Ocean Dynamics (2006) 56: 394–415DOI 10.1007/s10236-006-0086-x

Florent Lyard . Fabien Lefevre . Thierry Letellier .Olivier Francis

Modelling the global ocean tides: modern insights from FES2004

Received: 25 February 2006 / Accepted: 9 June 2006 / Published online: 27 September 2006# Springer-Verlag 2006

Abstract During the 1990s, a large number of new tidalatlases were developed, primarily to provide accurate tidalcorrections for satellite altimetry applications. During thisdecade, the French tidal group (FTG), led by C. Le Provost,produced a series of finite element solutions (FES) tidalatlases, among which FES2004 is the latest release,computed from the tidal hydrodynamic equations anddata assimilation. The aim of this paper is to review thestate of the art of tidal modelling and the progress achievedduring this past decade. The first sections summarise thegeneral FTG approach to modelling the global tides. In thefollowing sections, we introduce the FES2004 tidal atlasand validate the model against in situ and satellite data. Wedemonstrate the higher accuracy of the FES2004 releasecompared to earlier FES tidal atlases, and we recommendits use in tidal applications. The final section focuses on thenew dissipation term added to the equations, which aims toaccount for the conversion of barotropic energy intointernal tidal energy. There is a huge improvement in thehydrodynamic tidal solution and energy budget obtainedwhen this term is taken into account.

Keywords Tidal atlas . Finite element modelling .FES2004 . Data assimilation

1 Introduction

The description, understanding, and quantitative determi-nation of the ocean tides has attracted a long series ofbrilliant mathematicians and physicians in the history ofscience. In the early eighties, computer technology allowedus, for the first time, to generate global, realistic tidalmodels, notably the Naval Surface Weapon Center(NSWC) model, which was based on the solution of thehydrodynamic equation of the tides constrained with tidalgauge observation (Schwiderski 1980). Although it hadunprecedented accuracy, the NSWC model was notaccurate enough for the forthcoming satellite altimetrymissions, which depended on precise tidal estimates fororbit determination and for ocean sea level de-aliasing.Within the framework of the Topex/Poseidon (T/P)mission, the international tidal community has undertakena huge effort to improve or develop new tidal models, withthe objective of attaining a centimetre accuracy level intidal prediction, necessary to meet the altimetry productrequirements (Le Provost et al. 1995). This internationaleffort quickly split into two main approaches: the so-calledempirical approach based on the direct analysis of thealtimetry sea level time series (initiated by Cartwright andRay 1991), and a modelling approach based on hydrody-namic and assimilation models. Later on, the interactionbetween the two approaches (i.e. data assimilation based onaltimetry analysis on one hand, and hydrodynamic/assim-ilation modelling on the other hand) was a key factor forthe overall success in improving tidal prediction accuracyand reaching the T/P requirements.

Within the tidal community, the Grenoble groupModélisation des Ecoulements Océaniques à Moyenne etgrande échelle, led by C. Le Provost, followed the secondapproach by developing the hydrodynamic model Codeaux Eléments Finis pour la Marée Océanique (CEFMO)and the associated assimilation model Code d’Assimilationde Données Orienté Représenteur (CADOR). The tidalatlases produced from those models received the genericname of finite element solutions (FES). The major releasesformed a nearly continuous bi-annual production between

Responsible editor: Bernard Barnier

F. Lyard (*) . T. LetellierLaboratoire LEGOS, UMR 5566, Observatoire Midi-Pyrénées,Toulouse, Francee-mail: Florent.Lyard@cnes.fr

F. LefevreCLS,Ramonville St. Agnes,Toulouse, France

O. FrancisUniversité du Luxembourg, ECGS,Walferdange, Luxembourg, Belgium

1992 and 2004, including the FES95 (Le Provost et al.1998), FES99 (Lefevre et al. 2002) and FES2004 tidalatlases. The spectral and finite element characteristics ofthe CEFMO model proved to be key factors for the successof the FES atlases. The relatively cheap computational costof the spectral system allowed us to optimise the number offinite elements in the ocean discretisation, hence improvingthe tidal energy dissipation in shelf regions and local tidalconditions with unchallenged precision for a global oceanmodel. In addition, the early introduction of loading andself-attraction (LSA) terms in the tidal equation allowed usto compute highly accurate hydrodynamic solutions forbasin scale applications, especially in the Atlantic Ocean.In parallel, the CADOR assimilation code was developedfrom the original Bennett and McIntosh (1982) representerformulation (see also Egbert et al. 1994). Due to computerlimitations, the global tides are obtained from basinsolutions obtained from a Schur (or block resolution)-likemethod developed in 1997. Because of the quasi-linearityof the CEFMO equations (see Section 2.1), this approach ishighly efficient and is equivalent to the resolution of thetrue global problem. The first published tidal atlasproduced by using these techniques was the FES99 atlas.Continuous improvements in the model and data havemotivated the production of further atlases, of which themost recent release is the FES2004 atlas.

The aim of this paper is to provide a global overview ofthe modelling and data assimilation approach as well as therecent FES solutions. Because the FES2004 atlas is likelyto be the last tidal atlas produced from the CEFMO andCADOR models, the authors wish to highlight sometechnical issues that were not addressed in previouspublications. Sections 2 and 3 will briefly introduce thetidal equations for the hydrodynamic and assimilationcodes. The FES2004 atlas and its validation are developedin Section 4. Details of the atlas production are given andsome particular aspects of the tidal spectrum componentsare also discussed. Section 5 is devoted to tidal energydissipation, introducing the newly added parameterisationof barotropic to baroclinic energy conversion. This lastpoint is one of the major breakthroughs of the altimetryobservation mission (Egbert and Ray 2000), not only forthe tidal modelling itself but also because of its implica-tions for the general ocean circulation. This paper isdedicated to Christian Le Provost, who passed away in2004. His leadership and vision in tidal science has broadlycontributed to the great success of modelling ocean tidesduring the past decades and, consequently, of the satelliteocean altimetry mission.

2 CEFMO tidal equations

2.1 Fundamental hydrodynamic equations

The purpose of this section is to briefly introduce thedifferential equations used in the hydrodynamic solvers

and data assimilation software. The detailed developmentscan be found in previous publications (Le Provost et al.1981; Le Provost and Vincent 1986). The CEFMO tidalequations are derived from the classical shallow waterequations where the advection and horizontal viscosityterm is neglected. Assuming the existence of a dominanttidal constituent (in terms of currents), the friction terms areobtained in a tensor form as a linear development of thenon-linear bottom drag given by:

τ ¼ C

Huk ku (1)

where u is the depth-averaged tidal velocity, H the meanlocal depth and C a dimensionless friction coefficientusually taken as 2.5×10−3. Once the problem is fullylinearised, we can solve complex equations in the spectraldomain. The quasi-linearised, complex tidal equations canthen be written for any astronomical tidal constituent as:

jωαþr � Hu ¼ 0 (2)

jωuþ f � u ¼ �gr � αþ δð Þ þ grΠ�Du (3)

where j ¼ ffiffiffiffiffiffiffi�1p

, ω is the tidal frequency in rad/s, δ theocean bottom radial displacement, α the ocean tideelevation, Π the total tidal potential (i.e. astronomicalplus loading/self-attraction potential), D the drag tensor,f ¼ 2Ωk the Earth rotation vector and g the gravityconstant. The boundary conditions are the classical zero-flux Neuman conditions on tidal currents along the rigid (orclosed) boundaries ∂Ωo, and clamped (or Dirichlet)conditions on tidal elevations at the open ocean boundaries∂Ωo.

In the early version of CEFMO, the complex tensor

D ¼ r r0

r00 r000

� �is limited to the bottom drag contribution,

and its coefficients depend only on the dominant wavecurrents (as discussed in Section 5, additional terms havebeen added later to account for the wave drag energydissipation). The bottom friction tensor is identical for alltidal components, except the dominant wave itself.Because the tidal velocities for the dominant componentare not known before the tidal equations are resolved, weuse an iterative method initialised with approximatecurrents (about 1 m/s). Quasi-convergence is reachedafter a limited number of iterations, typically less than 10(i.e. convergence of the currents nearly occurs everywhereexcept in some very non-linear, shallow water regions).Once the dominant constituent is resolved, the rest of thetidal spectrum can be solved in a one-step manner. Inpractice, we extend the method to a dominant wavecouple, namely, M2 and K1, which is solved in anembedded iterative scheme, to account for the few localregions where the diurnal tidal currents are larger than thesemi-diurnal ones.

