Empirical singular vectors of baroclinic flows deduced from experimental data of a differentially heated rotating annulus
Instability is related to exponentially growing eigenmodes. Interestingly, when finite time intervals are considered, growth rates of certain initial perturbations can exceed the growth rates of the most unstable modes. Moreover, even when all modes are damped, such particular initial perturbations...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Borntraeger
2015-01-01
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| Series: | Meteorologische Zeitschrift |
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| Online Access: | http://dx.doi.org/10.1127/metz/2014/0563 |
| _version_ | 1818344783086616576 |
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| author | Michael Hoff Uwe Harlander Christoph Egbers |
| author_facet | Michael Hoff Uwe Harlander Christoph Egbers |
| author_sort | Michael Hoff |
| collection | DOAJ |
| description | Instability is related to exponentially growing eigenmodes. Interestingly, when finite time intervals are considered, growth rates of certain initial perturbations can exceed the growth rates of the most unstable modes. Moreover, even when all modes are damped, such particular initial perturbations can still grow during finite time intervals. The perturbations with the largest growth rates are called singular vectors (SVs) or optimal perturbations. They not only play an important role in atmospheric ensemble predictions, but also for the theory of instability and turbulence. Starting point for a classical SV-analysis is a linear dynamical system with a known system matrix. In contrast to this traditional approach, measured data are used here to estimate the linear propagator. For this estimation, a method is applied that uses the covariances of the measured time series to find the principal oscillation patterns (POPs) that are the empirically estimated linear eigenmodes of the system. By using the singular value decomposition (SVD), we can estimate the modes of maximal growth of the propagator which are thus the empirically estimated SVs. These modes can be understood as a superposition of POPs that form a complete but in general non-orthogonal basis. The data used, originate from a differentially heated rotating annulus laboratory experiment. This experiment is an analogue of the earth's atmosphere and is used to study the development of baroclinic waves in a well controlled and reproducible way without the need of numerical approximations. Baroclinic waves form the background for many studies on SV growth and it is thus straight forward to apply the technique of empirical SV estimation to these laboratory data. To test the method of SV estimation, we use a quasi-geostrophic barotropic model and compare the known SVs from that model with SVs estimated from a surrogate data set that was generated with the help of the exact model propagator and some random noise. In that context, we also address the question of the appropriate filter technique to remove noise from the data prior to the empirical SV-analysis. We ask whether there is an objective mean to distinguish between noise and signal. Finally, we compare the results with earlier findings from a numerical low-order model of baroclinic waves for which the system matrix is known. The results from the low-order model suggested that irregular flows have in general larger SV growth rates. These findings have been used to explain the gradual increase of irregularity when the rotation rate of the annulus is increased while keeping the radial temperature contrast constant. This simple picture cannot be confirmed by the laboratory data. The singular value spectrum becomes rather broad for irregular flows similar to the SV spectrum of atmospheric models. Thus the irregularity might be related to the presence of a large number of SVs with similar growth rates and not to few SVs with exceptional large growth rates. |
| first_indexed | 2024-12-13T16:51:57Z |
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| id | doaj.art-89e3c9dd9a9941f9b08722afb70458a2 |
| institution | Directory Open Access Journal |
| issn | 0941-2948 |
| language | English |
| last_indexed | 2024-12-13T16:51:57Z |
| publishDate | 2015-01-01 |
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| spelling | doaj.art-89e3c9dd9a9941f9b08722afb70458a22022-12-21T23:38:01ZengBorntraegerMeteorologische Zeitschrift0941-29482015-01-0123658159710.1127/metz/2014/056384476Empirical singular vectors of baroclinic flows deduced from experimental data of a differentially heated rotating annulusMichael HoffUwe HarlanderChristoph EgbersInstability is related to exponentially growing eigenmodes. Interestingly, when finite time intervals are considered, growth rates of certain initial perturbations can exceed the growth rates of the most unstable modes. Moreover, even when all modes are damped, such particular initial perturbations can still grow during finite time intervals. The perturbations with the largest growth rates are called singular vectors (SVs) or optimal perturbations. They not only play an important role in atmospheric ensemble predictions, but also for the theory of instability and turbulence. Starting point for a classical SV-analysis is a linear dynamical system with a known system matrix. In contrast to this traditional approach, measured data are used here to estimate the linear propagator. For this estimation, a method is applied that uses the covariances of the measured time series to find the principal oscillation patterns (POPs) that are the empirically estimated linear eigenmodes of the system. By using the singular value decomposition (SVD), we can estimate the modes of maximal growth of the propagator which are thus the empirically estimated SVs. These modes can be understood as a superposition of POPs that form a complete but in general non-orthogonal basis. The data used, originate from a differentially heated rotating annulus laboratory experiment. This experiment is an analogue of the earth's atmosphere and is used to study the development of baroclinic waves in a well controlled and reproducible way without the need of numerical approximations. Baroclinic waves form the background for many studies on SV growth and it is thus straight forward to apply the technique of empirical SV estimation to these laboratory data. To test the method of SV estimation, we use a quasi-geostrophic barotropic model and compare the known SVs from that model with SVs estimated from a surrogate data set that was generated with the help of the exact model propagator and some random noise. In that context, we also address the question of the appropriate filter technique to remove noise from the data prior to the empirical SV-analysis. We ask whether there is an objective mean to distinguish between noise and signal. Finally, we compare the results with earlier findings from a numerical low-order model of baroclinic waves for which the system matrix is known. The results from the low-order model suggested that irregular flows have in general larger SV growth rates. These findings have been used to explain the gradual increase of irregularity when the rotation rate of the annulus is increased while keeping the radial temperature contrast constant. This simple picture cannot be confirmed by the laboratory data. The singular value spectrum becomes rather broad for irregular flows similar to the SV spectrum of atmospheric models. Thus the irregularity might be related to the presence of a large number of SVs with similar growth rates and not to few SVs with exceptional large growth rates.http://dx.doi.org/10.1127/metz/2014/0563baroclinic wavenon-modal growthsingular vector |
| spellingShingle | Michael Hoff Uwe Harlander Christoph Egbers Empirical singular vectors of baroclinic flows deduced from experimental data of a differentially heated rotating annulus Meteorologische Zeitschrift baroclinic wave non-modal growth singular vector |
| title | Empirical singular vectors of baroclinic flows deduced from experimental data of a differentially heated rotating annulus |
| title_full | Empirical singular vectors of baroclinic flows deduced from experimental data of a differentially heated rotating annulus |
| title_fullStr | Empirical singular vectors of baroclinic flows deduced from experimental data of a differentially heated rotating annulus |
| title_full_unstemmed | Empirical singular vectors of baroclinic flows deduced from experimental data of a differentially heated rotating annulus |
| title_short | Empirical singular vectors of baroclinic flows deduced from experimental data of a differentially heated rotating annulus |
| title_sort | empirical singular vectors of baroclinic flows deduced from experimental data of a differentially heated rotating annulus |
| topic | baroclinic wave non-modal growth singular vector |
| url | http://dx.doi.org/10.1127/metz/2014/0563 |
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