We present a mathematical model to investigate heat transfer and mass transport dynamics in the wave-driven free-surface boundary layer of the ocean under the influence of long-crested progressive surface gravity waves. The continuity, momentum and convection–diffusion equations for fluid temperature are solved within a Lagrangian framework. We assume that eddy viscosity and thermometric conductivity are dependent on Lagrangian coordinates, and derive a new form of the second-order Lagrangian mass transport velocity, applicable across the entire finite water depth. We then analyse the convective heat dynamics influenced by the free-surface boundary layer. Rectangular distributions of free-surface temperature (i.e. a Dirichlet boundary condition) are considered, and analytical solutions for thermal boundary layer temperature fields are provided to offer insights into free-surface heat transfer mechanisms affected by ocean waves. Our results suggest the need to improve existing models that neglect the effects of free-surface waves and the free-surface boundary layer on ocean mass transport and heat transfer.

A mathematical model is developed to investigate seabed heat transfer processes under long-crested ocean waves. The unsteady convection–diffusion equation for water temperature includes terms depending on the velocity field in the laminar boundary layer, analogous to mass transfer near the seabed. Here we consider regular progressive waves and standing waves reflected from a vertical structure, which complicate the convective term in the governing equation. Rectangular and Gaussian distributions of seabed temperature and heat flux are considered. Approximate analytical solutions are derived for uniform and trapezoidal currents, and compared against predictions from a numerical solver of the full equations. The effects of heat source profile, location and strength on heat transfer dynamics in the thermal boundary layer are explained, providing insights into seabed temperature forced convection mechanisms enhanced by free-surface waves.