Characteristics of a Terrain-Following Sigma Coordinate

Size: px

Start display at page:

Download "Characteristics of a Terrain-Following Sigma Coordinate"

Eugene Tucker
5 years ago
Views:

1 ATMOSPERIC AND OCEANIC SCIENCE LETTERS, 0, VOL. 4, NO., 576 Caractristics of a Trrain-Following Sigma Coordinat LI Yi-Yuan, WANG Bin,, WANG Dong-ai Stat K Laborator of Numrical Modling for Atmospric Scincs Gopsical Fluid Dnamics, Institut of Atmospric Psics, Cins Acadm of Scincs, Bing 0009, Cina Cntr for Eart Sstm Scinc, Tsingua Univrsit, Bing 00084, Cina Stat K Laborator of Svr Watr, Cins Acadm of Mtorological Scincs, Bing 0008, Cina Rcivd Januar 0; rvisd 7 Marc 0; accptd 7 Marc 0; publisd 6 Ma 0 Abstract Tis stud quantifis t main caractristics of a trrain-following, -coordinat troug matmatical analss of its covariant contravariant basis vctors as wll as t vrtical coordinat of. A -D scmatic of t -coordinat in a curvilinar coordinat sstm is providd in tis stud. T caractristics of t basis vctors wr brokn down into tir local vctor caractristics spatial distribution caractristics, t act prssions of t covariant; in addition, t contravariant basis vctors of t -coordinat usd to lucidat tir dtaild caractristics wr proprl solvd. Troug rwriting t prssion of t vrtical coordinat of, a matmatical prssion of all t -coordinat surfacs was found, trb quantifing t socalld trrain-following caractristics lack of flibilit to adjust t slop variation of -coordinat surfacs for t classic dfinition of. Finall, an analsis on t rang valu of t vrtical coordinat dmonstratd tat t gnral valu rang of could b obtaind b liminating t -coordinat surfacs blow t Eart s surfac. All ts quantitativ dscriptions of t caractristics of -coordinat wr t foundation for improving t -coordinat or crating a nw on. Kwords: quantitativ dscription, sigma coordinat, -D scmatic, basis vctors, non-ortogonal Citation: Li, Y.-Y., B. Wang, D.-. Wang, 0: Caractristics of a trrain-following sigma coordinat, Atmos. Ocanic Sci. Ltt., 4, Introduction A trrain-following, -coordinat initiatd b Pillips (957) is widl applid to numrical modls bcaus of its advantag in implmnting boundar conditions. owvr, som caractristics of σ-coordinat, spciall bcaus it is non-ortogonal curvilinar, ar likl to rsult in computational problms in a modl. Clark (977) proposd tat t Cristoffl smbols in t tird momntum quation of t -coordinat aris from t non-ortogonal caractristics, wic brougt about t non-consrvativ problm in a modl. It as bn rportd tat t curvilinar -coordinat surfacs could caus computational rrors in oriontal gradint trms abov stp topograp (Corb t al., 97; Lin, 997; Klmp Skamarock, 00). Stpplr t al. (00) notd tat Corrsponding autor: WANG Bin, wab@lasg.iap.ac.cn t -coordinat was non-ortogonal strongl dformd ovr stp trrain, wic can rsult in a sris of potntial problms, suc as spurious flows. Mor rcntl, Ji t al. (005) pointd out tat analss on problms associatd wit t non-ortogonal caractristic of -coordinat wr still lacking. Altoug t main caractristics of -coordinat would caus a sris of problms in a modl, t wr usuall dscribd in a qualitativ wa, suc as a twodimnsional (-D) scmatic of -coordinat surfacs drawn b Pilk (00) basis vctors at crtain location illustratd b Zdundowski Bott (00b). Morovr, fw studis providd a complt -D scmatic of t -coordinat, littl was don to quantif tos caractristics. W rvisit t main caractristics of t -coordinat in dtail troug a comprnsiv matmatical analsis of spcific prssions of t covariant contravariant basis vctors t vrtical coordinat of. Basd on ts quantitativ analss, fasibl approacs ar proposd in tis stud to improv t trrain-following coordinat to ovrcom its disadvantags. In tis stud, a complt -D scmatic of t -coordinat in a viw of a curvilinar coordinat sstm is prsntd. T caractristics of t basis vctors wr brokn down into two aspcts in addition, matmatical prssions of t covariant contravariant basis vctors of t -coordinat t -coordinat surfacs wr solvd. Basd on ts computations, t dtaild caractristics of t basis vctors wr dtrmind, t caractristics of t -coordinat surfacs wr quantifid, a gnral rang of was validatd. Caractristics of t -coordinat in a -D viw A coordinat sstm is composd of an origin, coordinat as, coordinat lins, coordinat surfacs. Coordinat surfacs ar t surfacs on wic coordinats ar constant, coordinat lins ar t curvs wr two coordinat surfacs intrsct, coordinat as ar tangntial lins of t coordinat lins. Figur provids a -D viw of ts caractristics for t -coordinat, Tabl summaris t caractristics. Basis vctors ar anotr important lmnt of a coordinat sstm wit two dfinitions: t on is tat covariant basis vctors ar tangnt to coordinat lins; t otr

