Fourier Representation of Quadratic Friction

Abstract

If a current is composed of a number of constituents with different frequencies, then quadratic friction may be analyzed at the same frequencies. The ratios of the constituents of the friction differ from the ratios for the current itself, with a classic result being that for unidirectional flow a very weak current constituent experiences proportionately 50% more friction than a strong constituent. Here, exact results for the magnitude of the friction constituents are derived and confirmed numerically. The results are applied to the tidal currents in Juan de Fuca Strait and the Strait of Georgia, showing that minor constituents experience proportionately more friction than the main constituent by an amount that varies spatially but is typically less than the classic result of 50%. For two-dimensional currents it is shown that, if there are two current constituents with the same ellipticity and major axis direction, the friction coefficients are separable functions of the current constituent ratio and the ellipticity. Some results are derived for two constituents with different ellipticity and major axis direction. For the case of two constituents with rectilinear but misaligned currents, each constituent experiences friction inclined at an angle to its current. Last, the effect of a tidal current on the bottom friction experienced by a steady flow is investigated for arbitrary relative magnitudes and directions of the tide and steady flow. In particular, the inclination of the mean friction to the mean flow is quantified.