Lyapunov, Floquet, and singular vectors for baroclinic waves

The dynamics of the growth of linear disturbances to a chaotic basic state is analyzed in an asymptotic model of weakly nonlinear, baroclinic wave-mean interaction. In this model, an ordinary differential equation for the wave amplitude is coupled to a partial differential equation for the zonal flow correction. The leading Lyapunov vector is nearly parallel to the leading Floquet vector <font face='Symbol'><i>f</i></font>₁ of the lowest-order unstable periodic orbit over most of the attractor. Departures of the Lyapunov vector from this orientation are primarily rotations of the vector in an approximate tangent plane to the large-scale attractor structure. Exponential growth and decay rates of the Lyapunov vector during individual Poincaré section returns are an order of magnitude larger than the Lyapunov exponent <font face='Symbol'>l</font> ≈ 0.016. Relatively large deviations of the Lyapunov vector from parallel to <font face='Symbol'><i>f</i></font>₁ are generally associated with relatively large transient decays. The transient growth and decay of the Lyapunov vector is well described by the transient growth and decay of the leading Floquet vectors of the set of unstable periodic orbits associated with the attractor. Each of these vectors is also nearly parallel to <font face='Symbol'><i>f</i></font>₁. The dynamical splitting of the complete sets of Floquet vectors for the higher-order cycles follows the previous results on the lowest-order cycle, with the vectors divided into wave-dynamical and decaying zonal flow modes. Singular vectors and singular values also generally follow this split. The primary difference between the leading Lyapunov and singular vectors is the contribution of decaying, inviscidly-damped wave-dynamical structures to the singular vectors

Unstable periodic orbits in a chaotic meandering jet flow

We study the origin and bifurcations of typical classes of unstable periodic orbits in a jet flow that was introduced before as a kinematic model of chaotic advection, transport and mixing of passive scalars in meandering oceanic and atmospheric currents. A method to detect and locate the unstable periodic orbits and classify them by the origin and bifurcations is developed. We consider in detail period-1 and period-4 orbits playing an important role in chaotic advection. We introduce five classes of period-4 orbits: western and eastern ballistic ones, whose origin is associated with ballistic resonances of the fourth order, rotational ones, associated with rotational resonances of the second and fourth orders, and rotational-ballistic ones associated with a rotational-ballistic resonance. It is a new kind of nonlinear resonances that may occur in chaotic flow with jets and/or circulation cells. Varying the perturbation amplitude, we track out the origin and bifurcations of the orbits for each class

Kinematic studies of transport across an island wake, with application to the Canary islands

Transport from nutrient-rich coastal upwellings is a key factor influencing biological activity in surrounding waters and even in the open ocean. The rich upwelling in the North-Western African coast is known to interact strongly with the wake of the Canary islands, giving rise to filaments and other mesoscale structures of increased productivity. Motivated by this scenario, we introduce a simplified two-dimensional kinematic flow describing the wake of an island in a stream, and study the conditions under which there is a net transport of substances across the wake. For small vorticity values in the wake, it acts as a barrier, but there is a transition when increasing vorticity so that for values appropriate to the Canary area, it entrains fluid and enhances cross-wake transport.Comment: 28 pages, 13 figure

The homotopy type of the loops on (n−1)(n-1)-connected (2n+1)(2n+1)-manifolds

For $n\geq 2$ we compute the homotopy groups of $(n-1)$-connected closed manifolds of dimension $(2n+1)$. Away from the finite set of primes dividing the order of the torsion subgroup in homology, the $p$-local homotopy groups of $M$ are determined by the rank of the free Abelian part of the homology. Moreover, we show that these $p$-local homotopy groups can be expressed as a direct sum of $p$-local homotopy groups of spheres. The integral homotopy type of the loop space is also computed and shown to depend only on the rank of the free Abelian part and the torsion subgroup.Comment: Trends in Algebraic Topology and Related Topics, Trends Math., Birkhauser/Springer, 2018. arXiv admin note: text overlap with arXiv:1510.0519

