{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T01:22:02Z","timestamp":1767835322639,"version":"3.49.0"},"reference-count":44,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,10,21]],"date-time":"2023-10-21T00:00:00Z","timestamp":1697846400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["12072012"],"award-info":[{"award-number":["12072012"]}]},{"name":"National Natural Science Foundation of China","award":["92252201"],"award-info":[{"award-number":["92252201"]}]},{"name":"National Natural Science Foundation of China","award":["91952302"],"award-info":[{"award-number":["91952302"]}]},{"name":"Fundamental Research Funds for the Central Universities","award":["12072012"],"award-info":[{"award-number":["12072012"]}]},{"name":"Fundamental Research Funds for the Central Universities","award":["92252201"],"award-info":[{"award-number":["92252201"]}]},{"name":"Fundamental Research Funds for the Central Universities","award":["91952302"],"award-info":[{"award-number":["91952302"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The mean velocity distributions of unstably and stably stratified atmospheric surface layers (ASLs) are investigated here using the symmetry approach. Symmetry groups for the mean momentum and the Reynolds stress equations of ASL are searched under random dilation transformations, which, with different leading order balances in different flow regions, lead to a set of specific scalings for the characteristic length \u211313 (defined by Reynolds shear stress and mean shear). In particular, symmetry analysis shows that in the shear-dominated region, \u211313 scales linearly with the surface height z, which corresponds to the classical log law of mean velocity. In the buoyancy-dominated region, \u211313\/L\u223cz\/L4\/3 for unstably stratified ASL and \u211313\/L\u223cconst for stably stratified ASL, where L is the Obukhov length. The specific formula of the celebrated Monin\u2013Obukhov similarity function is obtained, and hence an algebraic model of mean velocity profiles in ASL is derived, showing good agreement with the datum from the QingTu Lake observation array (QLOA) in China.<\/jats:p>","DOI":"10.3390\/sym15101951","type":"journal-article","created":{"date-parts":[[2023,10,21]],"date-time":"2023-10-21T12:59:01Z","timestamp":1697893141000},"page":"1951","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Symmetry Analysis of Mean Velocity Distribution in Stratified Atmospheric Surface Layers"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1128-6599","authenticated-orcid":false,"given":"Yong","family":"Ji","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4702-8735","authenticated-orcid":false,"given":"Xi","family":"Chen","sequence":"additional","affiliation":[{"name":"Key Laboratory of Fluid Mechanics of Ministry of Education, Beihang University (Beijing University of Aeronautics and Astronautics), Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1146\/annurev.fl.06.010174.001051","article-title":"The Atmospheric Boundary Layer Below 150 Meters","volume":"6","author":"Panofsky","year":"1974","journal-title":"Annu. 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