Pub Date : 2023-11-06DOI: 10.1080/03091929.2023.2268817
Azza M. Algatheem, Andrew D. Gilbert, Andrew S. Hillier
A classic stability problem relevant to many applications in geophysical and astrophysical fluid mechanics is that of Kolmogorov flow, a unidirectional purely sinusoidal velocity field written here as u=(0,sinx) in the infinite (x,y)-plane. Near onset, instabilities take the form of large-scale transverse flows, in other words flows in the x-direction with a small wavenumber k in the y-direction. This is similar to the phenomenon known as zonostrophic instability, found in many examples of randomly forced fluid flows modelling geophysical and planetary systems. The present paper studies the effect of incorporating a magnetic field B0, in particular a y-directed “vertical” field or an x-directed “horizontal” field. The linear stability problem is truncated to determining the eigenvalues of finite matrices numerically, allowing exploration of the instability growth rate p as a function of the wavenumber k in the y-direction and a Bloch wavenumber ℓ in the x-direction, with −1/2<ℓ≤ 1/2. In parallel, asymptotic approximations are developed, valid in the limits k→0, ℓ→0, using matrix eigenvalue perturbation theory. Results are presented showing the robust suppression of the hydrodynamic Kolmogorov flow instability as the imposed magnetic field B0 is increased from zero. However with increasing B0, further branches of instability become evident. For vertical field there is a strong-field branch of destabilised Alfvén waves present when the magnetic Prandtl number Pm<1, as found recently by A.E. Fraser, I.G. Cresswell and P. Garaud (J. Fluid Mech. 949, A43, 2022), and a further branch for Pm>1 in the presence of an additional imposed x-directed fluid flow U0. For horizontal magnetic field, a branch of field-driven, tearing mode instabilities emerges as B0 increases. The above instabilities are present for Bloch wavenumber ℓ=0; however allowing ℓ to be non-zero gives rise to a further branch of instabilities in the case of horizontal field. In some circumstances, even when the system is hydrodynamically stable arbitrarily weak magnetic fields can give growing modes, via the instability taking place on large scales in x and y. Detailed comparisons are given between theory for small k and ℓ, and numerical results.
{"title":"Zonostrophic instabilities in magnetohydrodynamic Kolmogorov flow","authors":"Azza M. Algatheem, Andrew D. Gilbert, Andrew S. Hillier","doi":"10.1080/03091929.2023.2268817","DOIUrl":"https://doi.org/10.1080/03091929.2023.2268817","url":null,"abstract":"A classic stability problem relevant to many applications in geophysical and astrophysical fluid mechanics is that of Kolmogorov flow, a unidirectional purely sinusoidal velocity field written here as u=(0,sinx) in the infinite (x,y)-plane. Near onset, instabilities take the form of large-scale transverse flows, in other words flows in the x-direction with a small wavenumber k in the y-direction. This is similar to the phenomenon known as zonostrophic instability, found in many examples of randomly forced fluid flows modelling geophysical and planetary systems. The present paper studies the effect of incorporating a magnetic field B0, in particular a y-directed “vertical” field or an x-directed “horizontal” field. The linear stability problem is truncated to determining the eigenvalues of finite matrices numerically, allowing exploration of the instability growth rate p as a function of the wavenumber k in the y-direction and a Bloch wavenumber ℓ in the x-direction, with −1/2<ℓ≤ 1/2. In parallel, asymptotic approximations are developed, valid in the limits k→0, ℓ→0, using matrix eigenvalue perturbation theory. Results are presented showing the robust suppression of the hydrodynamic Kolmogorov flow instability as the imposed magnetic field B0 is increased from zero. However with increasing B0, further branches of instability become evident. For vertical field there is a strong-field branch of destabilised Alfvén waves present when the magnetic Prandtl number Pm<1, as found recently by A.E. Fraser, I.G. Cresswell and P. Garaud (J. Fluid Mech. 949, A43, 2022), and a further branch for Pm>1 in the presence of an additional imposed x-directed fluid flow U0. For horizontal magnetic field, a branch of field-driven, tearing mode instabilities emerges as B0 increases. The above instabilities are present for Bloch wavenumber ℓ=0; however allowing ℓ to be non-zero gives rise to a further branch of instabilities in the case of horizontal field. In some circumstances, even when the system is hydrodynamically stable arbitrarily weak magnetic fields can give growing modes, via the instability taking place on large scales in x and y. Detailed comparisons are given between theory for small k and ℓ, and numerical results.","PeriodicalId":56132,"journal":{"name":"Geophysical and Astrophysical Fluid Dynamics","volume":"204 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135679716","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 : 2023-10-24DOI: 10.1080/03091929.2023.2262100
William J. McKiver
