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Jeffrey S. Whitaker

Abstract

The balance equations are an approximate set of equations that reduce to gradient wind balance under steady, circular flow conditions on an f plane. Scale analysis indicates that these equations are potentially quite accurate over a wide variety of atmospheric conditions. Motivated by an apparent lack of numerical solutions of this set, we compare simulations of nonlinear baroclinic waves under adiabatic conditions with the primitive and balance equations. As predicted by the scale analysis, the balance equations describe the wave evolution with very high accuracy in situations where the errors made by approximate equation sets based on geostrophy are significant.

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Jeffrey S. Whitaker and Albert Barcilon

Abstract

Stability calculations on basic-state velocity profiles representative of the preferred regions for the development of the upper-level disturbances active in Type B cyclogenesis show that conditions in these regions (weak low-level baroclinicity, large low-level static stability, and large surface roughness) are favorable for the growth of baroclinic waves with maximum amplitude near the tropopause. The structure of these waves compares favorably with observations of developing short-wavelength upper-level troughs in the atmosphere. Basic states characteristic of the storm track regions (strong low-level baroclinicity and small surface roughness) favor the development of baroclinic waves with maximum amplitude at the surface. The dynamics of both the surface-trapped and the upper-tropospheric waves can be interpreted concisely using concepts of potential vorticity. Based on these results, a possible mechanism for Type B cyclogenesis in the storm track regions is proposed that involves the propagation and structural modification of baroclinic wave packets in a zonally varying basic flow.

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Jeffrey S. Whitaker and Chris Snyder

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The effects of spherical geometry on the nonlinear evolution of baroclinic waves are investigated by comparing integrations of a two-layer primitive equation (PE) model in spherical and Cartesian geometry. To isolate geometrical effects, the integrations use basic states with nearly identical potential vorticity (PV) structure.

Although the linear normal modes are very similar, significant differences develop at finite amplitude. Anticyclones (cyclones) in spherical geometry are relatively stronger (weaker) than those in Cartesian geometry. For this basic state, the strong anticyclones on the sphere are associated with anticyclonic wrapping of high PV in the upper layer (i.e., high PV air is advected southward and westward relative to the wave). In Cartesian geometry, large quasi-barotropic cyclonic vortices develop, and no anticyclonic wrapping of PV occurs. Because of their influence on the synoptic-scale flow, spherical geometric effects also lead to significant differences in the structure of mesoscale frontal features.

A standard midlatitude scale analysis indicates that the effects of sphericity enter in the next-order correction to β-plane quasigeostrophic (QG) dynamics. At leading order these spherical terms only affect the PV inversion operator (through the horizontal Laplacian) and the advection of PV by the nondivergent wind. Scaling arguments suggest, and numerical integrations of the barotropic vorticity equation confirm, that the dominant geometric effects are in the PV inversion operator. The dominant metric in the PV inversion operator is associated with the equatorward spreading of meridians on the sphere, and causes the anticyclonic (cyclonic) circulations in the spherical integration to become relatively stronger (weaker) than those in the Cartesian integration.

This study demonstrates that the effects of spherical geometry can be as important as the leading-order ageostrophic effects in determining the structure of evolution of dry baroclinic waves and their embedded mesoscale structures.

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Jeffrey S. Whitaker and Albert Barcilon

Abstract

It is hypothesized that surface cyclogenesis in the Northern Hemisphere storm-track regions can be described by the structural modification of baroclinic wave packets traversing a zonally varying flow field. We test this hypothesis using a linear, quasigeostrophic model with a zonally varying basic state and zonally varying Ekman layer eddy viscosity. At midchannel, the basic state consists of a region of strong low-level baroclinicity and weak Ekman dissipation, surrounded by regions of weak low-level baroclinicity, strong Ekman dissipation, and enhanced low-level static stability. Eigenanalyses and initial-value integrations support this model of Type B cyclogenesis. The results can be summarized as follows:1) A disturbance initiated upstream of the midchannel baroclinic zone rapidly evolves into a wave packet with maximum amplitude near the tropopause. The wave packet undergoes a structural modification upon entering the low-level baroclinic zone, developing maximum amplitude at the surface. The storm track in this model results from the transient amplification and structural modification of wave packets passing through the midchannel baroclinic zone.2) The effective growth rate of the surface disturbance exceeds those of the most unstable mode of the zonally varying basic state, and of the most unstable mode of zonally homogeneous basic-state characteristic of the midchannel baroclinic zone.3) The transient evolution of the wave packet is a result of the superposition and interference between the many global eigenmodes with different structures and frequencies excited by the initial condition. The surface cyclogenesis can be interpreted as a local constructive interference between these eigenmodes.4) From a potential vorticity perspective, the evolution of the baroclinic wave packet is a two-stage process. Initially, the growth of upper-level disturbances results from the mutual interaction of potential vorticity anomalies near the tropopause and in the lower troposphere. After the wave packet enters the storm-track region, the growth of surface cyclones is associated with the interaction between tropospheric potential vorticity anomalies and surface-temperature anomalies.5) The addition of a simple parameterization of moist physics in the midchannel baroclinic zone does not significantly alter the initial stages of surface cyclogenesis, but results in a longer period of rapid development and a reduction in the characteristic scale of the disturbance.

