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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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Thomas M. Hamill
and
Jeffrey S. Whitaker

Abstract

The spread of an ensemble of weather predictions initialized from an ensemble Kalman filter may grow slowly relative to other methods for initializing ensemble predictions, degrading its skill. Several possible causes of the slow spread growth were evaluated in perfect- and imperfect-model experiments with a two-layer primitive equation spectral model of the atmosphere. The causes examined were the covariance localization, the additive noise used to stabilize the assimilation method and parameterize the system error, and the model error itself. In these experiments, the flow-independent additive noise was the biggest factor in constraining spread growth. Preevolving additive noise perturbations were tested as a way to make the additive noise more flow dependent. This modestly improved the data assimilation and ensemble predictions, both in the two-layer model results and in a brief test of the assimilation of real observations into a global multilevel spectral primitive equation model. More generally, these results suggest that methods for treating model error in ensemble Kalman filters that greatly reduce the flow dependency of the background-error covariances may increase the filter analysis error and decrease the rate of forecast spread growth.

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Jeffrey S. Whitaker
and
Andrew F. Loughe

Abstract

Statistical considerations suggest that 1) even for a perfect ensemble (one in which all sources of forecast error are sampled correctly) there need not be a high correlation between spread and skill, 2) the correlation between spread and skill should be larger where the day-to-day variability of spread is large, and 3) the spread is likely to be most useful as a predictor of skill when it is “extreme,” that is, when it is either very large or very small compared to its climatological mean value. The authors investigate the relationship between spread and skill in an operational setting by analyzing ensemble predictions produced by the National Centers for Environmental Prediction. The geographical dependence of the spread–skill relationship is found to be related to the geographical dependence of day-to-day variability of spread. Dynamical mechanisms for spread variability are investigated using a linear quasigeostrophic model. Problems associated with the sample size needed to define what constitutes an extreme value of spread at a given location are discussed.

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Thomas M. Hamill
and
Jeffrey S. Whitaker

Abstract

Insufficient model resolution is one source of model error in numerical weather predictions. Methods for parameterizing this error in ensemble data assimilations are explored here. Experiments were conducted with a two-layer primitive equation model, where the assumed true state was a T127 forecast simulation. Ensemble data assimilations were performed with the same model at T31 resolution, assimilating imperfect observations drawn from the T127 forecast. By design, the magnitude of errors due to model truncation was much larger than the error growth due to initial condition uncertainty, making this a stringent test of the ability of an ensemble-based data assimilation to deal with model error. Two general methods, “covariance inflation” and “additive error,” were considered for parameterizing the model error at the resolved scales (T31 and larger) due to interaction with the unresolved scales (T32 to T127). Covariance inflation expanded the background forecast members’ deviations about the ensemble mean, while additive error added specially structured noise to each ensemble member forecast before the update step.

The method of parameterizing this model error had a substantial effect on the accuracy of the ensemble data assimilation. Covariance inflation produced ensembles with analysis errors that were no lower than the analysis errors from three-dimensional variational (3D-Var) assimilation, and for the method to avoid filter divergence, the assimilations had to be periodically reseeded. Covariance inflation uniformly expanded the model spread; however, the actual growth of model errors depended on the dynamics, growing proportionally more in the midlatitudes. The inappropriately uniform inflation progressively degradated the capacity of the ensemble to span the actual forecast error.

The most accurate model-error parameterization was an additive model-error parameterization, which reduced the error difference between 3D-Var and a near-perfect assimilation system by ∼40%. In the lowest-error simulations, additive errors were parameterized using samples of model error from a time series of differences between T63 and T31 forecasts. Scaled samples of differences between model forecast states separated by 24 h were also tested as additive error parameterizations, as well as scaled samples of the T31 model state’s anomaly from the T31 model climatology. The latter two methods produced analyses that were progressively less accurate. The decrease in accuracy was likely due to their inappropriately long spatial correlation length scales.

