Laplace's Tidal Equations Research Articles

Local identification of equatorial Kelvin waves in real‐time operational forecasts

AbstractEquatorial Kelvin waves (KWs) are associated with a variety of equatorial atmospheric phenomena, namely, tropical convection, cloud and precipitation variability, tropical cyclogenesis, the onset of monsoon season over Africa and India, and even the quasi‐biennial oscillation and the Madden–Julian oscillation. As such, if operational forecasts are to improve near the Tropics, it is important that they correctly represent KWs. This article aims at developing a novel methodology for identifying KWs in real‐time operational forecasts at specific longitudes using only the meridional structures of the solutions to the free Laplace tidal equations, known as Hough vector functions. The main advantages of this newly proposed methodology are that (a) it can identify KWs at a specific longitude and (b) very low computational cost is needed to apply it to real‐time operational forecasts. The application of the method to 2015–2017 European Centre for Medium‐range Weather Forecasts (ECMWF) analysis and operational forecast data reveals a systematic bias in the representation of KWs by the ECMWF operational forecast model.

On the indices of the normal modes of Laplace's tidal equations for zonal wavenumber zero

AbstractThe normal modes of Laplace's tidal equations seem to be a consolidated subject in the literature. However, up until now, no mode for the zonal wavenumber zero, (i.e., independent of longitude) had been identified as a mixed Rossby–gravity mode with physical meaning. Moreover, the meridional structures of westward gravity modes were not considered to be similar to the meridional structures of the corresponding inertio‐gravity modes for non‐zero wavenumbers (k>0). Here, a mixed Rossby–gravity mode with zero zonal wavenumber () is identified for the first time, and a new classification of the normal modes for the zonal wavenumber zero is proposed. Under the new classification, all the meridional structures corresponding to zero wavenumber modes are similar to the meridional structures of their respective non‐zero wavenumber modes. Moreover, using the new classification, all modes of a given type show smooth frequency variations with the zonal wavenumber, which approach asymptotically the analytic dispersion curves from the ‐plane shallow‐water equations as the fluid height tends to zero.

Impact of the Core Deformation on the Tidal Heating and Flow in Enceladus' Subsurface Ocean

AbstractWe present a novel approach to modeling the tidal response of icy moons with subsurface oceans. The problem is solved in the time domain and the flow in the ocean is calculated simultaneously with the deformation of the core and the ice shell. To simplify the calculations, we assume that the internal density interfaces are spherical and the effective viscosity of water is equal to or greater than 100 Pa s. The method is used to study the effect of an unconsolidated core on tidal dissipation in Enceladus' ocean. We show that the partitioning of tidal heating between the core and the ocean strongly depends on the thickness of the ocean layer. If the ocean thickness is significantly greater than 1 km, heat production is dominated by tidal dissipation in the core and the amount of heat produced in the ocean is negligible. In contrast, when the ocean thickness is less than about 1 km, tidal heating in the core diminishes and dissipation in the ocean increases, leaving the total heat production unchanged. Extrapolation of our results to realistic conditions indicates that tidal flow is turbulent which suggests that the linearized Navier‐Stokes equation may not be appropriate for modeling the tidal response of icy moons. Finally, we compare our results with those obtained by solving the Laplace tidal equations and discuss the limitations of the two‐dimensional models of ocean circulation.

Hough Mode Extensions (HMEs) and Solar Tide Behavior in the Dissipative Thermosphere

