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## Unstable Modes (unstable + mode)
## Selected Abstracts## Analytical approach for the toroidal relaxation of viscoelastic earth GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 1 2006Hansheng WangSUMMARY This paper is concerned with post-seismic toroidal deformation in a spherically symmetric, non-rotating, linear-viscoelastic, isotropic Maxwell earth model. Analytical expressions for characteristic relaxation times and relaxation strengths are found for viscoelastic toroidal deformation, associated with surface tangential stress, when there are two to five layers between the core,mantle boundary and Earth's surface. The multilayered models can include lithosphere, asthenosphere, upper and lower mantles and even low-viscosity ductile layer in the lithosphere. The analytical approach is self-consistent in that the Heaviside isostatic solution agrees with fluid limit. The analytical solution can be used for high-precision simulation of the toroidal relaxation in five-layer earths and the results can also be considered as a benchmark for numerical methods. Analytical solution gives only stable decaying modes,unstable mode, conjugate complex mode and modes of relevant poles with orders larger than 1, are all excluded, and the total number of modes is found to be just the number of viscoelastic layers between the core,mantle boundary and Earth's surface,however, any elastic layer between two viscoelastic layers is also counted. This confirms previous finding where numerical method (i.e. propagator matrix method) is used. We have studied the relaxation times of a lot of models and found the propagator matrix method to agree very well with those from analytical results. In addition, the asthenosphere and lithospheric ductile layer are found to have large effects on the amplitude of post-seismic deformation. This also confirms the findings of previous works. [source] ## Linear instability of ideal flows on a sphere MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2009Yuri N. SkibaAbstract A unified approach to the normal mode instability study of steady solutions to the vorticity equation governing the motion of an ideal incompressible fluid on a rotating sphere is considered. The four types of well-known solutions are considered, namely, the Legendre-polynomial (LP) flows, Rossby,Haurwitz (RH) waves, Wu,Verkley (WV) waves and modons. A conservation law for disturbances to each solution is derived and used to obtain a necessary condition for its exponential instability. By these conditions, Fjörtoft's (Tellus 1953; 5:225,230) average spectral number of the amplitude of an unstable mode must be equal to a special value. In the case of LP flows or RH waves, this value is related only with the basic flow degree. For the WV waves and modons, it depends both on the basic flow degree and on the spectral distribution of the mode energy in the inner and outer regions of the flow. Peculiarities of the instability conditions for different types of modons are discussed. The new instability conditions specify the spectral structure of growing disturbances localizing them in the phase space. For the LP flows, this condition complements the well-known Rayleigh,Kuo and Fjörtoft conditions related to the zonal flow profile. Some analytical and numerical examples are considered. The maximum growth rate of unstable modes is also estimated, and the orthogonality of any unstable, decaying and non-stationary mode to the basic flow is shown in the energy inner product. The analytical instability results obtained here can also be applied for testing the accuracy of computational programs and algorithms used for the numerical stability study. It should be stressed that Fjörtoft's spectral number appearing both in the instability conditions and in the maximum growth rate estimates is the parameter of paramount importance in the linear instability problem of ideal flows on a sphere. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Barotropic instability in the tropical cyclone outer region THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 641 2009Jiayi PengAbstract The growth of asymmetric perturbations and their interactions with the symmetric flow are investigated for wind profiles in a tropical cyclone with instability in its outer region. Three tangential wind profiles are examined: TC1, a strong barotropic instability profile in the outer region; TC2, a stable wind profile; and TC3, a weaker instability profile comparing to TC1 with a larger distance between the inner negative and the outer positive vorticity gradient centres. An eigenvalue analysis indicates that azimuthal wave-number two is the most unstable mode in both TC1 and TC3, with an e-folding time-scale of about 1 and 9 days, respectively. Numerical simulations using a linear barotropic model, with an initial asymmetry specified in the outer region, confirm the eigenvalue analysis. A mechanism is provided to explain the difference between simulations in TC1 and TC2. In both the