Sixth-Order Difference Scheme for Sigma Coordinate Ocean Models
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1 064 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 7 Sxth-Order Dfference Scheme for Sgma Coordnate Ocean Models PETER C. CHU AND CHENWU FAN Department of Oceanography, Naval Postgraduate School, Monterey, Calforna 15 May 1996 and 14 March 1997 ABSTRACT How to reduce the horzontal pressure gradent error s a key ssue of usng -coordnate ocean models, especally of usng prmtve equaton models for coastal regons. The error s caused by the splttng of the horzontal pressure gradent term nto two parts and the subsequent ncomplete cancellaton of the truncaton errors of those parts. Due to the fact that the hgher the order of the dfference scheme, the less the truncaton error and the more complcated the computaton, a sxth-order dfference scheme for the -coordnate ocean models s proposed n order to reduce error wthout ncreasng complexty of the computaton. After the analytcal error estmaton, the Sem-spectral Prmtve Equaton Model s used to demonstrate the beneft of usng ths scheme. The stablty and accuracy are compared wth those of the second-order and fourth-order schemes n a seres of calculatons of unforced flow n the vcnty of an solated seamount. The sxth-order scheme s shown to have error reductons by factors of 5 compared to the fourth-order dfference scheme and by factors of 50 compared to the second-order dfference scheme over a wde range of parameter space as well as a great parametrc doman of numercal stablty. 1. Introducton In shallow-water predcton models the effects of bottom topography must be taken nto account. Ths can be done by usng a terran-followng -coordnate system, where the water column s dvded nto the same number of grd cells ndependent of depth. Let (x *, y *, z) denote Cartesan coordnates and (x, y, ) sgma coordnates. In most sgma coordnate ocean models the relatonshp between the two coordnate systems s z x x *, y y *, 1, (1) H(x, y) where z and ncrease vertcally upward such that z 0, 1 at the surface and z Hand 1at the bottom; H H(x, y) s the bottom topography. We restrct our attenton to two dmensons (x, ) for smplcty. Horzontal gradents n the z- and -coordnates are related by p p H 1 p (1 ). () x * x x H Most ocean models are hydrostatc balanced; that s, p g, (3) z Correspondng author address: Dr. Peter C. Chu, Department of Oceanography, Naval Postgraduate School, Code OC/CU, Monterey, CA E-mal: chu@nps.navy.ml where g s the acceleraton due to gravty and s the densty of the flud. Substtuton of (3) nto () and use of z H yeld p p 1 H (1 ) g, (4) x * x x whch s used for computng the horzontal pressure gradent. A problem has long been recognzed n computng the horzontal pressure gradent n the -coordnate system (e.g., Gary 1973; Haney 1991; Mellor et al. 1994; McCalpn 1994) the horzontal pressure gradent becomes a dfference between two terms, whch leads to a large truncaton error at steep topography. We refer the reader to a comprehensve revew by Haney (1991). Mellor et al. (1994) obtaned an error estmaton scheme and showed that the pressure gradent error s advectvely elmnated after a long tme ntegraton. McCalpn (1994) employed a fourth-order dfference scheme to elmnate the pressure gradent error by a factor of Followng McCalpn s path, we propose a sxthorder dfference scheme for a further reducton of pressure gradent error.. Second-, fourth-, and sxth-order schemes In ocean models, the most common spatal dfference scheme used to approxmate the frst dervatve s a two
