Pub Date : 2021-07-28DOI: 10.1080/03091929.2021.1946803
W. Ye, Zhenfeng Zhai, Hua-Jau Huang
This paper investigates solitary wave diffraction around concentric porous cylindrical structure in front of a vertical wall, where the exterior cylinder is permeable and the interior cylinder is impermeable. This problem is transformed into an issue of bi-directional incident waves diffraction around two concentric cylindrical systems based on the image theory. An analytical solution of the problem is obtained by applying the eigenfunction expansion approach and Graf's addition theorem. Unlike previous studies using Airy wave, this paper uses solitary wave as incident wave. The hydrodynamic loads and wave elevations on the concentric cylindrical system are calculated and compared with existing work. An excellent agreement is obtained between the model and data. Parametric studies on porosity, annular spacing, incident wave angle, distance between the concentric structure and wall were also explored. The results indicated that wave loads on the exterior cylinder near a vertical wall could reach twice as large as the force on it in open water, while the magnitudes of wave loads on the interior cylinder are similar for the two cases. In addition, we compared the modelling results of wave surface elevations using solitary wave and Airy waves. It shows that the amplitude of the wave surface elevations caused by the solitary wave is significantly higher than that of the Airy wave, which means the wave effects on offshore structures will be underestimated when using the Airy wave model in the same shallow water conditions.
{"title":"Solitary wave diffraction around a concentric porous cylindrical structure in front of a vertical wall","authors":"W. Ye, Zhenfeng Zhai, Hua-Jau Huang","doi":"10.1080/03091929.2021.1946803","DOIUrl":"https://doi.org/10.1080/03091929.2021.1946803","url":null,"abstract":"This paper investigates solitary wave diffraction around concentric porous cylindrical structure in front of a vertical wall, where the exterior cylinder is permeable and the interior cylinder is impermeable. This problem is transformed into an issue of bi-directional incident waves diffraction around two concentric cylindrical systems based on the image theory. An analytical solution of the problem is obtained by applying the eigenfunction expansion approach and Graf's addition theorem. Unlike previous studies using Airy wave, this paper uses solitary wave as incident wave. The hydrodynamic loads and wave elevations on the concentric cylindrical system are calculated and compared with existing work. An excellent agreement is obtained between the model and data. Parametric studies on porosity, annular spacing, incident wave angle, distance between the concentric structure and wall were also explored. The results indicated that wave loads on the exterior cylinder near a vertical wall could reach twice as large as the force on it in open water, while the magnitudes of wave loads on the interior cylinder are similar for the two cases. In addition, we compared the modelling results of wave surface elevations using solitary wave and Airy waves. It shows that the amplitude of the wave surface elevations caused by the solitary wave is significantly higher than that of the Airy wave, which means the wave effects on offshore structures will be underestimated when using the Airy wave model in the same shallow water conditions.","PeriodicalId":56132,"journal":{"name":"Geophysical and Astrophysical Fluid Dynamics","volume":"4 1","pages":"78 - 100"},"PeriodicalIF":1.3,"publicationDate":"2021-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79624668","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-13DOI: 10.1080/03091929.2021.1990912
Melanie Kobras, M. Ambaum, V. Lucarini
Eddy saturation describes the nonlinear mechanism in geophysical flows whereby, when average conditions are considered, direct forcing of the zonal flow increases the eddy kinetic energy, while the energy associated with the zonal flow does not increase. Here, we present a minimal baroclinic model that exhibits complete eddy saturation. Starting from Phillips' classical quasi-geostrophic two-level model on the beta channel of the mid-latitudes, we derive a reduced order model comprising of six ordinary differential equations including parameterised eddies. This model features two physically realisable steady state solutions, one a purely zonal flow and one where, additionally, finite eddy motions are present. As the baroclinic forcing in the form of diabatic heating is increased, the zonal solution loses stability and the eddy solution becomes attracting. After this bifurcation, the zonal components of the solution are independent of the baroclinic forcing, and the excess of heat in the low latitudes is efficiently transported northwards by finite eddies, in the spirit of baroclinic adjustment.
