Figure 1 . The operational ECMWF analysis of height and wind at 500hPa for 12 UTC 13 April 1997.

Faster wave motions are, however, excited if the atmosphere is suddenly disturbed. The massive volcanic eruption of Krakatau (between Java and Sumatra) in1883 excited a wave whose propagation could be traced in the surface pressure field around to the antipodal point and back to the source (Fig. 2 ). The signal took about 35 hours to complete its outward and return trip, moving at about 320 ms^-1, around the speed of sound. This was what has come to be known as a Lamb wave, or in numerical weather prediction as an external gravity wave. More commonly, and on a much smaller scale, propagating internal gravity waves may be excited by intense convective systems. Fig. 3 presents observations of wave structure in mid-tropospheric temperature and wind fields over Japan apparently associated with the occurrence of exceptionally heavy rainfall.

Figure 2 . Position at two-hourly intervals of the surface-pressure wave excited by the volcanic eruption of Krakatau, from Taylor(1929).

Figure 3 . Fine-scale structure in temperature at 500hPa (left) and wind at 550hPa (right) over Japan at 12UTC on 1 July 1971, from Ninomiya (1983)

Numerical weather prediction models mostly assume hydrostatic balance, and thus do not support the existence of sound waves. They are generally based, however, on the primitive equations of motion, and Lamb and internal gravity waves can be excited in these models. In particular, they will be excited in data assimilation if the system uses observations in such a way as to produce analyses in which the balance present in the background state is disturbed. The waves in this case contaminate the short-range forecast that provides the background state for the next analysis, adding to the difficulty of extracting useful information from the observations.

The primary control of gravity waves in a modern data assimilation system is through the multivariate formulation of the analysis. In such a system, an observation of surface pressure or temperature that differs from the background will generally result in an analysis which differs from the background not only in surface pressure and temperature, but also in wind, thus preventing an unrealistic deviation from geostrophic balance. This aspect of the control of gravity waves is not the main concern of these lecture notes, although we shall return to the topic briefly later. Rather, we shall be concerned mostly with what is known as the process of initialization.

The term initialization is usually used to refer to the adjustment of analysed initial model conditions to prevent excessive gravity-wave activity in the subsequent forecast. This dynamical initialization is needed if the multivariate formulation of the analysis is unable to provide an adequate balance throughout the model domain. This is the type of initialization that will be discussed here. More widely, initialization may be defined as an adjustment of the initial conditions either to maintain dynamical or physical balance or to satisfy some physical constraint. Thus it could include the adjustment of fields to ensure that relative humidities lie in the range from 0 to 100%, or the adjustment of fields to ensure that initial rainfall rates match estimates from observations. This is referred to as physical initialization.

Although the principal use of dynamical initialization is to remove imbalance introduced during the analysis step in data assimilation, it has additional applications in numerical weather prediction. It is often used to remove imbalance in an initial state in which the model resolution or orography has been changed, for example in:

• Incremental variational data assimilation (as discussed later);

• Creating initial conditions for ensemble prediction based on a model with resolution lower than that of the model used in the data assimilation;

• Testing of a new higher resolution version of a model or new specification of orography.

Initialization may also be used to remove imbalance when running forecasts from analyses produced by other forecasting centres. This is done occasionally to investigate forecasting failures and is under investigation in the context of ensemble prediction.

Some further characteristics of atmospheric motion must be taken into account in considering the topic of initialization. A rich structure of largely stationary gravity-wave motion may be excited by flow over orography. Fig. 4 shows a classical example. These waves can have an important direct effect on the tropospheric circulation, and together with the convectively-excited waves play a critical role in aspects of the circulation of the stratosphere and mesosphere. Their effects are in general parametrized in large-scale models, but they are coming increasingly to be resolved directly in limited-area and finer-resolution global models, and are a target for short-range mesoscale prediction. They should not be unduly suppressed by initialization.

Figure 4 . Section of potential temperature crossing the Rocky Mountains in Colorado, for 17 February 1970, from Lilly and Kennedy (1973).

Tidal motion in the atmosphere is a form of forced gravity-wave motion which also should not be suppressed by initialization. The strong semi-diurnal component at the surface is forced by the daily variation in solar heating of the stratosphere, dominating the more strongly-forced diurnal component due to differences in downward propagation. It is seen in quite pronounced oscillations in surface pressure in the tropics and subtropics (Fig. 5 and Fig. 17 ) and needs to be handled well if other signals in the observations are to be interpreted effectively. Moreover, the quasi-steady tropical circulation systems (as illustrated schematically in Fig. 6 ) characteristically involve a balance between diabatic heating (or cooling) and the adiabatic cooling (or heating) associated with vertical motion. Care is needed to ensure that initialization does not produce analyses in which these circulations are weakened.

Figure 5 . The semi-diurnal tide. The upper plot shows surface pressure (hPa) and the lower plot the tendency of pressure (hPa/hour) at the lowest model level. The quantity plotted is where denotes the mean ECMWF analysis at UTC for January 1997, truncated spectrally at T10 to remove local orographic and station-specific features. Solid contours denote positive values and dashed contours negative values. The nature of atmospheric tides is described in Chapman and Lindzen (1970).

Figure 6 . Schematic view of the mean tropospheric circulation in the equatorial plane, from Webster(1983).

The method of non-linear normal-mode initialization that has been used extensively at ECMWF to initialize its global models is described in the following section. The way gravity waves are controlled in the Centre's variational data assimilation system is described and illustrated in section 3. An alternative approach to initialization, that of digital filtering, has been developed in recent years, particularly for application in limited-area models. An introduction to this method is given in section 4.

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07.06.2002

The control of gravity waves in data assimilation 27 April 1999 By Adrian Simmons European Centre for Medium-Range Weather Forecasts

The control of gravity waves in data assimilation
27 April 1999

By Adrian Simmons
European Centre for Medium-Range Weather Forecasts