Modern numerical methods in climate analysis and modelling

1
Modern numerical methods in climate analysis and
modelling

• Andreas Hense and colleagues
• (to be announced)
• Universität Bonn

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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

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The arctic water budget problem
with Martin Göber (Met Office), Reinhard
Hagenbrock , Felix Amendt
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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

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The arctic water budget problem
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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

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A mass consistent windfield
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A mass consistent wind field
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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

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The grid (horizontal)
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The Grid (vertical)
e.g. 500 hPa
e.g.700 hPa
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Result for the Arctic moisture balance
Effect of mass modification on Radiosonde and
ERA15 (subsampled) 1979-94
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Result for the Arctic moisture balance
Effects of doubled resolution in subsampled ERA15
doubled
Radiosonde network
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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

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The equations - coordinate free version
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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

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The elliptic equation
Discretization with finite elements on a
triangular mesh leads to the linear equation
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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

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The algorithm
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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

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The macrotriangulation
C
or Soccerball
60
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Results
Initial field and topography
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Results
After 10 days
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Mass- and Energy conservation?
(Mass)
(Energy)
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Numerical stability?
Grid size ca. 150 km
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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