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The present paper considers numerical properties of two different approaches to discretization of the isoneutral diffusion. The necessity of an alternative treatment of the isoneutral diffusion in a terrain-following climate ocean model as opposed to the more convenient rotated tensor formalism is studied. A new method of the approximation of the isoneutral diffusion based on a non-local computational stencil is formulated. The validity of the non-local discretization of the isoneutral diffusion operator with regard to a terrain-following vertical coordinate in the INMCM ocean model is demonstrated. [ABSTRACT FROM AUTHOR]

Many numerical weather prediction models employ terrain‐following vertical coordinates. As a consequence, over orography, flat tops of stratus clouds are intersected by the vertical coordinate surfaces. We conduct idealised two‐dimensional simulations of a stratus cloud with the COSMO model to study the effect of such sloping vertical coordinate surfaces. The evolution of the stratus cloud above a flat surface within a horizontally homogeneous atmosphere serves as a reference. During night‐time, the cloud thickens, driven by radiative cooling at the cloud top. Adding a sinusoidal perturbation to the vertical coordinate surfaces reduces the growth of the stratus cloud. With strong perturbations, the cloud starts to dissipate. The physical processes in the two simulations are identical, hence this behaviour is purely driven by numerical diffusion. The cloud is similarly thinned when sinusoidal orographic features are introduced. The reduction depends on the amplitude and wavelength of the perturbations and hills. Increasing the horizontal resolution partly mitigates the numerical diffusion. However, this is a very costly measure for an operational weather model. We suggest conducting further research on a new vertical coordinate with additional local smoothing of the orographic signal. [ABSTRACT FROM AUTHOR]

Tracer advection schemes used by ocean models are susceptible to artificial oscillations: a form of numerical error whereby the advected field alternates between overshooting and undershooting the exact solution, producing false extrema. Here we show that these oscillations have undesirable interactions with a coupled sea ice model. When oscillations cause the near-surface ocean temperature to fall below the freezing point, sea ice forms for no reason other than numerical error. This spurious sea ice formation has significant and wide-ranging impacts on Southern Ocean simulations, including the disappearance of coastal polynyas, stratification of the water column, erosion of Winter Water, and upwelling of warm Circumpolar Deep Water. This significantly limits the model’s suitability for coupled ocean-ice and climate studies. Using the terrain-following-coordinate ocean model ROMS (Regional Ocean Modelling System) coupled to the sea ice model CICE (Community Ice CodE) on a circumpolar Antarctic domain, we compare the performance of three different tracer advection schemes, as well as two levels of parameterised diffusion and the addition of flux limiters to prevent numerical oscillations. The upwind third-order advection scheme performs better than the centered fourth-order and Akima fourth-order advection schemes, with far fewer incidents of spurious sea ice formation. The latter two schemes are less problematic with higher parameterised diffusion, although some supercooling artifacts persist. Spurious supercooling was eliminated by adding flux limiters to the upwind third-order scheme. We present this comparison as evidence of the problematic nature of oscillatory advection schemes in sea ice formation regions, and urge other ocean/sea-ice modellers to exercise caution when using such schemes. [ABSTRACT FROM AUTHOR]

Discretisation of the horizontal pressure gradient force in layered ocean models is a challenging task, with non-trivial interactions between the thermodynamics of the fluid and the geometry of the layers often leading to numerical difficulties. We present two new finite-volume schemes for the pressure gradient operator designed to address these issues. In each case, the horizontal acceleration is computed as an integration of the contact pressure force that acts along the perimeter of an associated momentum control-volume. A pair of new schemes are developed by exploring different control-volume geometries. Non-linearities in the underlying equation-of-state definitions and thermodynamic profiles are treated using a high-order accurate numerical integration framework, designed to preserve hydrostatic balance in a non-linear manner. Numerical experiments show that the new methods achieve high levels of consistency, maintaining hydrostatic and thermobaric equilibrium in the presence of strongly-sloping layer geometries, non-linear equations-of-state and non-uniform vertical stratification profiles. These results suggest that the new pressure gradient formulations may be appropriate for general circulation models that employ hybrid vertical coordinates and/or terrain-following representations. [ABSTRACT FROM AUTHOR]

