A complex analytic approach to some fluid flows

Over the last decades considerable progress towards the understanding of water flows has been achieved. Since typically the horizontal velocities are much larger than the vertical velocity, most approaches neglect the latter. However, the vertical flow is very important. For example, in coastal regions the movement of cold, deep, nutrient-rich water to the surface layer occurs when the wind is deflected by the Earths rotation (Coriolis force) parallel to the coast. Such regions account for about one quarter of the global fish catch. On the other hand, the downward movement of surface water is essential for the Arctic Ocean, transporting oxygen-rich surface water to the deep seafloor. The vertical water flow can only be understood through its interaction with the horizontal one, in the setting of three-dimensional water flows. In this context, exact explicit solutions are important because they enable detailed insight into the dynamics. Typically, perturbations of such exact solutions lead to more general flows that can be studied in detail. Large-scale ocean flows have to account for the three-dimensional Coriolis effect due to the Earths rotation. Moreover, while viscosity is negligible for flows at intermediate depths, it cannot be neglected for near-surface flows (that are typically wind-generated). We aim to investigate three types of geophysical flows for which we expect to be able to find several families of explicit exact solutions by describing the individual particle paths: 1. Inviscid homogeneous flows at intermediate depths 2. Wind-driven arctic flows in the near-surface layer 3. Arctic flows at intermediate depths The effect of the wind is limited in arctic regions to a subsurface ocean layer that is about 50 m deep and sea ice attenuates incoming ocean waves. While for tropical ocean regions the flow at intermediate depths is typically homogeneous, for arctic ocean flow at intermediate depths vertical stratification has to be accounted for, due to temperature and salinity effects. Since vertical motion is also important in smaller-scale water flows, a fourth topic to be addressed is the vertical structure of the most regular two-dimensional water flows, the Stokes waves. For these studies it is necessary to develop a framework suitable for incorporating nonlinear effects. While for weak nonlinear effects, effective approximation methods analytical as well as numerical are available, for typically nonlinear flows these approximations remain unreliable and advances require finding and exploiting structural properties. These are revealed by experiments, field data or numerical simulations, which under specific circumstances indicate recurring features that might be valid in a broader context. A theoretical framework will be developed with the purpose of producing a systematic way to sort out the dominant factors, thus providing a solid starting point for more detailed subsequent investigations.