a comparison of grid quality of optimized spherical

A Comparison of Grid Quality of Optimized Spherical Hexagonal–PentagonalGeodesic Grids

HIROAKI MIURA

Frontier Research Center for Global Change, Japan Agency for Marine-Earth Science and Technology, Kanazawa-ku,Yokohama-city, Kanagawa, Japan

MASAHIDE KIMOTO

Center for Climate System Research, University of Tokyo, Tokyo, Japan

(Manuscript received 15 November 2004, in final form 16 February 2005)

ABSTRACT

Construction and optimization methods of spherical hexagonal–pentagonal geodesic grids are investi-gated. The objective is to compare grid structures on common ground.

The distinction between two types of hexagonal–pentagonal grids is made. Three conventional gridoptimization methods are summarized. In addition, three new optimization methods are proposed. Sixdesirable conditions for an ideal grid are described, and the grid optimization methods are organized in viewof such conditions.

Interval uniformity, area uniformity, isotropy, and bisection of cell faces are systematically investigatedfor optimized grids. There are compensations of preferable grid features in each optimization method, andan optimal method cannot be decided based only on the research of grid features. It is suggested that gridoptimization methods should be selected based on research of numerical schemes.

1. Introduction

Geodesic dome, which was first developed by Buck-minster Fuller, is a generic name of polyhedra, whichare composed of plane triangles inscribed inside asphere. It is desirable that component triangles are asequilateral as possible to construct geodesic domes withsufficient strength. Sadourny et al. (1968) and William-son (1968) independently recognized the possibility ofusing the structure, which approximates a sphere, as aspherical grid system for meteorological applications.They proposed integration schemes of the barotropicvorticity equation on such triangular grid systems. Thegrid systems they used are composed of geodesics—inother words, arcs of great circles. We call such grid sys-tems, which are composed of geodesics and approximatea sphere, the “spherical geodesic grid” in this paper.

A regular polyhedron is ideal for the construction ofspherical geodesic grids since distortion of constructedtriangular grid systems possibly impairs accuracy of nu-merical schemes (e.g., Tomita et al. 2001). Only thetetrahedron, octahedron, and icosahedron are the regu-lar polyhedra constituted of regular triangles. It is con-sidered that the icosahedron is the best for the startingpoint because it has more faces than the others.

Various grid construction methods based on theicosahedron have been proposed (e.g., Sadourny et al.1968; Williamson 1968; Cullen 1974; Baumgardner andFrederickson 1985; Heikes and Randall 1995a). Con-structed triangular grid systems were directly used forshallow-water models by several researchers (e.g., Stu-hne and Peltier 1996; Giraldo 2000). On the other hand,some researchers adopted the dual-mesh approach andgenerated hexagonal–pentagonal grid systems, whichare the dual grid systems of the triangular ones (e.g.,Sadourny and Morel 1969; Masuda and Ohnishi 1986;Thuburn 1997; Tomita et al. 2001). Sadourny et al.(1968) and Williamson (1968) used triangular grid sys-tems to derive a finite-difference (FD) form of the Ja-cobian operator and partially used the dual hexagonal–

Corresponding author address: Dr. Hiroaki Miura, Frontier Re-search Center for Global Change, Japan Agency for Marine-Earth Science and Technology, 3173-25 Showamachi, Kanazawa-ku, Yokohama-city, Kanagawa 236-0001, Japan.E-mail: [email protected]

VOLUME 133 M O N T H L Y W E A T H E R R E V I E W OCTOBER 2005

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pentagonal grid systems to derive a discrete form of theLaplacian or the curl operator by, what is now called,the finite-volume (FV) method. Ringler and Randall(2002) used both triangular and hexagonal–pentagonalgrid systems for their shallow-water model based on astaggered arrangement of variables.

In the FV method, a solution domain is subdividedinto contiguous control volumes (CVs). It is known thatpoor grid quality leads to larger truncation errors (e.g.,Ferziger and Peric 2001); that is, irregularities and im-perfect shapes of CVs are sources of truncation errors.Grid optimizations are widely used to restore accuracyof numerical schemes. For spherical geodesic grids, sev-eral grid optimization methods have been proposed(Heikes and Randall 1995b; Tomita et al. 2001; Du etal. 2003a).

Heikes and Randall (1995b) slightly adjusted the in-tersection points of a triangular grid to improve accu-racy of their shallow-water model that is based on thestreamfunction-velocity potential form of equations(Masuda and Ohnishi 1986). The intersection points aredenoted by generators since they are used to configurea dual hexagonal–pentagonal grid system. Tomita et al.(2001) proposed a two-step procedure; the first step isan arrangement of hexagonal shapes of CVs and thesecond one is the relocation of computational nodes, atwhich means of variables over CVs are represented inthe FV method, to the centroids of CVs. In the methodof Du et al. (2003a), the positions of generators areforced to locate at the centroids of Voronoi cells, theconfiguration of which is summarized in section 2.

Ringler (2003) compared three optimized grids con-structed by the methods of Heikes and Randall(1995b), Du et al. (1999), and by the MESQUITE soft-ware library in terms of accuracy of the Laplacian op-erator and of solutions of the Poisson equation. Theypointed out that the truncation error, in their case theerror of the discrete Laplacian operator, provides onlyan upper bound to the solution error—that is, the errorof the Laplacian inversion—in terms of convergencerate. In their paper, attention was limited to theVoronoi type (VT) grid, and grids of higher resolutionshad not been available by the method of Heikes andRandall (1995b).

