Adaptive Meshes on the Sphere: Cubed-Spheres versus

Latitude-Longitude Grids

Christiane JablonowskiUniversity of Michigan

Dec/8/2006

AcknowledgmentsAcknowledgments The AMR comparison is based on a joint paper with The AMR comparison is based on a joint paper with

Amik St-Cyr and collaborators from NCAR, submitted to Amik St-Cyr and collaborators from NCAR, submitted to Monthly Weather Review in November 2006Monthly Weather Review in November 2006

The AMR Spectral Element Model was mainly developed by The AMR Spectral Element Model was mainly developed by Amik St-Cyr, John Dennis & Steve Thomas (NCAR)Amik St-Cyr, John Dennis & Steve Thomas (NCAR)

The AMR FV model is documented inThe AMR FV model is documented inJablonowski (2004), Jablonowski et al. (2004, 2006)Jablonowski (2004), Jablonowski et al. (2004, 2006)

Contributors to the AMR FV model areContributors to the AMR FV model areMichael Herzog (GFDL) & Joyce Penner (UM)Michael Herzog (GFDL) & Joyce Penner (UM)Robert Oehmke (NCAR) & Quentin Stout (UM)Robert Oehmke (NCAR) & Quentin Stout (UM)Bram van Leer (UM) & Ken Powell (UM)Bram van Leer (UM) & Ken Powell (UM)

The AMR comparison is based on a joint paper with The AMR comparison is based on a joint paper with Amik St-Cyr and collaborators from NCAR, submitted to Amik St-Cyr and collaborators from NCAR, submitted to Monthly Weather Review in November 2006Monthly Weather Review in November 2006

The AMR Spectral Element Model was mainly developed by The AMR Spectral Element Model was mainly developed by Amik St-Cyr, John Dennis & Steve Thomas (NCAR)Amik St-Cyr, John Dennis & Steve Thomas (NCAR)

The AMR FV model is documented inThe AMR FV model is documented inJablonowski (2004), Jablonowski et al. (2004, 2006)Jablonowski (2004), Jablonowski et al. (2004, 2006)

Contributors to the AMR FV model areContributors to the AMR FV model areMichael Herzog (GFDL) & Joyce Penner (UM)Michael Herzog (GFDL) & Joyce Penner (UM)Robert Oehmke (NCAR) & Quentin Stout (UM)Robert Oehmke (NCAR) & Quentin Stout (UM)Bram van Leer (UM) & Ken Powell (UM)Bram van Leer (UM) & Ken Powell (UM)

OverviewOverview

Computational Grids on the SphereComputational Grids on the Sphere

Adaptive mesh refinement (AMR) Adaptive mesh refinement (AMR)

techniquestechniques Why are we interested in variable Why are we interested in variable resolutions / resolutions / multi-scales?multi-scales?

Overview of two AMR shallow water modelsOverview of two AMR shallow water models

Finite volume (FV) modelFinite volume (FV) model

Spectral element model (SEM)Spectral element model (SEM)

Results: Static and dynamic adaptationsResults: Static and dynamic adaptations

2D shallow water experiments 2D shallow water experiments

Conclusions and OutlookConclusions and Outlook

Computational Grids on the SphereComputational Grids on the Sphere

Adaptive mesh refinement (AMR) Adaptive mesh refinement (AMR)

techniquestechniques Why are we interested in variable Why are we interested in variable resolutions / resolutions / multi-scales?multi-scales?

Overview of two AMR shallow water modelsOverview of two AMR shallow water models

Finite volume (FV) modelFinite volume (FV) model

Spectral element model (SEM)Spectral element model (SEM)

Results: Static and dynamic adaptationsResults: Static and dynamic adaptations

2D shallow water experiments 2D shallow water experiments

Conclusions and OutlookConclusions and Outlook

Latitude-Longitude GridLatitude-Longitude Grid

Popular choice Meridians converge:polar filters or/andtime steps

Orthogonal

Popular choice Meridians converge:polar filters or/andtime steps

Orthogonal

Platonic solids - Regular grid structures

Platonic solids - Regular grid structures

Platonic solids can be enclosed in a sphere

Platonic solids can be enclosed in a sphere

Cubed Sphere Geometry

Courtesy of Ram Nair (NCAR)

