Uncertainties in Numerical Weather Prediction

Uncertainties in Numerical Weather Prediction is a comprehensive work on the most current understandings of uncertainties and predictability in numerical simulations of the atmosphere. It provides general knowledge on all aspects of uncertainties in the weather prediction models in a single, easy to use reference. The book illustrates particular uncertainties in observations and data assimilation, as well as the errors associated with numerical integration methods. Stochastic methods in parameterization of subgrid processes are also assessed, as are uncertainties associated with surface-atmosphere exchange, orographic flows and processes in the atmospheric boundary layer.

Through a better understanding of the uncertainties to watch for, readers will be able to produce more precise and accurate forecasts. This is an essential work for anyone who wants to improve the accuracy of weather and climate forecasting and interested parties developing tools to enhance the quality of such forecasts.

• Provides a comprehensive overview of the state of numerical weather prediction at spatial scales, from hundreds of meters, to thousands of kilometers
• Focuses on short-term 1-15 day atmospheric predictions, with some coverage appropriate for longer-term forecasts
• Includes references to climate prediction models to allow applications of these techniques for climate simulations

• Language: English
• Publisher: Elsevier Science
• Release date: Nov 25, 2020
• ISBN: 9780128157107

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Preface

Haraldur Ólafsson

Jian-Wen Bao

This book project was motivated by our observation that on the one hand it has become widely accepted that numerical weather prediction (NWP) has uncertainties, but on the other hand, the main-steam textbooks rarely go deep to discuss in detail where the uncertainties lie. Most freshly-graduated students may even feel reluctant to accept the notion that there is a degree of empiricism in the development of NWP models. Such empiricism is required in NWP model development because there remain questions unanswerable using current theories and observations about the actual dynamical and physical processes in nature that govern the evolution of weather-related atmospheric circulations. There is a gap between what is being taught in schools about the uncertainty in NWP associated with empiricism and the reality in the operational field of numerical weather prediction. This book is an experimental effort to fill the gap by inviting actual practitioners of NWP model development and applications to contribute in-depth overviews of the real uncertainties in NWP models.

The book aims to present a comprehensive overview of some key issues in numerical weather prediction with emphasis on uncertainties and aspects of predictability of elements of the atmosphere. The book starts with an introduction to the governing equations and to numerical methods applied to solve these equations by discretization in time and space. The perspective of improvements in solving the dynamic equations is discussed in general and in particular concerning the development of supercomputing techniques. As a continuation of Chapter 1, Chapter 2 discusses uncertainties arising from the discretization of the shallow-water equations and addresses numerical techniques to maintain stability and increasing calculation efficiency. In Chapter 3, the governing equations are revisited, and the elements of a numerical weather prediction system are explained. There is a discussion of errors and chaos and the concept of probabilistic weather prediction is introduced. The perturbation of initial conditions is explained, leading the reader to ensemble forecasting and presentation of products of probabilistic forecasts. Chapter 4 continues in the line of probability. It addresses the sources and the growth of errors, limits of prediction, and the dependence of predictability on the scales at which the atmospheric processes occur. One of the large uncertainties of numerical weather prediction stems from water in the atmosphere. Chapter 5 presents perspectives of the parameterization of unresolved moist processes and interaction of physics and dynamics, waves at different scales, turbulent mixing, convection, and radiation.

Observations used in the initialization of NWP models have both instrumentational and representational uncertainties. Chapter 6 provides a theoretical overview of how to deal with these uncertainties in the process of data assimilation used to initialize NWP models. Chapter 7 provides an overview of the treatment of subgrid turbulent mixing in NWP models and the associated uncertainty in the treatment due to the very fact that the modeling of turbulence remains semiempirical. The uncertainty in the lower boundary conditions of NWP models at the earth’s surface is concisely but pointedly discussed in Chapter 8. The ultimate energy source of the atmospheric motion is solar radiation. Chapter 9 describes the processes governing the radiation-atmosphere interaction and their uncertainty related to the uncertain aspect of the cloud modeling in NWP models. Chapter 10 further discusses the uncertainty in the process modeling of cloud microphysics that is widely used in NWP models.

