Laplace Tidal Research Articles

Solving the Laplace Tidal Equations using Freely Available, Easily Extensible Finite Element Software

The Laplace Tidal Equations (LTEs) play a basic role for investigating the dissipation of long-term orbital energy as heat in surface oceans of icy satellites. Here we address solving LTEs by means of finite elements using PDE2D, a general-purpose finite element program for solving multidimensional PDEs. We use PDE2D to solve the equations for eccentricity and obliquity gravitational potentials for a shallow water approximation, and calculate ocean dissipation in three icy satellites, Enceladus, Europa and Titan. Compared with previous modeling procedures, PDE2D solves for unknowns that are approximated by piecewise cubic polynomials, thus attaining a higher O(dφ4) + O(dθ4) numerical precision. Our modeling results show, within few percent errors, a good agreement with earlier semi-analytical models and numerical solutions. We provide an open access to our PDE2D codes which are intended to be simple and easy to adjust for a wide range of tidal heating related problems. Our modeling results show that PDE2D can be a useful tool for researchers studying tidal heating related problems.

Tides in subsurface oceans with meridional varying thickness

Tidal heating can play an important role in the formation and evolution of subsurface oceans of outer-planet moons. Up until now tidal heating has only been studied in subsurface oceans of spatially uniform thickness. We develop a numerical model to consider oceans of spatially variable thickness. We use the Laplace Tidal Equations for the ocean and model the ice shell using membrane theory. The problem is solved using the commercial Finite Element software Comsol Multiphysics®. We use this new model to study the tidal response of Enceladus' ocean with a twofold objective: to understand how ocean thickness variations modify the tidal response of a subsurface ocean and to assess if tidal dissipation in an Enceladan ocean with varying ocean thickness can explain the high heat flux emanating from Enceladus' South Polar Terrain and the perdurance of a subsurface ocean. We consider the effect of meridional ocean thickness changes of spherical harmonic degree two and three as suggested by topography and gravimetry data. We observe that an ocean with degree two topography responds with the same eigenmodes as an ocean of constant thickness but resonances occur for thicker oceans. However, resonant ocean thicknesses are still thin compared to current estimates for Enceladus ocean thickness. Rossby-Haurwitz waves, excited by the obliquity tide for thick oceans of constant thickness, are not excited at the tidal frequency when oceans of variable thickness are considered. This result implies that the role of the obliquity tide in ocean tidal-dissipation might have been overestimated for Enceladus and other icy worlds. An antisymmetric, degree-three ocean thickness variation mixes the ocean modes excited in a constant thickness ocean by the eccentricity and obliquity tide.

Simultaneous Estimation of Ocean Tides and Underwater Topography in the Weddell Sea

AbstractA new model for the M2, S2, K1, and O1 tides in the Weddell Sea is developed by assimilating CryoSat‐2 data into a barotropic tide model. A variational approach is used, which explicitly allows for errors in the water depth, that is, the bottom topography in open water and the water column thickness under floating ice shelves, so that an optimized estimate of the topography is obtained together with the tidal fields. In preparation for assimilation, the sensitivity of the tidal elevation to the interfacial drag at the sea floor and the ice‐water interface (under the floating ice shelves) is investigated; this motivates the development of a new drag parameterization which is more accurate and physically plausible in comparison with the interfacial drag alone. The assimilation of CryoSat‐2 data into the model results in tidal elevations with essentially the same accuracy as previous estimates, which is demonstrated by comparisons with independent in situ data and withheld CryoSat‐2 data. The novelty of the present tidal estimates is that they are consistent with well‐defined dynamics based on the Laplace Tidal Equations—augmented with the new parameterization of drag—and modifications of the prior estimate of underwater topography and water column thickness. Analysis of the sensitivity to the topography finds that, at this level of precision, the topography is not uniquely determined by the observed data.

