Tales from topological oceans

Thesis

Functional relationships between variables can offer great illumination to hard problems. In geophysical fluid dynamics these often arise via alignment of gradients, in which case the functional relationship is multivalued; its branches are determined by a topological analysis of the connectedness of level sets, accomplished by the Reeb graph.

The neutral tangent plane, along which mixing in the ocean predominantly occurs, is defined by ∇_nρ = ρ_p ∇_n p, thus a multivalued function relates the in-situ density ρ and the pressure p. With topological analysis, a new approximately new surface is presented, called a topobaric surface. This surface possesses an exact geostrophic stream function, namely the exact geostrophic stream function for neutral surfaces, which ΜcDougall (1989) proved must exist. A closed form expression for this stream function is presented.

For a steady, inviscid, buoyancy conserving fluid, a multivalued function relates the potential vorticity Q and the Bernoulli potential B on buoyancy surfaces. In the Antarctic Circumpolar Current (ACC), level sets of B are dominated by one circumpolar contour, so Q is studied as a single-valued function of B. A tight linear relationship is discovered. Its slope, appropriately non-dimensionalised, provides information about the degree of PV homogenisation on a buoyancy surface, and about the shear stability of the flow regime via Arnol'd's stability theorems.

The real ocean does not obey these assumptions, and Q is not exactly materially conserved. In the time-mean ACC, it is found that Q can vary along streamlines by up to 50%. The causes for these fluctuations are studied, revealing a balance between two large and opposite-signed forcings---mean vertical advection and eddy interfacial form stress---with friction significant near topography. Variation of Q along B contours adds complexity on top of a background state defined by constant Q along B contours.

Authors: Stanley, G

• DOI: 10.5287/ora-mzbngqxnr
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Topology; Southern Ocean; Rossby waves; Multivalued function; Instability; Reeb graph; Potential vorticity; Geostrophic stream function; Neutral surfaces
• Subjects: Oceanography