Linear Waves in Midlatitudes on the Rotating Spherical Earth

2000

Zonally propagating wave solutions of the linearized shallow water equations (LSWE) in a zonal channel on the rotating spherical earth are constructed from numerical solutions of the eigenvalue equation for the meridional variation of the waves' amplitudes, which also yield the phase speeds of these waves. An approximate Schrödinger equation whose potential depends on one parameter only is derived

Dynamics of Atmospheres and Oceans, 2012

The Linearized Shallow Water Equations (LSWE) are formulated on an oblate spheroid (ellipsoid of revolution) that approximates Earth's geopotential surface more accurately than a sphere. The application of a previously developed invariant theory (i.e. applied to an arbitrary smooth surface) to oblate spheroid yields exact equations for the meridional structure function of zonally propagating wave solutions such as Planetary (Rossby) waves and Inertia-Gravity (Poinacré) waves. Approximate equations (that are accurate to first order only of the spheroid's eccentricity) are derived for the meridional structure of Poincaré (Inertia-Gravity) and Rossby (Planetary) and the solutions of these equations yield expressions in terms of prolate spheroidal wave functions. The eigenvalues of the approximate equations provide explicit expressions for the dispersion relations of these waves. Comparing our expressions for the dispersion relations on a spheroid to the known solutions of the same problem on a sphere shows that the relative error in the dispersion relations on a sphere is of the order of the square of spheroid's eccentricity (i.e. about 0.006 for Earth) for both Poincaré and Rossby waves.

Journal of the Atmospheric Sciences, 2015

The general 1D theory of waves propagating on a zonally varying flow is developed from basic wave theory, and equations are derived for the variation of wavenumber and energy along ray paths. Different categories of behavior are found, depending on the sign of the group velocity cg and a wave property B. For B positive, the wave energy and the wavenumber vary in the same sense, with maxima in relative easterlies or westerlies, depending on the sign of cg. Also the wave accumulation of Webster and Chang occurs where cg goes to zero. However, for B negative, they behave in opposite senses and wave accumulation does not occur. The zonal propagation of the gravest equatorial waves is analyzed in detail using the theory. For nondispersive Kelvin waves, B reduces to 2, and an analytic solution is possible. For all the waves considered, B is positive, except for the westward-moving mixed Rossby–gravity (WMRG) wave, which can have negative B as well as positive B. Comparison is made between...

J Fluid Mech, 2008

The Kelvin wave is the lowest eigenmode of Laplace's Tidal Equation and is widely observed in both the ocean and the atmosphere. In this work, we neglect mean currents and continuous stratification, but instead include the full effects of the earth's sphericity and the wave dispersion it induces in a one-and-a-half-layer model. In the first part, we derive a new asymptotic approximations for linear Kelvin waves on the sphere when (Lamb's parameter/nondimensional reciprocal depth) and integer zonal wavenumber s are of similar magnitude; we show that for large s, the Kelvin wave is equatorially-confined even in the barotropic limit (= 0). In the second part, through a mix of perturbation theory and numerical computations using a Fourier/Newton iteration/continuation method, we show that for sufficiently small amplitude, there are Kelvin traveling waves (cnoidal waves). As the amplitude increases, the branch of traveling waves terminates in a so-called "corner wave" with a discontinuous first derivative. All waves larger than the corner wave evolve to fronts and break. The singularity is a point singularity in which only the longitudinal derivative is discontinuous. As we solve the nonlinear shallow water equations on the sphere with increasing ("Lamb's parameter"), dispersion weakens, the amplitude of the corner wave decreases rapidly, and the longitudinal profile of the corner wave narrows dramatically.