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Jeffrey S. Whitaker and Albert Barcilon

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The authors consider a two-layer quasigeostrophic model with linear surface drag and forcing that relaxes to a zonal baroclinically unstable equilibrium state consisting of a meridionally confined temperature gradient. It is observed that the most energetic wave in the time-mean climate has near zero frequency and is not driven by upscale nonlinear energy transfers. This wave has a zonal scale near the long-wave cutoff of the equilibrium state, and its energy balance is mainly between baroclinic generation and dissipation. This maintenance mechanism is different from that suggested by, β-plane, two-dimensional, and quasigeostrophic turbulence arguments and may be relevant to the dynamics of zonally asymmetric low-frequency variability in the atmosphere, particularly in the Southern Hemisphere.

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Shiling Peng and Jeffrey S. Whitaker

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Previous GCM experiments demonstrated that a model atmosphere produces two different responses to a midlatitude warm SST anomaly over the Pacific under perpetual January and February conditions. To elucidate the mechanisms responsible for the different GCM responses and their dependence on the background flow, experiments with two idealized models are conducted. Experiments with a linear baroclinic model reveal that the GCM responses at equilibrium are primarily maintained by the anomalous eddy forcing. The anomalous flow induced directly by an idealized initial heat source exhibits little sensitivity to the background flow. Eddy feedbacks on the heating-induced anomalous flow are examined using a linear storm track model. The anomalous eddy forcing produced by the storm track model is sensitive to the basic state. The eddy forcing in January acts to shift the heating-induced upper-level ridge toward the northeast of the Gulf of Alaska, while in February it acts to reinforce the ridge. This suggests that the differences in the GCM responses are primarily associated with differences in the response of synoptic eddies to the presence of an anomalous ridge at the end of the Pacific storm track.

The idealized model experiments are also performed with the observed winter mean flow. The eddy feedbacks depend on the position of the heating relative to the storm track. With the heating centered over the western Pacific the eddy-driven anomalous flow reinforces the ridge over the Pacific, similar to that in GCM February, but much stronger. No such reinforcement by the transients is found with the heating shifted over the eastern Pacific. These results suggest that SST anomalies over the western Pacific perhaps play a more active role in midlatitude atmosphere–ocean interactions.

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Lili Lei and Jeffrey S. Whitaker

Abstract

The analysis produced by the ensemble Kalman filter (EnKF) may be dynamically inconsistent and contain unbalanced gravity waves that are absent in the real atmosphere. These imbalances can be exacerbated by covariance localization and inflation. One strategy to combat the imbalance in the analyses is the incremental analysis update (IAU), which uses the dynamic model to distribute the analyses increments over a time window. The IAU has been widely used in atmospheric and oceanic applications. However, the analysis increment that is gradually introduced during a model integration is often computed once and assumed to be constant for an assimilation window, which can be seen as a three-dimensional IAU (3DIAU). Thus, the propagation of the analysis increment in the assimilation window is neglected, yet this propagation may be important, especially for moving weather systems.

To take into account the propagation of the analysis increment during an assimilation window, a four-dimensional IAU (4DIAU) used with the EnKF is presented. It constructs time-varying analysis increments by applying all observations in an assimilation window to state variables at different times during the assimilation window. It then gradually applies these time-varying analysis increments through the assimilation window. Results from a dry two-layer primitive equation model and the NCEP GFS show that EnKF with 4DIAU (EnKF-4DIAU) and 3DIAU (EnKF-3DIAU) reduce imbalances in the analysis compared to EnKF without initialization (EnKF-RAW). EnKF-4DIAU retains the time-varying information in the analysis increments better than EnKF-3DIAU, and produces better analysis and forecast than either EnKF-RAW or EnKF-3DIAU.