stable and unstable case, an inner asymmetry is induced by the initial outer asymmetry acting on the symmetric vorticity gradient. Subsequently, the newly generated inner asymmetry feeds back positively to the outer asymmetry with the unstable profile. Because of this positive feedback, the inner and the outer asymmetries maintain an up-shear phase tilting, leading to a continuous energy transfer from the symmetric flow to the asymmetric perturbation. In the stable TC2, the inner asymmetry could not amplify the outer initial asymmetry as there is no basic-state radial vorticity gradient there. Also due to this feedback process, disturbances grow faster where the (absolute) basic-state vorticity gradients are large. Therefore, the position of an initial disturbance plays a minor role in determining the outcome of the system. Simulations with a nonlinear barotropic model and a primitive equation model further confirm the significant weakening of the maximum tangential wind due to the positive feedback process in TC1. Simulations for TC3 show a smaller change of the symmetric tangential wind, as expected. Copyright © 2009 Royal Meteorological Society [source] ## A finite-strain quadrilateral shell element based on discrete Kirchhoff,Love constraints INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2005Pedro M. A. AreiasAbstract This paper improves the 16 degrees-of-freedom quadrilateral shell element based on pointwise Kirchhoff,Love constraints and introduces a consistent large strain formulation for this element. The model is based on classical shell kinematics combined with continuum constitutive laws. The resulting element is valid for large rotations and displacements. The degrees-of-freedom are the displacements at the corner nodes and one rotation at each mid-side node. The formulation is free of enhancements, it is almost fully integrated and is found to be immune to locking or unstable modes. The patch test is satisfied. In addition, the formulation is simple and amenable to efficient incorporation in large-scale codes as no internal degrees-of-freedom are employed, and the overall calculations are very efficient. Results are presented for linear and non-linear problems. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## Instabilities of Boussinesq models in non-uniform depth INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2009F. LøvholtAbstract The von Neumann method for stability analysis of linear waves in a uniform medium is a widely applied procedure. However, the method does not apply to stability of linear waves in a variable medium. Herein we describe instabilities due to variable depth for different Boussinesq equations, including the standard model by Peregrine and the popular generalization by Nwogu. Eigenmodes are first found for bathymetric features on the grid scale. For certain combinations of Boussinesq formulations and bottom profiles stability limits are found in closed form, otherwise numerical techniques are used for the eigenvalue problems. Naturally, the unstable modes in such settings must be considered to be as much a result of the difference method as of the underlying differential (Boussinesq) equations. Hence, modes are also computed for smooth depth profiles that are well resolved. Generally, the instabilities do not vanish with refined resolution. In some cases convergence is observed and we thus have indications of unstable solutions of the differential equations themselves. The stability properties differ strongly. While the standard Boussinesq equations seem perfectly stable, all the other formulations do display unstable modes. In most cases the instabilities are linked to steep bottom gradients and small grid increments. However, while a certain formulation, based on velocity potentials, is very prone to instability, the Boussinesq equations of Nwogu become unstable only under quite demanding conditions. Still, for the formulation of Nwogu, instabilities are probably inherent in the differential equations and are not a result of the numerical model. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Stability and slightly supercritical oscillatory regimes of natural convection in a 8:1 cavity: solution of the benchmark problem by a global Galerkin method INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2004Alexander Yu.Abstract The global Galerkin method is applied to the benchmark problem that considers an oscillatory regime of convection of air in a tall two-dimensional rectangular cavity. The three most unstable modes of the linearized system of the Boussinesq equations are studied. The converged values of the critical Rayleigh numbers together with the corresponding oscillation frequencies are calculated for each mode. The oscillatory flow regimes corresponding to each of the three modes are approximated asymptotically. No direct time integration is applied. Good agreement with the previously published results obtained by solution of the time-dependent Boussinesq equations is reported. