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Keke Zhang - One of the best experts on this subject based on the ideXlab platform.
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precessing cylinders at the second and third resonance turbulence controlled by Geostrophic Flow
Physical Review E, 2015Co-Authors: Jianfei Jiang, Dali Kong, Keke ZhangAbstract:We investigate, via both asymptotic analysis and direct numerical simulation, precessionally driven Flow of a homogeneous fluid confined in fluid-filled circular cylinders that rotate rapidly about their symmetry axis and precess about a different axis and that are marked by radius-height aspect ratios $\mathrm{\ensuremath{\Gamma}}=1.045\phantom{\rule{0.16em}{0ex}}945$ and $\mathrm{\ensuremath{\Gamma}}=1.611\phantom{\rule{0.16em}{0ex}}089$. At these radius-height aspect ratios, the Poincar\'e force resonates directly with the two special inertial modes that have the simplest vertical structure. An asymptotic analytical solution in closed form describing weakly precessing Flow is derived in the mantle frame of reference for asymptotically small Ekman numbers, showing quantitative agreement with the result of direct nonlinear numerical simulation. Our numerical simulation makes use of a finite-element method with the three-dimensional tetrahedralization of a cylindrical cavity that allows the construction of dense nodes in the vicinity of the bounding surface of the cavity for resolving the thin viscous boundary layer. It is found that axisymmetric Geostrophic Flow in the alternating eastward and westward direction can be generated and maintained by nonlinear and viscous effects in the viscous boundary layer. It is also found that, when the precessing rate is moderate and, consequently, the Geostrophic Flow is weak, nonlinear interaction between the resonant inertial mode and the nonesonant inertial modes driven by the Poincar\'e force and the boundary-layer influx leads to strongly turbulent Flow with irregular temporal-spatial fluctuation. When the cylinders are strongly precessing such that the Geostrophic Flow becomes predominant, however, the effect of the Geostrophic Flow controls/stabilizes its nonlinear dynamics, leading to weakly turbulent Flow that can be largely described by a dominant quasisteady Geostrophic component and a weak nonaxisymmetric component localized in the region where the Geostrophic Flow is weak.
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a new legendre type polynomial and its application to Geostrophic Flow in rotating fluid spheres
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2010Co-Authors: Xinhao Liao, Keke ZhangAbstract:In rapidly rotating spheres, the whole fluid column, extending from the southern to northern spherical boundary along the rotation axis, moves like a single fluid element, which is usually referred to as Geostrophic Flow. A new Legendre-type polynomial is discovered in undertaking the asymptotic analysis of Geostrophic Flow in spherical geometry. Three essential properties characterize the new polynomial: (i) it is a function of r and theta but takes a single argument (r sin theta), which is restricted by 0 <= r <= 1 and 0 <= theta = pi, where (r, theta, phi) denote spherical polar coordinates with theta = 0 at the rotation axis; (ii) it is odd and vanishes at the axis of rotation theta = 0, and (iii) it is defined within-and orthogonal over-the full sphere. As an example of its application, we employ the new polynomial in the asymptotic analysis of forced Geostrophic Flows in rotating fluid spheres for small Ekman and Rossby numbers. Fully numerical analysis of the same problem is also carried out, showing satisfactory agreement between the asymptotic solution and the numerical solution.
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A new Legendre-type polynomial and its application to Geostrophic Flow in rotating fluid spheres
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2010Co-Authors: Xinhao Liao, Keke ZhangAbstract:In rapidly rotating spheres, the whole fluid column, extending from the southern to northern spherical boundary along the rotation axis, moves like a single fluid element, which is usually referred to as Geostrophic Flow. A new Legendre-type polynomial is discovered in undertaking the asymptotic analysis of Geostrophic Flow in spherical geometry. Three essential properties characterize the new polynomial: (i) it is a function of r and theta but takes a single argument (r sin theta), which is restricted by 0
Gregory L Wagner - One of the best experts on this subject based on the ideXlab platform.
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an asymptotic model for the propagation of oceanic internal tides through quasi Geostrophic Flow
Journal of Fluid Mechanics, 2017Co-Authors: Gregory L Wagner, G Ferrando, W R YoungAbstract:Starting from the hydrostatic Boussinesq equations, we derive a time-averaged `hydrostatic wave equation' that describes the propagation of inertia-gravity internal waves through quasi-Geostrophic Flow. The derivation uses a multiple-time-scale asymptotic method to isolate wave field evolution over intervals much longer than a wave period, assumes that the wave field has a well-defined and non-inertial frequency such as that of the mid-latitude semi-diurnal lunar tide, neglects nonlinear wave-wave interactions and makes no restriction on either the background density stratification or the relative spatial scales between the wave field and quasi-Geostrophic Flow. As a result the hydrostatic wave equation is a reduced model applicable to the propagation of large scale internal tides through the inhomogeneous and moving ocean. A numerical comparison with the linearized and hydrostatic Boussinesq equations demonstrates the validity of the hydrostatic wave equation and illustrates the manners of model failure when the quasi-Geostrophic Flow is too strong and the wave frequency is too close to inertial.
