Jeffrey Early | Searcharxiv

arXiv2026

Toward a Universal Framework for the Internal Gravity Wave Spectrum

The Garrett-Munk (GM) spectrum has long provided a canonical model of the oceanic internal gravity wave field. However, it relies on hydrostatic assumptions and idealized stratification that limit its applicability where non-hydrostatic dynamics, vertical boundary effects, or non-monotonic stratification are important. Here we develop a generalized framework for the internal wave spectrum based on non-hydrostatic vertical modes formulated in horizontal wavenumber-vertical mode space. Energetic orthogonality among wave modes requires that such a formulation be cast in horizontal wavenumber space rather than frequency space. In this formulation, the deformation radius associated with each vertical mode provides a proxy for distinguishing hydrostatic and non-hydrostatic regimes. Vertical modes are obtained numerically from the fixed-K Sturm-Liouville problem, allowing arbitrary stratification and multiple turning depths. Combined with a generalized spectral function, the formulation yields expected distributions of horizontal kinetic, vertical kinetic, and potential energy as functions of depth, frequency, and horizontal wavenumber. Example applications illustrate departures from GM theory associated with boundary effects and non-hydrostatic dynamics, including improved representation of vertical variance and high-frequency vertical kinetic energy, while reproducing observed features of horizontal wavenumber spectra.

physics.flu-dyn↗

arXiv2024

Integral theorems for the gradient of a vector field, with a fluid dynamical application

The familiar divergence and Kelvin-Stokes theorem are generalized by a tensor-valued identity that relates the volume integral of the gradient of a vector field to the integral over the bounding surface of the outer product of the vector field with the exterior normal. The importance of this long-established yet little-known result is discussed. In flat two-dimensional space, it reduces to a relationship between an integral over an area and that over its bounding curve, combining the 2D divergence and Kelvin-Stokes theorems together with two related theorems involving the strain, as is shown through a decomposition using a suitable tensor basis. A fluid dynamical application to oceanic observations along the trajectory of a moving platform is given. The potential generalization of the generalized identity to curved two-dimensional surfaces is considered and is shown not to hold. Finally, the paper includes a substantial background section on tensor analysis, and presents results in both symbolic notation and index notation in order to emphasize the correspondence between these two notational systems.

physics.flu-dyn↗