AbstractHere we consider a model of an isolated vortex to understand the vertical dynamics induced by mesoscale eddies in the ocean. We use the analytical solutions to a balanced model for an ellipsoid of uniform potential vorticity to examine how the vertical motions induced depend on the vortex shape and its orientation, i.e. whether the vortex is vertically upright or tilted with respect to the vertical axis. The motion induced by the vortex can be divided into two kinds: (1) the interior flow which acts on the vortex itself and (2) the exterior flow which acts on its surroundings. For an upright ellipsoid, there are no self-induced vertical motions and the vortex rotates steadily about the vertical axis. However, for a tilted ellipsoid we find solutions exist where the vortex rotates about the vertical axis, while the vertical motions cause the tilt angle of the vortex to oscillate. This effect is stronger as the tilt angle is increased. Considering the exterior flow, there exists an exterior vertical velocity for the upright and tilted ellipsoids. However, the dynamics induced by the exterior vertical velocity is very different for the upright and tilted cases. We find that for an upright ellipsoidal vortex, the vertical motions are largest for vortices with high horizontal eccentricity and a vertical height-to-width aspect ratio near unity, vanishing as the horizontal cross-section of the vortex becomes circular. Instead for the tilted case, the vertical motions are largest when the horizontal cross section is circular, and for strongly prolate vortices, with the largest vertical motions occurring when the tilt angle is 45∘.Keywords: Vortexgeophysicalbalancedvertical motions Disclosure statementNo potential conflict of interest was reported by the author(s).
{"title":"Scales of vertical motions due to an isolated vortex in ageostrophic balanced flows","authors":"William J. McKiver","doi":"10.1080/03091929.2023.2262100","DOIUrl":"https://doi.org/10.1080/03091929.2023.2262100","url":null,"abstract":"AbstractHere we consider a model of an isolated vortex to understand the vertical dynamics induced by mesoscale eddies in the ocean. We use the analytical solutions to a balanced model for an ellipsoid of uniform potential vorticity to examine how the vertical motions induced depend on the vortex shape and its orientation, i.e. whether the vortex is vertically upright or tilted with respect to the vertical axis. The motion induced by the vortex can be divided into two kinds: (1) the interior flow which acts on the vortex itself and (2) the exterior flow which acts on its surroundings. For an upright ellipsoid, there are no self-induced vertical motions and the vortex rotates steadily about the vertical axis. However, for a tilted ellipsoid we find solutions exist where the vortex rotates about the vertical axis, while the vertical motions cause the tilt angle of the vortex to oscillate. This effect is stronger as the tilt angle is increased. Considering the exterior flow, there exists an exterior vertical velocity for the upright and tilted ellipsoids. However, the dynamics induced by the exterior vertical velocity is very different for the upright and tilted cases. We find that for an upright ellipsoidal vortex, the vertical motions are largest for vortices with high horizontal eccentricity and a vertical height-to-width aspect ratio near unity, vanishing as the horizontal cross-section of the vortex becomes circular. Instead for the tilted case, the vertical motions are largest when the horizontal cross section is circular, and for strongly prolate vortices, with the largest vertical motions occurring when the tilt angle is 45∘.Keywords: Vortexgeophysicalbalancedvertical motions Disclosure statementNo potential conflict of interest was reported by the author(s).","PeriodicalId":56132,"journal":{"name":"Geophysical and Astrophysical Fluid Dynamics","volume":"71 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135273239","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 : 2023-10-20DOI: 10.1080/03091929.2023.2257372
R. Okatev, P. Frick, D. Sokoloff
AbstractThe temporal spectrum of the solar activity is more than just the main cycle. It contains different timescales, which can be considered as continuous components of the activity spectrum. The possibility of finding a realistic spectrum of the solar magnetic activity variation is analysed for several versions of a simple model for solar activity based on the original idea of E. Parker. In particular, we study the original set of partial differential equations with two versions of suppression of the dynamo action and the fourth-order dynamical system obtained by truncating the Parker equations. We show that the effects included in the models, i.e. the nonlinear dynamo suppression and the dynamical chaos, as well as random fluctuations of the dynamo drivers, are quite sufficient to obtain the main solar cycle and the continuous components of the spectrum similar to the observed ones. However, the capabilities of the approach under consideration substantially vary from one model to another. Each model reproduces a continuous component of the spectrum in a specific parameter range. This study has confirmed the view that the examination of various solar dynamo models with the aim to find a reasonable combination of main activity cycle and continuous spectrum of solar activity can be used as an additional test of their validity.Keywords: Solar cycledynamo modelssolar activity spectrum Disclosure statementNo potential conflict of interest was reported by the author(s).Notes1 We are grateful to the anonymous referee who attracted our attention to this important problem.Additional informationFundingThis study has been performed within the framework of the Russian Academy of Sciences project AAAA-A19-119012290101-5. DS is grateful for support of the BASIS foundation under grant 21-1-1-4-1.
