The heated rotating annulus: PIV- and LDV-measurements of...

14th Int Symp on Applications of Laser Techniques to Fluid MechanicsLisbon, Portugal, 07-10 July, 2008

The heated rotating annulus: PIV- and LDV-measurements of multiple-scaleinteractions in a weakly turbulent flow

Uwe Harlander1, Thomas von Larcher2, Yongtai Wang1, Christoph Egbers1

1: Dept. of Aerodynamics and Fluid Mechanics, Brandenburg University of Technology Cottbus,Cottbus, Germany, [email protected]

2: Institute for Mathematics, Free University Berlin, Berlin, Germany, [email protected]

Abstract Already in the fifties, an elegant laboratory experiment had been designed to studybaroclinic waves. It consists of a cooled inner and heated outer cylinder mounted on a rotatingplatform, mimicking the heated tropical and cooled polar regions of the earth’s atmosphere.Depending on the strength of the heating and the rate of rotation, different flow regimes had beenidentified: wave-regimes that can be classified by prograde propagating waves of differentwavenumbers, and quasi-chaotic regimes where waves and small scale vortices coexist. In the presentpaper we will use multivariate statistical techniques to understand better the variability of the heatedrotating annulus flow in a quasi-stable regime. In the experiment this regime is quantified by usingboth, the PIV- as well as the LDV-method. The PIV-data are analyzed by employing the so calledEmpirical Orthogonal Function (EOF) method. This technique reveals the main patterns ofvariability of the transient flow field. In a subsequent step, the Multi-Channel Singular SpectrumAnalysis (M-SSA) is applied to the LDV-data.In the annulus, interactions between the dominant mode and so called weaker modes can lead tolow-frequency amplitude and structure vacillations of the dominant mode. The M-SSA shows thatmore than 87% of the total variance is cumulated in the first and second eigenvalue corresponding tothe leading wave pattern. However, the EOF analysis reveals a weaker mode from the PIV data. It issuggested that this weak mode indicates wave mean flow interactions leading to the mentionedvacillations. Moreover, the weak mode might also be viewed as a precursor of a 4 wave regime,existing in proximity to the 3 wave regime.

1 Introduction

When Edward N. Lorenz (1963) discussed the problem of deducing the climate from thegoverning equations, he used the heated rotating annulus as an analogy to the complexdynamics of the large-scale weather. Baroclinic instability is one of the main reasons for thevariability of the mid-latitude weather, and indeed this instability is also the dominant processin the heated rotating annulus, steadily transforming available potential to kinetic energy.Since the flow regime that develops in the cylindrical gap of the annulus depends on theradial temperature gradient between the inner and outer cylinder, and on the rotation rate ofthe apparatus, a 2D parameter space (called regime diagram), spanned by the Taylor and thethermal Rossby number includes all possible flow regimes. Regime diagrams have beeninvestigated in detail, already by Fowlis and Hide (1965), and later by von Larcher andEgbers (2005). Critical parameters have been found, separating different regimes that can be

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characterized as wave-like jets with dominant zonal wavenumber m, where typicallym = 0, 1, 2, · · · , 5 was found. For wide gaps, wavenumbers large than 5 have not beenobserved since the flow becomes more and more turbulent for higher wavenumbers.A phenomenon that has over many years attracted attention is the so called amplitude andstructural vacillation. What is meant is a modulation of the amplitude and the dominantwavenumber in actually regular wave regimes. In particular Früh (1996), and Früh and Read(1997) have investigated this phenomenon in detail. Roughly speaking, they suggested thatresonant wave triads are responsible for the vacillations. Such triads, beside the dominantmode, involve three other, weaker modes (the triad). Energy is redistributed betweenmembers of such a trio and the dominant pattern vacillates with a characteristic time. On theother hand, Yang (1990) proposed that a wave packet shows structural vacillations due tozonal and meridional variations of the mean flow. In contrast to the wave triad mechanismthat seems to create burst like amplitude/structure changes, the latter mechanism leads to amore regular oscillation of the dominant wave mode. Note that both explanations foramplitude/structure vacillations assume the existence of weak modes that interact with thedominant mode and thus leads to its vacillation.In the present study we analyze velocity data from heated rotating annulus experiments,recovered by the PIV and LDV technique. The data are sampled twice/once per rotation ofthe annulus and cover time periods that allow for the application of multivariate statisticaltechniques. With these techniques, the dominant modes of variability are isolated. Our mainresult is that besides the dominant m = 3 mode, a weaker m = 4 mode is present in theactually regular flow regime. Although no other weak mode was detected and the ideassketched above cannot be verified directly, it still appears to be the first time that a weakmode was found by PIV observations of the heated rotating annulus flow.The paper is organized as follows. In the next section we give a brief outline of theexperimental setup and the data acquisition. In section 3 the multivariate statistical methodsemployed to analyze the data are described. Results are given in section 4. In section 5 webriefly discuss our findings. Finally, section 6 gives an outlook on future work.

