predictability in science and society || numerical weather prediction

Numerical Weather PredictionAuthor(s): John MasonSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 407, No. 1832, Predictability in Science and Society (Sep. 8, 1986), pp. 51-60Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/2397781 .

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Proc. R. Soc. Lond. A 407, 51-60 (1986)

Printed in Great Britain

Numerical weather prediction

BY SIR JOHN MASON, TREAS.R.S.

Centre for Environmental Technology, Imperial College, London SW7 lLU, U.K.

The use of physico-mathematical models for the numerical prediction of weather changes on the global scale is described. The accuracy of the predictions is assessed in relation to the limitations of both the observa- tional data and the representation in the model of the many interactive physical and dynamical processes that govern the evolution of the major features of the atmospheric circulation. The concept of predictability in relation to the evolution of atmospheric disturbances is discussed.

Given an adequate global coverage of observations and continued improvement in the models, it should be possible to extend the range of useful forecasts up to about 14 days; this may prove to be the limit of deterministic predictability set by the random nature of atmospheric fluctuations. However, some relatively stable atmospheric states, such as those that produce long dry summers, may possess greater predictability. Furthermore, it may well be possible to predict the general character of the weather for some weeks or months ahead even if the day-to-day variations are unpredictable this far ahead.

Because the longer-range forecasts are bound to show considerable variability in skill and reliability, it will be desirable to assign them a confidence rating based on the rate at which an ensemble of model forecasts diverge when starting from slightly different initial states.

1. INTRODUCTION

It had already become apparent, a quarter of a century ago, that the traditional, empirical methods of weather forecasting, based mainly on the extrapolation of very recent developments and the experience of individual human forecasters, were unlikely to improve significantly or produce reliable forecasts for more than about 24 hours ahead. Fortunately, with the arrival of powerful digital computers, it became possible to replace these highly subjective methods by objective mathematical predictions based on a firm structure of physical theory that treats the atmosphere as a rotating turbulent fluid with energy sources and sinks.

This involves the building of very large and complex mathematical models of the atmosphere based on the physical and dynamical laws that govern the birth, growth, decay and movement of the main weather systems that can be resolved by the model. In other words, the models must properly represent the relevant or significant scales of motion and their nonlinear interactions, but smooth out all the smaller scale motions that cannot be adequately observed or represented individually while allowing for their overall contribution to transport and energy conversion processes by representing their statistically averaged properties in terms of larger-scale parameters that can be measured.

[51]



52 Sir John Mason

2. THE STRUCTURE OF A WEATHER PREDICTION MODEL

The models incorporate the principles of conservation of mass, momentum, energy and of water in all its phases, the Newtonian (Navier-Stokes) equations of motion applied to a parcel of air, the laws of thermodynamics and radiation, and the equation of state of a gas. The set of predictive equations is as follows.

Equations (1) and (2). Two Navier-Stokes equations describing the horizontal motion of the air in which the time rates of change of the E-W and N-S components of the wind are related to the forces exerted on the air by the rotation of the Earth, by horizontal pressure gradients, by dissipative forces such as friction and turbulence, and the sources and sinks of momentum.

Equation (3). A similar equation describing the vertical motion of the air under the influence of forces that arise from vertical pressure gradients, gravity, rotation of the Earth and from frictional and turbulent stresses.

Equation (4). An equation of continuity (mass conservation) relating changes in vertical motion to the divergence of the horizontal wind field. Integration of this gives the vertical motion and a predictive equation for the atmospheric pressure at the Earth's surface.

Equation (5). A thermodynamic equation relating changes in the potential temperature of the air to the heat supplied by radiation, condensation, etc.

Equation (6). An equation of conservation of total water content in all its phases relating changes in humidity of the air to sources and sinks of moisture (condensation, evaporation, precipitation).

Parameters specified in advance include the size, rotation, geography and topography of the Earth, the incoming solar radiation and its diurnal and seasonal variations, the radiative and heat conductive properties of the land surface according to the nature of the soil, vegetation and snow or ice cover, and also the surface temperature of the oceans updated every five days.

The predictive equations which subsume the equation of state of the atmospheric gases and equations representing the vertical transfer of heat by radiation and convection, for the fractional drag of the land, ocean waves and mountains on the air, and for the conversion of water vapour into rain and snow, have six forecast variables. These are the E-W, N-S and vertical components of the wind, potential temperature, changes in the height of constant pressure levels and the specific humidity. Also predicted are the surface pressure and temperature and the precipitation reaching the ground.