Chapter 11 presents the main patterns of orographic flows, with emphasis on mesoscale structures, referring to spatial scales and the governing momentum equation. The patterns are presented concerning a mountain wind diagram, based on dimensionless parameters. The challenge of forecasting the individual patterns is addressed together with an assessment of flow pattern related uncertainties.

NWP practitioners need to understand the degree of uncertainty in the product of NWP. Chapter 12 introduces a few numerical methods for quantifying the uncertainty.

The book concludes with Chapter 13, presenting a dynamically based method of analyzing and tracing errors in numerical models of the atmosphere. The method is applied to various cases of erroneous forecasts of mean sea level pressure, surface winds, and precipitation.

We wholeheartedly thank all the contributors to this book, and it is our sincere hope that it will be of good use and enjoyment to research students and scientists in the flourishing field of atmospheric sciences.

Chapter 1: Dynamical cores for NWP: An uncertain landscape*

Nigel Wood    Met Office, Dynamics Research, Exeter, United Kingdom

Abstract

In an numerical weather prediction (NWP) model, the complex interplay, in a rotating frame of reference, between the winds, the highs and lows of pressure, and the transport of mass and thermal energy are the responsibility of the dynamical core.

A recent review lists 26 existing NWP dynamical cores. So why, after more than 50 years of operational NWP, does there remain such a plethora of dynamical cores with little hint of convergence in the choice of which schemes to use? Indeed, as the architectures of supercomputers are on the brink of changing significantly, what convergence there might have been seems set to dissipate. The same review lists 28 models under development.

By exploring some of the physical properties of the equations that lie at the heart of the dynamical core and revealing the challenges for the numerical schemes used to discretize those equations, this chapter attempts to uncover some of the reasons for this uncertain landscape in such a critical component of NWP. In particular, the chapter introduces and discusses in a little detail a variety of different numerical approaches to discretizing the equations in both the temporal and spatial dimensions. Discussing these in terms of their accuracy and numerical stability, as well as aspects of their efficiency on supercomputer architectures, it hopefully emerges why there is not a dominant strategy for designing a dynamical core.

Keywords

Dynamical core; Temporal discretization; Spatial discretization; Semi-Lagrangian; Finite difference; Finite element; Finite volume; Numerical stability; Numerical accuracy; Grids

Acknowledgments

The author would like to thank Christine Johnson of the Met Office for her careful reading of an early version of this chapter, which led to many improvements. It is also his pleasure to acknowledge the insights patiently and generously shared with him by Andrew Staniforth, Andy White (both formerly of the Met Office), and John Thuburn (University of Exeter).

1: Introduction

The dynamical core of a numerical weather prediction model is that part of the model that is responsible for the fluid-dynamical processes that can be represented (with whatever degree of accuracy) by the spatial and temporal discretization of the governing equations. This is distinct from the physics packages, which represent the effects of nonfluid-dynamical processes, such as radiation, and of fluid-dynamical processes that occur at scales that cannot be resolved by the smallest scales represented by the discretized equations, typically such as boundary-layer turbulence.

The first operational numerical weather forecast was created in the summer of 1954 at the Swedish Hydrometeorological Institute, followed less than a year later by the US Joint Numerical Weather Prediction Unit (e.g., Persson, 2005). Creating those forecasts required enormous amounts of insight into the underpinning physics of the problem and, using that insight, ingenuity as to how to derive equations that could be successfully solved using the limited and emerging computer resources available. Those first models were based on manipulations to, and approximations of, the inviscid, Euler equations in a rotating frame of reference. Those first models neglected the effects of unresolved processes and so represent what will be referred to as an integration of a dynamical core, that is, the inviscid, energy-conserving aspects of the complete system. Over the last 60 years, the development of the dynamical core parts of numerical weather (and climate) prediction models can be regarded as the progressive relaxation of the approximations made in those early days. This has been made possible principally by the exponential increase in the speed of computers as well as improved methods of deriving the initial conditions for those models; in a sense the cunning of numerical modelers has been replaced by the cunning of computational scientists! However, we are perhaps at the end of that journey and need again to explore how to squeeze the most out of a limited resource.