In this paper, we summarize three conventional op-timization methods mentioned above. There remainpossibilities of other grid optimization methods. Weprovide three new optimization methods; one of themimitates well the quality of grids produced by themethod of Heikes and Randall (1995b) and works ef-fectively even in higher resolutions. These six grid op-timization methods are organized and compared on acommon ground. Two types of hexagonal–pentagonal

grid systems are analyzed; one is the VT and the otheris called in this paper the “barycentric type” (BT), thedefinition of which is given in section 2. The BT wasused for the spherical geodesic grids in Tomita et al.(2001).

In meteorological models, there are a variety ofneeds. Accuracy and stability are required in terms ofnumerical schemes. The consistency of physical param-eterizations is desired for atmospheric simulations. Inaddition to these physical requirements, computationalefficiency is necessary for long-term simulations withhigher horizontal resolutions. It is not obvious thatthese requirements are simultaneously achieved onspherical geodesic grids. It is useful that we are aware ofeffects and side effects of a grid optimization methodwhen we choose it. As discussed in Miura (2004), im-pacts of grid optimizations on accuracy and stabilitydepend on numerical schemes. The objective of thispaper is to discuss grid features for the convenience indesigning numerical schemes or selecting grid optimi-zations. Assessments of the accuracy of discretized op-erators and shallow-water models (e.g., Heikes andRandall 1995a,b) will be the subject of a future paper.

Section 2 summarizes a grid construction method andgives definitions of the VT and BT grid systems. Wediscuss six important conditions for an ideal grid in sec-tion 3. In section 4, grid optimization methods are de-scribed. In section 5, comparisons of grid properties aremade with respect to intervals of generators, cell areas,distortions of hexagonal cells, and condition of cellfaces. Section 6 gives summary and discussions.

2. Grid construction

a. Simple spherical geodesic grid

Until now, various construction methods of sphericalgeodesic grids have been proposed as mentioned in theprevious section. Among them, we follow the methodof Heikes and Randall (1995a) in this paper because ofits simplicity. This method was employed by severalresearchers (e.g., Stuhne and Peltier 1996; Thuburn1997; Tomita et al. 2001). We briefly summarize themethod below.

The starting point is a plane icosahedron inscribedinto a unit sphere (Fig. 1a). First, the edges of planetriangles are bisected and four new triangles are createdin each triangle face. Next, bisection points are pro-jected onto the surface of the unit sphere and a newpolyhedron is constructed (Fig. 1b). By repeatedly op-erating the bisection and the projection, the number oftriangle faces increases and constructed polyhedracome to approximate the sphere. The two-step processis recursively repeated until we obtain a needed reso-

2818 M O N T H L Y W E A T H E R R E V I E W VOLUME 133


lution. Figures 1b and 1c are examples of constructedpolyhedra after one and two operations of the two-stepprocess, respectively. After a necessary resolution is ob-tained, the edges of a constructed polyhedron are pro-jected onto the sphere.

A constructed polyhedron consists of quasi-homogeneous triangles. The dual grid of the triangulargrid is a hexagonal–pentagonal grid. An example ofsuch dual hexagonal–pentagonal grids is shown in Fig.1d, which is the dual grid of the triangular grid indicatedin Fig. 1c. Both the triangular and hexagonal–pentagonal grid systems constructed by “mth” recur-sions are denoted by “glevel-m” in this paper followingTomita et al. (2001). This has the same meaning withthe notation “level-m” in Stuhne and Peltier (1999).Note that the initial icosahedron corresponds to glevel-

0. Both Figs. 1c and 1d are designated by glevel-2. Thenumber of hexagonal–pentagonal cells after mth recur-sions is 10 � 22m � 2.

b. Two types of hexagonal–pentagonal grids

Sadourny et al. (1968) and Williamson (1968) used aconfiguration as shown in Fig. 2a to define hexagonalgrid cells. A similar configuration was also used by Ma-suda and Ohnishi (1986) to derive needed operators bythe FV method. Heikes and Randall (1995a) intro-duced a notation “Voronoi grid” to represent such ahexagonal–pentagonal grid.

A construction method of VT grid systems was sum-marized by Heikes and Randall (1995a) following Au-genbaum and Peskin (1985). The method is briefly de-scribed here. Let S2 denote the surface of a unit sphere.

FIG. 1. (a) An icosahedron inscribed inside a unit sphere. (b) A polyhedron constructed by the first operation ofthe bisection and the projection. (c) A polyhedron after the second operation. (d) A dual hexagonal–pentagonalgrid of the triangular grid in (c).

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Given a set of N points on the sphere, which are de-noted by position vectors {G1, G2. · · · , Gn}, the“Voronoi region” around Gi is defined by

Vi � �x ∈ S2:|x � Gi| � |x � Gj| for

j � 1, 2, · · · , N, j � i�, �1�

where |x � Gi| is the spherical distance between twopoints x and Gi.

In Fig. 2a, the symbols Gi(i � 0, 1. · · · , Ns) denote thepositions of generators, where NS is the number of seg-ments surrounding a hexagonal or pentagonal cell; Nc

� 5 for pentagons and Nc � 6 for hexagons. The posi-tion vector Qi is the center of the circumscribed circleof the triangle configured by the generators G0, Gi�1,and Gi. From the definition, Qi is computed on the unitsphere using the outer products as

Qi ��Gi�1 � G0� � �Gi � G0�

|�Gi�1 � G0� � �Gi � G0�| . �2�

The circumcenter Qi is the corner shared by three hex-agonal cells with generators G0, Gi�l, and Gi. The neigh-boring corners are mutually connected with geodesicarcs, and a hexagonal–pentagonal grid is generated.