Each forecast is treated as an initial value problem with specified boundary conditions. The physical state of the atmosphere itself is updated every 12 hours from observations made simultaneously over the whole globe, both at the surface from thousands of land stations, ships and buoys, and in the upper air from several hundred balloon-borne sondes. A great deal of non-synoptic data, on winds and air temperature especially, are provided by satellites and aircraft. The Meteorolo- gical Office model atmosphere is divided into 15 layers from the ground up to an altitude of 25 km (about 80000 feet) and each level is divided into a network of points about 150 km apart, about 350000 points in all. Each of these grid points is assigned new values of temperature, pressure (or heights of constant pressure



Numerical weather prediction 53

surfaces), wind and humidity every 12 hours and the governing differential equations are integrated in 15-minute time steps at each point to provide forecast values for up to six days ahead, twice daily. A forecast for 24 hours ahead involves about one hundred thousand million (1011) calculations, but with the CYBER 205 performing about 400 million operations per second, these are completed in about

42 minutes. The whole operation results in the automatic production of hundreds of different forecast charts of pressure, temperature, wind, humidity, vertical motion and rainfall that form the basis of forecasts issued to the general public and to almost every weather-sensitive industry besides world-wide forecasts for other countries and for the world's airlines. Examples of very successful six-day forecasts of the onset of the recent (February 1986) prolonged cold spell over the U.K. and the breakdown of the fine summer of 1983 were presented at the meeting but are not reproduced here.

3. ASSESSMENT OF NUMERICAL FORECASTS

The performance of numerical models and the accuracy of their predictions are assessed by comparing the forecast values of field quantities such as air pressure, wind and temperature at a large number of grid points at each level with the verifying values derived from the corresponding observations. The differences are expressed in terms of root-mean square errors or as correlation coefficients.

Figure 1 shows a marked improvement in all of the Meteorological Office forecast parameters for Europe and the North Atlantic after the introduction of the 10-level, northern-hemisphere model in August 1972, and again after the current 15-level global model was introduced in September 1982. The RMS errors in the forecast heights of all the pressure surfaces have roughly halved since 1972. In this respect the 72-hour, 500 mbt (5.4 km) forecasts are now as good as the 48-hour forecasts were seven years ago, and the 48-hour forecasts are now as good as the 24-hour forecasts were then. Similar 24-hour improvements in performance are also apparent in the wind fields. The correlation coefficients between the forecast and actual height changes of the 1000 mb level now exceed 0.8 for 72-hour forecasts and 0.9 for 24-hour forecasts. This is shown in figure 2. On this criterion the 72-hour forecasts are now as good as the 24-hour forecasts were 7 years ago. A similar analysis for the Europe-North Atlantic area shows increases in predictive skill ranging from 48 hours at 3 days to 40 hours at 6 days were achieved over the period 1976-85.

Although RMS errors and correlation coefficients are useful indicators of the performance of different models for the same area and period, they are only partial indicators of the models' predictive skill. A better judgement is obtained by comparing the forecasts RMS errors with the long-term climatological variance or with the errors of a persistence (zero-skill) forecast based on persistence (no change) from the initial conditions. In 1974, the RMS errors of the three-day forecasts at 500 mb were 80 % of the persistence forecast errors; in 1984 the corresponding figure was 48 %, and the 80 % level was not reached until the sixth day, suggesting a gain of three days in predictive skill.

t 1 mb = 1 millibar - 102 Pa.



54 Sir John Mason

120

BMO - Europe - North Atlantic

10-level model 15-level model

100 _ / - O~~~~~~~ mb, 72 h

80 -<200mb,48h

N 6500 mb, 48 h

200mb, 24h i;\ \

40 - <

1000 mb, 24 h --f

-2()6l I l _ 1972 1976 1980 1984

FIGURE 1. Annual average RMS error in the predicted heights of the 200, 500 and 1000 mb surfaces produced by the British Meteorological Office from 1968 to 1984 for Europe and the North Atlantic area.

Experience shows that numerical forecasts are unlikely to provide good or useful guidance for the issue of surface weather forecasts if their RMS errors exceed 75 % of the persistence error. Adopting this criterion, the average useful predictive range of forecasts issued from Bracknell in 1984 was 5.3 days at 500 mb for extratropical regions in the Northern Hemisphere, 5.0 days in the Southern Hemisphere and one day less for surface forecasts in both hemispheres.

