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Chaos and weather
prediction
January 2000
By Roberto Buizza
European Centre for Medium-Range Weather
Forecasts
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1 . Introduction
A dynamical system shows a chaotic behavior if most orbits exhibit sensitive dependence (Lorenz 1993). An orbit is characterized by sensitive dependence if most other orbits that pass close to it at some point do not remain close to it as time advances.
Figure 1 . (a)-(c) Forecast for the geopotential height at 1000 hPa (this field illustrates the atmospheric state close to the surface) given by three forecasts started from very similar initial conditions, and (d) verifying analysis. Contour interval is 5 m, with only values smaller than 5 m shown.
The atmosphere exhibits this behavior. Fig. 1 shows three different weather forecasts, all started from very similar initial conditions. The differences among the three initial conditions were smaller than estimated analysis errors, and each of the three initial conditions could be considered as an equally probable estimate of the "true" initial state of the atmosphere. After 5 days of numerical integration, the three forecasts evolved into very different atmospheric situations. In particular, note the different positions of the cyclone forecast in the Eastern Atlantic approaching United Kingdom (Fig. 1 (a)-(c) ). The first forecast indicated two areas of weak cyclonic circulation west and south of the British Isles; the second forecast positioned a more intense cyclone southwest of Cornwall, and the third forecast kept the cyclone in the open seas. This latter turned out to be the most accurate when compared to the observed atmospheric state (Fig. 1 (d). This is a typical example of orbits initially close together and then diverging during time evolution.
The atmosphere is an intricate dynamical system with many degrees of freedom. The state of the atmosphere is described by the spatial distribution of wind, temperature, and other weather variables (e.g. specific humidity and surface pressure). The mathematical differential equations describing the system time evolution include Newton's laws of motion used in the form `acceleration equals force divided by mass', and the laws of thermodynamics which describe the behavior of temperature and the other weather variables. Thus, generally speaking, there is a set of differential equations that describe the weather evolution, at least, in an approximate form.
Richardson (1922) can be considered the first to have shown that the weather could be predicted numerically. In his work, he approximated the differential equations governing the atmospheric motions with a set of algebraic difference equations for the tendencies of various field variables at a finite number of grid points in space. By extrapolating the computed tendencies ahead in time, he could predict the field variables in the future. Unfortunately, his results were very poor, both because of deficient initial data, and because of serious problems in his approach.
After World War II the interest in numerical weather prediction revived, partly because of an expansion of the meteorological observation network, but also because of the development of digital computers. Charney (1947, 1948) developed a model applying an essential filtering approximation of the Richardson's equations, based on the so-called geostrophic and hydrostatic equations. In 1950, an electronic computer (ENIAC) was installed at Princeton University, and Charney, Fjørtoft and Von Neumann & Ritchmeyer (1950) made the first numerical prediction using the equivalent barotropic version of Charney's model. This model provided forecasts of the geopotential height near 500 hPa, and could be used as an aid to provide explicit predictions of other variables as surface pressure and temperature distributions. Charney's results led to the developments of more complex models of the atmospheric circulation, the so-called global circulation models.
With the introduction of powerful computers in meteorology, the meteorological community invested more time and efforts to develop more complex numerical models of the atmosphere. One of the most complex models used routinely for operational weather prediction is the one implemented at the European Centre for Medium-Range Weather Forecasts (ECMWF). At the time of writing (December 1999), its is based on a horizontal spectral triangular truncation T319 with 60 vertical levels formulation (Simmons et al. 1989, Courtier et al. 1991, Simmons et al. 1995). It includes a parameterization of many physical processes such as surface and boundary layer processes (Viterbo & Beljaars 1995) radiation (Morcrette 1990), and moist processes (Tiedtke 1993, Jacob 1994).
The starting point, in mathematical terms the initial conditions, of any numerical integration is given by very complex assimilation procedures that estimate the state of the atmosphere by considering all available observations. The fact that a limited number of observations are available (limited compared to the degrees of freedom of the system) and that part of the globe is characterized by a very poor coverage introduces uncertainties in the initial conditions. The presence of uncertainties in the initial conditions is the first source of forecast errors.
A requirement for skilful predictions is that numerical models are able to accurately simulate the dominant atmospheric phenomena. The fact that the description of some physical processes has only a certain degree of accuracy, and the fact that numerical models simulate only processes with certain spatial and temporal, is the second source of forecast errors. Computer resources contribute to limit the complexity and the resolution of numerical models and assimilation, since, to be useful, numerical predictions must be produced in a reasonable amount of time.
These two sources of forecast errors cause weather forecasts to deteriorate with forecast time.
