Grid Generation for OGCMs with Schwarz‐ChristoffelConformal Mappings

Shiming Xu, Yanluan Lin, Wenyu Huang, Xiaomeng Huang, Fanghua Xu

Department of Earth System Sciences Tsinghua University

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23rd CESM Annual Meeting, Boulder, CO, June 2017

Outline

• Motivation• Schwarz‐Christoffel mapping based grid generation• Sample grid & applications• Updates on new ocean grid in CIESM development

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State‐of‐the‐Art OGCM Grids

Global Models Regional Models Models w/ Irregular Meshes

Grid Analytical Forms Coastline Following Coastline Fitting

Precision of Bathemetry Effects* BAD GOOD ACHIEVABLE

Land Occupation HIGH LOW NONE

Computational Efficiency LOW HIGH HIGH

Multi‐Scale Modeling NONE NONE ACHIEVABLE

Friendliness w/ Existing Models GOOD GOOD BAD

3* isobath alignment, river discharge & plumes, etc.

V.S.

Tripolar Dipolar

&

POP, MOM, NEMO … POM, ROMS, … FVCOM, ICOM, …

Lateral Boundaries As Key Regions in (High‐Resolution) Ocean Models

• Smaller Rossby radii due to shallower bathymetry• Strong model biases, due to complex air‐sea processes• A key area of kinematic energy cascading of mesoscale eddies• Biogeochemistry processes• Closely related to human activities

4[Zhai et al, 2010] [Wang et al, 2014]

Coastal Kelvin Waves – A Case Study

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[Griffith, JCP, 2013]

[Frischknecht et al, JGR, 2015]

10 2 10 1 10010 2

10 1

tan

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Motivations

• Computation: remove continents from grid index space • Science‐I: aligning (large‐scale) continental boundaries & isobaths to grid lines

• Science‐II: multi‐scale simulation support with higher spatial resolution on continental boundaries

• Practicality: compatible with existing Bryan‐Cox‐Semtner type OGCMs

• POP, MOM, NEMO | POM, ROMS

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Geophysical Fluid Dynamics

Models & Algorithms

Parallel Computers

Outline

• Motivation• Schwarz‐Christoffel mapping based grid generation• Sample grid & applications• Updates on new grid in CIESM development

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Conformal Mappings

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f

Schwarz‐Christoffel Mapping Deals with Irregular Boundaries …

• Riemann Mapping Theorem for the existence of a conformal mapping between:

• Schwarz‐Christoffel mapping• User‐specified boundary information and SC Mapping

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[Driscoll and Trefethen, 2002]

Schwarz‐Christoffel Conformal Mapping for Multiply Connected Regions

** *

(c) Canonical slits I - parallel lines

(d) Canonical slits II -radial lines

(e) Canonical slits III -circular lines

*

(a) A sample region (connectivity=5)

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(b) Canonical circles

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• Canonical forms• Construction of maps:

• Reflection principle [DeLillo and Kropf, 2011]

• Laurent series expansion [DeLillo et al, 2013]

SCC‐Grid – Multiply‐Connected Schwarz‐ChristoffelConformal Mapping based OGCM Grid

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Shiming Xu, Bin Wang and Jiping Liu, On the use of Schwarz–Christoffel conformal mappings to the grid generation for global ocean models, Geoscientific Model Development, 8, 3471‐3485, doi:10.5194/gmd‐8‐3471‐2015, 2015.

Outline

• Motivation• Schwarz‐Christoffel mapping based grid generation• Sample grid & applications• Updates on new grid in CIESM development

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180oW 135oW 90oW 45oW 0o 45oE 90oE 135oE 180oW

75oS

30oS

0o

30oN

75oN

Sample Global Grid (1/4 deg.) 

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* 1 in every 10 points shown

Cell Sizes in Grid Index Space

Meridional Zonal

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15

135

o W

90 oW

45oW

0o

45o E

60

o N

160 oW 150oW 140oW 130oW 120oW

110o W

25 oN

35 oN

45 oN

55 oN

65 oN

100oE 110oE 120oE

15oN

25oN

35oN

10 oW 0o

10oE 20oE 30oE 40o E

30 oN

40 oN

50 oN

Spin‐Up Run with POP model

• Idealized forcings• Actions needed: replacement of grid files• 50‐yrs spin‐up run• Climate simulations w/ CESM underway

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Outline

• Motivation• Schwarz‐Christoffel mapping based grid generation• Sample grid & applications• Updates on new grid in model development

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Ocean Model for CMIP6

•New 0.5 deg. dipolar SCCGrid (720x560)•Canuto vertical mixing scheme•A scalable barotropic solver (already in POP2)•dT‐dependent wind stress scheme

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New Medium‐Resolution Ocean/Ice Grid 

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• Schwarz‐Christoffel mapping for the North Atlantic‐Arctic region• Up to 94% of the ocean is pure Lat.‐Lon. 

• Higher than any other global model grid

Comparisons

Grid Model Type Turning Lat.

Non. Lat-Lon (Oceanic Area)

Scaling FactorTransition

(Oceanic Area)

ORCA025 NEMO Tripolar 20.31 24.5% 3.21

HYCOM HYCOM Tripolar 46.99 8.8% 2.09

GX1V6POP -CESM

Dipolar 0 42.9% 1.02

TX0.1V2 Tripolar 28.55 18.7% 1.23

SX0.5V3 POP -CESM Dipolar Non-

Uniform 6.2% 1.10

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New Medium‐Resolution Ocean/Ice Grid 

GX1V6 (320x384x60)

SX0.5V2 (720x560x46)

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Spin‐Up of OI w/ CORE‐2 Forcings (1)

• AMOC

• SST

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Spin‐Up of OI w/ CORE‐2 Forcings (2)

SST (yr 351~400) AMOC (yr 351~400)

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Spin‐up of OI w/ CORE‐2 Forcings (3)

Spin‐Up (yr 0400) Mean of Yr 351~400

24Sea ice

Atmosphere model for CMIP6

• MG1.5 single ice scheme

• Park‐RH/Zhang condensation a diagnostic PDF cloud scheme

• ZM new ZM

• Four stream shortwave radiation

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Surface net SW flux

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Preliminary Coupled Run (PI‐Control, 1870‐79)

New

Old

Diff

Double ITCZ problem alleviated

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New

Old

Diff

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Seasonal variation of P over the Pacific

Enhanced Walker circulation [10S‐5S]

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New

Old

Diff

Summary & Conclusion

• New grid generation for current OGCMs• Alignment of grid lines w/ large‐scale coastlines• Removal of continental areas, free computational load balancing• Better coastal spatial resolution, enabling easier spatial refinement

• Application as easy as replacing grid files• No need to modify the model

• Preliminary application shows reasonable results• Coupled runs underway

• Future work includes high‐resolution (0.1‐deg, tripolar) grid generation and simulation

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Thanks. Any questions?

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Computational Infrastructure for CIESM

• TODO

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