A numerical discretization of the three-dimensional (3D) diffusion equation for the scalar case and the vector case (i.e. for the momentum equation) in terrain-following coordinates on the sphere is described. The discretization uses the horizontally explicit-vertically implicit (HE-VI) approach, which is often applied in atmospheric simulation models. Firstly, a spatially second-order discretization is proposed, which treats the metric terms of the terrain-following coordinates in a stable manner even for very steep terrain. A von Neumann stability analysis calculates the maximum stable diffusion Courant number for different implicitness weights as a function of slope angle and grid anisotropy. Secondly, simple analytic solutions of the diffusion equation for both scalar and vector cases are proposed for testing and validation purposes. The implementations in the two atmospheric model systems Icosahedral Non-hydrostatic (ICON) (global) and Consortium for Small-scale Modelling (COSMO) (regional) are compared against these exact solutions. [ABSTRACT FROM AUTHOR]

Abstract: Although our names appear as co-authors in the above article (Haidvogel et al. (2008) , hereafter H2008), we were not aware of its existence until after it was published. In reading the article, we discovered that a significant portion of it (, or 10 pages) repeats three large fragments from our own previously published work, Shchepetkin and McWilliams (2005) (hereafter SM2005), but now presented in such a way that the motivation for the specific algorithmic choices made in ROMS and the relations among the different model components are no longer clear. The model equations appearing in H2008, Section 2.1 (taken from an earlier article, Haidvogel et al. (2000) ) are not entirely consistent with the actual equations solved in the ROMS code, resulting in contradictions within H2008 itself. In our view the description in H2008 does not constitute a mathematically accurate statement about the hydrodynamic core of ROMS. The purpose of this note is to clarify and correct this, as well as to explain some of the algorithmic differences among ROMS versions now in use. [Copyright &y& Elsevier]

Abstract: In this paper, we identify a crucial numerical problem in sigma coordinate models, leading to unacceptable spurious diapycnal mixing. This error is a by-product of recent advances in numerical methods, namely the implementation of high-order diffusive advection schemes. In the case of ROMS, spurious mixing is produced by its third-order upwind advection scheme, but our analysis suggests that all diffusive advection schemes would behave similarly in all sigma models. We show that the common idea that spurious mixing decreases with resolution is generally false. In a coarse-resolution regime, spurious mixing increases as resolution is refined, and may reach its peak value when eddy-driven lateral mixing becomes explicitly resolved. At finer resolution, diffusivities are expected to decrease but with values that only become acceptable at resolutions finer than the kilometer. The solution to this problem requires a specifically designed advection scheme. We propose and validate the RSUP3 scheme, where diffusion is split from advection and is represented by a rotated biharmonic diffusion scheme with flow-dependent hyperdiffusivity satisfying the Peclet constraint. The rotated diffusion operator is designed for numerical stability, which includes improvements of linear stability limits and a clipping method adapted to the sigma-coordinate. Realistic model experiments in a southwest Pacific configuration show that RSUP3 is able to preserve low dispersion and diffusion capabilities of the original third-order upwind scheme, while preserving water mass characteristics. There are residual errors from the rotated diffusion operator, but they remain acceptable. The use of a constant diffusivity rather than the Peclet hyperdiffusivity tends to increase these residual errors which become unacceptable with Laplacian diffusion. Finally, we have left some options open concerning the use of time filters as an alternative to spatial diffusion. A temporal discretization approach to the present problem (including implicit discretization) will be reported in a following paper. [Copyright &y& Elsevier]

Abstract: We perform large-eddy simulations (LES) of the flow past a scale model of a complex building. Calculations are accomplished using two different methods to represent the edifice. The first method employs the standard Gal-Chen and Somerville terrain-following coordinate transformation, common in mesoscale atmospheric simulations. The second method uses an immersed boundary approach, in which fictitious body forces in the equations of motion are used to represent the building by attenuating the flow to stagnation within a time comparable to the time step of the model. Both methods are implemented in the same hydrodynamical code (EULAG) using the same nonoscillatory forward-in-time (NFT) incompressible flow solver based on the multidimensional positive definite advection transport algorithms (MPDATA). The two solution methods are compared to wind tunnel data collected for neutral stratification. Profiles of the first- and second-order moments at various locations around the model building show good agreement with the wind tunnel data. Although both methods appear to be viable tools for LES of urban flows, the immersed boundary approach is computationally more efficient. The results of these simulations demonstrate that, contrary to popular opinion, continuous mappings such as the Gal-Chen and Somerville transformation are not inherently limited to gentle slopes. Calculations for a strongly stratified case are also presented to point out the substantial differences from the neutral boundary layer flows. [Copyright &y& Elsevier]