Initial conditions will always be known approximately, since each item of data is characterized by an error that depends on the instrumental accuracy. In other words, small uncertainties related to the characteristics of the atmospheric observing system will always characterize the initial conditions. As a consequence, even if the system equations were well known, two initial states only slightly differing would depart one from the other very rapidly as time progresses (Lorenz 1965). Observational errors, usually in the smaller scales, amplify and through non-linear interactions spread to longer scales, eventually affecting the skill of these latter ones (Somerville 1979).
The error growth of the 10-day forecast of the ECMWF model from 1 December 1980 to 31 May 1994 was analyzed in great detail by Simmons et al. (1995). It was concluded that 15 years of research had improved substantially the accuracy over the first half of the forecast range (say up to forecast day 5), but that there had been little error reduction in the late forecast range. While this applied on average, it was also pointed out that there had been improvements in the skill of the good forecasts. In other words, good forecasts had higher skill in the nineties than before. The problem was that it was difficult to assess a-priori whether a forecast would be skilful or unskillful using only a deterministic approach to weather prediction.
Figure 2 . The deterministic approach to numerical weather prediction provides one single forecast (blue line) for the "true" time evolution of the system (red line). The ensemble approach to numerical weather prediction tries to estimate the probability density function of forecast states (magenta shapes). Ideally, the ensemble probability density function estimate includes the true state of the system as a possible solution.
Generally speaking, a complete description of the weather prediction problem can be stated in terms of the time evolution of an appropriate probability density function (PDF) in the atmosphere's phase space (Fig. 2 ). Although this problem can be formulated exactly through the continuity equation for probability (Liouville equation, see e.g. Ehrendorfer 1994), ensemble prediction based on a finite number of deterministic integrations appears to be the only feasible method to predict the PDF beyond the range of linear error growth. Ensemble prediction provided a way to overcome one of the problems highlighted by Simmons et al. (1995), since it can be used to estimate the forecast skill of a deterministic forecast, or, in other words, to forecast the forecast skill.
Since December 1992, both the US National Center for Environmental Predictions (NCEP, previously NMC) and ECMWF have integrated their deterministic high-resolution prediction with medium-range ensemble prediction (Tracton & Kalnay 1993, Palmer et al. 1993). These developments followed the theoretical and experimental work of, among others, Epstein (1969), Gleeson (1970), Fleming (1971a-b) and Leith (1974).
Both centres followed the same strategy of providing an ensemble of forecasts computed with the same model, one started with unperturbed initial conditions referred to as the "control" forecast and the others with initial conditions defined adding small perturbations to the control initial condition. Generally speaking, the two ensemble systems differ in the ensemble size, in the fact that at NCEP a combination of lagged forecasts is used, and in the definition of the perturbed initial. The reader is referred to Toth & Kalnay (1993) for the description of the 'breeding' method applied at NMC and to Buizza & Palmer (1995) for a thorough discussion of the singular vector approach followed at ECMWF.
A different methodology was developed few years later at the Atmospheric Environment Service (Canada), where a system simulation approach was followed to generate an ensemble of initial perturbations (Houtekamer et al. 1996). A number of parallel data assimilation cycles is run randomly perturbing the observations, and using different parameterisation schemes for some physical processes in each run. The ensemble of initial states generated by the different data assimilation cycles defines the initial conditions of the Canadian ensemble system. Moreover, forecasts started from such an ensemble of initial conditions are used to estimate forecast-error statistics (Evensen 1994, Houtekamer & Mitchell 1998).
Ensemble prediction, which can be considered one of the most recent advances in numerical weather prediction, is the first topic discussed in this work. The development of objective procedures to target adaptive observations is the second topic on which attention will be focused.
The idea of targeting adaptive observations is based on the fact that weather forecasting can be improved by adding extra observations only in sensitive regions. These sensitive regions can be identified using tangent forward and adjoint versions of numerical weather prediction models (Thorpe et al. 1998, Buizza & Montani 1999). Once the sensitive regions have been localised, instruments can be sent to those locations to take the required observations using pilot-less aircraft, or energy-intensive satellite instruments can be switched on to sample them with greater accuracy.
After this Introduction, section 2 describes some early results by Lorenz, and illustrates the chaotic behavior of a simple 3-dimension system. In section 3 the main steps of numerical weather prediction are delineated. The impact of initial condition and model uncertainties on numerical integration is discussed in section 4. The ECMWF Ensemble Prediction System is described in section 5. Targeting adaptive observations using singular vectors is discussed in section 6. Some conclusions are reported in section 7. Some mathematical details are reported in two Appendices.
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