The Geometry of Neutral Paths

Caratheodory's axiomatic development of thermodynamics is applied here to the thermohaline circulation of the ocean. The helicity of the differential form dD for the density change in an isentropic and isohaline ascent, relative to the change in the water column, does not vanish since the ratio of the thermal and haline compressibilities is a function of pressure. The form for relative density consequently lacks an integrating denominator and so there are no surfaces of constant relative density, or so-called neutral surfaces (McDougall and Jackett). As a consequence of a remarkable theorem (Caratheodory), any two points in the thermohaline state space or equivalently in the real space of the ocean are mutually accessible, in the sense that they can be joined by a neutral path. Many of the paths between any two points and possibly all may only be piecewise smooth. The theorem is supported here with analytical examples of neutral paths in state space, and a numerical example of an idealized ocean in real space, for all of which the seawater obeys a relatively simple equation of state. The existence of multiple neutral paths for pairs of close points is explicitly demonstrated. Some such neutral paths take large excursions throughout state space and throughout the ocean basin. The implications for hydrography and for ocean modeling are discussed.