GENERATION OF ZONAL TRANSPORT BY WEAKLY NONLINEAR ROSSBY WAVES

 

TARMO SOOMERE

 

Senior Research Fellow, Estonian Marine Institute, Paldiski mnt. 1, 10137 Tallinn, Estonia

 

TIIT KOPPEL

 

Professor, Tallinn Technical University, Ehitajate tee 5, 19086 Tallinn, Estonia

Correspondence to: T. Soomere, Estonian Marine Institute, Paldiski mnt. 1, Tallinn, Estonia, ph. +(372) 6 328 011, fax +(372) 6 311 069, e-mail soomere@anet.ee

 

 

ABSTRACT

Rossby waves and their interactions were studied experimentally in the 13-m "Coriolis'' rotating tank. Practically linear travelling waves (typical velocities of fluid particles ~1 mm/s, phase velocity ~10 mm/s) were excited by a radially oscillating paddle. A stochastic wave field (expected to arise owing to reflections of waves as well as owing to non-homogeneities of the bottom and the walls) with a broad spectrum is subject to evolve according to the kinetic theory and it will support zonal flow components. In many experiments a noticeable zonal transport in the form of weak jets, with a typical width 0.4...1.0 m and velocity ~0.04 mm/s, was detected after ca 10 wave periods. In some cases it lead to water displacements to a distance comparable to the paddle amplitude (0.3 m).

 

Keywords: Rossby waves, wave-wave interactions, zonal transport

 

LINEAR ROSSBY WAVES AND NONLINEAR SYNOPTIC VORTICES

The Rossby wave - a transversal low frequency oscillation in rotating systems - exists owing to the variation of the vertical component of the background rotation (e.g. the North-South variation of the Coriolis parameter called b-effect). Rossby (planetary) waves (linear disturbances propagating along medium) and synoptic vortices (cyclones and anticyclones in the atmosphere, synoptic rings in the oceans - nonlinear structures carrying medium with them) are the main examples of synoptic(-scale) motions in the nature (called geostrophic turbulence). A remarkable property of the latter is energy transfer to an anisotropic flow directed along parallels (zonal flow). This peculiarity usually characterises the nonlinear flow regime. However, after a long time geostrophic turbulence must become spectrally isotropic (Carnevale, 1984). In the nature, this is not the case: the overwhelming domination of the zonal wind component in the atmosphere is well known and the ocean currents mostly flow in the East-West direction.

The anizotropization might be, at least, partially created by Rossby wave interactions since systems of Rossby waves always evolve towards a special equilibrated state, consisting of a superposition of a zonal flow and a spectrally isotropic wave system (Reznik and Soomere, 1984; Reznik, 1986, Soomere, 1995; 1996). Since one cannot separate Rossby waves from other synoptic-scale motions in natural conditions, one can only verify the hypothesis by setting up a proper experiment.

Experimental studies of Rossby waves have always been connected with serious technical problems (for an overview see Faller, 1981). It has been quite difficult to excite a free long-living Rossby wave or isolate a travelling wave from eddies and mean flow (see e.g. Ibbetson and Phillips, 1967; Colin de Verdiere, 1977). Typically, owing to large viscous and Ekman damping (caused by large wave amplitudes), only one or two wave crests, nearest to the wavemaker, were detectable. Observations were mostly made in the near-wave field, from which one can establish main features of waves but cannot capture fine details of the wave evolution.

In this experimental study dynamics of Rossby waves was isolated from geostrophic turbulence through keeping the wave amplitudes as small as possible. Small-scale nonhomogeneties of the bottom and the walls of the experimental device redistributed the initial wave energy to a spectrally broad wave system. The kinetic theory predicts that the latter should give rise to low-frequency harmonics propagating along the tank radius. The main goal of the experiment was to detect these harmonics in the form of striped flow along bottom isolines (equivalent to zonal flow).

 

EXPERIMENTAL SETUP AND GOVERNING EQUATIONS

Experiments were performed in the "Coriolis" Laboratory of Geophysical Flows, Joseph Fourier Grenoble University, on the counterclockwards-rotating platform with the diameter 13 m. Its rotation period was mostly 50 s; the fresh water depth - 50 cm.

