A Potential Enstrophy and Energy Conserving Numerical Scheme For Solution of the Shallow-Water Equations on a Geodesic Grid

Transcription

1 8:56 am A Potential Enstrophy and Energy Conserving Numerial Sheme For Solution of the Shallow-Water Equations on a Geodesi Grid Todd D. Ringler and David A. Randall Department of Atmospheri Siee Colorado State University Fort Collins, CO Otober 29, 200 revised for Monthly Weather Review Corresponding author: Todd Ringler Department of Atmospheri Siee Colorado State University Fort Collins, CO todd@atmos.olostate.edu

2 Abstrat Using the shallow water equations, we develop a numerial framework on a spherial geodesi grid that onserves domain-integrated mass, potential vortiity, potential enstrophy, and total energy. The numerial sheme is equally appliable to hexagonal grids on a plane and to spherial geodesi grids. We ompare this new numerial sheme to its predeessor and show that the new sheme does onsiderably better in onserving potential enstrophy and energy. Furthermore, in a simulation of geostrophi turbulee, the new numerial sheme produes energy and enstrophy spetra with slopes of approximately K 3 and K, respetively, where K is the total wave number. These slopes are in agreement with theoretial preditions. This work also exhibits a disrete momentum equation that is ompatible with the Z-grid vortiity-divergee equation.

3 . Introdution A spherial geodesi grid is a tessellation of the sphere that is generated by using the iosahedron as a starting point (e.g., Heikes and Randall, 995 a). The potential appliability of spherial geodesi grids to the simulation of the atmospheri general irulation has been reognized sie the 960s (Sadourny et al. 968 and Williamson 968, Sadourny and Morel 969, Williamson 969). Reently there has been renewed interest in this idea (Baumgardner and Frederikson 985; Masuda and Ohnishi 986; Heikes 993; Heikes and Randall 995 a, b; Stuhne and Peltier 996, 999; Thuburn 997; Giraldo 2000; Randall et al. 2000; Ringler et al. 2000). Sadourny and Morel (969) proposed that an optimal grid struture on the sphere should be as uniform as possible while preserving the isotropy of the spherial geometry. Spherial geodesi grids ome lose to meeting these ideal speifiations. In terms of uniformity, a spherial geodesi grid an be onstruted suh that grid ell areas vary by less than 5% over the entire sphere (Heikes, 993). In terms of isotropy, relative to any given grid ell enter all ell neighbors are nearly equidistant, and all the neighbors lie aross ell walls (Randall et al. 2000). While Sadourny and Morel (969) hose the spherial geodesi grid beause of its uniformity, Williamson (969) hose a spherial geodesi grid beause the grid ould be gradually distorted in spae to produe ireased resolution in ritial regions suh as the Gulf Stream. In ontrast to spherial geodesi grids, onventional latitude-longitude grids are neither uniform nor isotropi. The grid-pole singularities of latitude-longitude grids result in grid-ell areas that vary by O() over the globe. Furthermore, sie latitude-longitude grids have ell neighbors that lie aross both ell walls and ell orners, the grids are not isotropi. While the merits of the spherial geodesi grid may have been appreiated deades ago, only reently have these merits been realized in simulations with full atmospheri general irulation models (AGCMs). Attempts to use spherial geodesi grids to solve the nondivergent shallow water equations 2

4 were suessful (Sadourny et al. 968 and Williamson 968). The diffiulties involved in modeling the divergent shallow water equations on the spherial geodesi grid were not overome, however, until Masuda and Ohnishi (986) proposed using the vortiity-divergee form of the equations. The vortiitydivergee formulation requires inverting ellipti equations at every time step to obtain the veloity field. The omputational overhead required to do this with finite-differee models was generally onsidered prohibitive. Heikes and Randall (995a,b; hereafter HR) overame this signifiant diffiulty by implementing a multi-grid method to invert the ellipti equations in a omputationally effiient manner. Ringler et al. (2000) extended this framework from the shallow water equations to the full 3-D primitive equations and iorporated the physial parameterizations required for limate simulations. Williamson (969) hose to tile the sphere with triangles by onneting the grid points of the spherial geodesi grid (see Fig. ). Alternatively, Sadourny and Morel (969) used the inverse, or dual, of the triangular grid whih results in hexagonal grid ells also shown in Fig.. The lineage of work leading to the reation of a full AGCM based on a spherial geodesi grid in Ringler et al. (2000) emphasized the use of the hexagonal grid ells. An alternative line of researh that emphasized the use of the triangular grid ells originated with both Williamson (969) and Baumgardner and Frederikson (985). The more reent works by Stuhne and Peltier (996 and 999) and Giraldo (2000) use triangular finite elements to onstrut their numerial shemes. The present study makes extensive use of both the hexagonal and triangular grids. As disussed by Arakawa and Lamb (977), the problem of designing a numerial sheme for a general irulation model an be oeptually separated into the design of the linear properties of the sheme, and the design of its nonlinear properties. The numerial simulation of linear wave phenomena in geophysial fluid dynamis an be analyzed in terms of the geostrophi adjustment proess (Winninghoff 968, Arakawa and Lamb 977, Randall 994). The atmosphere is ontinually adjusting to forings, suh as diabati heating, by the radiation of inertia-gravity waves, whih leave behind an adjusted state that is 3

5 lose to geostrophi balae. In order to produe realisti results, a numerial sheme must faithfully reprodue this adjustment proess. Following the lead of Masuda and Ohnishi (986), HR used the Z grid, on whih vortiity, divergee, and mass are prognosed, in part beause, as shown by Randall (994), the Z-grid simulates geostrophi adjustment very well, i.e., shemes based on the Z grid have good linear properties. HR used a finite-volume method whih guarantees onservation of mass, iluding traer mass, as well as potential vortiity. HR did not disuss the onservation of nonlinear futions of the prognosti fields, suh as the kineti energy, total energy, and potential enstrophy. The present study is aimed at modifying the sheme of HR so as to guarantee onservation of kineti energy, total energy, and potential enstrophy under fritionless proesses. Suh onservation properties are partiularly important in long-term simulations suh as those that are needed for the study of limate. On long time sales, soures and sinks of energy and other quantities are fundamental to the irulation. Small but systemati spurious soures and sinks of fundamental quantities, suh as energy, an lead to unrealisti irulation regimes. The ontinuous two-dimensional nonlinear shallow water equations are highly onstrained in terms of their energy and enstrophy asades. The net effet of advetion is to transfer energy from shorter to longer spatial sales, while transferring enstrophy from longer to shorter spatial sales. Arakawa and Lamb (977, 98) point out that we have little hope of aurately simulating suh energy asades and the assoiated energy spetra if the numerial shemes do not onserve energy and potential enstrophy. They show that onserving basi quantities, suh as mass, potential vortiity, potential enstrophy, and energy, an dramatially irease the overall auray of long numerial simulations, even with no hange in the loal order of auray of the shemes. While shemes of higher-order auray are generally to be preferred, all other things being equal, a formal irease in order of auray should not be hosen at the expense of the onservation priiples. Arakawa and Lamb (977) provide an example of a lower-order onservative numerial sheme that produes a more realisti simulation than a higher-order non- 4