2 Experiment set-up and data acquistion

In this section we briefly describe the experimental setup and the settings of the experimentalparameters. We also inform the reader about the methods used to collect the data and theway the data are conditioned for further analysis.

2.1 Experimental setup

The set-up (Fig. 1) described in detail by von Larcher and Egbers (2005) consists of a tankwith three concentric cylinders mounted on a turntable. While the inner cylinder is made ofanodised aluminium and cooled by a thermostate, the middle and outer one is made ofborosilicate glass. The outer side-wall of the experiment gap is heated by a heating coil that

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ab

d

coldwarm

Ω

g

inner cylinder

drift

inner side wall ofouter cylinder

a) geometric dimensions of the set-upinner radius, a [mm] 45outer radius, b [mm] 120fluid depth, d [mm] 135gap width, b − a [mm] 75radius ratio, η = a/b 0.38aspect ratio, Γ = d/(b−a) 1.8

b) experimental conditionsrotation rate, Ω [rad s−1] 0.47 . . . 0.50mean temp, <T> [C] 23.0ref. temp. diff., ∆T [K] 7.5 . . . 7.6

c) physical properties of the fluiddensity, ρ [kg m−3] 998.21kin. viscosity, ν [m2 s−1] 1.004 × 10−6

th. conductivity, κ [m2 s−1] 0.1434 × 10−6

exp. coefficent, α [K−1] 0.207 × 10−3

Figure 1: Left: sketch of the rotating annulus with illustration of a typical large-scale jet-stream structure that has a drift relative to the rotating reference system. Right:experimental parameters and physical properties of the working fluid.

is mounted at the bottom of the outer cylinder bath. In our setup, the experiments has a freesurface and a flat bottom. Deionised water is used as working fluid.While the most important dynamic parameters, the rotation rate of the annulus, Ω, and thetemperature difference in the cylindrical gap, ∆T , are characterised by the Taylor number Taand thermal Rossby number Ro, the physical properties of the working fluid are characterisedby the Prandtl number Pr:

Ta =4 · Ω2 · (b− a)5

ν2 · dRo =

g · d · α∆T

Ω2 · (b− a)2Pr =

ν

κ.

A description of the parameters is given in Figure 1 that also shows a sketch of theexperimental setup. In our experiments, the Taylor number is varied between1.55× 107 ≤ Ta ≤ 1.74× 107, the thermal Rossby number is in the range of 1.48 ≤ Ro ≤ 1.57,and the Prandtl number of water is Pr = 7.0.

2.2 PIV data acquisition

A PIV system is used to measure the horizontal velocity components about 15 mm below thefluid surface. The experimental parameters are Ω = 0.5 rad/s (4.77 rpm), ∆T = 7.6 K leadingto a Taylor and Rossby number of Ta = 1.74× 107 and Ro = 1.48. The PIV camera ismounted in an inertial frame above the cylinder, that is the camera does not co-rotate with

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Figure 2: Ta = 1.55 × 107(4.50 rpm), Ro = 1.57, m = 3: space-time plot of the multi-channelLDV data set. Radial velocity is shown. Note the stretching of the time axis.

