{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T18:38:35Z","timestamp":1770748715699,"version":"3.50.0"},"reference-count":34,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,4]],"date-time":"2023-09-04T00:00:00Z","timestamp":1693785600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11975143"],"award-info":[{"award-number":["11975143"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, the propagation of Rossby waves under barotropic-baroclinic interaction in polar co-ordinates is studied. By starting from the two-layer quasi-geotropic potential vorticity equation (of equal depth) with the \u03b2 effect, the coupled KdV equations describing barotropic-baroclinic waves are derived using multi-scale analysis and the perturbation expansion method. Furthermore, in order to more accurately describe the propagation characteristics of barotropic-baroclinic waves, fifth-order coupled KdV-mKdV equations were obtained for the first time. On this basis, the Lie symmetry and conservation laws of the fifth-order coupled KdV-mKdV equations are analyzed in terms of their properties. Then, the elliptic function expansion method is applied to find the soliton solutions of the fifth-order coupled KdV-mKdV equations. Based on the solutions, we further simulate the evolution of Rossby wave amplitudes and investigate the influence of the high-order terms\u2014time and wave number\u2014on the propagation of barotropic waves and baroclinic waves. The results show that the appearance of the higher-order effect makes the amplitude of the wave lower, the width of the wave larger, and the whole wave flatter, which is obviously closer to actual Rossby wave propagation. The time and wave number will also influence wave amplitude and wave width.<\/jats:p>","DOI":"10.3390\/axioms12090856","type":"journal-article","created":{"date-parts":[[2023,9,4]],"date-time":"2023-09-04T02:59:55Z","timestamp":1693796395000},"page":"856","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Barotropic-Baroclinic Coherent-Structure Rossby Waves in Two-Layer Cylindrical Fluids"],"prefix":"10.3390","volume":"12","author":[{"given":"Jing","family":"Xu","sequence":"first","affiliation":[{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5515-3758","authenticated-orcid":false,"given":"Yong","family":"Fang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jingxuan","family":"Geng","sequence":"additional","affiliation":[{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huanhe","family":"Dong","sequence":"additional","affiliation":[{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1357\/002224039806649023","article-title":"Relation between Variations in the Intensity of the Zonal Circulation of the Atmosphere and the Displacements of the Semi-permanent Centers of Action","volume":"2","author":"Rossby","year":"1939","journal-title":"J. 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