Equatorial water waves with underlying currents in the f-plane approximation

We present an exact solution to the governing equations for equatorial geophysical water waves which admit a constant underlying zonal current and a variable meridional current in the f-plane approximation. The solution is three-dimensional, nonlinear and explicit in Lagrangian formulation. We provide an analysis of the mean flow velocities and the related mass transport.     

Finite element methods for an acoustic well-logging problem associated with a porous medium saturated by a two-phase immiscible fluid

A mathematical model is presented concerning wave propagation in a domain that arises in geophysical well-logging problems. The domain consists of a borehole Ω_f surrounded by a porous medium Ω_p. Ω_f is filled with a compressible inviscid fluid, and Ω_p is saturated by a two-phase immiscible fluid. Absorbing boundary conditions for artificial boundaries and boundary conditions on the interface between Ω_f and Ω_p are used. The existence and uniqueness theorems are stated for the resulting initial-boundary value problem. Stability and convergence estimates for a finite element method are also studied. © 1993 John Wiley & Sons, Inc.     

Some new solutions of the problems of wave propagation in a two-layer fluid

We present new analytic solutions of the problem of wave propagation in a continuously stratified fluid in the Boussinesq approximation. We study the propagation of internal waves in an ideal fluid in systems of homogeneous-layer/continuously stratified layer and homogeneous-layer/continuously stratified half-space type. We obtain the dispersion equations and study several limiting cases. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, Vol. 27, 1997, pp. 132–137.     

Internal Kelvin waves in a two-layer liquid model

The subinertial internal Kelvin wave solutions of a linearized system of the ocean dynamics equations for a semi-infinite two-layer f-plane model basin of constant depth bordering a straight, vertical coast are imposed. A rigid lid surface condition and no-slip wall boundary condition are imposed. Some trapped wave equations are presented and approximate solutions using an asymptotic method are constructed. In the absence of bottom friction, the solution consists of a frictionally modified Kelvin wave and a vertical viscous boundary layer. With a no-slip bottom boundary condition, the solution consists of a modified Kelvin wave, two vertical viscous boundary layers, and a large cross-section scale component. The numerical solutions for Kelvin waves are obtained for model parameters that take account of a joint effect of lateral viscosity, bottom friction, and friction between the layers.     

The Heat Equation with Initial Data Corrupted by Measurement Error and Missing Data

In this work the Cauchy problem for the one-dimensional heat equation is considered. In contrast to existing literature it is assumed that the initial state f is unknown and that information regarding f is obtained by some process of measurement. To enhance realism, both measurement errors and missing data are allowed for. Under assumptions on f in the Fourier-domain first an approximation to f is derived from the data by means of a novel uncertainty principle. Then, it is studied how this perturbation in the initial state propagates with time.      

On uniform approximation by harmonic and almost harmonic vector fields

Three- dimensional analogs of rational uniform approximation in \mathbbC \mathbb{C} are considered. These analogs are related to approximation properties of harmonic (i. e., curl-free and solenoidal) vector fields. The usual uniform approximation by fields harmonic near a given compact set K ⊂ \mathbbR³ \mathbb{R}^3 is compared with the uniform approximation by smooth fields whose curls and divergences tends to zero uniformly on K. A similar two-dimensional modification of the uniform approximation by functions f that are complex analytic near a given compact set K ⊂ \mathbbC \mathbb{C} (when f is assumed to be in C ¹ with [`(?)] f\bar \partial {\kern 1pt}f small on K) results in a problem equivalent to the original one. In the three-dimensional settings, the two problems (of harmonic and of almost harmonic approximation) are different. The first problem is nonlocal whereas the second one is local (i. e., an analog of the Bishop theorem on the locality of R(K) is still valid for almost harmonic approximation). Almost curl-free approximation is also considered. Bibliography: 7 titles.     