the cylinder. To obtain the velocity components in the co-rotating frame we subtract the rigidbody velocity v = Ω× r from each observed PIV velocity field. We sample the PIV data witha rate twice the rotation rate, i.e. the sampling period is 6.28 s. In total we did 98 PIVmeasurements. That is, in total we observed 49 annulus rotations during a time span of615.75s.The inner cylinder of the rotating annulus is not transparent for laser light. To avoid a gap inthe PIV velocity field due to the shadow behind the inner cylinder we cut two consecutiveobservations through the common center of the cylinders, rotated the later observation by πand then glued the lower half of the earlier observation together with the upper half of thelater observation. This procedure gives good results as long as the velocity field does notevolve much during a half rotation of the annulus. Indeed, for the parameters given above,this assumption holds quite well.Any PIV observation contains erroneous vectors that are removed from the data. As a resultgaps occur within the data. Moreover, the size of the PIV field can differ for differentobservations. To homogenize the data we applied a linear interpolation by using theMATLAB function griddata.

2.3 LDV data acquisition

The radial velocity component is measured with a LDV that is fixed in the inertial frame (i.e.it does not co-rotate with the cylinder). The measurement takes place about 2 mm below thefluid surface at the mid radius of the research cavity (r = 82.5mm). It should be noted thatthe radial velocity component is the same for the inertial and the co-rotating frame. Duringone rotation of the cylinder, 20 LDV observations are made. This gives a spatial resolution of18 in the azimuthal direction. At each of these 20 points we observe the radial velocity onceper rotation period, that is a sample rate of 13.33 s. The time gap of neighboring pointsamounts 0.67 s. To obtain data on a regular grid with respect to space and time, we apply a

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linear interpolation to each of the 20 time series covering a period of about three hours. Afterthis pre-processing we obtain a homogeneous data set with grid distance ∆φ = 18 and∆t = 13.33 s. These pre-processed LDV data are then analyzed by using the Multi-channelSingular Spectrum Analysis (M-SSA) software toolkit (Dettinger et al., 1995) as described insection 3.2.The LDV data set is taken at the parameters Ta = 1.55× 107 (4.50 rpm), Ro = 1.57.Figure 2 shows a typical cut of the multi-channel data matrix.

3 Multivariate data analysis methods

Two different statistical techniques have been applied to the data. In the following, bothmethods are discribed briefly.

3.1 Empirical Orthogonal Functions (EOFs)

The EOF method is the method of choice for analyzing the variability of fields and is thuswidely used in the geosciences (Lorenz, 1956; v. Storch and Zwiers, 1999). The method isable to find the spatial patterns of variability, their time variation, and provides a measure forthe ’relevance’ of each pattern. Simply speaking, the EOF method breaks the data into’modes of variability’ that might (as is the case for our data) be interpreted as physical modesof the system. How can the EOFs of the PIV observations be computed?Let us assume that any PIV analysis gives a number of velocity vectors (ui, vi) at pointslabeled with i = 1, · · · , p. Further we assume that in total we did n PIV observations. Thenwe store the velocity components u and v in two matrices

Fu =

u11 u12 · · · u1p

u21 · · ·· ·· ·

un1 · · · unp

. (1)

and Fv (or simply F for short). Note that the columns of F are the time series for eachobservation point, and any row corresponds with a PIV observation at a certain time. Thecovariance matrix of F is defined as R = FTF. The eigenvectors vi of R are the EOFs, andthe corresponding eigenvalues λi define the variance explained by vi.To see how the ith EOF vi evolves in time we have to compute the expansion coefficientsai = Fvi. Finally, the data matrix F can be reconstructed by the series

F =p∑

j=1

ajvj. (2)

It should be noted that the EOF analysis sketched above is most useful for single scalar fields.In general, coupled variability is obtained from a coupled EOF analysis or a corresponding

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singular value decomposition (SVD). Moreover, propagating patterns of variability are usuallyfound from complex EOFs. However, as will be obvious below, straightforward EOF analysisfor u and v gives good results that can be interpreted physically. Thus, for the PIV dataanalyzed here, standard EOF analysis is sufficient.

