A multi-scale asymptotic theory is derived for the evolution and interaction of currents and surface gravity waves in water of finite depth, under conditions typical of coastal shelf waters outside the surf zone. The theory provides a practical and useful model with which wave–current coupling may be explored without the necessity of resolving features of the flow on space and time scales of the primary gravity-wave oscillations. The essential nature of the dynamical interaction is currents modulating the slowly evolving phase of the wave field and waves providing both phase-averaged forcing of long infra-gravity waves and wave-averaged vortex and Bernoulli-head forces and hydrostatic static set-up for the low-frequency current and sea-level evolution equations. Analogous relations are derived for wave-averaged material tracers and density stratification that include advection by horizontal Stokes drift and by a vertical Stokes pseudo-velocity that is the incompressible companion to the horizontal Stokes velocity. Illustrative solutions are analysed for the special case of depth-independent currents and tracers associated with an incident surface wave field and a vortex with O(1) Rossby number above continental shelf topography.

The effects of wind-generated surface gravity waves on more slowly evolving long waves, currents and material distributions in stratified coastal waters are investigated using the wave-averaged, asymptotic equations developed in McWilliams et al. (2004), based on small wave slope and on scale separations in both time and horizontal space. Excluding non-conservative effects such as wave breaking, the lowest order radiation stress, introduced by Longuet-Higgins & Stewart (1960) and Hasselmann (1971), can be completely characterized in terms of wave set-up, forcing of long (infra-gravity) waves and an Eulerian current whose divergence cancels that of the primary waves' Stokes drift. The vortex force of Craik & Leibovich (1976) and its generalization for inhomogeneous waves and Earth's rotation are shown to be the dominant wave-averaged effects on currents, and these effects can occur at higher order than the apparent leading order for the radiation stress. The leading-order, wave-averaged dynamical effects are completed with material advection by Stokes drift, modified pressure-continuity and kinematic surface boundary conditions, and parameterized representations of wave generation by the wind and breaking near the shoreline.

[1] There is not yet widespread agreement as to the underlying cause of the 80-100 ppmv roughly 100-kyr-duration glacial-interglacial cycles in atmospheric pCO 2 . Most of the mechanisms which have been proposed to account for the observed pCO 2 variations appear to in some way violate interpretations of paleo proxy data. The inability of a single mechanism to explain the observed cycles in atmospheric CO 2 (which show amazing similarity over the past 430,000 years) is perplexing, and leads us to consider whether a combination of mechanisms might be consistent with available evidence. Consistent with previous work, we find that physical changes (ocean circulation, temperature, mixing) can only explain part of the observed atmospheric pCO 2 variability; changes in ocean chemistry are invoked to explain the remainder. In order to account for the initial pCO 2 drawdown (from ''interglacial'' to ''intermediate'' levels), we invoke physical changes in the ocean (mixing, temperature). The transition from intermediate atmospheric pCO 2 levels to full glacial conditions involves a small increase in mean ocean nutrient levels and mean ocean alkalinity, accomplished by falling sea level and subsequent erosion of organic-rich shelf sediments. The first part of the transition out of full glacial conditions is achieved through increased temperature and increased mixing in the Southern Ocean. The final part of the atmospheric pCO 2 rise up to full interglacial conditions is accomplished through rising sea level and the subsequent change in mean ocean alkalinity and phosphate, and a rise in the Northern Hemisphere temperature and ocean mixing. The proposed sequence of events is consistent with most existing proxy evidence for paleo-nutrient levels and changes in export production over the last glacial-interglacial cycle. Furthermore, it is consistent with evidence for a whole-ocean shift in d 13C toward significantly more negative values in the late glacial. The proposed scenario is also consistent with ice core-based timing constraints, as summarized by Broecker and Henderson (1998). We show that we are able to explain the full magnitude of the glacial-interglacial cycle in atmospheric pCO 2 without the need to invoke iron-fertilization in the Southern Ocean.