NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-23-21-2016Complex environmental β-plane turbulence: laboratory experiments with altimetric imaging velocimetryMatulkaA. M.ZhangY.AfanasyevY. D.afanai@mun.caMemorial University of Newfoundland, St. John's, CanadaY. D. Afanasyev (afanai@mun.ca)28January201623121299October20159November20158January201612January2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://npg.copernicus.org/articles/23/21/2016/npg-23-21-2016.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/23/21/2016/npg-23-21-2016.pdf
Results from the spectral analyses of
the flows in two experiments where turbulent flows were generated in a
rotating tank with a topographic β-effect are presented. The flows were
forced either by heating water from below or supplying fresh water at the top
of a saline layer. The flow was essentially barotropic in the first
experiment and baroclinic in the second experiment. The gradient of the
surface elevation was measured using optical altimetry (altimetric imaging
velocimetry). Multiple zonal jets of alternating direction were observed in
both experiments. Turbulent cascades of energy exhibit certain universal
properties in spite of the different natures of flows in the experiments.
Introduction
Two-dimensional β-plane turbulence is an important concept in complex
environmental flows where planetary rotation and the effect of the variation
of the Coriolis parameter with latitude (β-effect) are significant
factors. Kraichnan (1967) formulated a Kolmogorov-type theory which predicted
the slope -5/3 of the energy spectrum in the energy range and the
slope -3 in the enstrophy range for two-dimensional turbulence. The
β-effect modifies two-dimensional turbulence towards anisotropy. The
energy spectra in wavenumber space become a figure-of-eight with most of the
energy concentrated at a zonal wavenumber close to zero. In physical space
this effect manifests itself in the creation of zonal jets. Oceanographic
observations of this phenomenon have been extensively discussed in the
literature (Maximenko et al., 2005, 2008; Centurioni at al., 2008; Ivanov et
al., 2009).
Experiments on the Coriolis rotating platform by Read et al. (2007) confirmed
the theoretical prediction of a -5/3 slope in the energy range. The authors
used convective forcing in their experiments which generated motions of very
small scale. Since the size of the domain was large, the scale separation was
large enough for the development of the inverse cascade of energy. A recent
study on two-dimensional turbulence (in the absence of β-effect) by
Afanasyev and Craig (2013) gave the experimental evidence of dual cascade
with the spectral slopes of -5/3 and -3. Further experiments with
barotropic turbulence on the β-plane by Zhang and Afanasyev (2014)
demonstrated the dual cascade in the presence of β-effect as well as
the figure-of-eight energy spectrum in the wavenumber space.
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 β-plane fluid
(e.g. Sommeria et al., 1988, 1989; Marcus and Lee, 1998; Afanasyev et al.,
2011; Slavin and Afanasyev, 2012). However, a
distributed forcing such as that provided by baroclinic instability also
creates jets. In the ocean, the regions where baroclinic instability is
dynamically important include the Antarctic Circumpolar Current (ACC) as well
as western boundary currents and their extensions. Multiple jets as well as
mesoscale eddies are created there by the baroclinic instability, thus
forming a dynamically complex turbulent flow. A classic model for a
baroclinically unstable system is a rotating annulus where heating/cooling is
provided at the outer wall/centre of
the tank respectively. Different aspects of the dynamics of this system were
studied in a number of experiments (Hide and Mason, 1975; Mason, 1975; Bastin
and Read, 1997, 1998; Wordsworth et al., 2008; Smith et al., 2014). A
somewhat different experimental approach to modelling a baroclinically
unstable two-layer flow with vertical shear was used in a recent study by
Matulka and Afanasyev (2015). It was shown that the meridional scale of the
jets is determined to a large extent by the radius of deformation and, at the
same time, is in good agreement with the Rhines scale (Rhines, 1975). The
jets are driven by (non-linear) Reynolds stresses due to baroclinic meanders.
This is in agreement with a scenario described by Berloff et al. (2009a, b).
The authors describe the formation of jets as a secondary instability of the
primary instability of the baroclinic flow in the form of mainly meridional
motions (so-called “noodles”).
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.
