Mesh refinement methods in ROMS

Laurent Debreu

INRIA, Grenoble, France

In collaboration with

Patrick Marchesiello and Pierrick Penven (IRD, Brest, France)

Outline

Principles of mesh refinement Computational aspects Integration in the ROMS kernel Applications

Outline

Principles of mesh refinement Computational aspects Integration in the ROMS kernel Applications

4

Principles of mesh refinement

• Improve a global solution

two way (fixed or adaptive) mesh refinement

for a given computation cost

Will a multiresolution model performs better than a uniform grid model ?

• Improve a local solution

one or two way (fixed or adaptive mesh refinement)

Is it necessary to use two way nesting ?

• Improve the tracking of a particular structure

Adaptive mesh refinement

5

Principles of mesh refinement

Run the same model on grids with different space/time resolutions

Required for the embedding:

• A time integration algorithm

• Grid’s interactions

Required for the adaptivity:

• A refinement criterion

• An efficient grid’s initialization procedure

6

Principles of mesh refinement

G0

G1

G2

interpolation

1

6

543

2

11

10

987 1312

update

Time integration algorithm

Outline

Principles of mesh refinement Computational aspects Integration in the ROMS kernel Applications

Computational Aspects: the AGRIF software

AGRIF: Adaptive Grid Refinement In Fortran

Goal: « easy » integration of (fixed or adaptive) mesh refinement features in an existing numerical model

• automatic changes of data structures at compile time

• provides interpolation and update operators

• Fortran 77/90, 1D/2D/3D refinement, Staggered grids, Masked fields, parallelization(MPI)

• fixed and/or adaptive grids, clustering algorithm, restoring algorithm

Some features:

Computational aspects: ROMS_AGRIF

http://www.brest.ird.fr/Roms_tools

AGRIF in ROMS:

• each grid has it own input file and outputs

• grid’s locations specified in AGRIF_FixedGrids.In

• works in OPENMP/MPI

• forcings, initial conditions made through the « nesting gui »

2

20 45 34 59 3 3 3

30 55 70 89 3 3 2

0

1

10 30 20 40 5 3 5

0

10

Computational aspects: AGRIF in other ocean models

AGRIF in the OPA model

Outline

Principles of mesh refinement Computational aspects Integration in the ROMS kernel Applications

Integration in the ROMS numerical kernel: Roms: Time step, Boundary conditions

adjust to

• pre_step3D 1/ 2 1/ 2,n nu T BC on 1/ 2 1/ 2,n nu T

• step2D 1/ 2 1 1, ,n n nU U

• step3D_uv1

adjust to 1/ 2nu

1 1/ 2n n nu u t rhs

• step3D_uv2

• set_HUV2

1nu BC on 1nu

* 11

2n nu u u

adjust to *u

1 * 1/ 2( , )n n nT T t DIV u t BC on 1nT

• step3D_t

1/ 2nU

1nU

1/ 2nU

UP 1/ 2 1 1, ,n n nU U

UP 1 *,nu u

UP1nT

13

Integration in the ROMS numerical kernel: barotropic mode, boundary conditions

Characteristic variables :

0 0f f c c

g gU U

H H

0

gU

H

On a western boundary :

0

gU

H

0

gU

H

is the incoming characteristic

is the outgoing characteristic

0 0f f upwind upwind

g gU U

H H

0

1

2f c upwind c upwind

gU U U

H

(at speed )0 0U gH

14

Integration in the ROMS numerical kernel: barotropic mode

• One way

Enforces volume conservation :*

U Uc f

f f c fU U U U

• Two – way : , (no free surface update) Uf

c fU U

(including boundary points)

Update area

15

Integration in the ROMS numerical kernel: 3D velocities

3D :

0

1ˆ ˆ ˆ( ) ( )

2f f c c upwind upwind c upwindH u H u H u gH H H

• Two – way :uf

c fu u

(including boundary points)

ˆf f fu u u

16

Integration in the ROMS numerical kernel: conservation

• Let be a conserved quantity:

Define by

• At initial time :

• conservation of flux equality at fine/coarse grid interfaces

• (in one way interaction) two solutions

• correct or

• correct such that then correct

• (in two way interaction) two other solutions:

• correct

• correct (in ) such that

K

( ) 0K

uKt

K

\H hK K K

0 \ 0 0 0( ) ( ) ( ) ( ( )!)H h HK t t K t t K t t K t t

hK

K

0 0H

H h

t T t T

H h ht t t tu K u K

hu

huh H

h Hu u

hK

Hu

HK 1 1

h H

n n n n n nH H h h H HK K t u K u K

\

17

Integration in the ROMS numerical kernel: 3D tracers

1 1 1, , ,( ) /n n n

c W c W f c c WT T t Flux Flux H

fc f

c

HT T

H

• Two-way:

*Uf

f f f fFlux H u T

W

At (first two) interior grid points

18

Integration in the ROMS numerical kernel: topography construction

Topography and initial (tracers) fields

(1 ) Ifine HR coarseK K K

IcoarseK satisfying

,

, ,1 1

2 2

Icoarse coarse

i j

I W I E W Ecoarse coarse coarse coarse

j

K K

K K K K

19

Integration in the ROMS numerical kernel: summary

• Boundary conditions

• 2D velocities : Characteristics variables method

• 3D velocities : boundary conditions consistents with 2D BC

• 3D tracers : clamped

• Update (two way)

• conservative updates (two first cells only)

• flux correction for tracers

• topography definition

• identical volume and faces area in first two cells

Outline

Principles of mesh refinement Computational aspects Integration in the ROMS kernel Applications

21

Applications: (One/Two way comparison)

Peru application: Coarse grid domain results

Coarse grid Run Nested Run

Surface temperature and velocites

22

Applications: (One/Two way comparison)

Peru application: Fine grid domain results

23

Mesh refinement methods in Roms: conclusions and perspectives

• Different applications have been done in one way nesting

• Two-way nesting shoud now be extensively tested

• « fully » two way scheme

• differents topographies on coase and fine grids

• exact conservation of volume and tracers

• Future two way developments

• Time refinement

• sponge layer on instead of

• treatment of momentum fluxes

( )f cu ufu