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A three-component model for the coupled evolution of near-inertial waves, quasi-Geostrophic Flow and the near-inertial second harmonic
Journal of Fluid Mechanics, 2016Co-Authors: Gregory L Wagner, William R. YoungAbstract:We derive an asymptotic model that describes the nonlinear coupled evolution of (i) near-inertial waves (NIWs), (ii) balanced quasi-Geostrophic Flow and (iii) near-inertial second harmonic waves with frequency near $2f_{0}$, where $f_{0}$ is the local inertial frequency. This ‘three-component’ model extends the two-component model derived by Xie & Vanneste (J. Fluid Mech., vol. 774, 2015, pp. 143–169) to include interactions between near-inertial and $2f_{0}$ waves. Both models possess two conservation laws which together imply that oceanic NIWs forced by winds, tides or Flow over bathymetry can extract energy from quasi-Geostrophic Flows. A second and separate implication of the three-component model is that quasi-Geostrophic Flow catalyses a loss of NIW energy to freely propagating waves with near-$2f_{0}$ frequency that propagate rapidly to depth and transfer energy back to the NIW field at very small vertical scales. The upshot of near-$2f_{0}$ generation is a two-step mechanism whereby quasi-Geostrophic Flow catalyses a nonlinear transfer of near-inertial energy to the small scales of wave breaking and diapycnal mixing. A comparison of numerical solutions with both Boussinesq and three-component models for a two-dimensional initial value problem reveals strengths and weaknesses of the model while demonstrating the extraction of quasi-Geostrophic energy and production of small vertical scales.
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on the coupled evolution of oceanic internal waves and quasi Geostrophic Flow
2016Co-Authors: Gregory L WagnerAbstract:Author(s): Wagner, Gregory | Advisor(s): Young, William R | Abstract: Oceanic motion outside thin boundary layers is primarily a mixture of quasi-Geostrophic Flow and internal waves with either near-inertial frequencies or the frequency of the semidiurnal lunar tide. This dissertation seeks a deeper understanding of waves and Flow through reduced models that isolate their nonlinear and coupled evolution from the Boussinesq equations. Three physical-space models are developed: an equation that describes quasi-Geostrophic evolution in an arbitrary and prescribed field of hydrostatic internal waves; a three-component model that couples quasi-Geostrophic Flow to both near-inertial waves and the near-inertial second harmonic; and a model for the slow evolution of hydrostatic internal tides in quasi-Geostrophic Flow of near-arbitrary scale. This slow internal tide equation opens the path to a coupled model for the energetic interaction of quasi-Geostrophic Flow and oceanic internal tides. Four results emerge. First, the wave-averaged quasi-Geostrophic equation reveals that finite-amplitude waves give rise to a mean Flow that advects quasi-Geostrophic potential vorticity. Second is the definition of a new material invariant: Available Potential Vorticity, or APV. APV isolates the part of Ertel potential vorticity available for balanced-Flow evolution in Eulerian frames and proves necessary in the separating waves and quasi-Geostrophic Flow. The third result, hashed out for near-inertial waves and quasi-Geostrophic Flow, is that wave-Flow interaction leads to energy exchange even under conditions of weak nonlinearity. For storm-forced oceanic near-inertial waves the interaction often energizes waves at the expense of Flow. We call this extraction of balanced quasi-Geostrophic energy `stimulated generation' since it requires externally-forced rather than spontaneously-generated waves. The fourth result is that quasi-Geostrophic Flow can encourage or `catalyze' a nonlinear interaction between a near-inertial wave field and its second harmonic that transfers energy to the small near-inertial vertical scales of wave breaking and mixing.
Liu Yongming - One of the best experts on this subject based on the ideXlab platform.
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nonlinear stability of zonally symmetric quasi Geostrophic Flow
Advances in Atmospheric Sciences, 1999Co-Authors: Liu YongmingAbstract:By using the conservation laws and the method of variational principle, an improved Arnol'd's second nonlinear stability theorem for the two-dimensional multilayer quasi-Geostrophic model in periodic channel is obtained.
