Tidal Equations Research Articles

Spectral Analysis of Symmetric Operators: Application to the Laplace Tidal Model

A finite-storage iterative algorithm is proposed for performing spectral analysis and synthesis with respect to an arbitrary symmetric operator. This operator can be of any nature, and it is only necessary to define its action. The whole analysis may be performed by a repeated application of the operator. Compared with traditional methods, our approach allows us to treat discrete systems of very high dimension. As an example, we consider in the framework of an ocean governed by the Laplace tidal equations the problem of separation of large-scale geostrophic modes and surface-waves contributing to a given flow. As a second example we apply the algorithm to a problem in satellite remote sensing. We reconstruct the mean large-scale oceanic circulation from observed sea surface height data. In the case of a synthetic data set the agreement of analytical and numerical results is satisfactory.

On the dynamical effects of an inhomogeneous liquid core in the theory of nutation

As shown in previous work, dynamical effects of a realistic model of a heterogeneous, compressible, stably stratified liquid core may be obtained by means of a simple analysis of the generalized two-dimensional Laplace tidal equation which describes tidal flows of an incompressible and non-gravitating fluid in a thin spherical layer with mobile boundaries. The solution was presented in the form of expansions in powers of a small parameter κ being the ratio of nutational motion frequency in space to the frequency of the Earth's diurnal rotation. Whereas in an earlier paper only first-order terms were taken into account, our present approach includes not only main second-order terms in the spherical harmonic expansions of the solutions, but also the terms of higher orders. These effects are calculated numerically for realistic models of the Earth's outer liquid core, solid inner core and anelastic mantle (PREM model). All tables are found in electronic version at http://www.tu-darmstadt.de/fb/vw/ipg/Welcome2.html

Tides in the Sea of Okhotsk

Eight major tidal constituents in the Sea of Okhotsk have been investigated using a numerical solution of tidal equations on a 59 space grid. The tides are dominated by the diurnal constituents. Diurnal tidal currents are enhanced in Shelikhov Bay and Penzhinskaya Guba, at Kashevarov Bank, in proximity to the Kuril Islands and at a few smaller locations. The major energy sink for diurnal tides (over 60% of the total energy) is Shelikhov Bay and Penzhinskaya Guba. The major portion of semidiurnal tide energy is dissipated in the northwestern region of the Sea of Okhotsk and in Shelikhov Bay and Penzhinskaya Guba. Nonlinear interactions of diurnal currents are investigated through K1 and O1 constituent behavior over Kashevarov Bank. These interactions generate residual circulation of the order of 10 cm s 21, major oscillations at semidiurnal and fortnightly periods (13.66 days), and higher harmonics of basic tidal periods. The M2 tidal current, caused by the nonlinear interaction of the diurnal constituents over Kashevarov Bank, constitutes approximately a half of the total M2 tide current there. The fortnightly current, through nonlinear interactions, also influences basic diurnal tidal currents by inducing fortnightly variations in the amplitude of these currents.

On Divergent Barotropic and Inertial Instability in Zonal-Mean Flow Profiles

Abstract Impacts of atmospheric mean zonal flows on equatorial and subequatorial waves are examined using the generalized Laplace tidal equation, a horizontal structure equation that explicitly includes the effect of horizontal divergence on disturbances via the Lamb parameter e = (2Ωa)2/(ghe). Two important consequences of the mean zonal flow profile are the excitation of barotropically unstable waves associated with a reversed meridional gradient of potential vorticity and the excitation of inertially unstable waves related to anomalous potential vorticity values. Investigations show that both types of instabilities occur in realistic wind fields of the upper troposphere around 200 hPa. The barotropically unstable waves are divided into weakly divergent waves located in the middle latitudes and the more important class of strongly divergent waves located at the equatorward flank of the jet stream in the subtropics. The strongly divergent inertially unstable waves are identified by equatorially trapped w...

