The Quarterly Journal of the Royal Meteorological Society

doi: 10.1002/qj.49712455001pmid: N/A

Courtier, P.; Andersson, E.; Heckley, W.; Vasiljevic, D.; Hamrud, M.; Hollingsworth, A.; Rabier, F.; Fisher, M.; Pailleux, J.

doi: 10.1002/qj.49712455002pmid: N/A

In the first of this set of three papers, the formulation of the European Centre for Medium‐Range Weather Forecasts (ECMWF) implementation of 3D‐Var is described. In the second, the specification of the structure function is presented, and the last is devoted to the results of the extensive numerical experimentation programme which was conducted. The 3D‐Var formulation uses a spherical‐harmonic expansion, much as the ECMWF optimal interpolation (OI) scheme used an expansion of Bessel functions. This formulation is introduced using a convolution algebra over the sphere expressed directly in spectral space. It is shown that all features of the OI statistical model can be implemented within 3D‐Var. Furthermore, a non‐separable statistical model is described. In the present formulation, geostrophy is accounted for through a Hough‐modes separation of the gravity and Rossby components of the analysis increments. As in OI, the tropical analysis remains essentially non‐divergent and with a weak mass‐wind coupling. The observations used, as well as their specified statistics of errors, are presented, together with some implementation details. In the light of the results, 3D‐Var was implemented operationally at the end of January 1996.

Rabier, F.; McNally, A.; Andersson, E.; Courtier, P.; Undén, P.; Eyre, J.; Hollingsworth, A.; Bouttier, F.

doi: 10.1002/qj.49712455003pmid: N/A

Structure functions for the 3D‐Var assimilation scheme of the European Centre for Medium‐Range Weather Forecasts are evaluated from statistics of the differences between two forecasts valid at the same time. Results compare satisfactorily with those reported in the existing literature. Non‐separability of the correlation functions is a pervasive feature. Accounting for non‐separability in 3D‐Var is necessary to reproduce geostrophic characteristics of the statistics, such as the increase of length‐scale with height for the horizontal correlation of the mass variable, sharper vertical correlations for wind than for mass and shorter horizontal length‐scales for temperature than for mass. In our non‐separable 3D‐Var, the vertical correlations vary with total wave‐number and the horizontal correlation functions vary with vertical level.

Andersson, Erik; Haseler, Jan; Undén, Per; Courtier, Philippe; Kelly, Graeme; Vasiljevic, Drasko; Brankovic, Cedo; Gaffard, Catherine; Hollingsworth, Anthony; Jakob, Christian; Janssen, Peter; Klinker, Ernst; Lanzinger, Andreas; Miller, Martin; Rabier, Florence;

Rabier, Florence; Thépaut, Jean‐Noel; Courtier, Philippe

doi: 10.1002/qj.49712455005pmid: N/A

Results of four‐dimensional variational assimilations, 4D‐Var, in cycling mode, over a few two‐week assimilation periods are presented. 4D‐Var is implemented in its incremental formulation, with a high‐resolution model with the full physical parametrization package to compare the atmospheric states with the observations, and a low‐resolution model with simplified physics to minimize the cost‐function. The comparison of 4D‐Var using several assimilation windows (6, 12 and 24 hours) with 3D‐Var (the equivalent of 4D‐Var with no time‐dimension) over a two‐week period shows a clear benefit from using 4D‐Var over a 6 or 12—hour window compared to the static 3D‐Var scheme. It also exhibits some problems with the forecasts started using 4D‐Var over a 24‐hour window. The poorer performance of 4D‐Var over a relatively long assimilation window can be partly explained by the fact that, in these experiments, the tangent‐linear and adjoint models used in the minimization are only approximations of the assimilating model (having lower resolution and crude physics). The error these approximations introduce in the time evolution of a perturbation affects the convergence of the incremental 4D‐Var, with larger discontinuities in the values of the cost‐function when going from low to high resolution for longer assimilation windows. Additional experiments are performed comparing 4D‐Var using a 6‐hour window with the 3D‐Var system. Two additional 2‐week periods show a consistent improvement in extratropical forecast scores with the 4D‐Var system. The main 4D‐Var improvements occur in areas where the 3D‐Var errors were the largest. Local improvement can be as large as 35% for the root‐mean‐square of the 5‐day‐forecast error, averaged over a two‐week period. A comparison of key analysis errors shows that, indeed, 4D‐Var using a 6‐hour window is able to reduce substantially the amplitude of its fast‐growing error components. The overall fit to observations of analyses and short‐range forecasts from 3D‐Var and 4D‐Var is comparable. In active baroclinic areas, the fit of the background to the data is considerably better for the 4D‐Var system, resulting in smaller increments. It appears that in these areas (and in particular over the west Atlantic), 4D‐Var is able to better use the information contained in the observations. The ability of 4D‐Var to extrapolate some aircraft data in the vertical with a baroclinic tilt is illustrated. Problems exist in the tropics and mountainous areas due partly to a lack of physics in the tangent‐linear model. Possible improvements to the system (the introduction of more physics; better behaviour of the incremental approach owing to a line search at high resolution) are also discussed.

