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

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

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

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

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

Internal Boundary Layer Scaling in “Two Layer” Solutions of the Thermocline Equations

The diffusivity dependence of internal boundary layers in solutions of the continuously stratified, diffusive thermocline equations is revisited. If a solution exists that approaches a two-layer solution of the ideal thermocline equations in the limit of small vertical diffusivity kᵥ, it must contain an internal boundary layer that collapses to a discontinuity as kᵥ → 0. An asymptotic internal boundary layer equation is derived for this case, and the associated boundary layer thickness is proportional to kᵥ¹/². In general, the boundary layer remains three-dimensional and the thermodynamic equation does not reduce to a vertical advective–diffusive balance even as the boundary layer thickness becomes arbitrarily small. If the vertical convergence varies sufficiently slowly with horizontal position, a one-dimensional boundary layer equation does arise, and an explicit example is given for this case. The same one-dimensional equation arose previously in a related analysis of a similarity solution that does not itself approach a two-layer solution in the limit kᵥ → 0

Geostrophic Circulation in a Rectangular Basin with a Circumpolar Connection

A simple theory is presented for steady geostrophic circulation of a stratified fluid in a rectangular basin with a circumpolar connection. The interior flow obeys the β-plane Sverdrup vorticity balance, and the circulation is closed by geostrophic boundary currents. The circulation is forced by surface thermal gradients and wind-driven Ekman transport near the latitudes of the circumpolar connection. A thermal circumpolar current arises in response to imposed surface thermal gradients and northward Ekman transport across the gap latitudes. The transport of this model circumpolar current depends on the imposed surface thermal gradients and the gap geometry, but not on the strength of the wind forcing. In contrast, the circulation induced in a related reduced-gravity model by Sverdrup transport into the gap latitudes has zero zonally integrated zonal transport. The thermal current arises as a consequence of the geostrophic constraint, which requires that the northern region fill with warm fluid until it reaches the sill depth, where return geostrophic flow can be supported. Thus, the structure of the middepth, midlatitude thermocline is directly influenced by the geometry of the gap. A similar constraint evidently operates in the Southern Ocean