Surface-intensified Rossby waves over rough topography

Observations and numerical experiments that suggest that sea-floor roughness can enhance the ratio of thermocline to abyssal eddy kinetic energy, motivate the study of linear free wave modes in a two layer quasi-geostrophic model for several eases of idealized variable bottom topography. The foeus is on topography with horizontal seale comparable to that of the waves, that is, on rough small-amplitude topography. Surface-intensified modes are found to exist at frequencies greater than the flat-bottom baroclinic cut-off frequency. These modes exist for topography that varies in both one and two horizontal dimensions. An approximate bound indicates that the maximum frequency of the surface-intensified modes is greater than the baroclinic cut-off by a factor equal to the total fluid depth divided by the lower layer depth. For fixed topographic wavenumber, there is not a simple dependence of the degree of surface-intensification on topographic amplitude, but rather a resonant structure with peaks at certain topographic amplitudes. These modes may be resonantly excited by surface forcing

Corrigendum to Surface-intensified Rossby waves over rough topography (J. Mar. Res., 50, 367–384)

In Samelson (1992), Eq. (13) should be changed to… respectively. The same error appears in the Abstract

Linking and causality in globally hyperbolic spacetimes

The linking number $lk$ is defined if link components are zero homologous. Our affine linking invariant $alk$ generalizes $lk$ to the case of linked submanifolds with arbitrary homology classes. We apply $alk$ to the study of causality in Lorentz manifolds. Let $M^m$ be a spacelike Cauchy surface in a globally hyperbolic spacetime $(X^{m+1}, g)$. The spherical cotangent bundle $ST^*M$ is identified with the space $N$ of all null geodesics in $(X,g).$ Hence the set of null geodesics passing through a point $x\in X$ gives an embedded $(m-1)$-sphere $S_x$ in $N=ST^*M$ called the sky of $x.$ Low observed that if the link $(S_x, S_y)$ is nontrivial, then $x,y\in X$ are causally related. This motivated the problem (communicated by Penrose) on the Arnold's 1998 problem list to apply link theory to the study of causality. The spheres $S_x$ are isotopic to fibers of $(ST^*M)^{2m-1}\to M^m.$ They are nonzero homologous and $lk(S_x,S_y)$ is undefined when $M$ is closed, while $alk(S_x, S_y)$ is well defined. Moreover, $alk(S_x, S_y)\in Z$ if $M$ is not an odd-dimensional rational homology sphere. We give a formula for the increment of \alk under passages through Arnold dangerous tangencies. If $(X,g)$ is such that $alk$ takes values in $\Z$ and $g$ is conformal to $g'$ having all the timelike sectional curvatures nonnegative, then $x, y\in X$ are causally related if and only if $alk(S_x,S_y)\neq 0$. We show that $x,y$ in nonrefocussing $(X, g)$ are causally unrelated iff $(S_x, S_y)$ can be deformed to a pair of $S^{m-1}$-fibers of $ST^*M\to M$ by an isotopy through skies. Low showed that if (\ss, g) is refocussing, then $M$ is compact. We show that the universal cover of $M$ is also compact.Comment: We added: Theorem 11.5 saying that a Cauchy surface in a refocussing space time has finite pi_1; changed Theorem 7.5 to be in terms of conformal classes of Lorentz metrics and did a few more changes. 45 pages, 3 figures. A part of the paper (several results of sections 4,5,6,9,10) is an extension and development of our work math.GT/0207219 in the context of Lorentzian geometry. The results of sections 7,8,11,12 and Appendix B are ne

Large-scale circulation with small diapycnal diffusion: The two-thermocline limit

