category:Physical oceanography]]
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| [[Category:Water waves]]
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