{"title":"Can the observable solar activity spectrum be reproduced by a simple dynamo model?","authors":"R. Okatev, P. Frick, D. Sokoloff","doi":"10.1080/03091929.2023.2257372","DOIUrl":"https://doi.org/10.1080/03091929.2023.2257372","url":null,"abstract":"AbstractThe temporal spectrum of the solar activity is more than just the main cycle. It contains different timescales, which can be considered as continuous components of the activity spectrum. The possibility of finding a realistic spectrum of the solar magnetic activity variation is analysed for several versions of a simple model for solar activity based on the original idea of E. Parker. In particular, we study the original set of partial differential equations with two versions of suppression of the dynamo action and the fourth-order dynamical system obtained by truncating the Parker equations. We show that the effects included in the models, i.e. the nonlinear dynamo suppression and the dynamical chaos, as well as random fluctuations of the dynamo drivers, are quite sufficient to obtain the main solar cycle and the continuous components of the spectrum similar to the observed ones. However, the capabilities of the approach under consideration substantially vary from one model to another. Each model reproduces a continuous component of the spectrum in a specific parameter range. This study has confirmed the view that the examination of various solar dynamo models with the aim to find a reasonable combination of main activity cycle and continuous spectrum of solar activity can be used as an additional test of their validity.Keywords: Solar cycledynamo modelssolar activity spectrum Disclosure statementNo potential conflict of interest was reported by the author(s).Notes1 We are grateful to the anonymous referee who attracted our attention to this important problem.Additional informationFundingThis study has been performed within the framework of the Russian Academy of Sciences project AAAA-A19-119012290101-5. DS is grateful for support of the BASIS foundation under grant 21-1-1-4-1.","PeriodicalId":56132,"journal":{"name":"Geophysical and Astrophysical Fluid Dynamics","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135617928","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 : 2023-10-19DOI: 10.1080/03091929.2023.2256024
Jianming Miao, Yu Han, Sen Liu, Zhenfeng Zhai
AbstractAn analytical solution is presented to study a plane solitary wave propagating past a concentric segmented arc-shaped breakwater using the matched eigenfunction and separation of variable methods. The undetermined potential coefficients are obtained based on matching conditions. The numerical results of this study are found to agree well with previous calculation results. The major factors (including the number of arc-shaped breakwaters, opening angle, incident angle, and gap width) that affecting the hydrodynamic loads and diffracted wave surface are discussed. The results indicate that the shielding effect of a segmented two-arc breakwater with a small gap width is better than that of an unsegmented arc-shaped breakwater. However, the arrangement of the segmented arc impacts the sheltering effect. Numerical results provide a valuable reference for the hydrodynamic analyses and structural design of segmented arc-shaped breakwaters.Keywords: Solitary wavesegmented arc-shaped breakwateranalytical derivationhydrodynamic force Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work is supported by the National Natural Science Foundation of China (42227901, 52371358), Innovation Group Project of Southern Marine Science and Engineering Guangdong Laboratory (Zhuhai) (311020011), Key-Area Research and Development Program of Guangdong Province (2020B1111010004), and the Special project for marine economy development of Guangdong Province (GDNRC [2022] 31).
