{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T15:21:43Z","timestamp":1772119303893,"version":"3.50.1"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2023,8,16]],"date-time":"2023-08-16T00:00:00Z","timestamp":1692144000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,8,16]],"date-time":"2023-08-16T00:00:00Z","timestamp":1692144000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000781","name":"European Research Council","doi-asserted-by":"publisher","award":["STUOD-DLV-856408"],"award-info":[{"award-number":["STUOD-DLV-856408"]}],"id":[{"id":"10.13039\/501100000781","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2023,10]]},"abstract":"Abstract<\/jats:title>\n \n This work involves theoretical and numerical analysis of the thermal quasi-geostrophic (TQG) model of submesoscale geophysical fluid dynamics (GFD). Physically, the TQG model involves thermal geostrophic balance, in which the Rossby number, the Froude number and the stratification parameter are all of the same asymptotic order. The main analytical contribution of this paper is to construct local-in-time unique strong solutions for the TQG model. For this, we show that solutions of its regularised version\n \n \n $$\\alpha $$<\/jats:tex-math>\n \n \u03b1<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -TQG converge to solutions of TQG as its smoothing parameter\n \n \n $$\\alpha \\rightarrow 0$$<\/jats:tex-math>\n \n \n \u03b1<\/mml:mi>\n \u2192<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and we obtain blow-up criteria for the\n \n \n $$\\alpha $$<\/jats:tex-math>\n \n \u03b1<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -TQG model. The main contribution of the computational analysis is to verify the rate of convergence of\n \n \n $$\\alpha $$<\/jats:tex-math>\n \n \u03b1<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -TQG solutions to TQG solutions as\n \n \n $$\\alpha \\rightarrow 0$$<\/jats:tex-math>\n \n \n \u03b1<\/mml:mi>\n \u2192<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , for example, simulations in appropriate GFD regimes.\n <\/jats:p>","DOI":"10.1007\/s00332-023-09943-9","type":"journal-article","created":{"date-parts":[[2023,8,16]],"date-time":"2023-08-16T07:02:15Z","timestamp":1692169335000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Theoretical and Computational Analysis of the Thermal Quasi-Geostrophic Model"],"prefix":"10.1007","volume":"33","author":[{"given":"D.","family":"Crisan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"D. D.","family":"Holm","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"E.","family":"Luesink","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"P. R.","family":"Mensah","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"W.","family":"Pan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,8,16]]},"reference":[{"issue":"22","key":"9943_CR1","doi-asserted-by":"publisher","first-page":"2439","DOI":"10.1175\/1520-0469(1985)042<2439:OTROTI>2.0.CO;2","volume":"42","author":"DLT Anderson","year":"1985","unstructured":"Anderson, D.L.T., McCreary, J.P., Jr.: On the role of the Indian Ocean in a coupled ocean-atmosphere model of El Ni\u00f1o and the Southern Oscillation. J. Atmos. Sci. 42(22), 2439\u20132442 (1985)","journal-title":"J. Atmos. Sci."},{"issue":"1","key":"9943_CR2","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1007\/BF01212349","volume":"94","author":"JT Beale","year":"1984","unstructured":"Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94(1), 61\u201366 (1984)","journal-title":"Commun. Math. Phys."},{"issue":"5","key":"9943_CR3","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1175\/1520-0485(1997)027<0615:ANIOTA>2.0.CO;2","volume":"27","author":"E Beier","year":"1997","unstructured":"Beier, E.: A numerical investigation of the annual variability in the Gulf of California. J. Phys. Oceanogr. 27(5), 615\u2013632 (1997)","journal-title":"J. Phys. Oceanogr."