Formulation of a Three-Dimensional Spectral Model for the Primitive Equations Using Laguerre Functions

Formulation of a Three-Dimensional Spectral Model for the Primitive Equations Using Laguerre Functions | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Formulation of a Three-Dimensional Spectral Model for the Primitive Equations Using Laguerre Functions Shun Fujita, Keiichi Ishioka This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7919343/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 9 You are reading this latest preprint version Abstract This study proposes a novel formulation of a three-dimensional spectral model for the primitive equations, where the spectral method is applied in both the horizontal and vertical directions. We utilize scaled Laguerre functions as the vertical basis and introduce a scaling parameter that enables flexible control over the distribution of vertical grid points. We demonstrate that the optimal setting of this parameter allows the model top to be placed at significantly higher altitudes while maintaining adequate grid spacing in the upper atmosphere, thereby addressing a practical limitation of previous three-dimensional spectral models. The proposed formulation is implemented as a numerical model and validated through several standard atmospheric benchmark experiments. Comparative experiments reveal that the numerical error of the three-dimensional spectral models converges significantly faster than that of a conventional vertical finite-difference model, exhibiting the rapid error reduction characteristic of spectral methods when the vertical degrees of freedom are increased. Furthermore, we investigate and compare the properties of gravity wave propagation using the proposed discretization and the finite-difference method using linearized two-dimensional primitive equations. This confirms that the proposed discretization method can accurately represent upward-propagating waves. three-dimensional spectral model atmospheric general circulation model dynamical core Laguerre function semi-implicit time integration Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 12 Jan, 2026 Reviews received at journal 31 Dec, 2025 Reviews received at journal 30 Nov, 2025 Reviewers agreed at journal 28 Oct, 2025 Reviewers agreed at journal 27 Oct, 2025 Reviewers invited by journal 27 Oct, 2025 Editor assigned by journal 23 Oct, 2025 Submission checks completed at journal 23 Oct, 2025 First submitted to journal 21 Oct, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7919343","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":536120870,"identity":"ef3eb98e-27a5-4512-b909-f28c51d48f31","order_by":0,"name":"Shun Fujita","email":"data:image/png;base64,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","orcid":"","institution":"Kyoto University","correspondingAuthor":true,"prefix":"","firstName":"Shun","middleName":"","lastName":"Fujita","suffix":""},{"id":536120872,"identity":"8a18e5c1-3500-4f3d-ae2f-9b20c69b326e","order_by":1,"name":"Keiichi Ishioka","email":"","orcid":"","institution":"Kyoto University","correspondingAuthor":false,"prefix":"","firstName":"Keiichi","middleName":"","lastName":"Ishioka","suffix":""}],"badges":[],"createdAt":"2025-10-22 08:27:54","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7919343/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7919343/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":95002222,"identity":"bce8b37d-2bf3-477c-82b8-dbc162ef6aa8","added_by":"auto","created_at":"2025-11-03 09:04:24","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":4396,"visible":true,"origin":"","legend":"","description":"","filename":"89b30d98848a4a0f9678c94aa0f048ee.json","url":"https://assets-eu.researchsquare.com/files/rs-7919343/v1/de067a396d3a472ada0f7d1d.json"},{"id":95002530,"identity":"76b13cdc-6199-4439-a545-d7d9cd1cdd7a","added_by":"auto","created_at":"2025-11-03 09:04:52","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2394574,"visible":true,"origin":"","legend":"","description":"","filename":"submissionfile.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7919343/v1_covered_feb863c1-8613-450d-9cf5-328f9f3aaf4c.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Formulation of a Three-Dimensional Spectral Model for the Primitive Equations Using Laguerre Functions","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"journal-of-the-meteorological-society-of-japan","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"","snPcode":"44394","submissionUrl":"https://submission.springernature.com/new-submission/44394/3","title":"Journal of the Meteorological Society of Japan","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Open","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"three-dimensional spectral model, atmospheric general circulation model dynamical core, Laguerre function, semi-implicit time integration","lastPublishedDoi":"10.21203/rs.3.rs-7919343/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7919343/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":" This study proposes a novel formulation of a three-dimensional spectral model for the primitive equations, where the spectral method is applied in both the horizontal and vertical directions. 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