Observations of spatial flow patterns at the coral colony scale on a shallow reef flat

Although small-scale spatial flow variability can affect both larger-scale circulation patterns and biological processes on coral reefs, there are few direct measurements of spatial flow patterns across horizontal scales <100 m. Here flow patterns on a shallow reef flat were measured at scales from a single colony to several adjacent colonies using an array of acoustic Doppler velocimeters on a diver-operated traverse. We observed recirculation zones immediately behind colonies, reduced currents and elevated dissipation rates in turbulent wakes up to 2 colony diameters downstream and enhanced Reynolds stresses in shear layers around wake peripheries. Flow acceleration zones were observed above and between colonies. Coherent flow structures varied with incident flow speeds; recirculation zones were stronger and wakes were more turbulent in faster flows. Low-frequency (<0.03 Hz) flow variations, for which water excursions were large compared with the colony diameters (Keulegan-Carpenter number, KC >1), had similarspatial patterns to wakes, while higher-frequency variations (0.05-0.1 Hz, KC<1) had no observable spatial structure. On the reef flat, both drag and inertial forces exerted by coral colonies could have significant effects on flow, but within different frequency ranges; drag dominates for low-frequency flow variations and inertial forces dominate for higher frequency variations, including the wave band. Our scaling analyses suggest that spatial flow patterns at colony and patch scales could have important implications or both physical and biological processes at larger reef scales through their effects on forces exerted on the flow, turbulent mixing, and dispersion. © 2013. American Geophysical Union. All Rights Reserved.

Levi-flat Minimal Hypersurfaces in Two-dimensional Complex Space Forms

The purpose of this article is to classify the real hypersurfaces in complex space forms of dimension 2 that are both Levi-flat and minimal. The main results are as follows: When the curvature of the complex space form is nonzero, there is a 1-parameter family of such hypersurfaces. Specifically, for each one-parameter subgroup of the isometry group of the complex space form, there is an essentially unique example that is invariant under this one-parameter subgroup. On the other hand, when the curvature of the space form is zero, i.e., when the space form is complex 2-space with its standard flat metric, there is an additional `exceptional' example that has no continuous symmetries but is invariant under a lattice of translations. Up to isometry and homothety, this is the unique example with no continuous symmetries.

Some examples of special Lagrangian tori

A short paper giving some examples of smooth hypersurfaces M of degree n+1 in complex projective n-space that are defined by real polynomial equations and whose real slice contains a component diffeomorphic to an n-1 torus, which is then special Lagrangian with respect to the Calabi-Yau metric on M.