An important property of VT grid systems is that cellwalls are the perpendicular bisectors of geodesics thatconnect pairs of generators. Consider an arbitrary sca-lar , which is continuous over a solution domain. Sup-pose that we define computational nodes at the posi-tions of generators. The derivative normal to the cellface QiQi�l can be approximated at the center of thearc G0Gi, denoted by mi , by the central finite-differ-ence scheme as

��

�n�m �i

��Gi

� �G0

Li, �3�

where Li is the spherical distance between G0 and Gi.This is the second-order-accurate approximation at mi .

Tomita et al. (2001) used another type of hexagonal–pentagonal grid, a schematic figure of which is shown inFig. 2b. The notations are the same with Fig. 2a, but thistime the position of the corner Qi is defined at thebarycenter of the triangle configured by the generatorsG0, Gi�l, and Gi. It is computed on the sphere by thefollowing formula:

Qi �G0 � Gi�1 � Gi

|G0 � Gi�1 � Gi|. �4�

Connecting neighboring corners by geodesics, we cangenerate a hexagonal–pentagonal grid. Such hexago-nal–pentagonal grids are denoted by BT grid systems inthis paper.

The differences of grid properties between VT andBT grid systems are discussed in section 5.

3. Grid quality

Consider a regular square grid. Suppose that compu-tational nodes are located at the centroids of squareCVs; this is called the (i) “collocation” condition in thispaper. The line that connects a pair of computationalnodes and the cell face shared by the nodes orthogo-nally bisect each other; we call this the (ii) “orthogo-nality” and (iii) “bisection” conditions. If the colloca-tion condition is satisfied, values at the computationalnodes are the second-order representations of themeans over CVs. In addition, if the orthogonality and

FIG. 2. Configurations of hexagonal cells: (a) Voronoi cell and (b) barycentric cell.



bisection conditions are satisfied, normal derivatives atcell faces are approximated by (3) to the second-orderaccuracy. On such ideal grids, numerical scheme withthe second-order accuracy or more may be constructed.

On spherical geodesic grids, however, hexagonal CVsare not the perfect hexagons; that is, their shapes andsizes are somewhat distorted. Therefore, the three con-ditions are not necessarily satisfied simultaneously. ForVT grid systems, the orthogonality condition is satisfiedif we set computational nodes at the positions of gen-erators, while the collocation and bisection conditionsare generally not assured. If computational nodes arepositioned at the centroids of CVs, the collocation con-dition is satisfied for both VT and BT grid systems;however, the other conditions are not necessarily real-ized.

Tomita et al. (2001) showed a proof that the collo-cation condition is necessary for their discrete diver-gence operator to be a second-order-accurate approxi-mation. If either the bisection or the orthogonality con-dition is broken, an approximation of the normalderivative at cell faces in the form of (3) no longerattains the second-order accuracy. In Fig. 2a, the dis-agreement between the center of the segment G0Gi,denoted by m, and the center of the segment QiQi�l,denoted by m, is a source of truncation error since (3)is defined at m and it does not represent the mean overthe segment QiQi�l. The distance between m and m, e,relative to the length of the cell face, l, is a measure ofgrid quality (Heikes and Randall 1995b; Ferziger andPeric 2001).

If all of the three conditions mentioned above aresatisfied, there is a possibility that a scale mismatchhappens. Consider a rectangular grid box, the scale ofwhich in the x direction is much larger than that in they direction. In such a case, operators in the x directionare computed with more distant nodes compared to they direction. In general, meteorological phenomena donot strongly depend on the horizontal axis directions.Ideally, then, the scale in each direction should be iden-tical with the other; that is, a grid cell should be isotro-pic. We denote this as (iv) “isotropy.”

A variety of assumptions are made in physical pa-rameterizations in meteorological models, and some ofthem may depend on horizontal resolution. If cell sizevaries from place to place, the adequacy of size-dependent assumptions differs from one location to an-other. Then, it is required that cell sizes are homoge-neously distributed around the computational domain;this is designated by (v) “area uniformity” here.

If computational nodes are distributed nonuniformly,the time interval of a numerical simulation is limited bythe shortest grid interval because it defines the Cou-

rant–Friedrichs–Lewy (CFL) condition of the simula-tion. Therefore, for computational efficiency, grid in-tervals should be identical; this condition is called (vi)“interval uniformity” in this paper.

In following sections, six grid optimization methodsare represented and comparisons are made in terms ofconditions (i)–(vi) mentioned above.

4. Optimization methods

a. SMPL method

Consider a polyhedron that is constructed by thesimple recursion method; that is, no modification is ap-plied to the positions of generators. We refer to thismethod as the “SMPL” (simple) method. Both triangu-lar and hexagonal grids constructed by the SMPLmethod are denoted by the SMPL grids. We distinguishVT and BT grid systems by denoting “VT SMPL” and“BT SMPL,” respectively. This notational convention iscommon for the following optimized grid systems.Table 1 shows some basic features of the SMPL gridsystems and it is similar to the result of Thuburn (1997).

b. HR95 method

Masuda and Ohnishi (1986) derived discrete forms ofoperators on a VT grid system and applied them to theshallow-water model on a VT SMPL grid system on asphere. The VT grids are desirable for their methodbecause the normal derivative (3) should be computedat cell faces. Heikes and Randall (1995a,b) used thesame definitions of operators and they showed that ac-curacy of the discrete Laplacian and flux-divergenceoperators is not improved by the simple refinement ofthe grid. They suggested a grid modification methodthat aims at improving the bisection condition, while

TABLE 1. Properties of the SMPL grids as a function of recur-sive subdivisions. Level is the number of recursions. The symbolNc is the number of hexagonal and pentagonal cells, which is alsothe number of vertices of a constructed polyhedron. The symbolLave is the average distance between neighboring generators, andAave is the average area of hexagonal and pentagonal cells. It isnoted that Aave is common to both the VT and BT grids.