Initially, of course there were only a very small number of different models. As time progressed that number increased as differences in both approach (numerical scheme and level of approximation) and application (weather, climate, global, and regional) increased. But as the number of approximations applied has reduced and the understanding of the numerical methods available has increased, one might have anticipated that the number of different models would now be reducing. That is not the case. An indication of this is given by Tables 1 and 2 of Marras et al. (2016) in which are listed, respectively, 26 existing NWP systems and 28 that are under development. In fact, in the early decades of the 21st century, we seem to be in an era where the number of approaches and hence different models appear to be increasing. The following sections are aimed at giving a potential architect of a new dynamical core an idea of the range of choices that are available and that need to be made. From this, it will perhaps become clear why we appear to not yet be converging on one approach to modeling of the dynamical core.

The discussion covers a range of topics each of which is covered in a mostly qualitative, nonanalytical manner, with the intention of giving a broad overview of the issues involved rather than specific detail and derivation. The interested reader is referred to any of the excellent text books available that cover each of the topics in much more analytic detail. Two of particular relevance to this topic are Durran (2010) and Lauritzen et al. (2011).

2: Governing equations

The unapproximated Euler equations can be written symbolically in terms of the physical terms. Starting with the wind field:

•Horizontal winds:

(1)

•Vertical winds:

(2)

The earliest dynamical cores applied four key approximations to these equations. The first is the geostrophic approximation. This assumes that the Coriolis terms are in exact balance with the pressure gradient terms and, therefore, neglect the tendency and transport of the horizontal winds (the left-hand side of Eq. 1). Subsequent models relaxed this by allowing for some aspects of the transport term: the quasigeostrophic models allow for transport of the geostrophic wind by the geostrophic wind and semigeostrophic models allow for transport of the geostrophic wind by the full wind. But no contemporary NWP models make any of these approximations; they solve the full Eq. (1).

The second is the shallow-atmosphere approximation. This assumes that the depth of the atmosphere is small compared to the radius of the Earth. This allows certain simplifications in the equations to be made. Many models still make this approximation (together with the traditional approximation discussed later) though there are exceptions such as the series of dynamical cores used in the Met Office's Unified Model (White and Bromley, 1995; Davies et al., 2005; Wood et al., 2014).

The third is the hydrostatic approximation. This neglects the tendency and transport of the vertical component of the wind. Models employing this approximation often, but not always, also apply the shallow-atmosphere approximation (and the resulting equations are termed as the primitive equations). In this case, it is assumed that the vertical pressure gradient balances the acceleration due to gravity. When the shallow-approximation is not made, the hydrostatic approximation is referred to as the quasihydrostatic approximation. The hydrostatic approximation is only valid for horizontal scales that are significantly longer than vertical scales (see, e.g., Holton, 1992; Davies et al., 2003). Therefore, as horizontal resolution continues to increase, most new model formulations are not hydrostatic (but it is notable that the European Center's model remains hydrostatic despite running very successfully at the highest, global, operational resolution).

The fourth is the traditional approximation. White et al. (2005) discuss the application (or not) of the shallow and hydrostatic approximations and associated issues of consistency of the equation sets with various physical conservation principles (namely conservation of mass, energy, axial angular momentum, and potential vorticity). In particular, the authors found that when the shallow approximation is made then the Coriolis terms also need to be approximated in order for the equations to retain the principle of conservation of axial angular momentum. Specifically, axial angular momentum is conserved if the horizontal component of the Earth's rotation vector is neglected, meaning that the Coriolis term in Eq. (2) vanishes completely and the term in Eq. (1) is modified. This approximation is referred to as the traditional approximation and has generally been made in conjunction with the shallow-atmosphere approximation. It followed that there were four choices of consistent equation sets, determined by whether each of the hydrostatic and shallow approximations are or are not made independently (with the traditional approximation being made when the shallow approximation is made).

However, there has been something of a revival of interest in this area in the recent years with a number of recent papers revisiting and generalizing this work such as Charron and Zadra (2014) and Staniforth (2014a). In particular, Tort and Dubos (2014) showed that consistency can in fact be achieved while retaining a horizontal component of the Earth's rotation vector, albeit in an approximate form. This means that there are then six consistent equation sets, determined by whether the hydrostatic approximation is or is not made, together with one of: deep atmosphere and full Coriolis terms; shallow-atmosphere and traditional approximation; and shallow-atmosphere and approximated full Coriolis.