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Jeffrey S. Whitaker
and
Thomas M. Hamill

Abstract

The ensemble Kalman filter (EnKF) is a data assimilation scheme based on the traditional Kalman filter update equation. An ensemble of forecasts are used to estimate the background-error covariances needed to compute the Kalman gain. It is known that if the same observations and the same gain are used to update each member of the ensemble, the ensemble will systematically underestimate analysis-error covariances. This will cause a degradation of subsequent analyses and may lead to filter divergence. For large ensembles, it is known that this problem can be alleviated by treating the observations as random variables, adding random perturbations to them with the correct statistics.

Two important consequences of sampling error in the estimate of analysis-error covariances in the EnKF are discussed here. The first results from the analysis-error covariance being a nonlinear function of the background-error covariance in the Kalman filter. Due to this nonlinearity, analysis-error covariance estimates may be negatively biased, even if the ensemble background-error covariance estimates are unbiased. This problem must be dealt with in any Kalman filter–based ensemble data assimilation scheme.

A second consequence of sampling error is particular to schemes like the EnKF that use perturbed observations. While this procedure gives asymptotically correct analysis-error covariance estimates for large ensembles, the addition of perturbed observations adds an additional source of sampling error related to the estimation of the observation-error covariances. In addition to reducing the accuracy of the analysis-error covariance estimate, this extra source of sampling error increases the probability that the analysis-error covariance will be underestimated. Because of this, ensemble data assimilation methods that use perturbed observations are expected to be less accurate than those which do not.

Several ensemble filter formulations have recently been proposed that do not require perturbed observations. This study examines a particularly simple implementation called the ensemble square root filter, or EnSRF. The EnSRF uses the traditional Kalman gain for updating the ensemble mean but uses a “reduced” Kalman gain to update deviations from the ensemble mean. There is no additional computational cost incurred by the EnSRF relative to the EnKF when the observations have independent errors and are processed one at a time. Using a hierarchy of perfect model assimilation experiments, it is demonstrated that the elimination of the sampling error associated with the perturbed observations makes the EnSRF more accurate than the EnKF for the same ensemble size.

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Thomas M. Hamill
and
Jeffrey S. Whitaker

Abstract

A general theory is proposed for the statistical correction of weather forecasts based on observed analogs. An estimate is sought for the probability density function (pdf) of the observed state, given today’s numerical forecast. Assume that an infinite set of reforecasts (hindcasts) and associated observations are available and that the climate is stable. Assume that it is possible to find a set of past model forecast states that are nearly identical to the current forecast state. With the dates of these past forecasts, the asymptotically correct probabilistic forecast can be formed from the distribution of observed states on those dates.

Unfortunately, this general theory of analogs is not useful for estimating the global pdf with a limited set of reforecasts, for the chance of finding even one effectively identical forecast analog in that limited set is vanishingly small, and the climate is not stable. Nonetheless, approximations can be made to this theory to make it useful for statistically correcting weather forecasts. For instance, when estimating the state in a local region, choose the dates of analogs based on a pattern match of the local weather forecast; with a few decades of reforecasts, there are usually many close analogs.

Several approximate analog techniques are then tested for their ability to skillfully calibrate probabilistic forecasts of 24-h precipitation amount. A 25-yr set of reforecasts from a reduced-resolution global forecast model is used. The analog techniques find past ensemble-mean forecasts in a local region that are similar to today’s ensemble-mean forecasts in that region. Probabilistic forecasts are formed from the analyzed weather on the dates of the past analogs. All of the analog techniques provide dramatic improvements in the Brier skill score relative to basing probabilities on the raw ensemble counts or the counts corrected for bias. However, the analog techniques did not produce guidance that was much more skillful than that produced by a logistic regression technique. Among the analog techniques tested, it was determined that small improvements to the baseline analog technique that matches ensemble-mean precipitation forecasts are possible. Forecast skill can be improved slightly by matching the ranks of the mean forecasts rather than the raw mean forecasts by using highly localized search regions for shorter-term forecasts and larger search regions for longer forecasts, by matching precipitable water in addition to precipitation amount, and by spatially smoothing the probabilities.

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Jeffrey S. Whitaker
and
Chris Snyder

Abstract

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

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
Randall M. Dole

Abstract

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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