2 SEPTEMBER 1997 NOTES AND CORRESPONDENCE 065 pont, centered, second-order staggered scheme such as the C-grd scheme (Hadvogel et al. 1991) and the B-grd scheme (Robnson 1993): 3 p p 1/ p 1/ 1 p 3 x 4 x 1/ 1/ O( ), (5) where s the grd spacng. The truncaton error of ths scheme s O( ). For the fourth-order C-grd scheme, p p 3/ 7p 1/ 7p 1/ p 3/ x p x 9 1 (1/ 1/) (3/ 3/) O( ); (6) the truncaton error s O( 4 ). Comparng (6) wth (5), the second-order dfference, the error rato between the fourth-order and second-order schemes s estmated by p p x x r4, (7) p p x x Snce the truncaton error decreases wth the ncrease of the order of the dfference scheme, t mght be benefcal to use an even hgher-order dfference scheme. Thus, we propose a sxth-order dfference scheme 7 p 9p 5/ 15p 3/ 50p 1/ 50p 1/ 15p 3/ 9p 5/ 5 p 6 7 x x, (1/ 1/) (3/ 3/) (5/ 5/) O( ) (8) to compute the horzontal pressure gradent. For the two ponts nearest to the boundary, we use the lower-order schemes (takng one-sde boundary as an example), p p3/ p1/, x 1 p p1/ 7p3/ 7p5/ p7/, x 4 1/ 3/ 1, 9 1 ( 3/ 5/) ( 1/ 7/) Comparng (8) wth (6), the error rato between the sxth-order and fourth-order schemes s estmated by p p x x r6, (9) p p x x 3. Seamount test case a. Model descrpton Suppose a seamount located nsde a perodc f-plane ( f s 1 ) channel wth two sold, free-slp boundares along constant y. Unforced flow over a seamount n the presence of restng, level sopycnals s an deal test case for the assessment of pressure gradent errors n smulatng stratfed flow over topography. The flow s assumed to be reentrant (perodc) n the alongchannel coordnate (.e., x axs). We use ths seamount case of the Sem-spectral Prmtve Equaton Model (SPEM) verson 3.9 to test the new dfference scheme. The reader s referred to the orgnal reference (Hadvogel et al. 1991) and the SPEM 3.9 User s Manuel (Hedstrom 1995) for dataled nformaton. In the horzontal drectons the model uses the C-grd and the second-order fnte dfference dscretzaton except for the horzontal pressure gradent for whch the user has a choce of ether second-order or fourth-order dfference dscretzaton (McCalpn 1994). In the vertcal drecton the model uses a boundary-ftted -coordnate system. The dscretzaton s by spectral collocaton usng Chebyshev polynomals. Our model confguraton s smlar to that of Beckmann and Hadvogel (1993) and Mc- Calpn (1994). The tme step and grd sze used here are
3 066 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 7 FIG. 1. The seamount geometry. t 675 s, x y 5 km. b. Topography The doman s a perodc channel, 30 km long and 30 km wde. The channel walls are sold (no normal flow) wth free-slp vscous boundary condtons. The channel has a far-feld depth h max and n the center ncludes an solated Gaussan-shape seamount wth a wdth L and an ampltude h s (Fg. 1), [ ] (x x 0) (y y 0) h(x, y) hmax hsexp, (10) L where (x 0, y 0 ) are the longtude and lattude of the seamount center. The far-feld depth (h max ) s fxed as 5000 m. But the seamount ampltude (h s ) changes from 500 to 4500 m, and the lateral scale of the seamount (L) vares from 6 to 40 km for the study. c. Densty feld Suppose that the background flud s at rest and wth an exponentally stratfed mean densty z (z) 8 exp, (11) H where z s the vertcal coordnate, and H 1000 m. Here a constant densty, 1000 kg m 3, has been subtracted for error reducton. When a densty dsturbance ˆ exp(z/h ) was ntroduced ntally, (z), (1) the flud was slghtly less stably stratfed than the reference feld for each computaton. Here ˆ s called the densty anomaly. The larger the ˆ, the less stable the flud wll be. In ths study, ˆ vares from 0.1 to 1 kg m 3. Followng Beckmann and Hadvogel (1993) and McCalpn (1994), we subtract the mean densty feld (z) before ntegratng the densty feld to obtan pressure from the hydrostatc equaton. d. Lateral vscosty Ideally, the new dfference scheme should be tested wth no lateral dffuson of densty. Ths s due to the fact that the densty dffuson along surfaces generates horzontal gradents wherever the surfaces are not flat, and then produces horzontal pressure gradents that drve currents n much the same way as the pressure gradent errors (McCalpn 1994). Unfortunately, the absence of the horzontal dffuson keeps the small-scale pressure dsturbances generated by topographc-scale densty advecton and the nduced small-scale velocty felds, whch n turn cause the computatonal nstablty problem. Thus, some lateral vscosty on surfaces s requred n the momentum equatons to mantan stablty. A constant coeffcent (A H ) bharmonc formulaton s used here for the lateral vscosty, whch vares from to 10 8 m 4 s 1 n ths study. 4. Standard case a. Model parameters In ths secton, we assess the sxth-order scheme (8) by the cumulatve effects of the pressure gradent errors.
