Search Results

You are looking at 1 - 10 of 22 items for

  • Author or Editor: Jorgen S. Frederiksen x
  • All content x
Clear All Modify Search
Jorgen S. Frederiksen

Abstract

The finite-time normal mode instability of four-dimensional space–time basic states has been studied for cases of block development over the Gulf of Alaska, over the North Atlantic, and over southern Greenland using a two-level tangent linear model. The authors find three generic types of finite-time normal modes, denoted as “recurring,” “traveling,” and “flip” modes. The dominant finite-time normal modes associated with block development have large-scale structures in the respective blocking regions; they tend to closely reflect the structures of the developing blocks.

The time evolution in the tangent linear model of finite-time adjoint modes has been examined for each of the three cases of block development. These adjoint modes have faster than normal mode exponential growth. The initial structures of the dominant adjoint modes are characterized by small-scale baroclinic wave trains located primarily upstream of the blocking region. As the disturbances grow explosively, they increase their scale and propagate eastward into the blocking region where, within a few days, they take up large-scale structures similar to the respective fastest growing finite-time normal modes.

The structures and time evolution of maximum sensitivity perturbations during blocking have been analyzed. Dominant normal mode structures focused in the blocking regions have been chosen as weight functions in response functions measuring forecast sensitivity. The maximum sensitivity perturbations are found to be very similar to respective finite-time adjoint modes in their structures and time developments.

It is suggested that, during periods of rapid regime transition, the error structure at the end of a 2–4-day period of weather prediction is likely to resemble the dominant finite-time normal modes for the period in question.

Full access
Jorgen S. Frederiksen

Abstract

The evolution of finite-time singular vectors growing on four-dimensional space–time basic states is studied for cases of block development over the Gulf of Alaska and over the North Atlantic, using a two-level tangent linear model. The initial singular vectors depend quite sensitively on the choice of norm with the streamfunction norm characterized by small-scale baroclinic disturbances, the kinetic energy norm giving intermediate-scale baroclinic disturbances, and the enstophy norm typified by large-scale disturbances with large zonal flow contributions. In all cases, the final evolved singular vectors consist of large-scale equivalent barotropic wave trains across the respective blocking regions. There are close similarities between the evolved singular vectors in each of the norms, particularly for the longer time periods considered, and with corresponding evolved finite-time adjoint modes and evolved maximum sensitivity perturbations. For the longer time periods considered, each of these evolved perturbations also closely resembles some of the dominant finite-time normal mode disturbances, which are norm independent. For periods between about two weeks and a month, the convergence of the evolved leading singular vector and leading finite-time normal mode toward the leading left Lyapunov vector has been examined.

The evolution of errors, represented by singular vectors, is also considered in the space of finite-time normal modes. In all cases the evolved error dynamics contracts onto a low-dimensional subspace characterized by the dominant finite-time normal modes. The growth of norms based on streamfunction, kinetic energy, or enstrophy is compared with the growth of a norm based on the projection coefficients of the disturbance onto the dominant finite-time normal modes.

The prospect of ensemble prediction schemes in which the control initial conditions are perturbed by superpositions of the dominant finite-time normal modes is discussed.

Full access
Jorgen S. Frederiksen

Abstract

The development of perturbations or errors is considered during periods of observed Northern Hemisphere blocking. The evolution of initial onset-of-blocking mode perturbations is studied using a two-level tangent linear model with time-dependent basic states taken from observations in the first half of November 1979. During this period a blocking high formed in the Gulf of Alaska between 5 and 12 November followed by decay and the subsequent development of a block in the North Atlantic. Initial baroclinic wave train perturbations, with large amplitudes stretching from East Asia into the central Pacific, propagate eastward as they grow and become equivalent barotropic. Large-amplitude large-scale perturbations form over the Gulf of Alaska as the observed block amplifies. Subsequently, downstream development occurs with rapid amplification of the perturbation near the east coast of North America followed by the formation of large-scale tripole anomalies in the blocking region over the North Atlantic. As the Gulf of Alaska and North Atlantic blocks amplify, there are close similarities between the simulated anomalies and dominant normal mode instabilities of the instantaneous observed flows.

The inverse problem of what are the precursors to a given observed or specified forecast error is examined. Anomalies are specified as equivalent barotropic normal modes or similar structures with large amplitudes in the Gulf of Alaska. The precursors are smaller-scale wave train disturbances emanating from the East Asia–west Pacific sector. The sensitivity of the precursors to the structure of the prescribed error in the blocking region is examined and condition numbers for the propagator are calculated.

Full access
Jorgen S. Frederiksen

Abstract

General expressions for the eddy-topographic force, eddy viscosity, and stochastic backscatter, as well as a residual Jacobian term, are derived for barotopic flow over mean (single realization) topography. These subgrid-scale parameterizations are established on the basis of a quasi-diagonal direct interaction closure model, incorporating equations for the mean vorticity, vorticity covariance, and response functions. In general, the subgrid-scale parameterizations have a time–history integral representation, which reflects memory effects associated with turbulent eddies. In the Markov limit, the truncated equations for the ensemble mean and fluctuating parts of the vorticity have the same form as the full resolution equations but with the original “bare” viscosity and bare mean and fluctuating forcings renormalized by eddy drain viscosities, eddy-topographic force, and stochastic backscatter terms.

