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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.

To meet the challenges posed by future generations of massively parallel supercomputers a reformulation of the dynamical core for the Met Office’s weather and climate model is presented. This new dynamical core uses explicit finite‐volume type discretisations for the transport of scalar fields coupled with an iterated‐implicit, mixed finite‐element discretisation for all other terms. The target model aims to maintain the accuracy, stability and mimetic properties of the existing Met Office model independent of the chosen mesh while improving the conservation properties of the model. This paper details that proposed formulation and, as a first step towards complete testing, demonstrates its performance for a number of test cases in (the context of) a Cartesian domain. The new model is shown to produce similar results to both the existing semi‐implicit semi‐Lagrangian model used at the Met Office and other models in the literature on a range of bubble tests and orographically forced flows in two and three dimensions.

In a recent paper, a conservative semi-Lagrangian mass transport scheme SLICE has been coupled to a semi-implicit semi-Lagrangian scheme for the shallow-water equations. The algorithm involves the solution at each timestep of a nonlinear Helmholtz problem, which is achieved by iterative solution of a linear ‘inner’ Helmholtz problem; this framework, as well as the linear Helmholtz operator itself, are the same as would be used with a non-conservative interpolating semi-Lagrangian scheme for mass transport. However, in order to do this, a reference value of geopotential was introduced into the discretization. It is shown here that this results in a weak dependence of the results on that reference value. An alternative coupling is therefore proposed that preserves the same solution framework and linear Helmholtz operator but, at convergence of the nonlinear solver, has no dependence on the reference value. However, in order to maintain accuracy at large timesteps, this approach requires a modification to how SLICE performs its remapping. An advantage of removing the dependence on the reference value is that the scheme then gives consistent tracer transport. Copyright © 2010 Royal Meteorological Society and Crown Copyright. Published by John Wiley & Sons, Ltd.

When the trajectory equation is discretised in semi-Lagrangian models, the departure points can be spuriously located outside the model domain. Historically this has been addressed in an ad hoc manner by relocating any such departure point to the domain boundary. This procedure, whilst convenient, is not physically justified and is inaccurate. It is of particular concern for mass-conserving, finite-volume, semi-Lagrangian schemes that use the locations of departure points to determine upstream volumes, since it can lead to spuriously massless or near-massless volumes, and large errors in mass transport. This weakness of current discretisations is discussed. An improved near-boundary discretisation of the trajectory equation is then proposed and analysed via an example. Crown Copyright © 2009. Reproduced with the permission of HMSO. Published by John Wiley & Sons Ltd.

For the shallow-water equations on the sphere, an inherently mass-conserving semi-Lagrangian discretisation (SLICE) of the continuity equation is coupled with a semi-implicit semi-Lagrangian discretisation of the momentum equations. Various tests from the literature (two with analytical nonlinear solutions) are used to assess the model's performance and also to compare it with that of a variant model that instead employs a standard non-conserving semi-implicit semi-Lagrangian discretisation of the continuity equation. The mass-conserving version gives results that are overall somewhat better than the non-conserving one.

The dynamical core, which governs the evolution of resolved fluid-dynamical processes, is a critical element of any atmospheric model. Its governing equations must include all relevant dynamical terms, and the numerical formulae used to approximate them must be accurate, stable and efficient. This is particularly so in a unified modeling environment in which the same dynamical core is used for both operational weather prediction and long term climate simulations. Recent research at the Met Office on unified dynamical core issues is reviewed. Aspects covered include: properties of various equation sets; vertical coordinates; semi-Lagrangian advection and conservation; trajectory computation and dynamical equivalence; horizontal and vertical discretization; and coupling of physical parameterizations to a dynamical core.

We present a new two-step temporal discretization of the diffusion equation, which is formally second-order-accurate and unconditionally stable. A novel aspect of the scheme is that it is monotonically damping: the damping rate is a monotonically-increasing function of the diffusion coefficient, independent of the size of the time step, when the diffusion coefficient is independent of the variable being diffused. Furthermore, the damping rate increases without bound as the diffusion coefficient similarly increases. We discuss the nonlinear behaviour of the scheme when the diffusion coefficient is a function of the diffused variable. The scheme is designed to maintain any steady-state solution. We present examples of the performance of the scheme. © Crown Copyright 2007. Reproduced with the permission of the Controller of HMSO. Published by John Wiley & Sons, Ltd.