Numerically modelling tidal dissipation with bottom drag in the oceans of Titan and Enceladus

Icy satellites that contain subsurface oceans require sufficient thermal energy to prevent the liquid portion of their interiors from freezing. We develop a numerical finite difference model to solve the Laplace Tidal Equations on a sphere in order to simulate tidal flow and thermal energy dissipation in these oceans, neglecting the presence of an icy lid. The model is applied to Titan and Enceladus, where we explore how Rayleigh (linear) and bottom (quadratic) drag terms affect dissipation. The latter drag regime can only be applied numerically. We find excellent agreement between our results and recent analytical work. Obliquity tide Rossby-wave resonant features become independent of ocean thickness under the bottom drag regime for thick oceans. We show that for Titan, dissipation from this Rossby-wave resonance can act to dampen the rate of outward orbital migration by up to 40% for Earth-like values of bottom drag coefficient. Gravity-wave resonances can act to cause inward migration, although this is unlikely due to the thin oceans required to form such resonances. The same is true of all eccentricity tide resonances on Enceladus, such that dissipation becomes negligible for thick oceans under the bottom drag regime.

Crustal control of dissipative ocean tides in Enceladus and other icy moons

Could tidal dissipation within Enceladus’ subsurface ocean account for the observed heat flow? Earthlike models of dynamical tides give no definitive answer because they neglect the influence of the crust. I propose here the first model of dissipative tides in a subsurface ocean, by combining the Laplace Tidal Equations with the membrane approach. For the first time, it is possible to compute tidal dissipation rates within the crust, ocean, and mantle in one go. I show that oceanic dissipation is strongly reduced by the crustal constraint, and thus contributes little to Enceladus’ present heat budget. Tidal resonances could have played a role in a forming or freezing ocean less than 100 m deep. The model is general: it applies to all icy satellites with a thin crust and a shallow ocean. Scaling rules relate the resonances and dissipation rate of a subsurface ocean to the ones of a surface ocean. If the ocean has low viscosity, the westward obliquity tide does not move the crust. Therefore, crustal dissipation due to dynamical obliquity tides can differ from the static prediction by up to a factor of two.

TIDAL HEATING IN A MAGMA OCEAN WITHIN JUPITER’S MOON Io

Active volcanism observed on Io is thought to be driven by the temporally periodic, spatially differential projection of Jupiter's gravitational field over the moon. Previous theoretical estimates of the tidal heat have all treated Io as essentially a solid, with fluids addressed only through adjustment of rheological parameters rather than through appropriate extension of the dynamics. These previous estimates of the tidal response and associated heat generation on Io are therefore incomplete and possibly erroneous because dynamical aspects of the fluid behavior are not permitted in the modeling approach. Here we address this by modeling the partial-melt asthenosphere as a global layer of fluid governed by the Laplace Tidal Equations. Solutions for the tidal response are then compared with solutions obtained following the traditional solid-material approach. It is found that the tidal heat in the solid can match that of the average observed heat flux (nominally 2.25 W m−2), though only over a very restricted range of plausible parameters, and that the distribution of the solid tidal heat flux cannot readily explain a longitudinal shift in the observed (inferred) low-latitude heat fluxes. The tidal heat in the fluid reaches that observed over a wider range of plausible parameters, and can also readily provide the longitudinal offset. Finally, expected feedbacks and coupling between the solid/fluid tides are discussed. Most broadly, the results suggest that both solid and fluid tidal-response estimates must be considered in exoplanet studies, particularly where orbital migration under tidal dissipation is addressed.

Resonant tidal excitation of internal waves in the Earth’s fluid core

It has long been speculated that there is a stably stratified layer below the core-mantle boundary, and two recent studies have improved the constraints on the parameters describing this stratification. Here we consider the dynamical implications of this layer using a simplified model. We first show that the stratification in this surface layer has sensitive control over the rate at which tidal energy is transferred to the core. We then show that when the stratification parameters from the recent studies are used in this model, a resonant configuration arrives whereby tidal forces perform elevated rates of work in exciting core flow. Specifically, the internal wave speed derived from the two independent studies (150 and 155m/s) are in remarkable agreement with the speed (152m/s) required for excitation of the primary normal mode of oscillation as calculated from full solutions of the Laplace Tidal Equations applied to a reduced-gravity idealized model representing the stratified layer. In evaluating this agreement it is noteworthy that the idealized model assumed may be regarded as the most reduced representation of the stratified dynamics of the layer, in that there are no non-essential dynamical terms in the governing equations assumed. While it is certainly possible that a more realistic treatment may require additional dynamical terms or coupling, it is also clear that this reduced representation includes no freedom for coercing the correlation described. This suggests that one must accept either (1) that tidal forces resonantly excite core flow and this is predicted by a simple model or (2) that either the independent estimates or the dynamical model does not accurately portray the core surface layer and there has simply been an unlikely coincidence between three estimates of a stratification parameter which would otherwise have a broad plausible range.