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nonlinear stability of continuously stratified quasi Geostrophic Flow
Journal of Fluid Mechanics, 1996Co-Authors: Liu Yongming, Mu Mu, Theodore G ShepherdAbstract:Nonlinear stability theorems analogous to Arnol'd's second stability theorem are established for continuously stratified quasi-Geostrophic Flow with general nonlinear boundary conditions in a vertically and horizontally confined domain. Both the standard quasi-Geostrophic model and the modified quasi-Geostrophic model (incorporating effects of hydrostatic compressibility) are treated. The results establish explicit upper bounds on the disturbance energy, the disturbance potential enstrophy, and the disturbance available potential energy on the horizontal boundaries, in terms of the initial disturbance fields. Nonlinear stability in the sense of Liapunov is also established.
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nonlinear stability of multilayer quasi Geostrophic Flow
Journal of Fluid Mechanics, 1994Co-Authors: Mu Mu, Theodore G Shepherd, Zeng Qingcun, Liu YongmingAbstract:New nonlinear stability theorems are derived for disturbances to steady basic Flows in the context of the multilayer quasi-Geostrophic equations. These theorems are analogues of Arnol’d's second stability theorem, the latter applying to the two-dimensional Euler equations. Explicit upper bounds are obtained on both the disturbance energy and disturbance potential enstrophy in terms of the initial disturbance fields. An important feature of the present analysis is that the disturbances are allowed to have non-zero circulation. While Arnol’d's stability method relies on the energy–Casimir invariant being sign-definite, the new criteria can be applied to cases where it is sign-indefinite because of the disturbance circulations. A version of Andrews’ theorem is established for this problem, and uniform potential vorticity Flow is shown to be nonlinearly stable. The special case of two-layer Flow is treated in detail, with particular attention paid to the Phillips model of baroclinic instability. It is found that the short-wave portion of the marginal stability curve found in linear theory is precisely captured by the new nonlinear stability criteria.
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a problem related to nonlinear stability criteria for multi layer quasi Geostrophic Flow
Advances in Atmospheric Sciences, 1992Co-Authors: Liu Yongming, Mu MuAbstract:The second author studied the nonlinear stability of N-layer quasi-Geostrophic Flow subject to perturbations of parameters and initial data, and established the stability criteria for the Flow in question, which involve finding out the lowest eigenvalue of an elliptic boundary value problem. In this paper when the domain is a periodic zonal channel, a formula of the lowest eigenvalue is established, which is useful for further studies and practical applications.
W R Young - One of the best experts on this subject based on the ideXlab platform.
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an asymptotic model for the propagation of oceanic internal tides through quasi Geostrophic Flow
Journal of Fluid Mechanics, 2017Co-Authors: Gregory L Wagner, G Ferrando, W R YoungAbstract:Starting from the hydrostatic Boussinesq equations, we derive a time-averaged `hydrostatic wave equation' that describes the propagation of inertia-gravity internal waves through quasi-Geostrophic Flow. The derivation uses a multiple-time-scale asymptotic method to isolate wave field evolution over intervals much longer than a wave period, assumes that the wave field has a well-defined and non-inertial frequency such as that of the mid-latitude semi-diurnal lunar tide, neglects nonlinear wave-wave interactions and makes no restriction on either the background density stratification or the relative spatial scales between the wave field and quasi-Geostrophic Flow. As a result the hydrostatic wave equation is a reduced model applicable to the propagation of large scale internal tides through the inhomogeneous and moving ocean. A numerical comparison with the linearized and hydrostatic Boussinesq equations demonstrates the validity of the hydrostatic wave equation and illustrates the manners of model failure when the quasi-Geostrophic Flow is too strong and the wave frequency is too close to inertial.
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propagation of near inertial oscillations through a Geostrophic Flow
Journal of Marine Research, 1997Co-Authors: W R Young, Ben M JelloulAbstract:The method of multiple time scales is used to obtain an approximate description of the linear propagation of near-inertial oscillations (NIOs) through a three-dimensional Geostrophic Flow. This 'NIO equation' uses a complex field, M(x, y, z, t), related to the demodulated horizontal velocity by M z = exp (if 0 t)(u + iv), where f 0 is the inertial frequency. The three processes of wave dispersion, advection by Geostrophic velocity and refraction (Geostrophic vorticity slightly shifts the local inertial frequency) are all included in the formulation. The NIO equation has an energy conservation law, so that there is no transfer of energy between NIOs and the Geostrophic Flow in the approximation scheme. As an application, the NIO equation is used to examine propagation of waves through a field of smaller scale, Geostrophic eddies. The spatially local ζ/2 frequency shift, identified by earlier WKB calculations (ζ is the vertical vorticity of the Geostrophic eddies), is not expressed directly in the wave field: the large-scale NIO samples regions of both positive and negative ζ so that there is cancellation. Instead, the ζ/2 frequency shift is rectified to produce an average dispersive effect. The calculation predicts that an NIO with infinite horizontal scale has a frequency shift -Kf 0 m 2 /N 2 where K is average kinetic energy density of the Geostrophic eddies, m the vertical wavenumber of the NIO, f 0 the inertial frequency and N the buoyancy frequency. Because of the dependence of the frequency shift on m 2 , there is an effective vertical dispersion, whose strength is proportional to the eddy kinetic energy. This process greatly increases the vertical propagation rate of synoptic scale NIOs.