For thermodynamic variables, the equations are simpler, all having the form:

   (3)

This equation (and its simplified form in which the divergence term is not present) applies to any of the thermodynamic variables such as temperature, T, pressure, p, and density, ρ and any function of any of these, for example, Π, θ, etc., where θ denotes the potential temperature and Π is the Exner function. These are related to each other by the equation of state

   (4)

where R is the specific gas constant, and the definitions

   (5)

where p0 is a reference pressure, typically taken to be 1000 hPa and κ ≡ R/cp with cp the specific heat capacity of air at constant pressure, and

   (6)

The equations have been given symbolically at this stage to avoid having to make any specific choices about exactly which variables are used in the numerical implementation of the equations. For the analyst working with the continuous equations, the choice of variables is one of convenience, the solution is independent of how the equations are written. However, numerical modelers and discrete analysts do not have this luxury; when the equations are discretized only a (small) finite number of the properties of the continuous equations are preserved by any particular discretization. And, the properties that are preserved are strongly related to the choice of how the discrete equations are written. So, to tie down exactly the design of the dynamical core, a choice has to be made for how the winds will be represented and which thermodynamic variables will be used as prognostic variables.

For the winds (and for the moment assuming the horizontal surface is flat), there are broadly two choices:

•The components of the velocityrepresent the horizontal components (perpendicular to gravity) and w represents the vertical component parallel to gravity.

•The components of the momentum. These are simply the components of the velocity multiplied by the density, ρ.

The advantage of using the momentum is that it simplifies the pressure gradient term (it is a linear term involving only the pressure) and it allows the transport terms to be written in a form that allows exact numerical conservation of linear momentum. The latter property is useful for local regional models for which linear momentum is a natural quantity. This form is used, for example, in the Weather Research and Forecasting model (see Skamarock et al., 2019 and also the WRF web-based resources at https://doi.org/10.5065/D6MK6B4K). However, on the sphere, it is axial angular momentum that is the conserved quantity. Although some consideration to using angular momentum has been given, the author is not aware of any model that uses this for the wind prognostic variables. The disadvantage of using momentum is that the velocity needs to be diagnosed from it to evaluate the transport term. How much of an issue this is depends on the specific discretization used.

The advantage of using velocity is that this is directly the quantity needed to evaluate the transport. This enables the use of the semi-Lagrangian scheme, of which more later, and it also enables the equations to be written directly in vector form (often referred to as the vector invariant form), which avoids the need for the explicit inclusion of potentially complicated terms that are specific to the specific choice of coordinates used. The disadvantages of the velocity form are a nonlinear pressure gradient term, that is, one that involves the product of the gradient of the pressure variable (e.g., p or Π) with another thermodynamic quantity (e.g., 1/ρ or θ), and the lack of a straightforward way of obtaining discrete conservation of linear, and angular, momentum.

represent (other than choices of scaling factors). When the coordinates are not orthogonal, though, there are further choices to be made about what wind components to use. The two natural ones are: the contravariant components in which the wind vector is expressed in terms of base vectors that are parallel to the coordinate lines and the covariant components in which the base vector of one coordinate direction is perpendicular to the plane defined by the other coordinate directions. However, in meteorology, a third alternative is often used. When the Earth's surface is not flat because of orography (hills and mountains), the vertical coordinate is often chosen such that, near the surface at least, surfaces of constant vertical coordinate are parallel to the orography. Such a coordinate is termed a terrain-following coordinate, for example, that of Gal-Chen and Somerville (1975). Although either the contra- or covariant components could be used, Clark (1977) found that good numerical conservation of linear momentum and energy was much easier to achieve if a set of orthogonal base vectors are used to describe the wind field, with the vertical base vector parallel to the gravity vector, and this is a common practice in many contemporary NWP models (e.g., Wood et al., 2014).