{"title":"Eddy saturation in a reduced two-level model of the atmosphere","authors":"Melanie Kobras, M. Ambaum, V. Lucarini","doi":"10.1080/03091929.2021.1990912","DOIUrl":"https://doi.org/10.1080/03091929.2021.1990912","url":null,"abstract":"Eddy saturation describes the nonlinear mechanism in geophysical flows whereby, when average conditions are considered, direct forcing of the zonal flow increases the eddy kinetic energy, while the energy associated with the zonal flow does not increase. Here, we present a minimal baroclinic model that exhibits complete eddy saturation. Starting from Phillips' classical quasi-geostrophic two-level model on the beta channel of the mid-latitudes, we derive a reduced order model comprising of six ordinary differential equations including parameterised eddies. This model features two physically realisable steady state solutions, one a purely zonal flow and one where, additionally, finite eddy motions are present. As the baroclinic forcing in the form of diabatic heating is increased, the zonal solution loses stability and the eddy solution becomes attracting. After this bifurcation, the zonal components of the solution are independent of the baroclinic forcing, and the excess of heat in the low latitudes is efficiently transported northwards by finite eddies, in the spirit of baroclinic adjustment.","PeriodicalId":56132,"journal":{"name":"Geophysical and Astrophysical Fluid Dynamics","volume":"60 1","pages":"38 - 55"},"PeriodicalIF":1.3,"publicationDate":"2021-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90280142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-08DOI: 10.1080/03091929.2021.1941921
R. Yellin-Bergovoy, O. Umurhan, E. Heifetz
The vertical shear instability is an axisymmetric effect suggested to drive turbulence in the magnetically inactive zones of protoplanetary accretion disks. Here we examine its physical mechanism in analytically tractable “minimal models” in three settings that include a uniform density fluid, a stratified atmosphere, and a shearing-box section of a protoplanetary disk. Each of these analyses show that the vertical shear instability's essence is similar to the slantwise convective symmetric instability in the mid-latitude Earth atmosphere, in the presence of vertical shear of the baroclinic jet stream, as well as mixing in the top layers of the Gulf Stream. We show that in order to obtain instability, the fluid parcels' slope should exceed the slope of the mean absolute momentum in the disk radial-vertical plane. We provide a detailed and mutually self-consistent physical explanation from three perspectives: in terms of angular momentum conservation, as a dynamical interplay between a fluid's radial and azimuthal vorticity components, and from an energy perspective involving a generalised Solberg-Høiland Rayleigh condition. Furthermore, we explain why anelastic dynamics yields oscillatory unstable modes and isolate the oscillation mechanism from the instability one.
