Peter B. Rhines

Waves and turbulence on a beta-plane

Two-dimensional eddies in a homogeneous fluid at large Reynolds number, if closely packed, are known to evolve towards larger scales. In the presence of a restoring force, the geophysical beta-effect, this cascade produces a field of waves without loss of energy, and the turbulent migration of the dominant scale nearly ceases at a wavenumber k β = (β/2 U ) ½ independent of the initial conditions other than U , the r.m.s. particle speed, and β, the northward gradient of the Coriolis frequency. The conversion of turbulence into waves yields, in addition, more narrowly peaked wavenumber spectra and less fine-structure in the spatial maps, while smoothly distributing the energy about physical space. The theory is discussed, using known integral constraints and similarity solutions, model equations, weak-interaction wave theory (which provides the terminus for the cascade) and other linearized instability theory. Computer experiments with both finite-difference and spectral codes are reported. The central quantity is the cascade rate, defined as \[ T = 2\int_0^{\infty} kF(k)dk/U^3\langle k\rangle , \] where F is the nonlinear transfer spectrum and 〈 k 〉 the mean wavenumber of the energy spectrum. (In unforced inviscid flow T is simply U −1 d 〈 k 〉 −1 / dt , or the rate at which the dominant scale expands in time t .) T is shown to have a mean value of 3·0 × 10 −2 for pure two-dimensional turbulence, but this decreases by a factor of five at the transition to wave motion. We infer from weak-interaction theory even smaller values for k [Lt ] k β . After passing through a state of propagating waves, the homogeneous cascade tends towards a flow of alternating zonal jets which, we suggest, are almost perfectly steady. When the energy is intermittent in space, however, model equations show that the cascade is halted simply by the spreading of energy about space, and then the end state of a zonal flow is probably not achieved. The geophysical application is that the cascade of pure turbulence to large scales is defeated by wave propagation, helping to explain why the energy-containing eddies in the ocean and atmosphere, though significantly nonlinear, fail to reach the size of their respective domains, and are much smaller. For typical ocean flows,

Long-Term Coordinated Changes in the Convective Activity of the North Atlantic

k_{\beta}^{-1} = 70\,{\rm km}

Homogenization of potential vorticity in planetary gyres

, while for the atmosphere,

How rapidly is a passive scalar mixed within closed streamlines

k_{\beta}^{-1} = 1000\,{\rm km}

The Labrador Sea Deep Convection Experiment

. In addition the cascade generates, by itself, zonal flow (or more generally, flow along geostrophic contours).

Atmospheric Blocking and Atlantic Multidecadal Ocean Variability

The North Atlantic is a peculiarly convective ocean. The convective renewal of intermediate and deep waters in the Labrador Sea and Greenland/Iceland Sea both contribute significantly to the production and export of North Atlantic Deep Water, thus helping to drive the global thermohaline circulation, while the formation and spreading of 18-degree water at shallow-to-intermediate depths off the US eastern seaboard is a major element in the circulation and hydrographic character of the west Atlantic. For as long as time-series of adequate precision have been available to us, it has been apparent that the intensity of convection at each of these sites, and the hydrographic character of their products have been subject to major interannual change, as shown by Aagaard (1968), Clarke et al (1990), and Meincke et al (1992) for the Greenland Sea, in the OWS BRAVO record from the Labrador Sea, (eg Lazier,1980 et seq.), and at the PANULIRUS / Hydrostation “S” site in the Northern Sargasso off Bermuda (eg Jenkins, 1982, Talley and Raymer, 1982). This paper reviews the recent history of these changes showing that the major convective centres of the Greenland- and Labrador Seas are currently at opposite convective extrema in our postwar record, with vertical exchange at the former site limited to 1000 m or so, but with Labrador Sea convection reaching deeper than previously observed, to over 2300 m. As a result, Greenland Sea Deep Water has become progressively warmer and more saline since the early ‘70’s due to increased horizontal exchange with the Arctic Ocean through Fram Strait, while the Labrador Sea Water has become progressively colder and fresher over the same period through increased vertical exchange; most recently, convection has become deep enough there to reach into the more saline NADW which underlies it, so that cooler, but now saltier and denser LSW has resulted.

Shear-Flow Dispersion, Internal Waves and Horizontal Mixing in the Ocean

The mean circulation of planetary fluids tends to develop uniform potential vorticity q in regions where closed time-mean streamlines or closed isolines of mean potential vorticity exist. This state is established in statistically steady flows by geostrophic turbulence or by wave-induced potential-vorticity flux. At the outer edge of the closed contours the expelled gradients of q are concentrated. Beyond this transition lies motionless fluid, or a different flow regime in which the planetary gradient of q may be dominant. The homogenized regions occur where direct forcing by external stress or heating within the closed isoline is negligible, upon the potential-density surface under consideration. In the stably stratified ocean such regions are found at depths greater than those of direct wind-induced stress or penetrative cooling. In ‘channel’ models of the atmosphere we again find constant q when mesoscale eddies cause the dominant potential-vorticity flux. In the real atmosphere the results here can apply only where internal heating is negligible. The derivations given here build upon the Prandtl–Batchelor theorem, which applies to non-rotating, steady two-dimensional flow. Supporting evidence is found in numerical circulation models and oceanic observations.

Labrador Sea boundary currents and the fate of the Irminger Sea water

The homogenization of a passive ‘tracer’ in a flow with closed mean streamlines occurs in two stages: first, a rapid phase dominated by shear-augmented diffusion over a time ≈ P 1/3 ( L / U ), where the Peclet number P = LU /κ ( L,U and κ are lengthscale, velocity scale and diffusivity), in which initial values of the tracer are replaced by their (generalized) average about a streamline; second, a slow phase requiring the full diffusion time ≈ L 2 /κ. The diffusion problem for the second phase, where tracer isopleths are held to streamlines by shear diffusion, involves a generalized diffusivity which is proportional to κ, but exceeds it if the streamlines are not circular. Expressions are also given for flow fields that are oscillatory rather than steady.

An Example of Eddy-Induced Ocean Circulation

In the autumn of 1996 the field component of an experiment designed to observe water mass transformation began in the Labrador Sea. Intense observations of ocean convection were taken in the following two winters. The purpose of the experiment was, by a combination of meteorological and oceanographic field observations, laboratory studies, theory, and modeling, to improve understanding of the convective process in the ocean and its representation in models. The dataset that has been gathered far exceeds previous efforts to observe the convective process anywhere in the ocean, both in its scope and range of techniques deployed. Combined with a comprehensive set of meteorological and air-sea flux measurements, it is giving unprecedented insights into the dynamics and thermodynamics of a closely coupled, semienclosed system known to have direct influence on the processes that control global climate.

Observing Deep Convection in the Labrador Sea during Winter 1994/95

Changing ocean circulation patterns and sea surface temperatures affect atmospheric flow in the North Atlantic region. Atmospheric blocking over the northern North Atlantic, which involves isolation of large regions of air from the westerly circulation for 5 days or more, influences fundamentally the ocean circulation and upper ocean properties by affecting wind patterns. Winters with clusters of more frequent blocking between Greenland and western Europe correspond to a warmer, more saline subpolar ocean. The correspondence between blocked westerly winds and warm ocean holds in recent decadal episodes (especially 1996 to 2010). It also describes much longer time scale Atlantic multidecadal ocean variability (AMV), including the extreme pre–greenhouse-gas northern warming of the 1930s to 1960s. The space-time structure of the wind forcing associated with a blocked regime leads to weaker ocean gyres and weaker heat exchange, both of which contribute to the warm phase of AMV.