1. F. Lyard, Y. Soufflet, D. Allain, L. Roblou, LEGOS/CNRS
Application of the T-UGOm spectral solver for the new global tidal atlas FES2012
F. Lyard, Y. Soufflet, D. Allain, L. Roblou, LEGOS/CNRS
M. Cancet, Noveltis
L. Carrère, CLS
Abstract:
The spectral (i.e. harmonic) solvers embedded in T-UGOm have been initially developed to provide tidal currents for open boundary conditions in tidal, time-stepping applications for the case where elevations only are available along open limits. It can also be used for 2D and 3D tidal modeling when computational cost must be kept minimal. It is quite the case for the processing of the new FES2012 global 2D tides atlas, aimed to replace the FES2004 one in the operational altimetry detiding, as data assimilation is based on ensemble technique, thus requiring 100 to 200 simulations to build a suitable members set.
Quasi-linearized spectral equations are solved by using an implicit wave equation form. Horizontal discretization is based on discontinuous non-conforming ncP1 velocities and continuous LGP2 elevation. Non-linearities (bottom friction, advection and non-linear divergence in the continuity equation) are solved through an iterative process. Performances (numerical cost, accuracy), some notable issues, such as resolution impact on velocity solution, and further 3D developments (including 3D baroclinic for internal tides) will be discussed.

2. Motivations FES2004 global tidal atlas is getting out-dated
Nowadays computer power and model capabilities allows for higher resolution
Bathymetry/shorelines databases improvements
Data assimilation improvements:
SpENOI ensemble method (harmonic, mono/polychromatic) more adequate than variational approach (FES2004)
Topex-Poseidon/Jason sea level time series are now close to 20 years of continuous, coherent and high quality observations
20 years is enough to properly analyze deep ocean and shelf seas tidal spectrum
less than 10 years available for FES2004
Challenging tidal correction needs for the SWOT mission (2020)
wide swath altimetry → nearly full coverage of ocean (~1 km resolution)
Internal tide surface signature issue
SWOT mission by it-self will not provide much help for tidal correction along its ground- track

3. Global bathymetry construction
Building the most accurate global bathymetry is the starting issue:
It constrains mesh resolution
It is the first accuracy
Available digital bathymetry:
Global :
GEBCO (IHO) gebco_08 and gridone
ETOPO 1 (found to be inappropriate)
Smith & Sandwell (release 14.X)
Regional:
Hydrographic services (European countries, Australia, Japan, USA...)
Pangea (Antatctic)
IBCAO-IHO (Arctic Sea)
All quality databases used to build a composite global bathymetry

4. Mesh generation Starting from FES2004 mesh Strategy:
Quicker than starting from scratch
Allows for an easier control of final mesh size (less than elements to comply with project's computer ressources for data assimilation)
Strategy:
Keep FES2004 coastal resolution (5 to 7 km in P2 discretisation)
Locally re-sample FES2004 coastlines (Antarctic, Baltic Sea,...)
Diminish open ocean element max size (now 75 km)
Increase mesh resolution above bathymetry slopes (ridges, continental shelves)
Iterative approach (based on test simulation to assess true model accracy improvements)
About 60 iterations to reach FES2012 mesh
Efficient automatic tools developed for mesh splitting/merging and regional (polygon-wise) re-meshing

5. Mesh refurbishing Large size triangle splitting FES2004
Local re-meshing:
major shelves
major ridges
offshore Antarctic
Baltic Sea
FES2004
FES2012
re-meshing regions: mostly ocean ridges and shelves have been regionally reprocessed 5

6. Antarctic processing:
free water beneath ice shelves (Weddell Sea, Ross Sea, Amery Ice Shelf, etc…) deduced from bottom topography and ice draft thickness shorelines follows the grounding line must be consistent with bathymetry
FES2012
shoreline in FES2004
shoreline in
FES2012
FES2004

7. Spectral equations astronomical tides
Quasi-linearised, spectral SW equations
Momentum equation
continuity equation
see Lyard et al., 2006, Modelling the global ocean tides: modern insights from FES2004, Ocean Dynamics
Discrete equations
Wave equation
discrete velocity (more precisely transport) and elevation solution are fully consistent

8. Discrete system Implicit wave equation
Generic coding: only B, M, G and Δ need discretisation-specific routines
Wave equation solved first then momentum
10 iterations cycle (M2, K1) used to solve bottom friction non-linearities
Double complex sparse matrix solver: PASTIX
OpenMP optimized
Stable for large numbers of DoF (UMFPACK is not)
FES2012 performances
~10 Go RAM needed
~1mn / inversion
~30 mn for a typical spectrum (12 constituents, 30 inversions)