{"title":"A minimal model for vertical shear instability in protoplanetary accretion disks","authors":"R. Yellin-Bergovoy, O. Umurhan, E. Heifetz","doi":"10.1080/03091929.2021.1941921","DOIUrl":"https://doi.org/10.1080/03091929.2021.1941921","url":null,"abstract":"The vertical shear instability is an axisymmetric effect suggested to drive turbulence in the magnetically inactive zones of protoplanetary accretion disks. Here we examine its physical mechanism in analytically tractable “minimal models” in three settings that include a uniform density fluid, a stratified atmosphere, and a shearing-box section of a protoplanetary disk. Each of these analyses show that the vertical shear instability's essence is similar to the slantwise convective symmetric instability in the mid-latitude Earth atmosphere, in the presence of vertical shear of the baroclinic jet stream, as well as mixing in the top layers of the Gulf Stream. We show that in order to obtain instability, the fluid parcels' slope should exceed the slope of the mean absolute momentum in the disk radial-vertical plane. We provide a detailed and mutually self-consistent physical explanation from three perspectives: in terms of angular momentum conservation, as a dynamical interplay between a fluid's radial and azimuthal vorticity components, and from an energy perspective involving a generalised Solberg-Høiland Rayleigh condition. Furthermore, we explain why anelastic dynamics yields oscillatory unstable modes and isolate the oscillation mechanism from the instability one.","PeriodicalId":56132,"journal":{"name":"Geophysical and Astrophysical Fluid Dynamics","volume":"26 1","pages":"674 - 695"},"PeriodicalIF":1.3,"publicationDate":"2021-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87243812","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-31DOI: 10.1080/03091929.2021.2011269
A. Prugger, J. Rademacher, Junliang Yang
Motivated by numerical schemes for large-scale geophysical flow, we consider the rotating shallow water and Boussinesq equations on the whole space with horizontal kinetic energy backscatter source terms built from negative viscosity and stabilising hyperviscosity with constant parameters. We study the impact of this energy input through various explicit flows, which are simultaneously solving the nonlinear equations and the linear equations that arise upon dropping the transport nonlinearity, i.e. the linearisation in the zero state. These include barotropic, parallel and Kolmogorov flows as well as monochromatic inertia gravity waves. With focus on stable stratification, we find that the backscatter generates numerous solutions of this type that grow exponentially and unboundedly, also with vertical structure. This signifies the possibility of undesired energy concentration into specific modes due to the backscatter. Families of steady-state flows of this type arise as well and superposition principles in the nonlinear equations provide explicit sufficient conditions for instability of some of these. For certain steady barotropic flows of this type, we provide numerical evidence of eigenmodes whose growth rates are proportional to the amplitude factor of the flow. For all other arising steady solutions, we prove that this is not possible.
{"title":"Geophysical fluid models with simple energy backscatter: explicit flows and unbounded exponential growth","authors":"A. Prugger, J. Rademacher, Junliang Yang","doi":"10.1080/03091929.2021.2011269","DOIUrl":"https://doi.org/10.1080/03091929.2021.2011269","url":null,"abstract":"Motivated by numerical schemes for large-scale geophysical flow, we consider the rotating shallow water and Boussinesq equations on the whole space with horizontal kinetic energy backscatter source terms built from negative viscosity and stabilising hyperviscosity with constant parameters. We study the impact of this energy input through various explicit flows, which are simultaneously solving the nonlinear equations and the linear equations that arise upon dropping the transport nonlinearity, i.e. the linearisation in the zero state. These include barotropic, parallel and Kolmogorov flows as well as monochromatic inertia gravity waves. With focus on stable stratification, we find that the backscatter generates numerous solutions of this type that grow exponentially and unboundedly, also with vertical structure. This signifies the possibility of undesired energy concentration into specific modes due to the backscatter. Families of steady-state flows of this type arise as well and superposition principles in the nonlinear equations provide explicit sufficient conditions for instability of some of these. For certain steady barotropic flows of this type, we provide numerical evidence of eigenmodes whose growth rates are proportional to the amplitude factor of the flow. For all other arising steady solutions, we prove that this is not possible.","PeriodicalId":56132,"journal":{"name":"Geophysical and Astrophysical Fluid Dynamics","volume":"21 1","pages":"374 - 410"},"PeriodicalIF":1.3,"publicationDate":"2021-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81576129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-02DOI: 10.1080/03091929.2021.1888375
J. A. Whitehead
Oceanographers use the term “differential diffusion” to express a greater value of bulk turbulent diffusivity of temperature within the ocean than the value of bulk diffusivity of salinity, the ratio quantified by Lewis number. Investigation of horizontal thermohaline convection at Prandtl number 1 and infinity over the range reveals a variety of new flow patterns. The chamber has a linearly changing temperature T and salinity S along the top extending from the cold, fresh “polar” end to the hot, salty “tropics” end. It has an aspect ratio of 8 and sides and bottom are insulated and impermeable. Five transition flow patterns occur with little hysteresis for a fixed salinity Rayleigh number Ras of order 105 as Rayleigh number Ra changes from 3.2 × 106 down to 1. They are: 1. A steady T-cell with sinking at the cold end flowing into a bottom flow that feeds up into a top thermal boundary layer. 2. Salty blobs in the boundary layer that amplify and move from the hot to cold end. Each cold end arrival triggers a sudden increase in overturning velocity. 3. A “stripes” pattern where top to bottom cells (alternating T and S cells) move toward the cold end. 4. An S-cell that is a mirror image of the T-cell near the top along with small T-cells lying at the bottom that move toward the cold end 5. A steady S-cell. Each pattern has a distinct volumetric signature in a T-S diagram. Ranges of Ra with various patterns are sizeable at Ras=7.5 × 105 if Le >4/3 but insensitive to Pr. Balanced convection at Ra = Ras >106 adopts a large unsteady supercell containing smaller T and S cells. Exact ranges of the supercell are unknown. Since differential diffusion produces a large collection of flows compared to thermal convection alone, it might produce unexpected new results if added into numerical models of the ocean.