{"title":"Solitary wave scattering by segmented arc-shaped breakwater","authors":"Jianming Miao, Yu Han, Sen Liu, Zhenfeng Zhai","doi":"10.1080/03091929.2023.2256024","DOIUrl":"https://doi.org/10.1080/03091929.2023.2256024","url":null,"abstract":"AbstractAn analytical solution is presented to study a plane solitary wave propagating past a concentric segmented arc-shaped breakwater using the matched eigenfunction and separation of variable methods. The undetermined potential coefficients are obtained based on matching conditions. The numerical results of this study are found to agree well with previous calculation results. The major factors (including the number of arc-shaped breakwaters, opening angle, incident angle, and gap width) that affecting the hydrodynamic loads and diffracted wave surface are discussed. The results indicate that the shielding effect of a segmented two-arc breakwater with a small gap width is better than that of an unsegmented arc-shaped breakwater. However, the arrangement of the segmented arc impacts the sheltering effect. Numerical results provide a valuable reference for the hydrodynamic analyses and structural design of segmented arc-shaped breakwaters.Keywords: Solitary wavesegmented arc-shaped breakwateranalytical derivationhydrodynamic force Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work is supported by the National Natural Science Foundation of China (42227901, 52371358), Innovation Group Project of Southern Marine Science and Engineering Guangdong Laboratory (Zhuhai) (311020011), Key-Area Research and Development Program of Guangdong Province (2020B1111010004), and the Special project for marine economy development of Guangdong Province (GDNRC [2022] 31).","PeriodicalId":56132,"journal":{"name":"Geophysical and Astrophysical Fluid Dynamics","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135779974","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 : 2023-07-04DOI: 10.1080/03091929.2023.2232939
J. Reinaud
Three-dimensional filaments of quasi-geostrophic potential vorticity are generic features of atmospheric and oceanic flows. They are often generated during the strong interactions between three-dimensional quasi-geostrophic vortices. They contribute to a direct cascade of enstrophy in spectral space. These filaments correspond to shear zones. Therefore they may be sensitive to shear instabilities akin to the Kelvin–Helmholtz instability of the classical two-dimensional vorticity strip. They are, however, often subjected to a straining flow induced by the surrounding vortices. This straining flow affects their robustness. This paper focuses on a simplified model of this situation. We consider the effect of a pure strain on a three-dimensional filament of uniform quasi-geostrophic potential vorticity. We first consider a quasi-static situation where the strain, assumed small, only affects the cross-sectional shape of the filament, but not the velocity field. We address the linear stability of the filament in that context and also show examples of the filament's nonlinear evolution. We then consider the linearised dynamics of the filament in pure strain. In particular we focus on the maximum perturbation amplification observed in the filament. We conclude that small to moderate strain rates are efficient at preventing a large perturbation growth. Nonlinear effects can nevertheless leads to the roll-up of weakly strained filaments.
