Consistency relations for gravity and vortical modes in the ocean

Abstract

An incomressible, Boussinesq fluid system on an ƒ-plane supports both the gravity and vortical modes with the vortical mode being the potential vorticity carrier of the system. In the linear limit, the gravity mode represents linear internal waves and the vortical mode zero-frequency geostrophic flows. Consistency relations among cross-spectra of horizontal velocity components and the vertical displacement are studied in projected Fourier spaces. Hypothetical models include the gravity mode, the vortical mode, the linear gravity mode, and the linear vortical mode. Consistency relations for linear internal waves (the linear gravity mode) in the frequency domain have been previously described by Fofonoff [1969, Deep-Sea Research, 16 (Suppl.), 59–71]. Here, additional consistency relations for linear internal waves are obtained in projected Fourier spaces containing the frequency, the orientation of the horizontal wavevector, and the direction of the vertical wavenumber. In addition, five independent consistency relations exist for the pure gravity mode which represents nonlinear, forced or dissipating internal waves. Consistency relations for the pure vortical mode are also obtained. Three exist in any projected Fourier space and can be applied easily to oceanic measurements. The horizontal isotropy and vertical symmetry conditions are also investigated. They are identical for the linear gravity mode and vortical mode.