2 58 ATMOSPERIC AND OCEANIC SCIENCE LETTERS VOL. 4 Figur A -D scmatic of coordinat surfacs, coordinat lins, coordinat as of a -coordinat. Rd, ligt-blu, grn ms surfacs rprsnt -, -, -coordinat surfacs, rspctivl. Dark-blu lins ar -, -, -coordinat lins, black arrows rprsnt -, -, -coordinat as. is tat contravariant basis vctors ar normal to t coordinat surfacs (Dutton, 976a). In a -coordinat, t oriontal covariant basis vctors t vrtical contravariant basis vctors var in t oriontal vrtical, rspctivl, wil t covariant contravariant basis vctors ar non-ortogonal wn t igt slop of trrain do not qual ro (Fig. ). Quantifing t caractristics of t -coordinat T covariant contravariant basis vctors of t -coordinat wr solvd t prssion of t vrtical coordinat of was rwrittn in a spcific form to quantif t caractristics of t basis vctors t -coordinat surfacs as wll as to anal t rang valus of. T classic prssions of t -coordinat dfind b Gal-Cn Somrvill (975) av bn usd b man numrical modls, suc as t Rgional Atmospric Modling Sstm (Pilk t al., 99), t Coupld Ocan/Atmospr Msoscal Prdiction Sstm (odur, 997), t Advancd Rgional Prdiction Sstm (Xu t al., 000). T dfinitions of ts classic -coordinat ar usd in t following computation as an ampl:, (), (), () wr ar two oriontal coordinats of t -coordinat, is t vrtical coordinat, is t top of t modl, =(, ) rprsnts t trrain.. Covariant contravariant basis vctors To obtain dtaild caractristics of ts vctors, tir caractristics wr brokn down into two aspcts: () t local vctor caractristics comprising t magnitud dirction of vr basis vctor at crtain location () t spatial distribution caractristics comprising t variation of all t basis vctors according to t oriontal vrtical. Dutton (976b) Pilk (00) solvd basis vctors of a gnralid vrtical coordinat; owvr, t did not prsnt t act prssions in t -coordinat. rin, Eqs. ()() wr usd to solv t covariant contravariant basis vctors of t -coordinat, illustratd b t vrtical coordinat trrain. T dfinitions of covariant contravariant basis vctors ar givn b t following: j i i i j, (4) q i j i j, (5) q wr i =,, or, j is a sum from to, i j ar basis vctors of t Cartsian coordinat, i i ar covariant contravariant basis vctors of t σ-coordinat, rspctivl, j q i ar coordinats in t Cartsian coordinat σ-coordinat, rspctivl. T valus of j i i j q q wr calculatd according to Eqs. ()() pd t summation; t covariant contravariant basis vctors of -coordinat wr tn obtaind as follows: i k, (6) j k, (7) k, (8) =i, (9) =j, (0) i j k, () T rigt- sid (RS) of Eqs. (6)() dmonstratd t spatial distribution caractristics of t basis vctors (Tabls ). Spcificall, wn t vrtical coordinat incrasd, t scond trm on t RS of Eq. (6) Eq. (7) dcrasd, t first trm stad constant; t vrtical componnts of t covariant basis vctors,, dcrasd, wil tir oriontal componnts wr constant, wit t rsult tat bcam flat, according to t igt (Fig. ). In addition, i Tabl Caractristics of t coordinat surfacs, lins, as in a -coordinat. Coordinat surfacs Coordinat lins Coordinat as Vrtical plans Curvilinar surfacs following t trrain Curvilinar lins following t trrain Vrtical lins Tangnt to t trrain at t ground Alwas vrtical

