Earth System Modelling - Volume 2 pp 19-24 | Cite as

# Numerical Algorithms for ESM: Future Perspectives for Atmospheric Modelling

## Abstract

In the past two decades, a number of factors have reenlivened the debate on the optimal numerical techniques for the highly demanding tasks of climate simulation. The increasing amount of computer power available has made viable options that would have appeared unfeasible not long ago. This growth in computational power has also been accompanied by major changes in computer architecture, such as the development of massively parallel computers. Therefore, great emphasis has been placed on the application of highly scalable techniques that can employ most efficiently large numbers of relatively small sized CPUs.

## Keywords

Spectral Element Spectral Element Method Primitive Equation Model Optimal Numerical Technique High Order Finite ElementIn the past two decades, a number of factors have reenlivened the debate on the optimal numerical techniques for the highly demanding tasks of climate simulation. The increasing amount of computer power available has made viable options that would have appeared unfeasible not long ago. This growth in computational power has also been accompanied by major changes in computer architecture, such as the development of massively parallel computers, discussed in Chap. 8. Therefore, great emphasis has been placed on the application of highly scalable techniques that can employ most efficiently large numbers of relatively small sized CPUs. The growth in computer power has also implied the possibility of achieving increasingly high spatial resolutions, which exposed some limitations of the basic discretization approaches reviewed in Chap. 2.

In particular, the search for alternatives to uniform Cartesian meshes for global atmospheric modelling has been one of the main motivations for research in this field. We will review here several of the finite element and finite volume approaches that have been proposed, which represent attractive possibilities for the future development of ESM. In particular, convincing high order alternatives to global spectral transform methods have emerged, based on spectral element or Discontinuous Galerkin approaches. Furthermore, traditional energy and enstrophy preserving methods have been extended successfully to quasi-uniform triangular and hexagonal meshes. Conservative versions of the semi-Lagrangian (SL) method have been developed that address the issue of the increasing computational burden of chemistry and biogeochemistry computations, whose full coupling to the hydrodynamical cores constitutes one of the main goals of the future ESM. Finally, the correct simulation of the interplay between adiabatic atmospheric dynamics and more complex physical processes is being devoted increasing attention.

The spectral transform method has been at the basis of the atmospheric components of most leading ESM. However, its potential limitations for high resolution simulations and massively parallel hardware have been highlighted already long ago. Some attempts have been made to further increase such efficiency by eliminating the need for Legendre transforms in the so called Double Fourier series approaches, see e.g. Cheong (2000, 2006), but these techniques require the use of uniform latitude longitude meshes and the application of some type of polar filter. Furthermore, advanced implementations of alternative discretization approaches such as the one presented by Tomita et al. (2008) show clearly that the efficiency of spectral transform models can be matched by gridpoint methods and that further increases in model resolution might actually lead to an inferior efficiency of the spectral transform approach.

The need to solve the pole problem and to achieve a more uniform gridpoint distribution over the sphere has been a relevant research topic in atmospheric modelling since the late 1960s. Several approaches have been studied in recent years which appear to constitute feasible alternatives to uniform latitude longitude grids. One such approach is the so called *cubed sphere* projection technique, which was originally proposed by Sadourny (1972). This technique is based on the non conformal projection of a spherical surface onto the faces of a cube. This results in describing Earth’s surface by means of six planar meshes, which are connected by appropriate flux matching and numerical discretizations along the boundaries. Some of the problems in the original Sadourny approach were overcome in the further developments of Rancic et al. (1996) and Ronchi et al. (1996) and a global, ocean and atmosphere coupled circulation model based on this type of grid has been proposed by Adcroft et al. (2004).

