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EGU General assembly 2014, AS 1.5. A three-dimensional Conservative Cascade semi-Lagrangian transport Scheme using the Reduced Grid on the sphere (CCS-RG). V. Shashkin 1,2 ( [email protected] ), R. Fadeev 1 , M. Tolstykh 1,2. April 29, 2014. Desirable features for transport schemes : - PowerPoint PPT Presentation
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EGU General assembly 2014, AS 1.5
A three-dimensional Conservative Cascade semi-Lagrangian transport Scheme using the
Reduced Grid on the sphere (CCS-RG)
V. Shashkin1,2([email protected]), R. Fadeev1, M. Tolstykh1,2
April 29, 2014
1 - Institute of Numerical Mathematics, Russian Academy of Sciences
2 - Hydrometeorological centre of Russia
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
Desirable features for transport schemes:(Rasch and Williamson, QJRMS, 1990 & Lauritzen et al. , GMD, 2010)
• Accurate• Transportive• Local• Invariant (mass etc.) conservation• Monotonicity preserving• Non-linear correlations preserving• Computationally efficient
No ideal scheme invented
A lot of schemes!
Let’s have a look at one more!
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
Semi-Lagrangian method (in GCMs)
To be … … or not to be?Stable for large CFL => large time-steps Large CFL => scalability problems
(efforts to make scalable)
Inherently multi-tracer efficientNon-conservative (mass, energy,
enstrophy etc)Spurious orographic resonance
…
solved!
The discussion is still open!
Our believe: SL is ideal at least for relatively low-resolution simulations with relatively low (104) number of cores (Russian reality for future decade?)
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
CCS-RG basics: Mass-conservative SL (finite-volume SL)
( )
0V t
dq dV
dt
Integral formulation of transport equation:
Air densityLagrangian air volume
Tracer mass conservation provided no physical sources/sinks
*
1
*
1
( )
( )
( ) ( )ijk
nijk
nijk
n nijkijk
V
V t V
V t V
q V q dV
- Arrival volume = Grid cell
- Departure volume
Prognostic variable: Tracer density
Time discretization:
Tracer specific concentration
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
CCS-RG basics: 1D finite-volume SL
*1/2
*1/2
1 1( ) ( ) ( )
i
i
xn ni
x
q q x dVx
( )n
iq
2( ) ( ) ( ) ( ) ( )4
nni c ci
bq x q a x x b x x x PPM, Colella &
Woodward, JCP, 1984
Subgrid reconstruction
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
CCS-RG: spatial approximation
*
1 1( ) ( )
ijk
n nijk
ijk V
q q dVV
- Integral over departure volume
• Approximation of departure cell geometry O(Δx2)• Tracer density approximation O(Δx3)
3D integral
3 x 1D integrals (remappings)(using cascade approach)(2D – Nair et al, MWR, 2002,
3D – Shashkin, HMC Proc, 2012)
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
CCS-RG Monotonicity
Diagnostic filter (DF)
Monotonicity violation
Tracer mass
LL
L
RR
R
Alternative option: Barth & Jespersen 1989 filter
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
Reduced grid
Regular lat-lon gridMeridian convergence
Reduced gridLess points in latitude row near the
poles
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
Reduced grid: How to build it?
Physical approach: keep longitudinal grid step constant (in length units) => works bad!
Spectral approach: use asymptotic properties of associated Legendre polyn. => good for spectral models
Interpolation accuracy approach (Fadeev, RCMMP, 2013)
Given the fixed ration of central symmetric function interpolation errors on the regular and reduced grids minimize number of grid points:
/2 /2
/2 /2
0, 0, 0,red reg regd d
RMS Interpolation error
2 2( , ) cosI exact
Sphere
f f a d d Function center
SL shallow water results with this rg design (Tolstykh, Shashkin, JCP, 2012)
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
Reduced grid, structure
reduced grid of 10x10 resolution (at the equator)
15% less points
20% less points
25% less points
30 % less points
… than in 10x10 regular grid
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
DCMIP 1-1 testcase, Deformational flow (Kent et al, QJRMS, 2014)
Tracer Q1. Cosine bells
T=6 days (maximum deformation)T=0 days, T=12 days (initial distribution, exact solution)
Tracer Q3. Slotted cylinder
T=0 days, (vertical cross-section at 1500 west)
Initial distribution
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
Deformational flow. Q1
No filter
Diagnostic filter
BJ filter
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
Deformational flow. Q1
grid Regular 10x10x60 levs, time-step 1800s Reduced 10x10 (equator) x 60 levs, 30% less points
filter l1 l2 l∞ max l1 l2 l∞ max
No .159 .132 .275 .078 .167 .134 .275 .078
DF .150 .140 .285 -0.015 .155 .143 .285 -0.014
BJ .223 .177 .305 -0.096 .232 .180 .305 -0.097
CAM-FV(from Kent et al.)