Experimental technique
The laboratory experiments were carried out in a cylindrical tank of radius
R= 55 cm filled with water of depth H0= 10–12 cm. The
tank was installed on a rotating table (Fig. 1a) and rotated in an
anticlockwise direction at a constant angular rate
Ω= 2.32 rad s-1. In this paper we compare two experiments
where the flows were forced in two different ways. The first experiment was
forced thermally with a heating wire at the bottom of the tank. The wire was
arranged in an approximately uniform pattern such that the distance between
the segments of the wire was 4.5 cm. Figure 1b shows the magnitude of
velocity in the range between 0 (black) and 0.5 cm s-1 (white) at the
very beginning of the experiment. The flow is initially along the wires such
that the wire pattern is visible. This wire arrangement (in contrast to, say,
a spiral) was chosen in order to avoid a direct forcing in the zonal
direction but yet to provide an approximately uniform mean heat flux across
the bottom. The heater supplied the total power of 2300 W.
In the second experiment the flow was forced by delivering fresh water at the
surface of a salt water layer of salinity S= 30 ppt. The fresh water
source was distributed along the wall of the tank and created a coastal
current flowing anticlockwise around the tank. The fresh water was
pumped into the tank by a pump which delivered 20 L of water in about
6 min. Figure 1c shows the camera view of the flow in the middle of the
forcing period when the fresh water from the source has not yet spread over
the entire surface of the tank but is concentrated mainly in the coastal
current.
Sketch of the experimental set-up (a) and view of the flow
at the beginning of the experiments with the thermal (b) and
saline (c) forcing: (1) cylindrical tank filled with water and
installed on a table rotating with angular velocity Ω; (2) light box
for the optical thickness measurements; (3) video camera; (4) light source
with colour mask; and (5) heating wire on the bottom.
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 r from the axis of rotation
h(r)=H0+Ω22gr2-R22,
where g is the gravitational acceleration. This creates a topographic (polar)
β-plane where the β-parameter at some distance r=r0
from the pole is given by
β=f0h(r)dh(r)drr=r0,
where f0= 2Ω is the Coriolis parameter. The values of
the β-parameter in our experiments were between 0.07 and 0.1 cm-1 s-1.
We use the Altimetric Imaging Velocimetry (AIV) system to measure the
gradient of the surface elevation η (for more details see Afanasyev et
al., 2009). AIV is based on optical altimetry first described in Rhines et
al. (2006). Laboratory altimetry is not unlike the satellite altimetry which
became an irreplaceable tool in oceanography. Apart from measuring
∇η, AIV can be used as a tool to visualise the entire surface of
the rotating fluid just like satellite altimetry provides a global coverage
of the Earth's oceans. AIV also provides an alternative to the well-known
particle imaging velocimetry (PIV) technique. To obtain the velocity field
using PIV one has to find correlations between small areas (typically
12–48 pixels in both dimensions) of two successive images of the flow. If a
camera has an imaging array of size say 1000 × 1000 pixels, the
resulting array of velocity vectors has dimensions less than
100 × 100. Thus, the PIV technique effectively reduces the spatial
resolution by a factor of 100 or more. AIV, on the other hand, allows one to
obtain the velocity vector in every pixel of the image, which makes its
spatial resolution practically unlimited given that the cameras with large
imaging arrays are now easily available. The limitation of the AIV is that it
can only be used in a relatively fast rotating fluid with free surface. It
is, however, perfectly suited for oceanographic fluid dynamics experiments on
the β-plane such as those described in this paper. The paraboloidal
surface of the rotating fluid is used like the mirror of a Newtonian
telescope. If the surface is disturbed by the pressure perturbations due to
the flow, the slope of the surface changes slightly. These perturbations of
the slope are detected by the AIV and measured using simple geometry and a
colour coding.
AIV measures an “exact” (within experimental accuracy) surface elevation
gradient, ∇η, which translates into the pressure gradient,
∇p=ρg∇η, at the surface. Note that the main
uncertainty is due to the colour noise of the camera sensor. To reduce the
noise we pass data through a median filter with a window size of
5 × 5 pixels (physical size of approximately
0.2 × 0.2 cm). The overall uncertainty of ∇η can be
estimated to be approximately 5%. The velocity field is not measured
directly by the AIV but is rather obtained from the measured ∇η
using a (quasi-)geostrophic approximation.