{"title":"Flows in horizontal thermohaline convection with differential diffusion","authors":"J. A. Whitehead","doi":"10.1080/03091929.2021.1888375","DOIUrl":"https://doi.org/10.1080/03091929.2021.1888375","url":null,"abstract":"Oceanographers use the term “differential diffusion” to express a greater value of bulk turbulent diffusivity of temperature within the ocean than the value of bulk diffusivity of salinity, the ratio quantified by Lewis number. Investigation of horizontal thermohaline convection at Prandtl number 1 and infinity over the range reveals a variety of new flow patterns. The chamber has a linearly changing temperature T and salinity S along the top extending from the cold, fresh “polar” end to the hot, salty “tropics” end. It has an aspect ratio of 8 and sides and bottom are insulated and impermeable. Five transition flow patterns occur with little hysteresis for a fixed salinity Rayleigh number Ras of order 105 as Rayleigh number Ra changes from 3.2 × 106 down to 1. They are: 1. A steady T-cell with sinking at the cold end flowing into a bottom flow that feeds up into a top thermal boundary layer. 2. Salty blobs in the boundary layer that amplify and move from the hot to cold end. Each cold end arrival triggers a sudden increase in overturning velocity. 3. A “stripes” pattern where top to bottom cells (alternating T and S cells) move toward the cold end. 4. An S-cell that is a mirror image of the T-cell near the top along with small T-cells lying at the bottom that move toward the cold end 5. A steady S-cell. Each pattern has a distinct volumetric signature in a T-S diagram. Ranges of Ra with various patterns are sizeable at Ras=7.5 × 105 if Le >4/3 but insensitive to Pr. Balanced convection at Ra = Ras >106 adopts a large unsteady supercell containing smaller T and S cells. Exact ranges of the supercell are unknown. Since differential diffusion produces a large collection of flows compared to thermal convection alone, it might produce unexpected new results if added into numerical models of the ocean.","PeriodicalId":56132,"journal":{"name":"Geophysical and Astrophysical Fluid Dynamics","volume":"96 1","pages":"473 - 498"},"PeriodicalIF":1.3,"publicationDate":"2021-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80428485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-26DOI: 10.1080/03091929.2021.1881780
J. Reinaud, X. Carton
We investigate the strong interaction between two baroclinic tripolar eddies in a three-dimensional, rapidly-rotating, continuously stratified flow under the quasi-geostrophic approximation. Each tripolar eddy consists of an anticyclonic central vortex with two oblate cyclonic vortices located above and below the anticyclone. The interaction depends on the vertical and horizontal offsets between the two tripolar eddies. For small and low PV oblate cyclones, each tripolar eddy alone is only weakly unstable to a baroclinic mode. The instability puts the three vortices out of alignment. Most of the eddy however survives the instability. When two tripolar eddies interact, their constituent vortices may merge. Merger occurs when the eddies are close enough together, and shows similarities with the merger of monopolar vortices. Vertically separated eddies do not align vertically. This suggests the importance of an external flow for the alignment, observed in the oceans, to occur. We finally show that the interaction between two tripolar eddies with intense oblate cyclones is very different and show similarities with the dynamics of dipolar baroclinic eddies known as hetons.