Forecasts for the tropical regions, which are rarely reliable beyond one or two days, show much less skill than those for middle latitudes, even though the magnitude of the random (RMs) errors at all levels is considerably less. The reason is mainly because the tropical forecasts contain relatively large systematic (mean) errors which result from the models' inadequate handling of the large-scale, quasi-stationary systems that dominate the tropical atmosphere. This is exacer- bated by poor coverage and quality of observations in the tropics, weak dynamical coupling of the mass and wind fields, and the more important role of physical processes such as deep convection, which are rather poorly represented in the model.

In summary, the best computer models now produce good four-five day forecasts




1.0_ 10-level model 15-level model

1000 mb, 24h

1000 mb, 72h

0./

0.6 -

10

04-

*> 0.4 | BMO U.K. area

0.2 1968 1972 1976 1980 1984

FIGURE 2. Correlation coefficients between the predicted and observed (analysed) values of the change in height of the 1000 mb surface for forecasts issued by the Meteorological Office from 1968 to 1984 for the United Kingdom.

of the evolution of the major weather systems in the extra-tropical regions of the globe and give a useful indication of major developments for a further one-two days ahead in the Northern Hemisphere. On some occasions, in rather stable weather regimes, good guidance may be given for up to seven days.

These performances represent an advance of two-three days in predictive skill over the last decade, during which time the proportion of poor 48-hour forecasts of surface weather over the U.K. has dropped from 25 to only 5 % and that of poor 72-hour forecasts from 40 to 20 %. Twenty years ago, forecasts beyond 24 hours were rarely attempted.

4. PRECISION, ACCURACY AND MODEL DEFECTS

The precision to be attempted and the accuracy likely to be achieved in predicting the location and timing of a particular weather feature will be dependent upon the time range of the prediction, on the scale and life-time of the weather systems involved, and on the spatial resolution of the forecasting model. Thus a fine-mesh version of the Meteorological Office global model, which covers western Europe and the North Atlantic ocean with a horizontal resolution of only 75 km, will predict the geographical position and central pressure of a major depression more than 1000 km in diameter and lasting for five or six days within 50 km and 2 mb respectively. For a forecast made only 24 hours ahead, it is reasonable to attempt this degree of precision in relation to the accuracy achieved.



56 Sir John Mason

However, the errors grow roughly linearly as the forecast is extended, so that although the model will continue to produce a deterministic and apparently precise forecast, accuracy progressively declines and beyond five to six days usually falls below useful limits.

At the other end of the scale, prediction of the location, movement and intensity of individual shower clouds, only 1-10 km in diameter and lasting for less than one hour, is impossible with present models which lack the requisite small-scale physics, observations and resolution. The best that can be done is to predict where and when the atmosphere is likely to become convectively unstable within a layer of restricted depth and so liable to sporadic outbreaks of showers; but there is no possibility of predicting exactly where and when a shower will actually appear. The forecast may then perforce take the form of, say, 'a high risk of scattered showers, mostly light and of short duration' since they are likely to be almost randomly distributed in space and time.

In summary, it is not possible, by the intrinsic nature of the problem, to predict accurately the location of a small weather system for more than a short time in advance; precision may be possible at the expense of range, but a long-range forecast will necessarily be lacking in precision and detail. It is just not possible to achieve both precision and range in the same forecast.

The achievable accuracy for prediction on a particular space-time scale is limited by:

(i) inadequacies in the coverage, frequency, accuracy and representativity of the observations used to define the initial state;

(ii) neglect in the models of some of the physical processes, especially small-scale processes, or their inadequate statistical representation;

(iii) computational errors which arise at each time step of the integration and so build up cumulatively;

(iv) random fluctuations in the real turbulent atmosphere which are not represented in the model.

-Serious errors in short-range forecasts (1-5 days) probably arise mainly from deficiencies in the models, where it is difficult to distinguish between the effects of numerical errors introduced by the finite difference approximations to the continuous dynamical equations, and those that arise because the equations themselves do not provide a complete and exact description of atmospheric behaviour but the latter appear to be the more important. Some of the major deficiencies arise from inadequate horizontal and vertical resolution, inadequate representation of topography, surface hydrology, the surface fluxes of heat, moisture and momentum, and of the radiative, convective and advective processes within the surface boundary layer, including their interaction with low-level clouds. Considerable effort is being devoted to the prediction of cloud cover and height from the humidity and vertical motion fields and to calculating the influence of cloud on the short-wave and long-wave radiative fluxes at all levels. Improved representations of both shallow and deep convection are required, not only to produce better forecasts of clouds and precipitation, but also to stimulate more realistically their role in the transfer of heat and moisture.

