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291 Short* and Medium-Range Numerical Weather Prediction Collection of Papers Presented at the WMO! 1UGG NWP Symposium, Tokyo, 4-8 August 1986

A Numerical Scheme to Compute the of the Wind Field

By Yuk-kwan Chan

Royal Observatory Hong Kong (Manuscript received 29 October 1986, in revised form 12 February 1987)

A new finite difference scheme is constructed for the divergence term with a view to decreasing the errors in its computation. A divergence scheme is designed such that it will give zero divergence for any non- divergent wind field. In other words, the computational error in applying this scheme to any non-divergent wind field is zero. Since any wind field can be separated into a non- divergent part and a divergent part, the computational error in applying this scheme to any wind field is equal to the error involved with the divergent part only. At the same time, the divergent part wind is an order of magnitude smaller than the non-divergent part. Hence the error is reduced. The new scheme is simple, especially in latitude-longitude or Mercator map projections. The scheme has been applied to a 3-Ievel limited area primitive equation model for Hong Kong and is found to give much improvement over the usual divergence schemes.

1. Introduction 2 It is a well known fact that the divergence term is usually the difference of two terms of an order of magnitude larger. Thus large error can result in the computation of the If (1.1) is used to compute the divergence finite difference approximation of the diver- a geostrophic wind field, the values of I gence. If ' V ' stands for the finite difference divergence obtained is comparable in ore analogue of the continuous ' Div of magnitude to those of an ordinary wind then, a simple representation such as field even though the former should be zero. To decrease the error in the computation S7v=nm{dx(^) +Sy(-^) } (1.1) of the divergence, Y. K. Chan (1984) proposed a finite difference scheme ' Vc' which gave is not satisfactory. Here, 'm', 'n' are map zero divergence for any non-divergent wind factors (Section 1-8, Haltiner and Williams, field. In other words, this scheme is exact 1980) in the x- and ^-directions respectively for any non-divergent wind field. The phi- and the operators '§' and '―', are defined losophy that this scheme will decrease the as follows: computational error for other divergent wind field is briefly outlined as follows. Since any ≪ Uj+i/2, j ^i-1/2, j wind field can be decomposed into the sum of the rotational component v^ and the divergent component vx, the application of 3v^i Vj,j+ 1/2 Vj.j- 1/2 i y Vc to v gives 292 A numerical scheme to compute the divergence of the wind field

VcU ― VcfejlT^l) ― Div L'x + S* Ckl(a{j) = 0.

where VcH equals zero and ex is the error The actual form of the expression for X7v associated with the divergent component. If depends on the distribution of the variables another finite difference divergence scheme in space. Among the common staggered grid V' is applied to the same wind field v, systems discussed by Arakawa (1972), the

＼J'v―£j,+Div vx-＼-sx A-grid is seldom used in numerical models and the D- and E-grids are rotations of the where sx and s £, are the errors associated C- and B-grids respectively. Therefore, the with the divergent and the rotational compo- detailed derivation of the scheme will be nents respectively. In general, vx is an order included in this article for the B- and C-grids of magnitude smaller than v$ and there is no only. The distribution of variables in the reason to assume that sx and b'x differ much B-grid and the C-grid is showh in Fig. 1 and in order of magnitude. Thus, sx is smaller Fig. 2 respectively. than sz+sj.. In other words, Vc approxi- The x- and ^-components of a non-diver- mates the divergence with smaller error than gent wind field are given by any other schemes. dj> d(b However, Chan derived the finite difference _ v ―m-^ ― dy ' ox scheme for the latitude-longitude projection only. This paper attempts to generalize the where

2. Formulation d(p nu~xy-z ~"dycPij s j x forf n-^ The principle of the derivation of the scheme V to give zero divergence for non- d

