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

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

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

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

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

The Vertical Structure of Linear Coastal-Trapped Disturbances

The vertical structure of coastal-trapped disturbances in several idealized models of a stably stratified lower atmosphere is examined. The vertical structure and phase speeds of the trapped modes depend on the resting stratification and the height of the orographic step. The presence of a stable layer above the boundary layer inversion increases the gravest-mode phase speed and supports the existence of higher vertical modes. Trapped wave solutions for the step orography are obtained for a lower atmosphere with constant buoyancy frequency. The modes are primarily concentrated below the step but penetrate weakly into the stratified region above the step. The phase speed of the gravest trapped mode is greater than the gravest-mode Kelvin wave speed based on the height of the step. Results from a linear two-layer model suggest that the observed vertical structure of isotherms at the leading edge of a 10–11 June 1994 event may arise during a transition from a directly forced, barotropic, alongshore velocity response to a regime influenced by wave propagation, as the coastal-trapped vertical modes excited by the mesoscale pressure gradients begin to disperse at their respective phase speeds. The results suggest also that the observed vertical structure of alongshore velocity, with largest velocities in the stable layer above the boundary layer, may arise from drag at the sea surface

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

Periodic Orbits and Disturbance Growth for Baroclinic Waves

The growth of linear disturbances to stable and unstable time-periodic basic states 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. Floquet vectors, the eigenmodes for linear disturbances to the oscillatory basic states, split into wave-dynamical and decaying zonal-flow modes. Singular vectors reflect the structure of the Floquet vectors: the most rapid amplification and decay are associated with the wave-dynamical Floquet vectors, while the intermediate singular vectors closely follow the decaying zonal-flow Floquet vectors. Singular values depend strongly on initial and optimization times. For initial times near wave amplitude maxima, the Floquet decomposition of the leading singular vector depends relatively weakly on optimization time. For the unstable oscillatory basic state in the chaotic regime, the leading Floquet vector is tangent to the large-scale structure of the attractor, while the leading singular vector is not. However, corresponding inferences about the accessibility of disturbed states rely on the simple attractor geometry, and may not easily generalize. The primary mechanism of disturbance growth on the wave timescale in this model involves a time-dependent phase shift along the basic wave cycle. The Floquet vectors illustrate that modal disturbances to time-dependent basic states can have time-dependent spatial structure, and that the latter need not indicate nonmodal dynamics. The dynamical splitting reduces the ‘‘butterfly effect,’’ the ability of small-scale disturbances to influence the evolution of an unstable large-scale flow

Potential and timescales for oxygen depletion in coastal upwelling systems: A box-model analysis

A simple box model is used to examine oxygen depletion in an idealized ocean-margin upwelling system. Near-bottom oxygen depletion is controlled by a competition between flushing with oxygenated offshore source waters and respiration of particulate organic matter produced near the surface and retained near the bottom. Upwelling-supplied nutrients are consumed in the surface box, and some surface particles sink to the bottom where they respire, consuming oxygen. Steady states characterize the potential for hypoxic near-bottom oxygen depletion; this potential is greatest for faster sinking rates, and largely independent of production timescales except in that faster production allows faster sinking. Timescales for oxygen depletion depend on upwelling and productivity differently, however, as oxygen depletion can only be reached in meaningfully short times when productivity is rapid. Hypoxia thus requires fast production, to capture upwelled nutrients, and fast sinking, to deliver the respiration potential to model bottom waters. Combining timescales allows generalizations about tendencies toward hypoxia. If timescales of sinking are comparable to or smaller than the sum of those for respiration and flushing, the steady state will generally be hypoxic, and results indicate optimal timescales and conditions exist to generate hypoxia. For example, the timescale for approach to hypoxia lengthens with stronger upwelling, since surface particle and nutrient are shunted off-shelf, in turn reducing subsurface respiration and oxygen depletion. This suggests that if upwelling winds intensify with climate change the increased forcing could offer mitigation of coastal hypoxia, even as the oxygen levels in upwelled source waters decline