A multi-envelope vertical coordinate system for numerical ocean modelling

Abstract

A multi-envelope generalised coordinate system for numerical ocean modelling is introduced. In this system, computational levels are curved and adjusted to multiple ‘virtual bottoms’ (aka envelopes) rather than following geopotential levels or the actual bathymetry. This allows defining computational levels which are optimised to best represent different physical processes in different sub-domains of the model. In particular, we show how it can be used to improve the representation of tracer advection in the ocean interior. The new vertical system is compared with a widely used z-partial step scheme. The modelling skill of the models is assessed by comparison with the analytical solutions or results produced by a model with a very high-resolution z-level grid. Three idealised process-oriented numerical experiments are carried out. Experiments show that numerical errors produced by the new scheme are much smaller than those produced by the standard z-partial step scheme at a comparable vertical resolution. In particular, the new scheme shows superiority in simulating the formation of a cold intermediate layer in the ocean interior and in representing dense water cascading down a steep topography.

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Acknowledgments

The authors are grateful to the valuable comments and suggestions of the two anonymous reviewers which have greatly contributed to improving the manuscript.

Funding

This work was funded by the EASME/EMFF/2014/ 1.3.1.3/LOT4/SI2.709436 - Seabasin Checkpoints - Lot 4 - ‘BLACK SEA’ project.

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Correspondence to Diego Bruciaferri.

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Responsible Editor: Pierre F.J. Lermusiaux

Appendix

Appendix

For each (x,y) of the horizontal domain the complete cubic spline \(P_{x,y,i}^{3}(\sigma _{i})\) of the vertical sub-zone Di can be written as

$$\begin{array}{@{}rcl@{}} P_{x,y,i}^{3}(\sigma_{i}) &=& a_{x,y,i} + b_{x,y,i}(-\sigma_{i}) + c_{x,y,i}(-\sigma_{i})^{2}\\ &&+ d_{x,y,i}(-\sigma_{i})^{3} \end{array} $$
(12)

where σi is given by Eq. 4 and − 1 < σi ≤ 0.

Applying the three constraints defined in Section 2 leads to a tridiagonal linear system of four equations for the four unknowns ax,y,i, bx,y,i, cx,y,i and dx,y,i (de Boor 1978).

A modified version of the Fortran90 numerical library pppack (de Boor 1978) has been introduced in the NEMO code to compute the four coefficients of the complete cubic spline \(P_{x,y,i}^{3}(\sigma _{i})\).

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Bruciaferri, D., Shapiro, G.I. & Wobus, F. A multi-envelope vertical coordinate system for numerical ocean modelling. Ocean Dynamics 68, 1239–1258 (2018). https://doi.org/10.1007/s10236-018-1189-x

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Keywords

  • Ocean modelling
  • Vertical coordinate
  • Oceanic transport