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J. S. Frederiksen

Abstract

A study is made of the instability properties of three-dimensional Northern Hemisphere flows for the months of January and July 1978. The growth rates and phase frequencies of growing baroclinic disturbances and the perturbation streamfunctions and eddy heat and momentum fluxes are obtained using a five-level spherical quasi-geostrophic spectral model. It is found that the presence of the planetary waves in the basic states cause a considerable increase in the growth rates of the fastest growing modes, compared with the case for zonally averaged January and July flows. The results of three-dimensional instability theory am compared with the observed regions of most actively developing baroclinic disturbances and eddy fluxes. For both months, there is reasonable agreement between linear theory and observations as far as the geographical locations of the largest disturbance amplitudes and eddy fluxes are concerned. However, for the January three-dimensional basic state, the usual vertical structure-problem of instability theory occurs with the disturbance quantities being too large at the surface compared with those at the tropopause. For the July three-dimensional basic state, where local horizontal sheer or barotropic wave instability plays an important role, the disturbance streamfunctions and eddy fluxes are too large in the middle troposphere, compared with those at the surface. For both months, the linear solutions are very sensitive to local potential vorticity gradients.

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J. S. Frederiksen

Abstract

The complete growth rate and phase spectra of unstable baroclinic disturbances are studied in multi-level quasi-geostrophic spherical models and in a two-level non-geostrophic model, for two basic zonal jet-profile flows. The eigenvalue method employed has the advantages over initial value approaches of determining all the disturbance modes, being more efficient computationally and avoiding problems of slow convergence or lack of convergence when there are two modes of similar growth rates. The method is used to re-examine the spectra for Simmons and Hoskins' 30° jet and Gall's 42.1° jet. It is found that the disrepancies between our two-level and multi-level results are not as great as found by Simmons and Hoskins using primitive equation models, particularly with respect to phase speeds and momentum fluxes. Further, the short-wave maximum at zonal wavenumber 16 in their five-level primitive equation model does not occur here. Instead, our growth rates for the fastest growing modes are very similar to those of both the primitive equation and quasi-geostrophic eight-level models and in qualitative agreement with beta-plane calculations. For the 42.1° jet, our growth rate maximum occurs at wavenumber 11, in contrast with Gall's wavenumber 14 for his primitive equation model but in closer agreement with beta-plane calculations. The effects of changing the static stability in the lower troposphere and of varying the vertical and horizontal resolutions are also studied.

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J. S. Frederiksen

Abstract

An analytical study is made of the number, stability and bifurcations of solutions of time-dependent Budyko-type climate models with various nonlinear albedo parameterizations. With Budyko's (1969) albedo, a general stability criterion is derived and it is found that, for the present value of the solar constant, the present climate and ice-covered earth solutions are stable, a spurious solution is unstable and there is an ice-free solution which is stable. A Seller's (1969) type albedo leads to a stable present climate and an ice-covered earth solution as well as an unstable climate. Faegre's (1972) albedo produces a present climate which is unstable and has an incorrect behavior as the solar constant or the infrared flux is changed, as well as a stable warmer climate and an ice-covered earth solution. It is found that latidutinal variations in the albedo way have a profound effect on the number and stability of the solutions.

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J. S. Frederiksen

Abstract

A study is made of the changes in growth rates and phase speeds of growing baroclinic waves, disturbance streamfunctions and eddy momentum and heat fluxes induced by superimposing long planetary waves on purely zonal flow basic states in a five-level spherical quasi-geostrophic model. Attention is focused on basic states consisting of two 30° zonal jets with different vertical structures together with long zonal wavenumber 4 planetary waves having a number of different horizontal and vertical variations. In many respects, the qualitative changes in the properties of the fastest growing modes produced by the long planetary waves are similar to those found in two-layer models. In particular, the positions of the regions of preferential development of cyclones and anticyclones and the qualitative horizontal structures of the upper level non-oscillatory momentum fluxes bear a strong similarity to two-layer results with similar basic states. However, there are considerable differences in the vertical structures of disturbance streamfunctions and fluxes in two- and five-level models. The presence of the long planetary waves in the five-level model increases the upper level amplitudes of the disturbance streamfunctions and the upper level, poleward, zonally averaged momentum fluxes and thus improves comparison between these observed quantities and results of linear studies.