AbstractThis paper quantifies and interprets how thermosphere dissipation in the form of molecular viscosity, molecular thermal conductivity and collisions with ions (“ion drag”) determines the height versus latitude structures of a variety of migrating and non‐migrating solar thermal tides, pole‐to‐pole up to 400 km altitude. This is done through computation of thermosphere Hough Mode Extensions (HMEs); that is, solutions to the linearized momentum, thermal energy, continuity, state and hydrostatic balance equations wherein the horizontal structure of troposphere forcing for each HME corresponds to the eigenfunction (Hough function) of Laplace's tidal equation for a particular tidal mode, and the background state and thermosphere dissipation are specified for nominal solar minimum, average and maximum activity conditions. The broad features revealed, different for each HME, include changes in vertical(horizontal) structure with latitude(height), degree of vertical penetration to the middle and upper thermosphere, changes in vertical wavelength (λz) with height due to the transition in background thermal gradient around the mesopause, and insights into role of ion drag in determining tidal amplitudes and their solar cycle variability. The HMEs are particularly useful for fitting measurements of tides, and for estimating tidal fields beyond those dependent variables actually fit and including those outside the fitting domain. The HMEs reported here were created as part of the Ionospheric CONnection (ICON) mission to serve as observation‐based lower boundary conditions for Thermosphere Ionosphere Mesosphere General Circulation Model‐ICON. Access to the full set of HMEs in tabular and graphical form is provided.

The traditional approximation of rotation for rapidly rotating stars and planets

Context. We examine the dynamics of low-frequency gravito-inertial waves (GIWs) in differentially rotating deformed radiation zones in stars and planets by generalising the traditional approximation of rotation (TAR). The TAR treatment was built on the assumptions that the star is spherical (i.e. its centrifugal deformation is neglected) and uniformly rotating. However, it has been generalised in our previous work by including the effects of the centrifugal deformation using a non-perturbative approach. In the meantime, TAR has been generalised in spherical geometry to take the differential rotation into account. Aims. We aim to carry out a new generalisation of the TAR treatment to account for the differential rotation and the strong centrifugal deformation simultaneously. Methods. We generalise our previous work by taking into account the differential rotation in the derivation of our complete analytical formalism that allows the study of the dynamics of GIWs in differentially and rapidly rotating stars. Results. We derived the complete set of equations that generalises the TAR, simultaneously taking the full centrifugal acceleration and the differential rotation into account. Within the validity domain of the TAR, we derived a generalised Laplace tidal equation for the horizontal eigenfunctions and asymptotic wave periods of the GIWs, which can be used to probe the structure and dynamics of differentially rotating deformed stars with asteroseismology. Conclusions. A new generalisation of the TAR, which simultaneously takes into account the differential rotation and the centrifugal acceleration in a non-perturbative way, was derived. This generalisation allowed us to study the detectability and the signature of the differential rotation on GIWs in rapidly rotating deformed stars and planets. We found that the effects of the differential rotation in early-type deformed stars on GIWs is theoretically largely detectable in modern space photometry using observations from Kepler and TESS.

Wave topology brought to the coast

Since the pioneering work of Kelvin on Laplace tidal equations, a zoology of trapped waves have been found in the context of coastal dynamics. Among them, the one originally computed by Kelvin plays a particular role, as it is a unidirectional mode filling a frequency gap between different wave bands. The existence of such Kelvin waves is robust to changes in the boundary shape and in changes of the underlying model for the coast. This suggests a topological interpretation that has yet up to now remained elusive. Here we rectify the situation, by taking advantage of a reformulation of the shallow water dynamics that highlights an analogy with the celebrated Haldane model in condensed matter physics. For any profile of bottom topography, the number of modes that transit from one wave band to another in the dispersion relation is predicted by computing a first Chern number describing the topology of complex eigenmodes in a dual, simpler wave problem.

