LAUSR

Abstract We present an analytical study of two-dimensional flow in a wedge driven by a time-dependent surface heat flux as a model problem to understand buoyancy-induced cross-shore flow. Besides the turbulent Prandtl number and the relevant Rayleigh number, both assumed to be of order unity, the solution is seen to depend on the geometry through a parameter $\beta$, representing the bottom slope. An analytic solution is sought in the asymptotic limit $\beta \ll 1$ for a water layer bounded by an adiabatic bottom surface subject to a harmonic heat flux on the upper surface. The analysis reveals that the motion at leading order can be expressed as the sum of a harmonic component and a steady component, the latter driven by nonlinear advection. This steady-streaming motion includes a near-shore vortex with associated recirculating motion that can affect cross-shore transport and dispersion in coastal environments. The analytical solution is compared with numerical solutions of the complete conservation equations for small values of $\beta$. Excellent quantitative agreement is found for values of the Rayleigh number below a critical value at which the periodic solution undergoes a period-doubling bifurcation, leading to the establishment of thermal-instability cells that dominate the offshore flow dynamics, while the near-shore dynamics remains well described by the analytical solution. The analysis illustrates that a periodic heat input that leads to a vertically inhomogeneous temperature distribution can result in residual motion, net heat fluxes and persistent temperature structure in the cross-shore direction.