A Coupled Model for Laplace's Tidal Equations in a Fluid with One Horizontal Dimension and Variable Depth

AbstractTide–topography interactions dominate the transfer of tidal energy from large to small scales. At present, it is poorly understood how low-mode internal tides reflect and scatter along the continental margins. Here, the coupling equations for linear tides model (CELT) are derived to determine the independent modal solutions to Laplace's Tidal Equations (LTE) over stepwise topography in one horizontal dimension. CELT is (i) applicable to arbitrary one-dimensional topography and realistic stratification without requiring numerically expensive simulations and (ii) formulated to quantify scattering because it implicitly separates incident and reflected waves. Energy fluxes and horizontal velocities obtained using CELT are shown to converge to analytical solutions, indicating that “flat bottom” modes, which evolve according to LTE, are also relevant in describing tides over sloping topography. The theoretical framework presented can then be used to quantify simultaneous incident and reflected energy fluxes in numerical simulations and observations of tidal flows that vary in one horizontal dimension. Thus, CELT can be used to diagnose internal-tide scattering on continental slopes. Here, semidiurnal mode-1 scattering is simulated on the Australian northwest, Brazil, and Oregon continental slopes. Energy-flux divergence and directional energy fluxes computed using CELT are shown to agree with results from a finite-volume model that is significantly more numerically expensive. Last, CELT is used to examine the dynamics of two-way surface–internal-tide coupling. Semidiurnal mode-1 internal tides are found to transmit about 5% of their incident energy flux to the surface tide where they impact the continental slope. It is hypothesized that this feedback may decrease the coherence of sea surface displacement on continental shelves.

Zonally propagating wave solutions of Laplace Tidal Equations in a baroclinic ocean of an aqua-planet

Zonally propagating wave solutions of Laplace Tidal Equations in a baroclinic ocean of an aqua-planet

Zonally propagating wave solutions of Laplace Tidal Equations in a baroclinic ocean of an aqua-planet

Despite the accurate formulation of Laplace’s Tidal Equations (LTE) nearly 250 years ago, analytic solutions of these equations on a spherical planet that yield explicit expressions for the dispersion relations of wave solutions have been found only for slowly rotating planets, so these solutions are of no relevance to Earth. Analytic solutions of the LTE in a symmetric equatorial channel on a rotating sphere were recently obtained by approximating the LTE by a Schr¨odinger equation whose energy levels yield the dispersion relations of zonally propagating waves and whose eigenfunctions determine the meridional structure of the amplitude of these waves. A similar approximation of the LTE on a sphere (with no channel walls) by a Schr¨odinger equation yields accurate analytic solutions for zonally propagating waves in the parameter range relevant to a baroclinic ocean, where the ratio between the radius of deformation and Earth’s radius is small. For sufficiently low (meridional) modes the amplitudes of the solutions vanish at some extra-tropical latitudes but this is not assumed abinitio. These newly found solutions do not restrict the value of the zonal wavenumber to be smaller than the meridional wavenumber as is the case in previous theories on a slowly rotating sphere.