{"title":"The merger of two three-dimensional quasi-geostrophic baroclinic tripolar eddies","authors":"J. Reinaud, X. Carton","doi":"10.1080/03091929.2021.1881780","DOIUrl":"https://doi.org/10.1080/03091929.2021.1881780","url":null,"abstract":"We investigate the strong interaction between two baroclinic tripolar eddies in a three-dimensional, rapidly-rotating, continuously stratified flow under the quasi-geostrophic approximation. Each tripolar eddy consists of an anticyclonic central vortex with two oblate cyclonic vortices located above and below the anticyclone. The interaction depends on the vertical and horizontal offsets between the two tripolar eddies. For small and low PV oblate cyclones, each tripolar eddy alone is only weakly unstable to a baroclinic mode. The instability puts the three vortices out of alignment. Most of the eddy however survives the instability. When two tripolar eddies interact, their constituent vortices may merge. Merger occurs when the eddies are close enough together, and shows similarities with the merger of monopolar vortices. Vertically separated eddies do not align vertically. This suggests the importance of an external flow for the alignment, observed in the oceans, to occur. We finally show that the interaction between two tripolar eddies with intense oblate cyclones is very different and show similarities with the dynamics of dipolar baroclinic eddies known as hetons.","PeriodicalId":56132,"journal":{"name":"Geophysical and Astrophysical Fluid Dynamics","volume":"104 1","pages":"523 - 550"},"PeriodicalIF":1.3,"publicationDate":"2021-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84661818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-04DOI: 10.1080/03091929.2021.1877698
S. London
We study a model developed by Showman and Polvani [Equatorial superrotation on tidally locked exoplanets. Astrophys. J. 2011, 738, 71–94] to study the behaviour of a thin, upper atmospheric layer of a tidally locked exoplanet; in particular, the behaviour of the atmospheres of “hot Jupiters”, gas giants orbiting close to their stars and tidally locked to them, having a dayside and a nightside. We are interested in trying to elucidate the mechanism found by Showman and Polvani for the generation on such planets of an equatorial superrotation, ie. a rapid easterly current in the region about the equator. We extend their analysis of the two simple linear cases studied in the appendices of their paper by including the (small) nonlinear terms which then interact to generate mean flows. In the first of these cases, we expand about the small radiative time scale and find that the mean flow at the equator is eastward, This case may be particularly relevant for the hottest of tidally locked exoplanets. In the other case, we use a multiple parameter asymptotic expansion where the drag time scale is very large but much less than the reciprocal of the small amplitude scale of the expansion. In this case, we again find that the nonlinear terms generate a mean flow that is eastward at the equator. These results may help to provide a possible explanation for the equatorial superrotation.
