Nonlinear effects on tides In systems where tides are distorted by geometry, we may see asymmetries between flood and ebb, as illustrated here: Sharp floods, broad ebbs Tidal Rectification = Overtides and compound tides

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

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

Nonlinear effects in estuaries (Parker, 1991, Tidal Hydrodynamics, p. 247) We will talk mainly about nonlinear tidal interactions Consider the tide: And the nonlinear term and i = M2 only overtide

If M2 interacts with S2: 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

Interactions of M2 with other constituents generate constituents with the following frequencies: σM2 - σx σM2 + σx 2σM2 - σx 2σM2 + σx 4σM2 - σx Nonlinear terms on tidal constituents effect a modulation and a distortion of that constituent Generating mechanisms arise from

M2 Overtides

M2 interactions with overtides symmetric distortion (by odd harmonic) asymmetric distortion (by even harmonic)

In systems where tides are distorted by geometry, we may see asymmetries between flood and ebb, as illustrated here: Sharp floods, broad ebbs Rectified Tide

This results from the interaction of the semidiurnal tide with a harmonic. The addition of those two waves causes the distortion Rectified Tide

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: Difference between sinusoid and distorted wave yields energy in the 2nd harmonic energy at M4 frequency This is the asymmetric effect of the nonlinear continuity term (mechanism A)

The tidal current amplitude may be approximated as: 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 + u0 and the wave propagation velocity at the trough is C - u0 which results in a similarly distorted wave profile (tidal wave interacting with tidal current): ebb flood This is the effect of the inertial term: C – u0 C + u0

Generating mechanisms arise from 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 (M4) 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 (M6) Therefore, there are symmetric effects and asymmetric effects Asymmetric Effects generate even harmonics (e.g. M4) because max C and minimum attenuation occurs at crest

Symmetric Effects u | u | extreme attenuation at flood and ebb, and minimum attenuation at slack waters Produce odd harmonics, e.g., M6 because there are 3 slack waters and two current maxima in one period symmetric distortion (by odd harmonic) asymmetric distortion (by even harmonic)

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 M4 Frictional generation of M6 will continue as long as uR < u0 so that there are still slack waters greatest attenuation t Flood Ebb Attenuation

When uR > u0 Flow becomes unidirectional (no more slack waters) and no generation of odd harmonics u t Flood Maximum attenuation Ebb Minimum attenuation Attenuation Flood Flood Ebb Ebb t

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

σM2 - σx σM2 + σx 2σM2 - σx 4σM2 - σx

Example of Overtides and Compound Tides Ensenada de la Paz

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

Power spectrum of Principal-axis ADCP bins O1,K1 N2,M2,S2 M4 M6 MK3,2MK3 2MK5,2MO5 4MK7,4MO7 Appreciable overtides and compound tides – tidal rectification