3.2 Multi-channel Singular Spectrum Analysis

The pre-processed multichannel LDV time series data are applied to the Singular SpectrumAnalysis-Multi Taper Method (SSA-MTM) Toolkit, a software package to detect(intermittent) oscillations in noisy time series as well as multivariate data (Dettinger et al.1995). M-SSA is a generalization of the single time-series SSA method to multiple time-series(Read, 1993). These time-series may contain observations of a certain variable at differentlocations (as in our case) or even observations of different variables.While an extended description of the M-SSA method is beyond the scope of the presentpaper, for the reader’s convenience we give a brief overview of the technique that followsDettinger et al. (1995).Assume, we have the L time series (so called L channels), Xl,n : l = 1, . . . , L; n = 1, . . . , N,each containing N measurements with time increment ∆t. The covariance matrix TX

contains matrix blocks Tl,l′ which estimate the lag covariance between channel l and l′ andhas the form

TX =

T1,1 T1,2 . . . T1,L

T2,1 T2,2 . .. . . . .. . . . Tl,l′ .. . . . .. . . TL−1,L

TL,1 . . . TL,L−1 TL,L

.

The entries (j,j′) of each block Tl,l′ can be written as

(Tl,l′)j,j′ =1

N

min(N,N+j−j′)∑n=max(1,1+j−j′)

Xl,nXl′,n+j−j′ ,

whereas N is a factor that depends on the range of summation

N = min(N, N + j − j′)−max(1, 1 + j − j′) + 1.

Diagonalizing the LM × LM matrix TX yields LM eigenvectors Ek : 1 < k ≤ LM that arenot necessarily distinct. Each eigenvector Ek is composed of L consecutive M -long segments,with its elements denoted by Ek

l,m. The associated space-time Principal Components(ST − PCs) Ak are single-channel time series that are computed by projecting X onto the

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Empirical Orthogonal Functions (ST − EOFs):

Akn =

M∑m=1

L∑l=1

Xl,n+mEkl,m,

where n varies from 1 to N ′.For a given set of indices K, reconstructed components (RCs) are obtained by convolving thecorresponding ST-PCs with the ST-EOFs. Thus, the kth RC at time n for channel l is givenby:

Rkl,n =

1

Mn

M∑m=1

Akn−mEk

l,m.

The normalization factor Mn equals M , except near the ends of the time series, and the sumof all the RCs recovers the original time series, as it does in the single-channel case.

4 Results

In this section we give the results of the present study. First we present what we found fromthe PIV data by applying the EOF analysis, and second we give preliminary findings from theLDA data set that has been analyzed by using the M-SSA.

4.1 PIV Measurements

In the following we first discuss the time mean flow and second the dominant patterns ofvariability for the experiment conducted with the parameters given in section 2.2. Asdiscussed in detail by Fowlis and Hide (1965) and von Larcher and Egbers (2005), the flowregime is determined by the Taylor and Rossby number. In our setting, the flow typicallyshows a meandering jet with three ridges and three troughs. This pattern propagatespro-grade. Under ideal conditions, the ridges would touch the outer cylinder and the troughsthe inner one. North of the ridge (i.e., towards the center of the annulus) a cyclone(anti-clockwise rotation) and to the south of the trough an anti-cyclone occurs.An actual PIV snapshot of the flow, taken 15 mm below the fluids free surface, is shown inFig. 3a. Obviously, this flow deviates from the ideal situation but all major features(meandering jet, 3 ridges and troughs, cyclones and anti-cyclones) are visible. Themeandering jet is wave-driven and the waves occur due to baroclinic instability of a verticallyand horizontally sheared mean flow. As was discussed by Williams (1971), away from theheated/cooled boundaries, the flow is almost two-dimensional (quasi-geostrophic). Byaveraging over time (i.e. over all 98 PIV observations) we retrieve an almost zonallysymmetric wave driven pro-grade flow (see Fig. 3b). In Fig. 3c we show the azimuthal part ofthis flow for Φ = 0 and Φ = π/2, where Φ denotes the azimuth angle of the cylindricalcoordinates. This flow corresponds well with mean flows found in other studies (e.g. Sommeriaet al. 1991) but does not show the fine structure discussed recently by Read et al. (2007).

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a b

c d

Figure 3: a) Single PIV observation for the parameters Ta = 1.74 × 107 and Ro = 1.48. b)Time-mean flow. c) Azimuthal velocity at Φ = 0, π/2. d) The first ten eigenvalues ofthe EOF covariance spectrum.

