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Pierre Gauthier

Abstract

This paper is concerned with the weakly nonlinear inviscid dynamics of a marginally unstable baroclinic wave near the point of minimum critical shear of a two-layer quasi-geostrophic model on the β-plane. In previous studies by Pedlosky and by Warn and Gauthier (WG), the parameters of the model were chosen in a specific way in order to be exactly at the minimum. They showed that at this particular point, a complete inviscid coarse-grain homogenization of the potential vorticity occurs in the bottom layer causing the amplitude of the unstable wave to equilibrate. It is the purpose of the present paper to investigate the behavior of the dynamics when the problem is not exactly at the minimum and more specifically, to establish how one goes from the analytical solution of WG to the single wave theory that one expects to be valid away from minimum critical shear. The nonlinear evolution equations of WG are extended in order to include a “detuning parameter” σ associated with a perturbation of the aspect ratio of the periodic channel. An analytical solution not being available when σ ≠ 0, a spectral form of these equations similar to those found in Pedlosky is integrated numerically at high resolution. The results show that for a fixed supercritical shear and arbitrary but sufficiently small initial conditions, the size of the vortices is decreasing with σ causing the potential vorticity to mix only in part of the domain and the amplitude of the unstable wave to oscillate around a nonzero mean. When σ is sufficiently large, closed streamlines are no longer possible and no vortices are developing. At that point, the single wave theory is becoming a better approximation to the dynamics.

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Kamel Chikhar and Pierre Gauthier

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Regional and global climate models are usually validated by comparison to derived observations or reanalyses. Using a model in data assimilation results in a direct comparison to observations to produce its own analyses that may reveal systematic errors. In this study, regional analyses over North America are produced based on the fifth-generation Canadian Regional Climate Model (CRCM5) combined with the variational data assimilation system of the Meteorological Service of Canada (MSC). CRCM5 is driven at its boundaries by global analyses from ERA-Interim or produced with the global configuration of the CRCM5. Assimilation cycles for the months of January and July 2011 revealed systematic errors in winter through large values in the mean analysis increments. This bias is attributed to the coupling of the lateral boundary conditions of the regional model with the driving data particularly over the northern boundary where a rapidly changing large-scale circulation created significant cross-boundary flows. Increasing the time frequency of the lateral driving and applying a large-scale spectral nudging significantly improved the circulation through the lateral boundaries, which translated in a much better agreement with observations.

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Cristina Lupu and Pierre Gauthier

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One of the objectives of data assimilation is to produce initial conditions that will improve the quality of forecasts. Studies on singular vectors and sensitivity studies have shown that small changes to the initial conditions can sometimes lead to exponential error growth. This has motivated research to include flow-dependent structures within the assimilation that would have the characteristics to correctly predict the growth or decay of meteorological systems. This relates to the characterization of precursors to atmospheric instability. In this paper, the observability of such structures by observations is discussed. Several studies have shown that deploying observations over regions where changes in the initial conditions may impact the forecast the most do not lead to the expected benefit. In this paper, it is shown that given the small magnitude of the signal to be detected, it is important to take into account the accuracy of the observations. If the signal-to-noise ratio is too low, observations cannot detect and characterize precursors to forecast error growth. From that perspective, the assimilation only has the possibility to extract information about evolved structures of error growth. Experiments with a simple one-dimensional variational data assimilation (1D-Var) system are presented and, then, an adapted three-dimensional variational data assimilation (3D-Var) system with different sensitivity structure functions is used. The results have been obtained by adapting the variational assimilation system of Environment Canada.

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Pierre Gauthier and Jean-Noël Thépaut

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In this paper, a weak constraint formulation of the digital filter based on the Dolph–Chebyshev window is introduced in a preoperational version of the 4DVAR analysis of Météo-France. The constraint is imposed only on the analysis increments to damp spurious fast oscillations associated with gravity–inertia waves. In the incremental formulation of 4DVAR, the analysis increments are obtained from a global model at a uniform low resolution with a simplified set of physical parameterizations, while the high-resolution forecast is obtained with a model that uses a variable-resolution grid having a higher resolution over France and the complete set of physical parameterizations. Both models have the same vertical resolution. In a set of preliminary experiments using the same background field and the same set of observations, it is shown that the weak constraint imposed only on the low-resolution increments manages to control efficiently the emergence of fast oscillations in the resulting high-resolution forecast while maintaining a closer fit to the observations than is possible if the digital filter initialization is applied externally on the final analysis increments. It is also shown that this weak constraint does not add any significant computer cost to the 4DVAR analysis. Finally, 4DVAR has been cycled over a period of 2 weeks and the results show that, compared to 3DVAR, the initial dynamical imbalances are significantly less in 4DVAR even if no constraint is imposed at all. However, it has been noted that the innovation statistics show a positive impact when a constraint is applied.