6 onserving sheme. The sheme presented in this paper is equally appliable to planar hexagonal grids (see Fig. ) and spherial geodesi grid. The spherial geodesi grid is quite similar to the hexagonal grid; all the grid ells are hexagons, with the exeption of twelve grid ells that are pentagons (Sadourny and Morel, 969). In Setion 2 we onstrut some of the building bloks of the numerial sheme, namely the divergee, and url operators, from their respetive definitions. We then use these basi operators, in Setion 3, to obtain a disrete form of the shallow water equations. In Setion 4 we show that this disrete system onserves a number of important quantities: mass, potential vortiity, potential enstropy, and total energy. In Setion 5 we demonstrate some properties of our finite-differee operators. This numerial sheme has been implemented in a shallow water model on a doubly-periodi plane. Results are shown in Setion 6, and these results are ompared and ontrasted to those produed with the numerial sheme urrently used in the Colorado State University (CSU) AGCM (Ringler et al. 2000). Setion 7 gives a disussion that plaes our numerial sheme into ontext. 2. The Divergee and Curl Operators 2. The grid and oordinate system The numerial sheme developed below is defined on a grid of polygons that are either hexagonal or pentagonal in shape. When disretizing the surfae of a plane, all the polygons are hexagons. When disretizing the sphere, all of the polygons are hexagons with the exeption of twelve polygons that are pentagons. All salars, suh as the fluid depth, are defined at the enters of the grid ells and are refereed with the subsript i, while all vetors, suh as the veloity vetor, will be defined at the ell orners and 5

7 refereed with the subsript. When we wish to analyze a speifi orner, we will set γ as shown in Fig. 2 and use the data from the surrounding numbered ell enters. In addition to ell enters and ell orners, we will need to referee segments of the ell walls defined by the symbol d. We will use the subsripts + and - to denote the ell wall segments on d in the ounter-lokwise and lokwise diretions from orner, respetively. The unit vetors ñ + and ñ - denote the diretions perpendiular to d + and d -, respetively, and point outward relative to ell enter 0. At eah orner we have plaed an arbitrary orthogonal oordinate system defined by the unit vetors ( ẽ, ẽ 2 ). The area fluxes aross the d + and d - wall segments are and Ṽ ñ + d + Ṽ ñ - d -, respetively. We use the symbols A and to represent the total areas assoiated with the i A grid ell enters and orners, respetively. The area of the shaded region in Fig. 2, denoted by R i, is the portion of A that is assoiated with ell enter i. 2.2 Vetor Operators With the oordinate system defined in Fig. 2, we an derive numerial approximations to the divergee and url operators from their respetive analyti definitions. The analyti form of the divergee operator is div( Ṽ) Ṽ lim A Ṽ ñ dl. () A The divergee operator yields a salar, whih will be defined at ell enters, by summing over the dot. We ould also derive an approximation to the gradient operator from its analytial definition. Instead, we will use the onstraint of onservation of total energy (Setion 4) to define the form of the gradient operator. 6

8 produt of vetor data defined at ell orners. Referring to Fig. 2, we an approximate the divergee operator at the ell enters as Ṽ ( Ṽ ) i ---- ( Ṽ A ñ + d + + Ṽ ñ - d -) i, (2) where the summation is over all ell orners assoiated with grid ell i. The variable is equal to 6 for hexagons and 5 for pentagons. If we define F + Ṽ and, (3) ñ + d + F - Ṽ ñ - d - then we an rewrite (2) in the more ompat form Ṽ ( Ṽ ) i ---- ( F + + F -) A i. (4) The definition of the url operator is url( Ṽ) Ṽ k k lim A ñ Ṽ dl A k (5) where we have expliitly seleted the vertial omponent of the url by dotting with. Sie the url is k a salar quantity, it will be defined at grid ell enters. We an approximate (5) as ( Ṽ ) k i ---- ( d + ñ + + d - ñ -) Ṽ A i, (6) k where, as with the divergee operator, the summation is over all ell orners assoiated with the ell 7

9 enter i. 3. Governing Equations The shallow water equations an be written as h t ( hṽ), (7) Ṽ t Ṽ [ K + g( h+ h, (8) ηk s )] ( τh) ( τhṽ), (9) t where h is the fluid depth, h s is the height of the surfae, Ṽ is the vetor veloity, K is the kineti energy, and η is the absolute vortiity. Equation (9) desribes the evolution of an arbitrary traer τ, where τ is a mixing ratio. Alternatively, we an take the url and divergee of (8) to generate equations for the vortiity and divergee: η t ( ηṽ ), (0) δ t ( ηṽ ). () k 2 [ K + g( h+ h s )] Here η f + Ṽ is the absolute vortiity and δ Ṽ is the divergee. In the ontinuous k equations, the vetor momentum formulation and the vortiity-divergee formulation are equivalent. 8

10 Given η and δ we an determine Ṽ in two steps. First, we solve the ellipti equations η 2 ψ, (2) δ 2 χ, (3) for the streamfution, ψ, and veloity potential, χ. We then ompute the veloity using the relation Ṽ ψ + χ. (4) k Given the disrete analogs to the divergee and url operators derived in Setion 2, we an write the disrete forms of (7), (8), and (9) as h i ---- h t ( +F + + h -F -) A i, (5) Ṽ t η Ṽ, (6) k ( K ) ( gh ( + h s )) τh ( ) i ---- h t ( +τ +F + + h -τ -F -) A i, (7) where (5) and (7) desribe the evolution of mass and mass-weighted traer within a grid ell. Equation (6) desribes the evolution of the veloity at a ell orner. For larity we reiterate our notation: terms with a subsript i are defined at ell enters, terms with a subsript are defined at ell orners, terms with a 9