Discretization grids obtained by inscription of the regular icosahedron in the sphere have also been widely investigated, since, apart from allowing for a quasi-uniform coverage of the sphere, their hierarchical structure provides a natural setting for multigrid and multiresolution approaches. A complete review of the early literature on this topic can be found in Williamson (1979). The attention devoted to these alternative meshes was indeed intertwined with the search for numerical methods possessing special discrete conservation properties, some of the best known of these methods being those published by Sadourny (1975) and Arakawa and Lamb (1981). Albeit dormant for more than a decade, interest in quasi-uniform triangular and hexagonal meshes on the sphere revived since it was shown in Heikes and Randall (1995) that proper numerical handling of gravity waves is feasible on these grids even by low order, finite volume based methods. This led to a number of further developments both in idealized studies and practical applications. It was shown by Ringler et al. (2000) that dynamical cores built with this techniques yield results comparable to those of other techniques on standard idealized climate dynamics tests, which led to the development of new generation ECMs based on this discretization approaches at the Colorado State University. Other discretization approaches based on these grids were proposed by Giraldo (1998), Thuburn (1997), and Giraldo (2001). The German Weather Forecasting Service has developed a hydrostatic primitive equation model based on this grid for its operational global forecasting (Majewski et al. 2002). In Tomita et al. (2001), an alternative discretization approach based on icosahedral meshes was proposed that was then used at the Frontier Research System for Global Change in Japan as the core of a nonhydrostatic global model (Tomita and Satoh 2004). Extensions of classical energy and/or enstrophy preserving numerical schemes to these meshes have been proposed by Ringler and Randall (2002), Bonaventura and Ringler (2005) and Ringler et al. (2010), which appear to be reproducing correct energy and enstrophy spectra in long term idealized simulations.

An intense research activity has also been aimed at the development of more local and more fully scalable high order discretization approaches. Spectral element formulations were first applied to the shallow water equations by Taylor et al. (1997), thus leading the way to a number of studies in which high order finite element techniques, that had been previously exclusively studied in a more classical CFD and engineering context, have been applied successfully to geophysical atmospheric flows. In particular, other spectral element methods were proposed by Giraldo (1998), Giraldo (2001), Thomas and Loft (2002), and Giraldo et al. (2003) in the simplified context of the shallow water equations and Discontinuous Galerkin methods were proposed by Giraldo et al. (2002) and Nair et al. (2005). A spectral element primitive equation model was presented by Giraldo and Rosmond (2004), showing that these techniques can cope accurately also with complex baroclinic flows.

Another important issue for future ESM is related to one of the most basic problems in computational fluid dynamics: the solution of advection diffusion equations. Atmospheric chemistry, aereosol and ocean biogeochemistry models require the solution of one advection—diffusion—reaction equation for each chemical or biological species involved. The number of these species can be quite high, if a detailed description of the processes of interest has to be achieved. Thus, they make up for an increasingly large share of ESM computational cost and very efficient numerical methods are crucial to perform long range simulations effectively.

The SL method is one of the most efficient and accurate advection schemes. Thanks to its very weak stability restrictions, it allows to use much longer time steps than standard Eulerian methods. However, all its earlier formulations have been intrinsically non conservative, thus presenting a serious problem for application to climate simulations. Starting with the work of Laprise and Plante (1995) on mass integrated SL methods and with the flux form SL techniques of Leonard et al. (1996) and Lin and Rood (1996), it has been shown how fully conservative SL methods can be derived. This has led to a series of new proposals that appear very promising for future ESM developments. The methods presented in Nair and Machenhauer (2002), Nair et al. (2002), Zerroukat et al. (2004), and Zerroukat et al. (2005) develop further the integrated mass approach. On the other hand, in Lipscomb and Ringler (2005) an extension of the flux form SL schemes to hexagonal meshes was proposed, while in Restelli et al. (2006) a combination of SL and Discontinuous Galerkin is proposed which allows naturally for high order extensions and *p*- adaptivity on arbitrary unstructured meshes. All these methods yield monotonic solutions, which is also extremely important in chemistry and biology computations.

Finally, the way in which the coupling of pure atmospheric dynamics and parameterised physical processes is realized in ESM has recently been analyzed in a much more careful way than in the previous decades of Earth System numerical modelling. This aspect of ESM is clearly essential for a correct reproduction of realistic atmospheric and oceanic circulations and climate scenarios. At the same time, as an increasingly detailed description of many physical processes is being included and, as the model resolution also keeps increasing, parameterized processes are often responsible for more than 50% of the total computational cost. The analyses and proposals presented, among others, in Caya et al. (1998), Cullen and Salmond (2003), Dubal et al. (2005), and Dubal et al. (2006), although dealing often with simplified equations sets and idealized models, have highlighted many limitations of common splitting approaches and the need for numerical methods in which the idealised dynamics and the parameterized physical processes are much more closely coupled than in the past. On the other hand, at the implementation level, a thorough revision of many existing parameterization codes is probably necessary for the next generation ESM. Indeed, it often happens that quantities that could be derived more accurately within the dynamical core are instead computed using crude approximations in the parameterization routines, possibly resulting in a loss of accuracy that is hard to trace back to the numerical methods used in the dynamical core itself.