The barotropic component of velocity can be calculated in geostrophic
approximation as follows:
V=gf0(n×∇η),
where n is the vertical unit vector. A next order of approximation is
provided by the quasi-geostrophy, which gives
V=gf0(n×∇η)-gf02∂∂t∇η-g2f03J(η,∇η),
where J is the Jacobian operator. The second and third terms on the RHS of
Eq. (4) are corrections to the geostrophic velocity which take into account
transient and non-linear effects. Their relative importance is determined by
the temporal Rossby number RoT= 1/(f0T) and
the Rossby number Ro=U/(f0L) respectively. Here T is
the timescale of the unsteady processes in the flow, while U and L are
velocity and length scales of the flow. Thus, the velocity field is
determined more accurately when the flow is closer to being
quasi-geostrophic. “Textbook” theory on the validity of the
quasi-geostrophic approximation applies here. Since the Rossby number did not
exceed unity even in the core of the eddies in our experiments and the mean
values of the Rossby number were of the order of 10-1, the velocity was,
on average, within 10 % of the “exact” velocity.
Relative vorticity, ζ=n⋅ curl V, was
calculated by differentiating the velocity field. Since numerical
differentiation amplifies noise in the original data, we used the Sobel
gradient operators with 5 × 5 kernels (e.g. Pratt, 2007). The
kernels are convolved with the velocity data to calculate the derivatives in
x and y directions.
Typical images from video sequences recorded in the experiments with
thermal (a) and saline (b) forcing. The flows are
visualised by optical altimetry (AIV); different colours indicate different
values (both in magnitude and direction) of the gradient of the surface
elevation.
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.
Results
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.
General flow evolution
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, ∇η. The colour intensity
indicates the magnitude of ∇η, while hue shows its direction. The
motions are initially of very small scale in this experiment (see also
Fig. 1b). Warm water heated by the wire rises to the surface in thin sheets
and forms long thin filaments. These filaments are unstable with respect to
baroclinic or frontal instability and break into small eddies. The size of
the eddies is most likely determined by the baroclinic radius of deformation
which can be defined as
Rd= (g′H0/2)1/2/(g′H0/2)1/2f0, where
g′ is the reduced gravity in the warm water filaments and eddies. The
reduced gravity is defined by the temperature difference between the water in
filaments and background temperature of the water in the tank. However, since
we did not measure the temperature field in this experiment, we cannot
pinpoint the exact value of Rd. It is important to establish here
the values of control parameters for the thermal convection in this
experiment. Flows created by buoyancy sources in the rotating fluid (in the
absence of the β-effect) were studied in laboratory experiments by
Fernando et al. (1991) and Maxworthy and Narimusa (1994). The energy flux per
unit area of the bottom of the tank is
Q= 2.5 × 103 Wm-2 in our experiments. This
translates into the buoyancy flux
B=αgQ/Cp= 1.2 × 10-6 m2 s-3,
where α is the thermal expansion coefficient and Cp is the
volumetric heat capacity of water at constant pressure. A dimensionless
parameter which controls the regime of the thermal convection is the Rayleigh
number. The flux Rayleigh number is quite high in our experiments,
Raflux=BH04/ν2κ= 1.8 × 109
(where κ is the thermal diffusivity of water), which indicates a
regime of turbulent convection. Comparing to the recent experiments by Read
et al. (2015) where similar heating was used but in an experiment of a larger
scale, we note that our buoyancy flux was almost 2 orders of magnitude
higher; the Rayleigh number was, however, an order of magnitude lower because
of the smaller depth of water in our experiments.