{"title":"Filaments of uniform quasi-geostrophic potential vorticity in pure strain","authors":"J. Reinaud","doi":"10.1080/03091929.2023.2232939","DOIUrl":"https://doi.org/10.1080/03091929.2023.2232939","url":null,"abstract":"Three-dimensional filaments of quasi-geostrophic potential vorticity are generic features of atmospheric and oceanic flows. They are often generated during the strong interactions between three-dimensional quasi-geostrophic vortices. They contribute to a direct cascade of enstrophy in spectral space. These filaments correspond to shear zones. Therefore they may be sensitive to shear instabilities akin to the Kelvin–Helmholtz instability of the classical two-dimensional vorticity strip. They are, however, often subjected to a straining flow induced by the surrounding vortices. This straining flow affects their robustness. This paper focuses on a simplified model of this situation. We consider the effect of a pure strain on a three-dimensional filament of uniform quasi-geostrophic potential vorticity. We first consider a quasi-static situation where the strain, assumed small, only affects the cross-sectional shape of the filament, but not the velocity field. We address the linear stability of the filament in that context and also show examples of the filament's nonlinear evolution. We then consider the linearised dynamics of the filament in pure strain. In particular we focus on the maximum perturbation amplification observed in the filament. We conclude that small to moderate strain rates are efficient at preventing a large perturbation growth. Nonlinear effects can nevertheless leads to the roll-up of weakly strained filaments.","PeriodicalId":56132,"journal":{"name":"Geophysical and Astrophysical Fluid Dynamics","volume":"1 1","pages":"225 - 242"},"PeriodicalIF":1.3,"publicationDate":"2023-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89827701","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 : 2023-07-04DOI: 10.1080/03091929.2023.2234596
J. Moss, T. Wood, P. Bushby
An ideal magneto-hydrodynamic fluid, whether fully compressible or incompressible, is a Hamiltonian system. This implies that the equations describing perturbations to any static state are self-adjoint, a fact that is useful in obtaining stability criteria. To describe weakly compressible flows, there are a number of “sound-proof” models that eliminate sound waves by making approximations to the governing equations. However, such approximations may violate the Hamiltonian structure of the system. In a recent work, we have introduced a very general sound-proof model and determined conditions under which it closely approximates the linear regime of magneto-buoyancy instability, motivated by conditions in the solar interior. In the present work, we take a complementary approach, by deriving constraints under which the linearised sound-proof system is self-adjoint. We show that there is a unique set of self-adjoint sound-proof equations that conserves the same energy as the fully compressible system.
{"title":"Self-adjointness of sound-proof models for magnetic buoyancy","authors":"J. Moss, T. Wood, P. Bushby","doi":"10.1080/03091929.2023.2234596","DOIUrl":"https://doi.org/10.1080/03091929.2023.2234596","url":null,"abstract":"An ideal magneto-hydrodynamic fluid, whether fully compressible or incompressible, is a Hamiltonian system. This implies that the equations describing perturbations to any static state are self-adjoint, a fact that is useful in obtaining stability criteria. To describe weakly compressible flows, there are a number of “sound-proof” models that eliminate sound waves by making approximations to the governing equations. However, such approximations may violate the Hamiltonian structure of the system. In a recent work, we have introduced a very general sound-proof model and determined conditions under which it closely approximates the linear regime of magneto-buoyancy instability, motivated by conditions in the solar interior. In the present work, we take a complementary approach, by deriving constraints under which the linearised sound-proof system is self-adjoint. We show that there is a unique set of self-adjoint sound-proof equations that conserves the same energy as the fully compressible system.","PeriodicalId":56132,"journal":{"name":"Geophysical and Astrophysical Fluid Dynamics","volume":"28 1","pages":"263 - 277"},"PeriodicalIF":1.3,"publicationDate":"2023-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85088372","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 : 2023-07-04DOI: 10.1080/03091929.2023.2231134
N. Prasad, R. M. Prasad, Prashant Kumar, Pulkit Kumar, Chandra Mani Prasad
The prime intention of this article is to investigate the effect of variable bottom topography and a bottom-sitting porous barrier on the hydroelastic response of an elastic plate floating on a two-layer fluid using small amplitude wave theory. Galerkin's single-mode approximation in each layer for variable bottom topography and the method of eigenfunction expansion for the fluid region of uniform bottom topography are used as mathematical tools to detail the phenomena. In the variable bottom topography, a system of differential equations is solved. By applying matching conditions, jump conditions, and the appropriate boundary conditions, the solution is expressed as an algebraic linear system from which all the unknown constants are evaluated. The effects of different parameters related to the fluid, bottom topography, and porous barrier on the bending moment, shear force, and deflection of an elastic plate are explored. The variations in the bending moments, shear forces, and plate deflection with respect to fluid density are found to be in opposite trends, caused by surface and interfacial waves, respectively. Further, as the density ratio becomes closer to one, the bending moments, shear forces, and plate deflection tend to diminish for interfacial waves. The bottom effect on bending moments, shear forces, and plate deflection is minimal due to surface waves but significant due to interface waves with maximum amplitude in a concave up bottom. The plate deformation can be further reduced using a suitable barrier, as investigated in this article. The findings hold good potential for furthering our understanding of installing an elastic plate in a stratified fluid with fluctuating water depth.