It is worth noting that, for hydrostatic models (for which the vertical velocity is not a prognostic variable), there is a third choice for the wind variables. In place of the two horizontal components of velocity, this approach uses the vertical component of vorticity and the divergence of the horizontal wind (i.e., the horizontal wind is split into its rotational and nonrotational components). This is a particularly attractive choice for models that are focused on the large-scale dynamics. Due to the thermal stratification of the atmosphere, the large-scale dynamics are dominated by the effects of the planetary rotation which are captured by the evolution of the vertical vorticity, and in particular the potential vorticity which judiciously combines the vorticity with the potential temperature (the interested reader is referred to any books on dynamical meteorology such as Holton, 1992 and Vallis, 2006). The divergence of the horizontal wind plays an essential role in the dynamics of the gravity waves (and neglecting the tendency of the divergence is a way of filtering out the gravity waves). However, in such models, the wind components are still required in order, for example, to evaluate how the vorticity is transported. In a vorticity-divergence-based model, these are obtained by inverting the relations between the winds and the vorticity and divergence. These operations are expensive for all but spectral models (for which the cost is effectively already paid for in the Fourier and Legendre transformations). For nonhydrostatic models, for which the interest includes much smaller scales than just planetary scales, the vertical velocity is a prognostic variable and all three components of the vorticity become relevant. The split into the vertical vorticity and divergence becomes a much less natural choice.

For the thermodynamic variables, there is a large range of choices. Essentially, any two independent choices out of T, p, ρ, and any function of any of these, for example, Π, θ, etc.

, where z denotes height above the surface. Together with the equation of state and the assumption that the atmosphere is in hydrostatic balance the vertical variation of all the other thermodynamic variables can then be obtained. It is found that

and

These all, therefore, decay exponentially (except for θ which increases exponentially) with the exponent of Π and θ being rather smaller (slower decay/increase) than that for p and ρ by the factor κ, which has the value 2/7 for ideal dry air. The scale height over which p and ρ decay is given by RT0/g which has a typical value of 7 km.

What then of the choices for thermodynamic variable?

An important question is whether the dynamical core should preserve the mass of the atmosphere. If it is required to do so, then one of the prognostic thermodynamic variables needs to be density and the density needs to be evolved using a numerically conservative scheme. If a quantity other than a linear function of density is used, or if a nonconservative numerical scheme is used, then mass conservation is only achieved by some form of global fixer that adds or removes the requisite amount of mass to restore the total mass. Such schemes inevitably lead to varying degrees of nonphysical transport of mass. An early example of a scheme that tried to localize that transport as much as possible is the scheme of Priestley (1993) with recent variants on that by, for example, Zerroukat and Allen (2015).

If density is not chosen as a prognostic variable, then a natural choice for one of the thermodynamic variables is pressure, either directly as p or as Π .

There seem to be two alternative routes to follow for which other variable to choose. Numerical schemes are most accurate when applied to quantities that vary smoothly. It has already been noted that temperature does not vary dramatically with height so this might be a natural choice that avoids the exponential changes. The other is potential temperature, θis proportional (in a dry atmosphere) to the entropy of the system, it is preserved in the absence of any diabatic processes, that is, the divergence term in Eq. (3) vanishes for θ and it is purely transported. The disadvantage is the exponential variation of θ , which varies in a manner closer to temperature.

It is worth noting that a tempting choice is to use ρθ as the transported variable since the equation for its evolution can be written in conservative form (i.e., a form that a numerical scheme can straightforwardly ensure the global integral of ρθ is exactly conserved). However, it is important to note that from Eqs. (4)–(6)ρθ is a function only of p (or equivalently Π). Therefore, the only remaining choice of independent thermodynamic variable would be a function only of T.

These choices have to be considered also in a broader sense, in terms of what impact they have on the design as a whole. A particular aspect is the form of the pressure gradient term in Eqs. (directly (where Grad denotes evaluation of the spatial gradient of its argument). Therefore, use of p might be more natural as one of the thermodynamic variables rather than using a derived quantity such as Π , both of which would reintroduce an unnecessary nonlinearity into the scheme. If velocity is used, then Pressure Gradient = ρ. Then, perhaps ρ and p are natural choices. However, using the equation of state this can also be written as either RTor cpθ. The choices seem to be increasing not reducing! However, help is at hand.