Tidal dynamical considerations constrain the state of an ocean on Enceladus

In previous work, solutions to the non-dissipative Laplace Tidal Equations (LTE) were used to provide bounds on the heat generated by the response of a subsurface ocean on Enceladus to an obliquity component of tidal forces. Here we improve these bounds using solutions from the LTE with a generic dissipation term explicitly added. We find solutions for a wider range of ocean tidal responses that include both unstratified (barotropic) and stratified (baroclinic) flow responses to obliquity as well as eccentricity components of the tidal forces. We consolidate the results in three ocean tidal scenarios on Enceladus that can explain the high heat fluxes (∼7 mW/m 2 globally averaged) inferred from measurements by the Cassini spacecraft: (1) a deep (1–50 km) barotropic ocean responding to obliquity tidal forces, where obliquity is at least 0.1°; (2) a shallow (∼360 m) barotropic ocean responding to eccentricity tidal forces; (3) a stratified (baroclinic) ocean responding to eccentricity tidal forces where the density-weighted “equivalent depth” (typically much smaller than the ocean’s physical depth) is near 360 m. The ocean is assumed to be global, but extensions for a semi-global case are also described. A more general result which is independent of the specific scenarios proposed is that an ocean attempting to freeze (with an associated decrease in its liquid depth, which affects the ocean’s dynamical response to the tidal forcing) must first pass through resonant configurations with a greatly increased generation of ocean tidal heat (exceeding 1 W/m 2 to 1 kW/m 2) that would act to halt further freezing and stagnate the ocean state in this configuration so long as there is still orbital energy to provide the tidal forces. With an additional assumption that the ocean has evolved from a more energetic state where the depth of the liquid ocean was greater, we obtain the three scenarios proposed.

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 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.

On the domains of minimal and maximal operators for regularisable singular differential expressions

SynopsisCertain classical differential expressions which are singular at a finite end-point (or at an interior point) can be represented as regular, scalar quasi-differential expressions, the best-known examples being the Boyd Equation and Laplace Tidal Wave Equation. We show here that in all such cases the domains of the minimal and maximal operators in the appropriate weighted Hilbert space, for the regularised expression, coincide with the corresponding domains for the expression in its original, singular form.This is contrasted with a known property of the corresponding expression domains. Whereas for an expressionM, the operator domains contain only functions y for which bothyandMylie in the appropriate Hilbert space, the expression domain comprises a much larger set of functions with no such restrictions beyond those necessary forMyto exist as a function. In the second-order case, the expression domain of the regularisation of a singular expression is known to be a strict subset of the original expression domain, contrasting with the results proved here for the operator domains.

Asymptotic Behaviour of Solutions to Laplace's Tidal Equations At Low Frequencies

SUMMARY The asymptotic behaviour of solutions to Laplace’s tidal equations at low frequencies is considered. The method used is based on perturbation in small parameters, these being the ratios of tidal frequency and the coefficient of bottom friction to the angular frequency of the Earth’s rotation. It is shown that the resulting solutions are unstable in that the functions involved in the zero-order approximation are not uniquely determined by the zero-order equations, but depend on first-order terms as well. Because of this instability, direct methods of numerical integration are inefficient. We propose a different procedure, replacing the original set of equations in partial derivatives by ordinary differential equations that have a stable solution. The equations are examined qualitatively. It is shown, in particular, that for the case of an ocean of uniform depth over the whole Earth, they coincide with the well-known Lamb’s equations. The asymptotic behaviour of the solutions is examined as modified by basin shape, bottom topography and bottom friction.

M2, S2, K1 models of the global ocean tide on an elastic earth

Ocean tidal signals appear in many geophysical measurements. Geophysicists need realistic tidal models to aid in interpretation of their data. Because of the closeness to resonance of dissipa‐tionless ocean tides, it is difficult for numerical models to correctly represent the actual open ocean tide. As an approximate solution to this problem, test functions derived by solving Laplace's Tidal Equations with ocean loading and self gravitation are used as a basis for least squares dynamic interpolation of coastal and island tidal data for the constituents M2, S2, and K1. The resulting representations of the global tide are stable over at least a ±5% variation in the mean depth of the model basin, and they conserve mass. The paper presents maps of the geocentric tide, the induced free space potential, the induced vertical component of the solid earth tide, and the induced vertical component of the gravitational field for each constituent.

A simulation of global ocean tide recovery using altimeter data with systematic orbit error

A least‐squares analysis is performed on simulated satellite altimeter data to investigate the separability of global ocean tide information from a noisy signal containing systematic orbit error. Numerical solutions of the Laplace Tidal Equations for the principal tidal constituents are used to generate the observed measurements based on simulated four‐day SEASAT orbits over a 200‐day interval. The global ocean tide is modeled as a sum of constituents represented by harmonic expansions, and the orbit error model parameters and tide model parameters are adjusted simultaneously. Good recovery of principal features of the numerical ocean tide solutions are obtained only when the systematic orbit error is adequately modeled in the estimation procedure.

