Motivated by wind blowing over water, we use asymptotic methods to study the evolution of short wavelength interfacial waves driven by the combined action of these flows. We solve the Rayleigh equation for the stability of the shear flow, and construct a uniformly valid approximation for the perturbed streamfunction, or eigenfunction. We then expand the real part of the eigenvalue, the phase speed, in a power series of the inverse wavenumber and show that the imaginary part is exponentially small. We give expressions for the growth rates of the Miles (J. Fluid Mech., vol. 3, 1957, pp. 185–204) and rippling (e.g. Young & Wolfe, J. Fluid Mech., vol. 739, 2014, pp. 276–307) instabilities that are valid for an arbitrary shear flow. The accuracy of the results is demonstrated by a comparison with the exact solution of the eigenvalue problem in the case when both the wind and the current have an exponential profile.

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.

We examine the buckling shape and critical compression of confined inhomogeneous composite sheets lying on a liquid foundation. The buckling modes are controlled by the bending stiffness of the sheet, the density of the substrate, and the size and the spatially dependent elastic coefficients of the sheet. We solve the beam equation describing the mechanical equilibrium of a sheet when its bending stiffness varies parallel to the direction of confinement. The case of a homogeneous bending stiffness exhibits a degeneracy of wrinkled states for certain lengths of the confined sheet; we explain this degeneracy using an asymptotic analysis valid for long sheets, and show that it corresponds to the switching of the sheet between symmetric and antisymmetric buckling modes. This degeneracy disappears for spatially dependent elastic coefficients. Medium length sheets buckle similarly to their homogeneous counterparts, whereas the wrinkled states in large length sheets concentrate the bending energy towards the soft regions of the sheet. We examine the buckling shape and critical compression of confined inhomogeneous composite sheets lying on a liquid foundation.

When wind blows over water, ripples are generated on the water surface. These ripples can be regarded as perturbations of the wind field, which is modelled as a parallel inviscid flow. For a given wavenumber k, the perturbed streamfunction of the wind field and the complex phase speed are the eigenfunction and the eigenvalue of the so-called Rayleigh equation in a semi-infinite domain. Because of the small air–water density ratio, ρa/ρw≡ϵ≪1, the wind and the ripples are weakly coupled, and the eigenvalue problem can be solved perturbatively. At the leading order, the eigenvalue is equal to the phase speed c0 of surface waves. At order ϵ, the eigenvalue has a finite imaginary part, which implies growth. Miles (J. Fluid Mech., vol. 3, 1957, pp. 185–204) showed that the growth rate is proportional to the square modulus of the leading-order eigenfunction evaluated at the so-called critical level z=zc, where the wind speed is equal to c0 and the waves extract energy from the wind. Here, we construct uniform asymptotic approximations of the leading-order eigenfunction for long waves, which we use to calculate the growth rate as a function of k. In the strong wind limit, we find that the fastest growing wave is such that the aerodynamic pressure is in phase with the wave slope. The results are confirmed numerically.

In 1948, Casimir predicted a net attractive force between two perfectly conducting parallel plates due to electromagnetic vacuum fluctuations. By analogy, the interaction of two ships on a wavy sea has been named Maritime Casimir effect. This is an example of force generation in non-equilibrium systems. Lee, Vella and Wettlaufer showed it to be oscillatory as it is induced by the sharply peaked energy spectrum measured in the sixties by Pierson and Moskowitz for a fully developed sea; a sea whose state is independent of the distance over which the wind blows and the time for which it has been blowing. The aim of this project is to construct a theory for that spectrum and understand how the Maritime Casimir effect emerges from wind-wave interaction. Waves in the absence of wind, so-called water waves, are mainly characterized by dispersion and weak non-linearity. The coupling of both results in the instability of a wave packet to side-band perturbations in deep water. The growth rate can be calculated thanks to a non-linear Schrödinger equation, which is a universal model for weakly non-linear waves in a dispersive medium. Furthermore, this instability can be understood in the even more general framework of resonant wave-wave interaction. The evolution of deep water gravity waves is actually a sum of four-wave interaction processes and triadic interactions should be added for capillary waves. That evolution is strongly affected by the presence of turbulent wind because it transfers energy to the waves. The growth rate of wind waves was calculated by Miles in 1957 on the basis of the weak air-water coupling. His formula involves the solution of the hydrodynamic Rayleigh equation at the critical level, which is the height at which the phase speed of the wave is equal to the wind speed. We develop an efficient numerical scheme to compute it and then compare the theory with the observational data compiled by Plant. We eventually propose asymptotic solutions of the Rayleigh equation for a generic wind profile, which will be useful to get a better understanding of the experimental results.

We use asymptotic methods to study the evolution of short wavelength interfacial waves driven by the combined action of wind and current. We solve the Rayleigh equation for the stability of the shear flow, and construct a uniformly valid approximation for the perturbed streamfunction, or eigenfunction. We then expand the real part of the eigenvalue, the phase speed, in a power series of the inverse wavenumber and show that the imaginary part is exponentially small. We give expressions for the growth rates of the Miles (1957) and rippling (e.g., Young & Wolfe 2013) instabilities that are valid for an arbitrary shear flow. The accuracy of the results is demonstrated by a comparison with the exact solution of the eigenvalue problem in the case when both the wind and the current have an exponential profile.

We examine the buckling shape and critical compression of confined inhomogeneous composite sheets lying on a liquid foundation. The buckling modes are controlled by the bending stiffness of the sheet, the density of the substrate, and the size and the spatially dependent elastic coefficients of the sheet. We solve the (linearized) Föppl–von Kármán equations describing the mechanical equilibrium of a sheet when its bending stiffness varies parallel to the direction of confinement. The case of a homogeneous bending stiffness exhibits a degeneracy of wrinkled states for certain sizes of the confined sheet. This degeneracy disappears for spatially dependent elastic coefficients. Medium length sheets buckle similarly to their homogeneous counterparts, whereas the wrinkled states in large length sheets localize the bending energy towards the soft regions of the sheet.