A radially symmetric topography simulating the geographical b-effect was used. A sloping bottom (a curved trapezoid, the 8%-slope covered ca 52 m² and raised towards the outer edge of the tank) was built from plain segments of ca 2*2 m (Fig. 1). Direction to the South was towards the rotation center; wave crests propagated in the rotation direction (westwards); longer waves arose down(west)wards from the wavemaker and the shorter ones - upwards from it.

This configuration locally represented the topographical b-plane. The open boundaries of the slope absorbed the wave energy and prevented both the growth of standing waves and the meridional wavenumber selection. This setup was called mixed geometry. For several experiments, a rigid barrier was built along the internal side of the slope, converting the model to that of a zonal channel. The deviations of the resulting fluid dynamics from that on the b-plane did not change the qualitative behaviour of the waves in question. The actual dispersion relation only somewhat differed from the theoretical one.

Geostrophic motions both in the ocean and in the rotating basin are described by the quasigeostrophic vorticity equation (e.g. Gill, 1982). Let us briefly examine the main points of its derivation. The continuity equation can be taken in the form [ is the velocity] since the expected and measured velocities were of the order of 1mm/s; the phase velocities of the Rossby waves - of the order of a few dozens cm/s. Likewise, the vertical displacements were negligible as compared to the reduced depth of the basin.

The background rotation suppresses the vertical velocity component. It was not measured since it was extremely small and much below the resolution of the measurement techniques. Use of the hydrostatic approximation is justified (here p is pressure, r - density and g - gravity acceleration) makes it possible to eliminate the vertical velocity component from the general equation for inviscid motions of a homogeneous fluid (e.g. Gill, 1982, Eq. 4.10.11). The geostrophic approximation reads:

 

; , (1)

 

 

where are the local Cartesian coordinates and - the angular velocity of the Earth (platform). It is only valid if (1) , where L is the typical horizontal scale, , f - Coriolis parameter and - its typical value, (2) the typical time scale and (3) the Rossby (Kibel) number , where U is the typical velocity scale (e.g. Gill, 1982, Eq. 12.2.27). In the experiments, the first condition is accompanied by the requirement for bottom slope (a=0.08) to be small. The generated Rossby wave periods typically exceeded 4...6 times the tank rotation period. The actual velocities of fluid particles were of the order of a few mm/s. For the typical length scale L=5 m, the Rossby number in most of the experiments. Thus, the restrictions of the geostrophic approximation were satisfied. The latter reduces the equation of motions to the quasigeostrophic vorticity equation. For the rotating platform with a homogeneous sloping bottom (the x-axis being directed against rotation and the y-axis from the rotation centre, and the mean depth of the basin m), its linearized form is

 

. (2)

 

The dispersion relation for Rossby waves on the infinite slope reads:

 

. (3)

 

Here is the barotropic Rossby radius and - the wave vector. In the case of a zonal channel the meridional wave number k is arbitrary but the zonal one may take only the values , where and A is the channel width. For A=4 m, the Rossby waves have the minimum period  s. The case corresponds to Kelvin waves for which radial displacements vanish identically. Without the inner barrier, Rossby waves extend to the area with horizontal bottom. Assuming that the wave amplitudes decrease there exponentially, their dispersion relation qualitatively coincides with those on an infinite plane and in a zonal channel. The minimum period of Rossby waves in this setup was  s (Borghi, 1996).

 

WAVE EXPERIMENTS

Background motions in the basin were carefully minimised. The wind stress was neutralised through fences mounted at each 2 m of the basin perimeter, ca 1 m above the water. The average zonal velocity over the whole duration of the experiments was less than 0.005 mm/s. The local zonal flow did not exceed 0.01 mm/s and had no directional preference. Measurements were delayed until the water achieved the solid body rotation state and the rms. background velocity field decreased under 0.3 mm/s.