11 subsript + or - are defined at the ell walls. The terms h +, τ +, h -, and should be interpreted as τ - averages of mass and traer from the ell enters to the ell wall segment + and -. The term η is the average of absolute vortiity from the ell enters to the ell orners. All of these symbols are undefined at this point. The form of the kineti energy, K i, is undefined, as is the disrete form of the gradient operator. The disrete forms of the vortiity and divergee equations an be written down by analogy with the analyti forms shown in (0) and (): η i ---- η t ( +F + + η -F -) A i, (8) δ i ---- η t ( +d ñ + η + + -d ñ ) Ṽ - - A i. (9) k 2 [ K i + gh ( i + h is )] Sie we have not yet speified the form of the disrete gradient operator, the disrete Laplaian operator is unspeified. In Setion 5 we will show that (8) and (9) an, alternatively, be derived from the disrete form of the momentum equation. As a result, the disrete momentum formulation will be entirely onsistent with the disrete vortiity-divergee formulation. In the shallow water equations the potential vortiity, q, is equal to the absolute vortiity divided by the layer thikness, h, so we an rewrite (8) as ( hq) i ---- h + t q + F + h - ( + q - F -) A i (20) 0

12 whih is idential in form to the traer equation. February, Conservation Priiples While the ontinuous equations allow an infinite number of quantities to be onserved, this is not possible within the disrete system. In this setion we will show that with the appropriate hoies of h +, τ +, h -,,,, and the gradient operator, the disrete shallow water equations shown above an be τ - η K i implemented suh that the following quantities are onserved in the domain-mean: mass, mass-weighted traer, and mass-weighted potential vortiity, mass-weighted traer variae, mass-weighted potential enstrophy, and total energy. 4. Conservation of the Domain-Mean Mass and Traer The domain-integrated mass, mass-weighted traer, and mass-weighted potential vortiity are onserved simply by virtue of their flux-forms, written in (5), (7), and (20), respetively. The only stipulation is that h +, τ +, h -,,, and have the same values when refereed by either of the τ - q + q - ell enters that share a given ell wall. This is equivalent to the requirement that the flux out of one ell is idential to the flux into its neighbor. This will beome more expliit later when we more preisely define these quantities. 4.2 Conservation of the Traer Variae In order to show how the mass-weighted traer variae an be onserved, we must start with the

13 advetive form of the traer equation. The advetive form is obtained by taking (7) - τ i *(5) to yield h i τi ( ) t ---- [ h + F + ( τ + τ ) + h - F - ( τ - τ )] A i i i (2) Following Arakawa and Lamb (977) we introdue a fution G G( τ) whih depends only on τ. We dg an onstrut an equation for G by taking (5)* G( τ) and adding it to (2)*. This results in dτ ( Gi h t i ) dg ---- i dg i h A +F + G i τ + τ i + ( ) + h dτ i -F - G i + ( τ - τ ) dτ i. (22) If we hoose γ, as shown in Fig. 2, and reognize that F + represents the flux between ell 0 and ell 2, while F - represents the flux between ell 0 and ell, we an see that G will be onserved if G 0 dg 0 τ + + ( τ ) G dτ dτ dg 2 τ + ( τ ), (23) 2 G 0 dg 0 τ - + ( τ ) G dτ 0 + dτ dg τ - ( τ ). (24) Solving (23) for τ + and (24) for τ -, we obtain τ + d G2 d G0 ( G 2 G 0 ) τ 2 ( ) + τ dτ 0 ( ) dτ , (25) dg 0 dg 2 dτ dτ 2

14 τ - d G d G0 ( G G 0 ) τ ( ) + τ dτ 0 ( ) dτ (26) dg 0 dg dτ dτ As an example, if we hoose G( τ) τ 2 we an solve (25) and (26) for τ + and τ - to obtain τ + -- ( τ, and (27) τ 0 ) τ - -- ( τ. (28) 2 + τ 0 ) Note that no additional onstraints on h + or h - are required. Sie the potential vortiity equation, (20), is idential in form to the traer equation, (7), we an use the results of (27) and (28) to guarantee the onservation of potential enstrophy in the disrete system. We an write the vortiity equation by using the LHS of (8) and the RHS of (20) to yield 6 η i ---- h + t q + F + h - ( + q - F -) A i. (29) Instead of averaging vortiity to the ell walls, we average potential vortiity to the ell walls using (27) and (28) and then multiply by the averaged mass. 3

15 4.3 Conservation of Total Energy The total energy in the disrete shallow water equations is onserved. We show this in two steps: first, that kineti energy is onserved under the proess of advetion, and seond, that the energy onversion term neither reates nor destroys total energy Conservation of Kineti Energy under Advetion In this setion we will find the form of the gradient operator that guarantees that the proess of advetion neither reates nor destroys kineti energy. First, we must form the kineti energy equation by taking the salar produt of Ṽ and the momentum equation given in (6): Ṽ Ṽ [ η t 2 Ṽ k ] Ṽ ( K ) Ṽ [ g( h+ h s )] Ṽ. (30) This equation is valid for every ell orner. The first term on the RHS of (30) is identially zero at every ell orner beause Ṽ is perpendiular to Ṽ, even in the disrete ase. Reall that while the k momentum equation is defined at ell orners, the kineti energy is a salar and will, therefore, be defined at ell enters. To move (30) from the orners to the enters, we weight eah momentum point by the area, R i, as defined in Fig. 2 and sum over all orners assoiated with grid ell i to yield h i R Ṽ Ṽ i h, (3) t 2 i R i { Ṽ ( K ) + [ g( h+ h s )] Ṽ } where we have multiplied by the ell mass, h i, after summing over the orners. Eq. (3) holds at every ell enter. 4

