{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T00:38:27Z","timestamp":1771461507999,"version":"3.50.1"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2007,9,1]],"date-time":"2007-09-01T00:00:00Z","timestamp":1188604800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2007,9,25]]},"DOI":"10.1007\/s00211-007-0104-z","type":"journal-article","created":{"date-parts":[[2007,8,30]],"date-time":"2007-08-30T23:03:58Z","timestamp":1188515038000},"page":"669-705","source":"Crossref","is-referenced-by-count":23,"title":["A fourth-order numerical method for the planetary geostrophic equations with inviscid geostrophic balance"],"prefix":"10.1007","volume":"107","author":[{"given":"Roger","family":"Samelson","sequence":"first","affiliation":[]},{"given":"Roger","family":"Temam","sequence":"additional","affiliation":[]},{"given":"Cheng","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Shouhong","family":"Wang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2007,9,1]]},"reference":[{"key":"104_CR1","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1006\/jcph.1996.0089","volume":"125","author":"C. Anderson","year":"1996","unstructured":"Anderson C., Reider M. (1996). A high order explicit method for the computation of flow about a circular cylinder. J. Comput. Phys. 125: 207\u2013224","journal-title":"J. Comput. Phys."},{"key":"104_CR2","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1002\/cpa.10056","volume":"56","author":"C.-S. Cao","year":"2003","unstructured":"Cao C.-S., Titi E. (2003). Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model. Commun. Pure Appl. Math. 56: 198\u2013233","journal-title":"Commun. Pure Appl. Math."},{"key":"104_CR3","first-page":"135","volume":"47","author":"J.G. Charney","year":"1947","unstructured":"Charney J.G. (1947). The dynamics of long waves in a baraclinic westerly current. J. Meteorol. 47: 135\u2013163","journal-title":"J. Meteorol."},{"key":"104_CR4","doi-asserted-by":"crossref","first-page":"1981","DOI":"10.1175\/1520-0485(1986)016<1981:OMFIWT>2.0.CO;2","volume":"16","author":"A. Colin de Verdiere","year":"1986","unstructured":"Colin de Verdiere A. (1986). On mean flow instabilities within the planetary geostrophic equations. J. Phys. Oceanogr. 16: 1981\u20131984","journal-title":"J. Phys. Oceanogr."},{"key":"104_CR5","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1175\/1520-0426(1993)010<0195:ARAIOT>2.0.CO;2","volume":"10","author":"J. Dukowicz","year":"1993","unstructured":"Dukowicz J., Smith R., Malone R. (1993). A reformulation and implementation of the bryan-cox-semtner ocean model on the connection machine. J. Atmos. Ocean. Technol. 10: 195\u2013208","journal-title":"J. Atmos. Ocean. Technol."},{"key":"104_CR6","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1006\/jcph.1996.0125","volume":"126","author":"W. E","year":"1996","unstructured":"E W., Liu J.-G. (1996). Essentially compact schemes for unsteady viscous incompressible flows. J. Comput. Phys. 126: 122\u2013138","journal-title":"J. Comput. Phys."},{"key":"104_CR7","doi-asserted-by":"crossref","first-page":"2182","DOI":"10.1063\/1.1761178","volume":"8","author":"F.H. Harlow","year":"1965","unstructured":"Harlow F.H., Welch J.E. (1965). Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8: 2182\u20132189","journal-title":"Phys. Fluids"},{"key":"104_CR8","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1016\/0045-7930(94)90053-1","volume":"23","author":"W. Henshaw","year":"1994","unstructured":"Henshaw W., Kreiss H., Reyna L. (1994). A fourth-order accurate difference approximation for the incompressible Navier\u2013Stokes equations. Comput. Fluids 23: 575\u2013593","journal-title":"Comput. Fluids"},{"key":"104_CR9","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1088\/0951-7715\/5\/2\/001","volume":"5","author":"J.L. Lions","year":"1992","unstructured":"Lions J.L., Temam R., Wang S. (1992). New formulations of the primitive equations of the atmosphere and applications. Nonlinearity 5: 237\u2013288","journal-title":"Nonlinearity"},{"key":"104_CR10","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1080\/00036810500328299","volume":"85","author":"J.-G. Liu","year":"2006","unstructured":"Liu J.-G., Samelson R., Wang C. (2006). Global weak solution of planetary geostrophic equations with inviscid geostrophic balance. Appl. Anal. 85: 593\u2013606","journal-title":"Appl. Anal."},{"issue":"2","key":"104_CR11","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1023\/A:1021168924020","volume":"18","author":"J.-G. Liu","year":"2003","unstructured":"Liu J.-G., Wang C., Johnston H. (2003). A fourth order scheme for incompressible Boussinesq equations. J. Sci. Comput. 18(2): 253\u2013285","journal-title":"J. Sci. Comput."},{"key":"104_CR12","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1006\/jcph.2001.6772","volume":"174","author":"J.-G. Liu","year":"2001","unstructured":"Liu J.-G., Wang W.-C. (2001). An energy-preserving MAC-Yee scheme for the incompressible MHD equation. J. Comput. Phys. 174: 12\u201337","journal-title":"J. Comput. Phys."},{"key":"104_CR13","doi-asserted-by":"crossref","first-page":"448","DOI":"10.1175\/1520-0485(1984)014<0448:TEFGMI>2.0.CO;2","volume":"14","author":"J. Pedlosky","year":"1984","unstructured":"Pedlosky J. (1984). The equations for geostrophic motion in the ocean. J. Phys. Oceanogr. 14: 448\u2013455","journal-title":"J. Phys. Oceanogr."