Residual streaming flows in buoyancy-driven cross-shore exchange

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 β, representing the bottom slope. An analytic solution is sought in the asymptotic limit β 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 β. 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.