16 We now multiply the ontinuity equation, (5), by K i to obtain h i K i A i K, (32) t i [ h + F + + h - F -] where we have moved the ell area, A i, to the left-hand side (LHS) of the equation. Equations (3) and (32) are both defined at ell enters, so we an add them together to yield h i K i A i h t i R Ṽ Ṽ + i t 2 K i [( h +Ṽ ñ + )d + + ( h -Ṽ ñ -)d -] h i R i { Ṽ ( K ) + [ g( h+ h s )] Ṽ }, (33) where we have used (3) to express (33) ompletely in terms of the veloity. The last term on the right hand side (RHS) of (33) is an energy onversion term. At this point we will drop the last term on the RHS of (33). We return to this term in the next setion. Summing both sides of (33) over the entire domain yields n i 0 n i 0 h i K i A i h t i R Ṽ Ṽ t 2 K i [( h +Ṽ ñ + )d + + ( h -Ṽ ñ -)d -] h i R { Ṽ ( K ) } (34) 5

17 where it is understood that the outside summation is over all the grid ell enters and the inside summation is over the orners assoiated with a given ell enter i. We will now show that the RHS of (34) sums to zero at the orner labeled γ in Fig. 2. Symmetry then implies that the sum is zero at every orner. Note that in Fig. 2 there is no way to tell whether the grid ells that share the orner γ are hexagons or pentagons. Both types of polygons are aounted for in this numerial sheme without exeption. At this point it is onvenient adopt a new naming onvention as shown in Fig. 3. Sie we are hoosing a speifi orner, we will drop the subsript. The area of the triangle is partitioned into three sub-areas, R i, for 0 i 2. The vetors normal to eah ell wall segment, ñ i, point away from ell enter i towards ell enter i+, where i + is yli. The averaging of mass to ell walls, h i, denotes a yet-unspeified averaging from ell enters to the ell wall segment shared by ell enter i and i+. If we sum (34) over all grid ells but keep only the terms in the summation that have γ and expand we are left with K 0 [ d 0 h 0 ( Ṽ ñ 0 ) d 2 h 2 ( Ṽ ñ 2 )] + K [ d h ( Ṽ ñ ) d 0 h 0 ( Ṽ ñ 0 )] + K 2 [ d 2 h 2 ( Ṽ ñ 2 ) d h ( Ṽ ñ )] + ( h 0 R 0 + h R + h 2 R 2 )[ Ṽ ( K )] 0, (35) where we require that the expression will sum to zero. Now let h h 0 R 0 + h R + h 2 R 2 i ( R 0 + R + R 2 ) (36) 6

18 for all i between 0 and 2, and let S R 0 + R + R 2. We an then rewrite (35) in the ompat form of K i d i ( Ṽ ñ i ) K i d i ( Ṽ ñ i ) + SṼ ( K ) 0 i 02,, (37) where there is an implied sum over the index i, and i is yli. We an fator out the veloity and rewrite (37) as Ṽ ( K i d i ñ i K i d i ñ i + S K) 0. (38) We want (38) to be true for an arbitrary vetor veloity field, so (38) redues to K i d i ñ i K i d i + S K 0. (39) ñ i If we take the dot produt of (39) with the unit vetors that define the loal oordinate system and solve for the omponents of K, we find K ẽ K i [ d, (40) S ( i ñ i ẽ ) d ñ ( ẽ )] i i K ẽ 2 K i [ d. (4) S ( i ñ i ẽ ) d ñ 2 ( ẽ )] i i 2 Equations (40) and (4) serve as the definition of the gradient operator. Note that the form of K is still unspeified; the RHS of (34) sums to zero for arbitrary salar fields. We will return to the gradient operator in the next setion. We an now write (34) as 7

19 n h i K i A i h. (42) t i R Ṽ Ṽ i t 2 i 0 If we interpret (42) as two terms in a hain rule expansion, we have n h i K i A i h t i R Ṽ Ṽ + i t 2 i 0 n i 0 n i 0 ( Ki A. (43) t i h i ) This will be true if we define the kineti energy as K i A i R i Ṽ Ṽ. (44) 2 Equations (36), (39), and (44) are the onstraints that allow kineti energy to be onserved under advetion The energy onversion term In addition to showing that advetion does not reate or destroy kineti energy, we must show that the terms that represent the onversion of energy between its kineti and potential forms do not reate or destroy energy. In order to do this we must derive the potential energy equation by multiplying ga i ( h s + h i ) through (5) to obtain gai h t i h s + --h 2 i g( h s + h i ) ( h +F + + h -F ) -. (45) 8

20 In an approah similar to that used above, we will show that the RHS of (45) and the last term in (33) ael at every ell orner. Combining the RHS of (45) and the last term on the RHS of (33) and summing over the domain, we want to ensure that n i 0 n gh ( s + h i ) [( h +Ṽ ñ + )d + + ( h -Ṽ ñ -)d -] h i R i Ṽ [ gh ( + h s )] 0 i 0 (46) If we ompare (46) to the RHS of (34), we see that the two expressions are idential, exept that K in (34) is replaed by gh ( s + h) in (46). Sie we have already shown that the RHS of (34) sums to zero for an arbitrary salar, (46) is satisfied provided that we use an expression analogous to (39) to ompute the gradient of gh ( s + h). The form of total energy that is onserved in this disrete system is n i 0 A i h i K i g h s + --h + 2 i (47) whih is onsistent with the ontinuous shallow water equations. 5. Properties of the finite-differee operators We have shown that we an onserve total energy by hoosing (39) as the disrete gradient operator. In this setion we will show that (39) has an alternative geometri interpretation as the slope of the plane fit through the surrounding three data points. For larity, we will assume a regular hexagonal grid 9

21 in this setion, but note that we have verified our findings on both distorted hexagonal grids and spherial geodesi grids. Referring to Fig. 2 and assuming a regular hexagonal grid, if we take the dot produt of (39) with the unit vetor ñ +, we find that K ñ + K 2 K , (48) l where l is the distae between grid ells 2 and 0. Equation (48) an be interpreted as the slope of the plane fit through ( K 0, K, K 2 ) in the ñ + diretion. Taking the dot produt of (39) with ñ - yields K ñ - K K (49) l Equations (48) and (49) are suffiient to prove that (39) is the slope of the plane fit through ( K 0, K, K 2 ). identity, Oe we interpret (39) as the slope of the plane fit through the surrounding data, the vetor K 0, (50) follows immediately in the disrete system. The physial interpretation of the url operator shown in (5) and (6) is a line integral of the omponent of a vetor field along the path of integration. In Fig. 2, if we start at γ and follow K along the perimeter of grid ell 0, we return to where we started. Hee, (50) holds in the disrete system. While this is a rough argument, we have proven (50) rigorously. With the divergee and gradient operators speified, we an determine the form of the disrete 20

