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

Published online by Cambridge University Press:  20 October 2017

F. T. Mayer Open the ORCID record for F. T. Mayer [Opens in a new window]  and
O. B. Fringer
F. T. Mayer*
Affiliation:
The Bob and Norma Street Environmental Fluid Mechanics Laboratory, Department of Civil and Environmental Engineering, Stanford University, Stanford, CA 94305, USA
O. B. Fringer
Affiliation:
The Bob and Norma Street Environmental Fluid Mechanics Laboratory, Department of Civil and Environmental Engineering, Stanford University, Stanford, CA 94305, USA
*
†Email address for correspondence: [email protected]
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Abstract

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 $Fr_{\unicode[STIX]{x1D6FF}}=u_{0}/\sqrt{g^{\prime }\unicode[STIX]{x1D6FF}}$, where $u_{0}$, $g^{\prime }$, and $\unicode[STIX]{x1D6FF}$ 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, $Fr_{\unicode[STIX]{x1D6FF}}^{2}=Nh_{0}/U=u_{0}^{2}/(g^{\prime }\unicode[STIX]{x1D6FF})$. 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, $Fr_{lee}$.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Stratified flows Geophysical and Geological Flows: Topographic effects
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 831 , 25 November 2017 , R3
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Copyright
© 2017 Cambridge University Press 

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