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Linear instability of ideal flows on a sphere MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2009Yuri N. SkibaAbstract A unified approach to the normal mode instability study of steady solutions to the vorticity equation governing the motion of an ideal incompressible fluid on a rotating sphere is considered. The four types of well-known solutions are considered, namely, the Legendre-polynomial (LP) flows, Rossby,Haurwitz (RH) waves, Wu,Verkley (WV) waves and modons. A conservation law for disturbances to each solution is derived and used to obtain a necessary condition for its exponential instability. By these conditions, Fjörtoft's (Tellus 1953; 5:225,230) average spectral number of the amplitude of an unstable mode must be equal to a special value. In the case of LP flows or RH waves, this value is related only with the basic flow degree. For the WV waves and modons, it depends both on the basic flow degree and on the spectral distribution of the mode energy in the inner and outer regions of the flow. Peculiarities of the instability conditions for different types of modons are discussed. The new instability conditions specify the spectral structure of growing disturbances localizing them in the phase space. For the LP flows, this condition complements the well-known Rayleigh,Kuo and Fjörtoft conditions related to the zonal flow profile. Some analytical and numerical examples are considered. The maximum growth rate of unstable modes is also estimated, and the orthogonality of any unstable, decaying and non-stationary mode to the basic flow is shown in the energy inner product. The analytical instability results obtained here can also be applied for testing the accuracy of computational programs and algorithms used for the numerical stability study. It should be stressed that Fjörtoft's spectral number appearing both in the instability conditions and in the maximum growth rate estimates is the parameter of paramount importance in the linear instability problem of ideal flows on a sphere. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## A kinetic approach to cosmic-ray-induced streaming instability at supernova shocks MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Issue 4 2009E. AmatoABSTRACT We show that a purely kinetic approach to the excitation of waves by cosmic rays in the vicinity of a shock front leads to predict the appearance of a non-Alfvénic fast-growing mode which has the same dispersion relation as that previously found by Bell in 2004 by treating the plasma in the magnetohydrodynamic approximation. The kinetic approach allows us to investigate the dependence of the dispersion relation of these waves on the microphysics of the current which compensates the cosmic ray flow. We also show that a resonant and a non-resonant mode may appear at the same time and one of the two may become dominant on the other depending on the conditions in the acceleration region. We discuss the role of the unstable modes for magnetic field amplification and particle acceleration in supernova remnants at different stages of the remnant evolution. [source] ## A simple model of coupled synoptic waves in the land surface and atmosphere of the northern Sahel THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 637 2008Douglas J. ParkerAbstract A simple dynamic model is developed to describe the observed interactions between the atmosphere and the soil moisture patterns of the northern Sahel. In the model, the atmosphere follows quasi-geostrophic dynamics, while land-atmosphere coupling is described by simple linear relationships. Dry surfaces heat the atmospheric boundary layer, while wet surfaces cool the boundary layer, relative to the equilibrium state of the atmosphere and land surface. In turn, cloud processes, which are assumed to maximise in the cool, humid phase of an atmospheric disturbance, cool the land surface through wetting (rainfall) and reduction of the incoming solar flux. These assumptions lead to a linear system which can be solved numerically to obtain modal solutions, and the adjoints (optimal excitation) of these. Moist convective influences on the atmospheric state are not explored in detail. The coupling with the land surface leads to the existence of unstable modes, which do not exist in the atmosphere-only part of the system. Solutions can be easterly or westerly propagating, according to wave number, with the longer waves tending to be easterly. Propagation relies on a favourable configuration between the atmospheric and soil moisture anomalies: easterly propagation requires the surface temperature pattern to be shifted to the east of the atmospheric temperature pattern. In contrast, optimal excitation of the fastest-growing mode occurs when the atmospheric pattern has a thermal anomaly lying to the east of a strong surface temperature (and moisture) anomaly. These results have value for weather prediction, and indicate the usefulness of soil moisture data for forecasters. Copyright © 2008 Royal Meteorological Society [source] |