It will be seen below that an important role of the dynamical core is the propagation of a variety of different oscillatory modes, or waves. The seminal work of Thuburn and Woollings (2005) showed that the choice of thermodynamic variables (and how to represent them spatially) plays a critical role in how well the dynamical core represents those waves, and hence how accurate the scheme is. In particular, the authors identified (for a given choice of vertical coordinate) a single choice of thermodynamic variable that gave what they described as the optimal configuration. In a follow-up piece of work, Thuburn (2006) showed how, by using specific numerical forms of the pressure gradient, some of the suboptimal configurations can in fact be made to be optimal. Depending on one's perspective, this is either unfortunate (it does not help narrow the possibilities!) or fortunate (it leaves open more possibilities!).

It is clear that before one even gets close to defining a numerical scheme for a dynamical core, there is a very large range of options for the choice of continuous equation set and its constituent variables. To close this section, it is worth giving some examples of the range of choices that have been made in widely used and widely known models:

•The operational dynamical core of ECMWF's IFS model is a global NWP model (see ECMWF's web-based documentation at https://www.ecmwf.int/en/publications/ifs-documentation). It makes the traditional, shallow-atmosphere, and hydrostatic approximations. It uses a pressure-based vertical coordinate. The wind is represented using velocity components. Its two thermodynamic variables are temperature and geopotential height (a measure of the height associated with a given value of pressure).

•The WRF dynamical core (Skamarock et al., 2019) is principally used as a regional NWP model (but it does have a global capability). It makes a form of the shallow approximation but not the traditional nor hydrostatic approximations. It uses a pressure-based vertical coordinate. The wind is represented using momentum components. It predicts three thermodynamic variables: density and density-weighted potential temperature (both in a conservative form), together with the geopotential height which is used in the pressure gradient term. (The use of three prognostic thermodynamic variables means that instead of the equation of state being used to eliminate one of the thermodynamic variables from the equation set, or being used as a constraint on the evolution of the thermodynamic variables, its derivative in time is used to create an additional prognostic equation.)

•The dynamical core of the Met Office's Unified Model (e.g., Walters et al., 2017) is both an operational global and operational regional NWP model. It makes neither the shallow-atmosphere, the traditional, nor the hydrostatic approximations. It uses a height-based vertical coordinate. The wind is represented using velocity components. Its two thermodynamic variables are density (but not in a conservative form) and potential temperature. However, the pressure gradient term is written in terms of the Exner function, which is derived from density and potential temperature using the equation of state.

3: Some physical properties

Before considering what numerical schemes to apply to the governing equations, it is important to have a clear idea of the nature of the physical phenomenon represented by those equations. There are perhaps three key properties of the equations represented by Eqs. (1)–(3). Two of them are related: the equations are energy preserving (they can be derived from Hamilton's principle, e.g., Salmon, 1988; Shepherd, 1990; Staniforth, 2014a); and they constitute a hyperbolic system of equations which means that they admit wave-like solutions. The third property, and one that sets them apart from many other similar equation sets, is that the presence of rotation means that the equations admit nontrivial steady states, in particular steady states with nonzero wind fields. The large-scale states of the atmosphere can often be regarded as the evolution from one near-steady state to another. Further, the transition from one near-steady state to another is achieved by the wave-like solutions to the equations. Therefore, it is essential for the accuracy of a dynamical core (a) that those steady states are well represented and maintained by the numerical schemes employed and (b) that the adjustment by the waves is well captured by the numerical schemes.

The leading order steady states have already been discussed. In the horizontal, it is geostrophic balance which can symbolically be represented as

   (7)

Since the Coriolis terms act perpendicular to the flow direction, geostrophic balance requires the wind field to be perpendicular to the pressure gradient term so that flow is parallel to lines of constant pressure.

In the vertical, it is hydrostatic balance which can be represented as

   (8)

where the traditional approximation has been assumed so the Coriolis effect on vertical motion has been neglected. If the flow is in both geostrophic and hydrostatic balance, then there is also thermal wind balance which requires that any vertical shear in the horizontal wind must be balanced by a horizontal gradient of temperature perpendicular to the wind direction.

