Steady Large-Scale Ocean Flows in Spherical Coordinates

We show that rotating spherical coordinates can and should be used for the study of steady, large-scale ocean flows, and describe the role of the stream function in simplifying the governing equations and for generating solutions. INTRODUCTION Scientific advances typically evolve from the observation of natural phenomena, noting patterns and regularities; causal theories are then inferred. Such theories play an important role in the furtherance of knowledge by explaining the significance of what is already known and making predictions that might generate novel discoveries, which are subsequently subjected to experimental evaluation. Thus, scientific progress toward ever more accurate and comprehensive theories of nature occurs by experiment and observation, closely linked to theoretical developments. To date, physical oceanography has been primarily an observational subject, and spectacular technological improvements have, in recent years, allowed the gathering of ocean data on a scale that would have been unimaginable to early scientists. Moreover, the computing power that is available now enables researchers to produce high-resolution numerical simulations of large-scale ocean models, and these can be compared with the wealth of available field data. However, we must be mindful that numerical solutions, no matter how accurate they may appear, suffer from discretization errors that naturally arise when a fluid continuum is represented numerically. Mathematics, on the other hand, is not merely useful in making specific observations intelligible, but it is instrumental in providing fundamental insights that are essential when we are attempting to understand the general motion of the ocean or trying to interpret numerical simulations. Indeed, ocean-data gathering is far from comprehensive and remains remarkably difficult, while numerical simulations always involve some systematic errors, for example, due to the minimal spatial resolving power that is available. A feedback from the theoretical standpoint is essential if major failures in understanding are to be avoided, and to ameliorate the effects of systematic errors. Furthermore, while every location in the ocean is different, at least in some detail, it is important to move beyond regional features and aim to describe global ocean-flow characteristics that apply to long-duration features and change over fairly large (horizontal) distances (i.e., about 100 km and larger; see the discussion in Wunsch, 2015). Earth is nearly an oblate spheroid (an ellipse rotated about its minor axis), with the equatorial bulge a consequence of the planet’s rotation history. However, because no dynamical consequences of the small deviation from a perfect sphere—the polar radius being about 21 km shorter that the equatorial one (of length 6,378 km)—have been observed in ocean flows, a spherical Earth is an adequate model (see Wunsch, 2015). Depicting Earth as a sphere, and the fact that the ocean is a very thin layer on the spherical surface (with a mean thickness of about 0.1% of the radius: most oceanic depths are between 3 km and 5.5 km), has major consequences for the dynamics of ocean flows. The inherent complexity of ocean flows, mainly due to the occurrence of nonlinear interactions on a wide array of spatial and temporal scales, precludes any attempt to solve the full governing equations with associated boundary and initial conditions. Thus, the challenge is to develop mathematically accurate and consistent approximate models for specific phenomena. Because every model omits some physical processes, but aims to identify the fundamental mechanisms on the relevant scales, it is essential to highlight overarching structural properties of these flows (e.g., conservation principles and quantities whose space-time evolution can be estimated with accuracy). In particular, large-scale ocean flows are affected by Earth’s rotation