Dynamic Soaring in Finite-Thickness Wind Shears: an Asymptotic Solution

Abstract

Building upon our recent description of dynamic soaring as a succession of small amplitude arcs nearly crosswind, rather than a sequence of half-turns, we formulate an asymptotic expansion for the minimum-wind dynamic soaring cycle when the shear layer between the slow and fast regions has a thin but finite thickness. Our key assumption is that the trajectory remains approximately planar even in finite thickness shears. We obtain an analytical approximation for key flight parameters as a function of the shear layer thickness Δ. In particular we predict that the turn amplitude, maximum climb angle, and cycle altitude scale as Δ [superscript 1=5], Δ[superscript 2=5], and Δ[superscript 3=5], respectively. Our asymptotic expansion is validated against numerical trajectory optimizations and compared with recordings of albatross flights. While the model validity increases with wing loading, it appears to constitute an accurate description down to wing loadings as low as 4kg/m[superscript 2] for oceanic boundary layer soaring, a third that of the wandering albatross.