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The ocean bottom radial displacement δ(λ,φ) due to thesolid Earth tides and loading effects is:

δ λ;ϕð Þ ¼ h2Πa λ;ϕð Þ þZ

Gδ λ;ϕ;λ0;ϕ

0� �

α λ;ϕð Þds(4)

where h2 is the deformation Love number, Πa(λ,φ) theastronomical potential expressed in metres (i.e. divided bythe gravity constant g), and Gδ the elastic Green’sfunctions. The total tidal potential is the sum of theastronomical tidal potential plus the tidal self-attraction andgravity perturbation potential due to the ocean bottomdisplacement:

Π λ;ϕð Þ ¼ 1þ k2ð ÞΠa λ;ϕð ÞþZ

Gπ λ;ϕ;λ0;ϕ

0� �

α λ;ϕð Þds (5)

Here,RGδ λ;ϕ;λ

0;ϕ

0� �

α λ;ϕð Þds andRGπ λ;ϕ;λ

0;

�ϕ

0 Þα λ;ϕð Þds are the convolution integrals between a pre-

existing tidal solution and the appropriate Green’s func-tions (see Farrell 1972; Francis and Mazzega 1990). Inpractice, the convolution terms in Eqs. 4 and 5 are puttogether in a so-called LSA term. In theory, this linear termcould be integrated implicitly into the tidal equations.However, this would make the tidal dynamic system matrixdense instead of band, because of the systematic node-to-node connections coming from the global ocean convolu-tion, and thus making the matrix practically impossible toinvert even with heavy computational means. The explicitmethod (i.e. LSA computed from a pre-existing tidalsolution) is much more efficient, but at a price! Firstly, thenew tidal solution may be inconsistent with the LSAprescribed in the forcing terms, which can induce anerroneous non-zero global rate of work of the LSA forcing.Secondly, this approach is less accurate in shelf and coastalregions, because here the tides and LSA show very rapidspatial variability, which can be captured differently by thetidal model and the pre-existing solution used to computethe LSA terms. Using a standard iteration scheme tosuccessively solve the tides, and then estimate new LSAforcing from the so-obtained solution, and so forth, isknown to be poorly convergent and inefficient. However,this technique is more successful when iterating not on theLSA itself but on the departure of LSA from its linearapproximation [such as suggested by Accad and Pekeris(1978), and carried out by Egbert et al. (2004)]. This linearapproximation (i.e. LSA being taken as about 10% of thelocal tidal elevation) can be easily taken as implicit in thetidal system without damaging the matrix bandwidth, andthe convergence on the LSA is proved to be rapid.

A similar form of the equations can be derived for thenon-linear constituents, where the tidal potential forcing

terms are set equal to zero and are replaced by the non-linear terms that apply to astronomical waves. For instance,the M4 constituent linearised equations are given by:

jωαþr � Hu ¼ �r � ϕc ηM2;uM2

� �(6)

jωuþ f � u ¼ �gr α� δð Þ �Du� ϕm uM2 ;ruM2ð Þ(7)

where ϕc ηM2;uM2

� �and ϕm uM2 ;ruM2ð Þ are the non-

linear forcing functions expressing the complex contribu-tions of the M2 constituent at the M4 frequency in thecontinuity and the momentum equations, respectively.These functions show significant value only where uM2 ,ruM2 and ηM2

are substantial, namely in the shallow seas.In practice, the computation of non-linear constituents fromthe spectral model is a tedious task with only limitedbenefits for deep-ocean-tide prediction (which was theprimary objective of the FES series of atlases). Therefore,the non-linear tides were not included in the early FESatlases. With the growing focus on shelf and coastalregions, the non-linear constituents can no longer beignored. For the FES2004 model, the MOG2D time-stepping, non-linear, shallow-water, finite-element P1model (Carrère and Lyard 2003) was used to compute thenon-linear constituents on the FES2004 grid, and the M4

constituent has now been added to the FES2004 atlas (seesection below).

2.2 Discretised equations

The linearised momentum Eq. 3 can be expressed in amatrix form as:

Hu ¼ M rα� Fð Þ (8)

where M ¼ � gHΔ

iωþ r000 f � r0

�f � r00 iωþ r

� �and

Δ ¼ detiωþ r r0 � f

r00 þ f iωþ r000

!Substituting Eq. 8 in Eq. 2 yields the spectral wave

equation:

S α½ � xð Þ ¼ 1

κiωαþr �Mrαð Þ xð Þ ¼ Ψ xð Þ

¼ 1

κFα þr �MFð Þ xð Þ

(9)

where κ is a numerical normalisation factor and Fα is zerofor astronomical tides. The tidal equations are discretisedover a triangular mesh and solved under the weak

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formulation. For any computational node n in the domain(except the open-boundaries nodes, where clamped condi-tions in elevation are applied), the weak formulation of thewave equation takes the form:

S α½ �;βnh i ¼ZΩ

S α½ � xð Þβn xð Þds ¼ZΩ

Ψ xð Þβn xð Þds

(10)

where βn is the P2-Lagrange interpolation associated withnode n (piecewise second-order interpolation). Green’sformula is used to transform the wave equation in thefollowing manner:ZΩ

r � Huð Þβnds ¼ZΩ

r � βnHu� Hu � rβnð Þds

¼IΓ

βnHu � ndl �ZΩ

Hu � rβnds (11)

Finally, the discrete wave equation is solved in thefollowing form:

S α½ �;βnh i ¼ZΩ

iωαþrβn �Mrαð Þ xð Þds

¼ZΩ

rβn �MF xð Þds(12)

The boundary integral vanishes under the rigid (zero-flux) boundary conditions. Once the tidal elevation systemis solved, the tidal currents can be deduced from Eq. 8.Note that explicit computations with Eq. 8 are ill-defined atfinite element edges and vertices (cross-edge discontinu-ities coming from triangle dependant derivation), andtherefore a weak formulation approach is, in principle,needed to solve for the tidal currents.

2.3 Model finite element mesh

The two main advantages of using a finite-elementdiscretisation are to better follow the coastline geometryand to optimise the trade-off between computational costand model accuracy. Away from the coastal boundaries, theresolution is constrained by the classical wavelengthcriterion set on the finite element sides elements(Le Provost and Vincent 1986; Greenberg et al. 2006):

λη ¼ T

15

ffiffiffiffiffiffiffigH

p(13)

where λη is the element’s side length, T the tidal waveperiod, g the gravity constant and H the local depth. Theminimum number of Lagrange-P2 computational pointsover a tidal wavelength was set to 30, corresponding to 15finite elements. As semi-diurnal tides demand a higherresolution compared to the diurnal and long-period tides,T is taken to be the period of the M2 tide. This criterion willincrease the model resolution in the shallower areas, suchas the continental shelves. The FES94, FES95 and FES99atlases have been computed on the Global Finite ElementModel-1 (GFEM-1) triangular finite element mesh de-signed originally for early basin-wide studies at thebeginning of the T/P project. Its resolution is about20 km along the coastlines (i.e. corresponding to a 10-kmresolution in Lagrange-P2 discretisation), with maximum-elements side reaching 400 km in the deepest part of theglobal ocean (i.e. corresponding to a 200-km numericalresolution). The total number of computational nodes inGFEM-1 is about 340,000. However, due to its piecewiseconstruction, the GFEM-1 mesh was rather inhomoge-neous in terms of spatial resolution. It also did not includesome marginal seas, such as the Persian Gulf, the Red Seaand the Gulf of Maine, which were discarded to savesome computational CPU time. A decision was made torefurbish the global mesh for the tidal atlases beginningwith FES99 (Lefevre 2004). The GFEM-2 finite-elementmesh was re-designed to homogenise, improve andextend the finite element mesh and provide morecomplete ocean coverage, including some missing mar-ginal seas, such as the Black Sea, the Baltic Sea and theBay of Fundy. In addition, a new geometric constraintwas added to improve tidal current solutions over steeptopographic slopes:

λu ¼ 2π

15

H

rHj j (14)

where λu is the element’s side length, H the local depthand rHj j the modulus of the local slope. This criterionwill increase the model resolution at the shelf break andover steep volcanic ridges. This criterion is much moredemanding than the wavelength criterion, and it has beenonly partially applied to constrain the new mesh. The newGFEM-2 mesh, shown in Fig. 1, contains about 500,000elements and 1,000,000 computational nodes.

3 CADOR assimilation method

The CADOR assimilation software is based on therepresenter approach (Bennett and McIntosh 1982; Bennett1990; Egbert et al. 1994). The full development of theassimilation code is described by Lyard (1999), and herewe present the key points for assimilating pointwise,harmonic sea elevation data. We define the linearobservation operator, which corresponds to available sea-

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surface harmonic data dk, located strictly inside themodelling domain under the integral form:

Lk αð Þ ¼ZΩ

μk� xð Þα xð Þds (15)

where α is the ocean tide complex elevation and μk isequivalent to the kernel of the observation operator. In thecase of pointwise observations, μk is the Dirac functionassociated with the kth data spatial location xk. Consideringthe whole set of assimilated data, the model misfit vectorcan be expressed as e ¼ d� L α½ � ¼ dk � Lk α½ �½ �. Theassimilation problem is based on L2 norm (quadratic) costfunction, defined as follows:

J αð Þ ¼ e�C�1" eþ

ZΩ

@ψ� xð ÞC�1i @ψ½ � xð Þds

þZ@Ω0

@αo� xð ÞC�1

o @αo½ � xð Þdl

þZ@Ωc

@Φ� xð ÞC�1i δΦ½ � xð Þdl

(16)

where C" is the error covariance matrix for observations,Ci, Co and Cc the error covariance operators associatedwith interior forcing errors, open boundary conditionserrors and rigid boundary conditions errors, respectively.