Fig. 3d displays the eigenvalue spectra of the covariance matrices obtained from the twovelocity components u and v. Clearly, the spectra are dominated by the first two eigenvaluesthat contain almost 50% of the total variance of the flow. The third and forth eigenvaluesinclude about 10% of the total variance. Fig. 4 shows the corresponding EOFs and their timeevolution. The PCs of EOF1 and 2 as well as EOF3 and 4 are in quadrature (see Figs. 4a,b).Reconstructing the velocity field at time t = t0 by using just the first two EOFs

u(x, y, t0) = au1(t0)vu1(x, y) + au2(t0)vu2(x, y) (3)v(x, y, t0) = av1(t0)vv1(x, y) + av2(t0)vv2(x, y) (4)

gives a very regular counterclockwise propagating vortex train with wavenumber three (calledmode 3 in the following)(see Fig. 4c). This is not too surprising since this field, together withthe mean flow shown in Fig. 3b gives the generic wavenumber three jet flow that is visible e.g.in Fig. 3a. More interesting is the field that can be constructed from EOF3 and 4,

u(x, y, t0) = au3(t0)vu3(x, y) + au4(t0)vu4(x, y) (5)v(x, y, t0) = av3(t0)vv3(x, y) + av4(t0)vv4(x, y). (6)

This reconstruction gives a rather regular and slowly propagating mode 4 pattern (Fig. 4d).Taking all four EOFs together we find that the wavy jet flow with dominant wavenumber

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a b

c d

Figure 4: a) The principal components (PCs) of EOF1 (black) and EOF 2 (red). b) PCs ofEOF3 (black) and EOF4 (red). c) Reconstruction of the velocity field by EOF1 and2. d) Reconstruction of the velocity field by EOF3 and 4.

three shows slow vacillations due to the interaction of the two modes. From the PCs we canread off the phase speed of the waves in the co-rotating frame of reference. Mode 3 needsabout 315s to complete a full cycle. From Fig. 4a we see that its frequency isω = 2π/105s−1 = 0.06s−1 that gives a phase speed of c = ω/k = 1.71mm/s, where k = 2π/λ(λ = 2πr/3, r ≈ 85mm) denotes the azimuthal wavenumber. In contrast, the mode 4 patternpropagates slowly clockwise and it takes about 1000s to complete one cycle, giving afrequency and phase speed of ω = 2π/250s−1 = 0.025s−1 and c = −0.52mm/s. Thus, relativeto mode 3, mode 4 propagates clockwise with c = c3 − c4 = 2.23mm/s. It should bementioned that there is also a faster oscillation superposed to the slow time evolution of themode 4 pattern (Fig. 4b). Perhaps the mode 4 variance pattern is a mixture of severalphysical modes with different frequencies. Therefore mode 4 should be interpreted with care.To validate our findings that the wavy jet flow (a superposition of the mean flow and mode 3)is perturbed by a regular smaller-scale vortex train (the mode 4), we repeat the EOF analysisbut in a frame that co-rotates with mode 3. In this frame, mode 3 is part of the mean flow(see Fig.5a) and we expect that mode 4 shows up as the now dominant EOF. Indeed, this isexactly what we observe (Fig.5b,d). Reading off the frequency of mode 4 in the frameco-rotating with mode 3 by inspecting Fig.5c, we find ω = 2π/63s−1 = 0.1s−1 andc = 2.12mm/s. This phase speed is in good agreement with the result found in the previous

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ab

cd

Figure 5: a) Time-mean of the flow in the frame co-rotating with mode 3. b) The first teneigenvalues of the EOF covariance spectrum. c) The PCs of EOF1 (black) and EOF2 (red). d) EOF1.

paragraph.To summarize the results of the EOF analysis, we find that even in a rather stablewavenumber three regime, a higher mode seems to be present that slowly modulates the jetflow. We will come back to this phenomena in section 5.