Wave motions form when the inertia of an air parcel is opposed by a restoring force. By considering the governing equations in the presence of perturbations to a geostrophically and hydrostatically balanced state, it is found (see, e.g., the comprehensive text books of Gill, 1982, Holton, 1992, and Vallis, 2006) that the five equations (three for the wind components and two for each of the chosen thermodynamic variables) support five modes of oscillations, as three different types of wave:

•One Rossby wave. This is due to the variation of the Coriolis effect with latitude (there needs to be a horizontal gradient in the background potential vorticity field). Rossby waves have a propagation speed, relative to the mean wind, that is close to zero. Indeed, on an f-plane, for which the Coriolis term is constant, the Rossby wave is purely transported by the mean wind, that is, it is stationary relative to the mean wind. Rossby waves are the most important waves for determining the large-scale evolution of the weather and are predominantly responsible for the large-scale patterns seen at mid-latitudes in satellite images.

•Two gravity waves. These are due to the buoyancy created by the effects of gravity (there needs to be a vertical gradient in the background potential temperature field). They propagate anisotropically and come in pairs that propagate in opposite directions. They are important for the adjustment of flow due to the presence of orography and strong convective disturbances. They have a propagation speed determined by the stratification of the atmosphere. A typical speed might be ±50 ms−1 (relative to the wind speed). However, for the deepest modes that extend through a significant proportion of the troposphere, the maximum propagation speed is around 320 ms−1, similar to the speed of sound.

•Two acoustic waves. These are due to the compressibility of the atmosphere. They propagate isotropically and come in pairs that propagate in opposite directions. The direct effect of acoustic waves on weather forecast models is negligible. Indeed, the hydrostatic approximation filters out all acoustic modes except for one pair of modes that propagate purely horizontally. They appear in NWP models effectively as a by-product of not wanting to make the hydrostatic approximation because of how that approximation distorts the propagation of the gravity waves. (Acoustic waves can also be filtered by removing, or at least approximating, the compressibility of the atmosphere by making the anelastic approximation. However, this is of questionable accuracy for the largest scales. See Davies et al. (2003) for further discussion.) The speed of sound varies with height and temperature but according to the US Standard Atmosphere, it varies between a maximum of 340 ms−1 (relative to the wind speed) near the surface and a minimum of around 275 ms−1 at heights around 80–90 km.

Hence, a dynamical core has to handle transport by the mean wind as accurately as possible, while representing both accurately and stably the propagation of gravity waves. Unless the equation set has been appropriately approximated, the dynamical core also has to handle stably the propagation of the acoustic waves but the accuracy with which it does this is of secondary importance.

A further important aspect to consider for any system before trying to model it is the energy spectrum of its motions, that is, how much energy there is in a given range of spatial (usually horizontal) scales. Various efforts have been made to observe the atmosphere's spectrum, the most famous being the work of Nastrom and Gage (1985). The spectrum is important in deciding what range of scales it is important for a model to resolve. It is clearly important to capture accurately those scales where there is the most energy as those are likely to be the most important. In the atmosphere, these scales are very large, of the order of thousands of kilometers. There will also generally be a length and/or a time scale below which the flow has virtually no energy. These scales then place a lower bound on the space and/or time scales that it is sensible for a model to resolve. In the atmosphere, it is the molecular viscosity that ultimately puts a lower bound on the scales of motion; kinematic molecular viscosity of air has a value that is of order 10−5 m² s−1. As a result, a complete representation of the atmosphere would have to represent all scales from the molecular (at most of order millimeters) to the planetary (of order 10,000 km in the horizontal), that is, scales covering at least eight orders of magnitude. This is, and will for a long time, remain beyond the reach of supercomputers. Therefore, simulations have to parameterize the effects of motions below some nominally small scale. It is, therefore, important to understand how energy flows from one range of scales to another. For example, if energy only flows from large scales to small scales (called a downward cascade), then the small scales are said to be slaved to the larger ones and parameterizing the effects of the small scales in terms of the larger ones is sensible. It turns out that to a large extent this is what happens in the atmosphere and it is this that has allowed accurate weather predictions with relatively limited computational resource. But as ever more accurate forecasts are sought the fact that there is some upward energy cascade from small scales to larger scales starts to become important. It also turns out (Holdaway et al., 2008) that the slope of the spectrum can influence, and even determine, the rate at which numerical methods converge to the correct answer; if the slope of the spectrum is in some sense shallow then increasing the accuracy of the numerical scheme will not necessarily improve the accuracy of the results—it is, in that case, more effective to improve the resolution of the