A robust paddle (0.3*1 m) oscillated with a typical amplitude 35 cm and a period P from 65 s to 300 s (corresponding to wavelengths of ca 2.5...15 m). Only waves travelling along bottom isolines (with l=0) were produced. The novel CIV method (Fincham and Spedding, 1996) was first used for remote measurements of the two-dimensional (2D) Lagrangian velocity field in a large area (ca 2*2.5 m), with an actual resolution of ca 60*80 points. Additionally two Eulerian velocity components in 3 points were measured with ultrasonic probes.

In experiments with the paddle period no travelling waves existed. Starting from P=160 s (mixed geometry) and P=200 s (channel) the sensors recorded well-defined disturbances with the paddle frequency. Eastwards from the paddle (shorter waves) they rapidly (within one period after arriving the paddle signal) obtained a stationary amplitude. Westwards their amplitude continued to grow during 4...5 periods and remained lower as compared to the short waves.

A theoretical velocity field in a Rossby wave in the actual geometry consists of a sequence of virtual vortice-like structures. Figure 2 shows the velocity field of the shorter waves in the experiment with P=200 s. The diagrams demonstrate that the wave deviates slightly to the North as its propagates westwards. The wave trajectory is modified by the bottom geometry. It should be piecewise straight (instead of a circle in radially symmetric basins or a straight line in Mercator coordinates): the waves should keep their initial propagation direction within each bottom segment and turn at the bottom segment junctions only. Apart from the wavemaker, the direction of the wave vector may no longer coincide with the western direction.

 

NONLINEAR EFFECTS: WEAK NONLINEARITY AND ITS INFLUENCE

The velocities of fluid particles were, typically, much lower than phase velocities. Their ratio e (a traditional measure of nonlinearity) is less than or equal to 0.2 in most cases except (seldom) in the near-wave field.

Owing to inhomogeneous bottom geometry and the presence of borders, the monochromatic wave should soon excite a spectrally broad wave field. The latter evolves according to the kinetic theory and supports zonal motion components. Numerical experiments suggest that intensity of the (nearly) zonal flow generation depends on the structure of the initial wave field but in all cases it occurs relatively fast as compared to other spectral changes. A noticeable zonal anisotropy emerges within one "slow" time unit , where T is a typical wave period (Reznik, 1986; Soomere, 1996). For , the "slow" time unit ca 20 times exceeds the wave period. Zonal peak in numerical experiments corresponds to low-frequency harmonics (with wavelengths comparable with the most energetic initial ones) propagating in the North-South direction. Their superposition should become evident in the form of weak zonal jets.

In the long-wave experiments we always observed generation of striped zonal flow. Weak jets with their width 0.4...1 m were first visually detected from the temporal behaviour of the lines of dye. A typical behaviour of such a line is shown in Fig. 3. During the first few paddle periods there was no detectable distortion of it. Starting from the 4th...5th period the line bent indicating presence of weak striped zonal currents. The zonal velocity was of the order of 0.05 mm/s. The translatory velocity of the water was not detectable (less than 0.01 mm/s).

CIV measurements testified that the structure of the zonal transport showed a great variability and no two experiments were similar to each other. The width of the stripes varied from 0.3 m to 0.9 m, corresponding to somewhat shorter waves than the generated ones (Fig. 4). They became evident after ca 5 paddle periods and continued to grow until the end of the experiments. The maximum zonal velocity varied from 0.03 mm/s (relatively narrow stripes) to 0.3 mm/s (wide stripes; subsequent experiments, both with  s). The jets persisted at least for a quarter of an hour. In some cases they caused zonal transport of water up to the first dozens of centimetres, a distance comparable with the amplitude of the primary waves.

Zonal transport in the short-wave area was much weaker than in the long-wave region. Bending of a dye line was not visually detectable. The zonal velocity field consisted of extremely weak jets with typical width of 0.4...1 m. They slowly propagated north- or southwards but were practically not amplified during the experiments. An explanation to the distinction probably comes from the fact that the shorter waves "feel" the bottom inhomogeneities more. They skirt more along the bottom isolines, reflect less from the outer wall and, thus, generate less random wave components.

