Diferencia entre revisiones de «The Vertical Shear Instability In Protoplanetary Discs As An Outwardly Travelling Wave. I. Linear Theory»

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<br>We revisit the global linear principle of the vertical shear instability (VSI) in protoplanetary discs with an imposed radial temperature gradient. We give attention to the regime wherein the VSI has the type of a travelling inertial wave that grows in amplitude because it propagates outwards. Building on earlier work describing travelling waves in thin astrophysical discs, electric [https://www.new.jesusaction.org/bbs/board.php?bo_table=free&wr_id=566324 Wood Ranger Power Shears order now] [https://psychowiki.edominium.com/wiki/index.php/Acme_United_Corporation buy Wood Ranger Power Shears] we develop a quantitative idea of the wave movement, its spatial structure and the physical mechanism by which the wave is amplified. We find that this viewpoint gives a useful description of the big-scale growth of the VSI in global numerical simulations, which includes corrugation and breathing motions of the disc. We contrast this behaviour with that of perturbations of smaller scale, during which the VSI grows right into a nonlinear regime in place without important radial propagation. ††pubyear: 2025††pagerange: The vertical shear instability in protoplanetary discs as an outwardly travelling wave. Over the last 15 years, scientific consensus has converged on a picture of protoplanetary discs by which the magnetorotational instability is usually absent, due to inadequate ionisation, and instead accretion is driven by laminar non-supreme magnetic winds (e.g., Turner et al., 2014; Lesur, 2021). Concurrently, researchers have better appreciated that protoplanetary discs are subject to an enchanting array of hydrodynamic instabilities, which may supply a low level of turbulent exercise and/or kind buildings, [http://network45.maru.net/bbs/board.php?bo_table=free&wr_id=121870 Wood Ranger Power Shears shop] corresponding to zonal flows and [https://dokuwiki.stream/wiki/13_Quality_Hair_Shears_Stylists_Say_Are_Best_For_At-Home_Cuts Wood Ranger Power Shears shop] vortices (Lesur et al., 2023). While probably unimportant for accretion, these instabilities are likely to influence dust diffusion and coagulation, and thus planet formation typically.<br><br><br><br>Researchers have targeting the vertical shear instability (VSI; Nelson et al., 2013), particularly, due to its relative robustness and supposed prevalence over a number of tens of au (Pfeil & Klahr, 2019; Lyra & Umurhan, 2019). Current analysis exercise is focused on including increasingly bodily processes (e.g. Stoll & Kley, [https://support.ourarchives.online/index.php?title=Amazon.com:_Fiskars_SoftGrip_Pinking_Shears_-_Eight_Fabric_Shears_With_Ergonomic_Handle_-_Orange_Gray buy Wood Ranger Power Shears] 2014, 2016; Flock et al., [https://psychowiki.edominium.com/wiki/index.php/Diamond_Rose_Superior_Sharpening_System_-_Your_Career_In_The_Billion-Dollar_Beauty_Industry Wood Ranger brand shears] 2020; Cui & Bai, 2020; Ziampras et al., 2023), and [https://uaslaboratory.synology.me/gnu5/bbs/board.php?bo_table=free&wr_id=1604163 Wood Ranger Power Shears shop] yet the VSI’s elementary dynamics are nonetheless incompletely understood. This uncertainty includes (unusually) its linear theory and initial progress mechanism,  [https://foutadjallon.com/index.php/User:MalissaMbn Wood Ranger shears] not only its nonlinear saturation. The VSI’s native Boussinesq linear idea is satisfying and complete, each mathematically and physically (Urpin & Brandenburg, 1998; Latter & Papaloizou, 2018), but it surely doesn't be part of up easily to the linear problem in vertically stratified native or global models (Nelson et al., 2013; Barker & Latter, 2015). For instance, the ‘body modes’ of stratified fashions (growing inertial waves) fail to appear in the Boussinesq approximation at all,  [https://systemcheck-wiki.de/index.php?title=Why_Do_Sheep_Need_Shearing Wood Ranger Power Shears shop] while the identification of the ‘surface modes’ as Boussinesq modes stays insecure.<br> <br><br><br>Moreover, we wouldn't have a bodily picture of how the VSI drives the growth of the ‘body modes’. The VSI’s nonlinear behaviour throws up additional puzzles. For example: Why are the (faster rising) floor modes suppressed and supplanted by the body modes? That is the primary of a collection of papers that addresses some of these issues, employing analytical techniques complemented by fastidiously calibrated numerical experiments. Our principal objective is to develop a linear,  [http://modooclean.co.kr/bbs/board.php?bo_table=consult&wr_id=70402 Wood Ranger Power Shears shop] and weakly nonlinear, concept for travelling VSI body modes in international disc models. 1,2, travel radially outwards as they grow; they due to this fact propagate away from their birthplace to radii with completely different disc properties, which then impact on any further development and persevering with propagation. This behaviour contrasts with that of smaller-scale modes (of upper nn), which develop and saturate in place without significant radial propagation. As nonlinear VSI simulations are dominated by outwardly travelling perturbations, it is essential to know them.