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Pierre Gauthier, Philippe Courtier, and Patrick Moll

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The object of this paper is to present some results obtained with an extended Kalman fitter (EKF). First, a discussion is given of the way that the EKF has been implemented and tested for a global nondivergent barotropic model spectrally truncated at T21. In the present paper, the assimilation experiments focused solely on the time evolution of the forecast error covariances that are influenced by two factors: 1) their time integration performed here with the tangent linear model obtained from a linearization around the true trajectory and 2) the accuracy and distribution of the observations. Data from a simulated radiosonde network have been assimilated over a 24-h period. The results show that even though no model error has been considered, there can be a substantial forecast error growth, especially in regions where the flow is unstable and no data are available. The error growth is attributed to instability processes that are embedded within the complex flow configuration around which the nonlinear model is linearized to obtain the tangent linear model. The impact of different initial conditions for the forecast error covariance is also looked at. In an experiment where the time integration of the forecast error covariance is suppressed, the results show that error growth is suppressed, causing the analysis error variance to differ substantially from the variance field obtained with the EKF. Especially in regions where instability is present and no data are available, this “improved” optimal interpolation considers the forecast to be more accurate than it actually is.

In a second set of experiments, a mini-observing system simulation experiment has been conducted for which wind data from a proposed satellite-based lidar instrument have been simulated and added to the radiosonde data of the previous experiments. Two configurations of the instrument have been considered where the satellite is set on a polar orbit, at an altitude of 400 km in the first scenario and 800 km in the second. Compared to the results obtained with the radiosonde data alone, the global data coverage leads to an improvement in the analysis, especially in the Southern Hemisphere. Data being available in the regions of instability, the assimilation is now capable of putting a stop to the unlimited error growth observed in the previous experiments. Due to a degradation of the measurement when the instrument is at an altitude of 800 km, the analysis is more accurate for the 400-km case, but the low-altitude orbit (400 km) leaves holes in the tropical belt that the data assimilation scheme is not quite able to compensate for.

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Joël Bédard, Stéphane Laroche, and Pierre Gauthier

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This study examines the assimilation of near-surface wind observations over land to improve wind nowcasting and short-term tropospheric forecasts. A new geostatistical operator based on geophysical model output statistics (GMOS) is compared with a bilinear interpolation scheme (Bilin). The multivariate impact on forecasts and the temporal evolution of the analysis increments produced are examined as well as the influence of background error covariances on different components of the prediction system. Results show that Bilin significantly degrades surface and upper-air fields when assimilating only wind data from 4942 SYNOP stations. GMOS on the other hand produces smaller increments that are in better agreement with the model state. It leads to better short-term near-surface wind forecasts and does not deteriorate the upper-air forecasts. The information persists longer in the system with GMOS, although the local improvements do not propagate beyond 6-h lead time. Initial model tendencies indicate that the mass field is not significantly altered when using static error covariances and the boundary layer parameterizations damp the poorly balanced increment locally. Conversely, most of the analysis increment is propagated when using flow-dependent error statistics. It results in better balanced wind and mass fields and provides a more persistent impact on the forecasts. Forecast accuracy results from observing system experiments (assimilating SYNOP winds with all observations used operationally) are generally neutral. Nevertheless, forecasts and analyses from GMOS are more self-consistent than those from both Bilin and a control experiment (not assimilating near-surface winds over land) and the information from the observations persists up to 12-h lead time.

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Monique Tanguay, Saroja Polavarapu, and Pierre Gauthier

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The tangent linear model (TLM) is obtained by linearizing the governing equations around a space- and time-dependent basic state referred to as the trajectory. The TLM describes to first-order the evolution of perturbations in a nonlinear model and it is now used widely in many applications including four-dimensional data assimilation. This paper is concerned with the difficulties that arise when developing the tangent linear model for a semi-Lagrangian integration scheme. By permitting larger time steps than those of Eulerian advection schemes, the semi-Lagrangian treatment of advection improves the model efficiency. However, a potential difficulty in linearizing the interpolation algorithms commonly used in semi-Lagrangian advection schemes has been described by , who showed that for infinitesimal perturbations, the tangent linear approximation of an interpolation scheme is correct if and only if the first derivative of the interpolator is continuous at every grid point. Here, this study is extended by considering the impact of temporally accumulating first-order linearization errors on the limit of validity of the tangent linear approximation due to the use of small but finite perturbations. The results of this paper are based on the examination of the passive advection problem. In particular, the impact of using incorrect interpolation schemes is studied as a function of scale and Courant number.