22 Laplaian operator as 2 K K, where K is a salar field. If we define the vetor Ṽ in (2) as K and use (39) as the form of the gradient operator, we find that ( 2 K )0 6 t K, (5) A 0 l ( i K 0 ) where t is the length of the line segment onneting grid ell orners. Equation (5) an be simplified to ( 2 K ) ( K i K 0 ) 3A 0 (52) whih is exatly the same form as derived by HR. Along with the vetor identity shown in (50) and the definition shown in (52), we require two more equations to hold in the disrete system: ( Ṽ ) ( ηṽ ), (53) ηk ( Ṽ ) ηk ( ηṽ ). (54) We simply note here that (53) and (54) hold in the disrete ase solely beause we assume that the vetor Ṽ is perpendiular to the vetor. k Equations (50), (52), (53), and (54) suffie to derive the disrete vortiity and divergee equations [(8) and (9)] from the disrete momentum equation (6). Thus we olude that the disrete momentum formulation is onsistent with the disrete vortiity-divergee formulation. 2

23 6. Numerial Tests We will show results from three numerial models. Two of the models are based on the framework developed above. The first uses the disrete vortiity-divergee formulation [eqs. (5), (8), and (9)] and we will refer to this model as NS_Vor-Div. The seond uses the disrete vetor momentum formulation [eqs. (5) and (6)] and will be referred to as NS_Momentum. The third model is based on the sheme developed by Masuda and Ohnishi (986) and HR. This sheme uses the vortiity-divergee formulation of the shallow-water equations and will be referred to as OS_Vor-Div. A fairly omplete omparison of OS_Vor-Div and NS_Vor-Div is ontained in Appendix A. The models are situated on a doubly-periodi f-plane with f s. The plane is disretized using a regular hexagonal grid of 28 x 28 (see Fig. 7). The distae between ell enters, n, is 00 km. The time-stepping sheme for all three models is 3rd-order Adams-Bashforth with a time step of 00 seonds. All three models are started from the same initial ondition, of the form ht ( 0) 400 ± 50m, (55) η( t 0) f 0 ± s, (56) δ( t 0) 0 ± s. (57) The first values on the RHS of (55), (56), and (57) are the respetive domain-mean values and the seond values are the maximum perturbations from the means. The perturbations were generated by a random number generator. The initial onditions are highly unbalaed. Given the absolute vortiity and divergee fields, the disrete analogs of (2) and (3) are used to solve for the streamfution and veloity 22

24 potential, respetively. For the NS_Momentum model, we use the disrete gradient operator to differentiate the streamfution and veloity potential to obtain the initial veloity field. The time rates of hange of all prognosti variables are zero at t 0. The surfae topography is random and has the form h s 0 ± 20m. With gravity set to 9.8ms 2, we obtain a Rossby radius of deformation of approximately 700 km. With n 00km, we marginally resolve eddies on this sale. All three models are invisid in the sense that no expliit dissipation is introdued. Fig. 4a measures the error in domain-integrated total energy over a 40-day integration. The y-axis is log base 0 of the absolute value of the frational hange in domain-integrated total energy. All of the simulations show substantial variability during the first few days of integration. During this period the state of the system is rapidly hanging while it moves toward geostrophi balae. The time truation error is relatively large during this period and we have verified that it aounts for the variations of the total energy over the first several days of integration. After 40 days, the NS_Vor_Div and NS_Momentum simulations have a total energy that is within 0.5% of the initial value. The total energy in the OS_Vor-Div simulation ontinues to drift and after 40 days the total energy has doubled. Fig. 4b shows the frational hange in domain-integrated mass-weighted potential enstrophy. Both the NS_Vor-Div and NS_Momentum simulations onserve mass-weighted potential enstrophy to within 0.05% after 40 days of integration. The NS_Vor-Div simulation displays a slow modulation of this quantity that is due to truation error in the ellipti solver [(2) and (3)]. If we make the onvergee threshold in the ellipti solver suffiiently stringent, we an eliminate this modulation. The OS_Vor-Div simulation shows a steady drift in potential enstrophy and after 40 days of integration the potential enstrophy has also doubled. Fig. 5a shows the wave number spetrum of total energy at the end of the 40 day integration for eah simulation. Fig. 5a also shows the energy spetrum at 23 t 0, whih is the same of all simulations.

25 Consistent with the white noise initial ondition, the initial energy spetrum is also white. Fig. 5b shows the enstrophy spetrum of eah simulation after 40 days and also shows the initial enstrophy spetrum. The purpose of Fig. 5 is to show whether there is anomalous buildup of energy or enstrophy at any given wave number. In terms of energy, both the NS_Vor-Div simulation and NS_Momentum simulation show an appropriate upsale transport of energy with no apparent buildup of energy at the grid-sale. The OS_Vor- Div simulation also shows the upsale asade, but the lak of onservation of total energy is readily apparent. In terms of enstrophy, both the NS_Vor-Div simulation and NS_Momentum simulation show no buildup of enstrophy at the smallest sales. The OS_Vor-Div simulation shows a buildup of enstrophy at both the smallest and largest sales. Now we wish to determine whether the new numerial sheme produes the appropriate energy and enstrophy asades. In order to test this, we must ilude a sink of enstrophy at the smallest resolved sales in order to generate a down-sale asade of enstrophy. Using the NS_Momentum model, we do this by iluding a 6 diffusion in the momentum equation 2. The oeffiient on the diffusion is 0 24 m 6 s ; sale analysis suggests that this value is suffiient to dissipate a vortiity or divergee perturbation of size 0 4 s, with a spatial sale of 2 n, in several hours. The model is initialized with the same initial ondition shown above, but there is no topography. Fig. 6 shows the energy and enstrophy spetra averaged over days 950 to 050. Dimensional analysis suggests that, within the inertial range, the energy spetrum should deay as K 3, where K is the total wave number, and the enstrophy spetrum should deay as K (Pedlosky 987 and Salmon 998). Fig. 6 indiates that the numerial model is aurately apturing both the upsale asade of energy and the down-sale asade of enstrophy. After 000 days of integration, the total energy has hanged by only 0.5% (not shown). So while the 6 2. If we use the NS_Vor-Div model with a 6 diffusion operator of vortiity and divergee, our finding are the same. 24