The structure and dynamics of the large-scale circulation of a single-hemisphere, closed-basin ocean with small diapycnal diffusion are studied by numerical and analytical methods. The investigation is motivated in part by recent differing theoretical descriptions of the dynamics that control the stratification of the upper ocean, and in part by recent observational evidence that diapycnal diffusivities due to small-scale turbulence in the ocean thermocline are small (≈0.1 cm2 s−1). Numerical solutions of a computationally efficient, three-dimensional, planetary geostrophic ocean circulation model are obtained in a square basin on a mid-latitude β-plane. The forcing consists of a zonal wind stress (imposed meridional Ekman flow) and a surface heat flux proportional to the difference between surface temperature and an imposed air temperature. For small diapycnal diffusivities (vertical: κv ≈0.1 – 0.5 cm2 s−1, horizontal: κh ≈105 – 5 × 106 cm2 s−1), two distinct thermocline regimes occur. On isopycnals that outcrop in the subtropical gyre, in the region of Ekman downwelling, a ventilated thermocline forms. In this regime, advection dominates diapycnal diffusion, and the heat balance is closed by surface cooling and convection in the northwest part of the subtropical gyre. An ‘advective’ vertical scale describes the depth to which the wind-driven motion penetrates, that is, the thickness of the ventilated thermocline. At the base of the wind-driven fluid layer, a second thermocline forms beneath a layer of vertically homogeneous fluid (‘mode water’). This ‘internal’ thermocline is intrinsically diffusive. An ‘internal boundary layer’ vertical scale (proportional to κv1/2) describes the thickness of this internal thermocline. Two varieties of subtropical mode waters are distinguished. The temperature difference across the ventilated thermocline is determined to first order by the meridional air temperature difference across the subtropical gyre. The temperature difference across the internal thermocline is determined to first order by the temperature difference across the subpolar gyre. The diffusively-driven meridional overturning cell is effectively confined below the ventilated thermocline, and driven to first order by the temperature difference across the internal thermocline, not the basin-wide meridional air temperature difference. Consequently, for small diapycnal diffusion, the abyssal circulation depends to first order only on the wind-forcing and the subpolar gyre air temperatures. The numerical solutions have a qualitative resemblance to the observed structure of the North Atlantic in and above the main thermocline (that is, to a depth of roughly 1500 m). Below the main thermocline, the predicted stratification is much weaker than observed

Theory and computation of covariant Lyapunov vectors

Lyapunov exponents are well-known characteristic numbers that describe growth rates of perturbations applied to a trajectory of a dynamical system in different state space directions. Covariant (or characteristic) Lyapunov vectors indicate these directions. Though the concept of these vectors has been known for a long time, they became practically computable only recently due to algorithms suggested by Ginelli et al. [Phys. Rev. Lett. 99, 2007, 130601] and by Wolfe and Samelson [Tellus 59A, 2007, 355]. In view of the great interest in covariant Lyapunov vectors and their wide range of potential applications, in this article we summarize the available information related to Lyapunov vectors and provide a detailed explanation of both the theoretical basics and numerical algorithms. We introduce the notion of adjoint covariant Lyapunov vectors. The angles between these vectors and the original covariant vectors are norm-independent and can be considered as characteristic numbers. Moreover, we present and study in detail an improved approach for computing covariant Lyapunov vectors. Also we describe, how one can test for hyperbolicity of chaotic dynamics without explicitly computing covariant vectors.Comment: 21 pages, 5 figure

Simple Mechanistic Models of Middepth Meridional Overturning

Two idealized, three-dimensional, analytical models of middepth meridional overturning in a basin with a Southern Hemisphere circumpolar connection are described. In the first, the overturning circulation can be understood as a “pump and valve” system, in which the wind forcing at the latitudes of the circumpolar connection is the pump and surface thermodynamic exchange at high northern latitudes is the valve. When the valve is on, the overturning circulation extends to the extreme northern latitudes of the basin, and the middepth thermocline is cold. When the valve is off, the overturning circulation is short-circuited and confined near the circumpolar connection, and the middepth thermocline is warm. The meridional overturning cell in this first model is not driven by turbulent mixing, and the subsurface circulation is adiabatic. In contrast, the pump that primarily drives the overturning cell in the second model is turbulent mixing, at low and midlatitudes, in the ocean interior. In both models, however, the depth of the midlatitude deep layer is controlled by the sill depth of the circumpolar gap. The thermocline structures in these two models are nearly indistinguishable. These models suggest that Northern Hemisphere wind and surface buoyancy forcing may influence the strength and structure of the circumpolar current in the Southern Hemisphere