[1] Akira Kageyama and Tetsuya Sato, The "Yin-Yang Grid": An Overset Grid in Spherical Geometry, to be published Geochem. Geophys. Geosyst., E-print: physics/0403123

[2] D. L. Williamson et al., A Standard Test for Numerical Approximations to the Shallow Water Equation in Spherical Geometry, J. Comput. Phys., 102, pp.211-224(1992)

[3] Wicker L. J. and W. C. Skamarock, Time-Splitting Methods for Elastic Models Useing Forward Time Schemes, Mon. Wea. Rev., 130, pp.2088-2097(2002)

[4] Peng X., F. Xiao, K. Takahashi and T. Yabe, Conservative CIP transport in meteorological models, submitted to JSME International Journal, 2004

[5] R. Jakob, J. J. Hack and D. L. Williamson, Solutions to the Shallow Water Test Set Using the Spectral Transform Method., NCAR/TN-388+STR, 1993

References

Near Future WorkIn this study, we showed validation results of introduced

simulation code on Yin-Yang grid system with mass conservation. Furthermore, we will introduce conservation scheme of the total energy and potential enstrophy.

Figure 8 shows that Rossby-Haurwitz wave shape has been propagated from the west to the east without change from initial field after 14 days integration. Results shows that the height field does not have inferiority with the reference solution[5] (not shown).Fig.8. Height field at (a) day 0, (b) day 1, (c) day 7,

and (d) day 14. Contour interval is 100 m.

(b)

(d)

(a)

(c)

Test Case 6 : Rossby-Haurwitz Wave

The mountain is located at (30N,90E) as shown a circle in Figure 7(a) and has the shape of a cone. Initially, wave spreads all over the sphere as time integration progresses. The height field was indistinguishable from the reference solution[5] (not shown).

Fig.7. Height field at (a) day 0, (b) day 5, (c) day 10, and (d) day 15. Contour interval is 50 m.

(b)

(d)

(a)

(c)

Test Case 5 : Zonal Flow over an Isolated Mountain

The solid body rotation field is maintained as shown in Figure 5. The l2(h) norm is described in Figure 6. Although absolute value under the condition of α=π/2 in Fig.6(b) was larger compared with results from an experiment under condition of α=0 in Fig.6(a), the 2nd-orderaccuracy is maintained.

Fig.6. l2(h) norm.

(a)

(b)

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

0 1 2 3 4 5days

l 2(h)

np= 40 (263.4km)np= 80 (128.3km)np=160 ( 63.3km)

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

0 1 2 3 4 5days

l 2(h)

np= 40 (263.4km)np= 80 (128.3km)np=160 ( 63.3km)

Fig.5. Height & velocity field at day 5.

(a)

(b)

Test Case 2 : Global Steady State Nonlinear Zonal Geostrophic Flow

The bell structure is mostly maintained on Yin-Yang grid system. Even though the bell passes through overlapped grid boundary of Yin-Yang Grid, absolute error value is small.

Since the accuracy of conservative interpolation scheme with 2nd-order was used, the 2nd-order accuracy is maintained for any horizontal resolution as shown in Figure 3. In Figure 4, well mass conserved results were presented, because time evolution of relative error of the mass changed within computational rounding error.

Fig.4. Relative error of the mass within a

one-revolution advection.

-1.2E-15-1.0E-15-8.0E-16-6.0E-16-4.0E-16-2.0E-160.0E+002.0E-164.0E-166.0E-168.0E-161.0E-15

0 2 4 6 8 10 12days

Rel

ativ

e Er

ror

α=0α=π /2

Fig.3. l2(h) norm.

(a)

(b)

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0 2 4 6 8 10 12days

l 2(h)

np= 40 (263.4km)np= 80 (128.3km)np=160 ( 63.3km)

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0 2 4 6 8 10 12days

l 2(h)

np= 40 (263.4km)np= 80 (128.3km)np=160 ( 63.3km)

Fig.2. Height field at day 12.

(a)

(b)

Test Case 1 : Advection of Cosine Bell over the Pole

Williamson et al.[2] are shown as follows. Test case 1 and 2 were performed with horizontal resolution of 63 km, 128 km, and 263 km, and test case 5 and 6 were carried out with 63 km horizontal resolution.

Results

Variable Arrangement Advection Term Non-Advection Term

f E � � = f N � �

: Local and global mass conservation condition

f E(N) : Flux

� � : A part of boundary of Overlap region

* Refer to Dr. Peng's following presentation. Peng X., et al., Application of the CSLR on the "Yin-Yang Grid" in Spherical Geometry, PDE2004, pp21

Items

Coordinate System

Time Integration

Interpolation on Boundary

Differencing

Contents

Yin-Yang Grid + Spherical Coordinate System Arakawa-C Grid

5th-order FDM[3]

4th-order FDM, 4th-order Interpolation 4th-order Runge-Kutta Method

5th-order Lagrange's Interpolation Polynomial Local and Global Mass Conservative Interpolation Scheme[4]

Table.1. Numerical schemes.Numerical Schemes

where h* is the depth of the fluid and h is the height of the free surface above a reference surface height. hs denotes the height of the underlying topography such as h=h*+hs. v is the horizontal vector velocity of the fluid with components u and v in the longitudinal (λ) and latitudinal (θ) directions, respectively. g presents the gravitational constant and f the Coriolis parameter.

0,tan)(

0cos

tan)(

0,)(

***

*

***

*

**

=∂∂

+

++⋅∇+

∂∂

=∂∂

+

+−⋅∇+

∂∂

=⋅∇+∂

∂

θθ

λθθ

ha

ghuhaufvh

tvh

ha

ghvhaufuh

tuh

ht

h

v

v

v

The shallow water equations on a sphere whose radius is a can be written in flux form as follows.

Shallow Water Equations

,

Fig.1. Yin-Yang grid system.(a) view from (0N,135E) (b) view from (0N,180E) (c) view from (0N,135W)

Non-hydrostatic coupled atmosphere-ocean simulation code has been developed in the Earth Simulator Center. The coupled code is designed for use with several kilometers resolution for horizontal as a cloud resolving models. We adopted Yin-Yang Grid[1] system (Fig.1), which was developed by Solid Earth Simulation Group of the Earth Simulator Center.

In this study, we present validation results that proposed schemes applied on the introduced grid system perform well in standard test set introduced by Williamson (1992)[2]. In addition, we will discuss the features of this introduced grid system taking account into the accuracy and/or mass conserving schemes.

Introduction

* mail:ohdaira@jamstec.go.jp, http://www.es.jamstec.go.jp/

Mitsuru Ohdaira＊,Keiko Takahashi, Kunihiko WatanabeEarth Simulator Center, Japan Agency for Marine-Earth Science and Technology

Validation for the Solution of Shallow Water Equations in Spherical Geometry with Overset Grid System