Inertial wave

About: Inertial wave is a research topic. Over the lifetime, 1710 publications have been published within this topic receiving 40906 citations.

Mechanics of continua and wave dynamics

Abstract: I Theory of Elasticity.- 1. The Main Types of Strain in Elastic Solids.- 1.1 Equations of Linear Elasticity Theory.- 1.1.1 Hooke's Law.- 1.1.2 Differential Form of Hooke's Law. Principle of Superposition.- 1.2 Homogeneous Strains.- 1.2.1 An Elastic Body Under the Action of Hydrostatic Pressure.- 1.2.2 Longitudinal Strain with Lateral Displacements Forbidden.- 1.2.3 Pure Shear.- 1.3 Heterogeneous Strains.- 1.3.1 Torsion of a Rod.- 1.3.2 Bending of a Beam.- 1.3.3 Shape of a Beam Under Load.- 1.4 Exercises.- 2. Waves in Rods, Vibrations of Rods.- 2.1 Longitudinal Waves.- 2.1.1 Wave Equation.- 2.1.2 Harmonic Waves.- 2.2 Reflection of Longitudinal Waves.- 2.2.1 Boundary Conditions.- 2.2.2 Wave Reflection.- 2.3 Longitudinal Oscillations of Rods.- 2.4 Torsional Waves in a Rod. Torsional Vibrations.- 2.5 Bending Waves in Rods.- 2.5.1 The Equation for Bending Waves.- 2.5.2 Boundary Conditions. Harmonic Waves.- 2.5.3 Reflection of Waves. Bending Vibrations.- 2.6 Wave Dispersion and Group Velocity.- 2.6.1 Propagation of Nonharmonic Waves.- 2.6.2 Propagation of Narrow-Band Disturbances.- 2.7 Exercises.- 3. General Theory of Stress and Strain.- 3.1 Description of the State of a Deformed Solid.- 3.1.1 Stress Tensor.- 3.1.2 The Strain Tensor.- 3.1.3 The Physical Meaning of the Strain Tensor's Components.- 3.2 Equations of Motion for a Continuous Medium.- 3.2.1 Derivation of the Equation of Motion.- 3.2.2 Strain-Stress Relation. Elasticity Tensor.- 3.3 The Energy of a Deformed Body.- 3.3.1 The Energy Density.- 3.3.2 The Number of Independent Components of the Elasticity Tensor.- 3.4 The Elastic Behaviour of Isotropic Bodies.- 3.4.1 The Generalized Hooke's Law for an Isotropic Body.- 3.4.2 The Relationship Between Lame's Constants and E and v.- 3.4.3 The Equations of Motion for an Isotropic Medium.- 3.5 Exercises.- 4. Elastic Waves in Solids.- 4.1 Free Waves in a Homogeneous Isotropic Medium.- 4.1.1 Longitudinal and Transverse Waves.- 4.1.2 Boundary Conditions for Elastic Waves.- 4.2 Wave Reflection at a Stress-Free Boundary.- 4.2.1 Boundary Conditions.- 4.2.2 Reflection of a Horizontally Polarized Wave.- 4.2.3 The Reflection of Vertically Polarized Waves.- 4.2.4 Particular Cases of Reflection.- 4.2.5 Inhomogeneous Waves.- 4.3 Surface Waves.- 4.3.1 The Rayleigh Wave.- 4.3.2 The Surface Love Wave.- 4.3.3 Some Features of Love's Waves.- 4.4 Exercises.- 5. Waves in Plates.- 5.1 Classification of Waves.- 5.1.1 Dispersion Relations.- 5.1.2 Symmetric and Asymmetric Modes.- 5.1.3 Cut-Off Frequencies of the Modes.- 5.1.4 Some Special Cases.- 5.2 Normal Modes of the Lowest Order.- 5.2.1 Quasi-Rayleigh Waves at the Plate's Boundaries.- 5.2.2 The Young and Bending Waves.- 5.3 Equations Describing the Bending of a Thin Plate.- 5.3.1 Thin Plate Approximation.- 5.3.2 Sophie Germain Equation.- 5.3.3 Bending Waves in a Thin Plate.- 5.4 Exercises.