},{"issue":"2","key":"9943_CR4","doi-asserted-by":"publisher","first-page":"719","DOI":"10.1016\/j.jcp.2005.06.008","volume":"211","author":"E Bernsen","year":"2006","unstructured":"Bernsen, E., Bokhove, O., van der Vegt, J.J.W.: A (dis) continuous finite element model for generalized 2d vorticity dynamics. J. Comput. Phys. 211(2), 719\u2013747 (2006)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"9943_CR5","doi-asserted-by":"publisher","first-page":"036608","DOI":"10.1063\/5.0045191","volume":"33","author":"FJ Beron-Vera","year":"2021","unstructured":"Beron-Vera, F.J.: Nonlinear saturation of thermal instabilities. Phys. Fluids 33(3), 036608 (2021)","journal-title":"Phys. Fluids"},{"issue":"3","key":"9943_CR6","first-page":"351","volume":"67","author":"FJ Beron-Vera","year":"2021","unstructured":"Beron-Vera, F.J.: Multilayer shallow-water model with stratification and shear. Revista mexicana de f\u00edsica 67(3), 351\u2013364 (2021)","journal-title":"Revista mexicana de f\u00edsica"},{"key":"9943_CR7","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-75934-0","volume-title":"The Mathematical Theory of Finite Element Methods","author":"SC Brenner","year":"2008","unstructured":"Brenner, S.C., Scott, L.R., Scott, L.R.: The Mathematical Theory of Finite Element Methods, vol. 3. Springer, Berlin (2008)"},{"key":"9943_CR8","unstructured":"Brezis H, Gallouet T: Nonlinear Schr\u00f6dinger evolution equations. Technical report, Wisconsin Univ-Madison Mathematics Research Center, (1979)"},{"key":"9943_CR9","doi-asserted-by":"crossref","unstructured":"Crisan D, Holm DD, Lang O, Mensah PR, Wei P: Theoretical analysis and numerical approximation for the stochastic thermal quasi-geostrophic model. arXiv:2207.07457, (2022)","DOI":"10.1142\/S0219493723500399"},{"issue":"2","key":"9943_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11040-020-09339-1","volume":"23","author":"E Dinvay","year":"2020","unstructured":"Dinvay, E.: Well-posedness for a Whitham\u2013Boussinesq system with surface tension. Math. Phys. Anal. Geom. 23(2), 1\u201327 (2020)","journal-title":"Math. Phys. Anal. Geom."},{"key":"9943_CR11","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1016\/S0167-2789(01)00191-9","volume":"152","author":"C Foias","year":"2001","unstructured":"Foias, C., Holm, D.D., Titi, E.S.: The Navier\u2013Stokes-alpha model of fluid turbulence. Physica D Nonlinear Phenom. 152, 505\u2013519 (2001)","journal-title":"Physica D Nonlinear Phenom."},{"issue":"1","key":"9943_CR12","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1023\/A:1012984210582","volume":"14","author":"C Foias","year":"2002","unstructured":"Foias, C., Holm, D.D., Titi, E.S.: The three dimensional viscous Camassa\u2013Holm equations, and their relation to the Navier\u2013Stokes equations and turbulence theory. J. Dyn. Differ. Equ. 14(1), 1\u201335 (2002)","journal-title":"J. Dyn. Differ. Equ."},{"key":"9943_CR13","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-23957-2","volume-title":"Compatible Finite Element Methods for Geophysical Flows: Automation and Implementation Using Firedrake","author":"TH Gibson","year":"2019","unstructured":"Gibson, T.H., McRae, A.T.T., Cotter, C.J., Mitchell, L., Ham, D.A.: Compatible Finite Element Methods for Geophysical Flows: Automation and Implementation Using Firedrake. Springer, Berlin (2019)"},{"key":"9943_CR14","volume-title":"Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications","author":"JS Hesthaven","year":"2007","unstructured":"Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, Berlin (2007)"},{"issue":"2176","key":"9943_CR15","first-page":"20140963","volume":"471","author":"DD Holm","year":"2015","unstructured":"Holm, D.D.: Variational principles for stochastic fluid dynamics. Proc. R. Soc. A Math. Phys. Eng. Sci. 471(2176), 20140963 (2015)","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"issue":"2","key":"9943_CR16","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00332-021-09682-9","volume":"31","author":"DD Holm","year":"2021","unstructured":"Holm, D.D., Luesink, E.: Stochastic wave-current interaction in thermal shallow water dynamics. J. Nonlinear Sci. 31(2), 1\u201356 (2021)","journal-title":"J. Nonlinear Sci."