For a constant zonal wind leading to an integral value of the Courant number, the first-order linearization errors are seen to amplify linearly in time and to resemble the second-order derivative of the advected field for linear interpolation and the fourth-order derivative for cubic Lagrange interpolation. Solid-body rotation experiments on the sphere show that in situations where linear interpolation results in accurate integrations, the limit of validity of the TLM is nevertheless reduced. First-order cubic Lagrange linearization errors are smaller and affect small scales. For this to happen requires a wind configuration leading to a persistent integral value of the Courant number. Regions where sharp gradients of the advected tracer field are present are the most sensitive to this error, which is nevertheless observed to be small. Finally, passive tracers experiments driven by winds obtained from a shallow-water model integration confirm that higher-order interpolation schemes (whether correct or not) give similar negligible linearization errors since the probability of having the upstream point being located exactly on a grid point is vanishingly small.

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Cristina Lupu, Pierre Gauthier, and Stéphane Laroche

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Observing system experiments (OSEs) are commonly used to quantify the impact of different observation types on forecasts produced by a specific numerical weather prediction system. Recently, methods based on degree of freedom for signal (DFS) have been implemented to diagnose the impact of observations on the analyses. In this paper, the DFS is used as a diagnostic to estimate the amount of information brought by subsets of observations in the context of OSEs. This study is interested in the evaluation of the North American observing networks applied to OSEs performed at the Meteorological Service of Canada for the period of January and February 2007. The relative values of the main observing networks over North America derived from DFS calculations are compared with those from OSEs in which aircraft or radiosonde data have been removed. The results show that removing some observation types from the assimilation system influences the effective weight of the remaining assimilated observations, which may have an increased impact to compensate for the removal of other observations. The response of the remaining observations when a given set of observations is denied is illustrated comparing DFS calculations with the observations’ impact estimated from OSEs.

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Cristina Lupu, Pierre Gauthier, and Stéphane Laroche

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The degrees of freedom for signal (DFS) is used in data assimilation applications to measure the self-sensitivity of analysis to different observation types. This paper describes a practical method to estimate the DFS of observations from a posteriori statistics. The method does not require the consistency of the error statistics in the analysis system and it is shown that the observational impact on analyses can be estimated from observation departures with respect to analysis or the forecast. This method is first introduced to investigate the impact of a complete set, or subsets, of observations on the analysis for idealized one-dimensional variational data assimilation (1D-Var) analysis experiments and then applied in the framework of the three dimensional (3D)- and four-dimensional (4D)-Var schemes developed at Environment Canada.

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Stéphane Laroche, Pierre Gauthier, Monique Tanguay, Simon Pellerin, and Josée Morneau

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A four-dimensional variational data assimilation (4DVAR) scheme has recently been implemented in the medium-range weather forecast system of the Meteorological Service of Canada (MSC). The new scheme is now composed of several additional and improved features as compared with the three-dimensional variational data assimilation (3DVAR): the first guess at the appropriate time from the full-resolution model trajectory is used to calculate the misfit to the observations; the tangent linear of the forecast model and its adjoint are employed to propagate the analysis increment and the gradient of the cost function over the 6-h assimilation window; a comprehensive set of simplified physical parameterizations is used during the final minimization process; and the number of frequently reported data, in particular satellite data, has substantially increased. The impact of these 4DVAR components on the forecast skill is reported in this article. This is achieved by comparing data assimilation configurations that range in complexity from the former 3DVAR with the implemented 4DVAR over a 1-month period. It is shown that the implementation of the tangent-linear model and its adjoint as well as the increased number of observations are the two features of the new 4DVAR that contribute the most to the forecast improvement. All the other components provide marginal though positive impact. 4DVAR does not improve the medium-range forecast of tropical storms in general and tends to amplify the existing, too early extratropical transition often observed in the MSC global forecast system with 3DVAR. It is shown that this recurrent problem is, however, more sensitive to the forecast model than the data assimilation scheme employed in this system. Finally, the impact of using a shorter cutoff time for the reception of observations, as the one used in the operational context for the 0000 and 1200 UTC forecasts, is more detrimental with 4DVAR. This result indicates that 4DVAR is more sensitive to observations at the end of the assimilation window than 3DVAR.

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