A regularized time-staggered discretization of the shallow-water equations has recently been proposed. The form of the regularization operator is chosen such that linear equivalence with the semi-implicit method can be achieved. Here, we present a further generalization of the regularization that also takes into account nonlinear balance. The performance of the improved regularization procedure is demonstrated for a simple nonlinear test problem. © Crown Copyright 2007. Reproduced with the permission of the Controller of HMSO. Published by John Wiley & Sons, Ltd.

A recently proposed time-staggered semi-Lagrangian discretization of the unforced rotating shallow-water equations is extended to include orographic forcing. Linear analysis shows that, as for traditional semi-implicit semi-Lagrangian schemes, it is also susceptible to spurious orographic resonance for large Courant numbers. A solution is proposed which, as shown by further linear analysis, addresses spurious orographic resonance whilst also maintaining computational stability. © Crown copyright, 2006. Royal Meteorological Society

A time-staggered semi-Lagrangian discretization of the rotating shallow-water equations is proposed and analysed. Application of regularization to the geopotential field used in the momentum equations leads to an unconditionally stable scheme. The analysis, together with a fully nonlinear example application, suggests that this approach is a promising, efficient, and accurate alternative to traditional schemes. © Royal Meteorological Society, 2006. A. Staniforth's and N. Wood's contributions are Crown copyright material, reproduced with the permission of the Controller of Her Majesty's Stationery Office.

A regularized and time-staggered discretization of the two-dimensional, vertical slice Euler equation set is described and analysed. A linear normal mode analysis of the time-discrete system indicates that unconditional stability is obtained, for appropriate values of the regularization parameters, for both the hydrostatic and non-hydrostatic cases. Furthermore, when these parameters take their optimal values, the stability behaviour of the normal modes is identical to that obtained from a semi-implicit discretization of the unregularized equations. © Crown Copyright 2006. Reproduced with the permission of the Controller of HMSO. Published by John Wiley & Sons, Ltd.

A key aspect of the recently proposed Hamiltonian particle-mesh (HPM) method is its time-staggered discretization combined with a regularization of the continuous governing equations. In this article, the time discretization aspect of the HPM method is analysed for the linearized, rotating, shallow-water equations with orography, and the combined effect of time-staggering and regularization is compared analytically with the popular two-time-level semi-implicit time discretization of the unregularized equations. It is found that the two approaches are essentially equivalent, provided the regularization parameter is chosen appropriately in terms of the time step Δt. The article treats space as a continuum and, hence, its analysis is not limited to the HPM method. Copyright © 2005 Royal Meteorological Society

An important aspect of semi-Lagrangian models is the determination of departure points. This involves the calculation of particle displacements and is done by integrating a discrete form of the trajectory, or kinematic, equation. The way this equation is discretized affects the stability and accuracy of the equation set. Additionally, recent work in the context of a time-centred, semi-implicit, semi-Lagrangian scheme has shown that the form of the discrete kinematic equation reflects the preservation or violation of a dynamical equivalence between momentum and angular momentum formulations. Here that work is extended to the case of off-centred semi-implicit time weighting as used in most semi-implicit, semi-Lagrangian models. Two types of discretization of the trajectory equation are identified which preserve dynamical equivalence: some existing discretizations are found to be approximations to them. It is also shown how to incorporate physical forcings and/or predictor–corrector dynamics into the formulation. Two simple model problems (for solid-body rotation and wave motion) are used to provide some further insight into the accuracy and stability properties of various trajectory schemes, including: dynamically-equivalent schemes, approximations to these, and several existing schemes. Copyright © Crown copyright, 2003. Royal Meteorological Society

Recently an inherently mass-conserving semi-Lagrangian transport scheme has been successfully coupled to an iterative semi-implicit scheme in a global shallow-water-equation (SWE) model. Here that methodology is extended and applied to an iterative semi-implicit semi-Lagrangian (SISL) compressible, non-hydrostatic vertical-slice model, in which the constant reference state of the SWE model is now replaced by a vertically varying reference state. An advantage of this approach is that it preserves the same basic structure as the interpolating, non-mass-conserving, iterative SISL model. The resulting mass-conserving model is applied to a standard set of test problems for such models and compared with results from both the literature and the interpolating, iterative SISL version of the model. © Crown Copyright 2010. Reproduced with the permission of HMSO. Published by John Wiley & Sons, Ltd.