The first term quantifies the misfit between the α solutionand data and the last three terms quantify the misfitsbetween α and the prior solution taken as a function of theinternal forcing and boundary conditions. The assimilationsolution is the α field that minimises the above costfunction.

3.1 Representer approach

The minimisation of the cost function can be done bydifferent techniques. In CADOR, we follow the representerapproach which has the great advantage of being dimen-sioned by the number of assimilated data K. Ignoring non-trivial mathematic details and assuming that the linearoperators Ci, Co and Cc have an inverse, we define the C-scalar product as follows:

α1;α2h iC ¼ZΩ

S α1½ �ð Þ�C�1i S α2½ �½ �ds

þZ@Ω0

α1�C�1

o α2½ �dl

þZ@Ω0

Mrα1 � nð Þ�C�1c Mrα2 � n½ �dl

(17)

for each assimilated datum, the kth observation linear formcan be associated with a representer field rk defined over

Fig. 1 GFEM-2 finite element mesh (FES2002/2004 tidal atlas)

398

the domain so that the observation of the α field is equal toits C-scalar product with rk:

Lk αð Þ ¼ rk;αh iC ¼ZΩ

S r½ �ð Þ�C�1i S α½ �½ �ds

þZ@Ω0

r�C�1o α½ �dl

þZ@Ω0

Mrr � nð Þ�C�1c Mrα � n½ �dl

(18)

Then, the assimilation solution is sought as the sum of theprior solution plus a linear combination of the datarepresenters, i.e.:

α xð Þ ¼ αprior xð Þ þXKk¼1

bkrk xð Þ (19)

At this point, we have reduced the minimisation problemfrom an infinite dimension to a finite dimension, with Kdegrees of freedom for the bk coefficients. The original costfunction simplifies into two terms, the first one beingunchanged and the last three concentrated into a uniqueterm accounting for the entiremisfits with the prior solution:

J αð Þ ¼ d� L α½ �ð Þ�C�1" d� L α½ �ð Þ þ b�Rb (20)

where Rij ¼ ri; rj

C¼ Li rj

� � ¼ ri; rj �

C¼ L�j ri½ �:R is the

Hermitian representer matrix. The bk coefficients thatminimise Eq. 20 are solutions of the K×K system:

RþC"ð Þb ¼ eprior (21)

Therefore, the data assimilation solution burden consistsmainly in computing the representers associated with thedata.

3.2 Representer equations

It can be shown that the representer is the solution of a two-step system. To simplify the following equations, we setMrα ¼ α, Mrr ¼ r and M�rη ¼ η, and we drop thedata index. The first step involves developing a so-calledbackwards, or adjoint, model of the hydrodynamic waveequation model:

S} η½ � ¼ 1

κ�iωηþr �M�rηð Þ ¼ μ (22)

with open boundary conditions

η xð Þ ¼ 0 8x 2 @Ωo (23)

and rigid boundary conditions

η � n ¼ 0 on @Ωc (24)

where M* is the canonical adjoint matrix of the matrixoperator M, as defined by u;Mvh i ¼ M�u; vh i. Namely,M* is the transpose conjugate matrix ofM defined in Eq. 8.In our case, it can easily be seen that:

M� ¼ � gH

Δ��iωþ r000� f � r0�

�f � r00� �iωþ r�

� �(25)

Once the adjoint system has been solved for η using Eqs.22 – 24, the representer can then be determined by solvingthe following system (forward step):

S r½ � ¼ Ci η½ � (26)

with open boundary conditions

r ¼ Co ν þ 1 κη � n� �

on @Ωo (27)

and rigid boundary conditions

r � n ¼ 1=κCc η½ �on @Ωc (28)

As with the direct hydrodynamic problem, the η and rsolutions are computed basin-wise, and the blockresolution technique is used to obtain the global gridsolution (see Appendix).

4 FES2004 atlas

The CEFMO model was initially developed for the shelfregions of the European seas and it was later extended forbasin- and global-scale applications. The global tidalequations could not be solved at once with the bestavailable computer national resources (vector super-computers at the Institut du Développement et desRessources en Informatique Scientifique), and thus, theglobal tides were solved from basin-scale solutions withelevation boundary conditions mostly extracted from theNSWC model and from the geodetic satellite (GEOSAT)empirical model (Cartwright and Ray 1991), and from tide-gauge data in dynamically complex regions such as theIndonesian Archipelago. The first global atlas released forthe community was the FES94 and FES95 set. Theseglobal solutions were built from the separate basinsolutions using a global optimal inversion method (inter-basin open boundary conditions optimisation in FES94,basin-wide variational assimilation in FES95), to whichcontinuity conditions at the inter-basin open limits wereadded. Despite their remarkable accuracy, these atlasessuffered from the limitations of the simplified assimilation

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and continuity techniques used at that time. This problemwas solved in 1997, with the development of the fullvariational assimilation code CADOR and a more effectiveblock resolution technique (see Appendix). The FES98atlas [tide gauge assimilation, Lefevre (2004)] and theFES99 atlas (tide gauge and altimetry assimilation) wereproduced using these improved methods, showing again asignificant improvement compared to the previous atlases.A decision was made to refurbish the global mesh (fromGFEM-1 to GFEM-2) and to produce a new global atlas.FES2002 was the first released atlas computed on of theGFEM-2 mesh. Unfortunately, the altimetric data assimi-lated in this atlas suffered from an improper invertedbarometer correction that could have affected some tidalconstituents, especially K1. To fix this problem and to takeadvantage of a better de-aliasing correction (MOG2D-Gsea level simulations), the altimetric data were re-analysedand assimilated into the FES2004 atlas (Letellier 2004).Compared to the data assimilated in FES2002, thedifferences with the data assimilated in FES2004 aremainly concentrated in high-latitude regions, where themeteorologically forced sea surface variations are the mostenergetic. In addition, some data were removed from theassimilation set because of their strong contamination bynon-tidal dynamics.

4.1 Description of the FES2004 tidal spectrum

Four main tidal constituents (Mf, K1, M2 and M4) plus S1are shown in Fig. 2. The digital FES2004 atlas (tidalelevation and loading/self-attraction) is available at theLaboratoire d’Etudes en Géophysique et OcéanographieSpatiales web site, at http://www.legos.obs-mip.fr/en/soa/,in the “Sea level and tides” submenu. In the followingsections, we will describe some specific aspects of the tidalcomponents of the FES2004 atlas.

4.1.1 The diurnal and semi-diurnal tides

The main semi-diurnal and diurnal tides have beencalculated using the CEFMO and CADOR codes: M2,S2, N2, K2, 2N2 and K1, O1, Q1, P1. The S1 radiational tidewas not computed for the FES2004 atlas, but is availablefrom the FES99 atlas. The S2 and S1 components arediscussed in detail below. To solve for the bottom frictionnon-linearities, M2 and K1 are computed within an iterativeprocess until the M2 and K1 tidal currents converge, andhence, the bottom friction coefficients converge (typically,10 iterations are needed to reach convergence, startingfrom a uniform M2 tidal current of 1 m/s magnitude). Afterthe hydrodynamic solutions are obtained with CEFMO, anassimilation is performed with the CADOR code with datafrom T/P and European remote-sensing satellite (ERS)crossover points, plus the harmonic analysis of tide gaugetime series. Details of the assimilation will be given insection 4.2.

4.1.2 The long-period tides

For a long time, the long-period tides were modelled asequilibrium tides, which are the isostatic (or equilibrium)solutions of the tidal equations. This remains a goodapproximation, as their low frequencies will induce a weakdynamic ocean response to the gravitational forcing.Nevertheless, the presence of continents and the resultingtransport barrier between the ocean basins is responsiblefor a slight departure of the true long-period tides fromequilibrium, as demonstrated by Egbert and Ray (2003).Purely hydrodynamic models are able to capture, withreasonable accuracy, the dynamic parts of the long-periodtides. In FES2004, no assimilation was performed for thosetides. The reason is the unfavourable signal-to-noise ratioin the sea level observations in the long-period band, whichcontains a good deal of non-tidal ocean signals. Theharmonic constants obtained from the analysis of thoseobservations are meaningless except for the largestconstituents in tropical regions (Ponchaut et al. 2001).Even in those low latitudes, the data are of poor interest forthe assimilation, as they are significantly contaminatedwith non-tidal contributions. A good illustration of thisproblem is the unrealistic Schwiderski long-period atlasdue to the nudging of the hydrodynamic solutions towardsirrelevant data. Therefore, the four long-period tidesdistributed with the FES2004 atlas, namely, Mf, Mm,Mtm and Msqm, are the solutions of the purely hydrody-namic CEFMO model.