Low‐Frequency Nonradial Oscillations in Rotating Stars. I. Angular Dependence

We obtain the θ-dependence of the displacement vector of rotationally modulated low-frequency nonradial oscillations by numerically integrating Laplace's tidal equation as an eigenvalue problem with a relaxation method. This method of calculation is more tractable than our previous method in which the θ-dependence was represented by a truncated series of associated Legendre functions. Laplace's tidal equation has two families of eigenvalues. In one of these families, an eigenvalue λ coincides with l(l + 1) when rotation is absent, where l is the latitudinal degree of the associated Legendre function, Pml(cos θ). The value of λ changes as a function of ν ≡ 2Ω/ω, where Ω and ω are the angular frequencies of rotation and of oscillation (seen in the corotating frame), respectively. These eigenvalues correspond to rotationally modulated g-mode oscillations. In the domain of | ν | > 1, another family of eigenvalues exists. Eigenvalues belonging to this family have negative values for prograde oscillations, while they change signs from negative to positive for retrograde oscillations as | ν | increases. Negative λ's correspond to oscillatory convective modes. The solution associated with a λ that has a small positive value after changing its sign is identified as an r-mode (global Rossby wave) oscillation. Amplitudes of g-mode oscillations tend to be confined to the equatorial region as | ν | increases. This tendency is stronger for larger λ. On the other hand, amplitudes of oscillatory convective modes are small near the equator.

Nonequilibrium Response of the Global Ocean to the 5-Day Rossby–Haurwitz Wave in Atmospheric Surface Pressure

The response of the global ocean to the surface pressure signal associated with the well-known 5-day Rossby–Haurwitz atmospheric mode is explored using analytical and numerical tools. Solutions of the Laplace tidal equations for a flat-bottom, globe-covering ocean, point to a depth-independent nonequilibrium response related to the near-resonant excitation of the barotropic oceanic mode. Numerical experiments with a shallow-water model illustrate the effects of realistic continental boundaries, topography, and dissipation on the solutions. The character of the oceanic adjustment and the structure of resonances changes substantially, but a nonequilibrium response occurs in all cases studied. Besides the excitation of large-scale vorticity modes or waves, which becomes less important when topography and strong dissipation are present, basin-scale nonequilibrium signals are associated with gravity wave dynamics and the process of interbasin mass adjustment in the presence of global-scale forcing and continents that require interbasin mass fluxes to occur through the Southern Ocean. Solutions with forcing most representative of the observed atmospheric wave agree qualitatively with the results of analyses of Pacific and Atlantic tide gauge records by Luther and Woodworth et al. The observed nonequilibrium signals thus seem related to the Rossby–Haurwitz forcing mode.

On azimuthal eigenwavenumbers associated with Laplace's tidal equations

SUMMARY Laplace's tidal equations for the case of an ocean of constant depth bounded by meridians were considered by two authors at a specific frequency as an eigenvalue problem in the azimuthal wavenumber. A finite spectrum of eigenwavenumbers was found. That eigenvalue problem is re-examined by means of asymptotic techniques and numerical integration of the governing equation of the problem. At low frequencies a formula connecting the frequency and the number of eigensolutions is established. It is shown that at a given frequency the spectrum of eigenwavenumbers is wider than that reported, but (for this type of solution) the meridional boundary conditions are satisfied approximately only for the case of very low frequencies.

Laboratory studies of equatorially trapped waves using ferrofluid

We introduce a new technique to model spherical geophysical fluid dynamics in the terrestrial laboratory. The local vertical projection of planetary vorticity, f, varies with latitude on a rotating spherical planet and allows an important class of waves in large-scale atmospheric and oceanic flows. These Rossby waves have been extensively studied in the laboratory for middle and polar latitudes. At the equator f changes sign where gravity is perpendicular to the planetary rotation. This geometry has made laboratory studies of geophysical fluid dynamics near the equator very limited. We use ferrofluid and static magnetic fields to generate nearly spherical geopotentials in a rotating laboratory experiment. This system is the laboratory analogue of those large-scale atmospheric and oceanic flows whose horizontal motions are governed by the Laplace tidal equations. As the rotation rate in such a system increases, waves are trapped to latitudes near the equator and the dynamics can be formulated on the equatorial β-plane. This transition from planetary modes to equatorially trapped modes as the rotation rate increases is observed in the experiments. The equatorial β-plane solutions of non-dispersive Kelvin waves propagating eastward and non-dispersive Rossby waves propagating westward at low frequency are observed in the limit of rotation fast compared to gravity wave speed.