Improvements at longer ranges (more than six days) will require better observations especially from the oceans and remote land areas.




5. POTENTIAL IMPROVEMENTS IN PREDICTIVE SKILL

Figure 3 illustrates that the average RMS forecast errors grow linearly with time up to about six days, and thereafter more slowly as they approach the 'persistence forecast' errors which indicate zero predictive skill. The forecast error reaches 75 % of the persistence error on day 6 and 90 % on day 9 so that, although the actual errors do not grow much after day 9, the forecast skill is now practically nil. In attempting to reduce these errors, thereby improving forecasting skill and extending the useful forecast range (predictability), it is important to distinguish between that part of the total forecast error which is due to deficiencies of the model and that which arises from errors in the initial state derived from the

150 _

0 | persistence error,P - -

10 2 . 0 21

~~ forecast error,cast da

~~~ 50 / .. . .~~~~~~~~~~~~~~~~~~~~~~ - - - - ~ ~ ~ ~ ........

0 nalysis -f the observations. The growth in forecast errordueto analysis error,sA

indicate thattheranalsis error, tedt rwepnnill tfrt obigi

'.4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~4

IfMsisth mansqar mde ero, 2 hemen qureanalysis error, the

0 2 4 6 8 10 12 14 forecast day

FIGUre 3. The average Ms errors for the height of the 500 mb surface in numerical predictions for up to 10 days ahead for the extra-tropical north hemisphere in 1984, showing the contributions made to the total forecast error F, by errors in the initial analysis, A, and by the model errors, M. The errors of the corresponding persistence forecast are shown for comparison.

analysis of the observations. The growth in forecast error due to analysis errors has been studied by comparing the evolution and divergence of a series of forecasts using the same model but starting from slightly different initial analyses. These indicate that the analysis errors tend to grow exponentially at first, doubling in amplitude every two days, and thereafter more slowly to reach an asymptotic value after about 14 days.

If M2P is the mean square model error, A 2 the mean square analysis error, then the total mean square forecast error F2 iS

Y= M2+A2 -2 coy (MA),

where coy (MA) is the covariance of M with A. After a long time, in an unbiased model-, F2, J)2 AF)2 and coy (MA) should all approach asymptotic values of 252 where 52 is the long-term climatological variance. Figure 3, taken from Mason



58 Sir John Mason

(I986), shows the results of an illustrative calculation in which the RMS analysis error (A2)' doubles every 2 days up to 6 days and then more slowly to approach the same value as F2 at day 14. The corresponding calculated RMS model errors are shown by the broken curve, which indicates that the model errors exceed the analysis errors up to day 7, and that their elimination (a perfect model) would produce an increase of about 2 days in predictive skill at all ranges from 2-10 days. Beyond 8 days, the analysis errors are larger than the model errors and both would have to be reduced to produce skilful forecasts.

Perfection of the model would allow useful forecasts (F/P < 0.75) up to 8 days ahead but further improvements would depend on reducing the analysis errors, a 20 % reduction in which would increase the useful predictive range by a further 3 days. The forecast error of a perfect model with the analysis errors of figure 3 reduced by 20 % would attain 75 % of the persistence error after 13 days. These figures are illustrative (tho actual values depend a little on the precise shapes of the forecast and analysis error curves) but they suggest that it will, in general, be very difficult to produce useful deterministic forecasts of synoptic-scale developments for more than 14 days ahead although some relatively stable atmospheric states may well possess greater predictability for the reasons given in the following section.

6. ATMOSPHERIC PREDICTABILITY

We now raise the crucial question of whether there is, for each scale of motion, a time limit beyond which it is not possible to make a deterministic forecast. This question is of great importance for practical weather forecasting because the answer may set ultimate limits to what is achievable and therefore worth aiming at.

Given, then,

(i) a physically faithful model of the real atmosphere, (ii) the ability to specify exactly the initial conditions for all scales of motion,

and (iii) no computational errors in integrating the differential equations,

would it be possible to predict the atmospheric evolution from an initial state with infinite precision infinitely far ahead? This is really a rhetorical question, because none of these conditions can be fulfilled in practice. Even with a perfect model, in which all the physical processes were represented exactly as they occur in the real atmosphere, predictability would be limited because the initial state can never be observed completely and accurately on all scales of motion. Moreover, it is axiomatic that the behaviour of an aperiodic system cannot be predicted at long range unless the initial (or some past) state is known with no uncertainty whatever. An aperiodic system is inherently unstable, so that the imposition of a random disturbance will render it chaotic (i.e. unpredictable) in the long run.

Atmospheric behaviour has periodic components such as annual and diurnal variations, but when these are subtracted out a strong aperiodic component remains. This includes the progression of cyclones and anticyclones across middle- latitude continents and oceans so even these cannot be predicted accurately at