The traditional approximation of rotation for rapidly rotating stars and planets

Context. The traditional approximation of rotation (TAR) is a treatment of the hydrodynamic equations of rotating and stably stratified fluids in which the action of the Coriolis acceleration along the direction of the entropy and chemical stratifications is neglected because it is weak in comparison with the buoyancy Archimedean force. This leads to the neglect of the horizontal projection of the rotation vector in the equations for the dynamics of gravito-inertial waves (GIWs). The dependent variables in those equations then become separable into radial and horizontal parts as in the non-rotating case. The TAR is built on the assumptions that the star is spherical (i.e., its centrifugal deformation is neglected) and uniformly rotating. However, it has recently been generalised to include the effects of a moderate centrifugal deformation using a perturbative approach. Aims. We study the feasibility of carrying out a new generalisation to account for the centrifugal acceleration in the case of strongly deformed uniformly and rapidly rotating stars (and planets), and to identify the validity domain of this approximation. Methods. We built a complete formalism analytically that allows the study of the dynamics of GIWs in spheroidal coordinates which take the flattening of uniformly and rapidly rotating stars into account by assuming the hierarchies of frequencies adopted within the TAR in the spherical case. Results. Using 2D stellar models, we determine the validity domain of the generalised TAR as a function of the rotation rate of the star normalised by its critical angular velocity and its pseudo-radius. Assuming the anelastic and the two-dimensional Jeffreys-Wentzel-Kramers-Brillouin approximations, we derive a generalised Laplace tidal equation for the horizontal eigenfunctions of the GIWs and their asymptotic wave periods, which can be used to probe the structure and dynamics of rotating deformed stars with asteroseismology. The generalised TAR where the centrifugal deformation of a star (or planet) is taken into account non-perturbatively allows us to identify, within the framework of 2D Evolution STEllaire en Rotation models, the validity domain of this approximation which is reduced by increasing the rate of rotation. We can affirm with a level of confidence of 90% that the TAR remains applicable in all the space domain of deformed stars rotating at a rotation rate lower than 20% of the critical rotation rate. Conclusions. A new generalisation of the TAR, which takes the centrifugal acceleration into account in a non-perturbative way, is derived. This generalisation allows us to study the detectability and the signature of the centrifugal effects on GIWs in rapidly rotating deformed stars (and planets). We found that the effects of the centrifugal acceleration in rapidly rotating early-type stars on GIWs are theoretically detectable in modern space photometry using observations from Kepler. We found also, by comparing the period spacing pattern computed with the standard and the generalised TAR, that the centrifugal acceleration affects the period spacing by increasing its values for low radial orders and by decreasing them slightly for high radial orders.

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.

The traditional approximation of rotation, including the centrifugal acceleration for slightly deformed stars

Context. The traditional approximation of rotation (TAR) is a treatment of the dynamical equations of rotating and stably stratified fluids in which the action of the Coriolis acceleration along the direction of the entropy (and chemicals) stratification is neglected, while assuming that the fluid motions are mostly horizontal because of their inhibition in the vertical direction by the buoyancy force. This leads to the neglect of the horizontal projection of the rotation vector in the equations for the dynamics of gravito-inertial waves (GIWs) that become separable, such as in the non-rotating case, while they are not separable in the case in which the full Coriolis acceleration is taken into account. This approximation, first introduced in geophysical fluid dynamics for thin atmospheres and oceans, has been broadly applied in stellar (and planetary) astrophysics to study low-frequency GIWs that have short vertical wavelengths. The appoximation is now being tested thanks to direct 2D oscillation codes, which constrain its domain of validity. The mathematical flexibility of this treatment allows us to explore broad parameter spaces and to perform detailed seismic modelling of stars. Aims. The TAR treatment is built on the assumptions that the star is spherical (i.e. its centrifugal deformation is neglected) and uniformly rotating while an adiabatic treatment of the dynamics of the waves is adopted. In addition, their induced gravitational potential fluctuations is neglected. However, it has been recently generalised with including the effects of a differential rotation. We aim to carry out a new generalisation that takes into account the centrifugal acceleration in the case of deformed stars that are moderately and uniformly rotating. Methods. We construct an analytical expansion of the equations for the dynamics of GIWs in a spheroidal coordinates system by assuming the hierarchies of frequencies and amplitudes of the velocity components adopted within TAR in the spherical case. Results. We derive the complete set of equations that generalises TAR by taking the centrifugal acceleration into account. As in the case of a differentially rotating spherical star, the problem becomes 2D but can be treated analytically if we assume the anelastic and JWKB approximations, which are relevant for low-frequency GIWs. This allows us to derive a generalised Laplace tidal equation for the horizontal eigenfunctions and asymptotic wave periods, which can be used to probe the structure and dynamics of rotating deformed stars thanks to asteroseismology. A first numerical exploration of its eigenvalues and horizontal eigenfunctions shows their variation as a function of the pseudo-radius for different rotation rates and frequencies and the development of avoided crossings.