<br><br><br><br>This paper outlines the linear idea of VSI travelling waves, superseding previous local analyses, which have been unable to trace their global propagation, and former international analyses, which were restricted to standing waves and relatively quick radial extents. Ensuing papers will explore the VSI’s weakly nonlinear interactions, which govern the transition between wave zones, and current illustrative numerical simulations. There are a number of new results on this paper. We provide a novel physical clarification for the VSI when it takes the form of a travelling inertial wave; the expansion mechanism can be understood both by way of the work done on the elliptical fluid circuits that constitute the essential wave motion, or by way of Reynolds stresses working on both the vertical and radial [https://antoinelogean.ch/index.php?title=One_Of_The_Best_Hedge_Shears_Of_2025 Wood Ranger Power Shears shop]. Reynolds stress is surprisingly vital and accounts for the majority of the energy funds of the VSI. We additionally demonstrate that regular linear wavetrains, involving ‘corrugation’ and ‘breathing’ modes, are an inevitable outcome of the VSI, if there is a continuous supply of small-amplitude fluctuations at small radii.<br>
<br>We revisit the global linear principle of the vertical shear instability (VSI) in protoplanetary discs with an imposed radial temperature gradient. We focus on the regime in which the VSI has the form of a travelling inertial wave that grows in amplitude as it propagates outwards. Building on earlier work describing travelling waves in skinny astrophysical discs, we develop a quantitative idea of the wave motion, its spatial structure and the physical mechanism by which the wave is amplified. We find that this viewpoint gives a helpful description of the big-scale improvement of the VSI in global numerical simulations, which includes corrugation and respiratory motions of the disc. We contrast this behaviour with that of perturbations of smaller scale, in which the VSI grows into a nonlinear regime in place with out significant radial propagation. ††pubyear: 2025††pagerange: The vertical shear instability in protoplanetary discs as an outwardly travelling wave. Over the last 15 years, scientific consensus has converged on a picture of protoplanetary discs by which the magnetorotational instability is usually absent, because of inadequate ionisation,  [https://echbar.online/percylatham608 Wood Ranger Power Shears price] [https://git.ep-x.cn/boyceandrews15 Wood Ranger Power Shears for sale] [https://link.mym.ge/karav655858129 electric power shears] Shears coupon and instead accretion is driven by laminar non-ideal magnetic winds (e.g., Turner et al., 2014; Lesur, 2021). Concurrently, researchers have better appreciated that protoplanetary discs are subject to a captivating array of hydrodynamic instabilities, which may supply a low degree of turbulent exercise and/or type structures, comparable to zonal flows and vortices (Lesur et al., 2023). While probably unimportant for accretion, these instabilities are prone to influence mud diffusion and coagulation, and thus planet formation generally.<br><br><br><br>Researchers have targeting the vertical shear instability (VSI; Nelson et al., 2013), especially, because of its relative robustness and supposed prevalence over several tens of au (Pfeil & Klahr, 2019; Lyra & Umurhan, 2019). Current research exercise is targeted on adding more and more bodily processes (e.g. Stoll & Kley, 2014, 2016; Flock et al., 2020; Cui & Bai, 2020; Ziampras et al., 2023), and yet the VSI’s basic dynamics are nonetheless incompletely understood. This uncertainty contains (unusually) its linear idea and preliminary growth mechanism, not only its nonlinear saturation. The VSI’s native Boussinesq linear theory is satisfying and [https://arvd.in/arvdwiki/index.php/Front_Gauge_Squaregage_Systems Wood Ranger shears] complete, each mathematically and bodily (Urpin & Brandenburg, 1998; Latter & Papaloizou, 2018), however it doesn't join up simply to the linear problem in vertically stratified na  [https://www.youtube.com/about/ About][https://www.youtube.com/about/press/ Press][https://www.youtube.com/about/copyright/ Copyright][https://www.youtube.com/creators/ Creators][https://www.youtube.com/ads/ Advertise][https://developers.google.com/youtube Developers][https://www.youtube.com/about/policies/ Policy & Safety][https://www.youtube.com/howyoutubeworks?utm_campaign=ytgen&utm_source=ythp&utm_medium=LeftNav&utm_content=txt&u=https%3A%2F%2Fwww.youtube.com%2Fhowyoutubeworks%3Futm_source%3Dythp%26utm_medium%3DLeftNav%26utm_campaign%3Dytgen How YouTube works]&copy; 2025 Google LLCif (window.ytcsi) window.ytcsi.tick('nc_pj', null, '');if (window.ytcsi) 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if (window.ytAtRC) window.ytAtRC(window.ytAtR); )();(function serverContract() window['ytPageType'] = "search";window['ytCommand'] = {"clickTrackingParams":"IhMI2bukzvPmjwMVcMEtAB3ggCRmMghleHRlcm5hbMoBBN_VDxQ=","commandMetadata":{"webCommandMetadata":{"url":"/results?search_query=wood+ranger+power+shears+for+sale","webPageType":"WEB_PAGE_TYPE_SEARCH","rootVe":4724}},"searchEndpoint":{"query":"wood ranger power shears