arXiv subjects

Alex Doak

Publications and source records attributed to Alex Doak.

6 recordsLinked to original sources

arXiv2025

A classification of mode-1 internal solitary waves in a three-layer fluid

We explore the bifurcation structure of mode-1 solitary waves in a three-layer fluid confined between two rigid boundaries. A recent study (Lamb, J. Fluid Mech. 2023, 962, A17) proposed a method to predict the coexistence of solitary waves with opposite polarity in a continuously stratified fluid with a double pycnocline by examining the conjugate states for the Euler equations. We extend this line of inquiry to a piecewise-constant three-layer stratification, taking advantage of the fact that the conjugate states for the Euler equations are exactly preserved by the strongly nonlinear model that we will refer to as the three-layer Miyata-Maltseva-Choi-Camassa (MMCC3) equations. In this reduced setting, solitary waves are governed by a Hamiltonian system with two degrees of freedom, whose critical points are used to explain the bifurcation structure. Through this analysis, we also discover families of solutions that have not been previously reported. Using the shared conjugate state structure between the MMCC3 model and the full Euler equations, we propose criteria for distinguishing the full range of solution behaviours. This alignment between the reduced and full models provides strong evidence that partitioning the parameter space into regions associated with distinct solution types is valid within both theories. This classification is further substantiated by numerical solutions to both models, which show excellent agreement.

physics.flu-dyn↗

arXiv2025

Large-amplitude periodic solutions to the steady Euler equations with piecewise constant vorticity

We consider steady solutions to the incompressible Euler equations in a two-dimensional channel with rigid walls. The flow consists of two periodic layers of constant vorticity separated by an unknown interface. Using global bifurcation theory, we rigorously construct curves of solutions that terminate either with stagnation on the interface or when the conformal equivalence between one of the layers and a strip breaks down in a $C^1$ sense. We give numerical evidence that, depending on parameters, these occur either as a corner forming on the interface or as one of the layers developing regions of arbitrarily thin width. Our proof relies on a novel formulation of the problem as an elliptic system for the velocity components in each layer, conformal mappings for each layer, and a horizontal distortion which makes these mappings agree on the interface. This appears to be the first local formulation for a multi-layer problem which allows for both overhanging wave profiles and stagnation points.

math.AP↗

arXiv2025

Stokes waves in rotational flows: internal stagnation and overhanging profiles

Periodic travelling waves at the free surface of an incompressible inviscid fluid in two dimensions under gravity are numerically computed for an arbitrary vorticity distribution. The fluid domain over one period is conformally mapped from a fixed rectangular one, where the governing equations along with the conformal mapping are solved using a finite difference scheme. This approach accommodates internal stagnation points, critical layers, and overhanging profiles, thereby overcoming limitations of previous studies. The numerical method is validated through comparisons with known solutions for zero and constant vorticity. Novel solutions are presented for affine vorticity functions and a two-layer constant vorticity scenario.

physics.flu-dyn↗

arXiv2022

Flow structure beneath periodic waves with constant vorticity under normal electric fields

Waves with constant vorticity and electrohydrodynamics flows are two topics in fluid dynamics that have attracted much attention from scientists for both the mathematical challenge and their industrial applications. The coupling of electric fields and vorticity is of significant research interest. In this paper, we study the flow structure of steady periodic travelling waves with constant vorticity on a dielectric fluid under the effect of normal electric fields. Through the conformal mapping technique combined with pseudo-spectral numerical methods, we develop an approach that allows us to conclude that the flow can have zero, two or three stagnation points according to variations in the voltage potential. We describe in detail the recirculation zones that emerge together with the stagnation points. Besides, we show that the number of local maxima of the pressure on the bottom boundary is intrinsically connected to the saddle points.

physics.flu-dyn↗

arXiv2022

Large mode-2 internal solitary waves in three-layer flows

In this paper, we investigate mode-2 solitary waves in a three-layer stratified flow model. Localised travelling wave solutions to both the fully nonlinear problem (Euler equations), and the three-layer Miyata-Choi-Camassa equations are found numerically and compared. Mode-2 solitary waves with speeds slower than the linear mode-1 long-wave speed are typically generalised solitary waves with infinite tails consisting of a resonant mode-one periodic wave train. Herein we evidence the existence of mode-2 embedded solitary waves, that is, we show that for specific values of the parameters, the amplitude of the oscillations in the tail are zero. For sufficiently thick middle layers, we also find branches of mode-2 solitary waves with speeds that extend beyond the mode-1 linear waves and are no longer embedded. In addition, we show how large amplitude embedded solitary waves are intimately linked to the conjugate states of the problem.

physics.flu-dyn↗

arXiv2022

Nonlinear shallow-water waves with vertical odd viscosity

The breaking of detailed balance in fluids through Coriolis forces or odd-viscous stresses has profound effects on the dynamics of surface waves. Here we explore both weakly and strongly non-linear waves in a three-dimensional fluid with vertical odd viscosity with and without the Coriolis effect. Our model describes the free surface of a shallow fluid composed of nearly vertical vortex filaments, which all stand perpendicular to the surface. We find that the odd viscosity in this configuration induces previously unexplored non-linear effects in shallow-water waves, arising from both stresses on the surface and stress gradients in the bulk. By assuming weak nonlinearity, we find reduced equations including Korteweg-de Vries (KdV), Ostrovsky, and Kadomtsev-Petviashvilli (KP) equations with modified coefficients. At sufficiently large odd viscosity, the dispersion changes sign, allowing for compact two-dimensional solitary waves. We show that odd viscosity and surface tension have the same effect on the free surface, but distinct signatures in the fluid flow. Our results describe the collective dynamics of many-vortex systems, which can also occur in oceanic and atmospheric geophysics.

physics.flu-dyn↗