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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 instability properties of two instantaneous Southern Hemisphere synoptic flow fields. The growth rates, phase frequencies, perturbation streamfunctions and, for some modes, eddy heat and momentum fluxes of growing disturbances are obtained using a linear, spherical, five-level, quasi-geostrophic model. Both the zonally averaged and three-dimensional flow fields are studied and the eigenmodes are compared with the observed regions of the onset of blocking and of developing midlatitude cyclones. The two principal latitude bands of development are picked out by the instability solutions for the zonally averaged basic flows. For the three-dimensional basic states, the linear solutions reproduce the geographical locations of some of the principal regions of the onset of blocking and cyclogenesis reasonably well.

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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 instability characteristics of Southern Hemisphere zonally averaged flows are studied for January, May and August basic states in a spherical, inviscid, adiabatic, quasi-geostrophic multilevel model. The growing disturbances may have considerably more complex growth rate curves, structures and momentum and heat fluxes than found for idealized single jet and Northern Hemisphere flows. For August, the largest growth rate occurs at the largest zonal wavenumber studied (m = 16), while for January and May it occurs at a more conventional intermediate value (m = 10). The modes for January have many properties in common with modes found with idealized single jet basic states, but in addition to eastward-propagating disturbances long-wave westward propagating disturbances occur. For May and August, the presence of both the subtropical and polar jets is felt by the eastward-propagating disturbances. Some of these grow primarily on one jet or the other, while other modes have two maxima of the disturbance streamfunctions at latitudes near the two jet streams. The presence of the two jets may also result in more complex momentum fluxes than found previously.

For each month, appropriate modes do have maximum streamfunction amplitude and eddy fluxes at the correct latitudes, but the usual vertical structure problem of linear theory occurs, viz., the amplitudes of streamfunctions, momentum and heat fluxes are too large at the surface compared with at the tropopause in relation to observations. However, for May, the instability results appear to contradict the usual hypothesis of baroclinic instability theory that one of the members, lying on the growth rate curve, as a function of zonal wavenumber, with largest growth rates, should be of the most meteorological significance. In fact, the second fastest growing modes of intermediate zonal wavenumbers appear to correspond most closely with observations, emphasizing the importance of finding all the growing modes by using, for example, an eigenvalue approach.

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J. S. Frederiksen

Abstract

A study is made Of the three-dimensional instability properties of a sequence of daily instantaneous Northern Hemisphere flow fields during the period 1 November– 16 November 1979. The growth rates phase frequencies, and structures of the ten fastest growing modes have been analyzed on each day. We have concentrated on instability modes associated with the most dramatic developmental in particular, with the formation of a block in the Gulf of Alaska between 5 and 12 November. Associated with the later stages of this block formation are large-scale equivalent barotropic dipole modes which are stationary or slowly propagating and which have quite large growth rates. In particular, an 6 November the fastest growing mode is a mature blocking mode in the Gulf of Alaska with an e-folding time of only 1.6 days.

The early stages of block formation are characterized by the formation of large westward tilting dipole onset-or-blocking modes with largest amplitudes upstream of the blocking region. Substantial changes in the locations of the cyclogenesis modes take place with the regions of preferential development occurring in the winter climatological storm tracks over the Pacific, Atlantic and Siberian regions during early November. With the formation of a block in the Gulf of Alaska, there is a splitting and northward deflection of some cyclogenesis modes near the block between 5 and 12 November. During the later stages of the period, a block in the North Atlantic-Arctic region causes some Atlantic cyclogenesis modes to he deflected southward.

The roles of barotropic and topographic instability in the formation of mature blocking modes are examined. The implications of the results for different theories of blocking are discussed.

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J. S. Frederiksen

Abstract

The transient and stationary flow fields produced when initial zonal solid body rotation currents flow over topography is studied in spherical barotropic models for eastward and westward flow directions and with variable and constant Coriolis parameters (f). Fully nonlinear analytical solutions are obtained using the methods of equilibrium statistical mechanics and are compared with linearized solutions and with the qualitative results of previous laboratory and numerical experiments.

The equilibrium solutions show that westward flow with variable f is stable in the sense that only very small amplitude transients are produced and the initial flow is practically unchanged. The stationary stream-function dominates the total flow field and there is excellent agreement with linear steady state solutions and with the qualitative results of laboratory and numerical experiments.

For eastward flow with variable f the equilibrium solutions demonstrate that the flow is unstable; that is, large amplitude transients are produced and the zonal flow direction changes to westward. The transients dominate the total eddy flow field while the smaller amplitude stationary eddy streamfunction is essentially a filtered version of the topography and is very similar to that for initial westward flow. In contrast, the linear steady state solutions are resonant or near resonant and differ dramatically from the nonlinear stationary solutions. The meaning of the linear steady state solutions is discussed and comparisons are made of linear transient and nonlinear equilibrium solutions with laboratory and numerical experiments.

With constant f, the equilibrium solutions show that eastward flow again reverses to westward while westward flow remains that way but decreases its strength. The amplitudes of the transients for the two flow directions are now comparable and are considerably less than for the case of eastward flow with variable f. For both flow directions the transient and stationary eddy energy spectra are strongly peaked at zonal wavenumber |m| = 1 with the stationary spectrum being, dominant. The corresponding linear steady state solutions have a resonance at |m| = 1 when the flow is inviscid and large amplitude there when viscosity is included.

