1. Non-hydrostatic effects on internal waves and mixing in the coastal ocean
Jiuxing Xing and Alan M. Davies
(Proudman Oceanographic Laboratory, Liverpool)

2. Motivation
Understand small scale processes (e.g. solitary waves, convection)
Stratified (tidal) flows over the steep topography (e.g., lee waves, flow separation and eddies)
Are current coastal ocean models sufficient (e.g., is non-hydrostatic dynamics important)?
Jonsmod 2006-Plymouth

3. Examples of small scale processes: ISWs of elevation on the Oregon shelf
Klymak and Moum (2003)
Jonsmod 2006-Plymouth

4. ISWs in the Faeroe-Shetland Channel
Hosegood et al (2004)
Jonsmod 2006-Plymouth

5. Stratified tidal flow over sills (Loch Etive, Inall et al, 2004)
Jonsmod 2006-Plymouth

6. Models σ-following coordinate models (e.g., POLCOMS, POM, BOM)
Z-coordinate models (e.g., MITgcm)
The iterative method for non-hydrostatic pressure:
an elliptic equation for the non-hydrostatic pressure
Jonsmod 2006-Plymouth

7. Tests of the MIT model using Lab. exp
Tests of the MIT model using Lab. exp. (Internal solitary waves over a slope (Michallet and Ivey 1999)
Jonsmod 2006-Plymouth

8. Internal solitary waves (model results)
Jonsmod 2006-Plymouth

9. Internal solitary waves (lab experiments vs model results)
Jonsmod 2006-Plymouth

10. A dispersive ISW (small-amplitude)
Jonsmod 2006-Plymouth

11. Large amplitude ISWs on a slope
Jonsmod 2006-Plymouth

12. Tidal flow over a sill – lee wave generation and non-hydrostatic effects
Idealized model setup
M2 tide forced at the seaward boundary
Closed landward boundary
Resolution: dx=10 to 100m, dz=1m
Minimum viscosity (Av=10-3 m2s-1, Ah=10-1 m2s-1, no explicit diffusivity)
Initial zero velocity, N=0.01 s-1
Model domain
Jonsmod 2006-Plymouth

13. Interaction of tidal waves with bottom topography: key non-dimensional parameters
The key physical parameters:
U0, ω0, f, N, h0, L, H.
Non-dimensional parameters:
U0 /(ω0L) the tidal excursion parameter;
h0/H the relative height of the topography;
[(ω02 - f 2 )/(N2- ω02)]1/2 the internal wave ray slope;
h0 /L the topographic slope;
U0 /(Nh0) the Froude number (or h’=Nh0 /U0);
Jonsmod 2006-Plymouth

14. Snapshots of (T,u,w) at 23 mins (4,5,6,7/32 Tm2,non-hydro run)
Jonsmod 2006-Plymouth

15. Snapshots of (T,u,w) at 23 mins (4,5,6,7/32 Tm2,non-hydro run)
Jonsmod 2006-Plymouth

16. Snapshots of Ri number at 4,5,6,7/32T (non-hydrostatic run)
Jonsmod 2006-Plymouth

17. Snapshots of temp & velocity at 4,5,6,7/32T (hydrostatic run)
Jonsmod 2006-Plymouth

18. Temp & velocity at t=2/8T, 3/8T, 4/8T, 5/8T (non-hydrostatic run)
Jonsmod 2006-Plymouth

19. Temp & velocity at t=2/8T, 3/8T, 4/8T, 5/8T (hydrostatic run)
Jonsmod 2006-Plymouth

20. Vertical averaged internal wave energy flux (non-hydrostatic (left) and hydrostatic (right) )
Jonsmod 2006-Plymouth

21. Non-hydrostatic (top) and hydrostatic pressure
Ph and Pnh have a 180o phase shift
Jonsmod 2006-Plymouth

22. In a linear system, Ph & Pnh out phase
Jonsmod 2006-Plymouth

23. nonhydrostatic pressure seafloor value
indirect estimate of hydrostatic pressure by matching isopycnals to streamlines
seafloor value
Jonsmod 2006-Plymouth

24. Power spectral density (w) at two locations, non-hydrostatic (left) vs hydrostatic (right) (N=0.01)
At lower frequency, as predicted by Khatiwala (2003), but not higher frequency.
Jonsmod 2006-Plymouth

25. Power spectral density (w) for a steeper topography (N=0
Power spectral density (w) for a steeper topography (N=0.01), non-hydrostatic (left) vs hydrostatic (right) (N=0.01)
Significant departure from recent theory at both lower & higher frequency.
Jonsmod 2006-Plymouth

26. Summary
Including non-hydrostatic dynamics in the coastal ocean models is feasible.
Model results are encouraging comparing to the laboratory data.
Importance of non-hydrostatic dynamics to lee wave generation and breaking;
Strong kinetic energy spectral peak at higher (lee wave) frequency near the sill - a challenge to observationlists;
Enhanced mixing due to the smaller-scale ripple topography - a challenge to modellers;
More work is needed to assess the model quantitatively and quantify wave drag effects on mixing and circulation (3D effects may be important) .
Jonsmod 2006-Plymouth