Radiation stress

For uni-directional wave propagation – say in the x-coordinate direction – the component of the radiation stress tensor of dynamical importance is S_xx. It is defined as:[4]

${\displaystyle S_{xx}={\overline {\int _{-h}^{\eta }\left(p+\rho {\tilde {u}}^{2}\right)\;{\text{d}}z}}-{\frac {1}{2}}\rho g\left(h+{\overline {\eta }}\right)^{2},}$

where p(x,z,t) is the fluid pressure, ${\displaystyle {\tilde {u}}(x,z,t)}$ 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, ½ρg(h+η)², is the integral of the hydrostatic pressure over the still-water 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]

${\displaystyle S_{xx}=\left(2{\frac {c_{g}}{c_{p}}}-{\frac {1}{2}}\right)E,}$

where c_p is the phase speed and c_g 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:[7]

${\displaystyle E={\frac {1}{2}}\rho ga^{2}={\frac {1}{8}}\rho gH^{2},}$

with a the wave amplitude and H = 2a the wave height. Note this equation is for periodic waves: in random waves the root-mean-square wave height H_rms should be used with H_rms = H_m0 / √2, where H_m0 is the significant wave height. Then E = 1⁄16ρgH_m0².

For wave propagation in two horizontal dimensions the radiation stress ${\displaystyle \mathbf {S} }$ is a second-order tensor[8][9] with components:

${\displaystyle \mathbf {S} ={\begin{pmatrix}S_{xx}&S_{xy}\\S_{yx}&S_{yy}\end{pmatrix}}.}$

With, in a Cartesian coordinate system (x,y,z):[4]

${\displaystyle {\begin{aligned}S_{xx}&={\overline {\int _{-h}^{\eta }\left(p+\rho {\tilde {u}}^{2}\right)\;{\text{d}}z}}-{\frac {1}{2}}\rho g\left(h+{\overline {\eta }}\right)^{2},\\S_{xy}&={\overline {\int _{-h}^{\eta }\left(\rho {\tilde {u}}{\tilde {v}}\right)\;{\text{d}}z}}=S_{yx},\\S_{yy}&={\overline {\int _{-h}^{\eta }\left(p+\rho {\tilde {v}}^{2}\right)\;{\text{d}}z}}-{\frac {1}{2}}\rho g\left(h+{\overline {\eta }}\right)^{2},\end{aligned}}}$

where ${\displaystyle {\tilde {u}}}$ and ${\displaystyle {\tilde {v}}}$ are the horizontal x- and y-components of the oscillatory part ${\displaystyle {\tilde {u}}(x,y,z,t)}$ 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]

${\displaystyle {\begin{aligned}S_{xx}&=\left[{\frac {k_{x}^{2}}{k^{2}}}{\frac {c_{g}}{c_{p}}}+\left({\frac {c_{g}}{c_{p}}}-{\frac {1}{2}}\right)\right]E,\\S_{xy}&=\left({\frac {k_{x}k_{y}}{k^{2}}}{\frac {c_{g}}{c_{p}}}\right)E=S_{yx},\quad {\text{and}}\\S_{yy}&=\left[{\frac {k_{y}^{2}}{k^{2}}}{\frac {c_{g}}{c_{p}}}+\left({\frac {c_{g}}{c_{p}}}-{\frac {1}{2}}\right)\right]E,\end{aligned}}}$

where k_x and k_y are the x- and y-components of the wavenumber vector k, with length k = |k| = √k_x²+k_y² 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|.

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]

${\displaystyle {\boldsymbol {M}}_{w}={\frac {\boldsymbol {k}}{k}}{\frac {E}{c_{p}}},}$

which is exact for progressive waves of permanent form in irrotational flow. Above, c_p is the phase speed relative to the mean flow:

${\displaystyle c_{p}={\frac {\sigma }{k}}\qquad {\text{with}}\qquad \sigma =\omega -{\boldsymbol {k}}\cdot {\overline {\boldsymbol {v}}},}$

with σ the intrinsic angular frequency, as seen by an observer moving with the mean horizontal flow-velocity 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:

${\displaystyle {\boldsymbol {M}}={\overline {\int _{-h}^{\eta }\rho \,{\boldsymbol {v}}\;{\text{d}}z}}=\rho \,\left(h+{\overline {\eta }}\right){\overline {\boldsymbol {v}}}+{\boldsymbol {M}}_{w},}$

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_w) is due to the waves.

Now the mass transport velocity u is defined as:[14][15]

${\displaystyle {\overline {\boldsymbol {u}}}={\frac {\boldsymbol {M}}{\rho \,\left(h+{\overline {\eta }}\right)}}={\overline {\boldsymbol {v}}}+{\frac {{\boldsymbol {M}}_{w}}{\rho \,\left(h+{\overline {\eta }}\right)}}.}$

Observe that first the depth-integrated horizontal momentum is averaged, before the division by the mean water depth (h+η) is made.

The equation of mean mass conservation is, in vector notation:[14]

${\displaystyle {\frac {\partial }{\partial t}}\left[\rho \left(h+{\overline {\eta }}\right)\right]+\nabla \cdot \left[\rho \left(h+{\overline {\eta }}\right){\overline {\boldsymbol {u}}}\right]=0,}$

with u including the contribution of the wave momentum M_w.

