A cell-integrated semi-Lagrangian dynamical scheme based on a step-function representation

Eigil Kaas, Bennert Machenhauer and Peter Hjort Lauritzen

Danish Meteorological InstituteLyngbyvej 100, DK-2100 Copenhagen, Denmark

SRNWP-NT mini workshop inToulouse 12-13 December 2002

The goals

To construct a dynamical scheme for atmospheric dynamics and tracer transport with all the following properties:

• Indefinite order of accuracy for advection by a flow that is constant in time an space (except for initial truncation).

• Full local mass conservation.• Positive definite.• Monotonic.• Numerically effective.

Our solution: SF-CISLStep-Function Cell Integrated Semi-Lagragian scheme combined with a semi-implicit scheme for inertia-gravity wave terms.

OUTLINE• The basic idea behind step function advection.

• The basic idea behind CISL.

• 2-D passive test simulations.

• A new semi-implicit formulation of CISL for the shallow water equations.

• Tests simulations.

• Discussion: efficiency, generalisation to 3-D and to spherical geometry …

What is CISL ?

Nair and Machenhauer (2002)dd ji

njiji

1nji VhVh ,,,,

)()( tVttV

hdVdt

dhdV

dt

d

Integrate the continuity equation over a time dependent Lagrangian volume:

0hdVdt

dtV

)(

What is CISL ?

Nair and Machenhauer (2002)

jiy ,

jix ,

jijiji yxV ,,,

dd jinjiji

1nji VhVh ,,,,

djiV ,

The basic idea behind step-function advectionT

ime

x-direction

idx

ih ih

i i+1i-1

)(

)(

1ii

1ii1ii

ii

i

hhabsfor1

hhabsforhh

hh

dx 1iii hhh ,

1iiiii hdx1hdxh )(

10dxi ,

1iii1i

1iiii1i

hhhh

hhhhh

,

,

Spatial truncation (horizontal diffusion)

2-D step function representation

jih ,

y

jih ,

jih ,

1j1ij1iji1jijiji

j1ij1ijijijijiji

hdy1dx1hdy1dx

hdydx1hdydxh

,,,,,,

,,,,,,,

))(()(

)(

Order of calculations

7. Calculate for each departure cell (=(result of 4.)/ )nji d

h ,

6. Calculate for each ”north-south” intersecty

nji d

h ,

4. Perform the cell integration (result = )dd ji

nji Vh ,,

5. Calculate for each departure grid cell point corner pointnji d

h ,

8. Do the ”horizontal diffusion” (i.e. modify if needed)

2. Calculate the departure grid cell corner points.

1. Calculate dxi,j and dyi,j for each Eulerian grid cell.

3. Calculate the ‘s and define the re-mappings.djiV ,

djiV ,

nji d

h ,

10. Calculate the relative areal change for each step function

Order of calculations (cont.)

9. Calculate new values of dxi,j and dyi,j based on upstream

values of , and .n

ji dh ,

yn

ji dh ,

nji d

h ,

12. Calculate the final values of and from the values of

dxi,j , dyi,j and .

y1n

jih ,

1njih

,

1njih

,

11. Use this information to calculate the final value of )( ,,

,,

nji

ji

ji1nji d

d hh

))(()(

)(

,,,,,,

,,,,,,,

1j1i1j1id1j1ij1ij1idj1i

1ji1jid1jijijidjiji

dy1dx1Vdydx1V

dy1dxVdydxVd

)])(()(

)([

,,,,

,,,,,,,

1j1i1j1ij1ij1i

1ji1jijijijijiji

dy1dx1dydx1

dy1dxdydxyx

ji

ji d

,

,

2-D passive test simulationsSolid body rotation, 6 rotations, 96 time steps per rotation

100 x 100 grid points/cells

A new semi-implicit formulation of CISL for the shallow water equations.

y

v

x

uDFD

dt

d

yfu

dt

dvx

fvdt

du

s

s

,

)(

)(

vus

,

depth of fluid

height of topographyvelocity components

a

d

d

nn1nE A

AFt

~

A new semi-implicit 2-D CISL formulation (1)

)~

( 1na

1na0

1nE

1nideal 2

t

DD

)~

( 1n1n0

1nE

1n DD2

t

~ indicates time extrapolation from n and n-1

The traditional two-time level SL-scheme:

d

nn1nad

nnndd

nn Ft2

tFt

2

tFt

DD ~

CISL explicit forecast:

DD t1A

A

A

AAt

a

d

a

da

)(

Ideal semi-implicit CISL forecast:

a

1nd

a

nd

d

nn

A

A50

A

A50Ft

ad ~..

Elliptic equation too complicated !

1n1nd

nn

d

nn1nE D

2

tD

2

ttF

~~

A new semi-implicit 2-D CISL formulation (2)

a

dd

nna0

1n1na0

a

dd

nn1n

AAD

2

t

D2

t

AAFt

~ˆ

)~

(

~

D

D

The basic explicit forecast

Inconsistent implicit correction term

Correction of the inconsistency in the previous time step

Tests of the semi-implicit SF-CISL in a shallow water channel model

2000

0 km

20000 km

Spectral Eulerian model:3 time level spectral transform scheme (double Fourier series) Semi-implicit formulation (Coriolis explicit)Reasonable implicit horizontal diffusion

Eulerian grid-point model: 3 time level centered difference scheme Semi-implicit formulation (Coriolis explicit)Reasonable explicit horizontal diffusion

Interpolating semi-Lagrangian (IPSL) model:2 time level scheme based on bi-cubic interpolation Semi-implicit formulation (Coriolis implicit)No additional horizontal diffusion

SF-CISL model: 2 time level scheme based on step-function representation Semi-implicit formulation (Coriolis implicit)No additional horizontal diffusion

Four different model formulatoins

Spectral Eulerian modelEulerian grid-point model

Interpol. semi Lagrangian model SF-CISL model

48 hour ”forecasts” at low resolution.Parameter: height field

Spectral Eulerian modelEulerian grid-point model

Interpol. semi Lagrangian model SF-CISL model

48 hour ”forecasts” at high resolution.Parameter: height field

Spectral Eulerian modelEulerian grid-point model

Interpol. semi Lagrangian model SF-CISL model

48 hour ”forecasts” at low resolution.Parameter: passive tracer

Spectral Eulerian modelEulerian grid-point model

Interpol. semi Lagrangian model SF-CISL model

48 hour ”forecasts” at high resolution.Parameter: passive tracer

Interpol. semi Lagrangian model

SF-CISL model

10 day ”forecasts” at high resolution.Parameter: passive tracer

Discussion and conclusion

• Cost Passive advection 1.7 times IPSL.

• Truncation/horizontal diffusion This is a critical point

• Memory consumption When step function geometries are defined from the total mass field they could in principle be used for all prognostic variables (i.e. only one prognostic variable per tracer variable)

• The passive advection tests in realistic flow demonstrate the monotonicity, mass conservation and positive definiteness

• The shallow-model works with the new scheme !No noise due to step functions

”Bad”

”Good”

Discussion and conclusion

• Other possible formulations“horizontal diffusion/truncation”Choice of step-functions.

• Generalisation to 3-DCascade interpolation (Nair et al. 1999) for the vertical problem.Prognostic variables: 3-D cell averages, horizontal averages at model levels, vertical averages at grid points, grid point values.

• Spherical geometryNo real problem (reduced lat-lon (or Gaussian) grid).