},{"issue":"7","key":"9943_CR17","doi-asserted-by":"publisher","first-page":"1287","DOI":"10.1175\/JPO2741.1","volume":"35","author":"DD Holm","year":"2005","unstructured":"Holm, D.D., Wingate, B.A.: Baroclinic instabilities of the two-layer quasigeostrophic alpha model. J. Phys. Oceanogr. 35(7), 1287\u20131296 (2005)","journal-title":"J. Phys. Oceanogr."},{"issue":"1","key":"9943_CR18","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1006\/aima.1998.1721","volume":"137","author":"DD Holm","year":"1998","unstructured":"Holm, D.D., Marsden, J.E., Ratiu, T.S.: The Euler\u2013Poincar\u00e9 equations and semidirect products with applications to continuum theories. Adv. Math. 137(1), 1\u201381 (1998a)","journal-title":"Adv. Math."},{"issue":"19","key":"9943_CR19","doi-asserted-by":"publisher","first-page":"4173","DOI":"10.1103\/PhysRevLett.80.4173","volume":"80","author":"DD Holm","year":"1998","unstructured":"Holm, D.D., Marsden, J.E., Ratiu, T.S.: Euler\u2013Poincar\u00e9 models of ideal fluids with nonlinear dispersion. Phys. Rev. Lett. 80(19), 4173 (1998b)","journal-title":"Phys. Rev. Lett."},{"issue":"4","key":"9943_CR20","doi-asserted-by":"publisher","first-page":"046603","DOI":"10.1063\/5.0040026","volume":"33","author":"DD Holm","year":"2021","unstructured":"Holm, D.D., Luesink, E., Pan, W.: Stochastic mesoscale circulation dynamics in the thermal ocean. Phys. Fluids 33(4), 046603 (2021)","journal-title":"Phys. Fluids"},{"key":"9943_CR21","doi-asserted-by":"crossref","unstructured":"Kato, T.: Liapunov functions and monotonicity in the Navier\u2013Stokes equation. In: Functional-analytic Methods for Partial Differential Equations, pp. 53\u201363. Springer (1990)","DOI":"10.1007\/BFb0084898"},{"issue":"1","key":"9943_CR22","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1016\/0022-1236(84)90024-7","volume":"56","author":"T Kato","year":"1984","unstructured":"Kato, T., Lai, C.Y.: Nonlinear evolution equations and the Euler flow. J. Funct. Anal. 56(1), 15\u201328 (1984)","journal-title":"J. Funct. Anal."},{"issue":"7","key":"9943_CR23","doi-asserted-by":"publisher","first-page":"891","DOI":"10.1002\/cpa.3160410704","volume":"41","author":"T Kato","year":"1988","unstructured":"Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier\u2013Stokes equations. Commun. Pure Appl. Math. 41(7), 891\u2013907 (1988)","journal-title":"Commun. Pure Appl. Math."},{"issue":"4","key":"9943_CR24","doi-asserted-by":"publisher","first-page":"481","DOI":"10.1002\/cpa.3160340405","volume":"34","author":"S Klainerman","year":"1981","unstructured":"Klainerman, S., Majda, A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34(4), 481\u2013524 (1981)","journal-title":"Commun. Pure Appl. Math."},{"issue":"1","key":"9943_CR25","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1007\/s00205-002-0207-8","volume":"166","author":"JE Marsden","year":"2003","unstructured":"Marsden, J.E., Shkoller, S.: The anisotropic Lagrangian averaged Euler and Navier\u2013Stokes equations. Arch. Ration. Mech. Anal. 166(1), 27\u201346 (2003)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"C7","key":"9943_CR26","doi-asserted-by":"publisher","first-page":"15535","DOI":"10.1029\/97JC00985","volume":"102","author":"JP McCreary Jr.","year":"1997","unstructured":"McCreary, J.P., Jr., Zhang, S., Shetye, S.R.: Coastal circulations driven by river outflow in a variable-density 1$$1\/2$$-layer model. J. Geophys. Res. Oceans 102(C7), 15535\u201315554 (1997)","journal-title":"J. Geophys. Res. Oceans"},{"issue":"1","key":"9943_CR27","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/s40562-019-0133-3","volume":"6","author":"JC McWilliams","year":"2019","unstructured":"McWilliams, J.C.: A survey of submesoscale currents. Geosci. Lett. 6(1), 1\u201315 (2019)","journal-title":"Geosci. Lett."},{"issue":"2","key":"9943_CR28","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1175\/1520-0469(1967)024<0197:TNLROA>2.0.CO;2","volume":"24","author":"JJ O\u2019Brien","year":"1967","unstructured":"O\u2019Brien, J.J., Reid, R.O.: The non-linear response of a two-layer, baroclinic ocean to a stationary, axially-symmetric hurricane: part I. Upwelling induced by momentum transfer. J. Atmos. Sci. 24(2), 197\u2013207 (1967)","journal-title":"J. Atmos. Sci."