Substituting (2.2) into (2.1), Vv becomes an for the latitude-longitude and the Mercator projections respectively. The details of the expression in (pkt with coefficients Ckl(aij) which are expressions in atj: above computation are listed in Appendix A. For the C-grid (Fig. 2), the desired ex- Vy ― 2 Ckl(

which reduces to Y.-K. Chan 293

i, j+1 i-1,j+1 i.3+1 i+1,0+1

i+i.j+i

i-1, 5 i- £ ≫ 0 i+i. i i-1. 0 i ≫ 3 i+1 ≫ 3

･i-

i+1,j-1

points where the potential field and the nap factor taken on the values H*. .ae well as m, 'ao ij .andij n.. ; points where the potential field and the map factor taken on the values 4'. as m. n, 'ij . asiJ wellij . and .

points where the wind components taken on the points where the wind components taken values ultij and values u.i+5 i≫J . and i>3+S v. . i

Fig 1 A portion of the B-grid. Fig. 2 A portion of the C-grid

to give zero divergence may become too complicated. An example is quoted in Ap- pendix D.

(2.8) 3. Results and discussions We compare the results of 12- and 24-hour for the latitude-longitude and the Mercator forecasts produced by applying the following projections respectively. Some details of the scheme above computation are listed in Appendix B. M ＼ TT^TV A comparison among the expressions (2.5) _ VVij y OxUij+WtjOyi y J to (2.8) shows that the form of the divergence m , ^ in, ' scheme which will give zero divergence for where any non-divergent wind field is dependent on the choice of the finite difference scheme for (???/) =mj-V2mj+in the non-divergent wind field. An example is quoted in Appendix C which shows that for (which will give zero divergence for any non- some choice of the finite difference scheme divergent wind field) to those produced by an for the non-divergent wind field, the com- ordinary scheme puted values of the divergence never vanish for any divergence scheme. For other choice VUij=mA Sxut/ + dy(-=~ of the finite difference scheme for the non- ＼ trij divergent wind field, the divergent scheme (which may give non-zero divergence for 294 A numerical scheme to compute the divergence of the wind field some non-divergent wind field). The numerical model employed (Chan, 4. Summary 1984) is a 3 level limited area primitive A finite difference scheme is constructed equation one in 2.5 degree latitude-longitude such that it will give zero divergence for B-grid. Hybrid vertical coordinate is used any non-divergent wind field. The scheme with two cr-levels beneath and one />-level on is applied successfully to a limited area prim- top. The level a=0 corresponds to £=400 itive equation model and gives better results hPa. A parameterization scheme for cumulus than those from the application of an ordinary convection similar to that of Kuo (1965) is divergence scheme. employed in the model while radiation is neglected. Data are dynamically initialized. Acknowledgements Figs. 3 and 4 show the total mass vari- The author would like to express his ation of the 12- and 24-hour forecasts re- heartiest gratitude to Dr. K.Y. Chan in the spectively. The larger errors in the values Mathematics Department, University of Hong of the divergence when an ordinary scheme Kong for the stimulating discussions on the is used are reflected in the inaccuracy in the subject and the helpful suggestions. values of the forecast surface pressure. The Thanks are also due to Miss W. Y. Hui larger errors associated with the 12-hour for her excellent typing of the paper, and to forecasts show that quasi-nondivergence is Miss M. Wai for her beautiful diagrams. not attained for a 12-hour forecast but is attained once more at the end of a 24-hour Appendix A integration. Let

VVij ― AUi +1/2, J+1/2"I~ 960.0 1/2, j+ Du%-1/2, j- 1/2

955.5 + -A Vi+i/2,j+l/2~＼~ B Vi+i/2, j-l/2 b -＼~C Vi- 1/2, j+1/2 D'Vi-u2, j-m ■ (Al) 955.0 Substituting (2.4) for u and v in (Al) and

954.5 equating the coefficients C of cp in the result- ing expression to zero, we obtain the follow- JAN 1979 ing system of linear simultaneous equations 7 6 9 10 11 . 3 Variation of total mass (expressed by Ct+i, j+i―Xi ―― 0 mean surface pressure). The 'x' represents the observed values, the 'O' for applying Ci+lij=x1 ―x2+x6+xe=0 the designed scheme in a 12-hour forecasts and the 'A' for the ordinary scheme. Ci+i, j-i ― X2-＼- §X ―― 0 C(,j+1 = Xi + X 3+ Xs ― X7 = 0

