Wave radiation stress

Abstract

There are differences in the literature concerning the vertically dependent equations that couple currents and waves. In this paper, currents are purposely omitted until the end. Isolating waves from currents allows one to focus on two main topics: an explanation of Stokes drift with apparent mean vorticity obtained from an otherwise irrotational flow and the determination of vertically dependent wave radiation stress which, when vertically integrated, conforms to that obtained by Longuet-Higgins and Stewart (1964) and Phillips (1977) nearly 50 years ago and, more recently, by Smith (2006). Discussion begins with the simple case of nonlinear flow beneath a stationary wavy wall.

Appendices

Appendix 1

1.1 Derivation of the Stokes drift

The conventional “Lagrangian” derivation of Stokes drift velocity is

$$ {u_S} \cong \overline {\tilde{u} + \tilde{x}\frac{{\partial \tilde{u}}}{{\partial x}} + \tilde{z}\frac{{\partial \tilde{u}}}{{\partial z}}} \;;\quad \tilde{x} \equiv \int {\tilde{u}dt\,;\;} \;\tilde{z} \equiv \int {\tilde{w}dt.} $$

After insertion of Eqs. 10a and 10b, one obtains

$$ {u_S} = {(ka)^2}c{e^{{2kz}}} $$

Another approach is to consider the flow through a fixed element bounded by z and z + Δz. The velocity is $\tilde{u}(z) + \tilde{z}{(\partial \tilde{u}/\partial z)_z}$, and the wave distorted flow area relative to the undistorted area is $1 + {(\partial \tilde{z}/\partial z)_z}$; the product of the two terms after phase averaging is

$$ {u_S} = \overline {\left( {\tilde{u} + \tilde{z}\frac{{\partial \tilde{u}}}{{\partial z}}} \right)\left( {1 + \frac{{\partial \tilde{z}}}{{\partial z}}} \right)} = \frac{{\partial \overline {\tilde{z}\tilde{u}} }}{{\partial z}} = {(ka)^2}c{e^{{2kz}}} $$

Appendix 2

1.1 Alternate derivation of Eq. 17 following Phillips (1977)

Adjusting the origin of z so that $\hat{\eta } = 0$, the result of integrating Eq. 2b from arbitrary negative z to $z = \tilde{\eta }$ is,

$$ \frac{\partial }{{\partial t}}\int^{{\tilde{\eta }}} {\tilde{w}dz} - \tilde{w}(\eta )\frac{{\partial \tilde{\eta }}}{{\partial t}} + \frac{\partial }{{\partial x}}\int^{{\tilde{\eta }}} {uwdz} - {\left. {uw} \right|_{{\tilde{\eta }}}}\frac{{\partial \tilde{\eta }}}{{\partial x}} + {\tilde{w}^2}(\eta ) - {\tilde{w}^2}(z) = - p(\eta ) + p(z) - g(\tilde{\eta } - z) $$

where $p(\tilde{\eta }) = {p_{\rm{atm}}}$, i.e., the instantaneous pressure is continuous across the interface. Note that $\tilde{w}(\eta ) = \partial \tilde{\eta }/\partial t + \tilde{u}(\tilde{\eta })\partial \tilde{\eta }/\partial x$ so that the second, fourth, and fifth terms on the left cancel. After phase averaging, $\overline {\partial \int^{{\tilde{\eta }}} {\tilde{w}dz/\partial t} } = \overline {\partial \int^{{\tilde{\eta }}} {uwdz/\partial x} } = 0$ leaving

$$ \bar{p} + \overline {{{\tilde{w}}^2}} + gz = {p_{\rm{atm}}};\;z \leqslant 0. $$

Thus, whereas the instantaneous pressure is continuous across the air–sea interface at z = 0, the phase-averaged pressure is discontinuous.