Mu Mu - One of the best experts on this subject based on the ideXlab platform.
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nonlinear stability of continuously stratified quasi Geostrophic Flow
Journal of Fluid Mechanics, 1996Co-Authors: Liu Yongming, Mu Mu, Theodore G ShepherdAbstract:Nonlinear stability theorems analogous to Arnol'd's second stability theorem are established for continuously stratified quasi-Geostrophic Flow with general nonlinear boundary conditions in a vertically and horizontally confined domain. Both the standard quasi-Geostrophic model and the modified quasi-Geostrophic model (incorporating effects of hydrostatic compressibility) are treated. The results establish explicit upper bounds on the disturbance energy, the disturbance potential enstrophy, and the disturbance available potential energy on the horizontal boundaries, in terms of the initial disturbance fields. Nonlinear stability in the sense of Liapunov is also established.
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on arnol d s second nonlinear stability theorem for two dimensional quasi Geostrophic Flow
Geophysical and Astrophysical Fluid Dynamics, 1994Co-Authors: Mu Mu, Theodore G ShepherdAbstract:Abstract Arnol'd's second hydrodynamical stability theorem, proven originally for the two-dimensional Euler equations, can establish nonlinear stability of steady Flows that are maxima of a suitably chosen energy-Casimir invariant. The usual derivations of this theorem require an assumption of zero disturbance circulation. In the present work an analogue of Arnol'd's second theorem is developed in the more general case of two-dimensional quasi-Geostrophic Flow, with the important feature that the disturbances are allowed to have non-zero circulation. New nonlinear stability criteria are derived, and explicit bounds are obtained on both the disturbance energy and potential enstrophy which are expressed in terms of the initial disturbance fields. While Arnol'd's stability method relies on the second variation of the energy-Casimir invariant being sign-definite, the new criteria can be applied to cases where the second variation is sign-indefinite because of the disturbance circulations. A version of Andrews...
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nonlinear stability of multilayer quasi Geostrophic Flow
Journal of Fluid Mechanics, 1994Co-Authors: Mu Mu, Theodore G Shepherd, Zeng Qingcun, Liu YongmingAbstract:New nonlinear stability theorems are derived for disturbances to steady basic Flows in the context of the multilayer quasi-Geostrophic equations. These theorems are analogues of Arnol’d's second stability theorem, the latter applying to the two-dimensional Euler equations. Explicit upper bounds are obtained on both the disturbance energy and disturbance potential enstrophy in terms of the initial disturbance fields. An important feature of the present analysis is that the disturbances are allowed to have non-zero circulation. While Arnol’d's stability method relies on the energy–Casimir invariant being sign-definite, the new criteria can be applied to cases where it is sign-indefinite because of the disturbance circulations. A version of Andrews’ theorem is established for this problem, and uniform potential vorticity Flow is shown to be nonlinearly stable. The special case of two-layer Flow is treated in detail, with particular attention paid to the Phillips model of baroclinic instability. It is found that the short-wave portion of the marginal stability curve found in linear theory is precisely captured by the new nonlinear stability criteria.
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a problem related to nonlinear stability criteria for multi layer quasi Geostrophic Flow
Advances in Atmospheric Sciences, 1992Co-Authors: Liu Yongming, Mu MuAbstract:The second author studied the nonlinear stability of N-layer quasi-Geostrophic Flow subject to perturbations of parameters and initial data, and established the stability criteria for the Flow in question, which involve finding out the lowest eigenvalue of an elliptic boundary value problem. In this paper when the domain is a periodic zonal channel, a formula of the lowest eigenvalue is established, which is useful for further studies and practical applications.
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nonlinear stability criteria for motions of multilayer quasi Geostrophic Flow
中国科学B辑(英文版), 1991Co-Authors: Mu MuAbstract:Some nonlinear stability criteria for motions of multilayer quasi-Geostrophic Flow on a beta-plane are obtained by combining Arnold's method with an accurate estimate method. The criteria can be applied to perturbations of initial data and parameters; rather than the former only. Particularly a criterion corresponding to Arnold's second theorem is gained, which relies on some precise analyses and estimates.