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Lili Lei and Jeffrey S. Whitaker

Abstract

Covariance localization is an essential component of ensemble-based data assimilation systems for large geophysical applications with limited ensemble sizes. For integral observations like the satellite radiances, where the concepts of location or vertical distance are not well defined, vertical localization in observation space is not as straightforward as in model space. The detailed differences between model space and observation space localizations are examined using a real radiance observation. Counterintuitive analysis increments can be obtained with model space localization; the magnitude of the increment can increase and the increment can change sign when the localization scale decreases. This occurs when there are negative background-error covariances and a predominately positive forward operator. Too narrow model space localization can neglect the negative background-error covariances and result in the counterintuitive analysis increments. An idealized 1D model with integral observations and known true error covariance is then used to compare errors resulting from model space and observation space localizations. Although previous studies have suggested that observation space localization is inferior to model space localization for satellite radiances, the results from the 1D model reveal that observation space localization can have advantages over model space localization when there are negative background-error covariances. Differences between model space and observation space localizations disappear as ensemble size, observation error variance, and localization scale increase. Thus, large ensemble sizes and vertical localization length scales may be needed to more effectively assimilate radiance observations.

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Jeffrey S. Whitaker and Prashant D. Sardeshmukh

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This paper investigates the extent to which the statistics of extratropical synoptic eddies may be deduced from the assumption that the eddies are stochastically forced disturbances evolving on a baroclinically stable background flow. To this end, a two-level hemispheric quasigeostrophic model is linearized about the observed long-term mean flow and forced with Gaussian white noise. The mean flow is baroclinically stable for a reasonable choice of dissipation parameters. Synoptic-scale eddy disturbances can still grow on such a flow, albeit for a finite time, either in response to the stochastic forcing or through baroclinic and barotropic energy interactions with the background flow. In a statistically steady state, a fluctuation–dissipation relation (FDR) links the covariance of the eddies to the spatial structure of the background flow and the covariance of the forcing. Although not necessary, in this study the forcing is assumed to have always the same trivial statistics (white in both space and time). Under this assumption, the FDR states that the geographical distribution of synoptic eddy covariance is controlled solely by the spatial structure of the background flow, which can therefore be used to predict it. All other second-order eddy statistics, such as eddy kinetic energy, momentum and heat fluxes, and power spectra, can then also be predicted. Despite the apparently drastic underlying assumption of synoptic eddy evolution as a stable linear Markov process, the comparisons of the predicted and observed geographical distributions of eddy kinetic energy and momentum and heat fluxes are found to be encouraging. The FDR is also shown to be sensitive enough to basic-state changes that it is able to predict important aspects of observed storm-track variability associated with seasonal and interannual changes of the background flow. The success of these calculations suggests that it is not necessary to invoke either baroclinic instability or the details of the eddy forcing to understand much of the observed spatial and temporal structure of extratropical synoptic eddy statistics. Rather, the dynamics of nonmodal eddy growth in the Pacific and Atlantic jets, and the downstream propagation and dispersion of the eddy activity in the diffluent regions downstream of the jets, appear sufficient for this purpose.

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Jeffrey S. Whitaker and Randall M. Dole

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A simple two-layer quasigeostrophic model is employed to investigate the sensitivity of storm tracks to changes in an externally imposed, zonally varying large-scale flow. Zonally asymmetric temperature and horizontal deformation fields are varied systematically in order to compare the effects of baroclinicity and horizontal deformation on storm track dynamics. The sensitivity of the storm tracks to uniform barotropic zonal flows is also examined.

The results show two competing processes for storm track organization, one associated with a local maximum in baroclinicity and the other with a local minimum in horizontal deformation. When the equilibrium state consists of a zonally symmetric temperature field and a barotropic stationary wave, the maximum in synoptic-scale transient eddy energy (storm track) is located in the entrance region of the upper jet just downstream of the point of minimum horizontal deformation. As zonal variations in baroclinicity become large (keeping the upper-layer horizontal deformation constant), the storm track shifts to the jet exit region just downstream of the point of maximum baroclinicity. For flows intermediate between the above cases,that is, having weaker zonal variations in baroclinicity and the same upper-layer deformation, two storm track maxima appear, one located in the jet entrance and the other in the jet exit region.

The results also indicate that the storm tracks are sensitive to changes in a uniform barotropic zonal flow. The presence of a uniform westerly flow extends the storm track and strengthens eddy activity, while the addition of a uniform easterly flow shortens the storm track and dramatically weakens eddy activity. The changes in the magnitudes of eddy activity appear related to differences in the efficiency of nonlinear barotropic decay processes in weakening the eddies in the jet exit region.

Sensitivities of the location of the storm tracks to changes in large-scale flow parameters are well captured by linear calculations, although sensitivities of the strength of the storm tracks are not. For sufficiently strong zonal variations in baroclinicity, two coherent modes of low-frequency variability develop. They are characterized synoptically by 1) a meridional shift, and 2) an extension/contraction as well as a modulation in the strength of the upper-layer jet and storm track.

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