Figure 3a and b shows the fields obtained as a result of the velocity
calculation from measured ∇η in the thermal experiment. Azimuthal
velocity Vaz in a polar coordinate system with the origin at the
centre of the tank is shown in panel a, while panel b shows the relative
vorticity, ζ=n⋅ curl V. The relative
vorticity field shows a fine structure of the flow with multiple small-scale
cyclonic (red) and anticyclonic (blue) eddies. The relative vorticity is
normalised by the Coriolis parameter f0; thus, the image in panel b can
be interpreted as a map of the Rossby number,
Ro=ζ/f0. The values of Ro can reach unity in
the strongest eddies, while the rms value is approximately 0.2. An original
alignment of filaments with the heating wires can still be seen in the
central area of the tank, while closer to the wall where the water is deeper
and the β-effect is stronger, the bands of positive and negative
vorticity are aligned in the zonal direction, which indicates the presence of
zonal jets. The jets can be seen more clearly in the azimuthal velocity image
in Fig. 3a. The circulation is mainly in the anticlockwise (eastward)
direction (Vaz is positive, red colour). Maximum values of
Vaz in the jets are approximately 1 cm s-1.
Flows generated by thermal forcing at
t= 280 s (a, b) and by saline forcing at
t= 150 s (c, d). (a) and (c) show the
x component of velocity (azimuthal velocity) u, while (b) and
(d) show the dimensionless relative vorticity, ζ/f0. The
salinity difference between the layers for the experiment with saline
forcing (a, b) is S= 30 ppt. The centre of the tank
corresponds to the North Pole of the polar β-plane.
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
S= 30 ppt. When a source distributed along the wall of the tank
delivers fresh water, it creates a current along the wall. This current is
initially wedge-shaped in cross section and is approximately in geostrophic
balance such that it “leans” on the wall to its right. This coastal current
can be seen clearly in Figs. 1c, 2b and 3c; the velocity of the current is
initially in excess of 5 cm s-1. The current is baroclinically
unstable (e.g. Griffiths and Linden, 1981) and creates meanders which
penetrate into the interior of the tank. During the forcing period of the
experiment when the source continuously supplies fresh water, the entire
surface of the tank eventually becomes covered with a layer of fresh water.
Thus, a two-layer system is created. The forcing then stops and the flow is
allowed to develop freely. The depth of the upper layer is not uniform in the
radial direction after the period of forcing. The layer is much thicker at
the wall of the tank rather than at its centre. Thus, the system contains a
large amount of available potential energy which is released gradually and
maintains the flow for a very long time after the forcing stops. The
adjustment involves slow radial motion towards the centre in the upper layer
and the motion in the opposite direction in the lower layer. The radial
motion, in turn, causes zonal circulation. Measurements of barotropic and
baroclinic components of velocity (for details see Matulka and Afanasyev,
2015) show that the upper layer rotates cyclonically while the lower layer
rotates anticyclonically. The shear between the layers makes the system
baroclinically unstable. Conditions for the development of the baroclinic
instability are maintained over a long period of adjustment. Note that
measurements of mean energy and enstrophy of the system (not shown here)
during the long period of adjustment indicate that the system is
approximately steady in this experiment.
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 Rd. Another difference is
perhaps more significant in distinguishing between the forcing mechanisms.
The flow with saline forcing is characterised by thin filaments rather than
circular eddies. In fact, eddies appear only as a result of breaking of the
filaments and do not have a very long lifetime. The filaments are created by
the baroclinic instability; they protrude in the radial (meridional)
direction. We hypothesise that these filaments are the manifestation of the
so-called “noodles” which are the primary mode of the instability of the
baroclinic flow (Berloff et al., 2009a, b).
Energy spectra in wavenumber space
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 β-plane turbulence
in the space of azimuthal and radial wavenumbers and then to obtain
one-dimensional spectrum by sorting data in terms of a polar analogue of the
Cartesian isotropic wavenumber. However, it is easier to perform digital
Fourier decomposition in the Cartesian coordinates because fast Fourier
transform routines can be used. Moreover, usage of Cartesian coordinates and
a conventional β-plane (rather than a quadratic polar β-plane)
simplifies the further comparison with available theory. For these reasons
here we introduce local Cartesian coordinates (x, y) centred at the
reference radius r0= 25 cm such that the x axis is directed to
the east, x=r0θ (where θ is the polar angle), and
the y axis is directed to the north, y=r0-r. For the
spectral analyses we chose a domain of half width 17 cm centred at the
reference radius r0= 25 cm such that the polar part of the tank,
where β-plane approximation is inappropriate, and the wall area, where
the coastal jet dominates, were excluded.