 

CONCLUSION AND DISCUSSION

The experiments confirm the prediction of the kinetic theory that propagation of practically linear Rossby waves may cause in a long-time run a considerable zonal transport of water masses. The detected striped transport arose in some cases surprisingly fast and had different structure in the subsequent experiments with identical parameters. This feature suggests that it has indeed random nature and emerges as a result of Rossby-wave interactions. The amplitude of the zonal transport accounts for a few percent of the barotropic Rossby radius (i.e., of the order tens of kilometers in the oceans) and is comparable with that of generated by large-amplitude Rossby wave packets (e.g., LaCasce and Speer, 1997).

A possibility that a part of the zonal transport came from side effects is deeply improbable since external factors (in particular wind stress), possibly supporting mean or low-frequency motions, were carefully excluded. The paddle itself may add a certain angular component of flow (Faller, 1981). However, its influence should be regularly present in experiments with identical parameters. Corresponding zonal velocity fields show no correlation (cf. Fig. 4); thus, the possible role of the paddle in the mean flow excitation can be neglected.

Nonlinear (turbulent) energy cascade generally supports a global vortex or dipole. Detailed study of velocity fields confirmed the absence of any stable zonal transport in the middle of the slope. Careful analysis of dye translation within somewhat larger area, observed by the analogue video recorder, also confirmed the absence of any regular zonal transport.

 

Acknowledgements

The experiment was financed by the Scientific Commission of the European Community (Contracts ERBCIPD940092 and ERBCGHECT920015). The final stage of the study was sponsored by the Estonian Science Foundation (Grant 3504/98).

 

REFERENCES

Borghi, C. 1996. Ondes de Rossby: une étude expérimentale, Research report, Ecole Centrale Paris, Université Pierre et Marie Curie; Laboratoire CORIOLIS, Inst. de Méchanique de Grenoble, 57p.

 

Carnevale, G. F. 1982. Statistical features of the evolution of two-dimensional turbulence, J. Fluid Mech., 122, 143-153.

 

Colin de Verdière, A. 1977. Quasigeostrophic flow and turbulence in a rotating homogeneous fluid. Ph. D. Theses, Massachusetts Inst. of Technology/Woods Hole Oceanographic Institution, 171p.

 

Faller, A. J. 1981. The origin and development of laboratory models and analogues of the ocean circulation. In Evolution of physical oceanography, ed. by B. A. Warren and C. Wunsch, MIT Press, Cambridge, Massachusetts, 462-479.

 

Fincham, A. M. & Spedding, G. R. 1996. Low cost, high resolution DPIV for measurement of turbulent fluid flow, Exp. Fluids, 23, 449-462.

 

Gill, A. E. 1982. Atmosphere-Ocean Dynamics, Academic Press, New York, London.

Ibbetson, A. & Phillips, N. 1967. Some laboratory experiments on Rossby waves in a rotating annulus, Tellus, 19, 81-87.

 

Kozlov, O. V., Reznik, G. M. & Soomere, T. 1987. Weak turbulence on the b-plane in a two-layer ocean, Izvestiya, USSR Acad. of Sci., Atmospheric and Oceanic Physics, 23, 869-874.

 

Lacasce, J. H. & Speer, K. G. 1997. Lagrangian spectra in Rossby wave fields, Ann. Geophys, 15, Suppl., C561.

 

Reznik, G. M. 1986. Weak turbulence on a b-plane. In Synoptic eddies in the ocean. D.Reidel Publishing Company, Dordrecht, Holland, 73-108.

 

Reznik, G. M. & Soomere, T. 1984. On the evolution of an ensemble of Rossby waves to an anisotropic equilibrium state, Oceanology, 24, 424-429.

 

Soomere, T. 1995. Generation of zonal flow and meridional anisotropy in two-layer weak geostrophic turbulence, Phys. Rev. Lett., 75, 2440-2443.

 

Soomere, T. 1996. Spectral evolution of two-layer weak geostrophic turbulence. Part I: Typical scenarios, Nonlin. Proc. Geophys., 3, 166-195.