3 NO. LI ET AL.: CARACTERISTICS OF A TERRAIN-FOLLOWING SIGMA COORDINATE 59 Figur A -D scmatic of t basis vctors in a vrtical cross sction of a -coordinat. Growt lins rprsnt -coordinat surfacs, wil blu grn arrows ar covariant contravariant basis vctors of t -coordinat, rspctivl. T black arrows indicat t basis vctors of t Cartsian coordinat. wn t igt incrasd, t first scond trms on t RS of Eq. () dcrasd t tird trm rmaind constant; t oriontal componnt of t contravariant basis vctor dcrasd wil its vrtical componnt was constant; trfor, bcam stp according to t igt (Fig. ). T mtric tnsors of t -coordinat wr usd to dmonstrat t local vctor caractristics of t basis vctors. T dfinitions of t covariant contravariant mtric tnsors wr givn b Zdunkowski Bott (00a) as follows: g, () g. () Componnts on t diagonal of g g rprsntd t innr products of t covariant contravariant basis vctors, rspctivl; t otrs wr t mutual innr products of t two. Substituting Eqs. (6)() into Eq. () Eq. (), g g of t -coordinat wr obtaind: g 0 0 g In Eq. (4) Eq. (5), g =g =, g =0, onl two oriontal contravariant basis vctors ar unit vctors ortogonal to ac otr (Tabls ). Tus, t co- Tabl Caractristics of t covariant basis vctors in t -coordinat, wn t igt slop of t trrain did not qual ro. Spatial distribution caractristics Local vctor caractristics In t oriontal In t vrtical i Not unit vctors, non-ortogonal Tangnt to t trrain at t ground Alwas vrtical wit its magnitud linarl dcrasing, according to t trrain igt Bcom flat wit t igt incrass Tabl Caractristics of t contravariant basis vctors in t -coordinat, wn t igt slop of t trrain did not qual ro. Spatial distribution caractristics Local vctor caractristics In t oriontal In t vrtical Unit vctors, ortogonal to ac otr Not unit vctor, not ortogonal to t otr two Normal to t trrain at t ground Bcoms stp wn t igt incrass, (4). (5)