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J. S. Frederiksen

Abstract

A study is made of the effect of a long Planetary wave on the regions of preferential development, eddy momentum and heat fluxes, and the growth rates and phase speeds of growing baroclinic disturbances. The model used is a linear, spherical, two-layer quasi-geostrophic model with a basic state consisting of a 30° jet and an upper layer long planetary wave which together provide an approximate representation of the observed average Northern Hemisphere winter flow. The results of the baroclinic instability theory are compared with the observed regions of most actively developing baroclinic disturbances and eddy momentum and heat fluxes. Considering the crude vertical structure of two-layer models, the agreement between the theoretical and observed results is noteworthy. Details such as the observed latitudinal gaps between the jet-stream maxima and the regions of most actively developing eddies are reproduced in the model. Both these gaps and much of the horizontal variations of the theoretical and observed eddy momentum and heat fluxes are related to Phillips’ criterion for incipient instability. Finally, closure hypotheses, needed for statistical dynamical models in which long waves as well as zonally averaged quantities are to be predicted, are proposed.

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Meelis J. Zidikheri
and
Jorgen S. Frederiksen

Abstract

A stochastic subgrid modeling method is used to parameterize horizontal and vertical subgrid-scale transfers in large-eddy simulations (LESs) of baroclinic flows with large-scale jets and energy spectra typical of the atmosphere. The approach represents the subgrid-scale eddies for LES (at resolutions of T63 and T31) by a stochastic model that takes into account the memory effects of turbulent eddies. The statistics of the model are determined from a higher-resolution (T126) direct numerical simulation (DNS). The simulations use a quasigeostrophic two-level model and the subgrid terms are inhomogeneous in the vertical and anisotropic in the horizontal and are represented by 2 × 2 matrices at each wavenumber. The parameterizations have the largest magnitudes at a cusp near the largest total wavenumbers of the truncations. At T63 the off-diagonal elements of the matrices are negligible (corresponding to effectively decoupled levels) and the diagonal elements are almost isotropic. At the lower resolution of T31 the off-diagonal elements are more important and even the diagonal elements are more anisotropic. At both resolutions, and for anisotropic or isotropized subgrid terms, LESs are in excellent agreement with higher-resolution DNS.

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Meelis J. Zidikheri
and
Jorgen S. Frederiksen

Abstract

Inverse methods for determining the anomalous mean forcing functions responsible for climate change are investigated. First, an iterative method is considered, and it is shown to successfully reproduce forcing functions for various idealized and observed climate states using quasigeostrophic simulations. Second, a new inverse method that is more computationally efficient is presented. This method closes the mean-field equations by representing the second-order statistical moments, the transient eddy heat and momentum (or potential vorticity) fluxes, as linear functions of the mean field. The coefficients of the linear parameterization are determined by least squares regression. It is shown that the new method also successfully reproduces the anomalous forcing functions responsible for climatic changes in quasigeostrophic simulations.

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J. S. Frederiksen
and
R. C. Bell

Abstract

The instability characteristics of a three-dimensional Northern Hemisphere flow for January 1978 have been examined in quasi-geostrophic spherical five-level and barotropic models with and without explicit topography present. The structures of the 30 fastest growing modes in the baroclinic model have been analyzed and the modes have been separated into four classes based on their phase frequencies and structures. Class I consists of rapidly propagating monopole cyclogenesis modes, most of which are shallow westward-tilting disturbances with largest amplitudes in the centers of the North Pacific and Atlantic observed storm tracks. In addition, new modes, which correspond to storm tracks across Siberia and across the Middle East are found. Some of the larger-scale cyclogenesis modes also have amplitudes that peak near the tropopause.