4.1.3 The S2 and S1 tidal constituents case

Due to the thermal excitation by the solar radiation in theupper atmosphere, internal pressure waves are generatedthat propagate inside the atmosphere and down to theEarth’s surface. The main high-frequency spectral lines(i.e. the ones which concern us in terms of de-aliasing) arethe S2 and S1 atmospheric tides. The ocean response to theatmospheric tides, i.e. the ocean radiational tides, can reachseveral centimetres in amplitude in deep ocean regions, andeven more on some continental shelf seas (some otherastronomical constituents have a radiational contribution,such as K1, but it is usually considered as negligiblecompared to the gravitational contribution). Thus, thosetidal constituents are far from negligible, and theircontribution should be included in the tidal corrections.However, the surface pressure tides are not strictly constantin time, as both the excitation and the propagation of theatmospheric internal pressure waves can vary with time.The significant temporal variability of those tides meansthey should probably be removed from a spectral atlas andtheir correction left to atmospherically forced models (likeMOG2D-G). Unfortunately, the time resolution of the

"Fig. 2 From top to bottom, Mf, K1, S1, M2 and M4 FES2004components. Amplitude (left-hand side) in metres, phase lag (right-hand side) in degrees

400

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available global atmospheric surface field distributions[such as the European Centre for Medium-range WeatherForecasts (ECMWF)] is 6 h (at least for the operationalanalyses products). This allows an acceptable representa-tion of the 24-h pressure oscillations, but is the exactNyquist frequency for the S2 atmospheric tide, making itappear as a stationary oscillation, whereas it should exhibita westward propagation. Thus, ocean models with atmo-spheric forcing, such as MOG2D-G, contain an atmo-spherically driven semi-diurnal signal in their sea levelfields, due to the partial atmospheric tide contribution at the6-h periods of the ECMWF pressure fields. However, thissignal is a distorted representation of the atmosphericforcing ocean response at this period. Further discussion onthis radiational tide problem requires that we split S2and S1.

The S1 tide is mainly composed of its radiationalcomponent, and its gravitational component can beneglected. Estimating S1 using a spectral approach can bequite efficient, through either an empirical model or ahydrodynamic model with assimilation, in which the tidalforcing is obtained from a harmonic analysis of theatmospheric surface pressure. This provides a mean S1 tidethat is fitted over the time period from which theatmospheric tides and harmonic data have been extracted[such an approach was used to compute the S1 constituentin the FES99 atlas using the harmonically analysedECMWF operational analyses; see also Ray and Egbert(2004)]. In general, for spectral models, the overallaccuracy of the global time-stepping hydrodynamic modelsis much better at the diurnal frequency than it is at the moreresonant and dynamically dominated semi-diurnal frequen-cies. The S1 tide can be simulated with reasonable accuracyfrom a purely hydrodynamic time-stepping model, such asMOG2D-G, as long as the atmospheric forcing data arethemselves accurate. The authors strongly advocate for thisapproach in altimetric measurement de-aliasing. The use ofa S1 spectral component for de-aliasing purposes should belimited to data processing where the atmospheric loadingeffects are corrected by using an inverted barometerapproximation instead of sea level elevations computedfrom a storm surge model.

The S2 tide is composed of two contributions: one forcedby the solar gravitational potential and one forced by theradiational one. In the open ocean, the radiationalcontribution is rather small, especially compared to thegravitational one, but it still can reach several centimetres,especially where the S2 tide is amplified in the presence ofthe continental shelf. Unlike the S1 case, the S2 radiationalcomponent is badly represented in the MOG2D-G simula-tions due to the ECMWF products time sampling. Inaddition, the two S2 components are undistinguishable inthe sea level observations, whether altimetric or in-situ. Asassimilation is mandatory to reach the required level ofaccuracy, it would be extremely difficult, and uncertain, toproduce a purely gravitational S2 solution. The mostsensible choice consists of keeping both components

mixed into a unique S2 solution where the radiationalcontribution comes from the use of sea level observations.Reconstruction of a “climatological” S2 air tide can be usedto improve the tidal forcing in the prior hydrodynamicmodel (Carrère 2003; Ray and Ponte 2003; Arbic 2005).Nevertheless, one must keep in mind that the timevariability of this contribution would not be taken intoaccount in the tidal corrections. Further investigations arenecessary to estimate the impact of this time variability onthe tidal correction error budget.

4.1.4 The M4 tidal constituent

Due to the non-linearities in the tidal dynamics, extraconstituents, called overtides or compound tides, aregenerated mainly on the continental shelves. The non-linear constituents can reach large amplitudes in theseregions, and therefore attracted the attention of manyauthors, such as Le Provost and his fundamental contribu-tions for the shallow water tides in the European shelf,especially in the English Channel (Chabert d’Hières and LeProvost 1978; Le Provost and Fornerino 1985). Most ofthose tides are barely significant in the open ocean, and arenot considered to be needed to accurately predict the globalocean tides. Nevertheless, we have paid attention to thesetides for two reasons: firstly, altimetry applications arebeing developed over the coastal regions and oceanshelves, and a precise tidal correction is needed. Secondly,the M4 tide, due to the M2 interaction with itself, is the mostenergetic compound tide in the global ocean, and isexceptional in the sense that we have evidence that it canreach a significant amplitude (about 5 mm to 1 cm) in someparts of the Atlantic Ocean. Due to its non-linear origin, theM4 tide is quite tricky to model from the CEFMO spectralmodel. Assimilation is also a challenge. It can be extractedfrom altimetry only with great difficulty because of itsweak amplitude and short wavelength (Andersen et al.2006). It can be determined from coastal tide-gauge data,but this will generally reflect very local conditions thatcannot be captured by the present resolution of globalmodels, and thus are doomed to a large data representa-tivity error. In the deep ocean regions, the M4 tideamplitude is generally close to the observation noiselevel and, thus, rarely analysed. In summary, tide-gaugedata that are appropriate for assimilation purposes aredifficult to gather. Instead, we have used the MOG2Dmodel to produce a M4 tidal chart on the GFEM-2 mesh. Itmust be noted that the MOG2D basis functions areLagrange-P1, where the CEFMO basis functions areLagrange-P2, which means that the spatial resolution istwice as fine in the CEFMO simulations. However,comparisons between CEFMO and MOG2D simulationsfor the M2 tide show a net advantage for the MOG2D runs,probably due to the better representation of the non-linearinteractions in the time-stepping model.

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4.2 Data assimilation

4.2.1 General case

The altimetry missions have brought to us the mostsynoptic observation network for tidal elevations, and onecould designate the satellite altimetry missions as a globaltide gauge system (Schrama and Ray 1994). Today, theaccuracy of the empirical tidal models such as GOT00[which is a follow-up of the GOT99.2 model; see Ray(1999)] far exceeds that of the purely hydrodynamicmodels for the global ocean tides, and most of the recentprogress in tidal science is directly or indirectly linked tothe tidal analysis of satellite measurements. Nevertheless,some characteristics tend to limit the accuracy of suchanalyses. The major drawback comes from the aliasing ofthe tidal frequencies due to the poor repetitivity of thesatellite missions. For a given constituent, its apparentfrequency projects into the low frequency part of the oceanspectrum (including the zero frequency, as is the case forthe S2 and S1 tides in a sun-synchronous mission). Thealiased tidal components get mixed up with non-tidalsignals (thus penalising the analysis of small-amplitudetidal constituents). The aliased frequency can also be veryclose to another tidal-constituent-aliased frequency, yield-ing unusual tidal separation problems (such as the 10 yearsrequired to separate K1 and Ssa in the T/P time series). Theuse of admittance or ortho-tide techniques (Groves andReynolds 1975; Munk and Cartwright 1966) can partiallyimprove the analysis problem, as well as the use of thecrossover doubled time series. In fact, the tidal analysisneeds to be carefully chosen and tuned for each satellitemission, and the accuracy will be quite heterogeneous fromone tidal constituent to another and from one satellitemission to another. Additional errors can arise frominternal tide, which can generate a significant surfacesignature (which can reach a few centimetres for the M2

tide in numerous regions of the global ocean), or when thetidal distribution shows structures with scales of the orderof 100 km, as can be found in shelf and coastal seas.

