## Abstract

Previous attempts to derive the depth-dependent expression of the radiation stress have led to a debate concerning (i) the applicability of the Mellor approach to a sloping bottom, (ii) the introduction of the delta function at the mean sea surface in the later papers by Mellor, and (iii) a wave-induced pressure term derived in several recent studies. The authors use an equation system in vertically Lagrangian and horizontally Eulerian (VL) coordinates suitable for a concise treatment of the surface boundary and obtain an expression for the depth-dependent radiation stress that is consistent with the vertically integrated expression given by Longuet–Higgins and Stewart. Concerning (i)–(iii) above, the difficulty of handling a sloping bottom disappears when wave-averaged momentum equations in the VL coordinates are written for the development of (not the Lagrangian mean velocity but) the Eulerian mean velocity. There is also no delta function at the sea surface in the expression for the depth-dependent radiation stress. The connection between the wave-induced pressure term in the recent studies and the depth-dependent radiation stress term is easily shown by rewriting the pressure-based form stress term in the thickness-weighted-mean momentum equations as a velocity-based term that contains the time derivative of the pseudomomentum in the VL framework.

*Corresponding author address:*Hidenori Aiki, Research Institute for Global Change, Japan Agency for Marine-Earth Science and Technology, Yokohama, Japan. E-mail: aiki@jamstec.go.jp