A review on Lamb? atmospheric oscillations using initial value problem approach

摘要

Waves at a surface of discontinuity in the atmosphere were analysed in 1910 by Lamb, who derived, using normal mode approach, an analytical dispersion relation for a discrete mode (surface mode). Lamb examined the case of waves propagated along a horizontal plane where the equilibrium temperature is discontinuous. For simplicity, the upper and the lower regions are considered incompressible. The oscillations are treated in the ideal (dissipationless) limit and the uniform gravitational acceleration is taken to be co-aligned with the prevailing temperature gradient. In this work, in order to show how the modes appear in the response of a surface discontinuity to an initial perturbation, we consider the initial value problem (IPV). The main difference from the standard analysis is that solutions to the linearized equations of motion which satisfy general conditions are obtained in terms of Fourier-Laplace transform of the hydrodynamics variables. These transforms can be inverted explicitly to express the fluid variables as integrals of Green? functions multiplied by initial data. In addition to discrete mode (surface mode), sets of continuum modes due to branch cuts in the complex plane, not treated explicitly in the literature, appears.