},{"key":"104_CR14","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-4650-3","volume-title":"Geophysical Fluid Dynamics, 2nd edn","author":"J. Pedlosky","year":"1987","unstructured":"Pedlosky J. (1987). Geophysical Fluid Dynamics, 2nd edn. \u00a0Springer, New York"},{"key":"104_CR15","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1029\/RG001i002p00123","volume":"1","author":"N.A. Phillips","year":"1963","unstructured":"Phillips N.A. (1963). Geostrophic motion. Rev. Geophys. 1: 123\u2013176","journal-title":"Rev. Geophys."},{"key":"104_CR16","doi-asserted-by":"crossref","first-page":"295","DOI":"10.3402\/tellusa.v11i3.9317","volume":"11","author":"A. Robinson","year":"1959","unstructured":"Robinson A., Stommel H. (1959). The oceanic thermocline and associated thermohaline circulation. Tellus 11: 295\u2013308","journal-title":"Tellus"},{"key":"104_CR17","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1357\/002224090784984650","volume":"48","author":"R. Salmon","year":"1990","unstructured":"Salmon R. (1990). The thermocline as an \u201cinternal boundary layer\u201d. J. Mar. Res. 48: 437\u2013469","journal-title":"J. Mar. Res."},{"key":"104_CR18","doi-asserted-by":"crossref","first-page":"1163","DOI":"10.1137\/S0036142901396284","volume":"41","author":"R. Samelson","year":"2003","unstructured":"Samelson R., Temam R., Wang C., Wang S. (2003). Surface pressure Poisson equation formulation of the primitive equations: Numerical schemes. SIAM J. Numer. Anal. 41: 1163\u20131194","journal-title":"SIAM J. Numer. Anal."},{"key":"104_CR19","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1080\/00036819808840682","volume":"70","author":"R. Samelson","year":"1998","unstructured":"Samelson R., Temam R., Wang S. (1998). Some mathematical properties of the planetary geostrophic equations for large scale ocean circulation. Appl. Anal. 70: 147\u2013173","journal-title":"Appl. Anal."},{"key":"104_CR20","first-page":"1","volume":"13","author":"R. Samelson","year":"2000","unstructured":"Samelson R., Temam R., Wang S. (2000). Remarks on the planetary geostrophic model of grye scale ocean circulation. Differ. Integral Equ. 13: 1\u201314","journal-title":"Differ. Integral Equ."},{"key":"104_CR21","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1175\/1520-0485(1997)027<0186:ASFADS>2.0.CO;2","volume":"27","author":"R. Samelson","year":"1997","unstructured":"Samelson R., Vallis G. (1997). A simple friction and diffusion scheme for planetary geostrophic basin models. J. Phys. Oceanogr. 27: 186\u2013194","journal-title":"J. Phys. Oceanogr."},{"key":"104_CR22","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1016\/0167-2789(92)90225-C","volume":"60","author":"R. Smith","year":"1992","unstructured":"Smith R., Dukowicz J., Malone R. (1992). Parallel ocean general circulation modeling. Phys. D 60: 38\u201361","journal-title":"Phys. D"},{"key":"104_CR23","first-page":"695","volume":"44","author":"H. Stommel","year":"1962","unstructured":"Stommel H., Webster J. (1962). Some properties of the thermocline equations in a subtropical gyre. J. Mar. Res. 44: 695\u2013711","journal-title":"J. Mar. Res."},{"key":"104_CR24","volume-title":"Navier\u2013Stokes Equations: Theory and Numerical Analysis","author":"R. Temam","year":"1984","unstructured":"Temam R. (1984). Navier\u2013Stokes Equations: Theory and Numerical Analysis. North-Holland, Amsterdam"},{"key":"104_CR25","unstructured":"Wang, C.: The primitive equations formulated in mean vorticity, Discrete and Continuous Dynamical Systems. In: Proceedings of International Conference on Dynamical Systems and Differential Equations, pp. 880\u2013887 (2003)"},{"key":"104_CR26","doi-asserted-by":"crossref","first-page":"1143","DOI":"10.3934\/dcdsb.2004.4.1143","volume":"4","author":"C. Wang","year":"2004","unstructured":"Wang C. (2004). Convergence analysis of the numerical method for the primitive equations formulated in mean vorticity on a Cartesian grid. Discret. Contin. Dynam. Syst. Ser. B 4: 1143\u20131172","journal-title":"Discret. Contin. Dynam. Syst. Ser. B"},{"key":"104_CR27","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1007\/s002110100311","volume":"91","author":"C. Wang","year":"2002","unstructured":"Wang C., Liu J.-G. (2002). Analysis of finite difference schemes for unsteady Navier\u2013Stokes equations in vorticity formulation. Numer. Math. 91: 543\u2013576","journal-title":"Numer. Math."},{"key":"104_CR28","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1007\/s00211-003-0508-3","volume":"97","author":"C. Wang","year":"2004","unstructured":"Wang C., Liu J.-G., Johnston H. (2004). Analysis of a fourth order finite difference method for incompressible Boussinesq equations. Numer. Math. 97: 555\u2013594","journal-title":"Numer. Math."},{"key":"104_CR29","doi-asserted-by":"crossref","first-page":"309","DOI":"10.3402\/tellusa.v11i3.9316","volume":"11","author":"P. Welander","year":"1959","unstructured":"Welander P. (1959). An advective model of the ocean thermocline. Tellus 11: 309\u2013318","journal-title":"Tellus"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-007-0104-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-007-0104-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-007-0104-z","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,17]],"date-time":"2024-02-17T17:49:51Z","timestamp":1708192191000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-007-0104-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,9,1]]},"references-count":29,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2007,9,25]]}},"alternative-id":["104"],"URL":"https:\/\/doi.org\/10.1007\/s00211-007-0104-z","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,9,1]]}}}