O. Andersen, Danish National Space Center, Copenhagen, Denmark,

1. Non-linear tides in shallow water regions from multi-mission satellite altimetry(The Northwest European shelf). O. Andersen, Danish National Space Center, Copenhagen, Denmark, G. Egbert, L. Erofeeva, Oregon State University, Corvallis, OR, USA. R. Ray, NASA GSFC, Greenbelt, MD, USA

2. Overview. • Introduction • Satellite Altimetry and tidal modelling. • Importance of shallow water tides. • Shallow water tides in the English Channel • Tidal Inversion (OTIS) on the NW European Shelf • Summary ESA/CNES – 15 years of progress in Satellite Altimetry | Venice, Italy, March 2006 | OA | page 2

3. Satellite altimetry revolutionized tidal mapping. ESA/CNES – 15 years of progress in Satellite Altimetry | Venice, Italy, March 2006 | OA | page 3

4. Non-linear tides Shallow water Equation. Momentum Continuity Compound tides are generated by nonlinearities in the dynamical equations. Non-linearites in the Momentum Equation (friction, advection). Non-linearities in the Continuity Equation Plus a contribution from boundaries if this is non-zero (M4) The constituent can conveniently be expressed in terms of interacting astronomic constituents. ESA/CNES – 15 years of progress in Satellite Altimetry | Venice, Italy, March 2006 | OA | page 4

5. Importance of Shallow water tides. Shallow water tidal currents are important for Biology – sedimentation - tracer distributions. The interaction with astronomical tides (I.e.) M2 + M4 create the strongest tidal currrents. ESA/CNES – 15 years of progress in Satellite Altimetry | Venice, Italy, March 2006 | OA | page 5

6. Depth (m) Speed (m/s) Wavelength (km) Diurnal Semidiurnal Quarter diurnal Sixth diurnal 4000 1000 500 50 20 198 99 70 22 14 17700 8860 6260 1980 1250 8860 4430 3130 990 625 4430 2215 1565 495 312 2950 1476 1043 330 270 Pros and Cons of Satellite Altimetry • Track ”resolution” is problematic in shallow water. • T/P+JASON 280 km (Equator) – M4 can be accurately determined (>400 repeats) • T/P2 decrease this to 140 km - Few obs will give in-accurate M4 estimate (<75) • GFO 140 km (Equator) – alias problems • ERS/ENVISAT 75 km (Equator) – unsuitable – sun syncronous • Subsampling gives Alias periods/periods (T/P superior) • Signal to ”noise” (2-3 cm for T/P+JASON , larger for GFO,ERS/ENVISAT) • High oceanic variability increase noise – problem on NW European shelf. ESA/CNES – 15 years of progress in Satellite Altimetry | Venice, Italy, March 2006 | OA | page 6

7. Shallow water constituents ESA/CNES – 15 years of progress in Satellite Altimetry | Venice, Italy, March 2006 | OA | page 7

8. Empirical OTM – Importance of T/P 2. M4 from T/P M4 from T/P-2 Mission ESA/CNES – 15 years of progress in Satellite Altimetry | Venice, Italy, March 2006 | OA | page 8

9. Data and Region for OTIS forward model • OTIS modelling by Svetlana Erofeeva and Gary Egbert • Forward non-linear model used to create prior model (1/12º ). • Bathymetry problematic – merged from 6 different sources • Joint model with 11 constituents. (M2,S2,K2,N2,K1,O1,M4,MN4,MS4) • Boundary for all const: TPXO7 • OTIS M4 FES 2004 M4 ESA/CNES – 15 years of progress in Satellite Altimetry | Venice, Italy, March 2006 | OA | page 9

10. Inverse Models – Assimilating altimetry +TG. Use linearlized shallow-water eq For the data assimilation. Standard OTIS only allows for Error in momentum equation. Investigate possibility of error in Continuity equation. Inv-1:Uses all T/P-Jason data on original T/P tracks. Inv-2:As Inv-1, but deleting some data in EnCEC area. Inv-3:As Inv-2, plus Topex2 interleaved data. Inv-4:As Inv-3, plus a subset of gauge data. ESA/CNES – 15 years of progress in Satellite Altimetry | Venice, Italy, March 2006 | OA | page 10

11. Errors in continuity Equation continuity equation Error in in continuity equation for M4 due to M2 forcing: Forcing is highly concentrated in very shallow areas More realistic (localized) forcing errors better inverse solution? Errors in the continuity equation contribute only about 10% to the elevation error, and are thus probably negligible. Standard OTIS covariance was used ESA/CNES – 15 years of progress in Satellite Altimetry | Venice, Italy, March 2006 | OA | page 11

12. M4 comparison Tide Gauge MisfitRMS misfit to Topex Best Worst ESA/CNES – 15 years of progress in Satellite Altimetry | Venice, Italy, March 2006 | OA | page 12

13. INV-4 Model (Most accurate M4 model) ESA/CNES – 15 years of progress in Satellite Altimetry | Venice, Italy, March 2006 | OA | page 13

14. Summary • Assimilation very useful to obtain accurate M4 model • Simple interpolating of altimetry not adequate (because short scales) • Accurate bathymetry + high resolution required (and is a problem) • Fitting Topex improves overall fit to the tide gauges. • Fitting TP2 makes the English channel TG worse. • Fitting a few TGs in the channel leads to the best overall fit • Still outstanding problems with TOPEX versus Tide Gauge data/phases. ESA/CNES – 15 years of progress in Satellite Altimetry | Venice, Italy, March 2006 | OA | page 14