{"title":"Barrier and bottom topography effects on hydroelastic response of floating elastic plate in a two-layer fluid","authors":"N. Prasad, R. M. Prasad, Prashant Kumar, Pulkit Kumar, Chandra Mani Prasad","doi":"10.1080/03091929.2023.2231134","DOIUrl":"https://doi.org/10.1080/03091929.2023.2231134","url":null,"abstract":"The prime intention of this article is to investigate the effect of variable bottom topography and a bottom-sitting porous barrier on the hydroelastic response of an elastic plate floating on a two-layer fluid using small amplitude wave theory. Galerkin's single-mode approximation in each layer for variable bottom topography and the method of eigenfunction expansion for the fluid region of uniform bottom topography are used as mathematical tools to detail the phenomena. In the variable bottom topography, a system of differential equations is solved. By applying matching conditions, jump conditions, and the appropriate boundary conditions, the solution is expressed as an algebraic linear system from which all the unknown constants are evaluated. The effects of different parameters related to the fluid, bottom topography, and porous barrier on the bending moment, shear force, and deflection of an elastic plate are explored. The variations in the bending moments, shear forces, and plate deflection with respect to fluid density are found to be in opposite trends, caused by surface and interfacial waves, respectively. Further, as the density ratio becomes closer to one, the bending moments, shear forces, and plate deflection tend to diminish for interfacial waves. The bottom effect on bending moments, shear forces, and plate deflection is minimal due to surface waves but significant due to interface waves with maximum amplitude in a concave up bottom. The plate deformation can be further reduced using a suitable barrier, as investigated in this article. The findings hold good potential for furthering our understanding of installing an elastic plate in a stratified fluid with fluctuating water depth.","PeriodicalId":56132,"journal":{"name":"Geophysical and Astrophysical Fluid Dynamics","volume":"11 1","pages":"243 - 262"},"PeriodicalIF":1.3,"publicationDate":"2023-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87077788","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 : 2023-05-04DOI: 10.1080/03091929.2023.2207018
Selina Hossain, Arijit Das, S. De
The two-dimensional problem of wave generation by a time-harmonic pressure distribution on the surface in a finite-depth ocean is studied in this article. Here, it is considered that the ocean has a flexible base, which is modelled as a thin elastic plate and is governed by the Euler–Bernoulli beam equation. The effect of surface tension at the free surface is also taken into account. Within the framework of the linearised theory of water waves, the initial boundary value problem is solved using the Laplace–Fourier transform technique and the integral form of the free surface elevation is obtained. The method of stationary phase is used to evaluate the asymptotic solutions of the free surface elevation for large time and distance. Different forms of the surface elevation have been demonstrated graphically for variation of parameters in a number of figures, and appropriate conclusions are drawn. Moreover, the dispersion relation associated with the wave motion is also derived and analysed using contour plots to understand the characteristics of the roots. Further, the phase and group velocities of the wave motion have also been investigated in this study for both shallow and deep water.
{"title":"The influence of flexible bottom on wave generation by an oscillatory disturbance in the presence of surface tension","authors":"Selina Hossain, Arijit Das, S. De","doi":"10.1080/03091929.2023.2207018","DOIUrl":"https://doi.org/10.1080/03091929.2023.2207018","url":null,"abstract":"The two-dimensional problem of wave generation by a time-harmonic pressure distribution on the surface in a finite-depth ocean is studied in this article. Here, it is considered that the ocean has a flexible base, which is modelled as a thin elastic plate and is governed by the Euler–Bernoulli beam equation. The effect of surface tension at the free surface is also taken into account. Within the framework of the linearised theory of water waves, the initial boundary value problem is solved using the Laplace–Fourier transform technique and the integral form of the free surface elevation is obtained. The method of stationary phase is used to evaluate the asymptotic solutions of the free surface elevation for large time and distance. Different forms of the surface elevation have been demonstrated graphically for variation of parameters in a number of figures, and appropriate conclusions are drawn. Moreover, the dispersion relation associated with the wave motion is also derived and analysed using contour plots to understand the characteristics of the roots. Further, the phase and group velocities of the wave motion have also been investigated in this study for both shallow and deep water.","PeriodicalId":56132,"journal":{"name":"Geophysical and Astrophysical Fluid Dynamics","volume":"22 1","pages":"177 - 212"},"PeriodicalIF":1.3,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76668691","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 : 2023-05-04DOI: 10.1080/03091929.2023.2215390
B. Seyed-Mahmoud
In the conventional treatment of the Earth's rotational dynamics using the Earth's angular momentum description (AMD), it is customary to assume that the velocity/displacement of a mass element in the liquid core (LC) has a displacement as well as an explicit rigid rotation component in addition to the uniform (solid-body) rotation. This makes for a very complex set of non-linear differential equations in the treatment of the dynamics of this body. In this work, I will use a simple three-layer Earth model with a rigid mantle (MT), a rigid inner core (IC), and an incompressible and homogeneous LC to show that in the alternative linearised dynamics of this body, it is redundant to assign a rigid rotation component to the motion. Next, I will use an approximation commonly used in dealing with the Earth's rotational dynamics, and further assume that the MT rotates uniformly, to show that the linearised equations yield identical analytical results to those in the literature for the periods of the inner-core wobble (ICW) and the free inner-core nutation (FICN).