Level Nc Lave (km) Aave (km2)

1 42 3765.0 1.215 � 107

2 162 1914.4 3.149 � 106

3 642 961.3 7.945 � 105

4 2562 481.1 1.991 � 105

5 10 242 240.6 4.980 � 104

6 40 962 120.3 1.245 � 104

7 163 842 60.2 3.113 � 103

8 655 362 30.1 7.783 � 102



retaining the orthogonality condition, to reduce trun-cation error of (3) discussed in the previous section.

Figure 2a helps us to derive the discrete Laplacianoperator of Masuda and Ohnishi (1986). It is assumedthat computational nodes locate at positions of genera-tors. Therefore, the collocation condition is not alwaysensured for distorted cells. The area-integrated Lapla-cian operator can be transformed using the Gauss’theorem as

�A

�2� dA � �C

��

�nds, �5�

where is an arbitrary scalar, A is the area bounded bya closed circle C, �/�n is the derivative normal to thecurve. Let i be the value of evaluated at a node Gi,and Ai be the area of the CV with the node. The Lapla-cian operator at the node G0 is derived from (5) as

�2�|G0�

1A0

i�1

Ns �li ��i � �0�

Li�, �6�

where (3) is used to approximate the normal derivativeat the cell face. The expression of (3) is the second-order-accurate approximation at mi , while the meanvalue over the arc QiQi�l should be defined at m toobtain the second-order accuracy. The break of the bi-section condition is a source of truncation error.

Heikes and Randall (1995b) showed that the ratiori � ei /li is a dominant term for the convergence of theLaplacian operator (6). To control the grid optimiza-tion, they defined a global function by the summationof the fourth power of the ratio as follows:

R � cells

i�1

Ns

r i4. �7�

The objective of the grid optimization is to find thegenerator distribution that minimizes (7).

Because of the symmetry of generator positions onthe sphere, the application of the optimization can belimited to the tenth part of the sphere. Computationalcosts can be substantially reduced by this limitation.This restriction is also commonly used in the followinggrid optimization methods. Heikes and Randall (1995b)forced additional limitation that only newly createdgenerators at the mth time recursion are modified on aglevel-m triangular grid while formerly- created genera-tors are fixed.

A function named UMINF in the InternationalMathematical and Statistical Library (IMSL) was usedby Heikes and Randall (1995b) to perform this optimi-zation. This routine operates to minimize a function ofan arbitrary number of variables using a quasi-Newtonmethod and an FD gradient. We use the same routine;

however, there is a problem that convergence of calcu-lations cannot be attained on levels higher than 6. Toavoid this problem, we developed another approach toimitate grid quality realized by this method; the ap-proach is described in section 4f. We call the method ofHeikes and Randall (1995b) the “HR95” method andrefer to optimized grid systems as “HR95” grid systems.

c. SPRG method

Tomita et al. (2001) used different FV operatorsfrom those of Masuda and Ohnishi (1986). It is notnecessary in their method to assume grid types; that is,both VT and BT grid systems are possible. This is dueto the fact that their scheme does not require the de-rivative normal to each cell face. The orthogonalitycondition and the bisection condition are not so impor-tant in their scheme. It was suggested in their paper thatlarger truncation errors are induced by abrupt changesof grid properties. One of the reasons of the accuracyreduction can be explained as follows. The abruptchange of grid properties causes a sudden change ofaccuracy of interpolations and approximations requiredfor numerical calculations. In the FV method, similar tothe FV Laplacian operator (6), many of discretized op-erators are defined by replacing line integrals by a sum-mation over the segments that surround a CV. There-fore, the summation, or the subtraction, of values ofdifferent properties possibly magnifies truncation er-rors. Such “smoothness” may be an important factor ofgrid qualities; however, we cannot define any measureto evaluate smoothness of grid systems for the present.The isotropy and the two uniformities help partly esti-mate the smoothness.

To obtain a smooth distribution of generators, a gridoptimization method was proposed by Tomita et al.(2001). They focused on controlling generator intervals.The schematic figure of their method is shown in Fig.3a. The length of a segment G0Gi is denoted by Li. Pairsof neighboring generators are virtually connected withsprings; the spring constant is k and the natural springlength is L for the springs in common.

A glevel-m SMPL triangular grid has 10 � 2m–1 in-tersection points, or generators, around the equator. Ifgenerators are uniformly distributed on the equator ofa unit sphere, the generator interval on the equator canbe calculated as

� �2�

10 � 2m�1 . �8�

This is a characteristic length of the grid interval on thesphere. The natural spring length is defined using anondimensional parameter � as

L � �. �9�



Tomita et al. (2001) used � � 0.4 while Tomita et al.(2002) carried out further research and recommended� � 1.2. The interval and area uniformities are wellimproved with � � 1.2 compared to � � 0.4, althoughthe isotropy is impaired. With � � 0.4, we can obtainthe similar grid quality realized by the method de-scribed in section 4d. We use � � 1.2 in this paper.