Diurnal migrating tide as seen by the high‐resolution Doppler imager/UARS: 2. Monthly mean global zonal and vertical velocities, pressure, temperature, and inferred dissipation

Meridional diurnal tidal winds derived as described by Khattatov et al. [this issue] from the high‐resolution Doppler imager (HRDI) data are used in the linearized tidal equations to solve for the diurnal tidal oscillations in the zonal and vertical velocity, pressure, temperature, and dissipation in the mesosphere and lower thermosphere. The resulting monthly mean latitude‐altitude cross sections of the amplitudes of the tidal zonal, meridional, and vertical velocity, pressure, temperature, and Rayleigh friction and vertical diffusivity are presented for 12 months of the combined 1992/1993 year. The results show profound seasonal changes and interhemispherical differences in the calculated tidal amplitudes and dissipation. The derived monthly mean vertical diffusion coefficients are compared with the results of Garcia and Solomon [1985]. The Rayleigh friction coefficients and vertical diffusivities derived from HRDI measurements can potentially have important impact on modeling of the chemistry and composition of the mesosphere and lower thermosphere, in particular, on numerical modeling of atmospheric thermal tides.

Excitation of the 5-day Wave by Antarctica

Abstract The 5-day wave has been detected in composite maps of geopotential height fields of European Centre for Medium-Range Weather Forecasts dataset for 7 years (1984–91, except 1985) and shows a characteristic northwest–southeast meridional phase tilt in the Southern Hemisphere. A numerical integration of Laplace’s tidal equations with periodic forcing of zonal wavenumber 1 only produced the meridional phase tilt when the forcing is located in high latitudes. Such a forcing is created through coupling of the time-fluctuating westerlies with the topography of Antarctica. Numerical simulations that incorporated this mechanism reproduced the observed meridional phase tilt of the 5-day wave, which suggests that Antarctica is responsible for the observed phase tilt through the process of resonance. A linear theory on the meridional phase tilt is given with a nondivergent barotropic model that includes both forcing and dissipation.

The complex wavenumber eigenvalues of Laplace's tidal equations for oceans bounded by meridians: corrigendum

An error in the article (O’Connor 1995) was brought to my attention by Charles Sozou. This changes the values of the eigenvalues calculated in table 1, and the geophysical conclusions are modified. The introduction and §2 on Laplace’s tidal equations are correct. The mistake occured in §3 when the Legendre function expansion was substituted into the set of Laplace’s tidal equations. The term a in the expansion for the tide height ζ* in equation (12) must be zero, otherwise equation (7) cannot be satisfied. I did not see this at the time because I neglected to include the a term when substituting the expansions (12) into equation (7). This is why no a term appears in equation (18). The corrections are to omit the a terms in equations (12) and (17).

On the ionization of a Keplerian binary system by periodic gravitational radiation

The gravitational ionization of a Keplerian binary system via normally incident periodic gravitational radiation of definite helicity is discussed. The periodic orbits of the planar tidal equation are investigated on the basis of degenerate continuation theory. The relevance of the Kolmogorov-Arnold-Moser theory to the question of gravitational ionization is elucidated, and it is conjectured that the process of ionization is closely related to the Arnold diffusion of the perturbed system.