- II Fluid Mechanics.- 6. Basic Laws of Ideal Fluid Dynamics.- 6.1 Kinematics of Fluids.- 6.1.1 Eulerian and Lagrangian Representations of Fluid Motion.- 6.1.2 Transition from One Representation to Another.- 6.1.3 Convected and Local Time Derivatives.- 6.2 System of Equations of Hydrodynamics.- 6.2.1 Equation of Continuity.- 6.2.2 The Euler Equation.- 6.2.3 Completeness of the System of Equations.- 6.3 The Statics of Fluids.- 6.3.1 Basic Equations.- 6.3.2 Hydrostatic Equilibrium. Vaisala Frequency.- 6.4 Bernoulli's Theorem and the Energy Conservation Law.- 6.4.1 Bernoulli's Theorem.- 6.4.2 Some Applications of Bernoulli's Theorem.- 6.4.3 The Bernoulli Theorem as a Consequence of the Energy-Conservation Law.- 6.4.4 Energy Conservation Law in the General Case of Unsteady Flow.- 6.5 Conservation of Momentum.- 6.5.1 The Specific Momentum Flux Tensor.- 6.5.2 Euler's Theorem.- 6.5.3 Some Applications of Euler's Theorem.- 6.6 Vortex Flows of Ideal Fluids.- 6.6.1 The Circulation of Velocity.- 6.6.2 Kelvin's Circulation Theorem.- 6.6.3 Helmholtz Theorems.- 6.7 Exercises.- 7. Potential Flow.- 7.1 Equations for a Potential Flow.- 7.1.1 Velocity Potential.- 7.1.2 Two-Dimensional Flow. Stream Function.- 7.2 Applications of Analytical Functions to Problems of Hydrodynamics.- 7.2.1 The Complex Flow Potential.- 7.2.2 Some Examples of Two-Dimensional Flows.- 7.2.3 Conformal Mapping.- 7.3 Steady Flow Around a Cylinder.- 7.3.1 Application of Conformal Mapping.- 7.3.2 The Pressure Coefficient.- 7.3.3 The Paradox of d'Alembert and Euler.- 7.3.4 The Flow Around a Cylinder with Circulation.- 7.4 Irrotational Flow Around a Sphere.- 7.4.1 The Flow Potential and the Particle Velocity.- 7.4.2 The Induced Mass.- 7.5 Exercises.- 8. Flows of Viscous Fluids.- 8.1 Equations of Flow of Viscous Fluid.- 8.1.1 Newtonian Viscosity and Viscous Stresses.- 8.1.2 The Navier-Stokes Equation.- 8.1.3 The Viscous Force.- 8.2 Some Examples of Viscous Fluid Flow.- 8.2.1 Couette Flow.- 8.2.2 Plane Poiseuille Flow.- 8.2.3 Poiseuille Flow in a Cylindrical Pipe.- 8.2.4 Viscous Fluid Flow Around a Sphere.- 8.2.5 Stokes' Formula for Drag.- 8.3 Boundary Layer.- 8.3.1 Viscous Waves.- 8.3.2 The Boundary Layer. Qualitative Considerations.- 8.3.3 Prandl's Equation for a Boundary Layer.- 8.3.4 Approximate Theory of a Boundary Layer in a Simple Case.- 8.4 Exercises.- 9. Elements of the Theory of Turbulence.- 9.1 Qualitative Considerations. Hydrodynamic Similarity.- 9.1.1 Transition from a Laminar to Turbulent Flow.- 9.1.2 Similar Flows.- 9.1.3 Dimensional Analysis and Similarity Principle.- 9.1.4 Flow Around a Cylinder at Different Re.- 9.2 Statistical Description of Turbulent Flows.- 9.2.1 Reynolds' Equation for Mean Flow.- 9.2.2 Turbulent Viscosity.- 9.2.3 Turbulent Boundary Layer.- 9.3 Locally Isotropic Turbulence.- 9.3.1 Properties of Developed Turbulence.- 9.3.2 Statistical Properties of Locally Isotropic Turbulence.- 9.3.3 Kolmogorov's Similarity Hypothesis.- 9.4 Exercises.- 10. Surface and Internal Waves in Fluids.- 10.1 Linear Equations for Waves in Stratified Fluids.