long range. Apart from random fluctuations in the atmosphere itself, there may also be undetected or intractable fluctuations in the boundary conditions, e.g. in the radiative properties of the Earth's surface due to changes in vegetation or snow cover, changes in the sea-surface temperature, or changes in the input solar radiation. In fact, there is no method of determining the maximum range of useful prediction from first principles; this can be established only by experiment and observation as indicated in ?5.

Beyond this, it is doubtful whether further improvements in the models or in computing accuracy would significantly increase predictability for these scales. This is because even if the larger-scale systems, represented by smooth fields and smooth differential equations, were observed perfectly and represented perfectly in the models, their behaviour would eventually be affected by the action of much smaller disturbances such as thunderstorms, tornadoes and even smaller eddies, which cannot be adequately observed or represented and yet may double in amplitude within a few minutes. These may, within hours, induce uncertainties in the larger scales comparable to the initial errors resulting from inadequate observations. This is inherent in the nonlinear behaviour of atmospheric processes whereby energy is exchanged between all scales of motion. The rates at which random fluctuations (uncertainty) on one scale may contaminate other scales depends on the degree of coupling between them. If there were to be a gap in the energy spectrum of atmospheric motions, the propagation of 'error' energy from the smaller to the larger scales would be retarded and the predictability of the latter increased. There is some, but not entirely convincing, evidence that such a gap exists between the synoptic-scales and meso-scales at a wavelength of about 100 km, and this may partly explain the fact that the forecast errors on the synoptic scale do not continue to grow rapidly after the first few days but thereafter grow more slowly to approach an asymptotic value after about 14 days as indicated in figure 3.

Although deterministic forecasts on the synoptic (1000 km) scale may eventually show useful skill up to 14 days, at these longer ranges they are likely to show greater variability in skill and reliability depending upon the configuration of the large-scale flow, the nature and degree of forcing from the underlying land and ocean surfaces, and the geographical location relative to energy sources and sinks. It may therefore be desirable and useful to express the longer-range predictions in probabilistic terms and assign them a confidence rating. This might be determined by running the same forecast several times with slightly different initial conditions in order to test the sensitivity of the forecast to small perturba- tions and weight the confidence factor accordingly.

Although the range of useful predictions of individual, mobile, extra-tropical weather systems may, in general, be limited to about two weeks, we sometimes observe more stable systems embedded in the general flow which retain their character, coherence and predictability for considerably longer periods. (A supreme example is the Great Red Spot of Jupiter, an enormous anticyclonic vortex larger than the Earth, which has retained its essential character for more than 300 years!) Two memorable examples are the stable blocking anticyclones that produced the prolonged hot summer in western Europe in 1976 and the long cold



60 Sir John Mason

spell over north-western Europe in recent weeks. The stability of these configura- tions may require only a temporary and local gap in the energy spectrum of atmospheric motions to impede the exchange of energy with the smaller scales. They also tend to occur in mid-summer or mid-winter when the solar radiation is only slowly varying. Lorenz (I985) attributes the observed coherence and persistence of the meso-scale tropical cyclones over the oceans to a gap between them and the small-scale disturbances. Of course, the path of such a storm is not easily predicted, but this is under the influence of motions of larger scales which are more difficult to predict than those of middle latitudes.

Even though the range of deterministic predictions is limited by the random nature of small-scale of atmospheric motions, it may be possible to predict average conditions for considerably longer periods (some weeks) ahead. The Meteorological Office model is now being run for up to 50 days ahead and the forecasts averaged over 10-day periods. If these turn out to be not too sensitive to the initial conditions, i.e. one 10-day average forecast is not radically different from a similar forecast started one day later, then they may well provide useful guidance on the evolution of the large-scale features of the global atmospheric -circulation such as the long planetary waves and the major jet streams, the general tracks of mobile depressions and the location of blocking anticyclones and therefore the general character of the weather over periods of several weeks. Such predictions, though necessarily lacking in detail and precision, would, if reliable, be of considerable economic and social value.

REFERENCES

Lorenz, E. N. I985 Prospects for improved weather forecasting. Global Atmosphere Research Programme Spec. Report no. 43, vol. III, pp. 1-11.

Mason, B. J. I986 In Proc. Int. Conf. on Global Weather Experiment. World Meteorological Organization. (In the press.)



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