Traps and Snares in Eigenvalue Calculations with Application to Pseudospectral Computations of Ocean Tides in a Basin Bounded by Meridians

We make several observations about eigenvalue problems using, as examples, Laplace's tidal equations and the differential equation satisfied by the associated Legendre functions. Whatever the discretization, only some of the eigenvalues of theN-dimensional matrix eigenvalue problem will be good approximations to those of the differential equation—usuallytheN/2 eigenvalues of smallest magnitude. For the tidal problem, however, the “good” eigenvalues are scattered, so our first point is: It is important to plot the “drift” of eigenvalues with changes in resolution. We suggest plotting the difference between a low resolution eigenvalue and the nearest high resolution eigenvalue, divided by the magnitude of the eigenvalue or the intermodal separation, whichever is smaller. Second, as a final safeguard, it is important to look at the Chebyshev coefficients of the mode: We show a numerically computed “anti-Kelvin” wave which has little eigenvalue drift, but is completely spurious as is obvious from its spectral series. Third, inverting the roles of parameters can drastically modify the spectrum; Legendre's equation may have either an infinite number of discrete modes or only a handful, depending on which parameter is the eigenvalue. Fourth, when the modes are singular but decay to zero at the endpoints (as is true of tides), a tanh-mapping can retrieve the usual exponential accuracy of spectral methods. Fifth, the pseudospectral method is more reliable than deriving a banded Galerkin matrix by means of recurrence relations; the pseudospectral code is simple to check, whereas it is easy to make an untestable mistake with the intricate algebra required for the Galerkin method. Sixth, we offer a brief cautionary tale about overlooked modes. All these cautions are applicable toallforms of spatial discretization including finite difference and finite element methods. However, we limit our illustrations to spectral schemes, where these difficulties are most easily resolved. With a bit of care, the pseudospectral method is a very robust and efficient method for solving differential eigenproblems, even with singular eigenmodes.

Atmosphere and Earth's rotation

The theoretical aspects of the transfer of angular momentum between atmosphere and Earth are treated with particular emphasis on analytical solutions. This is made possible by the consequent usage of spherical harmonics of low degree and by the development of large-scale atmospheric dynamics in terms of orthogonal wave modes as solutions of Laplace's tidal equations. An outline of the theory of atmospheric ultralong planetary waves is given leading to analytical expressions for the meridional and height structure of such waves. The properties of the atmospheric boundary layer, where the exchange of atmospheric angular momentum with the solid Earth takes place, are briefly reviewed. The characteristic coupling time is the Ekman spin-down time of about one week. The axial component of the atmospheric angular momentum (AAM), consisting of a pressure loading component and a zonal wind component, can be described by only two spherical functions of latitude ϕ: the zonal harmonicP 2 0 (ϕ), responsible for pressure loading, and the spherical functionP 1 1 (ϕ) simulating supperrotation of the zonal wind. All other wind and pressure components merely redistributeAAM internally such that their contributions toAAM disappear if averaged over the globe. It is shown that both spherical harmonics belong to the meridional structure functions of the gravest symmetric Rossby-Haurwitz wave (0, −1)*. This wave describes retrograde rotation of the atmosphere within the tropics (the tropical easterlies), while the gravest symmetric external wave mode (0, −2) is responsible for the westerlies at midlatitudes. Applying appropriate lower boundary conditions and assuming that secular angular momentum exchange between solid Earth and atmosphere disappears, the sum of both waves leads to an analytical solution of the zonal mean flow which roughly simulates the observed zonal wind structure as a function of latitude and height. This formalism is used as a basis for a quantitative discussion of the seasonal variations of theAAM within the troposphere and middle atmosphere. Atmospheric excitation of polar motion is due to pressure loading configurations, which contain the antisymmetric functionP 2 1 (ϕ) exp(iλ) of zonal wavenumberm=1, while the winds must have a superrotation component in a coordinate system with the polar axis within the equator. The Rossby-Haurwitz wave (1, −3)* can simulate well the atmospheric excitation of the observed polar motion of all periods from the Chandler wobble down to normal modes with periods of about 10 days. Its superrotation component disappears so that only pressure loading contributes to polar motion. The solar gravitational semidiurnal tidal force acting on the thermally driven atmospheric solar semidiurnal tidal wave can accelerate the rotation rat of the Earth by about 0.2 ms per century. It is speculated that the viscous-like friction of the geomagnetic field at the boundary between magnetosphere and solar wind may be responsible for the westward drift of the dipole component of the internal geomagnetic field. Electromagnetic or mechanical coupling between outer core and mantle may then contribute to a decrease of the Earth's rotation rate.