Class II modes are dipole onset-of-blocking modes with longer periods and larger scales than the cyclogenesis forms. In this class we find disturbances that would initiate the formation of positive and negative anomalies such as blocks and high zonal index flows in the North Soviet Union region as well as in the Atlantic and Pacific.

Class III modes have periods and properties intermediate between the onset-of-blocking and the mature anomaly modes of class IV. They are large-scale, essentially equivalent barotropic modes and are similar to the structures in the middle and late stages of the formation of Dole's Pacific and Atlantic mature anomaly patterns.

Class IV modes have the longest periods and largest scales; they are equivalent barotropic and have amplitude peaks in the upper troposphere or lower stratosphere. Modes which resemble the Atlantic (or Eastern Atlantic), North Soviet Union (or Eurasian) teleconnection patterns, the North Atlantic oscillation, the Western Pacific and the Western Atlantic teleconnection patterns are found. Some modes also appear to have the right properties such that if they were produced with sufficient amplitude, they could produce a stratospheric sudden warming.

In the baroclinic model, the structures of most of the cyclogenesis modes are little changed by the inclusion of explicit topography and its effect, in general terms, is increasingly felt by the larger-scale modes of increasing periods. Modes similar to some of the class IV disturbances of the baroclinic model are also found in the barotropic model; however, there are quantitative differences in their structures and growth rates, and phase frequencies may differ significantly.

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J. S. Frederiksen
and
K. Puri

Abstract

The nonlinear interactions of an initial small amplitude instability mode with a three-dimensional climatological flow (for Northern Hemisphere, January 1978), an which it is growing are studied in a multilevel nonlinear primitive equation model incorporating spherical geometry. Numerical experiments in which there is either constant or no forcing are considered. For the forced runs the forcing is such that the climatological flow is an exact steady state solution which, however, is unstable when perturbed by the normal mode disturbance. For the unforced runs, we examine the divergence of two “forecasts,” one of which has the climatological flow as initial conditions and the other, the climatological flow perturbed by the instability solution. The structural changes that occur in the total and disturbance streamfunctions and kinetic energies and in the disturbance momentum and heat fluxes are examined. The initially small-scale shallow disturbance increases in wale and penetrates into the upper troposphere as nonlinear effects become increasingly important during the integrations. Similar vertical penetrations occur for the disturbance kinetic energies and heat and momentum fluxes giving more realistic distributions than does the initial three-dimensional instability mode.

For both forced and free runs, the formation of mature anomalies is preceded by the upstream development of westward tilting high-low dipoles. The results are compared with observations of the development of mature anomalies and with theoretical predictions based on three dimensional instability theory. It is found that there is qualitative agreement between the observations, three-dimensional instability theory and the numerical integrations.

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J. S. Frederiksen
and
B. L. Sawford

Abstract

The effects of topography in forcing the stationary eddy flow field of the atmosphere have been examined using the spherical equivalent barotropic model. Fully nonlinear solutions obtained using the methods of equilibrium statistical mechanics have been compared and contrasted with linearized steady-state solutions, and both of these have been compared with observed flows.

Incorporation of nonlinear effects eliminates the resonant behavior characteristic of the linear solutions and thus leads to wide differences between the two types of solutions. Whereas the linear solutions are strongly dependent on the strength of the zonal flow, the qualitative appearance of the nonlinear eddy fields is remarkably constant over a wide variation of the relevant parameters and is essentially a filtered version of the topographic field. The nonlinear fields also show no evidence of the wavetrains which are such a striking feature of the linear fields.

Comparison with observed fields shows nonlinear effects to be most important at low altitudes. The nonlinear stationary flow field at 850 mb gives as realistic a representation of the qualitative observed features as can be produced by much more complicated models, whereas the linear field shows far too much structure. At high levels, where the zonal flow is stronger, the linear approximation (including drag) is better than at low levels.

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