The transport in the Ekman surface layer on the spherical Earth

The modification of the transport in the Ekman layer on the f-plane due to the Coriolis parameter\u27s variation with latitude and the curvature of Earth\u27s surface is analyzed by considering the temporal changes in the angular momentum. The latter plays the role of a dynamical variable of the model, replacing the zonal velocity component, and drag is modeled by Rayleigh friction. The steady transport, which on an f-plane is perpendicular to the applied wind stress, is recovered on the Earth as a special solution where the meridional velocity is time-independent. For zonal wind stress, the trajectory on Earth is simply a great circle that passes through the poles while for meridional wind stress the special solution can have a time-independent nonzero meridional component so the trajectory does not have to be purely zonal. This asymmetry between zonal and meridional wind stresses on the Earth is due to the Coriolis parameter\u27s variation with latitude only—an effect that is completely neglected on the f-plane. For steady wind forcing, the dynamical system is three-dimensional and its fixed points are located at the latitudes of vanishing wind stress. In the drag-free case, when the curl of the wind stress does not vanish at the fixed points, these points are always unstable; namely there exists at least one repulsive direction in (the 3D) phase space. When drag is included, these steady states still prevail but become stable for realistic values of the wind forcing and drag. An additional steady state, located right on the equator, exists in this case and its zonal velocity attains a constant value determined by the balance between the applied stress and the drag force. Although drag is present, this steady state is unstable for negative wind stress (i.e. easterly winds) so any deviation from a purely westward, equatorial, trajectory will grow exponentially in time. Naturally, no similar instability of the steady states occurs on the f-plane. The curl of the zonal wind stress at the latitudes where the stress itself vanishes determines the trajectory of a water column originating there via the nonlinear interaction between the motion due to inertial oscillations and that due to the wind-forced changes of the angular momentum. Temporal or zonal dependence of the wind stress has a profound effect on the trajectories, especially near the unstable latitudes due to the increase in the dimensionality of the system that enables more complex trajectories. The present simple model can quantitatively reproduce the observed fast dispersal of nearby launched drifters with steady and smooth wind stress. It can also explain qualitatively the different spectra of clusters of drifters launched in two field experiments in the NE Pacific Ocean under similar winds and the highly variable angle between the wind and the observed trajectories of clusters of drifters

On the Mixing Enhancement in a Meandering Jet Due to the Interaction with an Eddy

The interaction between a simple meandering jet such as the Gulf Stream, and an eddy is shown to greatly enhance the mixing and dispersal of fluid parcels in the jet. This enhanced mixing is quantified by calculating the rate of increase of the root-mean-square pair separation of Lagrangian particles (e.g., floats) launched in the jet\u27s immediate vicinity. In the presence of an eddy, particles can escape from the regions in which they were initially launched. Comparisons with observations show a markedly improved qualitative agreement when the eddy is allowed to interact with the meandering jet

A Lagrangian theory of geostrophic adjustment for zonally-invariant flows on a rotating spherical earth

We examine the late-time evolution of an inviscid zonally symmetric shallow-water flow on the surface of a rotating spherical earth. An arbitrary initial condition radiates inertia–gravity waves that disperse across the spherical surface. The simpler problem of a uniformly rotating (f-plane) shallow-water flow on the plane radiates these waves to infinity, leaving behind a nontrivial steady flow in geostrophic balance (in which the Coriolis acceleration balances the horizontal hydrostatic pressure gradient). This is called “geostrophic adjustment.” On a sphere, the waves cannot propagate to infinity, and the flow can never become steady due to energy conservation (at least in the absence of shocks). Nonetheless, when energy is conserved a form of adjustment still takes place, in a time-averaged sense, and this flow satisfies an extended form of geostrophic balance dependent only on the conserved mass and angular momentum distributions of fluid particles, just as in the planar case. This study employs a conservative numerical scheme based on a Lagrangian form of the rotating shallow-water equations to substantiate the applicability of these general considerations on an idealized aqua-planet for an initial “dam” along the equator in a motionless ocean.Publisher PDFPeer reviewe

Linear instability of uniform shear zonal currents on the β-plane

A unified formulation of the instability of a mean zonal flow with uniform shear is proposed, which includes both the coupled density front and the coastal current. The unified formulation shows that the previously found instability of the coupled density front on the f-plane has natural extension to coastal currents, where the instability exists provided that the net transport of the current is sufficiently small. This extension of the coupled front instability to coastal currents implies that the instability originates from the interaction between Inertia-Gravity waves and a vorticity edge wave and not from the interaction of the two edge waves that exist at the two free streamlines due to the Potential Vorticity jump there. The present study also extends these instabilities to the β-plane and shows that β slightly destabilizes the currents by adding instabilities in wavelength ranges that are stable on the f-plane but has little effect on the growthrates in wavelength ranges that are unstable on the f-plane. An application of the β-plane instability theory to the generation of rings in the retroflection region of the Agulhas Current yields a very fast perturbation growth of the scale of 1 day and this fast growth rate is consistent with the observation that at any given time, as many as 10 Agulhas rings can exist in this region