- 10.1.1 Linearization of the Hydrodynamic Equations.- 10.1.2 Linear Boundary Conditions.- 10.1.3 Equations for an Incompressible Fluid.- 10.2 Surface Gravity Waves.- 10.2.1 Basic Equations.- 10.2.2 Harmonic Waves.- 10.2.3 Shallow- and Deep-Water Approximations.- 10.2.4 Wave Energy.- 10.3 Capillary Waves.- 10.3.1 "Pure" Capillary Waves.- 10.3.2 Gravity-Capillary Surface Waves.- 10.4 Internal Gravity Waves.- 10.4.1 Introductory Remarks.- 10.4.2 Basic Equation for Internal Waves. Boussinesq Approximation.- 10.4.3 Waves in an Unlimited Medium.- 10.5 Guided Propagation of Internal Waves.- 10.5.1 Qualitative Analysis of Guided Propagation.- 10.5.2 Simple Model of an Oceanic Waveguide.- 10.5.3 Surface Mode. "Rigid Cover" Condition.- 10.5.4 Internal Modes.- 10.6 Exercises.- 11. Waves in Rotating Fluids.- 11.1 Inertial (Gyroscopic) Waves.- 11.1.1 The Equation for Waves in a Homogeneous Rotating Fluid.- 11.1.2 Plane Harmonic Inertial Waves.- 11.1.3 Waves in a Fluid Layer. Application to Geophysics.- 11.2 Gyroscopic-Gravity Waves.- 11.2.1 General Equations. The Simplest Model of a Medium.- 11.2.2 Classification of Wave Modes.- 11.2.3 Gyroscopic-Gravity Waves in the Ocean.- 11.3 The Rossby Waves.- 11.3.1 The Tangent of ?-Plane Approximation.- 11.3.2 The Barotropic Rossby Waves.- 11.3.3 Joint Discussion of Stratification and the ?-Effect.- 11.3.4 The Rossby Waves in the Ocean.- 11.4 Exercises.- 12. Sound Waves.- 12.1 Plane Waves in Static Fluids.- 12.1.1 The System of Linear Acoustic Equations.- 12.1.2 Plane Waves.- 12.1.3 Generation of Plane Waves. Inhomogeneous Waves.- 12.1.4 Sound Energy.- 12.2 Sound Propagation in Inhomogeneous Media.- 12.2.1 Plane Wave Reflection at the Interface of Two Homogeneous Media.- 12.2.2 Some Special Cases. Complete Transparency and Total Reflection.- 12.2.3 Energy and Symmetry Considerations.- 12.2.4 A Slowly-Varying Medium. Geometrical-Acoustics Approximation.- 12.2.5 Acoustics Equations for Moving Media.- 12.2.6 Guided Propagation of Sound.- 12.3 Spherical Waves.- 12.3.1 Spherically-Symmetric Solution of the Wave Equation.- 12.3.2 Volume Velocity or the Strength of the Source. Reaction of the Medium.- 12.3.3 Acoustic Dipole.- 12.4 Exercises.- 13. Magnetohydrodynamics.- 13.1 Basic Concepts of Magnetohydrodynamics.- 13.1.1 Fundamental Equations.- 13.1.2 The Magnetic Pressure. Freezing of the Magnetic Field in a Fluid.- 13.1.3 The Poiseuille (Hartmann) Flow.- 13.2 Magnetohydrodynamic Waves.- 13.2.1 Alfven Waves.- 13.2.2 Magnetoacoustic Waves.- 13.2.3 Fast and Slow Magnetoacoustical Waves.- 13.3 Exercises.- 14. Nonlinear Effects in Wave Propagation.- 14.1 One-Dimensional Nonlinear Waves.- 14.1.1 The Nonlinearity Parameter.- 14.1.2 Model Equation. Generation of Second Harmonics.- 14.1.3 The Riemann Solution. Shock Waves.- 14.1.4 Dispersive Media. Solitons.- 14.2 Resonance Wave Interaction.- 14.2.1 Conditions of Synchronism.- 14.2.2 The Method of Slowly-Varying Amplitudes.- 14.2.3 Multiwave Interaction.- 14.2.4 Nonlinear Dispersion.- 14.3 Exercises.- Appendix: Tensors.- Bibliographical Sketch.