Numerical Modeling of Tides in Hudson Bay

A two-dimensional numerical model is developed to study the cooscillating and independent tides in Hudson Bay. Using centered differences (forward differences for the dissipative term) and conjugate Richardson lattices, the Laplace Tidal Equations in spherical polar coordinates are integrated in time until cyclic equilibrium is reached. For the cooscillating tide, the direct tidal forcing term is set to zero, and the observed tidal constituent is specified at the mouth of Hudson Bay. Separate runs are made for M2, S2, N2, and K1. For the independent tide, the closed mouth boundary condition of zero water transport is imposed, and the model run for the M2 and K1 direct tidal forcing. A number of experiments are carried out to test the sensitivity of the model to uncertainties in the input data and parameterization of some of the terms. It is shown that the tidal propagation is relatively insensitive to friction coefficient and island schematization, but very sensitive to depth representation in the Belcher Islands area and phase variation in the specified boundary conditions.Comparison of the results with previous work and shore-based gauge observations gives good amplitude and phase agreement for the M2, S2, and N2 cooscillating tidal constituents except in the vicinity of the degenerate amphidromic points in James Bay and the Belcher Islands where the amplitudes are very small. The amplitudes of the K1 independent tide, unlike the M2, are found to be upwards of 30% of the K1 cooscillating tide. The M2 cooscillating tidal currents, when compared with current meter results at two stations across the mouth of James Bay, show good agreement in west–east decrease in amplitude, reversal of direction of rotation, and increase in rotary character, but generally tended to underestimate the absolute magnitude of these single depth measurements. Overall, the model gives good qualitative agreement with shore-based data and can be used to interpret tidal propagation in the Hudson–James Bay system.

Unstable Solutions to Laplace's Tidal Equation with Negative Equivalent Depth

Abstract The complex eigenvalues of frequency are computed for Hough functions of negative equivalent depth, using the method of Longuet-Higgins. The positive kinetic wave energy and negative wave potential energy of unstable modes cancel. These unstable modes may be important in the solar convective zone as a means of sustaining solar differential rotation.

The Eigenfunctions of Laplace's Tidal Equations over a Sphere

Numerical calculations are presented for the eigenvalues of Laplace’s tidal equations governing a thin layer of fluid on a rotating sphere, for a complete range of the parameter e ( omega = rate of rotation, R = radius, g = gravity, h — depth of fluid layer). The corresponding eigenfunctions or ‘Hough functions’ are shown graphically for the lower modes of oscillation. Negative values of e, which have application in problems involving forced motions, are also considered The calculations reveal many asymptotic forms of the solution for various limiting values of e. The corresponding analytical expressions are derived in the present paper. Thus, as e0 through positive values we have the well-known waves of the first and second class respectively, which were found by Margules and Hough. These can be represented in terms of spherical harmonics. As e-> + oo there are three distinct asymptotic forms. In each of these the energy is concentrated near the equator. In the first type, the kinetic energy is three times the potential energy. In the other two types the kinetic and potential energies are equal. The waves of the second type are all propagated towards the west. The waves of the third type are Kelvin waves propagated eastwards along the equator. All three types are described in terms of Hermite polynomials. As e 0 through negative values there is only one asymptotic form of solution, representing motions which are analytically continuous with Hough’s ‘waves of the second class’. As e —> — oc there are three different asymptotic forms, in each of which the energy tends to be concentrated near the poles of rotation. In the first two types the energy is mainly kinetic and the motion is in inertial circles. In the third type the energy is mainly potential. The modes tend to occur in pairs of almost the same frequency, one being symmetric and the other antisymmetric about the equator. The analytical forms of the solutions involve generalized Laguerre polynomials. In the special case of zonal oscillations, the first two limiting forms as E -> — oo go over into a different form in which the frequency tends to zero as e tends to a finite negative value. In this case the third type does not occur. The way in which the various asymptotic solutions are connected can be traced in figures 1 to 6 (e > 0) and figures 16 to 21 (e 4 4 are tabulated in tables 1 to 10. The eigenfunctions for the lower modes are presented graphically.