{"title":"On spectral scaling laws for averaged turbulence on the sphere","authors":"Sagy Ephrati, Erik Jansson, Klas Modin","doi":"10.1016/j.physd.2025.134808","DOIUrl":null,"url":null,"abstract":"<div><div>Spectral analysis for a class of Lagrangian-averaged Navier–Stokes (LANS) equations on the sphere is carried out. The equations arise from the Navier–Stokes equations by applying a Helmholtz filter of width <span><math><mi>α</mi></math></span> to the advecting velocity <span><math><mi>β</mi></math></span> times, extending previous results on the Navier–Stokes-<span><math><mi>α</mi></math></span> model and enabling a precise selection of the smallest length scale in the flow. Power laws for the energy spectrum are derived and indicate a <span><math><mi>β</mi></math></span>-dependent scaling at wave numbers <span><math><mi>l</mi></math></span> with <span><math><mrow><mi>α</mi><mi>l</mi><mo>≫</mo><mn>1</mn></mrow></math></span>. The energy and enstrophy transfer rates distinctly depend on the averaging, allowing control over the energy flux and the enstrophy flux separately through the choice of averaging operator. A necessary condition on the averaging operator is derived for the existence of the inverse cascade in two-dimensional turbulence. Numerical experiments with a structure-preserving integrator based on Zeitlin’s self-consistent truncation for hydrodynamics confirm the expected energy spectrum scalings and the robustness of the double cascade under choices of the averaging operator. The derived results have potential applications in reduced-complexity numerical simulations of geophysical flows on spherical domains.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"481 ","pages":"Article 134808"},"PeriodicalIF":2.7000,"publicationDate":"2025-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physica D: Nonlinear Phenomena","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0167278925002854","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Spectral analysis for a class of Lagrangian-averaged Navier–Stokes (LANS) equations on the sphere is carried out. The equations arise from the Navier–Stokes equations by applying a Helmholtz filter of width to the advecting velocity times, extending previous results on the Navier–Stokes- model and enabling a precise selection of the smallest length scale in the flow. Power laws for the energy spectrum are derived and indicate a -dependent scaling at wave numbers with . The energy and enstrophy transfer rates distinctly depend on the averaging, allowing control over the energy flux and the enstrophy flux separately through the choice of averaging operator. A necessary condition on the averaging operator is derived for the existence of the inverse cascade in two-dimensional turbulence. Numerical experiments with a structure-preserving integrator based on Zeitlin’s self-consistent truncation for hydrodynamics confirm the expected energy spectrum scalings and the robustness of the double cascade under choices of the averaging operator. The derived results have potential applications in reduced-complexity numerical simulations of geophysical flows on spherical domains.
期刊介绍:
Physica D (Nonlinear Phenomena) publishes research and review articles reporting on experimental and theoretical works, techniques and ideas that advance the understanding of nonlinear phenomena. Topics encompass wave motion in physical, chemical and biological systems; physical or biological phenomena governed by nonlinear field equations, including hydrodynamics and turbulence; pattern formation and cooperative phenomena; instability, bifurcations, chaos, and space-time disorder; integrable/Hamiltonian systems; asymptotic analysis and, more generally, mathematical methods for nonlinear systems.