A Three-Dimensional Balance Theory for Rapidly Rotating Vortices

Abstract: A three-dimensional balance formulation for rapidly rotating vortices, such as hurricanes, is presented. The asymmetric balance (AB) theory represents a new mathematical framework for studying the slow evolution of rapidly rotating fluid systems. The AB theory is valid for large Rossby number; it makes no formal restriction on the magnitude of the divergence or vertical advection, which need not be small. The AB is an ordered expansion in the square of the ratio of orbital to inertial frequencies, the square of a local Rossby number. The approximation filters gravity and inertial waves from the system. Advantage is taken of the weak asymmetries near the vortex care as well as the tendency for low azimuthal wavenumber asymmetries to dominate. Linearization about a symmetric balanced vortex allows the three-dimensional asymmetric dynamics to be deduced properly. The AB formulation has a geopotential tendency equation with a three-dimensional elliptic operator. The AB system has a uniformly valid co...

Inertial waves in a rotating spherical shell : attractors and asymptotic spectrum

Abstract: We investigate the asymptotic properties of inertial modes confined in a spherical shell when viscosity tends to zero. We first consider the mapping made by the characteristics of the hyperbolic equation (Poincare's equation) satisfied by inviscid solutions. Charac- teristics are straight lines in a meridional section of the shell, and the mapping shows that, generically, these lines converge towards a periodic orbit which acts like an attrac- tor (the associated Lyapunov exponent is always negative or zero). We show that these attractors exist in bands of frequencies the size of which decreases with the number of reflection points of the attractor. At the bounding frequencies the associated Lyapunov exponent is generically either zero or minus infinity. We further show that for a given frequency the number of coexisting attractors is finite. We then examine the relation between this characteristic path and eigensolutions of the inviscid problem and show that in a purely two-dimensional problem, convergence towards an attractor means that the associated velocity field is not square-integrable. We give arguments which generalize this result to three dimensions. Then, using a sphere immersed in a fluid filling the whole space, we study the critical latitude singularity and show that the velocity field diverges as 1/ √ d, d being the distance to the characteristic grazing the inner sphere. We then consider the viscous problem and show how viscosity transforms singularities into internal shear layers which in general betray an attractor expected at the eigenfre- quency of the mode. Investigating the structure of these shear layers, we find that they are nested layers, the thinnest and most internal layer scaling with E 1/3 -scale, E being the Ekman number; for this latter layer, we give its analytical form and show its simi- larity to vertical 1 -shear layers of steady flows. Using an inertial wave packet traveling around an attractor, we give a lower bound on the thickness of shear layers and show how eigenfrequencies can be computed in principle. Finally, we show that as viscosity decreases, eigenfrequencies tend towards a set of values which is not dense in (0,2), contrary to the case of the full sphere ( is the angular velocity of the system). Hence, our geometrical approach opens the possibility of describing the eigenmodes and eigenvalues for astrophysical/geophysical Ekman numbers (10 −10 − 10 −20 ), which are out of reach numerically, and this for a wide class of containers.

A Conceptual View on Inertial Oscillations and Nocturnal Low-Level Jets

Abstract: In the present work Blackadar’s concept of nocturnal inertial oscillations is extended. Blackadar’s concept describes frictionless inertial oscillations above the nocturnal inversion layer. The current work includes frictional effects within the nocturnal boundary layer. It is shown that the nocturnal wind speed profile describes an oscillation around the nocturnal equilibrium wind vector, rather than around the geostrophic wind vector (as in the Blackadar case). By using this perspective, continuous time-dependent wind profiles are predicted. As such, information on both the height and the magnitude of the nocturnal low-level jet is available as a function of time. Preliminary analysis shows that the proposed extension performs well in comparison with observations when a simple Ekman model is used to represent the equilibrium state in combination with a realistic initial velocity profile. In addition to jet dynamics, backward inertial oscillations are predicted at lower levels close to the surface, which also appear to be present in observations. The backward oscillation forms an important mechanism behind weakening low-level winds during the afternoon transition. Both observational and theoretical modeling studies are needed to explore this phenomenon further.