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.

On a new type of solving procedure for Laplace tidal equation

In this paper, we present a new approach for solving Laplace tidal equations (LTE) which was formulated first in [S.V.Ershkov, A Riccati-type solution of Euler-Poisson equations of rigid body rotation over the fixed point, Acta Mechanica, 228(7), 2719 (2017)] for solving Poisson equations: a new type of the solving procedure for Euler-Poisson equations (rigid body rotation over the fixed point) is implemented here for solving momentum equation of LTE, Laplace tidal equations. Meanwhile, the system of Laplace tidal equations (including continuity equation) has been successfully explored with respect to the existence of analytical way for presentation of the solution. As the main result, the new ansatz is suggested here for solving LTE: solving momentum equation is reduced to solving system of 3 nonlinear ordinary differential equations of 1-st order in regard to 3 components of the flow velocity (depending on time t), along with the continuity equation which determines the spatial part of solution. Nevertheless, the proper elegant partial solution has been obtained due to invariant dependence between temporary components of the solution. In addition to this, it is proved here that the system of Laplace tidal equations has not the analytical presentation of solution (in quadratures) in case of nonzero fluid pressure in the Ocean, as well as nonzero total gravitational potential and the centrifugal potential (due to planetary rotation).

Convergent Power Series for Boundary Value Problems and Eigenproblems with Application to Atmospheric and Oceanic Tides

It is usually impossible to apply power series to solve boundary value and eigenvalue problems because the radius of convergence does not extend to the boundaries. Hough functions, which solve a second order eigenproblem, are always entire functions or can be computed wholly in terms of entire functions. The eigenvalues ϵ are approximated by the zeros of the highest computed power series coefficient, which is always a polynomial in ϵ. The rate of convergence with series truncation N is proportional to exp(-μN) for some positive constant A brief table contains the entire Maple code for the algorithm. The scheme can also be extended to eigenproblems on an unbounded domain as here illustrated by the quantum harmonic oscillator and quartic oscillator. Computer algebra is very useful; the power series method can also be implemented in a stable, purely numerical fashion by applying trigonometric interpolation on a disk in the complex x-plane and Chebyshev interpolation on a real interval in e. Power series are briefly compared with Chebyshev, Fourier, and spherical harmonic spectral methods, which are mostly superior. The success of power series raises an interesting history-of-science issue: Why was there so little progress in solving Laplace's tidal equations in the nineteenth century?

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.

On computation of Hough functions

Abstract. Hough functions are the eigenfunctions of the Laplace tidal equation governing fluid motion on a rotating sphere with a resting basic state. Several numerical methods have been used in the past. In this paper, we compare two of those methods: normalized associated Legendre polynomial expansion and Chebyshev collocation. Both methods are not widely used, but both have some advantages over the commonly used unnormalized associated Legendre polynomial expansion method. Comparable results are obtained using both methods. For the first method we note some details on numerical implementation. The Chebyshev collocation method was first used for the Laplace tidal problem by Boyd (1976) and is relatively easy to use. A compact MATLAB code is provided for this method. We also illustrate the importance and effect of including a parity factor in Chebyshev polynomial expansions for modes with odd zonal wave numbers.

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.

The tide predictions for D-Day

Based on the physics of Newton and Laplace, the big brass tide-predicting machine designed by Lord Kelvin was crucial for the success of the Normandy invasion in World War II.