MCore(from Kent et al.)
.121 .0998 .192 .177 .155 .263
•DF improves l1•BJ is more diffusive than DF•Reduced grid affect error norms slightly (in rotated test-variants too)
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
Deformational flow. Q3
No filter
Diagnostic filter
B&J filter
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
Deformational flow. Q3
No filter
Diagnostic filter BJ filter
Day 12 exact
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
Deformational flow. Q3
grid Regular 10x10x60 levs, time-step 1800s Reduced 10x10 (equator) x 60 levs, 30% less points
filter l1 l2 l∞ max l1 l2 l∞ max
No .022 .217 .843 .203 .022 .222 .849 .203
DF .026 .263 .817 -0.145 .026 .265 .835 -0.145
BJ .029 .281 .827 -0.245 .029 .282 .850 -0.249
CAM-FV(from Kent et al.)
MCore(from Kent et al.)
0.024 0.252 .859 0.025 0.235 0.844
•DF improves l∞•BJ is more diffusive than DF•Reduced grid affect error norms slightly (in rotated test versions too)
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
Deformational flow. Non-linear correlations Q1, Q2
No filter DF
BJ
22 10.9 0.8q q
real mixing range pres unmixing
overshooting
UNLIM 1.19e-3 2.78e-4 8.67e-4
DF 1.24e-3 3.19e-4 0.00
BJ 1.68e-3 1.68e-4 0.00
Т=6 days (maximum deformation)Correlation diagnostics (Lauritzen & Thuburn, QJRMS, 2012)
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
DCMIP test 1-2. Idealized Hadley cell
No filter DF BJ
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
DCMIP test 1-2. Idealized Hadley cell
CCS-RG UNLIM CCS-RG DF
l1 l2 l∞ max l1 l2 l∞ max
20, 30 levs
0.16 0.16 0.35 8.68E-3 0.18 0.21 0.49 2.06E-14
10, 60 levs
3.28E-2 4.05E-2 0.12 1.97E-3 4.14E-2 6.65E-2 0.23 4.75E-14
0.50, 120 levs
4.72E-3 6.70E-3 2.54E-2 4.59E-5 7.15E-3 1.32E-2 6.29E-2 7.21E-14
conv. 2.54 2.28 1.89 2.32 2.01 1.48
CCS-RG BJ MCore (from Kent et al.)
20, 30 levs
0.22 0.24 0.53 1.95E-14
0.1368 0.1659 0.4214
10, 60 levs
6.44E-2 9.18E-2 0.30 3.73E-14
0.0286 0.0462 0.1586
0.50, 120 levs
1.54E-2 2.56E-2 0.11 6.72E-14
0.0063 0.0113 0.0435
conv. 1.91 1.61 1.16 2.22 1.94 1.64
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
Conclusions
• CCS-RG performs well and is competitive to CAM-FV and MCore (in terms DCMIP 1-x testcase diagnostics)• Error norms grow only 5% when using reduced grid (maybe DCMIP case 1-1 even rotated is not a severe test for reduced grid desing) => reduced grid using isn’t limited by advection accuracy
Two monotonic options are tested:• Diagnostic filter is less diffusive and more accurate in terms of l1, l2, l∞ error norms
•Diagnostic filter is better for species with rough distribution (hydrometeors etc)• Barth & Jespersen filter is better when tracer correlation is important
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
Thank you for attention!
More CCS-RG results (error norms, pictures) including DCMIP test 1-3 results can be found at:
http://nwplab.inm.ras.ru/DCMIP-advResults-17.04.14.pdf
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
CCS-RG Monotonicity
Barth & Jespersen filter 1989 (BJ filter)
2( ) ( ) ( )4
nni c ci
bq x q a x x b x x x
Scaling factor
q x q x
1 1 1 1min , , max , ,ni i i i i i iq q q q x q q q => No spurious max/min =>
=> monotonic scheme
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
CCS-RG: tracer-mass coupling
( ) ( ) ( )q x q x x q
Barth & Jespersen filter:
Unlimited or Diagnostic filter:
( )q q q x
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
FV-SL scalability, overview
Scheme Publication Num of cores CFL
CCS-RG Work in progress ? ?
CSLAM-HOMME Erath et al. Proc. Comp. Science., 2012
Lauritzen et al, JCP, 2010
4056(16244 – high
res)
< 1 *
SPELT-HOMME Erath & Nair, JCP, 2013 16244 < 1 *
FARSIGHT White III & Dongarra, JCP, 2011 10000 ~ 10
* - CFL ~ 1 is still large from high-order Eulerian SE point of view
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
DCMIP test 1-3. Flow over orography
Hybrid coordinates
Sigma coordinates
top
s top
p p
p p
/ sp p
V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014
DCMIP test 1-3. Flow over orography
Regular grid 10x10x60 levs. Dt = 3600 sec.