960.0 Cij=x1 ―x2+x3 ―xi―xs―xe+x7+xs=0

Ci: j-i―x.j Xa~ ＼~ ^8=0 & 955.5 Ci- 1, j+l ― Xs-^- Xi = 0

955.0 Ci-1, j― X$ X.i Xi Xg ― 0

954.5 Ci-1, j-＼ ― X.i Jt8 = 0 . Simplification and rearrangement lead to: JAN 1979 7 e 9 to 11 Mx=0 Fig. 4 Same as Fig. 3 except that the compari- son is carried out for 24-hour forecasts.

i Y.-K. Chan 295

1 0 0 0 1 0 0 0 Appendix B 0 1 0 0 0 1 0 0 Let 0 0 1 0 0 0 1 0 0 0 0 0 M ― 1 0 0 -1 V^i; -^^i+1/2, j~＼~ Bui-J j 0 0 0 0 1 0 + Cvi: (Bl) 0 1 j+ Dvii j-1/2 ■ 0 0 0 0 0 1 0 -1 Using 0 0 0 0 0 0 1 1 ―r~xv "sox(pij for ( dtp m-~ 0=(0, 0, 0, 0, 0, 0, 0)r

x = (xlt x2, x3, xit x5, xs, x7, xa)T and rii}x8y

Substituting (A2) and (A3) into (Al), we ob- JrCviij+ii'1+Dvitj-ni. tain (2.5). The corresponding expression in Using the following expressions latitude-longitude projection is obtained by d

mt}=mj=sec({j ―l)y/a), ntj=1; andA *riijOyCpij 5

c (C2) + m

1.J- i,j+l £+1, j+1 ― 4 Ay AA

x where 'a' is the radius of the earth. 296 A numerical scheme to compute the divergence of the wind field

used for the non-divergent wind field, there will c. ■mi (C3) 4 Ay A A x be no divergence scheme giving zero diver- gence for non-divergent wind fields. D mi (C4) iJx Appendix D Thus In the B-grid, if we use the following ex- pressions : Ci+1. ＼~Ci j+l~ + l, j-l ―v x ] x t d

Unless ?7ii, (C4) and (C5) will be in and rnijX8x(f>ij for contradiction when Ci+1,J+1, Ci+lij-1 and vanish at the same time. Thus, if (CI) is the coefficient matrix would be in the form

y 0 0 0 0 0 0 fti-1/2, j+1

xy 0 Xy 0 0 0 0 0 ^<-1/2, ;'+l/2 fti+1/2, ./'+1/2

0 0 0 0 0 0 fti+i, + 1/2 .7 7^1-1/2, j+1 y 0 0 0 0 0 0 Tfli+1/2, j+1/2 Tftt + 1/2, j-1/2

x y 0 0 0 0 0 0 WH+1I2, j-1 ftt + l, j+l/z

xy 0 0 0 0 0 ni 0 +1/2, J ―1/2 1/2, 1/2

0 0 0 0 0 0 y Wit-1/2, j- 1

The resulting expression in the divergence is References

too clumsy to be quoted here. However, the Arakawa, A., 1972 : Design of the UCLA General corresponding expressions in latitude-longitude Circulation Model. Numerical Simulation of and Mercator projections are relatively sim- Weather and Climate. Technical Report, No. ple : 7, Dept of Meteorology, UCLA. Chan, Y. K., 1984: A limited area primitive equa- tion weather prediction model for Hong Kong. 7r^ M. Phil. Thesis, University of Hong Kong, VUij=v + ―=y-dxUij+mjdy[=y ＼ yyi . ＼ni . ' 1984. Haltiner, G. J. and R. T. Williams, 1980 : Numeri- / (wj/) cal Prediction and Dynamic Meteorology. John ( Wiley and Sons, New York. Vv(j=mj ＼ j~8xUij+nijdy =^) ＼mj+m m. ＼llj ' Kuo, H. L., 1965: On formulation and intensifi- cation of tropical cyclones through latent heat release by cumulus convection. /. Atmos. Sci., 22, 40-63.