The effect of density gradient on boundary flow

Based on Prandtl's mixing-length theory, this study proposes new theoretical formulae of velocity profiles resulting from the forcing by a constant horizontal buoyancy gradient in unstratified and stably stratified flows. Based on the one-dimensional water column momentum equation, the vertical turbulent shearing stress profile is found to deviate from a linear distribution and follow a parabolic distribution, differing from that in neutral flow. The shearing stress curves upward with the current following the density gradient, and curves downward with the currents opposite to the density gradient. For a constant eddy viscosity, the well-known estuarine circulation is obtained through the parabolic shearing distribution. For a vertically parabolic eddy viscosity, the new-proposed velocity profile by Burchard & Hetland (2010) is obtained. In this paper, we estimate the viscosity profile based on Prandtl's mixing-length theory and then derive the new formulae of the velocity profiles. Through comparison with the numerical turbulence model, the velocity profiles presented in the paper agree well with the numerical results. In addition, the new velocity profiles can be applied to determine the valid range and evaluate errors of the log-fit in baroclinic boundary flows.