Adaptive Meshes on the Sphere: CubedSpheres versus LatitudeLongitude Grids

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Adaptive Meshes on the Sphere Cubed-Spheres
versus Latitude-Longitude Grids

• Christiane JablonowskiUniversity of
  MichiganAmik St-Cyr
• NCARICON Friends Workshop, May/31/2007

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Acknowledgments

• The AMR comparison is based on a joint paper with
  Amik St-Cyr and collaborators from NCAR,
  submitted to Monthly Weather Review, revised in
  May 2007
• The AMR Spectral Element Model was mainly
  developed by Amik St-Cyr, John Dennis Steve
  Thomas (NCAR)
• The AMR FV model is documented inJablonowski
  (2004), Jablonowski et al. (2004, 2006)
• Contributors to the AMR FV model areMichael
  Herzog (GFDL) Joyce Penner (UM)Robert Oehmke
  (NCAR) Quentin Stout (UM)Bram van Leer (UM)
  Ken Powell (UM)

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Overview

• Computational Grids on the Sphere
• Adaptive mesh refinement (AMR) techniques
• Why are we interested in variable resolutions /
  multi-scales?
• Overview of two AMR shallow water models
• Finite volume (FV) model
• Spectral element model (SEM)
• Results Static and dynamic adaptations
• 2D shallow water experiments
• Conclusions and Outlook

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Latitude-Longitude Grid

• Popular choice
• Meridians convergepolar filters or/andtime
  steps
• Orthogonal

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Platonic solids - Regular grid structures

• Platonic solids can be enclosed in a
  sphereSource Wikipedia

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Cubed Sphere Geometry
Advection of a cosine bell around the sphere (12
days)at a 45o angle
Courtesy of Ram Nair (NCAR)
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Adaptive Mesh Refinements (AMR)
Latitude-Longitude gridModel FV
Cubed-sphere gridModel SEM
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SEM Grid Points within Spectral Elements
Circles Gauss-Lobatto-Legendre (GLL) points for
vectorsSquaresGauss-Lobatto (GL) points
forscalarsElements are split in case of
refinements
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FV Block-Structured Adaptive Mesh Refinement
Strategy
Self-similar blocks with 3 ghost cells in x y
direction
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Other AMR Grids
Model ICONIcosahedral grid with nested
high-resolution regionsunder development at
the German WeatherService (DWD) and MPI, Hamburg
Source DWD, MPI
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Features of Interest in a Multi-Scale Regime
Hurricane Frances
Hurricane Ivan
September/5/2004
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High Resolution Multi-Scale Interactions
10 km resolution
W. Ohfuchi, The Earth Simulator Center, Japan
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Shallow Water Equations
Momentum equation in vector-invariant form
Continuity equation
only in FV
vh horizontal velocity vector? relative
vorticityf Coriolis parameterK 0.5(u2 v2)
kinetic energyD horizontal divergence, ? damping
coefficienth depth of the fluid, hs height of
the orographyg gravitational acceleration
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Finite Volume (FV) Shallow Water Model

• Developed by Lin and Rood (1996), Lin and Rood
  (1997)
• 3D version available (Lin 2004), built upon the
  SW model
• hydrostatic dynamical core used for climate and
  weather predictions
• Currently part of NCARs, NASAs and GFDLs
  General Circulation Models
• Numerics Finite volume approach
• conservative and monotonic transport scheme
• upwind biased 1D fluxes, operator splitting
• van Leer second order scheme for time-averaged
  numerical fluxes
• PPM third order scheme (piecewise parabolic
  method)for prognostic variables
• Staggered grid (Arakawa D-grid), C grid for
  mid-time levels
• Orthogonal Latitude-Longitude computational grid

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Spectral Element (SEM) Shallow Water Model

• Documented in Thomas and Loft (2002), St-Cyr and
  Thomas (2005), St-Cyr et al. (2007)
• 3D version available
• Experimental tests within NCARs Climate Modeling
  Software Framework
• Numerics Spectral Elements
• Non-conservative and non-monotonic
• Allows high-order numerical method
• Spectral convergence for smooth flows
• GLL and GL collocation points
• Non-orthogonal cubed-sphere computational grid

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Overview of the AMR comparison

• 2D shallow water tests (Williamson et al., JCP
  1992)
• Dynamic refinements for pure advection
  experiments
• Slotted cylinder
• Cosine bell advection test (test case 1)
• Static refinements in regions of interest (test
  case 2)
• Dynamic refinements and refinement criteria
  Flow over a mountain (test case 5)
• Rossby-Haurwitz wave with static refinements
  (test case 6)
• Barotropic instability test (Galewsky et al.,
  Tellus 2004)

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AMR Transport of a Slotted Cylinder
Model FV
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Transport of a Slotted Cylinder
SEM
FV
5 x 5 deg base grid, 3 refinement levels
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Transport of a Slotted Cylinder
SEM
FV

• Slotted cylinder is reliably detected and
  tracked
• Over- and undershoots in SEM, FV monotonic

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Snapshots Advection of a Cosine Bell
SEM
FV
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Snapshots Advection of a Cosine Bell
SEM
FV
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Error norms Cosine Bell Advection
Days
Days
Rotation angle ? 45Errors in SEM are lower
than in FV
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Error norms after 12 days
Rotation angle ? 0SEM produces
undershootsErrors arecomparable
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Snapshots Cosine Bell at day 3
North-polar stereographic projection at day 3 for
a 90
Uniform distribution in SEM, Convergence of
blocks in FV
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2D Static Adaptations
Test case 2, ? 45

• Height field at day 14
• Smooth flow through refined region

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Error norms Test case 2
Days
Days
Rotation angle ? 45Errors in FV partly due to
errors at AMR interfaces
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2D Dynamic adaptations in FV
Vorticity-basedadaptation criterion
2D shallowwater test 515-day run
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Snapshots Flow over a mountain
Geopotential height field (test case 5)
Longitude
Longitude
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Snapshots Flow over a mountain
Geopotential height field (test case 5)
SEM
FV
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Error norms Test case 5
Hours
Hours
Errors in SEM converge quicker to the reference
solution (T426 NCAR spectral model, provided by
DWD)
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Snapshots Rossby-Haurwitz Wave
Geopotential height field (test case 6) at day 7
Smooth flow through static refinement regions
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Barotropic instability test case
Convergence analysis Relative vorticity at day 6
1.25 resolution is needed to get a good
representation of the wave
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Barotropic instability test case
Convergence analysis Relative vorticity at day 6
Second highest and highest resolutions are very
similar to each other, SEM and FV are similar
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Barotropic instability test case
Vorticity-based adaptation criterion Day 3 and 4
5 deg base grid, 4 refinement levels
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Barotropic instability test case
SEM and FV are very similar
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Alternative AMR Unstructured Triangular Grid
Hurricane Floyd (1999)
OMEGA model Courtesy ofA. Sarma (SAIC, NC, USA)
Colors indicate the wind speed
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Conclusions

• Both grids, cubed-sphere meshes and
  latitude-longitude grids, are options for AMR
  techniques
• SEM model shows lower error norms in comparison
  to FV
• Mainly due to high-order numerical method
• Partly due to different AMR approach that does
  not need interpolations of ghost cells in
  blocks
• But SEM is non-monotonic and non-conservative
• Cubed-sphere grid has clear advantages
• No convergence of the meridians, no polar filters
• But, GLL and GL points for numerical method in
  SEM are clustered along boundaries of spectral
  elements
• SEM requires very small time steps