1. 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

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

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

4. 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

5. 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

6. 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

7. M2 Overtides

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

9. 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

10. This results from the interaction of the semidiurnal tide with a harmonic. The addition of those two waves causes the distortion
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:
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)

12. 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

13. 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

14. 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)

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 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

16. 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

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

18. σM2 - σx
σM2 + σx
2σM2 - σx
4σM2 - σx

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

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

21. 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