Linear instabilities of a two-layer geostrophic surface front near a wall

The development of linear instabilities on a geostrophic surface front in a two-layer primitive equation model on an ƒ-plane is studied analytically and numerically using a highly accurate differential shooting method. The basic state is composed of an upper layer in which the mean flow has a constant potential vorticity, and a quiescent lower layer that outcrops between a vertical wall and the surface front (defined as the line of intersection between the interface that separates the two layers and the ocean\u27s surface). The characteristics of the linear instabilities found in the present work confirm earlier results regarding the strong dependence of the growth rate (σi) on the depth ratio r (defined as the ratio between the total ocean depth and the upper layer\u27s depth at infinity) for r ≥ 2 and their weak dependence on the distance L between the surface front and the wall. These earlier results of the large r limit were obtained using a much coarser, algebraic, method and had a single maximum of the growth rate curve at some large wavenumber k. Our new results, in the narrow range of 1.005 ≤ r ≤ 1.05, demonstrate that the growth rate curve displays a second lobe with a local (secondary) maximum at a nondimensional wavenumber (with the length scale given by the internal radius of deformation) of 1.05. A new fitting function 0.183 r-0.87is found for the growth rate of the most unstable wave (σimax ) for r ranging between 1.001 and 20, and for L \u3e 2 Rd (i.e.where the effect of the wall becomes negligible). Therefore, σimax converges to a finite value for |r — 1 | \u3c\u3c 1 (infinitely thin lower layer). This result differs from quasi-geostrophic, analytic solutions that obtain for the no wall case since the QG approximation is not valid for very thin layers. In addition, an analytical solution is derived for the lower-layer solutions in the region between the wall and the surface front where the upper layer is not present. The weak dependence of the growth rate on L that emerges from the numerical solution of the eigenvalue problem is substantiated analytically by the way L appears in the boundary conditions at the surface front. Applications of these results for internal radii of deformation of 35– 45 km show reasonable agreement with observed meander characteristics of the Gulf Stream downstream of Cape Hatteras. Wavelengths and phase speeds of (180 –212 km, 39 –51 km/day) in the vicinity of Cape Hatteras were also found to match with the predicted dispersion relationships for the depth-ratio range of 1+ \u3c r \u3c 2

The Ekman spiral for piecewise-uniform viscosity

Funding: Support for this research has come from the UK Engineering and Physical Sciences Research Council (grant no. EP/H001794/1).We re-visit Ekman's (1905) classic problem of wind-stress-induced ocean currents to help interpret observed deviations from Ekman's theory, in particular from the predicted surface current deflection of 45∘. While previous studies have shown that such deviations can be explained by a vertical eddy viscosity varying with depth, as opposed to the constant profile taken by Ekman, analytical progress has been impeded by the difficulty in solving Ekman's equation. Herein, we present a solution for piecewise-constant eddy viscosity which enables a comprehensive understanding of how the surface deflection angle depends on the vertical profile of eddy viscosity. For two layers, the dimensionless problem depends only on the depth of the upper layer and the ratio of layer viscosities. A single diagram then allows one to understand the dependence of the deflection angle on these two parameters.Publisher PDFPeer reviewe

Wave propagation and growth on a surface front in a two-layer geostrophic current

We study analytically and numerically small amplitude perturbations of a geostrophically balanced semi-infinite layer of light water having a surface front and lying above a heavier layer of finite vertical thickness which is at rest in the mean. In contrast with previous studies where the latter layer was infinitely deep we find that the equilibrium is always unstable regardless of the distribution of potential vorticity, and the maximum growth rates are generally much larger than in the one-layer case. The amplifying ageostrophic wave transfers kinetic energy from the basic shear flow as well as potential energy. Good quantitative agreement is found with the laboratory experiments of Griffiths and Linden (1982), and our model seems to be the simplest one for future investigations of cross frontal mixing processes by finite amplitude waves. The propagation speed of very low frequency and nondispersive frontal waves is also computed and is shown to decrease with increasing bottom layer depth

On the role of viscosity in ideal Hadley circulation models

[1] A comparison is made between inviscid and viscous solutions of an axially symmetric nonlinear Shallow Water Model (SWM) of the Hadley circulation on the spherical rotating Earth that includes vertical advection of momentum and Rayleigh friction. The results of the 1D SWM are compared with those of 2D (latitude-height) nonlinear axially symmetric models. It is shown that solutions obtained from inviscid models do not predict correctly the main features of the circulation outside the Tropics. Specifically, the latitudes of the subtropical jets depend strongly on viscosity and on the pole-to-equator temperature difference. The differences from inviscid theory are attributed to the increased rate of energy dissipation in the viscous atmosphere. At the Tropics, however, the inviscid solutions of the SWM predict well the latitude and strength of the easterly jet and the weak temperature gradient. Citation: Adam, O., and N. Paldor (2010), On the role of viscosity in ideal Hadley circulation models

The mechanics of eddy transport from one hemisphere to the other

Abstract The trajectory of a dense eddy that propagates along the bottom of a meridional channel of parabolic cross-section from the Southern to the Northern Hemisphere is described by a twodegrees-of-freedom, Hamiltonian, system. Two simplified types of motion exist in which to first order the meridinal acceleration vanishes. In mid-latitudes the motion is geostrophic, poleward (equatorward) directed along the channel's west (east) flank. On the other hand, right on the equator the motion is describable by linear oscillations along the potential-well generated by the channel's parabolic bottom cross-section. The propagation speed along the equator is much larger than that in mid-latitudes, which enhances the eddy's dissipation via its mixing with overlying ocean water. For motions that occur slightly off the equator the eastward segment is stable while the westward segment is unstable so an expulsion from the equatorial regime takes place during the latter. A dense eddy that arrives near the equator along the channel's west flank has to cross the channel to its east flank where it can either oscillate back (westward) to the other side of the channel or move poleward from the equator along the channel's east flank. The eddy's dissipation during the equatorial part of its trajectory is very large and the probability of the dissipated eddy leaving the equator to either of the two Hemispheres is identical. The non-integrability of the system manifests itself in the sensitive combination of the equatorial, and the mid-latitude, regimes that renders the dynamics of the transport of AABW eddies to the Northern Hemisphere -chaotic. This description explains both the sharp decrease in the amount of AABW water mass in the immediate vicinity of the equator in the Western Atlantic Ocean and the "splitter" effect of the equator encountered in numerical simulations