Energy spectra in the wavenumber space (kx, ky) for the
experiments with thermal (a, b) and saline (c, d) forcing:
t= 30 s (a), 630 s (b) from the beginning of the
thermal experiment; t= 10 s (c), 722 s (d) after
the fresh water source stopped in the experiment with saline forcing. Colour
shows energy in logarithmic scale. Black circles in (c) and
(d) show Rd-1. Black crosses in (b) and
(d) represent the Rhines scale wavenumber (5), while the
dumbbell-shaped curve is given by Eq. (6).
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
u(kx, ky) in the wavenumber space (kx, ky). The
two-dimensional energy spectrum is given by
Ekx,ky=12ukx,ky2.
Figure 4 shows two-dimensional spectra in the experiments with thermal and
saline forcing. The spectra are measured in the beginning of both experiments
when the spectral signature of the forcing is still strong, and at a later
stage, when the spectra have reached a certain “saturation” and the
turbulent cascades are developed. The spectral properties of forcing or
background turbulent flow can be inferred from the initial spectra. In the
experiment with thermal forcing the eddies generated by thermal plumes are
quite small; their scale can be estimated from the value of wavenumber
k≈ 2.5 cm-1 (outer ring in Fig. 4a) to be ∼ 2.5 cm.
Here k= (kx2+ky2)1/2 is the isotropic
wavenumber. These eddies are initially concentrated along the heating wires.
The separation of the heating wires (∼ 4.5 cm) determines a wavenumber
k≈ 1.4 cm-1 where energy concentration is observed as well
(inner ring in Fig. 4a). In the experiment with saline forcing the initial
spectrum is determined by the baroclinic instability which is sustained in
the two-layer system even when the actual forcing (the injection of fresh
water) is stopped. The energy is distributed in a wide range of wavenumbers,
but is mainly contained within a circle which corresponds to the reciprocal
of the radius of deformation, Rd-1= 1.2 cm-1. This is
in agreement with the prediction of the Phillips model (Phillips, 1951) of
baroclinic instability. The model predicts that all perturbations of
wavenumber k<Rd-1 are unstable, with the most unstable
(before non-linear saturation) wavenumber being
k= 0.64 Rd-1. Thus the initial spectra for both
experiments confirm our initial observation that the forcing scale is smaller
in the experiment with the thermal forcing. The spectra measured in much
later times in both experiments (Fig. 4b and d) show that energy cascades
towards smaller wavenumbers (larger scales). The distribution of energy in
the wavenumber space also becomes more anisotropic; the energy flows towards
the ky axis (zonal modes, kx= 0).
The inverse energy cascade is a well-known phenomenon in two-dimensional
turbulence; energy is transferred from small scales (large k) to large
scales (small k). In the presence of β-effect, this scenario is
modified. The pioneering work by Rhines (1975) demonstrated that there is a
certain scale, now known as the Rhines scale, which separates the
large-scale motions where β-effect dominates from a small-scale
turbulence. A wavenumber corresponding to the Rhines scale is given by
kR=β/Urms,
where Urms is root-mean-square velocity. On large scales Rossby
wave elasticity is important and the flow becomes strongly anisotropic as the
(linear) dispersion relation for Rossby waves suggests. The anisotropy is
manifested by the appearance of zonal jets. The wavenumber kR is also
widely used as an estimate for the meridional wavenumber of an arrangement of
zonal jets. It was shown to work well in different circumstances including
flows on gas giants or flows in the laboratory, although different
modifications of the Rhines scale were discussed as well. The values of
kR for each experiment are indicated by crosses on the y axis in
Fig. 4b and d.
The one-dimensional energy spectra in log-log scale for the
experiments with thermal (a) and saline (b) forcing: t= 630 s (a),
722 s (b) (as in Fig. 4b and d).