We use a somewhat different method from that ofTomita et al. (2001) for simplicity. In their method, theposition of each generator is modified like the motionof a solid body; that is, each generator has a virtuallydefined mass and velocity. We delete this assumptionand perform simple iterations until a convergence con-dition is met.

To control the optimization, we define a global mea-sure by

R � cells

i�1

Ns �12

�Li � L�2�. �10�

This is the summation of the potential energy of eachvirtual spring. We evaluate the value of (10) to measurethe convergence of the iteration. The positions of gen-erators are slightly modified in proportion to the springforcing F0 as follows:

G0new �

G0 � �F0

|G0 � �F0| , �11�

F0 � i�1

Ns ��Li � L�Gi � G0

|Gi � G0|�, �12�

where is a control coefficient to limit the displace-ment, and the spring constant is k � l. This modification

FIG. 3. Configurations of grid optimization methods for the (a) SPRG, (b) SCV and SCB, and (c) I-HR andM-HR methods. (d) A configuration of the centroid of a hexagonal cell.



is applied for all generators simultaneously in the tenthpart of the sphere.

The iteration is terminated when the rate of thechange of (10) calms down, solving a balanced distri-bution of generators. We can obtain almost the samegrid properties with those of Tomita et al. (2001). Wecall this optimization method the “SPRG” (spring)method, following Tomita et al. (2001).

d. SCV method

Du et al. (2003a) proposed an algorithm to producespecial VT grid systems on general surfaces. Theyforced the positions of generators to coincide with thecentroids of VT cells. Three deterministic and probabi-listic algorithms were presented in their paper. Theycalled the special VT grid systems the centroidalVoronoi tessellations (CVTs). The collocation and theorthogonality conditions are exactly satisfied on CVTs,if computational nodes are located at the positions ofgenerators. In turn, the bisection condition becomesworse compared to HR95 grid systems as discussed insection 5d. Du et al. (2003b) applied the FV method toCVTs and solved partial-differential equations on asphere. They call CVTs on a sphere spherical CVTs(SCVTs). The simplest algorithm to construct SCVTs isa deterministic algorithm that is known as Lloyd’smethod (Du et al. 2003a). We use this algorithm in thispaper, and it is explained as follows.

The schematic figure to explain this method is shownin Fig. 3b. The positions of generators are denoted byGi. The centroids of the VT cells, shown by solid lines,are denoted by Gi . The centroids configure a new set ofgenerators. With the new set of generators, a VT grid isrecomputed as illustrated by broken lines. Again, a newset of generators is configured by the centroids of theVT cells. The positions of generators are optimized byiteratively computing VT grid and mass centroids.

The centroid of a hexagonal or pentagonal CV iscomputed as follows. In Fig. 3d, position vector Qi is acorner of the CV, and Ti means the area of the sphericaltriangle configured by three points G0, Qi, and QiQi�l.The cell is subdivided into Nc spherical triangles. Thecentroid of the CV, denoted by G0, is computed on aunit sphere as

G�0 � i�1

Ns

�Ti�G0 � Qi � Qi�1��

i�1

Ns

�Ti�G0 � Qi � Qi�1��. �13�

If a generator distribution meets a convergence cri-terion, the iterations terminate. We define a global

function by the summation of the second power of thedistance between Gi. and Gi as

R � cells

|G�i � Gi|2. �14�

In preliminary tests, it is found that converged values of(14) increase by the increase of grid level because of theround-off error. Therefore, it is difficult to select a fixedcritical value, denoted by �, to set a criterion in the formof R � �. We make a choice to stop the modificationsafter a sufficient number of iterations; the number isdenoted by nite and we set nite � 2000 in this paper.Note that the computational costs are not so large forthe parameter. We call this optimization method the“SCV” method in this paper.

e. SCB method

We extend the SCV method to BT grids. A BT gridis assumed in this method while the SCV method as-sumes a VT grid. We force the positions of generatorsto coincide with the centroids of BT cells. The colloca-tion condition is satisfied while the orthogonality andthe bisection conditions are broken on optimized BTgrid systems.

As in the SCV method, Lloyd’s algorithm is used.The modification of generator positions is iterativelyperformed by computing BT grid and mass centroids ofthe cells. We set the number of iterations as nite � 2000also in this method. We call this optimization methodthe “SCB” method.

f. I-HR method

As pointed out in section 4b, the authors cannot op-timize grid systems by the HR95 method in higher reso-lutions. In addition, it is found that computational costspossibly become much larger on grids of higher levels.In grid levels lower than 7, the HR95 method is effec-tive to reduce truncation errors of the Laplacian andflux-divergence operators derived by the FV method(Heikes and Randall 1995b). However, it is uncertainthat the HR95 method is also functional in higher reso-lutions.

We develop an iterative algorithm to realize nearlythe same quality with HR95 grids. Even in higher reso-lutions, convergence can be achieved. As an analogy forthe SPRG method, (11) is used to modify the positionsof generators. The control parameter is set to � 0.3.The forcing term F0 is defined as follows.