4.2 LDV measurements

Let us next briefly present the results from the M-SSA Analysis of the LDV data. Asmentioned in section 3.2, a strong feature of the M-SSA is its ability to detectoscillating/propagating patterns in noisy data. Thus, in theory, the M-SSA is more suitableto find all excited propagating modes in a certain experimental setup defined by the Taylorand Rossby number than standard EOF analysis.Fig. 6 shows the eigenvalue spectrum of the M-SSA covariance matrix. Strikingly, just twospace time EOFs (ST-EOFs) explain already 80% of the total variance contained in the LDAdata set. It appears that the flow measured with the LDV is less perturbed than the oneobserved by PIV, although both experiments are done essentially for the same Taylor andRossby number. The first two ST-EOFs (Fig. 7 left) show a perfect wavenumber three mode

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0 5 10 15 20rank

0.01

0.1

1

10

100

Var

ianc

e [%

]

050606 4.5rpm, interpoliert, gedreht, cut: MSSAVariance, M=232

Figure 6: The first twenty eigenvalues of the M-SSA covariance spectrum. Note that a loga-rithmic scale is used.

0 50 100 150 200 250-0.03

-0.02

-0.01

0

0.01

0.02

1 2

050606 4.5rpm, interpoliert, gedreht, cut: MSSAEOF 1 (dashed line) and EOF 2(solid line), Channel 1

0 100 200 300 400 500 600 700 800 900 1000

-0.1

0

0.1 1 2

050606 4.5rpm, interpoliert, gedreht, cut: MSSAPC 1 (dashed line) and PC 2 (solid line), Channel 1

Figure 7: Left: ST-EOF1 and 2. Right: ST-PCs 1 and 2 applied to channel 1.

and from the corresponding ST-PCs (Fig. 7 right) we can read off a frequency ofω = 2π/90s−1 = 0.069s−1, in reasonable agreement with the result from the standard EOFanalysis.As can be expected, a reconstruction (not shown) of single time series (single M-SSA channel)by using the first two ST-EOFs is successful, although several maxima and minima in thetime series are not well captured by the reconstruction due to a low frequency modulation ofthe series with a period in the order of 103s.

5 Discussion

In the present study we applied multivariate statistical methods to data from a heatedrotating annulus experiment. The Taylor and Rossby number have been chosen such that theflow is in a quasi-stable regime that is characterized by a wavy jet flow with a wavelength thatequals one third of the annulus’ circumference. The statistical techniques (EOF analysis andM-SSA) are able to find patterns that contain most of the variance of the data. Isolatingthese patterns from the noisy background can help i) to understand the role weak modes play

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Figure 8: Ta = 1.55× 107(4.50 rpm), Ro = 1.57, m = 3: image of the surface temperature.

for regime transitions, and ii) to elucidate how the dominant mode interacts with weakermodes to generate amplitude vacillations. Here, weaker modes are defined as modes thatexplain less variance than the dominant mode but are still significant in the eigenvaluespectrum of the covariance matrix.Already Hide et al. (1977) observed strong interactions between dominant and weaker modesin a rotating annulus experiment. Such interactions can lead to amplitude vacillations, as wasdiscussed in detail by Früh (1996). In the present study we isolate a weaker mode from PIVmeasurements. In the co-rotating frame of reference, the mode 3 (vortex train withwavenumber 3) is dominant. Mode 4 is a weaker mode that contains only about 10% of thetotal variance but that clearly stands out from the eigenvalue spectrum. For the waveinteraction mechanisms discussed by Früh, other weak modes should be present in the flow.E.g., for the so called harmonic route, weak modes 6 and 2 should also be present. So far wehave not found these modes in our experiments. With the present resolution, mode 6 canhardly be resolved. Moreover, mode 6 is so far not observed in the experiment even for largerTaylor numbers due to the wide gap between the inner and outer cylinder. Flows of such widegap systems have the tendency to become irregular already before the wavenumber 6 regimesettles down. Nevertheless, mode 2 should be observable. This mode is important in thealready mentioned harmonic route but also in the sideband coupling route to amplitudevacillations (Fr"uh, 1996). It might well be that this mode is hidden in the noisy part of theEOF spectrum. More data will be available in the near future and then other weak modesmight come to the fore.We have shown that by doing the EOF analysis in a frame that co-rotates with the dominantmode 3, the relative contribution of the weak mode to the total variance can significantly beenhanced. In general, working in this frame appears to be the better practice to detect weakermodes since in this frame the dominant mode becomes a part of the mean field.In addition to the EOF analysis of PIV data we used LDA data for a M-SSA analysis. TheLDA data consist of 20 time series regularly distributed along a circle in the center of theannulus. The M-SSA is computational time consuming but 20 channels can be handled evenon a small workstation. Nevertheless, with respect to a weak mode 4, the M-SSA results areinconclusive. In the original time series, a slight low-frequency amplitude modulation isvisible even by eye. However, a ST-EOF with a mode 4 structure could not be found. We