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Pub Date : 2023-03-04DOI: 10.1080/03091929.2023.2187054
R. Garvey, A. Fowler
ABSTRACT We reconsider the theory of turbulent plume formation provided by Schmidt (1941a,b) and its integral formulation, particularly that of Morton et al. (1956). A particular issue for the correct formulation of a mathematical theory is whether the plume is taken to have finite or infinite width, and whether the entrainment rate is prescribed or deduced. Fox (1970) showed that the entrainment rate for a plume can be deduced from the governing partial differential equations by the use of integral moment theory, providing one assumes expressions for the velocity and buoyancy profiles, but it is less clear if entrainment needs to be prescribed if the plume is taken to be of infinite width (as might be appropriate for a laminar plume). Here we choose an eddy viscosity model of a plume which differs from those previously used by allowing the eddy viscosity to vanish with the vertical velocity. We then show that for the ordinary differential equations describing the similarity solution for such a plume rising in an unstratified medium, their solution implies that the plume is finite, and that the entrainment rate at the plume edge is a consequence of the model formulation, and does not need to be hypothesised; and we also show that the entrainment coefficient which is thus determined is consistent with values obtained by experiment. We also show that the resulting velocity profiles differ from those found experimentally by their omission of the Gaussian tail, and we suggest that this discrepancy may be resolved in the model by the inclusion of the small molecular kinematic viscosity.
{"title":"On the mathematical theory of plumes","authors":"R. Garvey, A. Fowler","doi":"10.1080/03091929.2023.2187054","DOIUrl":"https://doi.org/10.1080/03091929.2023.2187054","url":null,"abstract":"ABSTRACT We reconsider the theory of turbulent plume formation provided by Schmidt (1941a,b) and its integral formulation, particularly that of Morton et al. (1956). A particular issue for the correct formulation of a mathematical theory is whether the plume is taken to have finite or infinite width, and whether the entrainment rate is prescribed or deduced. Fox (1970) showed that the entrainment rate for a plume can be deduced from the governing partial differential equations by the use of integral moment theory, providing one assumes expressions for the velocity and buoyancy profiles, but it is less clear if entrainment needs to be prescribed if the plume is taken to be of infinite width (as might be appropriate for a laminar plume). Here we choose an eddy viscosity model of a plume which differs from those previously used by allowing the eddy viscosity to vanish with the vertical velocity. We then show that for the ordinary differential equations describing the similarity solution for such a plume rising in an unstratified medium, their solution implies that the plume is finite, and that the entrainment rate at the plume edge is a consequence of the model formulation, and does not need to be hypothesised; and we also show that the entrainment coefficient which is thus determined is consistent with values obtained by experiment. We also show that the resulting velocity profiles differ from those found experimentally by their omission of the Gaussian tail, and we suggest that this discrepancy may be resolved in the model by the inclusion of the small molecular kinematic viscosity.","PeriodicalId":56132,"journal":{"name":"Geophysical and Astrophysical Fluid Dynamics","volume":"19 1","pages":"79 - 106"},"PeriodicalIF":1.3,"publicationDate":"2023-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81623351","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}
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