A Representation of Bounded Viscous Flow Based on Hodge Decomposition of Wall Impulse

Impulse exchange at the surface of the ocean and the fractal dimension of drifter trajectories

“…The foregoing exchange of impulse is conducted in a way that is consistent with the boundary conditions to be satisfied at ∂D, where we require that the two fluids do not interpenetrate, and that there is no "slip" at their mutual boundary. This latter "no-slip" condition is consistent with the creation of an impulse in the form of a thin vortex doublet sheet, coincident with the interface (there is a close relationship between elements of impulse of compact support and vortex elements; see Summers, 2000aSummers, , b, 2001. Once an impulse is created, it proceeds to evolve from the interface into the surrounding fluid according to an equation of motion introduced by Oseledets (1988).…”

“…Considering a segment of length aligned in the x-direction and centered at (x o , z o ), the velocity it induces at (x, z) is determined in Eqs. (22) and (23) in Summers (2000a).…”

“…(2). In the following section, we develop a relevant gradient boundary condition for φ (see also Summers, 2000aSummers, , b, 2001). …”

“…The choice of partition parameter may be based on a physical argument: choosing ∼ O(ν) is consistent with a flow in which translation due to inertial contributions is comparable to viscous displacement (see Summers, 2000a). This would represent the condition usually associated with the formation of a laminar boundary layer, for example.…”

“…The Euler Equation (13) is modelled numerically by evaluating the interaction between each pair of impulse elements (including their images). Note that these require modification in order to express the relative rotation between sheet elements (Summers, 2000a). We determine the advection of the resulting n-body system of impulse elements by using Runge-Kutta integration.…”