Waves and instabilities through the lens of asymptotic analysis

Abstract: Understanding the interaction of water waves with winds and marine currents is a fundamental problem in geophysical fluid dynamics. From the point of view of hydrodynamic stability, surface waves are regarded as perturbations of an inviscid parallel shear flow modeling the wind in the air and the current in the water. For small two-dimensional perturbations, the linearization of the Euler equation of motion yields an eigenvalue problem to be solved for a given wavenumber k. The eigenfunction is a streamfunction obeying the so-called Rayleigh equation. The eigenvalue is a complex phase speed, c, whose real part is the actual phase speed of sheared waves while the imaginary part of kc is the growth rate of the wave amplitude. Using the smallness of the air/water density ratio and assuming no flow in the water, Miles solved this eigenvalue problem perturbatively in 1957. He uncovered an instability of the wind field due to a critical layer in the air, where the wind speed equals the phase speed of free surface waves, and showed that the growth rate of wind-waves is proportional to the square modulus of the solution of the Rayleigh equation at the critical level. This level is a regular singular point, which makes the resolution of the Rayleigh equation challenging. For that reason, an explicit expression of the growth rate of the Miles instability as a function of the wavenumber was lacking. Firstly, I designed a numerical scheme to solve the Rayleigh equation for an arbitrary monotonic wind profile. Secondly, I solved it analytically using asymptotic methods for long and short waves.In physical oceanography, a standard model for the mean turbulent wind field is the logarithmic profile, which contains only one length scale: the roughness length, z0 ~1 mm, accounting for the presence of waves on the water surface. I am interested in waves propagating due to gravity and surface tension, which have wavelengths ranging from a few millimeters to hundreds of meters. Hence, a natural small parameter is kz0, which I used to obtain long wave solutions of the Rayleigh equation, and subsequently the growth rate of the Miles instability. The comparison with both numerical and measured growth rates is excellent. Furthermore, I approximated the maximum growth rate in the strong wind limit, and inferred that the fastest growing wave is such that the aerodynamic pressure is in phase with the wave slope.I also considered the short wave limit of the eigenvalue problem. Using 1/(kL) as a small parameter, where L is a characteristic length scale of the shear, I found general asymptotic solutions for interfacial waves in presence of a wind and a current, where the density ratio does not need to be small. One application concerns the mixing of elements at the surface of white dwarfs. Moreover, short wave asymptotics provide insights on another instability. When waves have a phase speed that matches the current speed, there is another critical layer, in the water, which is responsible for the so-called rippling instability. I obtained a general asymptotic formula for the growth rate of this instability.Finally, I used my experience in solving eigenvalue problems to study, in collaboration with other researchers, wrinkles in thin elastic sheets floating on a liquid foundation. We had to solve a fourth order eigenvalue problem where the eigenvalue is the compressive load imposed on the sheet and the eigenfunction is the vertical displacement. For homogeneous sheets, the bending stiffness of the sheet is constant and the eigenvalue problem could be solved analytically. We found that the buckling shape has a symmetric and an antisymmetric mode. The mode associated with the minimum compressive load depends on the size of the confined sheet. Hence, there are changes of symmetry at certain confinement sizes for which the buckling shape is degenerate. We numerically showed that this degeneracy disappears for composite sheets, whose bending stiffness depends on space due to the presence of liquid inclusions.