26 diffusion operator ontinues to destroy enstrophy, it destroys very little energy. Fig. 7 shows the relative vortiity field at four stages of the integration: t 0, t 0, t 00, and t 000 days. This figure qualitatively onfirms the spetral analysis shown in Fig. 6. The flow ontinually evolves into larger strutures. Also onsistent with the Fig. 6, we do not see any spurious buildup of energy or enstrophy at the grid-sale. 7. Disussion and Colusions We have onstruted a shallow water model, for use on a spherial geodesi grid, that onserves mass, potential vortiity, potential enstrophy, and total energy. We have demonstrated these onservation properties in a numerial simulation of geostrophi turbulee on a f-plane. In addition to onserving energy and potential enstrophy to within the time truation error, our numerial sheme was able to aurately apture the spetral distributions of energy and enstrophy in a simulation of geostrophi turbulee. In the Introdution we stated that onserving these basi quantities is only one neessary omponent of a robust numerial sheme; faithful simulation of the geostrophi adjustment proess and numerial onvergee are the other two neessary omponents. Looking at these other two neessary omponents will provide some ontext for the work we have ompleted here. In a study of the geostrophi adjustment proess, Randall (994) disretizes the vortiitydivergee form of the shallow water equations on the Z-grid. The Z-grid entails the use of the vortiitydivergee formulation with all salar quantities defined at the grid ell enters. He ompares the disrete dispersion relation obtained using the Z-grid to the dispersion relations obtained using the A-, B-, C-, D-, and E-grids (Arakawa and Lamb 977). The A- through E-grids entail the use of the momentum 25

27 formulation. Randall (994) learly shows that the Z-grid more aurately simulates the geostrophi adjustment proess than any of the A- through E-grid systems. On the Z-grid, the disrete equations involve no spatial averaging and only a disrete Laplaian operator is required [see Randall (994) Eqs. 4-6]. As shown in (52), the Laplaian operator derived here is idential to that used by HR. Furthermore, this Laplaian operator is onsistent with the formulation used in Randall (994). We olude that the framework developed here not only onserves the quantities listed above, but is also onsistent with the Z- grid disretization and, thus, does a better job of simulating the geostrophi adjustment proess than the A- through E-grid systems. This topi is more fully explored in Ringler and Randall (200, under revision for Monthly Weather Review). The order of auray of this numerial sheme is determined by three aspets: the auray of the interpolations [Eqs. (27), (28), and (36)], the auray of the vetor operators [Eqs. (2), (6), and (39)], and the properties of the grid. On a regular hexagonal grid, all of these approximations are formally seondorder aurate. We interpreted the approximation to the gradient operator in two ways: first, the gradient insures onservation of kineti energy under the proess of advetion, and seond, the gradient operator measures the slope of the plane fit through the surrounding three data points. This work has shed light on the relationship between the Z-grid and its disrete analog in the momentum equations. We have exhibited a momentum equation that is ompatible with the Z-grid. In the ontinuous system, we an move between the momentum formulation and vortiity-divergee formulation in a seamless manner using basi vetor operators. Both forms ontain the same information and yield the same result. In this work we were able to manipulate the disrete form of the vetor momentum equation using disrete analogs of the gradient, divergee, and url operators to derive a disrete form of vortiity-divergee equations. Sie the derived vortiity-divergee equations are the Z- grid vortiity-divergee equations, we are guaranteed that the disrete momentum equation is onsistent 26

28 with the Z-grid. The momentum analog to the Z-grid (all it the ZM-grid) solves the full momentum equation at every grid ell-orner. On the hexagonal grid there are twie as many grid ell orners as there are grid ell enters. An easy way to see this is as follows. Eah ell enter is assoiated with an area, A i, and eah ell orner is assoiated with an area, A. On a regular hexagonal grid A i 2A, so in order to over the same area there must be twie as many ell orners as there are ell enters. Sie the ZM-grid solves the momentum equation at every ell orner and the ontinuity equation at every ell enter, there are twie as many momentum points as there are mass points. The redunday of the momentum equation allows for the existee of omputational modes. We define omputational modes as solutions to the disrete equations that have no analogs in the ontinuous system. We are addressing the redunday of the ZM-grid in separate work by analyzing the linear geostrophi adjustment proess on a hexagonal grid (Ringler and Randall, 200 under revision for Monthly Weather Review). We have implemented the numerial sheme outlined in this paper on a spherial geodesi grid and we are urrently onduting the standard suite of shallow water test ases as well as several other tests. Aknowledgments We would like to thank Ross Heikes and Alistair Adroft for useful disussions and for omments on an earlier draft of this paper. This work was supported by the U. S. Department of Energy s Climate Change Predition Program under grant DE-FG03-98ER626 to Colorado State University. 27

29 Appendix A: Comparison of numerial shemes In this appendix we ompare the OS_Vor-Div sheme developed by Masuda and Ohnishi (986) and HR to our new sheme, NS_Vor-Div. For this omparison we will assume a regular hexagonal grid where the distae between grid ell enters is n and the distae between grid ell orners is t. C. Mass Equation Both shemes express the mass equation as fluxes aross the ell walls and we will denote these fluxes as F j, where the flux points from ell 0 to ell j. For both shemes, the mass equation for the fluid thikness at ell enter 0 an be written as h F t j t A 0 6 j. (58) The shemes will differ in the formulation of the flux, F j. C.. OS_Vor-Div Sheme The flux aross ell walls an be express as F h j + h 0 j χ j χ n h j + h ψ j ψ j (59) 2 3 t The first term on the RHS of (59) represents the flux of mass aross the ell wall shared by grid ells 0 and j due to divergent motion. The seond term on the RHS of (59) represents the flux aross the ell wall due 28