For the FES2004 atlas, the assimilated data set iscomposed of 671 tides gauges (Fig. 3a), plus 337 T/P(Fig. 3b) and 1,254 ERS (Fig. 3c) altimetric crossoverpoints. The locations of these data sets are similar to theFES2002 assimilation, except that some altimetric pointshave been removed where the harmonic analysis is badlydamaged by strong non-tidal sea level signals (two T/P datasites in the southeast Indian Ocean, one T/P data site in theGulf Stream and one ERS data site along the Somalicoastline). A tidal spectrum with 20 constituents wasanalysed for all the T/P altimetric data and a tidal spectrumwith 19 constituents for the ERS data (S2 constituentexcluded, see Section 4.2.2). A direct harmonic analysiswas performed in the case of T/P data and a generalisedadmittance algorithm for the ERS data (i.e. additionalconstraints based on the assumption of the ocean responsesmoothness in frequency space, similar to the ortho-tidemethod). As stated earlier, the major difference between theFES2002 and FES2004 solutions is the de-aliasing

correction used before the tide analyses of the altimetricdata. An alternative correction was used instead of theclassical inverted barometer. MOG2D-G is a modelledestimate of the sea surface response to both the atmosphericpressure and the wind forcing. It has already demonstratedits capacity to reduce the altimetric signal variance (Carrèreand Lyard 2003). MOG2D-G has been chosen by theOcean Surface Topography altimetric community as astandard de-aliasing correction of the sea-level response tothe atmospheric forcing. MOG2D-G is needed to removehigh frequencies due to the atmospheric forcing, aliased atthe same frequencies as the tidal frequencies. Using thiscorrection leads to some significant changes in the resultsof the harmonic analyses, especially in the middle to highlatitudes, where the sea level variability due to theatmospheric forcing is larger (Fig. 4). For the M2

constituent, the differences observed in the North AtlanticOcean are about 2 cm for the T/P data and more than 4 cmfor the ERS data (which have more points located over theshelf seas and along the coasts).

4.2.2 Assimilation of the S2 tidal constituent

As stated earlier, the S2 tide is composed of twocontributions: the solar gravitational potential tide and theradiational tide. This particularity implies a differenttreatment for the assimilation procedure, compared to theother astronomical constituents. Firstly, ERS1 and ERS2have a sun-synchronous orbit, so that the S2 tidalconstituent appears as a locally constant sea levelcontribution in the satellite observations, and thus, cannotbe directly analysed harmonically. The ortho-tide orgeneralised admittance harmonic analysis, which, in the-ory, allows us to extract a S2 component from the ERS1/2data, did not yield sufficient accuracy, especially in theshelf and coastal regions where the smoothness assumptionis less justified. In addition, mixing the gravitational andradiational contributions is known to degrade this ap-proach, as it depends on tidal gravity potential consider-ations only. Thus, no ERS1/2 data are assimilated in theFES2004 solution for the S2 constituent. ERS1/2 data werealso discarded from the assimilation in the FES2002 K2

solution, as they appeared to be of poor accuracy (whichmay have been a side effect of the S2 analysis problem inERS). By mistake, ERS1/2 data were included in theFES2004 atlas computation of the K2 tide. Our recom-mendation is to use the FES2002 K2 instead of theFES2004 K2 tidal elevation solution.

Secondly, specific processing is also needed for theT/P altimetric data before tidal analysis. The S2 semi-diurnal signal was removed from the MOG2D-Gcorrection using a harmonic analysis. Consequently, theharmonic analysis of the T/P data, corrected withMOG2D-G, includes the full S2 radiational contribution,as measured by the altimeter. This is also the case for thetide-gauge data, which need no specific correction beforethe tidal analysis, except for the bottom pressure gaugedata, which need an atmospheric pressure correction to

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Fig. 3 a Positions of the 671tide-gauge data assimilated.b Positions of the 337 T/Pcrossover data assimilated.c Positions of the 1254 ERScrossover data assimilated

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convert pressure measurements into sea level data [suchcorrection is usually already done by the data providersfor the coastal tide gauges; it is, unfortunately, not thecase for deep ocean data, such as the InternationalAssociation of the Physical Sciences of the Ocean(IAPSO) data set]. The CEFMO model used to computethe hydrodynamic solution is forced with the gravita-tional potential and does not include the radiationalcomponent of the S2 tidal constituent, but after theassimilation process, the FES2004 solution fully in-cludes the radiational component of the semi-diurnal S2constituent.

4.3 Validation against tide-gauge data

Because most of the available valuable tide-gauge datahave been assimilated in the FES2004 solutions (except forthe IAPSO data set), it is impossible to perform a totallyindependent validation using tide-gauge data. Instead,three tide-gauge data sets have been arbitrarily chosen forthe comparisons. The first two were chosen because oftheir high quality: the world ocean circulation experiment(WOCE) data set composed of about 400 tide gaugeslocated along the coasts and on islands and the IAPSO dataset consisting of 352 tide gauges located in deep oceanregions especially in the North Atlantic and Pacific Oceans.The third one is the ST102 tide-gauge data set, with datamostly located in the deep ocean. It is the historicaldatabase used to evaluate the T/P project tidal models(Shum et al. 1997). Again, the ST102 and WOCEdata have generally been used in FES2004 data assimila-tion, and not the IAPSO database (except for the IAPSOdata selected in the ST102 data set). Model–data compar-ison scores for these three data sets were computed for themain tidal constituents and are presented in Table 1. TheGOT00 empirical tidal model [which is a follow-up ofthe GOT99.2 model; see Ray (1999)] has been included inthe comparison to provide an external reference.

The performance of the three tidal models is very similarin comparison to the ST102 data set, which indicates thatthey are very similar on average over the world’s deepoceans. The same conclusion can be drawn from theIAPSO data set. The differences are more significant on the

Table 1 The global mean rms difference of the tidal models againsttide-gauge data sets, as explained in the text

Model Wave IAPSO WOCE ST102

FES2004 3.8 8.5 1.7FES2002 M2 3.9 9.7 1.7GOT2000 4.6 11.1 1.6FES2004 S2 2.2 4.4 1.0FES2002 2.2 4.8 1.0GOT2000 2.3 5.3 1.1FES2004 N2 1.1 2.2 0.7FES2002 1.1 2.5 0.7GOT2000 1.1 2.4 0.6FES2004 K2 0.9 3.0 0.5FES2002 0.9 2.1 0.5GOT2000 0.9 1.7 0.4FES2004 2N2 – 0.6 0.3FES2002 – 0.7 0.2GOT2000 – – –FES2004 K1 1.4 2.7 1.0FES2002 1.5 2.7 1.0GOT2000 1.4 2.8 1.0FES2004 O1 1.0 1.9 0.8FES2002 1.1 1.9 0.8GOT2000 1.3 2.0 0.8FES2004 Q1 0.5 0.5 0.3FES2002 0.5 0.6 0.3GOT2000 0.5 0.5 0.3FES2004 P1 0.5 1.0 0.4FES2002 0.4 1.1 0.4GOT2000 0.4 1.0 0.4

Units are in centimetres

Fig. 4 Difference for the K2constituent between tide ana-lyses of T/P data correction withMOG2D-G or with the invertedbarometer. The circle surface isproportional to the difference,about 1 mm in the central andabout 1.5 cm in the southernAtlantic Ocean

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WOCE data set and the discrepancies are (not surprisingly)amplified in coastal regions. The comparison shows animprovement in FES2004 compared to FES2002, exceptfor the K2 tide (see above section). To further highlight thedifferences between FES2002 and FES2004, it would benecessary to show regional comparisons all over the globalocean, as is the general case for all the best recent tidalmodels. This is far beyond the scope of this paper (focusingon global aspects), and therefore will be left to futurepublications on regional applications.

4.4 Validation against altimetry data

To further validate the accuracy of the FES2004 atlas at theglobal scale, we compare the associated tidal predictionagainst sea level altimetric measurements. Though themain tidal constituents computed within the FES2004 atlas(hydrodynamic/assimilation solutions) represent the prom-inent part of the tidal spectrum, one must consider theimportance of the other tidal constituents of smalleramplitude (such as Mu2, Nu2, 2Q1, etc.), whose individualcontributions are small but become significant when addedtogether. Consequently, additional semi-diurnal and diurnalconstituents were deduced from a spline and linearadmittance method (described by Cartwright and Spencer1988; Le Provost et al. 1991; Lefevre et al. 2002) andadded to the prediction spectrum. To compute thegeocentric tides that will be compared with altimetrydata, it is necessary to include the ocean bottom verticaldisplacement resulting from the solid Earth tide and thetidal loading. The solid Earth tide can be easily deducedanalytically from the astronomic potential, but the tidalloading needs to be computed from an ocean tide model. Atthe time of writing this paper, the computation of theloading effects derived from FES2004 was not yetcompleted, and the loading effects deduced from FES99are used for validation against altimetry.

