{"id":830,"date":"2005-09-06T03:23:27","date_gmt":"2005-09-06T07:23:27","guid":{"rendered":"http:\/\/mseas.net16.net\/?p=830"},"modified":"2021-08-16T21:11:30","modified_gmt":"2021-08-17T01:11:30","slug":"dynamics-and-lagrangian-coherent-structures-in-the-ocean-and-their-uncertainty-2","status":"publish","type":"post","link":"https:\/\/mseas.mit.edu\/?p=830","title":{"rendered":"Dynamics and Lagrangian Coherent Structures in the Ocean and their Uncertainty"},"content":{"rendered":"The observation, computation and study of &#8220;Lagrangian Coherent Structures&#8221;\r\n(LCS) in turbulent geophysical \r\nflows have been active areas of research in \r\nfluid\r\nmechanics for the last 30 years. Growing evidence for the existence of LCSs in\r\ngeophysical \r\nflows (e.g., eddies, oscillating jets, chaotic mixing) and other \r\nfluid \r\nflows\r\n(e.g., separation pro\fle at the surface of an airfoil, entrainment and detrainment\r\nby a vortex) generates an increasing interest for the extraction and understanding\r\nof these structures as well as their properties.\r\nIn parallel, realistic ocean modeling with dense data assimilation has developed\r\nin the past decades and is now able to provide accurate nowcasts and predictions\r\nof ocean \r\nflow fields to study coherent structures. Robust numerical methods\r\nand sufficiently fast hardware are now available to compute real-time forecasts of\r\noceanographic states and render associated coherent structures. It is therefore\r\nnatural to expect the direct predictions of LCSs based on these advanced models.\r\nThe impact of uncertainties on the coherent structures is becoming an increasingly\r\nimportant question for practical applications. The transfer of these uncertainties\r\nfrom the ocean state to the LCSs is an unexplored but intriguing scientific\r\nproblem. These two questions are the motivation and focus of this presentation.\r\nUsing the classic formalism of continuous-discrete estimation [1], the spatially\r\ndiscretized dynamics of the ocean state vector x and observations are described\r\nby\r\n(1a) dx =M(x; t) + d\u0011\r\nyok\r\n(1b) = H(xk; tk) + \u000fk\r\nwhere M and H are the model and measurement model operator, respectively.\r\nThe stochastic forcings d\u0011 and \u000fk are Wiener\/Brownian motion processes, \u0011 \u0018\r\nN(0;Q(t)), and white Gaussian sequences, \u000fk \u0018 N(0;Rk), respectively. In other\r\nwords, Efd\u0011(t)d\u0011\r\nT\r\n(t)g\r\n:=\r\nQ(t) dt. The initial conditions are also uncertain and\r\nx(t0) is random with a prior PDF, p(x(t0)), i.e. x(t0) = bx0 + n(0) with n(0)\r\nrandom. Of course, vectors and operators in Eqs. (1a-b) are multivariate which\r\nimpacts the PDFs: e.g. their moments are also multivariate.\r\nThe estimation problem at time t consists of combining all available information\r\non x(t), the dynamics and data (Eqs. 1a-b), their prior distributions and the initial\r\nconditions p(x(t0)). Defining the set of all observations prior to time t by yt","protected":false},"excerpt":{"rendered":"<p>The observation, computation and study of &#8220;Lagrangian Coherent Structures&#8221; (LCS) in turbulent geophysical flows have been active areas of research in fluid mechanics for the last 30 years. Growing evidence for the existence of LCSs in geophysical flows (e.g., eddies, oscillating jets, chaotic mixing) and other fluid flows (e.g., separation pro\fle at the surface of [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[31,32,33,28,5,189,72,191,71],"tags":[],"class_list":["post-830","post","type-post","status-publish","format-standard","hentry","category-uncertainty-quantification-and-reduced-order-modeling","category-numerical-ocean-modeling","category-uncertainty-quantification-and-predictions","category-multiscale-ocean-modeling","category-publications","category-lagrangian-modeling","category-proceedings-of-refereed-conferences-multiscale-ocean-modeling","category-proceedings-of-refereed-conferences-lagrangian-modeling","category-proceedings-of-refereed-conferences-uncertainty-quantification-and-predictions"],"_links":{"self":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/830","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=830"}],"version-history":[{"count":3,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/830\/revisions"}],"predecessor-version":[{"id":4065,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/830\/revisions\/4065"}],"wp:attachment":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=830"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=830"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=830"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}