Nonlinear internal waves in Massachusetts Bay: Using a model to make sense of observations

1. Nonlinear internal waves in Massachusetts Bay: Using a model to make sense of observations A. Scotti University of North Carolina

2. Many thanks to • R. Beardsley • B. Butman • J. Pineda • R. Grimshaw • NSF and ONR

3. Outline • Geographical setting • Observations • Modeling strategy • Generation/propagation • Shoaling • Late stage propagation • 3D effects

4. Nonlinear internal waves in Massachusetts Bay

5. Observations in Massachusetts Bay • Halpern (J.G.R 1971, J. Mar. Res. 1971) • Haury et al. (Nature 1979, J. Mar. Res. 1983) • Trask and Briscoe (J.G.R. 1983) • Chereskin (J.G.R. 1983) • Scotti and Pineda (GRL, 2004) • MBIWE98 (Scotti et al., JFM, 2006; JGR 2007, 2008)

6. MBIWE 98: Experiment layout

7. Propagation as wave of depression Generation Shoaling and conversion to elevation, along gentle shoaling area

8. 2D Modeling approach • The model solves the Euler equation in 2D along the line joining the MBIWE98 stations. • Spectral discretization • Realistic topography and stratification • Forced with barotropic tide • Hydrostatic approximation recovered if cut-off imposed at large scales O(100 m) (Scotti and Mitran, Ocean Modeling, 2008).

9. Generation/Propagation Effects of environmental (heaving of thermocline) and forcing parameters (spring/neap cycle)

10. Generation: CTD observations (Geyer and Terray, unpublished) and model End of ebb phase Observation Model

11. From standing wave to undular bore Beginning of flood phase

12. Nonlinearity and dispersion effects during generation

13. Standard conditions Spring tide Evolution of the undular bore The model predicts the formation of the undular bore. However, the high-frequency oscillations develop more slowly than observed. Note that rank-ordering not always observed

14. Sample of T record 5 km west of SB (A)

15. Shoaling • Interaction with shoaling topography • Bottom Collision Events (BCEs)

16. Undular bores at the 45 m isobath: Examples from observations. Still well offshore of location where coefficient of KdV quadratic vanishes.

17. Larger amplitude Smaller amplitude Modeled shoaling

18. Modeled temperature field at different depths along the shoaling region =>Eulerian measurements taken at different depts show markedly different time series.

19. Nonlinearity vs. dispersion during BCEs Nonlinearity alone captures essential aspects of BCEs.

20. Nonlinear effects of interaction with topography with a 2-layer hydrostatic model • The propagation speed of a point on the interface depends on the total depth, the thickness of the lower and upper layer and the velocity difference across the layer Total speed Barotropic advection Buoyancy speed

21. In deep water the non linear speed is maximum at the trough thus nonlinearity steepens the front. Past a critical depth, the maximum in c shifts towards the front of the wave, nonlinearity steepens the back, while at the same time the front becomes parallel to the bottom. Water is forced downward along the topography and the flow becomes supercritical. Instabilities develop on the back side. Shoaling in a 2-layer hydrostatic model Speed along inshore-moving characteristics

22. Characteristics along the shoaling area: fully nonlinear vs. weakly nonlinear models. 2-layer, hydrostatic, fully nonlinear. Extended KdV KdV

23. When to expect BCEs. • The undular bore cannot propagate undisturbed past the point where the total depth equals twice the displacement of the pycnocline. • The shoaling bottom acts as a low-pass filter. The high-frequency content is lost to instabilities. The internal tide propagates inshore as a wave of rarefaction followed by a bore that restores the stratification. • The energy dissipated in the process is about 35% of the flux just before the shoaling. Thus, a significant fraction of baroclinic energy is radiated inshore.

24. Life after a BCE. NLIWs in the shallow end of Mass Bay “Square bore” “Triangular bore” • The model indicates that waves reorganize after a BCE. • Possible outcomes include “squared” and “triangular” waves. • Depth of pycnocline in shallow end determine outcome: • if still closer to surface, “square” bores. • if close to middepth, “triangular” bores.

25. Observations of NLIWs in the shallow reach of Massachusetts Bay (depth 25 m).

26. Trapped cores are sometimes found in the trailing edge waves

27. Three-dimensional effects

28. Conclusions • Nonlinearity alone captures essential aspects of physics in Massachusetts Bay during generation and shoaling. • Bottom collision events can be predicted based on 2-layer hydrostatic models. • Evolution after BCEs gives rise to triangular or rectangular bores in the shallow reach. • Trapped cores within waves of elevation are found sometimes in the trailing edge waves.

29. Outstanding issues • Composition of packets highly variable. What controls it? • Energy focusing. How to model it? • Effects of friction and instabilities on formation and propagation of waves with trapped cores. • Mixing and transport.