Results from the spectral analyses of
the flows in two experiments where turbulent flows were generated in a
rotating tank with a topographic

Two-dimensional

Experiments on the Coriolis rotating platform by Read et al. (2007) confirmed
the theoretical prediction of a

Zonal jets have a long history of investigation starting from the pioneering
experiments by Whitehead (1975) and Collin de Verdiere (1979). The jets
readily form when a spatially localised forcing is applied in the

In this study we perform spectral analyses of the flows described in Matulka and Afanasyev (2015) and compare them with the results obtained for a somewhat different flow generated by thermal forcing (Zhang and Afanasyev, 2014). The latter experiment although forced baroclinically was more barotropic in its dynamics, while the former experiment was purely baroclinic. In Sect. 2 of this paper, we describe the laboratory set-up for both experiments. In Sect. 3 the results of the spectral analyses are reported. Concluding remarks are made in Sect. 4.

The laboratory experiments were carried out in a cylindrical tank of radius

In the second experiment the flow was forced by delivering fresh water at the
surface of a salt water layer of salinity

Sketch of the experimental set-up

The free surface of the rotating fluid is a paraboloid when in solid-body
rotation. The height of the water surface varies quadratically with the
distance

We use the Altimetric Imaging Velocimetry (AIV) system to measure the
gradient of the surface elevation

AIV measures an “exact” (within experimental accuracy) surface elevation
gradient,

The barotropic component of velocity can be calculated in geostrophic
approximation as follows:

Relative vorticity,

Typical images from video sequences recorded in the experiments with
thermal

According to the Taylor–Proudman theorem, the surface velocity given by Eq. (4) is a good approximation for the velocity in the entire column of water except the Ekman layer at the bottom. Note that in a stratified fluid, as in our two-layer experiment with saline forcing, the velocity field obtained by altimetry is in fact the barotropic component of the total velocity in the entire layer of water. It is also the upper layer velocity. A baroclinic component which allows one to obtain the total velocity in the lower layer can be measured by a different technique (e.g. Afanasyev et al., 2009; Matulka and Afanasyev, 2015), but is not discussed here.

We performed two sets of experiments with different forcing, namely the thermal forcing by a wire heater at the bottom and the saline forcing by injection of fresh water at the wall. In what follows we discuss them in parallel, highlighting the similarities and differences between them.

Figure 2a shows a typical snapshot of the surface of water in the experiment
with the thermal forcing when the flow is fully developed. The flow is
visualised by the AIV technique such that colour shows the horizontal
gradient of the surface elevation,

Figure 3a and b shows the fields obtained as a result of the velocity
calculation from measured

Flows generated by thermal forcing at

Figures 2b and 3c and d (see also Fig. 1c) show the experiment with the
saline forcing. The water in the tank is initially of salinity

Baroclinic instability together with other instabilities including, perhaps, wave breaking and barotropic and frontal instabilities, continuously generates meanders over the entire area of the tank. The meanders move water parcels in the radial (meridional) direction. According to conservation of potential vorticity the parcels acquire additional relative vorticity and radiate Rossby waves. Motion of the meanders/parcels correlated via the global Rossby wave field creates the Reynolds stresses which drive zonal jets in the interior of the tank. Thus, although direct forcing was stopped, the system is forced by the baroclinic instability similar to that in the ocean. Measurements of the Reynolds stresses in this experiment showed that jets in the interior are dynamically different from the coastal jet which is affected by the presence of the wall. The jets in the interior are true eddy-forced jets, while the coastal current is not. In what follows we perform spectral analyses of the flow in the inner area which contains these “true” jets and excludes the coastal current.

Visual comparison between the fields in the experiments with different
forcing (Fig. 3) shows that the scales of the turbulent eddies generated by
the forcing are noticeably different. The eddies in the thermal experiment
are smaller, which indicates smaller

Herein we describe the results of the spectral analysis of the flows. For a
circular domain such as our tank, it is perhaps more natural to use polar
coordinates for the purpose of spectral decomposition. Afanasyev and
Wells (2005) used Fourier–Bessel transform to
obtain two-dimensional energy spectra of the polar

Energy spectra in the wavenumber space (

The AIV technique gives velocity field on a regular rectangular grid covering
the entire area of the tank. The velocity vector field was interpolated onto
the local Cartesian coordinate system and then projected to the eastward and
northward directions to obtain zonal and meridional velocity components.
Discrete Fourier transform of these velocity components then gives velocity

The inverse energy cascade is a well-known phenomenon in two-dimensional
turbulence; energy is transferred from small scales (large

The one-dimensional energy spectra in log-log scale for the
experiments with thermal

To extend the Rhines' argument to two dimensions one can equate the
frequency of turbulent eddies to that of Rossby waves to obtain (Vallis and
Maltrud, 1993) a dividing line in the form

To study general spectral characteristics of the turbulent cascade without
regard to the anisotropy one can average energy over the polar angle

The estimates of the Rhines wavenumber (accidently) give similar values for
both experiments,

In our experiments we observed the formation of zonal jets in the experiments where flows were forced using two different methods. Perhaps the main difference between the forcing was that the heater at the bottom created convectively unstable vertical temperature distribution which resulted in small-scale convective plumes. Vertical mixing must be significant in this system and the fluid remained mainly unstratified. The large-scale flow in this experiment is then approximately barotropic, although the nature of forcing is baroclinic. In the second experiment, on the other hand, we created statically stable two-layer stratification. The flow was baroclinic to a significant degree. Since this system was unstable with respect to baroclinic instability, the instability was a source of small-scale turbulence. Thus in both cases some small-scale turbulence was created, but in the former experiment the flow was mainly barotropic, while in the latter it was mainly baroclinic.

In spite of this significant difference between the flows in our two
experiments, we observed a definite universality in their spectral dynamics.
The energy cascaded from small scales to larger scales and towards zonal
motions. The two-dimensional spectra demonstrated that this cascade is in
reasonable agreement with the Rhines theory. One-dimensional spectra of
energy reliably demonstrated the existence of the energy interval with the

The authors are grateful to Alexander Slavin for his help with one of the experiments. Y. D. Afanasyev is supported by the Natural Sciences and Engineering Research Council of Canada. Experimental data are available on request from Y. D. Afanasyev. Edited by: J. M. Redondo Reviewed by: W.-G. Fruh and one anonymous referee