Modelling the global ocean tides: modern insights from FES2004

Abstract

During the 1990s, a large number of new tidal atlases were developed, primarily to provide accurate tidal corrections for satellite altimetry applications. During this decade, the French tidal group (FTG), led by C. Le Provost, produced a series of finite element solutions (FES) tidal atlases, among which FES2004 is the latest release, computed from the tidal hydrodynamic equations and data assimilation. The aim of this paper is to review the state of the art of tidal modelling and the progress achieved during this past decade. The first sections summarise the general FTG approach to modelling the global tides. In the following sections, we introduce the FES2004 tidal atlas and validate the model against in situ and satellite data. We demonstrate the higher accuracy of the FES2004 release compared to earlier FES tidal atlases, and we recommend its use in tidal applications. The final section focuses on the new dissipation term added to the equations, which aims to account for the conversion of barotropic energy into internal tidal energy. There is a huge improvement in the hydrodynamic tidal solution and energy budget obtained when this term is taken into account.

Appendix

1.1 Block resolution technique

As mentioned earlier, the tidal solutions are computed separately for each of the main oceanic basins. In the FES atlases prior to FES99, the continuity of the solutions along the basins’ open limits is obtained by adding some constraints in the assimilation procedures, which are performed on a global basis. This rather simple approach proved to be not totally satisfying; in particular it was responsible for some local inconsistencies (unbalanced mass or energy budget) and some undesirable side effects on the solution along the shorelines and the open ocean boundaries. To overtake this difficulty, we have developed an additional step in the tidal numerical model which allows us to retrieve the global solutions by merging together the basin-wide simulations. It is based on an approach similar to the block resolution technique, and the so-obtained solution is exactly similar to the one that would be computed on a global FE mesh (as far as the hydrodynamic model is linear).

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

$$ {\left[ {\begin{array}{*{20}c} { \ddots } & {0} & {0} & { \vdots } \\ {0} & {{A_{{i,i}} }} & {0} & {{A_{{i,N}} }} \\ {0} & {0} & { \ddots } & { \vdots } \\ { \cdots } & {{A_{{N,i}} }} & { \cdots } & {{A_{{N,N}} }} \\ \end{array} } \right]}{\left[ {\begin{array}{*{20}c} {{X_{1} }} \\ { \vdots } \\ {{X_{{N - 1}} }} \\ {{X_{N} }} \\ \end{array} } \right]}{\left[ {\begin{array}{*{20}c} {{Y_{1} }} \\ { \vdots } \\ {{Y_{{N - 1}} }} \\ {{Y_{N} }} \\ \end{array} } \right]} $$

(31)

where X _i is the interior unknown vector for oceanic basin i, X _N the shared open limits unknown vector for basin 1 to N-1, Y _i the tidal forcing for oceanic basin i and Y _N the tidal forcing along the shared open limits. The zero blocks in Eq. 30 denote the disconnection between inner nodes belonging to different basins.

The matrix Eq. 30 then yields:

$$ \left\{ {\begin{array}{*{20}c} {{X_{i} = A^{{ - 1}}_{{i,i}} {\left( {Y_{i} - A_{{i,N}} X_{N} } \right)}\;\forall i \in {\left\{ {1,...N - 1} \right\}}}} \\ {{X_{N} = A^{{ - 1}}_{{N,N}} {\left( {Y_{N} - {\sum\limits_1^{N - 1} {A_{{N,i}} X_{i} } }} \right)}}} \\ \end{array} } \right. $$

(32)

The block solution technique consists in eliminating the X _i in the Eq. 31.

$$ {\left( {A_{{N,N}} + {\sum\limits_{i = 1}^{N - 1} {A_{{N,i}} A^{{ - 1}}_{{i,i}} A_{{i,N}} } }} \right)}X_{N} = Y_{N} - {\sum\limits_1^{N - 1} {A_{{N,i}} A^{{ - 1}}_{{i,i}} Y_{1} } } $$

(33)

The difficulty here is that Eq. 32 involves the computation of the inverse of A _i,i , which would normally prevent us from using this formulation in a direct method. The solution to this problem would be rather to use an iterative solver instead. However, the inverse matrices are multiplied by matrices such that not all coefficients in the inverse matrices are needed. Actually, the interior unknowns of basin i (i.e. all unknowns except open boundary ones) can be separated into two groups. Group 1 contains unknowns with no direct interactions with the open boundary unknowns. Group 2 contains the “neighbours” of the open boundary unknowns. Sorting the basin interior unknowns into groups 1 and 2 allow us to redefine a simpler problem by rewriting $A_{{N,i}} A^{{ - 1}}_{{i,i}} ,A_{{i,N}}$ as follow:

$$A_{{N,i}} \times A^{{ - 1}}_{{i,i}} \times A_{{i,N}} = {\underbrace {{\left[ {\begin{array}{*{20}c} {0} & {{B_{{1,2}} }} \\ \end{array} } \right]}}_{{A_{{N,i}} }}} \times {\underbrace {{\left[ {\begin{array}{*{20}c} {{C_{{1,1}} }} & {{C_{{1,2}} }} \\ {{C_{{2,1}} }} & {{C_{{2,2}} }} \\ \end{array} } \right]}}_{{A^{{ - 1}}_{{i,i}} }}} \times {\underbrace {{\left[ {\begin{array}{*{20}c} {0} \\ {{D_{{2,1}} }} \\ \end{array} } \right]}}_{{A_{{i,N}} }}} = B_{{1,2}} \times C_{{2,2}} \times D_{{2,1}} $$