Advection of a cosine bell around the sphere (12 days)at a 45o angle

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

Adaptive Mesh Refinements (AMR)

Latitude-Longitude grid:Model FV

Cubed-sphere grid:Model SEM

SEM: Grid Points within Spectral Elements

Circles: Gauss-Lobatto-Legendre (GLL) points for vectors

Squares:Gauss-Lobatto (GL) points forscalars

Elements are split in case of refinements

FV: Block-Structured Adaptive Mesh Refinement Strategy

Self-similar blocks with 3 ghost cells in x & y direction

Other AMR Grids

Source: DWD

Model ICON

Icosahedral grid with nested high-resolution regions

under development at the German WeatherService (DWD) and MPI, Hamburg

Features of Interest in a Multi-Scale Regime

Hurricane Ivan

Hurricane Frances

September/5/2004

High Resolution: Multi-Scale Interactions

W. Ohfuchi, The Earth Simulator Center, Japan

10 km resolution

AMR Transport of a Slotted Cylinder

AMR Transport of a Slotted Cylinder

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

Model FV

Transport of a Slotted CylinderTransport of a Slotted Cylinder

Transport of a Slotted CylinderTransport of a Slotted Cylinder

Transport of a Slotted CylinderTransport of a Slotted Cylinder

Transport of a Slotted CylinderTransport of a Slotted Cylinder

Transport of a Slotted CylinderTransport of a Slotted Cylinder

• Slotted cylinder is reliably detected and trackedSlotted cylinder is reliably detected and tracked

Shallow Water Equations

€

∂ r

v h

∂t+ (ζ + f )

r k ×

r v h +

r ∇(K −νD + g(h + hs)) = 0

€

∂h

∂t+

r ∇ • h

r v ( ) = 0

Momentum equation in vector-invariant form

Continuity equation

vh horizontal velocity vector relative vorticityf Coriolis parameterK= 0.5*(u2 + v2) kinetic energyD horizontal divergence, damping coefficienth free surface height, hs height of the orographyg gravitational acceleration

€

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 NCAR’s, NASA’s and GFDL’s 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)

– Orthogonal Latitude-Longitude computational grid

Finite Volume (FV) Shallow Water Model

Documented in Thomas and Loft (2002), St-Cyr and Thomas (2005), St-Cyr et al. (2006) 3D version available Experimental tests within NCAR’s 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

Spectral Element (SEM) Shallow Water Model

Overview of the AMR comparisonOverview of the AMR comparison

2D shallow water tests: (Williamson et al., JCP 2D shallow water tests: (Williamson et al., JCP

1992)1992)

Dynamic refinements for pure advection Dynamic refinements for pure advection

experimentsexperiments

Cosine bell advection test (test case 1)Cosine bell advection test (test case 1)

Static refinements in regions of interest Static refinements in regions of interest

(test case 2)(test case 2)

Dynamic refinements and refinement criteria: Dynamic refinements and refinement criteria:

Flow over a mountain (test case 5) Flow over a mountain (test case 5)

Rossby-Haurwitz wave with static refinements Rossby-Haurwitz wave with static refinements

(test case 6) (test case 6)

2D shallow water tests: (Williamson et al., JCP 2D shallow water tests: (Williamson et al., JCP

1992)1992)

Dynamic refinements for pure advection Dynamic refinements for pure advection

experimentsexperiments

Cosine bell advection test (test case 1)Cosine bell advection test (test case 1)

Static refinements in regions of interest Static refinements in regions of interest

(test case 2)(test case 2)

Dynamic refinements and refinement criteria: Dynamic refinements and refinement criteria:

Flow over a mountain (test case 5) Flow over a mountain (test case 5)

Rossby-Haurwitz wave with static refinements Rossby-Haurwitz wave with static refinements

(test case 6) (test case 6)

