8 Balance and initialization I still consider the elimination or dampening of noise to be the crucial problem in weather analysis and prediction. (Hinkelmann, 1985) The spectrum of atmospheric motions is vast, encompassing phenomena having periods ranging from seconds to millennia. The motions of interest for numerical weather prediction have timescales greater than a day, but the mathematical equa- tions describe a broader span of dynamical features than those of direct concern. One of the long-standing problems in numerical weather prediction has been to overcome the problems associated with high frequency motions. In this chapter we will consider the history and development of methods that ensure that dynamic balance of the initial data is achieved. We first discuss the phenomenon of balance in the atmosphere. Then the valuable concept of the slow manifold is introduced. We briefly review the principal methods for achieving balance in initial data. Next we introduce a simple mechanical system, the swinging spring; the solutions to the balance problem for this system have much wider applicability and are relevant to the problems arising in general forecasting models. Finally, we discuss digital filter initialization, the technique that will be used to initialize Richardson’s forecast. 8.1 Balance in the atmosphere In Chapter 3 we examined the linear spectrum of atmospheric motions and saw how the natural oscillations fall into two classes. The ‘solutions of the first class’ are the rapidly-travelling high frequency gravity-inertia wave solutions, with phase speeds of up to hundreds of metres per second and large divergence. The ‘solutions of the second class’, which are the solutions of meteorological significance, are the low frequency motions with phase speeds of the order of ten metres per second and characteristic periods of a few days. The mass and wind fields of these solutions are close to geostrophic balance and they are also called rotational or vortical modes, 135 136 Balance and initialization since their vorticity is greater than their divergence; if divergence is ignored, these modes reduce to the Rossby-Haurwitz waves. For typical conditions of large scale atmospheric flow (when the Rossby and Froude numbers are small) the two types of motion are clearly separated and inter- actions between them are weak. The high frequency gravity-inertia waves may be locally significant in the vicinity of steep orography, where there is strong thermal forcing or where very rapid changes are occurring, but overall they are of minor importance and may be regarded as undesirable noise. A subtle and delicate state of balance exists in the atmosphere between the wind and pressure fields, ensuring that the fast gravity waves have much smaller ampli- tude than the slow rotational part of the flow. Observations show that the pressure and wind fields in regions not too near the equator are close to a state of geostrophic balance and the flow is quasi-nondivergent. The bulk of the energy is contained in the slow rotational motions and the amplitude of the high frequency components is small. The situation was described colourfully by Jule Charney, in a letter dated February 12, 1947 to Philip Thompson: We might say that the atmosphere is a musical instrument on which one can play many tunes. High notes are sound waves, low notes are long [rotational] inertial waves, and nature is a musician more of the Beethoven than of the Chopin type. He much prefers the low notes and only occasionally plays arpeggios in the treble and then only with a light hand (in Thompson, 1983). The prevalance of quasi-geostrophic balance is a perennial source of interest. It is a consequence of the forcing mechanisms and dominant modes of hydrodynamic instability and of the manner in which energy is dispersed and dissipated in the atmosphere. Observations show that the bulk of the energy in the troposphere is in rotational modes with advective time scales. The characteristic time scale of the ex- ternal (solar) forcing is longer than the inertial time scale, so the forcing of gravity waves is weak. Coupling between the rotational and gravity wave components is also generally weak. Gravity waves interact nonlinearly to generate smaller spatial scales, which are heavily damped. They can propagate vertically, with amplitude growing until they reach breaking-point, releasing their energy in the stratosphere. This process is parameterized in many models. However, in the troposphere, grav- ity waves have little impact on the flow and may generally be ignored. Classical reviews of geostrophic motion have been published by Phillips (1963) and Eliassen (1984). The gravity-inertia waves are instrumental in the process by which the balance is maintained, but the nature of the sources of energy ensures that the low frequency components predominate in the large scale flow. The at- mospheric balance is subtle, and difficult to