The schematic figure of this method is shown in Fig.3c. Generators are indicated by Gi, and the corners ofVT cells are denoted by Qi. The distance between thecenter of the arc G0Gi, and the center of the cell face



QiQi�l is denoted by ei. If G0 and Gi are positioned atG0 and Gi , respectively, the center of the arc G0Gi isidentical to the center of the segment QiQi�l; that is,ei. � 0 is achieved. The position of G0 is written as

G�0 � G0 � ei

Qi�1 � Qi

|Qi�1 � Qi|. �15�

The second term on the right-hand side of (15) canperform to move the arc G0Gi toward the arc G0Gi .Taking the summation over the segments that surroundthe cell with the generator G0, the term F0 is defined by

F0 � i�1

Ns �ei

Qi�1 � Qi

|Qi�1 � Qi|�. �16�

The positions of generators are iteratively modifiedusing (12) and (16). We restrict the number of com-puted generators by the same rule with Heikes andRandall (1995b), which has been described in section4b. The grid optimization is applied at each level of gridrefinements. We compute the global function (7) tomeasure the convergence of the iteration. The modifi-cation terminates when (7) attains the minimum value.

We develop this algorithm to construct grid systemsthat imitate the properties of the HR95 grids. There-fore, we call this method the “I-HR” (imitate HR95)method. Some properties of the I-HR grids are com-pared to those of the HR95 grids in the next section.

g. M-HR method

It is found from the results of Heikes and Randall(1995b) that the area uniformity is improved on theHR95 grid systems while the interval uniformity is im-paired. It is possible that the isotropy is also impaired.

From the preliminary tests, it is found that the inter-val uniformity can be restored by changing the expres-sion of the forcing term (16). Equation (12) is also usedto modify the positions of generators. We adopt aweighting average of the right-hand side of (16) as theforcing term F0. The configuration of Fig. 3c is also usedhere. The weight is defined by the length of the segmentQiQi�l relative to the length of the arc G0Gi, and thenwe get

F0 � � i�1

Ns � liLi

� ei

Qi�1 � Qi

|Qi�1 � Qi|�� i�1

Ns liLi

. �17�

We define a cost function by the global summation ofthe absolute value of (17) as

R � cells

|F0|, �18�

to measure the convergence of the optimization. Theiteration terminates when (18) attains the minimumvalue.

We call this method as “M-HR” (modified HR95)method. It is noted that the bisection condition maybecome worse on the M-HR grids, compared to theHR95 and the I-HR grids, in exchange for the correc-tions of the other conditions.

5. Grid properties

Several grid optimization methods assume a type ofhexagonal–pentagonal grid systems. The HR95, SCV,I-HR, and M-HR methods are specialized to the VT; incontrast, the SCB method is specialized to the BT. Itshould be emphasized that the modified ones are onlythe positions of generators. Both VT and BT hexago-nal–pentagonal grid systems are available from the op-timized positions of generators.

The HR95 and I-HR methods aim at improving thebisection condition of the VT grid. The objectives of theSCV and SCB methods are to improve the collocationcondition on VT and BT grid systems, respectively. TheSPRG and M-HR methods work for improving the in-terval uniformity. In this section, we investigate advan-tages of optimizations and shortcomings in return. Thesummary of results is preliminarily shown in Table 2.

a. The interval uniformity

The positions of generators can be defined as com-putational nodes in the FV or the FD method. As an

TABLE 2. Summary of grid properties. (a) Comparison of theVT and the BT grid systems. (b) Comparison of grid optimizationmethods. It is noted that the collocation condition is not shownhere, because it can be achieved for all the methods by recalcu-lating computational nodes. In (b), the bisection condition isevaluated only for the VT grids. The symbol “O” is marked forthe better of the two types in (a) and for the better two of the sixoptimization methods in (b), respectively.(a)

Area Isotropy Orthogonality

VT O OBT O —

(b)

Interval Area Isotropy Bisection (VT)

HR95 O OI-HR O OM-HR OSPRG OSCV OSCB O



alternative choice, we can recalculate the positions ofcomputational nodes at the centroids of hexagonal andpentagonal CVs by using (13). That is, the positions ofgenerators and computational nodes do not always co-incide. The distance between a pair of neighboringcomputational nodes represents the horizontal resolu-tion. For simplicity, we adopt the distance betweenneighboring generators as a measure of the horizontalresolution. The distance is in common for VT and BTgrid systems.

The ratio of the smallest distance to the largest is aglobal measure of the interval uniformity. The ratio isshown in Fig. 4. The values of the SMPL grids arenearly constant for all levels. The lines for the SCV andSCB methods have negative slopes. This means thatgrid interval is partly overshortened in higher resolu-tions. The negative slope is more notable for the SCBgrids. The HR95 and I-HR grids take nearly, but notexactly, the same values in each level; however, theyare smaller than those of the SMPL grids. Note that thevalues of the HR95 grids at levels 7 and 8 are not avail-

able for the present. The values of the M-HR grids areslightly larger than those of the SMPL grids. The valuesof the SPRG grids decrease slightly with the grid levelwhile they are still close to those of the SMPL grids.

The interval uniformity becomes worse by the gridoptimizations except for the M-HR and the SPRGmethods. The break of the condition is most serious inthe SCB method. In terms of the interval uniformity,the M-HR and the SPRG methods are better than theothers.

b. The area uniformity

In the FV method, a computational domain is subdi-vided into a finite number of CVs. In the FD method, aregion covered by a grid point is assumed when physicalparameterizations are applied in meteorological mod-els. The concept of the region is almost the same as thatof the CV. On the spherical geodesic grids, it is naturalthat physical parameterizations are applied inside eachhexagonal or pentagonal cell for both the FD and theFV methods. The area uniformity is an important factorfor the consistency of physical parameterizations.