## References

- Adcroft A, Campin J, Hill C, Marshall J (2004) Implementation of an atmosphere-ocean general circulation model on the expanded spherical cube. Mon Weather Rev 132:2845–2863CrossRefGoogle Scholar
- Arakawa A, Lamb V (1981) A potential enstrophy and energy conserving scheme for the shallow water equations. Mon Weather Rev 109:18–136CrossRefGoogle Scholar
- Bonaventura L, Ringler T (2005) Analysis of discrete shallow water models on geodesic Delaunay grids with C-type staggering. Mon Weather Rev 133:2351–2373CrossRefGoogle Scholar
- Caya A, Laprise R, Zwack P (1998) Consequences of using the splitting method for implementing physical forcings in a semi-implicit, semi-Lagrangian model. Mon Weather Rev 126:1707–1713CrossRefGoogle Scholar
- Cheong H (2000) Application of double Fourier series to the shallow-water equations on a sphere. J Comput Phys 165:261–287CrossRefGoogle Scholar
- Cheong H (2006) A dynamical core with double Fourier series: comparison with the spherical harmonics method. Mon Weather Rev 134:1299–1315CrossRefGoogle Scholar
- Cullen M, Salmond D (2003) On the use of a predictor–corrector scheme to couple the dynamics with the physical parametrizations in the ECMWF model. Q J Royal Meteorol Soc 129:1217–1236CrossRefGoogle Scholar
- Dubal M, Wood N, Staniforth A (2005) Mixed parallel–sequential-split schemes for time-stepping multiple physical parameterizations. Mon Weather Rev 133:989–1002CrossRefGoogle Scholar
- Dubal M, Wood N, Staniforth A (2006) Some numerical properties of approaches to physics–dynamics coupling for NWP. Q J Royal Meteorol Soc 132:27–42CrossRefGoogle Scholar
- Giraldo F (1998) The Lagrange–Galerkin spectral element method on unstructured quadrilateral grids. J Comput Phys 147:114–146CrossRefGoogle Scholar
- Giraldo F (2001) A spectral element shallow water model on spherical geodesic grids. Int J Numer Methods Fluids 35:869–901CrossRefGoogle Scholar
- Giraldo F, Rosmond T (2004) A scalable spectral element Eulerian atmospheric model (SEE-AM) for NWP: dynamical core tests. Mon Weather Rev 132:133–153CrossRefGoogle Scholar
- Giraldo F, Hesthaven J, Warburton T (2002) High-order discontinuous Galerkin methods for the spherical shallow water equations. J Comput Phys 181:499–525CrossRefGoogle Scholar
- Giraldo F, Perot J, Fischer P (2003) A spectral element semi-Lagrangian (SESL) method for the spherical shallow water equations. J Comput Phys 190:623–650CrossRefGoogle Scholar
- Heikes R, Randall D (1995) Numerical integration of the shallow-water equations on a twisted icosahedral grid. Part I: basic design and results of tests. Mon Weather Rev 123:1862–1880CrossRefGoogle Scholar
- Laprise J, Plante R (1995) A class of semi-Lagrangian integrated-mass (SLIM) numerical transport algorithms. Mon Weather Rev 123:553–565CrossRefGoogle Scholar
- Leonard B, Lock A, MacVean M (1996) Conservative explicit unrestricted-time-step multidimensional constancy-preserving advection schemes. Mon Weather Rev 124:2588–2606CrossRefGoogle Scholar
- Lin S, Rood R (1996) Multidimensional flux-form semi-Lagrangian transport schemes. Mon Weather Rev 124:2046–2070CrossRefGoogle Scholar
- Lipscomb W, Ringler T (2005) An incremental remapping transport scheme on a spherical geodesic grid. Mon Weather Rev 133:2335–2350CrossRefGoogle Scholar
- Majewski D, Liermann D, Prohl P, Ritter B, Buchhold M, Hanisch T, Paul G, Wergen W, Baumgardner J (2002) The operational global icosahedral-hexagonal gridpoint model GME: description and high resolution tests. Mon Weather Rev 130:319–338CrossRefGoogle Scholar