4 SEPTEMBER 1997 NOTES AND CORRESPONDENCE 067 FIG.. Instantaneous error pattern after 5 days of ntegraton for the unforced (exponental stratfcaton) experment wth sxth-order dfference scheme: (a) u, (b), (c) w, and (d) mass transport streamfuncton. Here u,, ware evaluated at the slce (facng upchannel) through the center of the seamount. A varety of confguraton were used for the test. These wll be descrbed n next secton. At frst, we set up a standard test case to compare errors among three dfferent (second, fourth, and sxth) schemes: L 40 km, hs 4500 m, ˆ 0. kg m, AH 10 m s. (13) The SPEM3.9 standard was ntegrated for 0 days for the standard test case usng ordnary and compact fourth- and sxth-order dfference schemes for computng horzontal pressure gradent. Fgure dsplays errors n the streamfuncton and velocty felds after performng 5 days of ntegraton usng the sxth-order scheme. The mass transport streamfuncton has a large-scale eght-lobe pattern centered on the seamount. Ths symmetrc structure can be found n all the felds. After 5 days of ntegraton, the model generates spurous currents of O(0.001 cm s 1 ).
5 068 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 7 TABLE 1. Ratos of CPU tme and varous errors between the second-order and the fourth (sxth)-order schemes. Rato CPU V 1 V PG 1 PG nd/4th nd/6th FIG. 3. Peak error velocty for the second-, fourth-, and sxth-order schemes. b. Temporal varatons of peak error velocty Owng to a very large number of calculatons performed, we dscuss the results exclusvely n terms of the maxmum absolute value of the spurous velocty (called peak error velocty) generated by the pressure gradent errors. Fgure 3 shows the tme evoluton of the peak error velocty for the frst 0 days of ntegraton wth the second-, fourth-, and sxth-order schemes. The peak error velocty fluctuates rapdly durng the frst few days ntegraton. After the 5 days of ntegraton, the peak error velocty shows the decayng nertal oscllaton supermposed on mean values: 0.4 cm s 1 (the second order), 0.04 cm s 1 (the fourth order), and 0.01 cm s 1 (the sxth order). The steady approach of the peak error veloctes to these values for the three schemes ndcates the stablty of the computaton. Furthermore, the tme-mean peak error velocty for the second-order case s roughly tmes that of the sxthorder case. c. Performance of the second-, fourth-, and sxth-order schemes Feasblty of usng the sxth-order scheme s twofold: 1) drastc error reducton and ) no drastc CPU tme ncrease. We use four quanttes to compare the errors: the maxmum peak error velocty durng the 10-day ntegraton (V 1 ), the averaged peak error velocty durng the 5 10 day ntegraton (V ), the maxmum peak error pressure gradent durng the 10-day ntegraton (PG 1 ), and averaged peak error pressure gradent durng the 5 10 day ntegraton (PG ). Here V 1 (or PG 1 ) ndcates the maxmum error durng the geostrophc adjustment stage, and V (or PG ) shows the mean error after the adjustment. Usually, V 1 V, and PG 1 PG. The performance of each scheme s lsted n Table 1. Both V 1 and V reduce ther magntude by factors around 10 from the second-order scheme to the fourth-order scheme (V 1 from to cm s 1,V from to cm s 1 ), and by factors around 5 from the fourth-order scheme to the sxth-order scheme (V 1 from to cm s 1, V from to cm s 1 ). The pressure gradent error reduces even more when the hgh-order schemes are used: PG 1 reduces from to N m 3 from the second-order scheme to the fourth-order scheme (a factor of 11.7) and from to Nm 3 from the fourth-order scheme to the sxth-order scheme (a factor of 6.5). PG reduces from to Nm 3 from the second-order scheme to the fourthorder scheme (a factor of 14.3) and from to Nm 3 from the fourth-order scheme to the sxth-order scheme (a factor of 5.7). The CPU tme ncreases 10% from the second-order to fourthorder and sxth-order scheme. Notce that there s no CPU ncrease from the fourth-order to the sxth-order scheme, whch shows the great potental of usng the sxth-order scheme. 