Distributions of copepod nauplii and turbulence on the southern flank of Georges Bank: implications for feeding by larval cod (Gadus morhua)

The vertical distributions of copecod nauplii and water properties were sampled at well-mixed and stratified sites on Georges Bank using a pumping system, CTD and in vivo fluorometer during a four day period in late May 1992. At each stratified station at least one sample was taken within the thermocline and the fluorescence maximum, which usually co-occurred. Well-mixed sites had low average concentrations of nauplii, ca 41−1, and showed little variation of abundance with depth. Stratified sites had from 4 to 16 times the integrated (0–50 m) abundance of nauplii compared to well-mixed sites and showed strong vertical patterns of distribution. Maximum concentrations of nauplii, up to 1601−1, were associated with the thermocline at 7 of the 9 stratified stations. At the two remaining stratified sites the naupliar maximum was in the upper mixed layer, sampled at 5 m depth. The encounter rate between early feeding cod (Gadus morhua) larvae and their naupliar prey was calculated with and without turbulence. Turbulence was estimated from two sources: wind stress in the upper layer (calculated from wind observations during our cruise) and tidal shear in the lower layer (estimated initially from a tidal mixing equation). We ultimately replaced the lower layer estimates with turbulence data from a series of measurements made in 1995. The latter are more robust and had the advantage of providing dissipation rates for the pycnocline as well as the lower layer. Theory predicts an increase in encounters between a predator and its prey with the addition of turbulence parameters into standard models of encounter. We combined turbulence profiles with the vertical distribution of nauplii to examine the potential contribution of turbulent kinetic energy to predator-prey encounter rates at various depths in stratified and mixed water columns. Our calculations suggest the following increases due to turbulence at stratified sites on Georges Bank during our cruise: from 34 to 219% in the upper mixed layer, depending on wind speed and depth; approximately 8% in the pycnocline; and approximately 110% below the pycnocline. Mixed sites experience increases of at least 110% (tide only), but greater increases (118–192% in this study) occur when the wind blows because of the combined (spatially overlapped) effects of wind and tidal mixing at these sites. The absolute values for encounter rates and their modification by turbulence are sensitive to a number of assumptions in the models. We used a series of stated assumptions to generate estimates that range from 0.6 to 26.5 prey h−1, depending on geographical location, physical forcing and depth.

The complex wavenumber eigenvalues of Laplace’s tidal equations for oceans bounded by meridians

The complex wavenumber eigenvalues of Laplace’s tidal equations are determined for an ocean of constant depth bounded by meridians. A Galerkin method is used to expand the tide height and velocities in series of associated Legendre functions. A homogeneous system of equations results from the continuity and momentum equations. The frequency and depth are fixed, so that the meridional wavenumbers are the eigenvalues. This gives rise to a generalized eigenvalue problem that must be solved numerically by iteration. The eigenvalues are not integers and represent inertia-gravity wave solutions at the specified tidal forcing frequency that can be excited by the presence of meridional boundaries. Those complex eigenvalues represent solutions that decay away from meridional boundaries. The eigenvalue spectrum is investigated for the semi-diurnal, fortnightly, and monthly tides. One complex wavenumber for the semi-diurnal tide explains the amphidromic systems within 20° of a north-south coastline. The fortnightly and monthly tides have only real wavenumber eigenvalues. The basin scale deviation of these tides from equilibrium is attributed to low wavenumber divergent inertia-gravity waves.

Excitation of the 5-day Wave.