9. Spectral solver discretisation
Wave equation constraint: momentum equation inversion
momentum inversion must be easily invertible (block diagonal)
element-discontinuous velocity works, can be chosen as the natural space of elevation gradients
then using orthogonal base functions make it naturally lumped (diagonal). Ex: use discontinuous NCP1 instead of LGP1
Available discretisations
LGP0xLGP1 : cheap, for preliminary testing only
(discontinuous LGP1)xLGP2: artificially lumped
(discontinuous NCP1)xLGP2: naturally lumped ; FES2012 discretisation
(discontinuous Gauss points)xLGP2: increased resolution at velocity points, using quadrature integration ; a good candidate for the next release
Q0xQ1 : quadrangle element, still under developments

10. Model resolution/DoF P0xP1 DGP1xP2 GIP7xP2 DNP1xP2 FES2004
GIP7xLGP2
vertices
triangles
elevation nodes
velocity nodes (not mapped)
4 types of discretisations available: DGP0xP1 DGP1xP2 DG-NCP1xP2 GIP(XX)xP2
FES2012 : DG-NCP1xP2
vertices
triangles
elevation nodes
velocity nodes
h,u
P0xP1 h u
DGP1xP2
GIP7xP2
DNP1xP2 10

11. FES2012 hydrodynamic simulation tuning
M2 bottom friction dissipation (RoW)
Bathymetry improvements (crucial)
Additional mesh resolution increase
Internal tide drag (crucial)
Bottom friction, marginal except for Atlantic Ocean
major dissipation regions used to tune bottom friction coefficient
Thanks to spectral solver low-cost, accuracy improvements have been widely checked during bathymetry construction, mesh refinement and tuning sequences
ocean basins partition used to tune internal drag coefficient

12. State of the art comparison with altimetry-derived harmonic constants

13. Data assimilation Spectral data assimilation code (SpEnOI)
ensemble method within representers approach: perturbations on bathymetry, friction coefficient, wave drag coefficient (~150 members)
same discretisations as in T-UGOm, elevation and currents
polychromatic assimilation available:
minor constituents, especially non-linear tides barely observable
strong correlation between astronomical tides and non-linear tides
astronomical tides data can constrain non-linear tides data assimilation. Ex: M2 and M4 simultaneous data assimilation
Data for assimilation (about 9 000) :
Topex/Jason cross-overs (deep ocean) and along track (shelf seas)
ERS/Envisat cross-overs in very high latitudes (>65°)
if needed, Topex/Jason interleaved (shelves)
if needed, tide gauge data (coastal seas)

14. Data processing
Harmonic analysis of AVISO altimetry sea level time-series (error bars from non-tidal signal contamination estimate)
along-track filtering to remove internal tide surface signature
Use of GLORYS 20 years re-analysys to remove non-tidal annual and semi-annual contaminations (K1 aliased frequancy is 6 months in Topex, 1 year in ERS/Envisat)
M2 data analysis error bars. Main ocean surface circulations show up (>3 cm)
1st baroclinic vertical mode phase celerity. Corresponding M2 internal tide wave-length constrains the along-track filtering window
semi-annual signal in GLORYS-V2, used to correct K1 analysis (cm)

15. M2 solution after data assimilation comparison with altimetry-derived harmonic constants
DTU10
0,4 mm
GOT4.8
0,5 mm
FES2004
0,7 mm
FES2012
0,4 mm
FES2012 assimilated solution close to latest empirical solutions (DTU10 and GOT4,8)
Full atlas release (including tidal currents) in october, 2012 (after Venice OSTST meeting)

16. Latest hydrodynamic experiments
comparisons against TP/J1/J2 (deep / shelf)
State of the art
FES2012
FES2013 M2
24 / 93
13 / 53 S2
10 / 28
9 / 22 K1
11 / 30
10 / 23 O1
12 / 30
7 / 19
FES2012 reference
June 2012 16

17. Bathymetry issue, again ?
Atlantic anomaly
too much energy dissipated in Hudson Strait and Bay
Ungava Bay bathymetry issue:
local M2 resonance ?
spectral solver limitation ?
apparently not…
regional model study underway 17

18. Mesh reworking FES2012 faulty mesh correction resolution increase18

19. resolution issue
more resolution might be necessary to capture velocity details on shelves
discontinuous velocity:
can show noisy solution along the shelf edge
problem disappears when resolution increases
250 m isobath
FES2004 mesh, M2 meridional velocity
NEA regional mesh, M2 meridional velocity
FES2012 mesh, M2 meridional velocity

20. New meshing constraint: true tidal horizontal scales
In addition to usual mesh criteria:
…use true tidal elevation (η) scales:
M2 amplitude
Horizontal length scale from M2, S2, K1, O1 tides (km)

21. Future developments FES2013:
Integrate latest T-UGOm and SpENOI developments
Improve members generation from FES2012 outcomes
Resolution increase
3D applications:
Global spectral-3D, barotropic
Basin/regional spectral-3D, baroclinic (internal tides)
Project website at LEGOS:

22. Tide gauge validation database
Linked with Stammer inter-comparison project
Retreated data:
Deep ocean: DARTS, ACCALAIM, GLOUPS
Shelf Seas: ROSAME, GLOUPS
Coastal Seas: GLOSS, SONEL, BODC
coastal gauges
shelf gauges
deep gauges 22