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

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(Página creada con «<br>We revisit the global linear theory of the vertical shear instability (VSI) in protoplanetary discs with an imposed radial temperature gradient. We deal with the regime through which the VSI has the type of a travelling inertial wave that grows in amplitude because it propagates outwards. Building on previous work describing travelling waves in skinny astrophysical discs, we develop a quantitative principle of the wave motion, its spatial structure and the bodily…»)
 
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<br>We revisit the global linear theory of the vertical shear instability (VSI) in protoplanetary discs with an imposed radial temperature gradient. We deal with the regime through which the VSI has the type of a travelling inertial wave that grows in amplitude because it propagates outwards. Building on previous work describing travelling waves in skinny astrophysical discs, we develop a quantitative principle of the wave motion, its spatial structure and the bodily mechanism by which the wave is amplified. We find that this viewpoint supplies a useful description of the large-scale improvement of the VSI in international numerical simulations, which involves corrugation and breathing motions of the disc. We distinction this behaviour with that of perturbations of smaller scale, through 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 an image of protoplanetary discs through which the magnetorotational instability is mostly absent, due to inadequate ionisation, and instead accretion is pushed by laminar non-splendid magnetic winds (e.g., Turner et al., 2014; Lesur,  [https://cadmaster.iitcsolution.com/bbs/board.php?bo_table=free&wr_id=499314 Wood Ranger shears] 2021). Concurrently, researchers have higher appreciated that protoplanetary discs are topic to an interesting array of hydrodynamic instabilities, which may supply a low level of turbulent exercise and/or kind structures, equivalent to zonal flows and vortices (Lesur et al., 2023). While in all probability unimportant for accretion, these instabilities are prone to influence mud diffusion and coagulation, and thus planet formation typically.<br><br><br><br>Researchers have concentrated on the vertical shear instability (VSI; Nelson et al., [https://harry.main.jp/mediawiki/index.php/%E5%88%A9%E7%94%A8%E8%80%85:ErnestineLhotsky Wood Ranger shears] 2013), particularly, because of its relative robustness and supposed prevalence over several tens of au (Pfeil & Klahr, 2019; Lyra & Umurhan, 2019). Current research activity is focused on including 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 elementary dynamics are nonetheless incompletely understood. This uncertainty includes (unusually) its linear principle and [https://301.tv/traceygarnsey Wood Ranger Power Shears price] Ranger Power Shears USA initial growth mechanism, not solely its nonlinear saturation. The VSI’s local Boussinesq linear theory is satisfying and complete, both mathematically and bodily (Urpin & Brandenburg, 1998; Latter & Papaloizou, 2018), but it does not be a part of up easily to the linear drawback in vertically stratified native or international models (Nelson et al., 2013; Barker & Latter, 2015). For example, the ‘body modes’ of stratified models (growing inertial waves) fail to look within the Boussinesq approximation at all, while the identification of the ‘surface modes’ as Boussinesq modes remains insecure.<br><br><br><br>Moreover, we would not have a bodily image of how the VSI drives the growth of the ‘body modes’. The VSI’s nonlinear behaviour throws up further puzzles. For example: Why are the (quicker rising) surface modes suppressed and  electric [https://short.martinapps.shop/beatrizbevan82 cordless power shears] shears supplanted by the physique modes? This is the first of a series of papers that addresses a few of these issues,  [https://worldbox.wiki/w/User:BettieArnett706 Wood Ranger shears] using analytical techniques complemented by carefully calibrated numerical experiments. Our fundamental goal is to develop a linear, and weakly nonlinear, theory for travelling VSI body modes in global disc fashions. 1,2, journey radially outwards as they grow; they subsequently propagate away from their birthplace to radii with different disc properties, which then impression on any further development and persevering with propagation. This behaviour contrasts with that of smaller-scale modes (of higher nn), which develop and saturate in place with out significant radial propagation. As nonlinear VSI simulations are dominated by outwardly travelling perturbations, it is important to grasp them.<br><br><br><br>This paper outlines the linear principle of VSI travelling waves, superseding previous local analyses, which had been unable to track their international propagation, and previous international analyses, which were limited to standing waves and comparatively 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 several new results in this paper. We provide a novel physical clarification for the VSI when it takes the form of a travelling inertial wave; the growth mechanism could be understood both by way of the work achieved on the elliptical fluid circuits that constitute the basic wave motion, or in terms of Reynolds stresses engaged on each the vertical and radial [https://marketingme.wiki/wiki/Revolutionizing_Gardening_With_Wood_Ranger_Power_Shears Wood Ranger shears]. Reynolds stress is surprisingly necessary and  [https://www.ebersbach.org/index.php?title=How_To_Shear_A_Sheep_The_Correct_Way Wood Ranger shears] accounts for the vast majority of the vitality budget of the VSI. We also demonstrate that steady linear wavetrains, involving ‘corrugation’ and ‘breathing’ modes, are an inevitable outcome of the VSI, [https://americanspeedways.net/index.php/Work_Includes_Field_Machinery_In_Agriculture Wood Ranger shears] if there's a continuous provide 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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