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Adrian Constantin

Publications and source records attributed to Adrian Constantin.

At least 19 recordsLinked to original sources
arXiv2026

Pressure beneath a periodic travelling water-wave in constant-vorticity flow over a flat bed

We investigate within the framework of linear theory the behaviour of the total (hydrodynamic) pressure and of the dynamic pressure in a regular wave train which propagates at the surface of water with a flat bed in a flow with constant vorticity. We show that nonzero vorticity, the hallmark of a non-uniform underlying current, may strongly alter the behaviour with respect to the case of irrotational flows, for which the maximum and minimum of the dynamic pressure always occur at the wave crest and at the wave trough, respectively (the extrema of the dynamic pressure may occur along the flat bed or along the critical level, depending on the vorticity strength). While vorticity does not modify the increase of the hydrodynamic pressure with depth, it can significantly alter the location of the extrema of the hydrodynamic pressure at a fixed depth level.

physics.flu-dyn↗
arXiv2025

Classification of atmospheric traveling waves at cloud level

We classify within the quasi-geostrophic framework all types of traveling waves in zonal bands of the planetary atmosphere at cloud level according to their wave speeds. This classification pertains to waves of all amplitudes, going beyond the small-amplitude perturbative regime. It provides a structurally robust criterion for determining which traveling-wave profiles are dynamically possible and we show that each wave classification type was observed on Jupiter or Saturn. Building on this classification, we also investigate the related rigidity issue for large-amplitude traveling waves and waves propagating near shear flows. Our study offers a unified quantitative characterization of the intrinsic constraints for traveling waves in the quasi-geostrophic regime of planetary atmospheric flow.

math.AP↗
arXiv2025

On the propagation of mountain waves: linear theory

We derive and establish a solution concept for the linear mountain wave problem in two dimensions. After linearizing the governing equations and a change of variables, the problem can be stated as a Dirichlet boundary value problem for a Helmholtz equation in terms of the vertical wind profile in the upper half-plane, with altitude-dependent potential (the Scorer parameter). To single out the correct solution, we have to make use of a radiation condition which is, due to the different physical situation, different from the classical Sommerfeld radiation condition for electromagnetic or acoustic waves. We rigorously develop a transform method and construct the physically correct solution, following Lyra's monotonicity criterion for mountain waves. In this procedure, we clearly recognize the two typical types of mountain waves: vertically propagating waves and trapped lee waves. This paper is the first rigorous work on Helmholtz-like equations in the upper half-plane subject to such a non-classical radiation condition.

math.AP↗
arXiv2025

The onset of instability for zonal stratospheric flows

We investigate some qualitative aspects of the dynamics of the Euler equation on a rotating sphere that are relevant or stratospheric flows. Zonal flow dominates the dynamics of the stratosphere and for most known planetary stratospheres the observed flow pattern is a small perturbation of an n-jet, which corresponds to choosing the Legendre polynomial of degree n as the stream function. Since the 1-jet and the 2-jet are stable, the main interest is the onset of instability for the 3-jet. We confirm long standing conjectures based on numerical simulations by proving that the 3-jet is linearly unstable if and only if the rotation rate belongs to a critical interval. Turning to the nonlinear problem, we prove that linear instability implies nonlinear instability and that, as the rotation rate goes to infinity, nearby traveling waves change gradually from a cat's eyes streamline pattern to a profile with no stagnation points.

math.AP↗
arXiv2024

The onset of filamentation on vorticity interfaces in two-dimensional Euler flows

Two-dimensional Euler flows, in the plane or on simple surfaces, possess a material invariant, namely the scalar vorticity normal to the surface. Consequently, flows with piecewise-uniform vorticity remain that way, and moreover evolve in a way which is entirely determined by the instantaneous shapes of the contours (interfaces) separating different regions of vorticity -- this is known as `Contour Dynamics'. Unsteady vorticity contours or interfaces often grow in complexity (lengthen and fold), either as a result of vortex interactions (like merger) or `filamentation'. In the latter, wave disturbances riding on a background, equilibrium contour shape appear to inevitably steepen and break, forming filaments, repeatedly -- and perhaps endlessly. Here, we revisit the onset of filamentation. Building upon previous work and using a weakly-nonlinear expansion to third order in wave amplitude, we derive a universal, parameter-free amplitude equation which applies (with a minor change) both to a straight interface and a circular patch in the plane, as well as circular vortex patches on the surface of a sphere. We show that this equation possesses a local, self-similar form describing the finite-time blow up of the wave slope (in a rescaled long time proportional to the inverse square of the initial wave amplitude). We present numerical evidence for this self-similar blow-up solution, and for the conjecture that almost all initial conditions lead to finite-time blow up. In the full Contour Dynamics equations, this corresponds to the onset of filamentation.

physics.flu-dyn↗
arXiv2024

On the onset of filamentation on two-dimensional vorticity interfaces

We study an asymptotic nonlinear model for filamention on two-dimensional vorticity interfaces. Different re-formulations of the model equation reveal its underlying structural properties. They enable us to construct global weak solutions and to prove the existence of traveling waves.

math.AP↗
arXiv2021

Stratospheric planetary flows from the perspective of the Euler equation on a rotating sphere