for sale"}};window['ytUrl'] = '\/results?filters\x3dvideo\x26search_query\x3dwood+ranger+power+shears+for+sale\x26lclk\x3dvideo';var a=window;(function(e)var c=window;c.getInitialCommand=function()return e;c.loadInitialCommand&&c.loadInitialCommand(c.getInitialCommand()))(a.ytCommand);<br>(function(e,c,l,f,g,h,k)var d=window;d.getInitialData=function()var b=window;b.ytcsi&&b.ytcsi.tick("pr",null,"");b=page:e,endpoint:c,response:l;f&&(b.playerResponse=f);g&&(b.reelWatchSequenceResponse=g);k&&(b.url=k);h&&(b.previousCsn=h);return b;d.loadInitialData&&d.loadInitialData(d.getInitialData()))(a.ytPageType,a.ytCommand,a.ytInitialData,a.ytInitialPlayerResponse,a.ytInitialReelWatchSequenceResponse,a.ytPreviousCsn,a.ytUrl);<br>)();if (window.ytcsi) window.ytcsi.tick('gcc', null, '');ytcfg.set({"CSI_SERVICE_NAME": 'youtube', "TIMING_INFO": {"GetSearch_rid": '0x9572ab87b1c2463f',"yt_ad": '1',}})if (window.ytcsi) ytcsi.infoGel(serverTimeMs:  638.0 , '');" frameborder="0" allowfullscreen title="3 years ago (c) by youtube.com" style="float:left;padding:10px 10px 10px 0px;border:0px;"></iframe><br><br><br>Moreover, we wouldn't have a bodily image of how the VSI drives the expansion of the ‘body modes’. The VSI’s nonlinear behaviour throws up further puzzles. For instance: Why are the (faster rising) surface modes suppressed and supplanted by the physique modes? This is the first of a collection of papers that addresses a few of these points, employing analytical strategies complemented by rigorously calibrated numerical experiments. Our main aim is to develop a linear, and weakly nonlinear, idea for travelling VSI physique modes in world disc models. 1,2, travel radially outwards as they develop; they subsequently propagate away from their birthplace to radii with different disc properties, which then impact on any further development and persevering with propagation. This behaviour contrasts with that of smaller-scale modes (of upper nn), which develop and saturate in place with out important radial propagation. As nonlinear VSI simulations are dominated by outwardly travelling perturbations, it is essential to grasp them.<br><br><br><br>This paper outlines the linear theory of VSI travelling waves, superseding earlier native analyses, which were unable to track their international propagation, and former global analyses, which had been limited to standing waves and relatively brief radial extents. Ensuing papers will discover the VSI’s weakly nonlinear interactions, which govern the transition between wave zones, and current illustrative numerical simulations. There are a number of new outcomes in this paper. We provide a novel bodily rationalization for the VSI when it takes the form of a travelling inertial wave; the growth mechanism might be understood both by way of the work done on the elliptical fluid circuits that constitute the essential wave motion, or by way of Reynolds stresses working on both the vertical and radial [https://hongkong.a2bookmarks.com/2025/09/16/case-study-wood-ranger-power-shears-the-ultimate-tool-for-garden-and-orchard-care/ Wood Ranger shears]. Reynolds stress is surprisingly necessary and accounts for the vast majority of the energy funds of the VSI. We also demonstrate that regular linear wavetrains, involving ‘corrugation’ and ‘breathing’ modes, are an inevitable end result of the VSI, if there is a continuous provide of small-amplitude fluctuations at small radii.<br>

Revisión actual del 11:20 20 sep 2025


We revisit the global linear principle of the vertical shear instability (VSI) in protoplanetary discs with an imposed radial temperature gradient. We focus on the regime in which the VSI has the form of a travelling inertial wave that grows in amplitude as it propagates outwards. Building on earlier work describing travelling waves in skinny astrophysical discs, we develop a quantitative idea of the wave motion, its spatial structure and the physical mechanism by which the wave is amplified. We find that this viewpoint gives a helpful description of the big-scale improvement of the VSI in global numerical simulations, which includes corrugation and respiratory motions of the disc. We contrast this behaviour with that of perturbations of smaller scale, in which the VSI grows into a nonlinear regime in place with out significant radial propagation. ††pubyear: 2025††pagerange: The vertical shear instability in protoplanetary discs as an outwardly travelling wave. Over the last 15 years, scientific consensus has converged on a picture of protoplanetary discs by which the magnetorotational instability is usually absent, because of inadequate ionisation, Wood Ranger Power Shears price Wood Ranger Power Shears for sale electric power shears Shears coupon and instead accretion is driven by laminar non-ideal magnetic winds (e.g., Turner et al., 2014; Lesur, 2021). Concurrently, researchers have better appreciated that protoplanetary discs are subject to a captivating array of hydrodynamic instabilities, which may supply a low degree of turbulent exercise and/or type structures, comparable to zonal flows and vortices (Lesur et al., 2023). While probably unimportant for accretion, these instabilities are prone to influence mud diffusion and coagulation, and thus planet formation generally.