2003

The linear instability of divergent perturbations that evolve on a cos 2 mean steady zonal jet embedded in a zonal channel on the ␤ plane and on a rotating sphere is studied for zonally propagating wavelike perturbations of the shallow-water equations. The complex phase speeds result from the imposition of the no-flow boundary conditions at the channel walls on the numerical solutions of the linear differential equations for the wave latitude-dependent amplitude. In addition, the same numerical method is applied to the traditional problem of linear instability of nondivergent perturbations on the ␤ plane where results reaffirm the classical, analytically derived, features. For these nondivergent perturbations, the present study shows that the growth rate increases monotonically with the jet maximal speed and that the classical result of a local maximum at some finite westward-directed speed results from scaling the growth rates on the jet's speed. In contrast to nondivergent perturbations, divergent perturbations on the ␤ plane have no shortwave cutoff, and so the nondivergent solution does not provide an estimate for the divergent solution, even when the ocean is 1000 km deep (i.e., when the speed of gravity waves exceeds 10 Mach). For realistic values of the ocean depth, the growth rates of divergent perturbations are smaller than those of nondivergent perturbations, but with the increase in the ocean depth they become larger than those of nondivergent perturbations. For both perturbations, a slight asymmetry exists between eastward-and westward-flowing jets. The growth rates of divergent perturbations on a sphere are similar to those on the ␤ plane for the same values of the model parameters, but the asymmetry between eastward and westward jets is more conspicuous on a sphere. The value of gЈHЈ (gЈ is the reduced gravity; HЈ is the equivalent mean layer thickness), which is filtered out in nondivergent theory, determines for divergent perturbations the relative magnitude of zonal velocity, meridional velocity, and height but has little effect on the growth rates.

Dynamics of Atmospheres and Oceans, 2011

Tellus A, 2011

... and Nathan Paldor* Fredy and Nadine Herrmann Institute of Earth Sciences Edmond J. Safra Campus, Givat Ram ... formulation was applied to the midlatitude β-plane by Paldor et al. (2007) and Paldor and Sigalov (2008) and to the equatorial β–plane by Erlick et al. (2007). ...

Advances in Geophysical and Environmental Mechanics and Mathematics, 2011

We have already given a detailed description of rotation affected external and internal waves in idealized containers of constant depth: straight channels, gulfs, rectangles and circular and elliptical cylinders. Pure Kelvin and Poincaré waves were shown to describe the oscillating motion in straight channels and their combination yielded the solution of the reflection of the rotation affected waves at the end wall of a rectangular gulf. The typical characterizations of Kelvin and Poincaré waves were seen to prevail (with some modification) in the fluid motion of rotating circular and elliptical cylinders with constant depth. The behaviour was termed Kelvin-type if for basin-scale dynamics the amplitudes of the surface and isopycnal displacements and velocities are shore-bound (i.e. large close to the boundaries and smaller in the interior of the basin), the motion cyclonic (that is counter-clockwise on the N.H.) and frequencies sub-or (less often) superinertial. Alternatively, for Poincaré-type behaviour, the surface and isopycnal displacements and velocities have large amplitudes in off-shore regions, their motion is anti-cyclonic and frequencies are strictly superinertial.

Dynamics of Atmospheres and Oceans, 2005

A three-dimensional spectral analysis of Topex altimeter data reveals a large meridional component k y of the wavevector k for baroclinic Rossby waves of all timescales. Its existence necessitates some refinements in our estimates of certain basic properties of the Rossby wave field. In particular, by taking into account an actual off-zonal direction of k (often exceeding 70 •), one finds that the wavelength, phase speed, and group velocity of mid-latitude Rossby waves (with periods less than 2 years) are much smaller than they appear to be on the assumption of a purely zonal wavenumber vector. Because of a shorter wavelength (yielding kL as high as 0.6, where L is the Rossby radius of deformation), these waves are essentially dispersive. Their group velocity vector may depart from zonal by more than 30 •. An important intrinsic feature of the wave spectrum confirmed by our analysis is a broadband distribution with respect to k y. Some of the dynamical implications of the large k y /k x ratio are discussed.

Journal of Fluid Mechanics, 2008

The Kelvin wave is the lowest eigenmode of Laplace's tidal equation and is widely observed in both the ocean and the atmosphere. In this work, we neglect mean currents and instead include the full effects of the Earth's sphericity and the wave dispersion it induces. Through a mix of perturbation theory and numerical computations using a Fourier/Newton iteration/continuation method, we show that for sufficiently small amplitude, there are Kelvin travelling waves (cnoidal waves). As the amplitude increases, the branch of travelling waves terminates in a so-called corner wave with a discontinuous first derivative. All waves larger than the corner wave evolve to fronts and break. The singularity is a point singularity in which only the longitudinal derivative is discontinuous. As we solve the nonlinear shallow water equations on the sphere, with increasing ε (‘Lamb's parameter’), dispersion weakens, the amplitude of the corner wave decreases rapidly, and the longitudinal pro...