Veerse, F.; Thepaut, J.‐N.

doi: 10.1002/qj.49712455006pmid: N/A

The incremental approach provides an approximate solution to four‐dimensional variational data assimilation (4D‐Var) at a reasonable CPU cost. An extension of this approach is studied, namely the multiple‐truncation incremental technique, and is compared to the standard and single‐truncation incremental implementations of 4D‐Var, using a two‐dimensional barotropic vorticity‐equation model.

Klinker, E.; Rabier, F.; Gelaro, R.

doi: 10.1002/qj.49712455007pmid: N/A

An iteration procedure minimizing the short‐range forecast error leads, after some iterations, to so‐called key analysis errors. These are estimates of the part of analysis errors that is largely responsible for the short‐range forecast errors. The first step of the minimization procedure provides a scaled gradient of the two‐day forecast errors for which the ‘energy’ inner‐product provides an efficient way of identifying the analysis errors at scales that are relevant for forecast error growth. By using an ‘enstrophy’ like inner‐product as an alternative to ‘energy’ the sensitivity gradient obtains an unrealistically large scale. Performing a few more steps in the minimization provides better estimates of the analysis error in the directions spanned by the leading singular vectors of the tangent‐linear model. On a case study it is shown that three steps provide key analysis increments which, when added to the analysis, both significantly improve the fit to the available data, and substantially improve the subsequent model integration. It does not appear to be beneficial to do more steps of the minimization because of the uncertainty in the definition of the short‐range forecast error, and of approximations in the tangent‐linear model.

Buizza, R.; Petroliagis, T.; Palmer, T.; Barkmeijer, J.; Hamrud, M.; Hollingsworth, A.; Simmons, A.; Wedi, N.

doi: 10.1002/qj.49712455008pmid: N/A

Ensemble integrations for 14 cases are described. These integrations test the relative impact of increase in ensemble size and in the resolution of the model used to integrate the ensemble. The ensembles are evaluated using a variety of statistical tests. Some of these indicate a relative advantage of an increase in ensemble size, whilst most tests suggest a relative advantage of an increase in model resolution. However, overall, the best performance was obtained by combining enhancement in model resolution (from T63L19 to T106L31) with an increase in ensemble size (from 32 to 50 members).

Snyder, C.; Joly, A.

doi: 10.1002/qj.49712455009pmid: N/A

We explore the linear stability of a growing, three‐dimensional baroclinic wave by calculating the perturbation that grows most rapidly over various time intervals and at various stages in the development of the parent wave and its fronts. Three norms are used to measure growth: volume‐integrated energy, enstrophy and stream function variance. The flow is assumed adiabatic and quasi‐geostrophic for simplicity, and perturbations are required to have uniform potential vorticity.

Torrence, Christopher; Webster, Peter J.

doi: 10.1002/qj.49712455010pmid: N/A

A spring ‘predictability barrier’ exists in both data and models of the El Niño/Southern Oscillation (ENSO) phenomenon. In statistical analyses this barrier manifests itself as a drop‐off in monthly persistence (lagged correlation) while in coupled ocean‐atmosphere models it appears as a decrease in forecast skill.