5. Senstvty studes We performed varous senstvty study to show the dependence of model results on the four parameters lsted n (13). For each senstvty study, we vary only one parameter and then ntegrate the model for 10 days usng the second-, fourth-, and sxth-order schemes, respectvely. a. Seamount wdth Seamount wdths (L) of 40, 0, 15, 1, and 10 km were used. A summary of V 1 and V after 10 days of ntegraton for each case s presented n Fg. 4. The seamount wth L larger than 10 km can be resolved by the grd spacng used here (5 km). The fgure clearly shows a great mprovement of usng hgh-order schemes. Ths mprovement enhances as L ncreases. At L 10 km, V 1 reduces by a factor of 1.45 from the second-order scheme to the fourth-order scheme and a factor of 1.18 from the fourth-order scheme to the sxthorder scheme. At L 40 km, V 1 reduces by a factor of 9.06 from the second-order scheme to the fourth-order scheme and a factor of 5.08 from the fourth-order scheme to the sxth-order scheme. Smlar reducton of V was found (Fg. 4) from the second-order scheme to
6 SEPTEMBER 1997 NOTES AND CORRESPONDENCE 069 FIG. 4. Senstvty of error on the seamount horzontal scale for the second-, fourth-, and sxth-order schemes: (a) V 1 and (b) V. FIG. 5. Senstvty of error on the seamount ampltude for the second-, fourth-, and sxth-order schemes: (a) V 1 and (b) V. the fourth-order scheme, and from the fourth-order scheme to the sxth-order scheme. b. Seamount ampltude Seamount ampltudes (h s ) of 4500, 4000, 3500, 3000, 000, 1000, and 500 m were used. A summary of V 1 and V after 10 days of ntegraton for each case s presented n Fg. 5. The fgure clearly shows a great mprovement usng hgh-order schemes. Ths mprovement enhances as h s ncreases. At h s 500 m (.e., a small seamount wth a slope of 0.015), V 1 reduces by a factor of.55 from the second-order scheme to the fourth-order scheme and a factor of 1.10 from the fourth-order scheme to the sxth-order scheme. At h s 4500 m (.e., a large seamount wth a slope of 0.115), V 1 reduces by a factor of 9.06 from the second-order scheme to the fourth-order scheme and of 5.08 from the fourth-order scheme to the sxth-order scheme. Smlar reducton of V was found (Fg. 5) from the secondorder scheme to the fourth-order scheme, and from the fourth-order scheme to the sxth-order scheme. c. Densty anomaly Densty anomales ( ˆ ) of 1, 0.8, 0.6, 0.4, 0., and 0.1 kg m 3 were used. A summary of V 1 and V after 10 days of ntegraton for each case s presented n Fg. 6. The fgure clearly shows a great mprovement usng hgh-order schemes. Ths mprovement s not senstve to the change of ˆ. At ˆ 0.1 kg m 3 (.e., a small densty anomaly), V 1 reduces by a factor of 9.05 from the second-order scheme to the fourth-order scheme and a factor of 5.08 from the fourth-order scheme to the sxth-order scheme. At ˆ 1kgm 3 (.e., a large densty anomaly), V 1 reduces by a factor of 9.10 from the second-order scheme to the fourth-order scheme and of 5.07 from the fourth-order scheme to the sxth-order scheme. Smlar reducton of V was found (Fg. 6) from
7 070 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 7 TABLE. Senstvty of V 1 (cm s 1 )ona H for the second-, fourth-, and sxth-order schemes. Scheme nd 4th 6th A H (m 4 s 1 ) TABLE 3. Senstvty of V (cm s 1 )ona H for the second-, fourth-, and sxth-order schemes. Scheme nd 4th 6th A H (m 4 s 1 ) FIG. 6. Senstvty of error on the densty anomaly for the second-, fourth-, and sxth-order schemes: (a) V 1 and (b) V. the second-order scheme to the fourth-order scheme, and from the fourth-order scheme to the sxth-order scheme. d. Horzontal vscosty Horzontal vscostes (A H )of10 10,10 9, and 10 8 m 4 s 1 were used. A summary of V 1 and V after 10 days of ntegraton for each case s presented n Tables and 3, whch