The 5-day wave was detected in composite maps of geopotential height fields of ECMWF data set of 8 years from 1984 to 1991. The structure was similar to that predicted by the Laplace's tidal equation except a characteristic NW-SE meridional phase tilt in the Southern Hemisphere. To understand the cause of this meridional phase tilt, numerical simulations were carried out with the shallow water equation on a sphere (Laplace's tidal equation) with a periodic forcin g in aform of a standing wave of zonal wavenumber 1. The result showed that the meridional phase tilt of the 5-day wave was produced only when the forcing was located in high latitudes. Comparison of this result with the observational results indicates that a forcing of the 5-day wave is located in the high latitude of the Southern Hemisphere. The standing wave is produced by coupling of the Antarctica with the fluctuating Westerly. Numerical simulations which incorporated this mechanism reproduced the realistic meridional phase tilt of the observed 5-day wave, suggesting that this coupling is an effective mechanism which excites the 5-day wave by resonance.

Solitary waves of vortices

We study the propagation of solitary waves of vortices within a spherical shell which constitutes the uppermost layer of a solid planet. This solid-liquid configuration rotates with constant angular velocity about an axis which is fixed with respect to the solid surface. The fluid within the shell is inviscid, incompressible, and of constant density. The motion imparted by the planetary rotation upon this fluid mass is governed by the Laplace tidal equation from which the potential of the extraplanetary forces has been deleted. Consistent with this ocean model, we establish that the stream function of a solitary wave of vortices must satisfy a third-order partial differential equation. We obtain solutions to this wave equation by imposing the condition that the vertical component of vorticity be functionally related to the stream function. We find that this dependence must necessarily be of the exponential type and that the solution to the wave equation then reduces to a quadrature depending on some arbitrary parameters. We prove that we can always choose the values of these parameters in order to approximate the integral in question by means of an analytic function: we reach a representation of the stream function of a solitary wave of vortices in terms of hyperbolic functions of time and position.

TOPEX/POSEIDON tides estimated using a global inverse model

Altimetric data from the TOPEX/POSEIDON mission will be used for studies of global ocean circulation and marine geophysics. However, it is first necessary to remove the ocean tides, which are aliased in the raw data. The tides are constrained by two distinct types of information: the hydrodynamic equations which the tidal fields of elevations and velocities must satisfy, and direct observational data from tide gauges and satellite altimetry. Here we develop and apply a generalized inverse method, which allows us to combine rationally all of this information into global tidal fields best fitting both the data and the dynamics, in a least squares sense. The resulting inverse solution is a sum of the direct solution to the astronomically forced Laplace tidal equations and a linear combination of the representers for the data functionals. The representer functions (one for each datum) are determined by the dynamical equations, and by our prior estimates of the statistics of errors in these equations. Our major task is a direct numerical calculation of these representers. This task is computationally intensive, but well suited to massively parallel processing. By calculating the representers we reduce the full (infinite dimensional) problem to a relatively low‐dimensional problem at the outset, allowing full control over the conditioning and hence the stability of the inverse solution. With the representers calculated we can easily update our model as additional TOPEX/POSEIDON data become available. As an initial illustration we invert harmonic constants from a set of 80 open‐ocean tide gauges. We then present a practical scheme for direct inversion of TOPEX/POSEIDON crossover data. We apply this method to 38 cycles of geophysical data records (GDR) data, computing preliminary global estimates of the four principal tidal constituents, M2, S2, K1, and O1. The inverse solution yields tidal fields which are simultaneously smoother, and in better agreement with altimetric and ground truth data, than previously proposed tidal models. Relative to the “default” tidal corrections provided with the TOPEX/POSEIDON GDR, the inverse solution reduces crossover difference variances significantly (≈20–30%), even though only a small number of free parameters (≈1000) are actually fit to the crossover data.

On Some Forced Oscillations Relating to Laplace's Tidal Equation

SUMMARY Laplace's tidal equation for some forced oscillations in a global ocean of constant depth is integrated numerically and then the solutions are expressed in terms of other functions, such as infinite series in Legendre polynomials, and compared with the corresponding solutions constructed by other authors. the case relating to the pole tide, that is to the oscillation associated with the small frequency of the Chandler wobble, is solved analytically to the third order in the frequency for the region far from the equator and to leading order for a thin region about the equator.