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J. S. Frederiksen

Abstract

The instability characteristics of three-dimensional Northern Hemisphere average winter tropospheric flow are examined in a two-layer spherical quasi-geostrophic model. All the growing modes from the fastest down to the first stationary but growing mode have been examined for three cases (1, 2a and 3) having increasingly larger static stability parameters. Comparisons with observations and with the results of the corresponding barotropic model are presented.

It is found that the modes for cases 1, 2a and 3 can be divided into six approximately distinct classes. Class A modes are rapidly propagating monopole cyclogenesis disturbances with largest amplitudes in the geographical locations of the storm tracks in the Pacific and Atlantic Oceans. Class B and C modes are respectively Pacific and Pacific and Atlantic onset-of-blocking dipole modes with longer periods than class A modes and which like class A disturbances, have westward tilts with height. They also have largest amplitudes upstream of the observed regions of mature blocks in the Pacific and Atlantic Oceans. Class D modes are stationary but growing modes with structures similar to the Pacific-North American teleconnection or anomaly pattern while class E modes have second largest periods of just over 40 days and have structures similar to the North Atlantic oscillation. Both class D and E modes are essentially equivalent barotropic and are mainly confined to the key regions of largest amplitude in the lower layer, while the upper layer patterns are more extensive and complex. Class F modes have periods intermediate between class C and E modes and properties intermediate between class B and C and D and E disturbances.

The structural changes that occur in the time evolution of observed anomalies such as blocks are compared with the instability solutions. It is found that the development of mature anomalies may be thought of as approximately consisting of two stages which, for the Pacific-North American pattern, are as follows. The formation of a rapidly growing and eastward propagating dipole disturbance of the class B type, which tilts westward with height and has largest amplitude upstream of the region of mature blocks, initiates the process. The mode then changes into a disturbance similar to class D modes, through the operation of nonlinear effects. Finally, the class D mode amplifies without phase propagation and through largely equivalent barotropic effects to form the mature anomaly. A similar process occurs for the North Atlantic pattern.

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J. S. Frederiksen

Abstract

The nonlinear interaction of initial single zonal wavenumber small-amplitude waves with Southern Hemisphere zonal flows characteristic of January and May is studied in a multilevel primitive equation spectral model with spherical geometry and viscous dissipation. For January, zonal wavenumber 7 and 10 waves are considered which occlude after a period of 8-10 days of almost exponential growth, with growth rates comparable with linear theory. Thereafter the eddy kinetic energy and wave structures vacillate with large-amplitude variations. Similar vacillations of the maximum mean zonal wind and temperature occur with the vertical shear considerably reduced as the flow becomes increasingly barotropic with the lower layers spinning up to a maximum speed comparable with that at the original jet center. As the waves amplify, their structures and poleward heat fluxes penetrate increasingly into the troposphere so that at the first peak in the vacillation cycle the streamfunction with dominant zonal wavenumber 7 is largest in the upper troposphere. The penetration of the shorter wavenumber 10 wave is less effective. At this stage and during the first barotropic decay period following the occlusion, the momentum fluxes also compare closely, in most respects, with observations with largest convergence in the upper troposphere.

For May, viscous and inviscid integrations with wavenumber 10 waves are studied to examine the role of viscosity in determining the structure of waves in the nonlinear regime. During the viscous simulation, an initial wave growing on the polar jet rearranges its structure during the first 11 days and thereafter grows exponentially on the subtropical jet, occludes and shows indications of a vacillation cycle. In contrast, the wave grows exponentially on the polar jet in the inviscid integration. The changes in structure that occur during the viscous integration produce a dramatic improvement in the comparison with observations compared with linear results.

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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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J. S. Frederiksen

Abstract

The instability characteristics of three-dimensional flows typical of the Northern Hemisphere winter stratosphere and mesosphere are examined in a multi-level spherical quasi-geostrophic model. The basic-state distorted polar night jets are obtained from a numerical simulation of the sudden warming in a nonlinear spectral model. The instability is studied at days 10 (Flow I) and 15 (Flow II), prior to the onset of the model sudden warming, in addition, a parameter study is carried out in which the basic state zonal flow, static stability, wave amplitude, Rayleigh friction and Newtonian cooling are changed. It is found that the zonally averaged jets are stable in the presence of Rayleigh friction and Newtonian cooling. However, the three-dimensional distorted flows are unstable and the fastest growing modes have growth rates between 0.0664 and 0.144 day−1 (doubling times between 10.44 and 4.80 days). In the presence of damping, the disturbance streamfunctions for Flows I and II have largest amplitude at 61.5 km in the polar regions and fall off above and below.

A mechanism of competition between the tendency toward zonal mean flow instability with eastward traveling disturbances and wave instability with stationary disturbances is identified. It is suggested that differences between warmings in the Northern and Southern Hemispheres may in part be influenced by this mechanism.

The zonally averaged zonal winds u of the perturbations are centered at the same locations as the disturbance streamfunctions; with the direction of the largest u taken to be easterly, the zonally averaged temperature of the disturbances corresponds to heating of the polar stratosphere and cooling of the polar mesosphere.

The results indicate that wave instability may be an important contributing factor in the sudden warming, which complements Matsuno's mechanism involving vertical wave propagation and wave-zonal flow interaction.

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