This article is devoted to stationary solutions of Euler's equation on a rotating sphere, and to their relevance to the dynamics of stratospheric flows in the atmosphere of the outer planets of our solar system and in polar regions of the Earth. For the Euler equation, under appropriate conditions, rigidity results are established, ensuring that the solutions are either zonal or rotated zonal solutions. A natural analogue of Arnold's stability criterion is proved. In both cases, the lowest mode Rossby-Haurwitz stationary solutions (more precisely, those whose stream functions belong to the sum of the first two eigenspaces of the Laplace-Beltrami operator) appear as limiting cases. We study the stability properties of these critical stationary solutions. Results on the local and global bifurcation of non-zonal stationary solutions from classical Rossby-Haurwitz waves are also obtained. Finally, we show that stationary solutions of the Euler equation on a rotating sphere are building blocks for travelling-wave solutions of the 3D system that describes the leading order dynamics of stratospheric planetary flows, capturing the characteristic decrease of density and increase of temperature with height in this region of the atmosphere.

math.AP↗
arXiv2020

Liouville chains: new hybrid vortex equilibria of the 2D Euler equation

A large class of new exact solutions to the steady, incompressible Euler equation on the plane is presented. These hybrid solutions consist of a set of stationary point vortices embedded in a background sea of Liouville-type vorticity that is exponentially related to the stream function. The input to the construction is a "pure" point vortex equilibrium in a background irrotational flow. Pure point vortex equilibria also appear as a parameter $A$ in the hybrid solutions approaches the limits $A\to 0,\infty$. While $A\to 0$ reproduces the input equilibrium, $A\to\infty$ produces a new pure point vortex equilibrium. We refer to the family of hybrid equilibria continuously parametrised by $A$ as a "Liouville link". In some cases, the emergent point vortex equilibrium as $A\to\infty$ can itself be the input for a second family of hybrid equilibria linking, in a limit, to yet another pure point vortex equilibrium. In this way, Liouville links together form a "Liouville chain". We discuss several examples of Liouville chains and demonstrate that they can have a finite or an infinite number of links. We show here that the class of hybrid solutions found by Crowdy (2003) and by Krishnamurthy et al. (2019) form the first two links in one such infinite chain. We also show that the stationary point vortex equilibria recently studied by Krishnamurthy et al. (2020) can be interpreted as the limits of a Liouville link. Our results point to a rich theoretical structure underlying this class of equilibria of the 2D Euler equation.

physics.flu-dyn↗
arXiv2018

Large-amplitude steady downstream water waves

We study wave-current interactions in two-dimensional water flows of constant vorticity over a flat bed. For large-amplitude periodic traveling waves that propagate at the water surface in the same direction as the underlying current (downstream waves), we prove explicit uniform bounds for their amplitude. In particular, our estimates show that the maximum amplitude of the waves becomes vanishingly small as the vorticity increases without limit. We also prove that the downstream waves on a global bifurcating branch are never overhanging, and that their mass flux and Bernoulli constant are uniformly bounded.

math.AP↗
arXiv2016

Dressing Method for the Degasperis-Procesi Equation

The soliton solutions of the Degasperis-Procesi equations are constructed by the implementation of the dressing method. The form of the one and two soliton solutions coincides with the form obtained by Hirota's method.

nlin.SI↗
arXiv2014

Global bifurcation of steady gravity water waves with critical layers

We construct large families of two-dimensional travelling water waves propagating under the influence of gravity in a flow of constant vorticity over a flat bed. A Riemann-Hilbert problem approach is used to recast the governing equations as a one-dimensional elliptic pseudo-differential equation with a scalar constraint. The structural properties of this formulation, which arises as the Euler-Lagrange equation of an energy functional, enable us to develop a theory of analytic global bifurcation.

math.AP↗
arXiv2012

Inverse Scattering Transform for the Degasperis-Procesi Equation

We develop the Inverse Scattering Transform (IST) method for the Degasperis-Procesi equation. The spectral problem is an $\mathfrak{sl}(3)$ Zakharov-Shabat problem with constant boundary conditions and finite reduction group. The basic aspects of the IST such as the construction of fundamental analytic solutions, the formulation of a Riemann-Hilbert problem, and the implementation of the dressing method are presented.

nlin.SI↗
arXiv2010

Discrete wave turbulence of rotational capillary water waves

We study the discrete wave turbulent regime of capillary water waves with constant non-zero vorticity. The explicit Hamiltonian formulation and the corresponding coupling coefficient are obtained. We also present the construction and investigation of resonance clustering. Some physical implications of the obtained results are discussed.

math-ph↗
arXiv2010

Steady periodic water waves with constant vorticity: regularity and local bifurcation

This paper studies periodic traveling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed. Using conformal mappings the free-boundary problem is transformed into a quasilinear pseudodifferential equation for a periodic function of one variable. The new formulation leads to a regularity result and, by use of bifurcation theory, to the existence of waves of small amplitude even in the presence of stagnation points in the flow.

math.AP↗
arXiv2009

Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves

The influence of an underlying current on 3-wave interactions of capillary water waves is studied. The fact that in irrotational flow resonant 3-wave interactions are not possible can be invalidated by the presence of an underlying current of constant non-zero vorticity. We show that: 1) wave trains in flows with constant non-zero vorticity are possible only for two-dimensional flows; 2) only positive constant vorticities can trigger the appearance of three-wave resonances; 3) the number of positive constant vorticities which do trigger a resonance is countable; 4) the magnitude of a positive constant vorticity triggering a resonance can not be too small.

physics.flu-dyn↗