{
  "abstract": "We study the spectral stability of periodic shear flows for the two-dimensional Navier--Stokes equations on the $β$-plane in the long-wave regime. While it is known that non-rotating periodic shear flows are generically unstable to sufficiently long-wave perturbations, our results show that planetary rotation suppresses this instability mechanism. Using a perturbative analysis based on Kato's reduction, we derive an asymptotic expansion for the principal eigenvalue of the linearized operator that is uniform in the Rossby parameter $β\\geq 0$, and establish an explicit stability criterion in terms of the shear profile, the viscosity, and the Rossby parameter. Under the critical scaling where the Rossby parameter $β$ and the rescaled longitudinal wavenumber $\\varepsilon$ are of comparable size, this criterion extends Yudovich's classical long-wave instability threshold to rotating flows and reveals a sharp transition between stability and instability governed by the range of the ratio $β/\\varepsilon$. These results provide a rigorous analysis of how viscosity, shear, and planetary rotation interact to determine long-wave stability on the $β$-plane.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.06899",
  "author_total": 4,
  "authors": [
    {
      "name": "Robin Ming Chen"
    },
    {
      "name": "Tian Dai"
    },
    {
      "name": "Dehua Wang"
    },
    {
      "name": "Weiqiang Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0a70fcybxazpy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.06899",
  "fulltext_url": null,
  "id": "0a70fcybxazpy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.06899",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7202123117",
  "pdf_url": "https://arxiv.org/pdf/2608.06899",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0a70fcybxazpy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2608.06899",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.06899"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Long-Wave Stability And Instability Of Periodic Shear Flows For The 2D Navier-Stokes Equations On The $β$-Plane",
  "title_text": "Long-Wave Stability And Instability Of Periodic Shear Flows For The 2D Navier-Stokes Equations On The β-Plane",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/0a70fcybxazpy",
  "venue": "arXiv",
  "versions": []
}