Snapshots: Advection of a Cosine Bell

Snapshots: Advection of a Cosine Bell

Snapshots: Advection of a Cosine Bell

Snapshots: Advection of a Cosine Bell

Error norms: Cosine Bell Advection

Error norms: Cosine Bell Advection

Days Days

Rotation angle = 45:Errors in SEM are lower than in FV

Error norms after 12 daysError norms after 12 days

Rotation angle = 0

SEM produces undershoots

Errors arecomparable

Snapshots: Cosine Bell at day 3Snapshots: Cosine Bell at day 3

North-polar stereographic projection at day 3 for a = 90

Convergence of blocks in FV

2D Static adaptations2D Static adaptations

• Smooth flow in regimes with strong gradientsSmooth flow in regimes with strong gradients

FV model:

Test case 2, = 45

Error norms: Test case 2Error norms: Test case 2

Days Days

Rotation angle = 45:Errors in FV partly due to errors at AMR interfaces

2D Dynamic adaptations in FV

Vorticity-basedadaptation criterion

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

2D shallowwater test #5:15-day run

Snapshots: Flow over a mountainSnapshots: Flow over a mountain

Longitude Longitude

Geopotential height field (test case 5)

Snapshots: Flow over a mountainSnapshots: Flow over a mountainGeopotential height field (test case 5)

SEM FV

Error norms: Test case 5Error norms: Test case 5

Hours Hours

Errors in SEM converge quicker to the NCAR reference solution

Snapshots: Rossby-Haurwitz WaveSnapshots: Rossby-Haurwitz Wave

Geopotential height field (test case 6) at day 7

Smooth flow through static refinement regions

Alternative AMR: Unstructured Triangular Grid

Alternative AMR: Unstructured Triangular Grid

QuickTime™ and aBMP decompressor

are needed to see this picture.

Hurricane Floyd(1999)

Colors indicate the wind speed

OMEGA model

Courtesy ofA. Sarma (SAIC, NC, USA)

Conclusions & OutlookConclusions & Outlook

Both grids, cubed-sphere meshes and latitude-Both grids, cubed-sphere meshes and latitude-

longitude grids, are options for AMR techniqueslongitude grids, are options for AMR techniques

SEM model shows lower error norms in comparison to SEM model shows lower error norms in comparison to

FV:FV:

Mainly due to high-order numerical methodMainly due to high-order numerical method

Partly due to different AMR approach that does Partly due to different AMR approach that does

not need interpolations of ‘ghost cells’ in not need interpolations of ‘ghost cells’ in

blocksblocks

But: SEM is non-monotonic and non-conservativeBut: SEM is non-monotonic and non-conservative Cubed-sphere grid has clear advantages:Cubed-sphere grid has clear advantages:

No convergence of the meridians, no polar filtersNo convergence of the meridians, no polar filters But, GLL and GL points for numerical method in SEM But, GLL and GL points for numerical method in SEM

are clustered along boundaries of spectral elementsare clustered along boundaries of spectral elements

Future interests: Finite-volume AMR method on a Future interests: Finite-volume AMR method on a

cubed-sphere gridcubed-sphere grid

Both grids, cubed-sphere meshes and latitude-Both grids, cubed-sphere meshes and latitude-

longitude grids, are options for AMR techniqueslongitude grids, are options for AMR techniques

SEM model shows lower error norms in comparison to SEM model shows lower error norms in comparison to

FV:FV:

Mainly due to high-order numerical methodMainly due to high-order numerical method

Partly due to different AMR approach that does Partly due to different AMR approach that does

not need interpolations of ‘ghost cells’ in not need interpolations of ‘ghost cells’ in

blocksblocks

But: SEM is non-monotonic and non-conservativeBut: SEM is non-monotonic and non-conservative Cubed-sphere grid has clear advantages:Cubed-sphere grid has clear advantages:

No convergence of the meridians, no polar filtersNo convergence of the meridians, no polar filters But, GLL and GL points for numerical method in SEM But, GLL and GL points for numerical method in SEM

are clustered along boundaries of spectral elementsare clustered along boundaries of spectral elements

Future interests: Finite-volume AMR method on a Future interests: Finite-volume AMR method on a

cubed-sphere gridcubed-sphere grid