Propagation of Waves in a Varying and Inhomogeneously Moving Medium

The three-part paper entitled “Forced Baroclinic Ocean Motions” by Cane and Sarachik in the latter half of the 1970s set the stage for the Cane-Zebiak model of ENSO. In this series of papers, waves are treated as carriers to redistribute the forced perturbation energy in space, emphasizing the group velocities of waves and the choice of positive frequency. In this talk, the speaker will first explain the rationales behind Cane and Sarachik’s insistence on positive frequency: the frequency of hydrodynamic waves in a constant medium can be used to represent spatial wave energy density, a result like a photon’s energy in quantum mechanics, and therefore, it must be a non-negative quantity. From there, the speaker will further introduce two principles of waves propagating in a varying and inhomogeneously moving medium. The first principle asserts that the local wave dispersion relationships for such waves remain invariant from the perspective of observers moving with the medium. By incorporating Doppler shift, the wave propagation in this varying medium can be simplified to the corresponding case of a non-moving medium. The second principle involves the conservation of wave energy in a non-dissipative wave system. Leveraging this conservation, a general equation for wave amplitude variation in multidimensional spatiotemporal domain can be derived. Both principles apply to the class of hydrodynamic waves within the linear regime and reconcile previous studies. Examples of applications are provided to demonstrate the power of these two principles.