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think that the main problem here is spatial resolution. In surface temperature infraredobservations (see Fig. 8), that were associated to the LDA measurements, slight wavevacillations could visually be observed. However, due to the poor spatial resolution, theyappear to be not properly recorded by the LDV observations.

6 Outlook

In the future we plan to investigate the occurrence of weaker modes in more detail. First,experiments with parameters closer to the values at a bifurcation point will be done. Here weaddress the question if more weak modes appear when the bifurcation point, separating theregular from the irregular flow regime, is approached. Second, the M-SSA method will beapplied to the PIV data. Due to computational time restrictions it is not possible to use thefull PIV data set. However, the leading 20 PCs of an EOF analysis could be used as theM-SSA channels (Plaut and Vautard, 1994). Third, PIV observations will be combined withobservations of the surface temperature. Previous experiments conducted by in von Larcherand Egbers (2005) have shown that fine scale structures can be resolved by infrared pictures(see Fig. 8). Technically, surface temperatures are easier to obtain and are also more accuratethan PIV derived velocity vectors. Therefore it will be instructive to see if weak modes can befound in the temperature fields, too.

References

Dettinger MD, Ghil M, Strong CM, Weibel W, Yiou P (1995) Software expeditessingular-spectrum analysis of noisy time seies. EOS Trans. AGU 76(2): 12,14,21

Fowlis WW, Hide R (1965) Thermal convection in a rotating annulus of liquid: effect ofviscosity on the transition between axissymmetric and non-axissymmetric flow regimes.J. Atmos. Sci. 22: 541-558

Früh W-G (1996) Low-order models of wave interactions in the transition to baroclinicchaos. Nonl. Proc. in Geophys. 3: 150-165

Früh W-G, Read PL (1997) Wave interactions and the transition to chaos of baroclinicwaves in a thermally driven rotating annulus. Phil. Trans. R. Soc. Lond. A 355: 101-153

Hide R, Mason PJ, Plumb RA (1977) Thermal convection in a rotating fluid subject to ahorizontal temperature gradient: spatial and temporal characteristics of fully developedbaroclinic waves. J. Atmos. Sci. 34: 930-950

von Larcher Th, Egbers C (2005) Experiments on transitions of baroclinic waves in adifferentially heated rotating annulus. Nonl. Proc. in Geophys. 12:1033-1041

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Lorenz EN (1964) The problem of deducing the climate from the governing equations.Tellus 16: 1Ű11

Lorenz EN (1956) Empirical orthogonal functions and statistical weather prediction.Scientific rep. no 1, statistical forecasting project, Dept. Met., M.I.T.

Plaut G, Vautard R (1994). Spells of low-frequency oscillationsand weather regimes inthe northern hemisphere. J. Atmos. Sci. 51: 210-236

Read PL (1992) Application of singular system analysis to ’baroclinic chaos’. Physica D58: 455-468

Read PL, Yamazaki YH, Lewis SR, Williams PD, Wordsworth AR, Miki-Yamazaki K,Sommeria J, Didelle H (2007) Dynamics of Convectively Driven Banded Jets in theLaboratory. J. Atmos. Sci. 64: 4031-4052

Sommeria J, Mayers SD, Swinney HL (1991) Experiments on vortices and Rossby wavesin eastward and westward jets. Nonlinear Topics in Ocean Physics, North-Holland,Amsterdam: 227-269

von Storch H, Zwiers FW (1999) Statistical analysis in climate research. CambridgeUniversity Press, Cambridge, pp.1-484

Williams GP (1971) Baroclinic annulus waves. J. Fluid Mech. 49: 417-449

Yang H (1990) Wave Packets and their Bifurcations in Geophysical Fluid Dynamics.Springer, New York, pp.1-247

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