30 to rotational motion. Substituting (59) into (58) yields h t 6 A o j t h j + h ( χ n 2 j χ 0 ) A 0 i h j + h ( ψ. (60) 6 j + ψ j ) C..2 NS_Vor_Div Sheme The flux aross eah ell wall, F j, is broken into two parts: one part assoiated with the orner j, and one part assoiated with the orner j. The mass equation an be written just like (58), but the fluxes are expressed as F j -- h j + h j + h χ j χ h j + + h j + h χ j χ n 2 3 n. (6) -- h j + h j + h ψ j -- ( ψ ψ j ) h j + + h j + h -- ( ψ ψ j ) ψ j t t 2 2 Substituting (6) into (58) and doing some algebra yields h t 6 A o j t h j + 2h j + 2h 0 + h j ( χi χ n 6 0 ) A 0 i h j + h ( ψ. (62) 6 j + ψ j ) Comparison with (60) shows that the rotational motion is modeled exatly as before, but that a slight modifiation to the flux due to divergent motion is introdued. 29

31 C.2 Vortiity Equation Similar to the mass equation, the vortiity equation for both shemes an be written as η H, (63) t j t A o 6 j where, similar to F j in the mass equation [(58)], H j is the flux of absolute vortiity aross the ell walls. In fat, the vortiity equation is idential to the mass equation, expet h is replaed by η. Furthermore, in both shemes the averaging of absolute vortiity is done in exatly the same manner as the averaging of mass. Instead of dupliating the same manipulations we just ompleted, we an simply state that the OS_Vor-Div sheme integrates the vortiity equation as shown in (60) and the NS_Vor-Div sheme integrates the vortiity equation as shown in (62). C.3 Divergee Equation For larity, we re-iterate the ontinuous form of the divergee equations: δ t ( ηṽ ), (64) k 2 ( K + g( h+ h s )) whih we an write symbolially as δ t Term + Term2 (65) 30

32 In Setion 5 we show that both the OS_Vor-Div and NS-Vor-Div use the same disrete Laplaian operator, therefore, regarding Term2, the only differees between the two shemes ould be the form of kineti energy, K. In omparing the two shemes we need to identify any differees in the treatment of Term or the formulation of K. C.3. OS_Vor-Div This sheme approximates Term as Term A o i t h j + h o ( ψ n 2 j ψ 0 ) A 0 i h j + h o ( χ. (66) 6 j + χ j ) Masuda and Ohnishi (986) express the kineti energy in ontinuous form as K -- [ ( ψ ψ) ψ 2 ψ + ( χ χ) χ 2 χ + ( ψ χ) ]. (67) 2 k Eq. (67) is derived in Heikes (993). If we use the formulation developed by Masuda and Ohnishi (986) we an write the disrete form of (67) at ell enter 0 as K j ψ j ψ χ j χ ψ j ψ χ j + χ j n n n 3 t χ j χ ψ j + ψ j n 3 t.(68) In the proess of manipulating the disrete analog of (67) into (68), we ombined the first two terms on the RHS of (67) into the first term on the RHS of (68). The third and fourth terms on the RHS of (67) were ombined into the seond term on the RHS of (68). And finally, the last term on the RHS of (67) was 3

33 expanded to form the last two terms on the RHS of (68). We an now write (68) in terms of the veloity omponents as K 0 -- R, (69) 6 [ T + D N + R T D T + R N D N ] j where R and D are the rotational and divergent veloities, respetively. The subsript N and T are the normal and tangent diretions relative to eah ell wall, respetively. For example, R T is the rotational part of the veloity vetor in the tangent diretion. There are several points to note. First, (69) is not positive definite; the last two terms an be negative. Seond, (69) is biased in the sense that ertain omponents of the veloity vetor, suh as R T and D N, more strongly ontrol the value of K 0 than other omponents of the veloity, suh as R N and D T. C.3.2 NS_Vor_Div This sheme approximates Term as Term A 0 j t η j + 2η + 2η j o + η j ( ψi ψ n 6 0 ) A 0 i η j + η o ( χj.(70) 6 + χ j ) In omparing this to OS_Vor-Div we see that the part of Term due to divergent motion is modeled exatly the same in the two shemes, but the part due to rotational motion is different. The form of kineti energy used in NS_Vor-Div is shown in (44). When we use the vortiitydivergee formulation, we an rewrite (44) using (4) as 32

34 K i j Ṽ j Ṽ j j (( ψ) k j + ( χ) j ) (( ψ) k j + ( χ) j ) (7) 2 In pratie, we use the gradient operator to ompute ( ψ) j and ( χ) j at eah orner, then we onstrut the veloity vetor as eah orner. We ompute the kineti energy as the sum of -- ( Ṽ over the ell 2 j Ṽ j ) orners. The important point to note is that our formulation uses the full, unbiased, veloity vetor to ompute the kineti energy. 33

35 Referees Arakawa, A., and V. R. Lamb, 977: Computational design of the basi dynamial proess of the UCLA general irulation model. Methods in Computational Physis, 7, Aademi Press, Arakawa, A. and V. R. Lamb, 98: A potential enstrophy and energy onserving sheme for the shallow water equations. Mon. Wea. Rev., 09, Baumgardner, J.R., and P. O. Frederikson, 985: Iosahedral disretization of the two-sphere. SIAM J. Numer. Anal., 22, Giraldo, F.X., 2000: Lagrange-Galerkin methods on spherial geodesi grids: the shallow water equations. J. Comp. Phys., 60, Heikes, R., 993: The shallow water equations on a spherial geodesi grid. Department of Atmospheri Siee, Colorado State University, Tehnial Report 524, pp. 75. Heikes, R. and D. A. Randall, 995 a: Numerial integration of the shallow-water equations on a twisted iosahedral grid. Part I: Basi design and results of tests. Mon. Wea. Rev., 23, Heikes, R. and D. A. Randall, 995 b: Numerial integration of the shallow-water equations on a twisted iosahedral grid. Part II: A detailed desription of the grid and an analysis of numerial auray. Mon. Wea. Rev., 23, Masuda, Y., and H. Ohnishi, 986: An integration sheme of the primitive equations model with an iosahedral-hexagonal grid system and its appliation to the shallow water equations. Short- and 34