{"title":"Weakly nonlinear analysis of mean flow generation for tidally locked exoplanets","authors":"S. London","doi":"10.1080/03091929.2021.1877698","DOIUrl":"https://doi.org/10.1080/03091929.2021.1877698","url":null,"abstract":"We study a model developed by Showman and Polvani [Equatorial superrotation on tidally locked exoplanets. Astrophys. J. 2011, 738, 71–94] to study the behaviour of a thin, upper atmospheric layer of a tidally locked exoplanet; in particular, the behaviour of the atmospheres of “hot Jupiters”, gas giants orbiting close to their stars and tidally locked to them, having a dayside and a nightside. We are interested in trying to elucidate the mechanism found by Showman and Polvani for the generation on such planets of an equatorial superrotation, ie. a rapid easterly current in the region about the equator. We extend their analysis of the two simple linear cases studied in the appendices of their paper by including the (small) nonlinear terms which then interact to generate mean flows. In the first of these cases, we expand about the small radiative time scale and find that the mean flow at the equator is eastward, This case may be particularly relevant for the hottest of tidally locked exoplanets. In the other case, we use a multiple parameter asymptotic expansion where the drag time scale is very large but much less than the reciprocal of the small amplitude scale of the expansion. In this case, we again find that the nonlinear terms generate a mean flow that is eastward at the equator. These results may help to provide a possible explanation for the equatorial superrotation.","PeriodicalId":56132,"journal":{"name":"Geophysical and Astrophysical Fluid Dynamics","volume":"10 1","pages":"696 - 709"},"PeriodicalIF":1.3,"publicationDate":"2021-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87628953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-03DOI: 10.1080/03091929.2021.1873319
Y. Bai, T. Vieu, O. Crumeyrolle, I. Mutabazi
Instability modes of viscoelastic Taylor–Couette flow in the Keplerian regime are investigated using both linear stability analysis and experimental detection of critical states. A generalised Rayleigh criterion has been derived and it allows to separate the zone of potential purely elastic instability modes and the zone of stability. The analogy between the instability of viscoelastic Taylor–Couette flow in the Keplerian regime and the magnetorotational instability (MRI) of conducting magnetic fluids is established. Viscoelastic stress tensor can be represented by a tensor product of magnetic-like vectors. This allows to use the Velikhov-Chandrasekhar criterion of the MRI to predict the elasto-rotational instability (ERI). ERI modes obtained in linear stability and in the experiments are assumed to represent the MRI analogue modes as they are observed in the stable zone according to generalised Rayleigh discriminant.
{"title":"Viscoelastic Taylor–Couette instability in the Keplerian regime","authors":"Y. Bai, T. Vieu, O. Crumeyrolle, I. Mutabazi","doi":"10.1080/03091929.2021.1873319","DOIUrl":"https://doi.org/10.1080/03091929.2021.1873319","url":null,"abstract":"Instability modes of viscoelastic Taylor–Couette flow in the Keplerian regime are investigated using both linear stability analysis and experimental detection of critical states. A generalised Rayleigh criterion has been derived and it allows to separate the zone of potential purely elastic instability modes and the zone of stability. The analogy between the instability of viscoelastic Taylor–Couette flow in the Keplerian regime and the magnetorotational instability (MRI) of conducting magnetic fluids is established. Viscoelastic stress tensor can be represented by a tensor product of magnetic-like vectors. This allows to use the Velikhov-Chandrasekhar criterion of the MRI to predict the elasto-rotational instability (ERI). ERI modes obtained in linear stability and in the experiments are assumed to represent the MRI analogue modes as they are observed in the stable zone according to generalised Rayleigh discriminant.","PeriodicalId":56132,"journal":{"name":"Geophysical and Astrophysical Fluid Dynamics","volume":"244 1","pages":"322 - 344"},"PeriodicalIF":1.3,"publicationDate":"2021-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73243415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-22DOI: 10.1080/03091929.2020.1867123
A. Arslan, A. J. Mestel
Dynamo action is considered in the region between two differentially rotating infinite discs. The boundaries may be insulating, perfectly conducting or ferromagnetic. In the absence of a magnetic field, various well-known self-similar flows arise, generalising that of von Kármán. Magnetic field instabilities with the same similarity structure are sought. The kinematic eigenvalue problem is found to have growing modes for . The growth rate is real for the perfectly conducting and ferromagnetic cases, but may be complex for insulating boundaries. As it is shown that the dynamo can be fast or slow, depending on the flow structure. In the slow case, the growth rate is governed by a magnetic boundary layer on one of the discs. The growing field saturates in a solution to the nonlinear dynamo problem. The bifurcation is found to be subcritical and nonlinear dynamos are found for . Finally, the flux of magnetic energy to large r is examined, to determine which solutions might generalise to dynamos between finite discs. It is found that the fast dynamos tend to have inward energy flux, and so are unlikely to be realised in practice. Slow dynamos with outward flux are found. It is suggested that the average rotation rate should be non-zero in practice.