The parameterizations are evaluated at canonical equilibrium states for comparison with G. Holloway’s heuristic expression for the eddy-topographic force, involving a product of the total viscosity and a canonical equilibrium expression for a mean vorticity. His functional form is recovered but with his total viscosity replaced by an eddy drain viscosity. For dynamical consistency, Holloway’s parameterization also needs to be supplemented with a stochastic backscatter parameterization, even at canonical equilibrium. Implications of the results for subgrid-scale parameterizations of turbulent eddies in ocean and atmospheric circulation models are discussed.

Full access
Jorgen S. Frederiksen

Abstract

The particular role of evaporation–wind feedback, and as well cumulus convection and dissipation, in the formation of the Madden–Julian 30–60-day intraseasonal oscillation (MJO) and equatorially trapped waves, including Kelvin, equatorial Rossby, mixed Rossby–gravity, and eastward inertio-gravity waves, has been studied using a global two-level primitive equation instability model. The evaporation has been specified through a bulk aerodynamic formula, and the convection through a generalized Kuo-type parameterization with a moist static stability that is positive everywhere so that wave-conditional instability of the second kind (CISK) is not possible. Both three-dimensional and zonally averaged basic states for January 1979 have been employed, and the e-folding times, periods, structures, and propagation characteristics of these waves have been analyzed and compared with their observed properties.

The MJO modes appear to be particularly realistic when the three-dimensional basic state includes evaporation–wind feedback, which promotes the zonal wavenumber-1 component of the eastward-propagating velocity potential, and cumulus convection. The tropical velocity potential propagates eastward relatively quickly around the globe (≈13.6 m s−1, corresponding to a period of 34.4 days), while in the convective region between the Indian Ocean and the date line the divergence propagates more slowly (≈4.5 m s−1). The MJO has a largely first internal mode structure in the Tropics with a more equivalent barotropic structure of the streamfunction in the extratropics, including distinct Pacific–North American and Eurasian patterns. The MJO modes do not lie along any of the theoretical dispersion curves of equatorially trapped waves and have distinctly longer periods than the model convectively coupled Kelvin waves, even at zonal wavenumber 1, as in the observational study of M. Wheeler and G. N. Kiladis. The Kelvin mode streamfunction fields also do not exhibit the largely equivalent barotropic teleconnection patterns in the extratropics typical of the MJO. With cumulus convection and evaporation–wind feedback, the model equatorially trapped Kelvin, equatorial Rossby, mixed Rossby–gravity, and eastward inertio-gravity waves have periods, structures, and propagation characteristics comparable to those of corresponding convectively coupled observed waves.

Full access
Carsten S. Frederiksen and Jorgen S. Frederiksen

Abstract

The linear instability properties of exact steady-state solutions is examined for barotropic flow over topography on a sphere, in both severely truncated and high-resolution formulations. In particular, we consider the instability of flows consisting of a solid-body rotation plus a single nonzonal spherical harmonic. We compare our results with previous beta-plane studies and find that, while the results are generally qualitatively similar, there are some differences. In particular, there are some disturbances in the beta-plane study, which show no form drag mechanism, although the analogous disturbances in the spherical case show that form drag instability plays some role, even if only a very subordinate one.

Results are presented that show that topography can promote the growth of perturbations either through a form drag mechanism, which modifies the globally averaged angular momentum of the basic state, or through its action as a catalyst that can initiate wave-wave interactions between the basic state wave and the disturbance, while leaving the globally average angular momentum unchanged. Both types of topographic instability are illustrated using simple triad calculations. It is shown that analogous beta-plane results hold for sufficiently large channel width. Further, we show that the effects of the inclusion of topography on the instability of basic states, which are normally barotropically unstable, are to (i) promote stationary flow patterns, (ii) stabilize the flow in the super-resonant region, (iii) destabilize subresonant flows, and (iv) excite smaller-scale disturbances as the solid-body rotation term of the basic state approaches zero from above.

Full access
Carsten S. Frederiksen and Jorgen S. Frederiksen

Abstract

A study is made of the importance of horizontally varying static stability and nongeostrophic effects upon the location of Northern Hemisphere storm track instability modes and mature anomaly teleconnection pattern modes during January 1979. The analysis has been conducted with a two-level primitive equation model, and the results compared with corresponding results from one-, two-, and five-level quasigeostrophic models, although the main emphasis is a comparison between the two-level results.January 1979 was a period of frequent and severe storm activity in the Northern Hemisphere and a time of transition from high-latitude blocking over northwestern Canada and the Beaufort Sea area to persistent and large-scale blocking in the North Atlantic region. The three-dimensional instability modes from all models are discussed in the context of these synoptic developments. In particular, it is found that the inclusion of a horizontally varying static stability and nongeostrophic effects influence the structure of the cyclogenesis modes and lead to significant changes in the geographical locations of some of the preferred regions of cyclogenesis in situations of large-scale anomalous flow, such as occurred during January 1979. It is shown that this result can be understood from a consideration of Phillips' criterion for instability generalized to primitive equation and quasigeostrophic models in spherical geometry. In contrast, for more normal climatological flows there is a close correspondence between primitive equation and quasigeostrophic dominant storm track modes, instability criteria, and observations.We have found few differences in the structures of the larger-scale onset-of-blocking, intermediate, and mature anomaly modes that result from the primitive equation and quasigeostrophic instability calculations.