The archiving, validation and interpretation of satelliteoceanographic (AVISO) data processing chain has beenused to study the reduction of the standard deviation insea level when the different tidal corrections are applied(all other corrections being identical) for two subsetsof altimetric data [Jason-1 and environmental satellite(ENVISAT)]. The FES2004 correction is compared withthe best former FES model, namely, FES99 (where coastaland pelagic tide-gauge data and deep ocean T/P and ERSdata were assimilated). Both atlases contain the same tidalconstituents. We consider that a lower standard deviation(or the variance) reflects a better tidal model. Note thatthese two altimetric data sets are not assimilated in the

FES99 and FES2004 solutions. For Jason-1, we considercycles 001 to 101 (10 January 2002 to 3 October 2004),and for ENVISAT, cycles 011 to 031 (5 November 2002 to5 October 2004). Nearly 3 years of data are taken intoaccount for Jason-1 (2 years for ENVISAT). Over such atime span, the contribution of long-period tides (such as Mf

or Sa) or tides aliased on a long period (like K1) will bedetectable in our comparisons. More generally, the timeperiod is long enough to provide statistically significantinformation on the tidal model accuracy. The altimetry dataare used in two ways: First, computations are made atcrossover points especially to demonstrate the improve-ments yielded by FES2004 in coastal areas. Second,statistics are calculated on along-track data so as to providean overview of the quality of FES2004 on the global scale.

4.4.1 Standard deviation reductions at crossoverpoints

For each Jason-1 cycle, about 4,000–8,000 (25,000–50,000for ENVISAT) crossover points are available for compar-isons. The difference in the number of points is theconsequence of the orbit characteristics of the two satellitesand the numerous corrections applied before calculatingsea surface heights from the altimeter data. The main defectof FES99 was located in shallow water and coastal areas.Tests with crossover points are undertaken to demonstratethe improvements of FES2004 in such areas. Threeestimates are performed. The first estimate uses the globalsets of crossover data. The second estimate only considerscrossover points located in the deep ocean (at depthsgreater than 1,000 m). The third estimate considerscrossover points in shallow water and coastal areas (atdepths lower than 1,000 m). The comparison results(Table 2) demonstrate the better performances of FES2004on the global scale. Compared to Jason-1 data, FES2004 is12.0% (7.1% for ENVISAT) better than FES99. However,these overall improvements are mainly due to the improvedaccuracy of FES2004 tidal constituents in coastal and shelfareas where it is 30.1% better than FES99 for Jason-1(23.4% for ENVISAT).

These results are confirmed if we look at the standarddeviations calculated cycle by cycle for the two FESmodels for Jason-1 and ENVISAT in shallow water.Indeed, very few Jason-1 cycles showed larger residualstandard deviations for FES2004 than for FES99 (Fig. 5a).However for ENVISAT, which provides nearly 10 timesmore data in shallow water than Jason-1, FES2004 alwaysperformed better than FES99 for the 21 considered cyclesof the satellite (Fig. 5b). The complete set of results at

Table 2 Reduction in standarddeviation at crossover points

Units are in centimetres

Satellite Solution Global ocean Depth>1,000 m Depth<1,000 m

Jason-1 FES99 9.3 6.8 20.8FES2004 8.3 6.8 15.9

ENVISAT FES99 10.6 8.8 17.9FES2004 9.9 8.8 14.5

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crossover points leads to the conclusion that FES2004performs better in coastal areas without decreasing theaccuracy in the deep ocean when compared to FES99.

4.4.2 Standard deviation reduction along tracks

In addition to the crossover analysis, a huge amount ofaltimetry data is also available along track: about 400,000data points per cycle for Jason-1 and about 1,400,000 forENVISAT. This allows for highly valuable statisticalstudies of the along-track standard deviation reduction.For the two satellites, the variances of the altimetricresiduals were computed with exactly the same geophysi-

cal data record corrections, except for the tidal corrections(either FES99 is used or FES2004). The difference betweenthe two sets of sea level anomaly (SLA) variance is shownin Fig. 6a for Jason-1 and in Fig. 6b for ENVISAT. Theblue colours in the maps correspond to lower sea levelvariances when corrected with FES2004 rather than FES99correction. The differences are around 0±0.05 cm. Areasshaded light green to yellow highlight the locations whereFES2004 and FES99 perform almost the same (essentially,the deep ocean).

The main differences occur in coastal areas andcontinental shelves. These areas are mainly in deep blue(variances lower than 1 cm2), which means that FES2004correction performs better than FES99 (Indonesian Seas,

Fig. 5 a Jason-1 SSH crossoverstandard deviations forFES2004 (red) and FES99(black) for shallow water(depths less than 1,000 m).Units in centimetres.b ENVISAT SSH crossoverstandard deviations forFES2004 (red) and FES99(black) for shallow water(depths less than 1,000 m).Units in centimetres

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China Seas, Okhotsk Sea, Aleutian Islands area, HudsonBay, Patagonian shelf, east and west coasts of the AtlanticOcean...). These results are similar for Jason-1 andENVISAT, and confirm the results of the crossover studies.Depending on the satellite, several areas are coloured indeep red (variances higher than 1 cm2): FES99 performsbetter in these locations (for Jason-1: northeast coast ofAustralia, eastern part of the equatorial Pacific Ocean,some parts of the Antarctic Ocean; for ENVISAT: theArabian Sea, some parts of the Antarctic and ArcticOceans). We do not have an explanation for theseobservations, but we make two points. First, the assimila-tion scheme is quite complex to adjust at the global scalebecause one data assimilated in a given location can modifythe tidal solution at the other side of the world ocean(typically, the open ocean data representers show, at least,basin-wide correlations). Second, the loading effects used

for tidal corrections are computed with the LSA deducedfrom FES99. Consequently, they are not fully coherentwith the FES2004 tidal elevations (contrary to FES99elevations). However, the FES2004 achieves its mainpurpose: improving the previous best available FES atlas(FES99) in coastal areas and on continental shelves whilekeeping the best possible accuracy in the deep ocean.

4.5 FES2004 LSA atlas

The loading radial deformation and gravity potentialperturbations have been recently computed from theFES2004 atlas gridded fields (by opposition to the finiteelement fields). The overall distribution of these terms isvery similar to that computed from the FES99 atlas.Nevertheless, significant differences appear in coastal and

Fig. 6 a Difference of variancesbetween FES2004 and FES99along Jason-1 tracks. Units arein centimetres squared. b Dif-ference of variances betweenFES2004 and FES99 alongENVISAT tracks. Units are incentimetres squared

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shelf regions. The first reason for this is the difference inthe tidal elevations themselves. The second reason, andprobably the more significant, is the grid resolution used tocompute the LSA maps. The FES99 mesh has an average10-km-alongshore resolution, whereas the FES2004 has a7.5-km resolution. However, the FES99 atlas was projectedonto a 0.25-degree grid for distribution, whereas theFES2004 was projected onto a 0.125-degree grid. The finerhorizontal scales at the ocean and continental limits arethus much better captured in the FES2004 SLA atlas. Forexample, the complex difference between FES99 andFES2004 SLA is very large at the entrance of the EnglishChannel. In the Northeast Atlantic, both SLAs showsimilar, large values due to the open ocean tide influence,which later decrease rapidly upon entering the EnglishChannel because of the diminishing wet/dry land ratio. TheSLA distribution seen in this region is strongly affected bythe tidal model details and resolution and differs sig-nificantly between FES99 and FES2004. In a hydrody-namic model, this effect will be amplified by the smallhorizontal scale of the discrepancy because the dynamicforcing contribution is the gradient of the SLA field.Improving the SLA resolution thus leads to a better coastaltide solution in regions such as the European shelf, with animpact on the global scale if the tidal currents anddissipation are significantly modified in those areas.

5 Tidal dissipation

Initially developed during the 1970s for the shelf regions ofthe European seas, the CEFMO model was speciallydesigned to deal with coastal ocean conditions. Despite aspectral formulation, the wave-to-wave non-linear interac-tions are carefully taken into account within the quasi-linearisation of the shallow water equations. In particular,the bottom friction terms have been developed under theassumption of the existence of a dominant tidal wave (interms of the tidal current magnitude). The modeldiscretisation is based on a finite element grid that allowsus to modulate the grid resolution as a function of the localdynamical constraints. Although the primary objective is toestablish the best possible global tidal elevation solution,the correct representation of the tidal dissipation is also amajor priority. This implies having enough accuracy inregions of strong dissipation, but also implies including allof the dissipation regions, including the Weddell Sea.Modelling of this region, which is mostly covered by athick, permanent ice-shelf, has demanded a tedious, butworthy, effort to compute the thickness of the free seawaterlocated between the bedrock and the bottom of the ice shelf(Genco et al. 1994). Although appropriate bottom frictionhas been implemented, the global tidal energy budget of theFES99 assimilation solutions has highlighted worryinganomalies, especially for tidal dissipation (Le Provost andLyard 1998).

