EGU General assembly 2014, AS 1.5

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!


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?)


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


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


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)


CCS-RG Monotonicity

Diagnostic filter (DF)

Monotonicity violation

Tracer mass

LL

L

qq

RR

R

qq

Alternative option: Barth & Jespersen 1989 filter


Reduced grid

Regular lat-lon gridMeridian convergence

Reduced gridLess points in latitude row near the

poles


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)


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


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


Deformational flow. Q1

No filter

Diagnostic filter

BJ filter



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)



No filter

Diagnostic filter

B&J filter



No filter

Diagnostic filter BJ filter

Day 12 exact



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)


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)


DCMIP test 1-2. Idealized Hadley cell

No filter DF BJ


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


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


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

http://nwplab.inm.ras.ru/DCMIP-advResults-17.04.14.pdf


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


CCS-RG: tracer-mass coupling

( ) ( ) ( )q x q x x q

Barth & Jespersen filter:

Unlimited or Diagnostic filter:

( )q q q x


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


DCMIP test 1-3. Flow over orography

Hybrid coordinates

Sigma coordinates

top

s top

p p

p p

/ sp p


DCMIP test 1-3. Flow over orography

Regular grid 10x10x60 levs. Dt = 3600 sec.

Documents

EGU General assembly 2014, AS 1.5