},{"issue":"3","key":"9943_CR29","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/2998441","volume":"43","author":"F Rathgeber","year":"2016","unstructured":"Rathgeber, F., Ham, D.A., Mitchell, L., Lange, M., Luporini, F., McRae, A.T.T., Bercea, G.-T., Markall, G.R., Kelly, P.H.J.: Firedrake: automating the finite element method by composing abstractions. ACM Trans. Math. Softw. (TOMS) 43(3), 1\u201327 (2016)","journal-title":"ACM Trans. Math. Softw. (TOMS)"},{"issue":"1\u20134","key":"9943_CR30","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1080\/03091929308203588","volume":"70","author":"P Ripa","year":"1993","unstructured":"Ripa, P.: Conservation laws for primitive equations models with inhomogeneous layers. Geophys. Astrophys. Fluid Dyn. 70(1\u20134), 85\u2013111 (1993)","journal-title":"Geophys. Astrophys. Fluid Dyn."},{"key":"9943_CR31","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1017\/S0022112095004228","volume":"303","author":"P Ripa","year":"1995","unstructured":"Ripa, P.: On improving a one-layer ocean model with thermodynamics. J. Fluid Mech. 303, 169\u2013201 (1995)","journal-title":"J. Fluid Mech."},{"issue":"1","key":"9943_CR32","first-page":"117","volume":"42","author":"P Ripa","year":"1996","unstructured":"Ripa, P.: Low frequency approximation of a vertically averaged ocean model with thermodynamics. Revista Mexicana de F\u00edsica 42(1), 117\u2013135 (1996)","journal-title":"Revista Mexicana de F\u00edsica"},{"issue":"1","key":"9943_CR33","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0377-0265(98)00056-6","volume":"29","author":"P Ripa","year":"1999","unstructured":"Ripa, P.: On the validity of layered models of ocean dynamics and thermodynamics with reduced vertical resolution. Dyn. Atmos. Oceans 29(1), 1\u201340 (1999)","journal-title":"Dyn. Atmos. Oceans"},{"issue":"6","key":"9943_CR34","doi-asserted-by":"publisher","first-page":"917","DOI":"10.1175\/1520-0485(1983)013<0917:OEDMLP>2.0.CO;2","volume":"13","author":"PS Schopf","year":"1983","unstructured":"Schopf, P.S., Cane, M.A.: On equatorial dynamics, mixed layer physics and sea surface temperature. J. Phys. Oceanogr. 13(6), 917\u2013935 (1983)","journal-title":"J. Phys. Oceanogr."},{"key":"9943_CR35","doi-asserted-by":"publisher","first-page":"142","DOI":"10.1016\/j.dsr.2015.09.001","volume":"105","author":"DL Volkov","year":"2015","unstructured":"Volkov, D.L., Kubryakov, A.A., Lumpkin, R.: Formation and variability of the lofoten basin vortex in a high-resolution ocean model. Deep Sea Res. Part I Oceanogr. Res. Pap. 105, 142\u2013157 (2015)","journal-title":"Deep Sea Res. Part I Oceanogr. Res. Pap."},{"issue":"1","key":"9943_CR36","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1016\/j.aim.2011.05.008","volume":"228","author":"C Wang","year":"2011","unstructured":"Wang, C., Zhang, Z.: Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity. Adv. Math. 228(1), 43\u201362 (2011)","journal-title":"Adv. Math."}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-023-09943-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-023-09943-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-023-09943-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,26]],"date-time":"2023-12-26T08:07:45Z","timestamp":1703578065000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-023-09943-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,16]]},"references-count":36,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2023,10]]}},"alternative-id":["9943"],"URL":"https:\/\/doi.org\/10.1007\/s00332-023-09943-9","relation":{"has-preprint":[{"id-type":"doi","id":"10.21203\/rs.3.rs-2139170\/v1","asserted-by":"object"}]},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,16]]},"assertion":[{"value":"6 October 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 July 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 August 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no competing interests to declare that are relevant to the content of this work. All authors certify that they have no affiliations with or involvement in any organisation or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"96"}}