Full access
Carsten S. Frederiksen and Jorgen S. Frederiksen

Abstract

The author examine the effects of an enhanced sea surface temperature gradient, between the central Indian Ocean and the Indonesian archipelago, on the structure of the monthly averaged three-dimensional July global circulation and particularly on its consequent instability properties. This study has been conducted using a two-level primitive-equation model including a wave-CISK cumulus heating parameterization and using basic states taken from two AGCM simulations of the July circulation, with and without an enhanced SST gradient. The growing disturbances have been analyzed for various strengths of the cumulus heating.

The study has focused on circulation changes associated with the development of Australian northwest cloudband disturbances and changes in Southern Hemisphere storm tracks. With an enhanced SST gradient, the authors have found a new group of modes that can be associated with circulation changes surrounding the onset of the simulated northwest cloudband events. Such modes are shown to compare favorably with simulated perturbations in the circulation and the results of an EOF analysis. With an enhanced gradient, there is also a discernible equatorward shift of the storm track instability modes over the Australian region, and the modes have much larger amplitude on the subtropical jet. The upper-level divergence is also more concentrated in the Australian region.

Full access
Carsten S. Frederiksen and Jorgen S. Frederiksen

Abstract

We examine the linear instability properties of exact steady solutions for barotropic flow over topography in a beta-plane channel using models with different expansion representations and truncations. We compare our results with those of previous studies and find that in many cases the models used previously contained deficiencies. These range from (i) the stationary basic states in the truncated system not being exact solutions of the original barotropic vorticity equation to (ii) the form drag zonal momentum equation not being satisfied by the truncated scheme to (iii) the boundary conditions not being satisfied by the truncated perturbation to (iv) form drag instability only occurring for basic states and topographies with odd meridional wavenumber. In our study, we use a truncation scheme and expansion of the streamfunction, in terms of a nonorthogonal large-scale flow &ndashUy and meridional sine expansions for both the zonal and wave terms, which ensure that none of the problems occur at each level of truncation. These results are compared with those in a model in which the perturbation has a purely orthogonal function representation including a zonal flow represented by just a cosine expansion. At severe truncation there are significant difference between the two. At high resolution, we find that in some parts of parameter space there are significant qualitative, as well as quantitative, differences between the models while in other parts of parameter space qualitatively similar results may be obtained.

The nonlinear stability properties of stationary flow in each model are also examined and differences even at infinite resolution are attributed to Gibbs phenomena with the purely orthogonal function representation.

Full access
Jorgen S. Frederiksen and Carsten S. Frederiksen

Abstract

The results of a study are presented that indicate that a wide variety of atmospheric disturbances, including those associated with storm tracks and blocking in both hemispheres, quasi-stationary global teleconnection patterns, and localized monsoon disturbances, as well as intraseasonal oscillations, may be generated through the instability of the three-dimensional global basic state for January 1979 including a wave-CISK cumulus heating parameterization. The analysis has been conducted with a two-level primitive equation eigenvalue model, and the growing disturbances for various specifications of the strengths of the cumulus heating have been analyzed.

Within the parameter range studied, inclusion of explicit moisture in the basic state has little effect on the structures of the storm track and onset-of-blocking modes in both hemispheres, but it increases growth rates as expected. It is also responsible for generating a significant amplitude of the tropical shear streamfunction of low-frequency and quasi-stationary teleconnection pattern modes, particularly in the Australian/South Pacific region where a coupling to the Australian monsoon appears to occur. Remarkable localized quasi-stationary monsoon disturbances of fairly small scale are found, which tend to produce either break or active periods of the Australian monsoon depending on their phase. The CISK heating focuses these modes in the Australian region and increases their growth rates.

A group of intraseasonal oscillation modes with quite complex structures and periods between about 20 and 60 days is also found. They have eastward-propagating velocity potentials, peaking in the equatorial regions with a zonal wavenumber 1, and possibly 2, envelope wave within which are embedded smaller-scale structures. They have equivalent barotropic streamfunctions in extratropical regions with a baroclinic structure, typical of the first internal mode, in the tropical regions. At certain phases they can produce break or active periods of the Australian monsoon. Both a three-dimensional large-scale basic-state flow and a cumulus heating parameterization appear to be necessary for generating, through instability, intraseasonal oscillation modes with realistic structures.

Full access