The equation for the conservation of horizontal mean momentum is:[14]

${\displaystyle {\frac {\partial }{\partial t}}\left[\rho \left(h+{\overline {\eta }}\right){\overline {\boldsymbol {u}}}\right]+\nabla \cdot \left[\rho \left(h+{\overline {\eta }}\right){\overline {\boldsymbol {u}}}\otimes {\overline {\boldsymbol {u}}}+\mathbf {S} +{\frac {1}{2}}\rho g(h+{\overline {\eta }})^{2}\,\mathbf {I} \right]=\rho g\left(h+{\overline {\eta }}\right)\nabla h+{\boldsymbol {\tau }}_{w}-{\boldsymbol {\tau }}_{b},}$

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 non-conservative 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]

${\displaystyle {\begin{aligned}&{\frac {\partial }{\partial t}}\left[\rho \left(h+{\overline {\eta }}\right)\right]+\nabla \cdot {\boldsymbol {M}}=0,\\&{\frac {\partial {\boldsymbol {M}}}{\partial t}}+\nabla \cdot \left[{\overline {\boldsymbol {u}}}\otimes {\boldsymbol {M}}+\mathbf {S} +{\frac {1}{2}}\rho g(h+{\overline {\eta }})^{2}\,\mathbf {I} \right]=\rho g\left(h+{\overline {\eta }}\right)\nabla h+{\boldsymbol {\tau }}_{w}-{\boldsymbol {\tau }}_{b}.\end{aligned}}}$

In a Cartesian coordinate system, the mass conservation equation becomes:

${\displaystyle {\frac {\partial }{\partial t}}\left[\rho \left(h+{\overline {\eta }}\right)\right]+{\frac {\partial }{\partial x}}\left[\rho \left(h+{\overline {\eta }}\right){\overline {u}}_{x}\right]+{\frac {\partial }{\partial y}}\left[\rho \left(h+{\overline {\eta }}\right){\overline {u}}_{y}\right]=0,}$

with u_x and u_y respectively the x and y components of the mass transport velocity u.

The horizontal momentum equations are:

${\displaystyle {\begin{aligned}{\frac {\partial }{\partial t}}\left[\rho \left(h+{\overline {\eta }}\right){\overline {u}}_{x}\right]&+{\frac {\partial }{\partial x}}\left[\rho \left(h+{\overline {\eta }}\right){\overline {u}}_{x}{\overline {u}}_{x}+S_{xx}+{\frac {1}{2}}\rho g(h+{\overline {\eta }})^{2}\right]+{\frac {\partial }{\partial y}}\left[\rho \left(h+{\overline {\eta }}\right){\overline {u}}_{x}{\overline {u}}_{y}+S_{xy}\right]\\&=\rho g\left(h+{\overline {\eta }}\right){\frac {\partial }{\partial x}}h+\tau _{w,x}-\tau _{b,x},\\{\frac {\partial }{\partial t}}\left[\rho \left(h+{\overline {\eta }}\right){\overline {u}}_{y}\right]&+{\frac {\partial }{\partial x}}\left[\rho \left(h+{\overline {\eta }}\right){\overline {u}}_{y}{\overline {u}}_{x}+S_{yx}\right]+{\frac {\partial }{\partial y}}\left[\rho \left(h+{\overline {\eta }}\right){\overline {u}}_{y}{\overline {u}}_{y}+S_{yy}+{\frac {1}{2}}\rho g(h+{\overline {\eta }})^{2}\right]\\&=\rho g\left(h+{\overline {\eta }}\right){\frac {\partial }{\partial y}}h+\tau _{w,y}-\tau _{b,y}.\end{aligned}}}$

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]

${\displaystyle {\frac {\partial E}{\partial t}}+\nabla \cdot \left[\left({\overline {\boldsymbol {u}}}+{\boldsymbol {c}}_{g}\right)E\right]+\mathbf {S}$ :\left(\nabla \otimes {\overline {\boldsymbol {u}}}\right)={\boldsymbol {\tau }}_{w}\cdot {\overline {\boldsymbol {u}}}-{\boldsymbol {\tau }}_{b}\cdot {\overline {\boldsymbol {u}}}-\varepsilon ,}

where ":" denotes the double-dot product, and ε denotes the dissipation of mean mechanical energy (for instance by wave breaking). The term ${\displaystyle \mathbf {S}$ :\left(\nabla \otimes {\overline {\boldsymbol {u}}}\right)} is the exchange of energy with the mean motion, due to wave–current interaction. The mean horizontal wave-energy 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 wave-energy transport velocity.

In a Cartesian coordinate system, the above equation for the mean energy E of the flow fluctuations becomes:

${\displaystyle {\begin{aligned}{\frac {\partial E}{\partial t}}&+{\frac {\partial }{\partial x}}\left[\left({\overline {u}}_{x}+c_{g,x}\right)E\right]+{\frac {\partial }{\partial y}}\left[\left({\overline {u}}_{y}+c_{g,y}\right)E\right]\\&+S_{xx}{\frac {\partial {\overline {u}}_{x}}{\partial x}}+S_{xy}\left({\frac {\partial {\overline {u}}_{y}}{\partial x}}+{\frac {\partial {\overline {u}}_{x}}{\partial y}}\right)+S_{yy}{\frac {\partial {\overline {u}}_{y}}{\partial y}}\\&=\left(\tau _{w,x}-\tau _{b,x}\right){\overline {u}}_{x}+\left(\tau _{w,y}-\tau _{b,y}\right){\overline {u}}_{y}-\varepsilon .\end{aligned}}}$

So the radiation stress changes the wave energy E only in case of a spatial-inhomogeneous current field (u_x,u_y).