(34)

The C_2,2 term is independently defined for each basin i and has the dimensions of the square of the number of computational nodes which are neighbours of at least one shared open boundary node. It can be efficiently computed using an impulse response technique from the basin, boundary condition constrained models. The basin problem with boundary conditions is given by:

$$ {\left[ {\begin{array}{*{20}c} {{A_{{i,i}} }} & {{\widetilde{A}_{{i,N}} }} \\ {0} & {1} \\ \end{array} } \right]}{\left[ {\begin{array}{*{20}c} {{X^{0}_{i} }} \\ {{\widetilde{X}^{0}_{N} }} \\ \end{array} } \right]} = {\left[ {\begin{array}{*{20}c} {{Y_{i} }} \\ {{\widetilde{X}_{N} }} \\ \end{array} } \right]} $$

(35)

where the superscript ∼ indicates that the vector/matrix are reduced to their blocks related with the unknowns of basin i. $\widetilde{X}^{0}_{N}$ is the restriction of the $X^{0}_{N}$ vector and represents the open boundary conditions for basin i. In our procedure, this system is solved for each basin using prior boundary conditions. We can notice that:

$${\left[ {\begin{array}{*{20}c} {{A_{{i,i}} }} & {{\widetilde{A}_{{i,N}} }} \\ {0} & {1} \\ \end{array} } \right]} \times {\left[ {\begin{array}{*{20}c} {{A^{{ - 1}}_{{i,i}} }} & {{ - A^{{ - 1}}_{{i,i}} \widetilde{A}_{{i,N}} }} \\ {0} & {1} \\ \end{array} } \right]} = {\left[ {\begin{array}{*{20}c} {{A^{{ - 1}}_{{i,i}} }} & {{ - A^{{ - 1}}_{{i,i}} \widetilde{A}_{{i,N}} }} \\ {0} & {1} \\ \end{array} } \right]} \times {\left[ {\begin{array}{*{20}c} {{A_{{i,i}} }} & {{\widetilde{A}_{{i,N}} }} \\ {0} & {1} \\ \end{array} } \right]} = I$$

(36)

Rewriting Eq. 34 yields:

$$ {\left[ {\begin{array}{*{20}c} {{A^{{ - 1}}_{{i,i}} }} & {{ - A^{{ - 1}}_{{i,i}} \widetilde{A}_{{i,N}} }} \\ {0} & {1} \\ \end{array} } \right]}{\left[ {\begin{array}{*{20}c} {{Y_{i} }} \\ {{\widetilde{X}_{N} }} \\ \end{array} } \right]} = {\left[ {\begin{array}{*{20}c} {{X^{0}_{i} }} \\ {{\widetilde{X}^{0}_{N} }} \\ \end{array} } \right]} $$

(37)

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

$$ A_{{N,i}} A^{{ - 1}}_{{i,i}} Y_{i} = A_{{N,i}} X^{0}_{i} - A_{{N,i}} A^{{ - 1}}_{{i,i}} \widetilde{A}_{{i,N}} \widetilde{X}^{0}_{N} $$

(38)

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

$$\widetilde{A}_{{i,N}} \widetilde{X}^{0}_{N} = A_{{i,N}} X^{0}_{N} $$

(39)

The solution vector of the shared open boundary nodes is given by:

$$ {\left( {A_{{N,N}} - {\sum\limits_1^{N - 1} {A_{{N,i}} A^{{ - 1}}_{{i,i}} A_{{i,N}} } }} \right)}X_{N} = Y_{N} - {\sum\limits_1^{N - 1} {A_{{N,i}} X^{0}_{i} } } - {\sum\limits_1^{N - 1} {A_{{N,i}} A^{{ - 1}}_{{i,i}} A_{{i,N}} X^{0}_{N} } } $$

(40)

Due to the integral formulation of the wave equation, the A ₋, _N blocks of the dynamic equations of the shared open boundary nodes can be easily obtained by adding the partial dynamic equations formed for the basin-wide problems. After Eq. 39 has been solved, the global solution is obtained by applying the following formula to each basin:

$$ X_{i} = A^{{ - 1}}_{{i,i}} {\left( {Y_{i} - A_{{i,N}} X_{N} } \right)} = X^{0}_{i} + A^{{ - 1}}_{{i,i}} A_{{i,N}} {\left( {X^{0}_{N} - X_{N} } \right)} $$

(41)

Again $A^{{ - 1}}_{{i,i}} A_{{i,N}}$ involves only a limited number of coefficients of the inverse matrix, i.e. the coefficients related to open boundary nodes. Those necessary coefficients can be efficiently computed using an impulse response technique from the basin, boundary condition constrained models. In practice, the tidal problem is solved for each basin, and impulse response is simultaneously computed for the shared open boundary nodes and their immediate neighbours. Then the block resolution is carried out. The data assimilation uses the same approach for the backward and forward systems, except that no specific nor additional impulse response computations are needed (the impulse responses of the adjoint system are the complex conjugate of the impulse responses of the direct system). Note that in a linear problem, the prior boundary conditions are of no influence at all, and could be set to zero. In practice, specifying realistic boundary conditions is necessary for the (non-linear) dominant wave case and for computing realistic friction coefficients. This step also allows a quality control of the basin’s computation for the other (linearized) waves before passing to the computation of the global solution.