The ratio of the smallest area to the largest of hex-agonal–pentagonal cells is shown in Fig. 5 as a functionof grid level. This is a global measure of the area uni-formity. It is noted that the lines for the HR95 methodterminate at glevel-6. It is found that the patterns aresimilar between Figs. 5a and 5b while values are largerfor the VT. In both panels, the lines for the SCV andthe SCB grids are downsloping as in Fig. 4. Cell areasare partly over reduced by the optimizations. The VTHR95 and the VT I-HR grids are highly homogeneous,compared to the other grids. The HR95 and the I-HRmethods also produce relatively homogeneous gridsamong the BT grid systems. The ratios of the M-HRand the SPRG grids are nearly constant at all the levels.The lines for the SMPL grids have a weak negative

FIG. 4. The ratio of smallest distance to largest distancebetween neighboring generators.

FIG. 5. The ratio of smallest area to largest area of hexagonal–pentagonal cells: (a) VT and (b) BT gridsystems.



gradient from glevel-2 to -4 while it becomes horizontalfor the levels higher than 4.

Spatial distributions of cell areas relative to the larg-est cell area are shown in Figs. 6 and 7, respectively forthe VT and the BT grids. The viewpoint is the samewith Fig. 1. The distributions of the HR95 grid are notshown, since they are very similar to those of the I-HRgrid. The largest area is located around the center oftriangles, which compose an original icosahedron inFig. 1a. We call such triangles as major triangles. Thereis a resemblance between the VT and the BT grid sys-tems. A large exception is that the areas of the pen-

tagonal cells are obviously smaller on the BT grid sys-tems. This is the cause of the difference between the VTand the BT in Fig. 5.

The results for the SMPL grids show a geometricalpattern composed of similar triangles, reflecting thegrid construction processes. On the I-HR grids, largerratios are distributed around the pentagonal cells inaddition to the center of major triangles. A feature ofthe SCB grids is a gradation, which changes concentri-cally with a central focus on the pentagonal cells. Thedistributions of the SCV grids are the intermediate be-tween the SPRG and the SCB grids.

FIG. 6. The spatial distributions of the ratio of cell areas to largest cell area for the glevel-5 VT grid systems. (a) TheSMPL grid, (b) the I-HR grid, (c) the M-HR grid, (d) the SPRG grid, (e) the SCV grid, and (f) the SCB grid.



A smooth distribution of cell areas can be obtainedby each optimization method, compared to the SMPLgrids. It is revealed that the ratio of the smallest area tothe largest is generally smaller on the BT grid systemsthan on the VT ones. This is due to the large differencein the areas of pentagonal cells. There are distinctchanges in cell areas around pentagonal cells for the BTgrid systems. In terms of the area uniformity, the HR95and the I-HR methods are more suitable than the others.

c. The isotropy

It is preferable for meteorological models that gridsare horizontally isotropic over a solution domain since

horizontal scales of meteorological phenomena do notstrongly depend on horizontal directions. The break ofthe isotropy causes the problem that the resolvablescales of operators depend on the axis directions. Theisotropy of the grid is an important factor for accuracyand stability of numerical schemes.

For a hexagon to be regular, it should have followingfeatures: lengths of six edges are the same and angles ofsix corners are 2�/3. Therefore, we introduce two mea-sures to evaluate the isotropy of hexagonal–pentagonalcells. One is the normalized standard deviation of edgelengths li. The definition can be given following Tomitaet al. (2001) as

FIG. 7. The same as Fig. 6 except for the BT grid systems.



Dl � �� i�1

Ns

�li � lmean�2��Ns12� lmean, �19�

lmean � i�1

Ns

li�Ns, �20�

where Nc is the number of segments that surround ahexagonal or pentagonal cell. The other is the standarddeviation of corner angles �l, which is given as

D� � �� i�1

Ns ��i ��Ns � 2��

Ns�2��Ns12

. �21�

The maximum values of Dl and D� on the VT and theBT grid systems are shown in Fig. 8 as a function of gridlevel. Values of the HR95 method are not available inlevels higher than 6. The BT grid systems are less dis-torted compared to the VT ones for each distribution ofgenerators. In all figures, the SMPL grids indicate larg-est values for both the VT and BT grids in levels higherthan 4. The lines for the HR95, I-HR, and M-HR gridsconverge to those of the SMPL grids with grid level.The slope of the SPRG method also has positive gra-dient; that is, grids become distorted as resolution in-creases. The SCB grids are the least distorted in termsof D� for both the VT and BT grids. The SCB grids arealso the least distorted about Dl for the VT grids, while

the SCV method shows the minimum values for the BTgrids.

Spatial distributions of Dl are shown in Figs. 9 and 10,respectively, for the VT and the BT grid systems. Theviewpoint is the same with Figs. 1, 6, and 7. The distri-butions of the HR95 grids are very close to those of theI-HR grids. Therefore, only the results of the I-HRgrids are shown. In all the panels, Dl values of the pen-tagonal cells are exactly zero because of the directionalsymmetry around the pentagonal cells on spherical geo-desic grids. Larger distortions are distributed aroundpentagonal cells, and smaller distortions are foundaround the centers of each major triangle.

The distributions for the VT and the BT grid systemsare somewhat similar while the values are obviouslysmaller for the BT grids. On the BT SCB grid, thereremain largely distorted cells around the pentagonalcells. Because of this feature, the BT SCB grids are notthe least distorted grids for the BT in Fig. 8b. The val-ues of the M-HR grids are globally smaller than thoseof the I-HR grids. It is found that the modification tothe I-HR method in deriving the M-HR method is ef-fective to reduce grid distortions.