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
k8=βcosθUrms,
where θ is the polar angle in the wavenumber space,
θ=arctan(ky/kx). The line given by Eq. (6) resembles
a figure-of-eight or a dumbbell and is shown in Fig. 4b and d. Note that
values of Urms are not given by theory but have to be measured in
the experiment. Here we used the rms values of the y component of velocity
instead of the total velocity in order to avoid the mean flow in the
azimuthal direction which occurred to a different extent in both of the
experiments. Since Rhines' theory assumes isotropic small-scale turbulence,
it seems that the y component of velocity represents a better measure of
turbulence in our case. Spectra in Fig. 4b and d show that the turbulent
cascade of energy due to non-linear interaction of modes does not continue in
the isotropic manner within the area bounded by the k8 line. In fact,
there is little energy inside this area. The energy follows instead a
(linear) Rossby wave dispersion relation flowing along the k8 line
towards the ky axis and concentrating there. It is eventually dissipated
by friction which in our case is mainly the Ekman bottom friction which acts
on all scales. The distribution of zonal energy indicates that the isotropic
Rhines scale is indeed a reasonable measure of the jet wavenumber in both
experiments.
To study general spectral characteristics of the turbulent cascade without
regard to the anisotropy one can average energy over the polar angle θ
in the wavenumber space. As a result, one obtains a one-dimensional energy
spectrum which is defined as
E(k)= 2πk〈E(k)〉, where k= (kx,
ky) and the average is over |k|=k. Figure 5 shows the
one-dimensional spectra for our two experiments. Theory of two-dimensional
turbulence without β-effect predicts the existence of the dual cascade
such that energy cascades to large scales in the range k<kF
with a Kolmogorov type spectral slope -5/3 and to small scales in the range
k>kF with a slope -3. Here kF is the forcing
wavenumber where energy is injected into the system. The former range is
called the energy range, while the latter is called the enstrophy range. The
forcing wavenumber can be estimated to be
kF≈ 2.5 cm-1 in the experiment with thermal
forcing and
kF= 0.64 Rd-1≈ 0.8 cm-1 in the
experiment with saline forcing. These values mark the boundary between the
energy and enstrophy ranges in both experiments. Figure 5 shows that the
spectral slope does change in the vicinity of the estimated forcing
wavenumbers that confirms that these estimates are meaningful. In both
experiments, a -5/3 slope was observed in the energy range; in the first
experiment (thermal forcing) this range was longer than that in the second
one (saline forcing). In the enstrophy range the -3 slope was observed in
the first experiment (thermal forcing), while the second experiment (saline
forcing) showed a much steeper slope. The presence of the -3 slope might be
another consequence of the mostly barotropic (and two-dimensional) character
of the flow in the first experiment. Indeed, the flow in this experiment is
convectively unstable and mixing is significant. As a result, the fluid is
not significantly stratified and the flow must be mainly barotropic. The
steeper slope in the second experiment is, perhaps, due to the fact that this
two-layer, statically stable flow is significantly baroclinic. Note that a
steeper than -3 slope was also measured in the experiments with shallow
water, f-plane turbulence by Afanasyev and Craig (2013) and in the
numerical simulations by Yuan and Hamilton (1994).
The estimates of the Rhines wavenumber (accidently) give similar values for
both experiments, kR≈ 0.45 cm-1. This value indicates
the place for zonal jets in the spectral cascade. The maximum energy achieved
by the cascade in the second experiment roughly corresponds to the Rhines
wavenumber, which indicates that zonal motions should possess a significant
portion of the total energy. In the first experiment, the cascade does not
quite reach the Rhines wavenumber. Note that the Rhines wavenumber does not
stop the cascade to even lower wavenumbers; the cascade is just redirected
towards zonal motions.
Conclusions
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
-5/3 slope. Note that although in our experiments one can infer the
direction of the energy cascade from the form of the spectrum assuming that
the forcing wavenumber is known, direct evidence of the cascade direction can
only be provided by the analyses of the spectral energy flux. Such evidence
was provided for shallow water rotating turbulence (without β-effect)
in the laboratory investigation by Afanasyev and Craig (2013). The analysis
of the energy flux for the large-scale oceanic turbulence by Scott and
Wang (2005) revealed the existence of the inverse cascade in agreement with
the two-dimensional turbulence theory. However, the interplay between
barotropic and baroclinic modes and the extent to which each mode contributes
to the cascade in the ocean still remains a subject of research. The analysis
of the flux for the experiments reported here is yet to be done and will be
reported elsewhere.
Acknowledgements
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
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