Post on 13-Jan-2016
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).