An unambiguous definition of the Froude number for lee waves in the deep ocean

There is a long-standing debate in the literature of stratified flows over topography concerning the correct dimensionless number to refer to as a Froude number. Common definitions using external quantities of the flow include U/(ND), U/(Nh(0)), and Uk/N, where U and N are, respectively, scales for the background velocity and buoyancy frequency, D is the depth, and h(0) and k(-1) are, respectively, height and width scales of the topography. It is also possible to define an internal Froude number Fr-delta = u(0)/root g'delta where u(0), g' and delta are, respectively, the characteristic velocity, reduced gravity, and vertical length scale of the perturbation above the topography. For the case of hydrostatic lee waves in a deep ocean, both U/(ND) and Uk/N are insignificantly small, rendering the dimensionless number Nh(0)/U the only relevant dynamical parameter. However, although it appears to be an inverse Froude number, such an interpretation is incorrect. By non-dimensionalizing the stratified Euler equations describing the flow of an infinitely deep fluid over topography, we show that Nh(0)/U is in fact the square of the internal Froude number because it can identically be written in terms of the inner variables, Fr-delta(2) = Nh(0)/U u(0)(2)/(g'delta). Our scaling also identifies Nh(0)/U as the ratio of the vertical velocity scale within the lee wave to the group velocity of the lee wave, which we term the vertical Froude number, Fr-vert = Nh(0)/U = w(0)/c(g). To encapsulate such behaviour, we suggest referring to Nh(0)/U as the lee-wave Froude number, Fr-lee.