Periodic Diffusional Mass Transfer near Sediment/Water Interface: Theory

Time-variable (periodic) flow over a lake bed, and the associated boundary layer development, have the potential to control or at least influence rates of mass transfer across the sediment/water interface. An analysis for instantaneous and time averaged flux of a material across the sediment/water interface for infinite supply in the water and infinite sink in the sediment is presented. The water flow above the interface is characterized by the shear velocity $\left(U_{*}\right)$ which is a periodic function of time with a maximum amplitude of $\left(U_{*0}\right)$ as may be typical of an internal seiche (internal standing wave) motion in a density stratified lake. The relationship between the shear velocity on the lake bed and the wind shear on the lake surface is illustrated for an extremely simplified two-layered lake of constant depth. For a less restrictive analysis, shear velocities on a lake bed have to be obtained either from field measurements or from a three-dimensional lake circulation model driven by atmospheric forcing including wind. Smaller and wind-sheltered lakes will have lower $\left(U_{*0}\right)$ and periodicities (T). The response of the diffusive boundary layer was related to the period of the periodic motion (T), Schmidt number $\left(\mathrm{Sc}\right)\text{,}$ and shear velocity $\left(U_{*}\right)\text{.}$ The vertical diffusive flux at the sediment/water interface was expressed by a Sherwood number $\left(\mathrm{Sh}\right)\text{,}$ either instantaneous or time averaged. The mean Sherwood number $\left({\mathrm{Sh}}_{\mathrm{ave}}\right)$ varies with shear velocity of the wave motion over the sediment bed, Schmidt number $\left(\mathrm{Sc}\right)$ and the period (T) due to the response of the diffusive boundary layer to the time variable water velocity. Effective diffusive boundary layers develop only at low shear velocities. Where they do, maximum and minimum boundary layer thickness depends on all three independent variables (T, $\mathrm{Sc}\text{,}$ and $U_{*0})\text{.}$ The diffusive boundary layer strongly affects sediment/water mass transfer, i.e., Sherwood numbers. Mass transfer averaged over a period can be substantially less than that produced by steady-state flow at the same $U_{*0}$ and $\mathrm{Sc}\text{.}$ At $\mathrm{Sc}=500\text{,}$ typical for dissolved oxygen, the mass transfer ratio can be reduced to 60% of steady state, depending on the internal wave period (T).