However, due to the coherence of the numerical model,the energy budget of the FES hydrodynamic solutions isfairly balanced. Considering the case of the M2 component,

the rate of work of the astronomical potential forces wasfound to be close to the well-constrained global total valueof 2.45 TW. However, the rate of work of total bottomfriction turned out to be too small (about −2.1 TW).Inversely, the estimated total LSA rate of work (whichshould be zero when integrating over the whole ocean)reached an unphysical significant value compared to theglobal budget (about −300 GW for the M2 component),compensating for the lack of dissipation. The reason forthese anomalies is the inconsistency between the prior tidalelevation used to compute the LSA, and the actual modelsolution, but this inconsistency itself is probably linked tothe dissipation problem. A series of simulations was carriedout to tune the bottom friction coefficient over a large rangeof values, globally and regionally. The overall resultsshowed very limited changes in total dissipation, whereasthe solutions were degraded when the non-dimensionalfriction coefficient in Eq. 1 departs significantly (by anorder 10) from the usual value of 2.5×10−3. The conclusionis that the bottom friction cannot compensate for the lack ofdissipation.

Estimating the energy budget of the M2 FES99assimilated solution is much trickier. The output of theassimilation code CADOR is a tidal elevation solution.Retrieving the velocity field cannot be done directly byapplying the momentum equations; instead, one needs tosolve an over-determined system that includes the conti-nuity equation, with a misfit associated with eachdynamical equation. Even so, no satisfactory velocityfield could be obtained. The rate of work of the totalpressure forces constantly shows a significant non-zerovalue (about 0.6 TW for the M2 component) due topathological current determination (whereas the SLA rateof work was close to zero). In summary, the hydrodynamicmodel lacked dissipation, while a consistent velocity fieldcould not be found for the assimilated solutions, showingthe lack of some physics in our model.

In separate work, evidence of internal tides that areubiquitous in the world oceans was demonstrated from theanalysis of T/P along-track measurements (Ray andMitchum 1996). These observations renewed the questionabout how much of the barotropic tidal energy is convertedinto internal tides and indicated that the conversion mightbe significant. The observed surface signatures of theinternal tide have wavelengths comparable with the firstinternal mode wavelength computed from the Levitusclimatology (Carrère et al. 2004). The internal wavesappear to be generated from the largest topographic slopes,such as the Hawaiian volcanic ridge, then propagate untilthey reach regions with a strong stratification contrast, suchas the equatorial band or the subtropical gyre front, wheretheir surface signature vanishes. Intriguingly, the maximumhydrodynamic inconsistency (between the FES99 assimi-lated elevations and tidal hydrodynamic equations) wasfound at the same ocean ridge slopes. These factsencouraged us to investigate the problem of barotropictidal energy dissipation through internal wave generationprocesses.

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As the CEFMO model is a depth-averaged barotropicmodel, we cannot use it to solve the 3D internal oceandynamics, so a parameterisation has to be added to accountfor the internal wave drag. A great deal of theorical workhas been dedicated to this issue, with a recently renewedinterest in the matter. In short, the various specificapproaches and the necessary prior assumptions restrainthe field of validity of the proposed parameterisation. Forinstance, in the case of small topographic structures (i.e.typical horizontal lengths smaller than the local tidalexcursion), the following parameterisation can be foundwith more or less sophisticated forms in classical literature(see Jayne and St. Laurent 2001; Lewellyn Smith andYoung 2002; Egbert et al. 2004):

τ1 ¼ c1ρ0κ

ωN2b � ω2

� �ω2 � f 2� �� �1

2h2u (29)

where Nb is the Brünt–Vassala frequency at the oceanbottom, κ the typical topography horizontal wave number,h the typical topography vertical dimension, ω the tidalwave frequency, f the Coriolis factor and u the barotropicvelocity. c1 is a tuning coefficient which accounts forparameterisation uncertainties, such as the Nb estimate orthe inadequacy of the model resolution above topographicfeatures. This parameterisation is valid for topographicscales that are typically of the order of O (1,000 m) andless. From altimetry observations, the main sources ofinternal waves are located on much larger topographicstructures, such as volcanic ridges, and an originalparameterisation was designed for the barotropic models,such as CEFMO, to deal with the large topographic scales:

τ2 ¼ c2ρ0κ�1

ωN2 � ω2� �

ω2 � f 2� �� �1

2 rH � uð ÞrH

(30)

where H is the ocean mean depth and N a depth-weightedaverage buoyancy frequency, with weights decreasinglinearly from bottom up to the surface to account for thevertical velocity upward linear decrease. Despite the factthat it was derived by using a different approach, thisparameterisation can be seen as a simplified version of theparameterisation proposed by Bell (1975). The majordifference with the first parameterisation in Eq. 29 is thatthe wave drag has the direction of the topography gradientand not the direction of the barotropic current. Sensitivityexperiences have shown that this second formulation wasbetter suited for the observed energy transfer.

Two independent experiments were used to calibrate thetuning coefficient c2 in Eq. 30 and its impact on the M2

wave energy budget. The first experiment retrieved thevelocity field associated with the FES99 assimilatedelevations, with the objective of reducing the global rateof work of pressure forces to zero. In the secondexperiment, we recomputed a hydrodynamic solution(using GFEM-2 mesh) with the objective of minimisingthe misfits with the ST95 tidal constants data set. The

resulting spatial distribution of the wave drag and bottomfriction dissipation, similar in experiments 1 and 2, areshown in Fig. 7 and Fig. 8. Both experiments led to thesame dissipation estimate, i.e. 1.8 TW bottom frictiondissipation and about 0.7 TW internal wave drag dissipa-tion, with a spatial rate of work distribution that isextremely coherent with the satellite altimetry observationsof internal wave generation. These estimates are also closeto those from other tidal groups (Egbert and Ray 2001). Inaddition, the averaged M2 FES hydrodynamic solutiondiscrepancy was divided by a factor of 2 (from 12 cm rmsdown to 6 cm rms when comparing with the ST95 tidalconstant database) when the wave drag parameterisationwas applied. Contrary to the bottom friction dissipation,which is confined to some limited shelf regions, the wavedrag dissipation is widely spread in the ocean and can bethe dominant dissipation mechanism in basins where thecontinental shelves are of relatively small extent, such asthe Indian Ocean.

As a conclusion, the proportion of M2 barotropic tidalenergy dissipated through the internal wave generation isabout 0.7 TW, or roughly 25–30% of the total M2

barotropic energy dissipation. Approaching or similarnumbers were obtained using different approaches (10%in Bell 1975; 25% in Morozov 1995). The parameterisationof this energy sink is a key point in global tidalhydrodynamic modelling. Many authors have recentlyexamined the problem, including Egbert et al. (2004), whoused a paleo-tidal simulation experiment, but furtherdevelopments are needed to improve the performance ofthe present schemes. Simmons et al. (2004a,b) haverecently investigated the M2 baroclinic tide from anidealised 3D model (i.e. with a uniform stratification),showing the dominance of the first baroclinic mode as itwould be expected from the altimetry observations analysis(still, one must keep in mind that possible higher-ordermodes would be much more difficult to observe fromaltimetry, as their surface signatures will show muchsmaller wavelengths). Their energy conversion budget issimilar to that provided in this paper and in other recentstudies; however, they underlined the limitation of theirexperiment. In particular, the energy balance between theenergy captured over the large topographic features such asthe Hawaii archipelago, or submarine rugged structuressuch as ocean ridges, was found to be highly dependent onthe number of vertical layers in their model. Also, thedissipation of the baroclinic tides was poorly resolved. Forinstance, the baroclinic waves propagating from theTuamotu Islands do not “feel” the equatorial band crossing,whereas the altimetry observations indicate a dissipation ora strong modification of the internal wave regime. In theirtwo layer experiments, the baroclinic tide energy wasconverted back into barotropic energy at a rate (−0.73 TW)comparable to the total barotropic/baroclinic energy con-version budget (+0.67 TW). This could result from the lackof a dissipation mechanism for the internal waves, andtherefore, may vanish in a more realistic simulation. Thedifficulty in undertaking more accurate experiments is dueto the high computational cost of a global, internal tide

410

model, especially if realistic stratification is to be taken intoaccount. Nevertheless, because of the possible role of theinternal tide in the ocean mixing (as described by Munkand Wunsch 1998), the ocean circulation and tidalcommunities have now joined forces to investigate thisimportant matter. Because the simultaneous resolution ofthe ocean circulation and tidal dynamics requires hugecomputational power (mainly because of the reduced timestep and the need for sub-kilometric horizontal resolution),the first step is to parametrise the tidal effects in oceancirculation models and the stratification effects in tidalmodels. Both analytical and numerical attempts have beencarried out, on global (Simmons et al. 2004a,b; Bessières,personal communication) or regional scales. Improvementsin our comprehension and quantitative estimates ofbarotropic to baroclinic energy conversion will allowmajor improvements in future tidal hydrodynamic models.

