Drazin inverse and its application to linear degenerate systems

In the thesis, we review some recent progresses on the study of Drazin inverses and the study of linear degenerate systems with nonsingular pencil. Properties of the Drazin inverse are discussed. An application of Drazin inverses to linear degenerate systems is presented. Furthermore, a numerical algorithm for calculating Drazin inverses and a FORTRAN program are provided

Linear and nonlinear dynamics in stratified shear flows

Stably stratified shear flows, in which a less dense layer of fluid lies above and moves counter to a more dense layer below, are ubiquitous in geophysical fluid dynamics. These are often found to be unstable if the non-dimensional Richardson number Ri, quantifying the strength of stratification to shear, is sufficiently low. This is of particular importance in oceanography, where shear instabilities are conjectured to be important in the generation of turbulence in the deep ocean, an area of huge uncertainty in contemporary climate models. The Miles-Howard theorem tells us that for a steady, inviscid, parallel shear flow, if the local Richardson number is everywhere greater than one quarter, the flow is stable to infinitesimal perturbations. Though an important result, the strong restrictions in the applicability of this theorem mean care must be used when applying the criterion of Ri > 1/4 for stability. This thesis explores some of these limitations, beginning with an overview in chapter 1. Chapter 2 explores the infinitesimal restriction of the Miles-Howard theorem, by asking whether finite-amplitude perturbations could lead to significant nonlinear behaviour, in a so-called subcritical instability. It is found that while the classical Kelvin-Helmholtz instability does indeed exhibit subcriticality, nonlinear steady states are found only just above Ri = 1/4. Chapter 3 investigates in detail a hitherto unknown linear instability, which was discovered in chapter 2. Behaving similarly to the classic Holmboe instability, it exists for Ri > 1/4 when viscosity is introduced, and reveals new insights into the possible physical interpretations of stratified shear instability. Chapter 4 revisits the results of chapter 2 but considers two cases of the Prandtl number Pr, the ratio of diffusivity of the momentum to density. When Pr = 0.7, as is approximately the case for air, a simple supercritical instability is found. However, for Pr = 7, corresponding approximately to water, strong subcritical behaviour is observed, and it is demonstrated that finite-amplitude perturbations can trigger Kelvin-Helmholtz-like behaviour well above Ri = 1/4. Chapter 5 considers the time-varying, non-parallel flow of an oblique internal gravity wave incident on a shear layer. Using direct-adjoint looping, it is shown that the disturbances which maximise energy after a certain time, so-called linear optimal perturbations, can be convective-like rolls in the spanwise direction, rather than a shear instability, calling into question the relevance of the classical shear instabilities in oceanography. Chapter 6 concludes the thesis with a discussion of the implications of the results

Planar inviscid flows in a channel of finite length : washout, trapping and self-oscillations of vorticity

The paper addresses the nonlinear dynamics of planar inviscid incompressible flows in the straight channel of a finite length. Our attention is focused on the effects of boundary conditions on vorticity dynamics. The renowned Yudovich's boundary conditions (YBC) are the normal component of velocity given at all boundaries, while vorticity is prescribed at an inlet only. The YBC are fully justified mathematically: the well posedness of the problem is proven. In this paper we study general nonlinear properties of channel flows with YBC. There are 10 main results in this paper: (i) the trapping phenomenon of a point vortex has been discovered, explained and generalized to continuously distributed vorticity such as vortex patches and harmonic perturbations; (ii) the conditions sufficient for decreasing Arnold's and enstrophy functionals have been found, these conditions lead us to the washout property of channel flows; (iii) we have shown that only YBC provide the decrease of Arnold's functional; (iv) three criteria of nonlinear stability of steady channel flows have been formulated and proven; (v) the counterbalance between the washout and trapping has been recognized as the main factor in the dynamics of vorticity; (vi) a physical analogy between the properties of inviscid channel flows with YBC, viscous flows and dissipative dynamical systems has been proposed; (vii) this analogy allows us to formulate two major conjectures (C1 and C2) which are related to the relaxation of arbitrary initial data to C1: steady flows, and C2: steady, self-oscillating or chaotic flows; (viii) a sufficient condition for the complete washout of fluid particles has been established; (ix) the nonlinear asymptotic stability of selected steady flows is proven and the related thresholds have been evaluated; (x) computational solutions that clarify C1 and C2 and discover three qualitatively different scenarios of flow relaxation have been obtained

Macroscopic dynamics of incoherent soliton ensembles: soliton-gas kinetics and direct numerical modeling

We undertake a detailed comparison of the results of direct numerical simulations of the integrable soliton gas dynamics with the analytical predictions inferred from the exact solutions of the relevant kinetic equation for solitons. We use the KdV soliton gas as a simplest analytically accessible model yielding major insight into the general properties of soliton gases in integrable systems. Two model problems are considered: (i) the propagation of a `trial' soliton through a one-component `cold' soliton gas consisting of randomly distributed solitons of approximately the same amplitude; and (ii) collision of two cold soliton gases of different amplitudes (soliton gas shock tube problem) leading to the formation of an incoherend dispersive shock wave. In both cases excellent agreement is observed between the analytical predictions of the soliton gas kinetics and the direct numerical simulations. Our results confirm relevance of the kinetic equation for solitons as a quantitatively accurate model for macroscopic non-equilibrium dynamics of incoherent soliton ensembles.Comment: 20 pages, 8 figures, 34 references. Other author's papers can be downloaded at http://www.denys-dutykh.com

Fluctuations of large-scale jets in the stochastic 2D Euler equation

Two-dimensional turbulence in a rectangular domain self-organises into large-scale unidirectional jets. While several results are present to characterize the mean jets velocity profile, much less is known about the fluctuations. We study jets dynamics in the stochastically forced two-dimensional Euler equations. In the limit where the average jets velocity profile evolves slowly with respect to turbulent fluctuations, we employ a multi-scale (kinetic theory) approach, which relates jet dynamics to the statistics of Reynolds stresses. We study analytically the Gaussian fluctuations of Reynolds stresses and predict the spatial structure of the jets velocity covariance. Our results agree qualitatively well with direct numerical simulations, clearly showing that the jets velocity profile are enhanced away from the stationary points of the average velocity profile. A numerical test of our predictions at quantitative level seems out of reach at the present day