- Nair R D, Machenhauer B (2002) The mass-conservative cell-integrated semi-Lagrangian advection scheme on the sphere. Mon Weather Rev 130:649–667CrossRefGoogle Scholar
- Nair R D, Scroggs J, Semazzi F (2002) Conservative global transport schemes for climate and atmospheric chemistry models. Mon Weather Rev 130:2059–2073CrossRefGoogle Scholar
- Nair R D, Thomas S, Loft R (2005) A Discontinuous Galerkin transport scheme on the cubed sphere. Mon Weather Rev 133:814–828CrossRefGoogle Scholar
- Rancic M, Purser R, Mesinger F (1996) A global shallow-water model using an expanded spherical cube: gnomonic versus conformal coordinates. Q J Royal Meteorol Soc 122:959–982Google Scholar
- Restelli M, Bonaventura L, Sacco R (2006) A semi-Lagrangian Discontinuous Galerkin method for scalar advection by incompressible flows. J Comput Phys 216:195–215CrossRefGoogle Scholar
- Ringler T, Randall D (2002) A potential enstrophy and energy conserving numerical scheme for solution of the shallow-water equations a geodesic grid. Mon Weather Rev 130:1397–1410CrossRefGoogle Scholar
- Ringler T, Heikes R, Randall D (2000) Modeling the atmospheric general circulation using a spherical geodesic grid: a new class of dynamical cores. Mon Weather Rev 128:2471–2490CrossRefGoogle Scholar
- Ringler T, Thuburn J, Klemp J, Skamarock W (2010) A unified approach to energy conservation and potential vorticity dynamics for arbitrarily-structured C-grids. J Comput Phys 229:3065–3090CrossRefGoogle Scholar
- Ronchi C, Iacono R, Paolucci R (1996) The “cubed sphere”: a new method for the solution of partial differential equations in spherical geometry. J Comput Phys 124:93–114Google Scholar
- Sadourny R (1972) Conservative finite-difference approximations of the primitive equations on quasi-uniform spherical grids. Mon Weather Rev 100:136–144CrossRefGoogle Scholar
- Sadourny R (1975) The dynamics of finite difference models of the shallow water equations. J Atmos Sci 32:680–689CrossRefGoogle Scholar
- Taylor M, Tribbia J, Iskandarani M (1997) The spectral element method for the shallow water equations on the sphere. Mon Weather Rev 130:92–108Google Scholar
- Thomas S, Loft R (2002) Semi-implicit spectral element method for the shallow water equations on the sphere. J Sci Comput 17:339–350CrossRefGoogle Scholar
- Thuburn J (1997) A PV-based shallow-water model on a hexagonal-icosahedral grid. Mon Weather Rev 125:2328–2347CrossRefGoogle Scholar
- Tomita H, Satoh M (2004) A new dynamical framework of nonhydrostatic global model using the icosahedral grid. Fluid Dyn Res 34:357–400CrossRefGoogle Scholar
- Tomita H, Tsugawa M, Satoh M, Goto K (2001) Shallow water model on a modified icosahedral geodesic grid by using spring dynamics. J Comput Phys 174:579–613CrossRefGoogle Scholar
- Tomita H, Goto K, Satoh M (2008) A new approach to atmospheric general circulation model: global cloud resolving model NICAM and its computational performance. SIAM J Sci Comput 30:2755–2776CrossRefGoogle Scholar
- Williamson D. (1979) Numerical methods used in atmospheric models, vol. II, WMO, GARP Publication series no.17Google Scholar
- Zerroukat M, Wood N, Staniforth A (2004) SLICE-S: a semi-Lagrangian inherently conserving and efficient scheme for transport problems on the sphere. Q J Royal Meteorol Soc 130:2649–2664CrossRefGoogle Scholar
- Zerroukat M, Wood N, Staniforth A (2005) A monotonic and positive-definite filter for a semi-Lagrangian inherently conserving and efficient (SLICE) scheme. Q J Royal Meteorol Soc 131:2923–2936CrossRefGoogle Scholar