6 Conclusions

Much progress has been made during the past 15 years inthe field, and our present knowledge of deep-ocean andcoastal tides is considerable compared to what it was beforethe altimetry satellite missions. Not only are the tides nowprecisely known in the global ocean, but we also havelearned and quantified new aspects of tidal dynamics. TheFES2004 atlas is the last and most accurate of the FESatlases. Its performance is slightly improved in the deepocean regions, and more significantly in the shelf andcoastal areas (except for the K2 component, which should

be taken from the FES2002 atlas). However, the spectralapproach has probably reached its limits with the FES2004atlas. It also seems very difficult now to continue toglobally improve tidal models, at least in the deep ocean, astheir accuracy is bounded by the altimetric data errorbudget and analysis limitations. For instance, the presenceof non-tidal ocean dynamics and internal waves, whichcreate a high-frequency surface signature, is an obstacle toimproving our data assimilation results, or indeed toincluding minor tidal constituents in the assimilation listwith sufficient confidence. Further steps based on the sameapproaches as those developed during the last 15 yearswould probably lead to minor changes in the global tidalmodel solution but at an unsustainable cost.

Still, the tidal challenge has not yet ended. The next-generation models will need to improve the tidal solutionsin the coastal and shelf regions, with the objective ofnarrowing the gap between the deep ocean and shallowwater range of prediction accuracy. At the first step,regional hydrodynamic models are needed to study andsolve properly the local, strongly non-linear dynamics overthe ocean shelves. In such regions, it is the authors’ opinionthat assimilation cannot be the ultimate answer, firstlybecause data are much sparser relative to the tidalwavelength, and secondly because the tidal spectrummust be extended to the non-linear constituents which arebarely observed with altimetry. Indeed, the accuracy of thepresent atlas FES2004 relies very much on the dataassimilation and does not truly represent our actual abilityto model the tidal dynamics. We still need to improve thephysics of the tidal models, their overall spatial resolution,

Fig. 7 M2 wave drag dissipation (W/m2)

411

their energy dissipation through the internal wave gener-ation mechanism and the contribution of LSA effects,which are still mostly computed from pre-existing solu-tions. Accurate bathymetry is also needed, and presentglobal and regional databases do not fulfil this requirement.New assimilation techniques are to be developed, compat-ible with time-stepping, non-linear, high-resolution globaltidal models. Indeed, the task of continuing to improveglobal tidal atlases remains.

Acknowledgements Numerical simulations of the hydrodynamicand assimilation codes were performed at the Institut du Développe-ment et des Ressources en Informatique Scientifique computationalcentre (Paris, France). The authors wish to thank all the people whohave collaborated in the development of the FES atlases during thepast 15 years, including the tidal group of the T/P mission, and theCentre National d’Etudes Spatiales for its constant support. Theyalso thank R. Morrow for her encouragement and help in correctingearlier drafts.

Appendix

Block resolution technique

As mentioned earlier, the tidal solutions are computedseparately for each of the main oceanic basins. In the FESatlases prior to FES99, the continuity of the solutions alongthe basins’ open limits is obtained by adding someconstraints in the assimilation procedures, which areperformed on a global basis. This rather simple approach

proved to be not totally satisfying; in particular it wasresponsible for some local inconsistencies (unbalancedmass or energy budget) and some undesirable side effectson the solution along the shorelines and the open oceanboundaries. To overtake this difficulty, we have developedan additional step in the tidal numerical model whichallows us to retrieve the global solutions by mergingtogether the basin-wide simulations. It is based on anapproach similar to the block resolution technique, and theso-obtained solution is exactly similar to the one that wouldbe computed on a global FE mesh (as far as thehydrodynamic model is linear).

Let us consider the global problem, obtained by mergingthe N-1 separate basins, with unknowns sorted by basins,except for the unknowns belonging to the open limitswhich are shared by at least two different basins, which areamalgamated in a Nth vector. Under this formalism, theglobal hydrodynamic problem of N-1 basins is equivalentto the following system:

. ..

0 0 ...

0 Ai;i 0 Ai;N

0 0 . .. ..

.

� � � AN ;i � � � AN ;N

266664377775

X1

..

.

XN�1

XN

2666437775

Y1

..

.

YN�1

YN

2666437775 (31)

where Xi is the interior unknown vector for oceanic basini, XN the shared open limits unknown vector for basin 1 toN-1, Yi the tidal forcing for oceanic basin i and YN thetidal forcing along the shared open limits. The zero blocks

Fig. 8 M2 bottom friction dissipation (W/m2)

412

in Eq. 30 denote the disconnection between inner nodesbelonging to different basins.

The matrix Eq. 30 then yields:

Xi ¼ A�1i;i Yi �Ai;NXNð Þ 8i 2 1; :::N � 1f g

XN ¼ A�1N ;N YN � PN�1

1AN ;iXi

� �8<: (32)

The block solution technique consists in eliminating theXi in the Eq. 31.

AN ;N þXN�1

i¼1

AN ;iA�1i;i Ai;N

!XN ¼YN �

XN�1

1

AN ;iA�1i;i Y1

(33)

The difficulty here is that Eq. 32 involves thecomputation of the inverse of Ai,i, which would normallyprevent us from using this formulation in a direct method.The solution to this problem would be rather to use aniterative solver instead. However, the inverse matrices aremultiplied by matrices such that not all coefficients in theinverse matrices are needed. Actually, the interior un-knowns of basin i (i.e. all unknowns except open boundaryones) can be separated into two groups. Group 1 containsunknowns with no direct interactions with the openboundary unknowns. Group 2 contains the “neighbours”of the open boundary unknowns. Sorting the basin interiorunknowns into groups 1 and 2 allow us to redefine asimpler problem by rewriting AN ;iA

�1i;i ;Ai;N as follow:

AN ;i �A�1i;i �Ai;N ¼ 0 B1;2

� �|fflfflfflfflffl{zfflfflfflfflffl}AN ;i

�C1;1 C1;2

C2;1 C2;2

" #|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

A�1i;i

�0

D2;1

" #|fflfflfflffl{zfflfflfflffl}

Ai;N

¼ B1;2 � C2;2 �D2;1

(34)

The C2,2 term is independently defined for each basin iand has the dimensions of the square of the number ofcomputational nodes which are neighbours of at least oneshared open boundary node. It can be efficiently computedusing an impulse response technique from the basin,boundary condition constrained models. The basin prob-lem with boundary conditions is given by:

Ai;ieAi;N

0 1

� �X0

ieX0N

� �¼ YieXN

� �(35)

where the superscript ∼ indicates that the vector/matrix arereduced to their blocks related with the unknowns of basini. eX0

N is the restriction of the X0N vector and represents the

open boundary conditions for basin i. In our procedure, this

system is solved for each basin using prior boundaryconditions. We can notice that:

Ai;ieAi;N

0 1

� �� A�1

i;i �A�1i;ieAi;N

0 1

� �¼ A�1

i;i �A�1i;ieAi;N

0 1

� �� Ai;i

eAi;N

0 1

� �¼ I (36)

Rewriting Eq. 34 yields:

A�1i;i �A�1

i;ieAi;N

0 1

� �YieXN

� �¼ X0

ieX0N

� �(37)

The most right terms of the right hand-side of Eq. 32 canbe computed from the N-1 solutions of the basin problems,computed in the preliminary step:

AN ;iA�1i;i Yi ¼ AN ;iX

0i �AN ;iA

�1i;ieAi;N

eX0N (38)

Due to the dynamical disconnection of the interior nodesfrom different basins, we can infer the following equality:

eAi;NeX0

N ¼ Ai;NX0N (39)

The solution vector of the shared open boundary nodes isgiven by:

AN ;N �XN�1

1

AN ;iA�1i;i Ai;N

!XN

¼ YN �XN�1

1

AN ;iX0i �

XN�1

1

AN ;iA�1i;i Ai;NX

0N (40)

Due to the integral formulation of the wave equation, theA−,N blocks of the dynamic equations of the shared openboundary nodes can be easily obtained by adding thepartial dynamic equations formed for the basin-wideproblems. After Eq. 39 has been solved, the global solutionis obtained by applying the following formula to eachbasin:

Xi ¼ A�1i;i Yi �Ai;NXNð Þ ¼ X0

i þA�1i;i Ai;N X0

N �XN

� �(41)

Again A�1i;i Ai;N involves only a limited number of

coefficients of the inverse matrix, i.e. the coefficientsrelated to open boundary nodes. Those necessary coeffi-cients can be efficiently computed using an impulseresponse technique from the basin, boundary conditionconstrained models. In practice, the tidal problem is solved

413

for each basin, and impulse response is simultaneouslycomputed for the shared open boundary nodes and theirimmediate neighbours. Then the block resolution is carriedout. The data assimilation uses the same approach for thebackward and forward systems, except that no specific noradditional impulse response computations are needed (theimpulse responses of the adjoint system are the complexconjugate of the impulse responses of the direct system).Note that in a linear problem, the prior boundary conditionsare of no influence at all, and could be set to zero. Inpractice, specifying realistic boundary conditions isnecessary for the (non-linear) dominant wave case andfor computing realistic friction coefficients. This step alsoallows a quality control of the basin’s computation for theother (linearized) waves before passing to the computationof the global solution.

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