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The lecture also contrasts sigma versus eta and summarizes key limitations of eta coordinates, such as reduced boundary-layer detail and challenges near steep terrain.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/template/","Template",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/template/presentations/","Presentations",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/lecture-20-vertical-coordinate-systems/177646/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/lecture-20-vertical-coordinate-systems/177646.png","ImageObject",442,249,{"name":42,"@type":43},"วิน","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/vnd.openxmlformats-officedocument.wordprocessingml.document","2026-10-19","2026-09-02",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",11,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"Why is selecting a vertical coordinate important in atmospheric models?","Question",{"text":63,"@type":64},"Vertical structure affects how well the model represents the atmosphere, requiring a suitable vertical coordinate and enough vertical resolution to capture vertical variations accurately.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What problem does the eta (η) coordinate address compared with sigma (σ)?",{"text":68,"@type":64},"Eta was created to reduce errors in computing the pressure gradient force when using sigma coordinate models, by using mean sea level pressure as the bottom reference instead of surface pressure.",{"name":70,"@type":61,"acceptedAnswer":71},"How are eta levels chosen and applied to the model terrain?",{"text":72,"@type":64},"Eta surfaces are predefined and standard-atmosphere pressures at specific heights are used to determine the eta level closest to each terrain height, while the model topography is set to the nearest eta surface.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},177646,1791478260,{"code":4,"msg":82,"data":83},"success",[84,87,92,97,102,107,112,117,122],{"id":55,"doc_module":22,"doc_module_name":25,"category_name":29,"show_sort_weight":85,"slug":86},90,"presentations",{"id":88,"doc_module":22,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},12,"Resumes",80,"resumes",{"id":93,"doc_module":22,"doc_module_name":25,"category_name":94,"show_sort_weight":95,"slug":96},14,"Invoices",70,"invoices",{"id":98,"doc_module":22,"doc_module_name":25,"category_name":99,"show_sort_weight":100,"slug":101},15,"Posters",60,"posters",{"id":103,"doc_module":22,"doc_module_name":25,"category_name":104,"show_sort_weight":105,"slug":106},16,"Social 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structure of the atmosphere properly requires selection of a suitable vertical coordinate and sufficient vertical resolution.\n•Unlike the horizontal structure of models where discrete or continuous (grid point or spectral) configurations can be used, virtually all operational models use discrete vertical structures.\n20.2 Sigma Vertical Coordinate\nThe equations of motion have their simplest form in pressure coordinates.\nUnfortunately, pressure coordinate systems are not particularly suited to solving the forecast equations because, like height surfaces, they can intersect mountains and consequently 'disappear' over parts of the forecast domain.\nTo deal with this problem Phillips (1957) developed a terrain-following coordinate called the sigma (σ) coordinate.\nThe sigma coordinate or variants are used in the NGM, GFS, ECMWF, NOGAPS, and UKMET models and appear in some mesoscale models, such as MM5, COAMPS, and RAMS.\nThe sigma coordinate is defined by\n𝜎=𝑝/\u001a𝑝\u001b𝑠\u001b                       (20.1)\nwhere p is the pressure on a forecast level within the model and \u001ap\u001bs\u001b is the pressure at the earth's surface, not mean sea level pressure.\nThe lowest coordinate surface (usually labeled σ = 1) follows a smoothed version of the actual terrain. Note that the terrain slopes used in sigma models are always smoothed to some degree.\nThe other sigma surfaces gradually transition from being nearly parallel to the smoothed terrain at the bottom of the model (σ= 1) to being nearly horizontal to the constant pressure surface at the top of the model (σ= 0).\nThe sigma vertical coordinate can also be formulated with respect to height (z), rather than pressure.\nFigure (20.1) Sigma vertical coordinate\n20.3 Eta Vertical Coordinate\nThe eta coordinate (η) was created in the early 1980s in an effort to reduce the errors incurred in calculating the pressure gradient force using sigma coordinate models.\nThe eta coordinate is, in fact, another form of the sigma coordinate, but uses mean sea level pressure instead of surface pressure as a bottom reference level. As such, eta is defined as\n\u001a𝜂\u001b𝑠\u001b=\u001a\u001a𝑃\u001b𝑟\u001b\u001a\u001a𝑧\u001b𝑠\u001b\u001b−\u001a𝑃\u001b𝑡\u001b\u001b\u001a𝑃\u001b𝑟\u001b\u001a𝑧=0\u001b−\u001a𝑃\u001b𝑡\u001b\u001b                      (20.2)\nwhere \u001a𝑃\u001b𝑡\u001b is pressure at the model top, \u001a𝑃\u001b𝑟\u001b\u001a𝑧=0\u001b is the standard atmosphere MSL pressure (1013 hPa), \u001a𝑃\u001b𝑟\u001b\u001a\u001a𝑧\u001b𝑠\u001b\u001b is the standard atmosphere pressure at the model terrain level \u001a𝑧\u001b𝑠\u001b.\nFigure (20.2) Eta vertical coordinate\n20.4 Calculating Eta Surfaces\nFirst, the heights at each model level must be defined. In this example, we have defined a model with 10 eta layers (only 3 shown) distributed evenly with respect to pressure from sea level to the top of the atmosphere. Standard atmosphere pressures are then determined at each of these heights.\nFigure (20.3) Eta vertical coordinate\nAt point 1, the actual terrain elevation is 848m. This is closest to the 1000-m height defined for the first eta level. The standard atmosphere pressure at that height is 900 hPa. What, then, is the eta level closest to this point?\nUsing the eta equation,\n𝜂=\u001a900−0\u001b1000−0\u001b=0.9\nIf we go to point 2, the actual terrain height is 1126 m and is also closest to the 1000-m height. Therefore, the eta level closest to this point is again 0.9.\nHowever, if we go up to point 3 (1832 m), the nearest eta surface in the model is at 2000 m. Here the standard atmospheric pressure is 800 hPa. The nearest eta level is therefore\n=1×\u001a800−\u001a0\u001b1000\u001b−0\u001b=0.8\nThis has been a simplified example. In reality, it is necessary to choose the intervals between eta levels in a way that both depicts the planetary boundary layer (PBL) with sufficient detail and yet represents the average changes in elevation over the entire forecast domain.\nNote that the eta levels are predefined and the model topography is set to the nearest eta surface even if it does not quite match the average or smoothed terrain hei","cbCaigONGkmIseb1","https://ap.wps.com/l/cbCaigONGkmIseb1","docx",150554,6,"English","# Lecture 20: Vertical Coordinate Systems\n## Preamble\n## Sigma Vertical Coordinate\n## Eta Vertical Coordinate\n## Calculating Eta Surfaces\n## Sigma vs Eta Surfaces\n## Limitations of the Eta Vertical Coordinate\n## Other coordinates","[{\"question\":\"Why is selecting a vertical coordinate important in atmospheric models?\",\"answer\":\"Vertical structure affects how well the model represents the atmosphere, requiring a suitable vertical coordinate and enough vertical resolution to capture vertical variations accurately.\"},{\"question\":\"What problem does the eta (η) coordinate address compared with sigma (σ)?\",\"answer\":\"Eta was created to reduce errors in computing the pressure gradient force when using sigma coordinate models, by using mean sea level pressure as the bottom reference instead of surface pressure.\"},{\"question\":\"How are eta levels chosen and applied to the model terrain?\",\"answer\":\"Eta surfaces are predefined and standard-atmosphere pressures at specific heights are used to determine the eta level closest to each terrain height, while the model topography is set to the nearest eta surface.\"}]","Lecture 20 - Vertical Coordinate Systems | DOCX",1788328724]