Greedy approximations for minimum submodular cover with submodular cost

It is well-known that a greedy approximation with an integer-valued polymatroid potential function f is H(γ)-approximation of the minimum submodular cover problem with linear cost where γ is the maximum value of f over all singletons and H(γ) is the γ-th harmonic number. In this paper, we establish similar results for the minimum submodular cover problem with a submodular cost (possibly nonlinear) and/or fractional submodular potential function f.     

On linear and nonlinear Rossby waves in an ocean

This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode (n=1) analysis. An extension of the higher modes (n?2) of this work will be made in a subsequent paper.     

A Note on the Theory of Low-Frequency Waves in a Rotating Stratified Channel

It is found that the possible low-frequency, quasigeostrophic motions in a rotating, stratified channel with a wave-maker at one end include: (i) “standing waves” whose amplitudes are damped exponentially away from the forcing, and (ii) baroclinic internal Kelvin waves, trapped to the right-hand wall when facing in the direction of phase propagation. The Kelvin waves are excited only if the wave-maker transfers mean energy to the fluid. The standing waves, on the other hand, carry no energy and thus serve mainly to provide continuity between the wave-maker and the fluid. When the bottom of the channel is inclined to the horizontal by a small angle θ, topographic oscillations are possible. These waves behave like topographic Rossby waves if the forcing frequency is greater than sN and if the ratio HN/fL is small, where s=tanθ, θ is the angle of the bottom slope, L is the width of the channel, H is the mean depth, f is the Coriolis parameter, and N is the Brunt-Vaisala (or buoyancy) frequency. It is determined that topographic Rossby waves cannot exist in the channel if HN/fL?0.65. If the wave-maker frequency is smaller than sN, and if HN/fL~1, the topographic oscillations become bottom-trapped, decaying away from the bottom boundary in a distance ~λf/N, where λ is the horizontal wavelength. The phase and energy of the bottom-trapped wave both move to the left of an observer who is facing shallow water. The Kelvin waves are basically unchanged when the bottom is inclined if their down-channel wavelength is large relative to the width of the channel. The standing oscillations of the flat-bottom case exist as complex-horizontal-wave-number solutions to the topographic wave dispersion relation. Although these waves have propagating phase when s≠0, they are still trapped to the forcing, and do not transfer net energy from the wave-maker to the fluid. All three modes are required to solve the general matching conditions for an arbitrary wave-maker when the channel has a sloping bottom.     

Forward speed motions of a submerged body. The cauchy problem

We consider the linearized problem for the ideal fluid flow induced by the horizontal motion of a fully immersed body. The system of equations is made up of an elliptic problem (P) and an initialvalue problem (R) which are coupled by a pseudo-differential operator T. We define a regularized Cauchy problem (R?) using the Yosida approximation of T; we give energy and wave resistance estimates and finally we obtain existence uniqueness and regularity of the weak solution of (R) by taking the limit when ? goes to zero.     

A confinement result on a quasi-geostrophic flow in thef-plane

We study the behavior of a quasi-geostrophic flow in thef-plane. We consider a positive initial potential vorticity with a compact support and we bound the growth in time of its support. We prove also that a fluid particle cannot go fast away from the initial position.     

Constrained minimax approximation and optimal preconditioned for Toeplitz matrices

A good preconditioner is extremely important in order for the conjugate gradients method to converge quickly. In the case of Toeplitz matrices, a number of recent studies were made to relate approximation of functions to good preconditioners. In this paper, we present a new result relating the quality of the Toeplitz preconditionerC for the Toeplitz matrixT to the Chebyshev norm (f– g)/f, wheref and g are the generating functions forT andC, respectively. In particular, the construction of band-Toeplitz preconditioners becomes a linear minimax approximation problem. The case whenf has zeros (but is nonnegative) is especially interesting and the corresponding approximation problem becomes constrained. We show how the Remez algorithm can be modified to handle the constraints. Numerical experiments confirming the theoretical results are presented.     