4 60 ATMOSPERIC AND OCEANIC SCIENCE LETTERS VOL. 4 variant basis vctors of t -coordinat wr nonortogonal t contravariant basis vctors could b dscribd as alf-ortogonal, wn t igt slop of trrain did not qual ro (Fig. ). Not tat t oriontal (vrtical) covariant basis vctors ar alwas ortogonal to t vrtical (oriontal) contravariant basis vctors (Fig. ), watvr t prssion of, according to t dfinition of covariant contravariant basis vctors. Tus, using t oriontal (vrtical) covariant basis vctor t vrtical (oriontal) contravariant basis vctor of -coordinat as t basis vctors of a coordinat, an ortogonal trrain-following coordinat can b obtaind, upon wic t quations will b as simpl as tos in t Cartsian coordinat, wic can potntiall avoid t associatd computational problms. Finall, substituting 0, 0, =0 into Eqs. (6)(), Eq. (4) Eq. (5), rspctivl, t following wr obtaind: = =i, (6) = =j, (7) = =k, (8) 0 0 g g 0 0. (9) 0 0 Eqs. (6)(9) sowd tat bot t covariant contravariant basis vctors wr idntical to tos of t Cartsian coordinat, wn t igt slop of t trrain quald ro (Fig. ). Tis was anotr important caractristic of t -coordinat, wic must b complid b an trrain-following coordinat.. -coordinat surfacs First, Eq. () was rwrittn as t following:. (0) Eq. (0) was t act matmatical prssion of all t -coordinat surfacs in a viw of t Cartsian coordinat. Mor prcisl, wn quald a constant, Eq. (0) rprsntd t sap of t crtain -coordinat surfac. T RS of Eq. (0) bcam a linar combination of trrain, trfor, vr -coordinat surfac was trrain-following. Scond, t partial drivativ wit rspct to in Eq. (0) is givn b t following:. () T trm rprsnting t slops of -coordinat surfacs in t -dirction is proportional to /. Tus, t slop of t -coordinat surfac was consistnt wit t slop of trrain dcrasd wit incrasing igt. Tn, = was dfind its partial drivativ was solvd wit rspct to in Eq. ():. () T trm, rprsnting t slop variation of t -coordinat surfacs in t vrtical dirction, dpndd on t. Tus, t valu can b adjustd b onl canging t top igt of a modl or t trrain function. owvr, ts two variabls ar alwas fid in a modl. Man mtods av bn dsignd to crat smoot σ-coordinat surfacs, suc as t smoot lvl vrtical coordinat dvlopd b Scär t al. (00), wic modifid t trrain igt to rduc t slop of σ-coordinat surfac in igr lvls. Furtrmor, a trrain-following coordinat can b dsignd via Eq. () variabls could b addd to mak t valu dcras wit t igt.. Rang of t vrtical coordinat Substituting t rang 0, into Eq. (), t following was obtaind:,. () owvr, tis was diffrnt from t gnral on, wic was 0,. Furtrmor, t valu of monotonousl incrasd according to t igt, t -coordinat surfac coincidd wit t ground surfac wn =0 (=). As a rsult, wn t igt was lss ), t corrsponding -coordinat surfacs wr curvilinar surfacs blow t Eart surfac (Fig. ). Basd on tis analsis, 0, was substitutd into Eq. () to obtain t following:, 0. (4) In practical applications, -coordinat surfacs blow t Eart surfac wr liminatd b taking t valus ranging from Eq. (4) out of Eq. (), tn obtain t gnral valu rang of. tan t valu of t trrain ( 0, 0. (5), Not tat t dfinition of σ is actuall a binar function of variabls. Bcaus of t uslssnss of t Figur A cross-sction of t -coordinat surfacs blow t Eart surfac, vrticall. T black lin rprsnts trrain, wil colord lins rprsnt t -coordinat surfacs. T dirctions of t Cartsian coordinat ar indicatd at t lowr lft cornr.

