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Modern numerical methods in climate analysis and modelling

This presentation by Andreas Hense and colleagues from the Universität Bonn offers an in-depth examination of the Arctic water budget problem. It delves into the challenges of accurately quantifying Arctic evaporation (E) and precipitation (P), discussing discrepancies between reanalyses and radiosonde data. The methods and results of solving the shallow water equations on a spherical grid are explained, along with the development of a mass-consistent wind field solution using advanced numerical methods. The presentation also touches on grid structures, numerical stability, and the benefits of finite element formulation for climate analysis. These insights shed light on the complex interplay of mass and energy conservation in climate modeling.

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Modern numerical methods in climate analysis and modelling

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  1. Modern numerical methods in climate analysis and modelling Andreas Hense and colleagues (to be announced) Universität Bonn Andreas Hense, Universität Bonn

  2. Overview • Introduction • The water budget of the Arctic from radiosonde data • method • results • Solution of the shallow water equations on the sphere • method • results • Conclusion Andreas Hense, Universität Bonn

  3. The arctic water budget problem with Martin Göber (Met Office), Reinhard Hagenbrock , Felix Amendt Andreas Hense, Universität Bonn

  4. The arctic water budget problem • Arctic Evaporation E and precipitation P almost completely unknown • Atmospheric moisture flux vq from • Reanalyses ERA15 and NCEP • Radiosonde (Serezze) • Discrepancy between Reanalyses and Radiosonde Andreas Hense, Universität Bonn

  5. The arctic water budget problem Andreas Hense, Universität Bonn

  6. Reasons of discrepancies • Reanalysis budget are not closed • Moisture cycle spin-up • Spatial sampling problems for radiosondes • Measurement errors • mass inconsistent wind fields from radiosondes Andreas Hense, Universität Bonn

  7. A mass consistent windfield Andreas Hense, Universität Bonn

  8. A mass consistent wind field Andreas Hense, Universität Bonn

  9. A mass consistent wind field • Standard procedure: • interpolation of observations to a grid • Differentiation and minimization on the grid • Our solution: • Discretization of the Minimization integral • three dimensional finite elements • on an irregular triangular grid Andreas Hense, Universität Bonn

  10. The grid (horizontal) Andreas Hense, Universität Bonn

  11. The Grid (vertical) e.g. 500 hPa e.g.700 hPa Andreas Hense, Universität Bonn

  12. Result for the Arctic moisture balance Effect of mass modification on Radiosonde and ERA15 (subsampled) 1979-94 Andreas Hense, Universität Bonn

  13. Result for the Arctic moisture balance Effects of doubled resolution in subsampled ERA15 doubled Radiosonde network Andreas Hense, Universität Bonn

  14. A semi-Lagrangian semi-implicit finite element model for the shallow water equations on the sphere • with Thomas Heinze Technical University Munich • Develope the dynamic code of a global atmospheric model suitable for MPP machines • and local refinement capability • Unstructured triangular grid • finite element or finite volume technique Andreas Hense, Universität Bonn

  15. The equations - coordinate free version Andreas Hense, Universität Bonn

  16. Discretization in time • Integration along backward trajectories • coupled equations for the new geopotential heights and velocities at the endpoints of the trajectories • reduce to a elliptic equation for the new geopotential height Andreas Hense, Universität Bonn

  17. The elliptic equation Discretization with finite elements on a triangular mesh leads to the linear equation: Andreas Hense, Universität Bonn

  18. The algorithm • Compute backward trajectories from a given flow • evaluate the right hand side through interpolation • solve the linear equation for the new geopotential • obtain the new flow field Andreas Hense, Universität Bonn

  19. The algorithm Andreas Hense, Universität Bonn

  20. How to distribute the triangles on the sphere? • Choose a macrotriangulation • divide each triangle side in equal parts • connect the midpoints • one triangle four new triangles Andreas Hense, Universität Bonn

  21. The macrotriangulation C or Soccerball 60 Andreas Hense, Universität Bonn

  22. Results Initial field and topography Andreas Hense, Universität Bonn

  23. Results After 10 days Andreas Hense, Universität Bonn

  24. Mass- and Energy conservation? (Mass) (Energy) Andreas Hense, Universität Bonn

  25. Numerical stability? Grid size ca. 150 km Andreas Hense, Universität Bonn

  26. Summary • Finite element formulation is a very elegant method for least squares problems • Difference between Radiosonde and Reanalysis related water balance resolvable at twice the radiosonde network resolution • Finite element formulation is a very efficient method for the shallow equations • MPP and local refinement possible Andreas Hense, Universität Bonn

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