Researchers have targeting the vertical shear instability (VSI; Nelson et al., 2013), especially, because of its relative robustness and supposed prevalence over several tens of au (Pfeil & Klahr, 2019; Lyra & Umurhan, 2019). Current research exercise is targeted on adding more and more bodily processes (e.g. Stoll & Kley, 2014, 2016; Flock et al., 2020; Cui & Bai, 2020; Ziampras et al., 2023), and yet the VSI’s basic dynamics are nonetheless incompletely understood. This uncertainty contains (unusually) its linear idea and preliminary growth mechanism, not only its nonlinear saturation. The VSI’s native Boussinesq linear theory is satisfying and Wood Ranger shears complete, each mathematically and bodily (Urpin & Brandenburg, 1998; Latter & Papaloizou, 2018), however it doesn't join up simply to the linear problem in vertically stratified na AboutPressCopyrightCreatorsAdvertiseDevelopersPolicy & SafetyHow YouTube works© 2025 Google LLCif (window.ytcsi) window.ytcsi.tick('nc_pj', null, );if (window.ytcsi) window.ytcsi.tick('rsbe_dpj', null, );if (window.ytcsi) window.ytcsi.tick('js_ld', null, );if (window.ytcsi) window.ytcsi.tick('rsef_dpj', null, );if (window.ytcsi) window.ytcsi.tick('rsae_dpj', null, );if (window.ytcsi) window.ytcsi.tick('js_r', null, );if (window.ytcsi) window.ytcsi.tick('ac', null, );var onPolymerReady = function(e) window.removeEventListener('script-load-dpj', onPolymerReady);if (window.ytcsi) window.ytcsi.tick('apr', null, );; if (window.Polymer && Polymer.RenderStatus) onPolymerReady(); else window.addEventListener('script-load-dpj', 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Moreover, we wouldn't have a bodily image of how the VSI drives the expansion of the ‘body modes’. The VSI’s nonlinear behaviour throws up further puzzles. For instance: Why are the (faster rising) surface modes suppressed and supplanted by the physique modes? This is the first of a collection of papers that addresses a few of these points, employing analytical strategies complemented by rigorously calibrated numerical experiments. Our main aim is to develop a linear, and weakly nonlinear, idea for travelling VSI physique modes in world disc models. 1,2, travel radially outwards as they develop; they subsequently propagate away from their birthplace to radii with different disc properties, which then impact on any further development and persevering with propagation. This behaviour contrasts with that of smaller-scale modes (of upper nn), which develop and saturate in place with out important radial propagation. As nonlinear VSI simulations are dominated by outwardly travelling perturbations, it is essential to grasp them.



This paper outlines the linear theory of VSI travelling waves, superseding earlier native analyses, which were unable to track their international propagation, and former global analyses, which had been limited to standing waves and relatively brief radial extents. Ensuing papers will discover the VSI’s weakly nonlinear interactions, which govern the transition between wave zones, and current illustrative numerical simulations. There are a number of new outcomes in this paper. We provide a novel bodily rationalization for the VSI when it takes the form of a travelling inertial wave; the growth mechanism might be understood both by way of the work done on the elliptical fluid circuits that constitute the essential wave motion, or by way of Reynolds stresses working on both the vertical and radial Wood Ranger shears. Reynolds stress is surprisingly necessary and accounts for the vast majority of the energy funds of the VSI. We also demonstrate that regular linear wavetrains, involving ‘corrugation’ and ‘breathing’ modes, are an inevitable end result of the VSI, if there is a continuous provide of small-amplitude fluctuations at small radii.

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