{"title":"Dynamo action between two rotating discs","authors":"A. Arslan, A. J. Mestel","doi":"10.1080/03091929.2020.1867123","DOIUrl":"https://doi.org/10.1080/03091929.2020.1867123","url":null,"abstract":"Dynamo action is considered in the region between two differentially rotating infinite discs. The boundaries may be insulating, perfectly conducting or ferromagnetic. In the absence of a magnetic field, various well-known self-similar flows arise, generalising that of von Kármán. Magnetic field instabilities with the same similarity structure are sought. The kinematic eigenvalue problem is found to have growing modes for . The growth rate is real for the perfectly conducting and ferromagnetic cases, but may be complex for insulating boundaries. As it is shown that the dynamo can be fast or slow, depending on the flow structure. In the slow case, the growth rate is governed by a magnetic boundary layer on one of the discs. The growing field saturates in a solution to the nonlinear dynamo problem. The bifurcation is found to be subcritical and nonlinear dynamos are found for . Finally, the flux of magnetic energy to large r is examined, to determine which solutions might generalise to dynamos between finite discs. It is found that the fast dynamos tend to have inward energy flux, and so are unlikely to be realised in practice. Slow dynamos with outward flux are found. It is suggested that the average rotation rate should be non-zero in practice.","PeriodicalId":56132,"journal":{"name":"Geophysical and Astrophysical Fluid Dynamics","volume":"107 5 1","pages":"710 - 727"},"PeriodicalIF":1.3,"publicationDate":"2021-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89701541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-11DOI: 10.1080/03091929.2020.1845327
B. Seyed-Mahmoud, Y. Rogister
ABSTRACT We study the following rotational modes of Poincaré Earth models: the tilt-over mode (TOM), the spin-over mode (SOM) and free core nutation (FCN), using first a simple Earth model with a homogeneous and incompressible liquid core (LC) and a rigid mantle (MT). We obtain analytical solutions for the periods of these modes as well as that of the Chandler wobble (CW). We show analytically the distinction between the TOM and the SOM and that the FCN is indeed the same mode as the SOM of a wobbling Earth. The reduced pressure, in terms of which the vector momentum equation is known to reduce to a scalar second-order partial differential equation called the Poincaré equation, is used as the independent variable. Analytical solutions are then found for the displacement eigenfunctions in a meridional plane of the liquid core for the aforementioned modes. We next consider a three-layer Earth model similar to above which also includes a rigid inner core (IC). We first show that analytical solutions exist for the period and eigenfunctions of the CW if the IC is locked to the MT, i.e. they have the same wobbling motion. We show that this is significant as it shows that the CW manifests itself for a Poincaré (incompressible and inviscid LC) wobbling Earth model. We further allow for the inner core to wobble independently and compute numerically the periods and displacement eigenfunctions of the TOM, SOM and FCN, as well as those for still another rotational mode, the inner-core wobble (ICW). Next we show that the presence of the characteristic surfaces intercepted by the inner-core, when computing the period and eigenfunctions of the free inner-core nutation (FICN), may be the reason for the slow (or lack of the) convergence of this mode. Finally, we show that even though the wobbling motion of the mantle is ignored when solving for the frequencies of the ICW and the FICN when Sasao's approximation is used, the analytical solutions for both these modes yield periods nearly identical to those in the literature for a similar Earth model with mantle allowed to wobble as well. We infer that the Sasao's approximation, or the severe truncation of the series solution of the field variables, the pressure, the gravitational potential and the components of the displacement vector, may not be adequate to accurately describe the motion in the liquid core during the excitation of the FICN.
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