1. Tidal Rectification = Overtides and compound tides Nonlinear effects on tides

2. From Parker (2007) simple sine wave asymmetry between flood and ebb double low waters extreme distortion: tidal bore

3. From Parker (2007) 8 7 5 6 1 2 3 4

4. Nonlinear effects in estuaries (Parker, 1991, Tidal Hydrodynamics, p. 247) We will talk mainly about nonlinear tidal interactions Consider the tide: overtide And the nonlinear term and i = M 2 only

5. If M 2 interacts with S 2 : Nonlinear interactions also arise from bottom friction, which yields: η u|u| and u|u| and from the divergence term in the continuity equation, which is proportional to η u (one dimensional, vertically and laterally integrated equation; b is estuary’s breadth) We then have four mechanisms that generate nonlinearities: Generating mechanisms arise from

6. Nonlinear terms on tidal constituents effect a modulation and a distortion of that constituent Interactions of M 2 with other constituents generate constituents with the following frequencies: σ M2 - σ x σ M2 + σ x 2σ M2 - σ x 2σ M2 + σ x 4σ M2 - σ x

7. M 2 Overtides

8. M 2 interactions with overtides symmetric distortion (by odd harmonic) asymmetric distortion (by even harmonic)

9. Rectified Tide

11. Physical explanation for nonlinear interactions For long waves without friction, the wave propagation velocity C is [ g H ] ½ This is approximately constant throughout the tidal cycle, only if the tidal amplitude η << H, i.e., if η / H << 1 In reality, η / H is not much smaller than 1 and the wave crest will travel faster (progressive wave in shallow water) than the trough, resulting in: energy at M 4 frequency This is the asymmetric effect of the nonlinear continuity term (mechanism A) Difference between sinusoid and distorted wave yields energy in the 2 nd harmonic

12. The tidal current amplitude may be approximated as: This is the effect of the inertial term: ebb flood For η / H > 0.1, u is not negligible with respect to C (as it usually is). Then, the wave propagation velocity at the crest is C + u 0 and the wave propagation velocity at the trough is C - u 0 which results in a similarly distorted wave profile (tidal wave interacting with tidal current): C – u 0 C + u 0

13. Frictional loss of momentum per unit volume is greater at the trough than at the crest. Then, crest will travel faster than the trough; will generate asymmetric distortion and even harmonics (M 4 ) Generating mechanisms arise from Quadratic friction u| u | causes a symmetric distortion, i.e., maximum attenuation at maximum flood and at maximum ebb; minimum attenuation at slack water. This will generate an odd harmonic (M 6 ) Therefore, there are symmetric effects and asymmetric effects Asymmetric Effects generate even harmonics (e.g. M 4 ) because max C and minimum attenuation occurs at crest

14. Symmetric Effects u | u | extreme attenuation at flood and ebb, and minimum attenuation at slack waters Produce odd harmonics, e.g., M 6 because there are 3 slack waters and two current maxima in one period symmetric distortion (by odd harmonic) asymmetric distortion (by even harmonic)

15. Effects of a mean flow (e.g. River Flow) Can be explained in terms of changes in C and frictional attenuation (u | u | ) Mean river flow makes ebb currents stronger  increased frictional loss flood currents weaker  decreased frictional loss This results in greater energy loss than if the river flow was not present, which translates into: reduced tidal range greater damping of tidal wave Friction will now produce asymmetric effects and generation of M 4 Frictional generation of M 6 will continue as long as u R < u 0 so that there are still slack waters greatest attenuation t Flood Ebb Attenuation

16. When u R > u 0 Flow becomes unidirectional (no more slack waters) and no generation of odd harmonics t Flood Maximum attenuation Ebb Minimum attenuation u t Flood Ebb Attenuation Ebb Flood

17. Current velocity data near Cape Henry, in the Chesapeake Bay January 20-June 9, 2000

18. σ M2 - σ x σ M2 + σ x 2σ M2 - σ x 4σ M2 - σ x

19. Spectrum for current velocity at Ponce de Leon Inlet Spectral energy (m 2 /s 2 /cpd) Cycles per day

20. Ensenada de la Paz Example of Overtides and Compound Tides

21. More evidence sought from time series with Moored Instruments Early March to Early May 2003

22. Power spectrum of Principal-axis ADCP bins Appreciable overtides and compound tides – tidal rectification O 1,K 1 N 2,M 2,S 2 MK 3,2MK 3 M4M4 2MK 5,2MO 5 M6M6 4MK 7,4MO 7

23. ADCP pointing downward 1-m bins recorded for ~2.5 days, i.e., ~ 5 cycles December 14.5 to 17, 2004 Deployed just seaward of bar