{"paper":{"title":"Navier--Stokes equations on the $\\beta$-plane: determining modes and nodes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["physics.geo-ph"],"primary_cat":"math.AP","authors_text":"Djoko Wirosoetisno, Naoko Miyajima","submitted_at":"2018-02-23T17:16:34Z","abstract_excerpt":"We revisit the 2d Navier--Stokes equations on the periodic $\\beta$-plane, with the Coriolis parameter varying as $\\beta y$, and obtain bounds on the number of determining modes and nodes of the flow. The number of modes {and nodes} scale as $cG_0^{1/2} + c'(M/\\beta)^{1/2}$ and $cG_0^{2/3} + c'(M/\\beta)^{1/2}$ respectively, where the Grashof number $G_0=|f_v|_{L^2}^{}/(\\mu^2\\kappa_0^2)$ and $M$ involves higher derivatives of the forcing $f_v$. For large $\\beta$ (strong rotation), this results in fewer degrees of freedom than the classical (non-rotating) bound that scales as $cG_0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.08644","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}