clearly shows a great mprovement usng hghorder schemes. Ths mprovement s not very senstve to the change of A H.AtA H m 4 s 1,V 1 reduces by a factor of 9.06 from the second-order scheme to the fourth-order scheme and a factor of 5.08 from the fourth-order scheme to the sxth-order scheme. At A H 10 8 m 4 s 1, V 1 reduces by a factor of 9.19 from the second-order scheme to the fourth-order scheme and of 5.10 from the fourth-order scheme to the sxth-order scheme. Smlar reducton of V was found (Table 3) from the second-order scheme to the fourth-order scheme, and from the fourth-order scheme to the sxthorder scheme. 6. Conclusons 1) The -coordnate, pressure gradent error depends on the choce of dfference schemes. By choosng an optmal scheme, we may reduce the error a great deal wthout ncreasng the horzontal resoluton. Analytcal analyss shows that the truncaton error of the fourthorder scheme may be 1 orders of magntude smaller than the second-order scheme, and the truncaton error of the sxth-order scheme may be 1 orders of magntude smaller than the fourth-order scheme. ) The SPEM Verson 3.9 s used to demonstrate the beneft of usng the sxth-order scheme. A seres of calculatons of unforced flow n the vcnty of an solated seamount are performed. The results show that the sxthorder scheme has error reductons by factors of 5 compared to the fourth-order dfference scheme, and by factors of 50 compared to the second-order dfference scheme over a wde range of parameter space as well as a great parametrc doman of numercal stablty. 3) Usng the sxth-order scheme does not requre much more CPU tme. Takng SPEM3.9 as an example, the CPU tme for the sxth-order scheme s almost the same as for the fourth-order scheme, and 10% more than for the second-order scheme. 4) Snce the fourth-order dfferent scheme has error reductons by factors of 10 compared to the secondorder dfference scheme, there s no real advantage to gong to a hgher-order scheme f the bottom topography s not too complcated. The need for more accuracy wll go up wth ncreasngly complex bottom topography on small scales, so one mght expect that future demand for accuracy wll ncrease as models strve for more realsm. Acknowledgments. The authors wsh to thank Dale Hadvogel and Kate Hedstrom of the Rutgers Unversty for most kndly provng us wth a copy of the SPEM code. Ths work was funded by the Offce of Naval
8 SEPTEMBER 1997 NOTES AND CORRESPONDENCE 071 Research NOMP Program, the Naval Oceanographc Offce, and the Naval Postgraduate School. REFERENCES Beckmann, A., and D. B. Hadvogel, 1993: Numercal smulaton of flow around a tall solated seamount. Part I: Problem formulaton and model accuracy. J. Phys. Oceanogr., 3, Gary, J. M., 1973: Estmate of truncaton error n transformed coordnate prmtve equaton atmospherc models. J. Atmos. Sc., 30, Hadvogel, D. B., J. L. Wkn, and R. Young, 1991: A sem-spectral prmtve equaton model usng vertcal sgma and orthogonal curvlnear coordnates. J. Comput. Phys., 94, Haney, R. L., 1991: On the pressure gradent force over steep topography n sgma coordnate ocean models. J. Phys. Oceanogr., 1, Hedstrom, K., 1995: User s Manual for a Sem-Spectral Prmtve Equaton Ocean Crculaton Model Verson 3.9. Rutgers Unversty, New Jersey, 131 pp. [Avalable from Insttute of Marne and Coastal Scences, Rutgers Unversty, New Brunswck, NJ ] McCalpn, J. D., 1994: A comparson of second-order and fourthorder pressure gradent algorthms n a -coordnate ocean model. Int. J. Numer. Methods Fluds, 18, Mellor, G. L., T. Ezer, and L.-Y. Oey, 1994: The pressure gradent conundrum of sgma coordnate ocean models. J. Atmos. Oceanc Technol., 11, Robnson, A. R., 1993: Physcal processes, feld estmaton and nterdscplnary ocean modelng. Harvard Open Ocean Model Reports, Harvard Unversty, 71 pp. [Avalable from Harvard Unversty, 9 Oxford St., Cambrdge, MA 0138.]
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