By the applications of the grid optimization methods,smoother distributions of the distortions are obtainedrelative to the SMPL grids. On the HR95, I-HR, andM-HR grids, there are distinct changes of the distor-

FIG. 8. Largest Dl on the (a) VT and (b) BT grid systems. Largest D� on the (c) VT and (d) BT grid systems.



tions on both sides of the edges of the major triangles.Regarding the cell distortion, the SCB and the SCVmethods are more reasonable compared to the othermethods.

d. The bisection of cell faces

As mentioned in section 2, an important feature ofVT grids is that cell faces perpendicularly bisect geo-desics that connect pairs of generators. Because of thisfeature, the derivative normal to cell faces can be ap-proximated by Eq. (3). In Fig. 2a, the disagreementbetween m and m� is a source of truncation error, and

the distance between m and m�, e, relative to the lengthof the cell face, l, is a measure of grid quality, as notedin sections 3 and 4b. This subsection is relevant only tothe VT grids, and computational nodes are located atthe positions of generators to ensure the orthogonality.

Figure 11a shows the largest ri � ei/li for all cell wallsas a function of grid level. The variations of R definedby (7), which is a global measure of the magnitude of ri,are plotted in Fig. 11b as a function of grid level. Notethat values for levels 7 and 8 are not available for theHR95 method. It is found that the ratio is most effi-ciently reduced in the HR95 grids as the resolution in-

FIG. 9. The spatial distributions of the distortions on the glevel-5 VT grid systems. (a) The SMPL grid, (b) theI-HR grid, (c) the M-HR grid, (d) the SPRG grid, (e) the SCV grid, and (f) the SCB grid.



creases. The lines for the I-HR and the M-HR methodsare also downsloping nearly parallel to the line of theHR95 method. The convergence rates are approxi-mately the first order in Fig. 11a; that is, the magnitudeof these values are reduced by half when the grid levelincreases by one. It is expected that the maximum errorof the normal derivative (3) at cell faces will be reducedby the I-HR and the M-HR methods closely to thesame extent with that by the HR95 method. The ratioof the SPRG method also decreases, while the conver-gence rate is smaller than those of the HR95, the I-HR,and the M-HR methods. The lines for the SMPL, theSCV, and the SCB methods are not convergent as reso-

lution increases. In addition, the line of the SMPLmethod is upward-sloping in Fig. 11b. This means thatgrid quality becomes worse over the entire sphere asresolution increases.

It is found that the I-HR method imitate well theHR95 method in both panels. In terms of the bisectionof cell faces, the HR95 and the I-HR methods are thereasonable choices.

6. Summary and discussions

We have constructed spherical geodesic grids follow-ing the method of Heikes and Randall (1995a). Starting

FIG. 10. The same as Fig. 9 except for the BT grid systems.



from an icosahedron inscribed inside a unit sphere, gridrefinement is recursively operated by the bisection andthe projection. A constructed polyhedron is composedof quasi-homogeneous triangles. Each edge of the tri-angular faces is projected onto the unit sphere and aspherical triangular grid can be obtained. Two types ofthe dual hexagonal–pentagonal grid have been used;one is the VT and the other is the BT. The VT grid isgenerated by connecting the centers of circumscribedcircles of the triangles configured by neighboring threegenerators. By connecting the barycenters of such tri-angles, the BT grid is generated.

We have summarized six important conditions re-quired for an ideal grid: (i) collocation condition, (ii)orthogonality condition, (iii) bisection condition, (iv)isotropy, (v) area uniformity, and (vi) interval unifor-mity. On the spherical geodesic grids, most of hexago-nal cells are not the perfect hexagons. The imperfectshapes of CVs induce truncation errors to discretizedoperators. It is known that grid optimizations are effec-tive to restore accuracy of operators. We have summa-rized three conventional grid optimization methods: theHR95, the SPRG, and the SCV methods. In addition,we have proposed three other optimization methods:the SCB, I-HR, and M-HR methods.

We have compared the six optimization methodswith respect to (vi) the interval uniformity, (v) the areauniformity, (iv) the isotropy, and (iii) the bisection con-dition in this order. We have organized strong andweak points of the optimization methods on the basis ofa common ground.

The M-HR and the SPRG methods are appropriatefor the interval uniformity. As regards the area unifor-mity, the HR95 and the I-HR methods are the best. Interms of the isotropy, the SCB and the SCV methodsare the reasonable choices. The HR95 and the I-HRmethods work well to improve the bisection of cellfaces on the VT grid systems. In each measure, the

I-HR grids imitate well the HR95 grids. The BT gridsystems are generally less distorted compared to the VTones. In contrast, cell areas are more homogeneous onthe VT grid systems than on the BT ones.

There is a resemblance between the I-HR and theM-HR grids in the spatial distributions of cell areas anddistortions. The SPRG, the SCV, and the SCB gridsalso share similar patterns in the spatial distributions.Each optimization method can produce the smootherspatial distributions of cell areas and distortions com-pared to the SMPL grids.

There are compensations of the six conditions in eachoptimization. Each optimization method has some pref-erable features and also has some defects for numericalsimulations. One cannot really discuss grid optimiza-tion independent of the numerical schemes to be used.We have investigated impacts of each optimizationmethod on accuracy and stability of shallow-watermodels in Miura (2004). Results of a numerical assess-ment, which uses a shallow-water model on the ZM-grid arrangement (Ringler and Randall 2002) with amodified gradient operator, will be reported in a com-panion paper.

Acknowledgments. We thank Hirofumi Tomita andMasaki Satoh for helpful advice. We appreciate ToddD. Ringler and an anonymous reviewer for their carefulcomments and useful suggestions.

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a comparison of grid quality of optimized spherical

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