Application of the CSLR on the “Yin-Yang” Grid in Spherical

Geometry

X. Peng 　 (Earth Simulator Center)F. Xiao 　　　 (Tokyo Institute of Technology)K. Takahashi (Earth Simulator Center)

PDE2004-15:10-15:30 July20,2004, Yokohama

Present requirements and Issues

• Model development (global, non-hydrostatic, high-resolution) on sphere

• Requirement of high-accuracy, high-efficiency and high-performance computation to save CPU time.

Problems in global high-resolution model– Singularity of ordinary latitude-longitude coordinate– Solid Courant number limitation– “Negative mass” and non-conservative advection

A possible solution: using positive-definite, conservative semi-Lagrangian scheme on quasi-uniform grid without singular point.

CIP-CSLR

• Conservative semi-Lagrangian scheme with rational function (Xiao et al. 2002) based CIP (Constrained Interpolation Profile, Yabe et al. 1991)

• Predicts both the cell-integrated and interface values, which makes it more accurate but increase little computation.

• Be conservative, oscillation-free, positive-definite but no additional limiter needed.

• A high-accuracy scheme over merely one cell.

CIP algorithm

1D advection

Differentiate (1) in x direction, we get

Here, (1) and (2) are advection equation in the same formation, the only difference is the forcing term (RHT).

0

x

fu

t

f(1)

gx

u

x

gu

t

g

x

fg

(2)

CIP-CSLR algorithm

1D advection (flux-form) is

Suppose

We have

Using the same stencil, we construct conservative scheme. Also the rational function make the scheme be positive, monotonic convexity preserving

0

x

ug

t

g(3)

dxxgxf )()(

0

x

fu

t

f(4)

Refer to Xiao et al. 2002, JGR,107(D22),4609

Yin-Yang grid【 Yi Jing: the Book of Changes 】The universe （ both space and time ） can be divided into Yin and Yang, Which is composed with metals( 金 ), water( 水 ), wood( 木 ), fire( 火 ) and soil( 土 ). For example, the moon is Yin, and the sun is Yang. The energy of the atmosphere comes from the sun.

Provided by Dr. Kageyama, ESC, who proposed the Yin-Yang grid

Yang (N) zone Yin (E) zone Yin-Yang composition

☯

＋ ＝

Yin-Yang grid structure

In the Mercator projection

Some features of Yin-Yang grid

• Overset grid• Orthogonal coordinates （ same as the lat-lon geometry ）

• No polar singularity-- high computational efficiency.

• The same grid structure of Yin and Yang components.

• Easy to nest• Easy to parallelize (with domain selecting)

• But need to take care of conservation law.

The application to latitude-longitude or Yin-Yang grid system

Dynamical equation in spherical geometry

Modified to fully-flux form in

For Yin-Yang grid system, the same equation is used for both Yin and Yang zones.

q

qqvqu

acos

coscos

cos

1

t

q 2

2

qqq

avq

au

t

qcos

cos

Solid advection in Yin-Yang grid (np=40,CFL=1)• Wilianmson et al. (1992) test case 1.• Linear interpolation for Yin&Yang boundary.• Initial condition is distributed to lat-lon grid first, then interpolate t

o Yin, Yang zone.• Yin, Yang is plotted separately.

dlat=dlon=2.25°

Zonal advection Meridional advection

YANG

YIN

YANG

YIN

Results （２）　 np=40 CFL= ３

• Wilianmson et al. (1992) test 1• Linear interpolation to Yin,Yang’s boundary• CFL=3

dlat=dlon=2.25°

Zonal advection Meridional advection

YIN YIN

YANGYANG

Global mass variation dlat=dlon=1.125°

α=0.0

α=π/2

The conservative scheme

EE

dgdg NE

NN

dgdg NE

The necessary and sufficient condition for global conservationis as

The sufficient condition is

dgdg NE

dΓ denotes any part of the boundary of N,E, e.g. EF.

Yang

Yin

Solid advection test np=80,CFL=1

α=0.0

α=π/2

Solid advection test With large Courant number

np=80

Summary Precise advection is achieved with the CIP-CSLR that is

positive-definite, shape-preserving. High-efficient computation is also successful using CIP-CSLR

on the Yin-Yang grid. The minimum and maximum grid intervals in the Yin-Yang system bears a proportion of 0.707. Much longer time step is available under the same Courant number, in comparing with ordinary Lat-Lon grid. It is 144 times larger at the resolution of 0.5625 degree.

Being orthogonal grid, it is easy to implement time splitting procedure

Accuracy in polar region is greatly improved. Large Courant number is available, which is a possible in high

resolution model. Global conservation is developed, and is confirmed with the

idealized advection.