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Juhani Rinne
and
Heikki Järvinen

Abstract

This paper presents an estimate of the Cressman term that appears in the divergent barotropic model as a corrector of planetary-wave retrogression. In contrast with earlier studies, the term varies as a function of geographical location. The estimation is performed using the adjoint model.

The results deviate from those earlier derived theoretically and applied in routine forecasting. The Cressman term can now be viewed as a corrector of the systematic error or as a baroclinity parameter. The proposed form of the Cressman term can also be interpreted as a forcing parameter, maintaining the troughs and ridges of the main circulation by affecting the free long waves.

Parameter estimation using the adjoint model has shown its potential in these experiments. Not only are the parameter values determined but new interpretations and approaches have been found.

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Juhani Rinne
and
Heikki Järvinen

Abstract

The chaotic structure of two-dimensional atmospheric flow is illustrated. It is shown that certain errors in numerical approximations can Prevent the correct prediction of chaotic processes. This is the case when the numerical approximations do not sufficiently allow air parcels to deviate from each other. The error mechanism is described with a case study and is proposed as one explanation for the errors observed when forecasting the development of blocking highs. It can explain why the errors in blocking highs are similarly found in different models from different centers, why they appear in medium-range forecasts but not in short-range forecasts, and why the error decreases only slowly with increasing resolution.

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Juhani Rinne
and
Simo Járvenoja

Abstract

The spatial autocorrelation of the error of the ECMWF 24-h forecast of 500 mb geopotential height during 1980–83, hereafter referred to as the “forecast error,” is studied. The leading EOF of the forecast error describes a “teleconnection” between the Himalayas and the Pacific/European sector. The third EOF resembles the analysis error. Approximations for the isotropic part of the autocorrelation are presented. They consist of two source terms having different spatial scales. A “random-like” term could be due to model errors in baroclinic processes or to random analysis errors. A “scale-dependent” term could be due to model errors in barotropic processes and over mountain areas, or to analysis errors over data-sparse areas or mountains. The terms do not contribute uniformly. The “scale-dependent” term is strongest over mountains The relative contribution of that term decreased from forecasts of 1980–81 to those of 1982–83.

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Juhani Rinne
and
Simo Järvenoja

Abstract

The structure of the analysis error field of the 500 mb height north of 20°N was estimated from the differences between analyses issued by two different centers. The resulting variance field is very similar to that computed with the aid of the optimum interpolation. The leading empirical orthogonal functions of the analysis error show large cells over areas with a sparse observational network.

Artificial analysis error fields are constructed with the aid of empirical orthogonal functions. The structure of these fields is the same as that derived from the analyses.

The model results indicate that the perturbation fields can be produced without correlating them with previous analyses or forecasts. The smoothing of the analyses contributes rather much to the forecasting error and should therefore be simulated. The model results, however, are valid only if the forecasting error can be measured with the root-mean-square error.

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Juhani Rinne
and
Simo Järvenoja

Abstract

A rapid method of computing the EOFs for a problem with a large covariance matrix is presented. The resulting EOFs are not the “true” ones because they do not guarantee that the maximum variance is explained in the dependent sample. However, in an independent sample they turn out to be very efficient when compared with the “true” EOFs computed from a very large dataset. The differences between different EOF sets do not necessarily indicate essential differences in the corresponding sets of the source data. A rotation for secondary reasons may produce very noticeable differences between the EOFs.

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Juhani Rinne
and
Simo Järvenoja

Abstract

A method of computing autocorrelation fields with the aid of empirical orthogonal functions (EOF) is applied. The isotropic parts of the fields are separated, and a one-parameter model of the isotropic autocorrelation field is constructed. The approximation is compared with the Bessel representations discussed by others. The present model requires fewer parameters (one as against four) and gives a more realistic representation at greater distances. In other respects, the results are quite similar. At midlatitudes, the present model is a good approximation of the autocorrelation fields. Over the Himalayas, the fields are exceptionally anisotropic. The results substantiate the importance of the Himalayas in teleconnectivity.

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Juhani Rinne
and
Simo Järvenoja

Abstract

Different methods of studying the random error variance of 500 mb geopotential height analyses are discussed and applied. It has not been possible to show that the random errors of the operational analyses during the FGGE year have been smaller than those of previous years. In contrast to this, however, it is concluded that there have never been analyses of a quality comparable with the delayed-mode FOGE analyses (leaves IIIb/ECMWF). This conclusion is based on the whiteness of the error spectrum and the small variance of the random error.

The variances of the random errors of the operational analyses over a dense observational network are small. The maxima are found over the Pacific, the Atlantic, the Himalayas and around the North Pole. Their intensity varies from one analysis set to another. In the delayed-mode analyses, the distribution of the error variance is flat. One operational analysis set shows a similar flat distribution. The reason for this flatness is believed to lie in the smoothing applied by the analysing center. A study of the smoothing is thus necessary in order to draw final conclusions.

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Juhani Rinne
and
Simo Järvenoja

Abstract

The smoothing of 500 mb geopotential analyses has been studied by comparing the variances of the EOF components. As an indicator of the total analysis error, the sums of the smoothing variances and random error variances have been studied. In most cases, no essential differences between the total analysis error variances of different centers are found. The smoothing efficiency in the FGGE analysts (level IIIb, analyzed at ECMWF) seems to be close to that of the operational analyses. However, the total error variance of the FGGE analyses is smaller because of the small random error variance. In operational analysts the variances of the first EOF are larger than in the FOGGE analyses. An explanation for this can be given by biases in the climates of the operational models used in data assimilation.

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Juhani Rinne
and
Simo Järvenoja

Abstract

The yearly variances of the analysis error of the 500 mb height north of 20°N are estimated from analyses covering the period 1946–79. A new method is presented. The error variances are computed directly from the analyses without any assumptions concerning the observational network, the observation error or the analysis system. The results show a general decrease of the analysis error variance up to 1970, since when it has been fairly constant. A weak increase is visible in analyses from 1976 onward. The present day error variance is about half of that for the years preceding the International Geophysical Year. Sudden peaks occur when the analysis center was changed. Local developments are discussed. When subsamples are compared with each other, the problematic area in 1955–62 turn out to be around the North Pole, around the Great Lakes, and the area westward from Kamchatka. Over these area some phenomena have been analyzed less accurately than expected. In 1962–70 a problematic area is found over the northern Atlantic.

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Juhani Rinne
and
Simo Järvenoja

Abstract

When the height of the 500 mb surface is represented with the aid of empirical orthogonal functions, the residual should consist mainly of analysis error. However, unlike the analysis error, the residual is larger over continents than over oceans. The reason for this disagreement is thought to be in the different structure of the analysis error over oceans and over continents. To overcome the disagreement, new functions are required. In this study the problem is formulated and a method for finding an approximate solution to the problem is presented.

The residual variance field of the derived functions resembles the estimated analysis error variance field. The new functions take in more details from over continents. A study using the new functions shows that a weak ridge over the United States is described satisfactorily, whereas the old functions failed to do this. The new representation is smoother over oceans.

The usefulness of the new functions in practice depends on applications.

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