5 NO. LI ET AL.: CARACTERISTICS OF A TERRAIN-FOLLOWING SIGMA COORDINATE 6 -coordinat surfacs blow t Eart surfac, onl t prssion of tis function abov t trrain was considrd wn dsigning t trrain-following coordinat. 4 Summar T main caractristics of a -coordinat wr dscribd in a mor quantitativ wa tan in prvious studis to quantif t known caractristics to lucidat som dtaild caractristics. Spcificall, a -D scmatic of t coordinat surfacs, lins, as of a -coordinat was providd in viw of a curvilinar coordinat sstm. T spatial variation of t covariant contravariant basis vctors of t -coordinat wr obtaind b braking down t caractristics of t basis vctors into t local vctor caractristics spatial distribution caractristics analing tir matmatical prssions. Particularl, a matmatical prssion of all t -coordinat surfacs was found as viwd in t Cartsian coordinat framwork to quantif t so-calld trrain-following caractristic lack of flibilit to adjust t slop variation of -coordinat surfacs in t classic dfinition of. In addition, t -coordinat surfacs blow t Eart surfac wr found, trb validating t gnral rang valu of t vrtical coordinat. T quantitativ dscriptions of t caractristics of t -coordinat providd dtaild suggstions to improv t classic -coordinat or crat a nw on, tus potntiall rsolving its associatd computational problms. First, t local vctor caractristics spatial distribution caractristics of basis vctors manifst t possibilit to crat an ortogonal trrain-following coordinat; scond, t matmatical prssion of -coordinat surfacs tir slop variation can b usd to gnrat a smoot -coordinat surfac in ig lvls, wil prsrving t trrain-following caractristic; tird, t analsis of t valu rang of allowd for t focus to b placd primaril on t dfinition of abov t trrain. Incidntall, improvmnts of t classic -coordinat, wic complid wit t quantitativ dscriptions, nd to b invstigatd via furtr numrical primnts. Acknowldgmnts. T two anonmous rviwrs lpd us to aciv t prsnt rsults. Tis work was jointl supportd b t National Natural Scinc Foundation of Cina undr Grant Nos , 40606, Rfrncs Clark, T. L., 977: A small-scal dnamic modl using a trrain-following coordinat transformation, J. Comp. Ps., 4, 865. Corb, G. A., A. Gilcrist, R. L. Nwson, 97: A gnral circulation modl of t atmospr suitabl for long priod intgrations, Quart. J. Ro. Mtor. Soc., 98, Dutton, J. A., 976a: Vctor tnsor analsis fundamntal kinmatics of fluid flow, in: T Caslss Wind: An Introduction to T Tor of Atmospric Motion, McGraw-ill, NwYork, 90. Dutton, J. A., 976b: Mtorological quations of motion, in: T Caslss Wind: An Introduction to T Tor of Atmospric Motion, McGraw-ill, NwYork, Gal-Cn, T., R. C. J. Somrvill, 975: On t us of a coordinat transformation for t solution of t Navir-stoks quations, J. Comp. Ps., 7, 098. odur, R. M., 997: T naval rsarc laborator s coupld ocan/atmospr msoscal prdiction sstm (COAMPS), Mon. Wa. Rv., 5, Ji, L., J. Cn, D. Zang, t al., 005: Rviw of som numrical aspcts of t dnamic framwork of NWP modl, Cins J. Atoms. Sci. (in Cins), 9, 00. Klmp, J. B., W. C. Skamarock, 00: Numrical consistnc of mtric trms in trrain-following coordinats, Mon. Wa. Rv.,, 99. Lin, S., 997: A finit-volum intgration mtod for computing prssur gradint forc in gnral vrtical coordinats, Quart. J. Ro. Mtor. Soc.,, Pillips, N. A., 957: A coordinat sstm aving som spcial advantags for numrical forcasting, J. Mtor., 4, Pilk, R. A., 00: Coordinat transformations, in: Msoscal Mtorological Modling, nd (d.), Acadmic Prss, San Digo, 09. Pilk, R. A., W. R. Cotton, R. L. Walko, t al., 99: A comprnsiv mtorological modling sstm RAMS, Mtor. Atmos. Ps., 49, 699. Scär, C., D. Lunbrgr, O. Furr, t al., 00: A nw trrain-following vrtical coordinat formulation for atmospric prdiction modls, Mon. Wa. Rv., 0, Stpplr, J., R. ss, U. Scättlr, t al., 00: Rviw of numrical mtods for nondrostatic watr prdiction modls, Mtor. Atmos. Ps., 8, 870. Xu, M., K. K. Drogmir, V. Wong, 000: T advancd rgional prdiction sstm (ARPS) A multi-scal nondrostatic atmospric simulation prdiction modl. Part I: modl dnamics vrification, Mtor. Atmos. Ps., 75, 69. Zdunkowski, W., A. Bott, 00a: Rciprocal coordinat sstms, in: Dnamics of t Atmospr: A Cours in Tortical Mtorolog, Cambridg Univrsit Prss, Cambridg, 56. Zdunkowski, W., A. Bott, 00b: Orograp-following coordinat sstms, in: Dnamics of t Atmospr: A Cours in Tortical Mtorolog, Cambridg Univrsit Prss, Cambridg,

3-2-1 ANN Architecture

3-2-1 ANN Architecture ARTIFICIAL NEURAL NETWORKS (ANNs) Profssor Tom Fomby Dpartmnt of Economics Soutrn Mtodist Univrsity Marc 008 Artificial Nural Ntworks (raftr ANNs) can b usd for itr prdiction or classification problms.