specify precisely. It is delicate in that minor perturbations may disrupt it but robust in that local imbalance tends to be rapidly removed through radiation of gravity-inertia waves in a spontaneous ad- 8.1 Balance in the atmosphere 137 Fig. 8.1. Surface pressure as a function of time for two 24 hour integrations of a primitive equation model. Solid line: unmodified analysis data. Dashed line: data initialized by the nonlinear normal mode technique (from Williamson and Temperton, 1981). justment between the mass and wind fields. For a recent review of balanced flow, see McIntyre (2003). When the primitive equations are used for numerical prediction the forecast may contain spurious large amplitude high frequency oscillations. These result from anomalously large gravity-inertia waves that occur because the balance between the mass and velocity fields is not reflected faithfully in the analysed fields. High frequency oscillations of large amplitude are engendered, and these may persist for a considerable time unless strong dissipative processes are incorporated in the forecast model. It was the presence of such imbalance in the initial fields that gave rise to the totally unrealistic pressure tendency of 145 hPa/6h obtained by Richardson. The noise problem is illustrated clearly in Fig. 8.1, from Williamson and Temperton (1981), which shows the evolution in time of the surface pressure at a particular location predicted by a primitive equation model. The solid line is for an integration from unmodified analysis data. There are oscillations with am- plitudes up to 10 hPa and periods of only a few hours. The greatest instantaneous 138 Balance and initialization tendencies are of the order of 100 hPa/6h, comparable to Richardson’s value.1 The dashed line is for a forecast starting from data balanced by means of the nonlin- ear normal mode technique (NNMI) described in 8.3 below. The contrast with the uninitialized forecast is spectacular: the spurious oscillations no longer appear, and the calculated tendencies are realistic. Balance in the initial data is achieved by the process known as initialization, the principal aim of which is to define the initial fields in such a way that the gravity- inertia waves remain small throughout the forecast. If the fields are not initialized the spurious oscillations that occur in the forecast can lead to serious problems. In particular, new observations are checked for accuracy against a short-range fore- cast; if this forecast is noisy, good observations may be rejected or erroneous ones accepted. Thus, initialization is essential for satisfactory data assimilation. An- other problem occurs with precipitation forecasting: a noisy forecast has unrealisti- cally large vertical velocity that interacts with the humidity field to give hopelessly inaccurate rainfall patterns. To avoid this spin-up, we must control the gravity wave oscillations. 8.2 The slow manifold The slow manifold, introduced by Leith (1980), provides a useful conceptual framework for discussing initialization and balance. The state of the atmosphere ¢ at a given time may be represented by a point ¡ in a phase space . This state evolves in time according to an equation of the form £ ¡¥¤§¦¨¡¥¤ © ¡ (8.1) © where ¦ is a linear operator and is a nonlinear function. The natural oscillations, or linear normal modes, of the atmosphere are obtained by spectral analysis of ¦ . We saw in Chapter 3 that they fall into two classes, so we can partition the state into orthogonal slow and fast components ¡¤ The system (8.1) may then be separated into two sub-systems £ ¤§¦ ¤ ©!" #$%&' (8.2) £ (¤§¦)*(¤ ©)+ #$%&' (8.3) The phase space ¢ is the direct sum of two linear subspaces: ¢,-/.102 1 946587:7 ;4 A gravity wave of amplitude 346587 Pa and period s would yield such a tendency: if BDC¨EGFAHJILKM9DN O PQ;RKAPSI>O TVU WX4YFAHZ3[KM9G\Y7D] ^`_`ab?dcfe 3<>=:?A@ then or about 130 hPa/6h. For a phase speed g 4ih:7:7`jX? kl4mh:7onQj cfe , the wavelength is . 8.2 The slow manifold 139 Z A S N L Y ¡ Fig. 8.2. Schematic diagram showing the slow and fast linear subspaces and and the £¥¤§¦©¨ ¤¦©¨ (nonlinear) slow manifold ¢ . Point represents the analysis. Point is the result of linear normal mode initialization and point ¤¦¨ is the result of nonlinear normal mode initialization (after Leith, 1980). If nonlinearity is neglected, a point in either subspace will remain therein as the 0 flow evolves. This property defines - and as invariant sets for linear flow. We - call - the linear slow subspace. For the nonlinear system, is no longer invariant: © interactions between slow waves generate fast oscillations through the term ) in (8.3). Is there a nonlinear counterpart of the linear slow subspace, that is invariant and contains only slowly varying flows? Leith proposed such a set, which may be thought of as a distortion of - , and he called it the slow manifold. We denote it by and depict it schematically in Fig.