Conductivity of a two-dimensional system with a periodic distribution of circular inclusions

Wave propagation in rotating shallow water in the presence of small-scale topography

“…The outer solution Φ + takes periodic boundary conditions on the edge of Ω. At the cylinder edge, the boundary conditions are ) is a canonical problem arising in many areas of mathematical physics, most classically heat conduction through a two-dimensional porous medium with cylindrical occlusions (Rayleigh 1892;Keller 1963;McPhedran, Poladian & Milton 1988;Balagurov & Kashin 2001) but also in electrostatics and optics (McPhedran & McKenzie 1980) and in determining dielectric permittivity (Godin 2013). Our approach to solving (3.6a,b) and (3.7), closely follows that of Godin (2013), and is based on a multipole expansion which exploits the rapid convergence of the Laurent series of the Weierstrass zeta function.…”

“…Godin's method (see also Balagurov & Kashin 2001) results in an infinite linear system which must be inverted in order to solve for Φ + and Φ − and thus obtain H eff exactly. However, it turns out that truncation of the system at very low order results in a sequence of increasingly accurate Padé approximants to the exact solution.…”

“…Here the multipole method used to solve the cell problems (3.6a,b)-(3.7) and (3.9)-(3.10) in § 3 is described. The method closely follows that of Godin (2013) (see also Balagurov & Kashin 2001). First, G ± and Ψ ± α in (3.9)-(3.10) are expanded into their real and imaginary parts G ± = G ± R + iG ± I , and…”

“…For two-dimensional topography H eff lies somewhere between the two bounds, and one aim here is to quantify its exact dependence on topographic height and area fraction for some idealised two-dimensional topographies, in particular arrays of periodic cylinders for which highly accurate asymptotic solutions exist (e.g. Balagurov & Kashin 2001;Godin 2013). Another key question is how the classical wave equation analysis is modified by the introduction of rotation, i.e.…”

“…Particular attention is given to cylindrical seamounts, because the multipole expansion method of e.g. Balagurov & Kashin (2001) and Godin (2013) can be used in this case to obtain highly accurate asymptotic solutions to the cell problems, including the new 'rotating' cell problem which arises from the rSWE. The outcome is various means to determine the topographically induced corrections to the dispersion relations of Kelvin, Poincaré and Rossby waves, including an explicit formula valid for Rossby waves in the presence of finite amplitude topography, complementing the quasi-geostrophic results of Benilov (2000) and Vanneste (2000b).…”