Tidal Rectification = Overtides and compound tides

Nonlinear effects on tides

From Parker (2007)

simple sine wave

asymmetry between flood and ebb

double low waters

extreme distortion: tidal bore

From Parker (2007)

HgC

88

77

55

66

11

2233

44

Nonlinear effects in estuaries

(Parker, 1991, Tidal Hydrodynamics, p. 247)

We will talk mainly about nonlinear tidal interactions

Consider the tide: tuu ii

i cos0

tututuu MMMMMM 222222

22000

2 coscoscos

tu MM 22

2cos1212

0

tuuxu

u

M

MMM 4

2222cos

21

21

α 20

f(t)not residual

20

42 h 21.62

h 42.122

22 MM

overtide

And the nonlinear term 2

21

uxx

uu

and i = M2 only

If M2 interacts with S2:

coscoscoscos

days 4.81 modulation freqlow

h 1.6 distortion freq high

0000 2222222222ttuututu SMSMSMSSMM

days 8.14)0177.0)(24(

2 ;0177.0

122

42.122

.,.22

Tei SM

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

0 1

uHbxbt

(one dimensional, vertically and laterally integrated equation; b is estuary’s breadth)

We then have four mechanisms that generate nonlinearities:

uuDuuCxu

uBux

A ); ); ); )

H

uuC

xg

xu

utu

bGenerating mechanisms arise from

uuD

uuCxu

uB

ux

A

)

; )

; )

; )

H

uuC

xg

xu

utu

bGenerating mechanisms arise from

Nonlinear terms on tidal constituents effect a modulation and a distortion of that constituent

M2 - x M2 + x 2 M2 - x 2 M2 + x 4 M2 - x

M2 Residual M4 - M6 M6

(12.42 h) - (6.21 h) - (4.14 h) (4.14 h)

N2 MN (Mm) MN4 2MN2 2MN6 4MN6

(12.66 h) (27.3 d) (6.27 h) (12.19 h) (4.17 h) (4. 11 h)

S2 MS (MSf) MS4 2MS2 ( 2) 2MS6 4MS6

(12 h) (14.8 d) (6.10 h) (12.87 h) (4.09 h) (4.19 h)

K1 MK1 (O1) MK3 2MK3 2MK5 4MK7

(23.93 h) (1.07 d) (8.17 h) (8.38 h) (4.93 h) (3.57 h)

O1 MO1(K1) MO3 2MO3 2MO5 4MO7

(25.82 h) (0.99 d) (8.38 h) (8.17 h) (5 h) (3.52 h)

1-1 1+1 2-1 2+1 4-1

even even odd odd odd

Mech A, B, C, D A, B, C,D D D D

M2 + x 2 M2 + x

M4 M6

Interactions of M2 with other constituents

generate constituents with the following frequencies:

σM2 - σx σM2 + σx 2σM2 - σx 2σM2 + σx 4σM2 - σx

M2 Overtides

M2 interactions with overtides

symmetric distortionsymmetric distortion(by odd harmonic)(by odd harmonic)

asymmetric distortionasymmetric distortion(by even harmonic)(by even harmonic)

Rectified Tide

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:

energy at M4 frequency

This is the asymmetric effect of the nonlinear continuity term (mechanism A)

) ux

A

Difference between Difference between sinusoid and distorted sinusoid and distorted wave yields energy in wave yields energy in the 2the 2ndnd harmonic harmonic

The tidal current amplitude may be approximated as:

xg

dtdu

xu

dtd

H

txaCg

utxa

CH

u

1

)sin()sin(

0

This is the effect of the inertial term:

)xu

uB

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 + 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):

CC – – uu00

CC + + uu00

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 andeven harmonics (M4)

H

uuC

xg

xu

utu

bGenerating 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 (M6)

Therefore, there are symmetric effects and asymmetric effects

Asymmetric Effects

morevslessuu

uCvsuCxu

u

HgCvsHgCux

00

)()(

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 symmetric distortiondistortion(by odd (by odd harmonic)harmonic)

asymmetric asymmetric distortiondistortion(by even (by even harmonic)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 rangegreater 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

greatestattenuation

t

Flood

Ebb

Attenuation

When uR > u0

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

Current velocity data near Cape Henry, in the Chesapeake Bay

January 20-June 9, 2000

σM 2 - σx σM 2 + σx 2 σM 2 - σx 2 σM 2 + σx 4 σM 2 - σx

M 2 R esid ual M 4 - M 6 M 6

(1 2.42 h) - (6 .21 h) - (4 .14 h) (4 .14 h)

N 2 M N (M m) MN 4 2MN 2 2MN 6 4MN 6

(1 2.66 h) (2 7.3 d ) (6 .27 h) (1 2.19 h) (4 .17 h) (4 . 1 1 h)

S 2 M S (MS f) MS 4 2 M S 2 (µ 2) 2 M S 6 4 M S 6

(1 2 h) (1 4.8 d ) (6 .10 h) (1 2.87 h) (4 .09 h) (4 .19 h)

K 1 M K 1 (O 1) MK 3 2MK 3 2MK 5 4MK 7

(2 3.93 h) (1 .07 d ) (8 .17 h) (8 .38 h) (4 .93 h) (3 .57 h)

O 1 M O 1(K 1) MO 3 2MO 3 2MO 5 4MO 7

(2 5.82 h) (0 .99 d ) (8 .38 h) (8 .17 h) (5 h) (3 .52 h)

1 -1 1 +1 2 -1 2 + 1 4 -1

e ve n e ve n o dd o dd o dd

M e c h A, B, C A, B, C C C C

σM 2 - σx σM 2 + σx 2 σM 2 - σx 2 σM 2 + σx 4 σM 2 - σx

R esid ual M 4 - M 6 M 6

σM2 - σx σM2 + σx 2σM2 - σx 4σM2 - σx

Spectrum for current velocity at Ponce de Leon Inlet

Spe

ctra

l ene

rgy

(m2 /

s2 /cp

d)

Cycles per day

Ensenada de la Paz

Example of Overtides and Compound Tides

More evidence sought from

time series with Moored Instruments

Early March to Early May 2003

Power spectrum of Principal-axis ADCP bins

Appreciable overtides and compound tides – tidal rectification

O1,K1 N2,M2,S2

MK3,2MK3

M4

2MK5,2MO5

M6

4MK7,4MO7

ADCP pointing downward1-m bins recorded for ~2.5 days, i.e., ~ 5 cyclesDecember 14.5 to 17, 2004Deployed just seaward of bar