A method for rational Chebyshev approximation of rational functions on the unit disk and on the unit interval

A method for construction of CF approximants in some cases of rational approximation of a rational function f on the unit disk and on the unit interval is presented. The inverted square root of the greatest positive eigenvalue and a corresponding eigenvector of an eigenvalue problem defined by the coefficients of f gives the solution.     

Some exact pseudo-plane solutions of the first kind for the Navier-Stokes equations

We study pseudo-plane flows of the first kind generated by a stream function =f(x, z, t) +g(y, z, t) which are generalized Beltrami flows in every plane parallel to thexy-plane. This problem is solved completely resulting in several new families of exact solutions for the Navier-Stokes equations.     

Resonant triad model for studying evolution of the energy spectrum among a large number of internal waves

Internal waves are generally accepted to be responsible for a large fraction of mixing in the deep ocean. Internal waves interact nonlinearly with one another, exchanging energy among themselves to create the background internal wave spectrum. The most important mechanism resulting in the transfer of energy from one wave to another is believed to be resonant triad interactions. In this paper we consider a large number of resonantly interacting triads in order to investigate the evolution of the energy spectrum due to solely resonant triad interactions. To this end we solve the evolution equations for a large number of resonant triads to determine the temporal evolution of the energy distribution among the various possible wave numbers and frequencies. Our model involves internal waves with frequencies spanning the range of possible frequencies, i.e., between a maximum of the buoyancy frequency N for horizontal wave vectors (vertical motion) to a minimum of the inertial frequency f for vertical wave vectors (horizontal motion) [two limiting cases]. Because of the inclusion of high-frequency waves we cannot make the hydrostatic approximation. We investigate the evolution of the wave’s amplitudes to predict the evolution of the internal wave energy spectrum.     

Transient planetary waves in semi-bounded channels extending along a meridian

The unsteady-state boundary-value problem of the propagation of planetary waves in semi-bounded channels running north-south is studied in the 13-plane approximation. An explicit solution is obtained and the behaviour of normal transient waves at long times is investigated.     

A-Statistical extension of the Korovkin type approximation theorem

In this paper, using the concept ofA-statistical convergence which is a regular (non-matrix) summability method, we obtain a general Korovkin type approximation theorem which concerns the problem of approximating a functionf by means of a sequenceL _n f of positive linear operators.     

On barotropic trapped wave solutions with no-slip boundary conditions

Barotropic trapped wave solutions of a linearized system of the ocean dynamics equations are described for a semi-infinite, f-plane model basin of constant depth bordering a straight, vertical coast, for some “typical” values of the model parameters. No-slip boundary conditions are considered. When the wave length is shorter than the Rossby deformation radius, the main features of the wave solutions are as follows: the Kelvin wave exponential offshore decay scale essentially decreases as the wave length decreases, and an additional wave solution propagating in the opposite direction appears.     

Markov chain Monte Carlo Using an Approximation

This article presents a method for generating samples from an unnormalized posterior distribution f(·) using Markov chain Monte Carlo (MCMC) in which the evaluation of f(·) is very difficult or computationally demanding. Commonly, a less computationally demanding, perhaps local, approximation to f(·) is available, say f^**_x(·). An algorithm is proposed to generate an MCMC that uses such an approximation to calculate acceptance probabilities at each step of a modified Metropolis–Hastings algorithm. Once a proposal is accepted using the approximation, f(·) is calculated with full precision ensuring convergence to the desired distribution. We give sufficient conditions for the algorithm to converge to f(·) and give both theoretical and practical justifications for its usage. Typical applications are in inverse problems using physical data models where computing time is dominated by complex model simulation. We outline Bayesian inference and computing for inverse problems. A stylized example is given of recovering resistor values in a network from electrical measurements made at the boundary. Although this inverse problem has appeared in studies of underground reservoirs, it has primarily been chosen for pedagogical value because model simulation has precisely the same computational structure as a finite element method solution of the complete electrode model used in conductivity imaging, or “electrical impedance tomography.” This example shows a dramatic decrease in CPU time, compared to a standard Metropolis–Hastings algorithm.