36 Medium-Range Numerial Weather Predition. Japan Meteorologial Soiety, Tokyo, Pedlosky, J, 987. Geophysial Fluid Dynamis, Seond Edition, Springer-Verlag, 70 pp. Randall, D.A., 994: Geostrophi adjustment and the finite-differee shallow water equations. Mon. Wea. Rev., 22, Randall, D. A., R. Heikes, and T. D. Ringler, 2000: Global atmospheri modeling using a geodesi grid with an isentropi vertial oordinate. In General Cirulation Model Development, Aademi Press, pp Ringler, T.D., and D.A. Randall, 200: The ZM-grid: an alternative to the Z-grid. Mon. Wea. Rev, revised. Ringler, T. D., R. P. Heikes, and D. A. Randall, 2000: Modeling the atmospheri general irulation using a spherial geodesi grid: A new lass of dynamial ores. Mon. Wea. Rev., 28, Sadourny, R. and P. Morel, 969: A finite-differee approximation of the primitive equations for a hexagonal grid on a plane. Mon. Wea. Rev., 97, Sadourny, R., A. Arakawa, and Y. Mintz, 968: Integration of the nondivergent barotropi vortiity equation with an iosahedal-hexagonal grid for the sphere. Mon. Wea. Rev., 96, Salmon, R Letures on Geophysial Fluid Dynamis, Oxford University Press, 375 pp. Stuhne, G. R. and W. R. Peltier, 996: Vortex erosion and amalgamation in a new model of large sale flow on a sphere. J. Comp. Phys., 28, 58-8 Stuhne, G. R. and W. R. Peltier, 999: New iosahedral grid-point disretizations of the shallow water equations on the sphere. J. Comp. Phys., 44, Thuburn, J., 997: A PV-based shallow-water model on a hexagonal-iosahedral grid. Mon. Wea. Rev., 35

37 25, Williamson, D. L., 968: Integration of the barotropi vortiity equation on a spherial geodesi grid. Tellus, 20, Williamson, D. L., 969: Numerial integration of fluid flow over triangular grids. Mon. Wea. Rev., 97,

38 Figure Captions Figure : Grid points, shown as irles, an be onneted to form a triangular grid shown by the dashed lines. Alternatively, ell walls an be positioned halfway between grid points to form a hexagonal grid shown by the solid lines. The hexagonal grid is the dual of the triangular grid Figure 2: The grid is omposed of hexagons (and possibly pentagons) with salars defined at the ell enters and vetors defined at the ell orners. Eah vetor is desribed in the ( ẽ, ẽ 2 ) oordinate system that is plaed at eah ell orner. The normal vetors to the ell walls are ñ + and ñ - where the + diretion is in the ounter-lokwise diretion from and - is in the lokwise diretion. The area assoiated with eah orner, A, is defined by the line segments onneting the grid ell enters. R 0 is the portion of A owned by ell 0. The area assoiated with the grid ell enter, A i, is defined by the perimeter of the hexagon. Figure 3: The symbol, h i for 0 i 2 represents an averaging of mass to the ell wall shared by grid ell i and i+. The area of the triangle is divided into 3 parts, R i. The unit vetors normal to eah ell wall segment are given as ñ i and point from ell enter i to ell enter i+. Figure 4: Log base 0 of the frational hange in total energy (top) and mass-weighted potential enstrophy (bottom) for eah numerial simulation. Figure 5: Energy power spetrum (top) and enstrophy power spetrum (bottom) at day 40 for eah numerial simulation, plotted as futions of total wave number. The open irles show the initial energy and enstrophy power spetra. 37

39 Figure 6: Total energy power spetrum (top) and enstrophy power spetrum (bottom) averaged over days 950 to 050. The open irles show the initial energy and enstrophy power spetra. Also shown in eah figure part by a dashed line is the theoretial predition of the slope the respetive spetrum. The solid line denotes the equivalent wave number of the Rossby radius of deformation Figure 7: The evolution of the relative vortiity field over the first 000 days of integration. 38

40 Figure : Grid points, shown as irles, an be onneted to form a triangular grid shown by the dashed lines. Alternatively, ell walls an be positioned halfway between grid points to form a hexagonal grid shown by the solid lines. The hexagonal grid is the dual of the triangular grid 39

41 2 ẽ 2 Ṽ Ṽ ñ + ẽ 3 γ Ṽ ñ - R d - 0 d Figure 2: 5 The grid is omposed of hexagons (and possibly pentagons) with salars defined at the ell enters and vetors defined at the ell orners. Eah vetor is desribed in the ( ẽ, ẽ 2 ) oordinate system that is plaed at eah ell orner. The normal vetors to the ell walls are ñ + and ñ - where the + diretion is in the ounter-lokwise diretion from and - is in the lokwise diretion. The area assoiated with eah orner, A, is defined by the line segments onneting the grid ell enters. R 0 is the portion of A owned by ell 0. The area assoiated with the grid ell enter, A i, is defined by the perimeter of the hexagon 40

42 2 ẽ 2 R 2 ñ h ẽ h 2 ñ 0 R ñ 2 h 0 R 0 0 Figure 3: The symbol, h i for 0 i 2 represents an averaging of mass to the ell wall shared by grid ell i and i+. The area of the triangle is divided into 3 parts, R i. The unit vetors normal to eah ell wall segment are given as ñ i and point from ell enter i to ell enter i+. 4

43 (A) (B) Figure 4: Log base 0 of the frational hange in total energy (top) and massweighted potential enstrophy (bottom) for eah numerial simulation. 42

44 (A) (B) Figure 5: Energy power spetrum (top) and enstrophy power spetrum (bottom) at day 40 for eah numerial simulation, plotted as futions of total wave number. The open irles show the initial energy and enstrophy power spetra. 43

45 (A) Rossby radius of deformation K 3 (B) K Rossby radius of deformation Figure 6: Total energy power spetrum (top) and enstrophy power spetrum (bottom) averaged over days 950 to 050. The open irles show the initial energy and enstrophy power spetra. Also shown in eah figure part by a dashed line is the theoretial predition of the slope the respetive spetrum. The solid line denotes the equivalent wave number of the Rossby radius of deformation. 44

46 t 0 days t 0 days t 00 days t 000 days Figure 7: Evolution of the relative vortiity field over the first 000 days of integration. 45