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Liu Yongming - One of the best experts on this subject based on the ideXlab platform.
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On nonlinear stability theorems of 3D Quasi-Geostrophic Flow
Advances in Atmospheric Sciences, 2006Co-Authors: Liu Yongming, Cai JingjingAbstract:Nonlinear stability criteria for Quasi-Geostrophic zonally symmetric Flow are improved by establishing an optimal Poincare inequality. The inequality is derived by a variational calculation considering the additional invariant of zonal momentum. When applied to the Eady model in a periodic channel with finite zonal length, the improved nonlinear stability criterion is identical to the linear normal-mode stability criterion provided the channel meridional width is no greater than 0.8605··· times its channel length (which is the geophysically relevant case).
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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, Zeng Qingcun, Theodore G Shepherd, 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.
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, Zeng Qingcun, Theodore G Shepherd, 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.
Theodore G Shepherd - 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, Zeng Qingcun, Theodore G Shepherd, 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.
B Parthasarathy - One of the best experts on this subject based on the ideXlab platform.
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tearing of an aligned vortex by a current difference in two layer quasi geostrophic Flow
Journal of Fluid Mechanics, 1993Co-Authors: J S Marshall, B ParthasarathyAbstract:A study of two-layer Quasi-Geostrophic vortex Flow is performed to determine the effect of a current difference between the layers on a vortex initially extending through both layers. In particular, the conditions under which the vortex can resist being torn by the current difference are examined. The vortex evolution is determined using versions of the contour dynamics and discrete vortex methods which are modified for two-layer Quasi-Geostrophic Flows. The vortex response is found to depend upon the way in which the current difference between the layers is maintained. In the first set of Flows studied, the current difference is generated by a (stronger) third vortex in the upper layer located at a large distance from the (weaker) vortex under study.
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, German 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.
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Available potential vorticity and wave-averaged Quasi-Geostrophic Flow
Journal of Fluid Mechanics, 2015Co-Authors: Gregory L Wagner, William R. YoungAbstract:We derive a wave-averaged potential vorticity equation describing the evolution of strongly stratified, rapidly rotating Quasi-Geostrophic (QG) Flow in a field of inertia-gravity internal waves. The derivation relies on a multiple-time-scale asymptotic expansion of the Eulerian Boussinesq equations. Our result confirms and extends the theory of Bühler & McIntyre (J. Fluid Mech., vol. 354, 1998, pp. 609–646) to non-uniform stratification with buoyancy frequency $N(z)$ and therefore non-uniform background potential vorticity $f_{0}N^{2}(z)$, and does not require spatial-scale separation between waves and balanced Flow. Our interest in non-uniform background potential vorticity motivates the introduction of a new quantity: ‘available potential vorticity’ (APV). Like Ertel potential vorticity, APV is exactly conserved on fluid particles. But unlike Ertel potential vorticity, linear internal waves have no signature in the Eulerian APV field, and the standard QG potential vorticity is a simple truncation of APV for low Rossby number. The definition of APV exactly eliminates the Ertel potential vorticity signal associated with advection of a non-uniform background state, thereby isolating the part of Ertel potential vorticity available for balanced-Flow evolution. The effect of internal waves on QG Flow is expressed concisely in a wave-averaged contribution to the materially conserved QG potential vorticity. We apply the theory by computing the wave-induced QG Flow for a vertically propagating wave packet and a mode-one wave field, both in vertically bounded domains.