Factors modifying the framework established:

Tides

Atmospheric Forcing - wind, barometric pressure

River Discharge

Bathymetry

Morphology

TIDES

Tide - generic term to define alternating rise and fall in sea level with respect to land and is produced by the balance between the gravitational force (of the moon and sun mainly) and the centrifugal acceleration.

Tide also occurs in large lakes, in the atmosphere, and within the solid crust

Gravitational Force (Newton’s Law of Gravitation):

F = GmM/R2

G = 6.67×10-11 N m2/kg2

Centrifugal Force

Center of mass of Earth-Moon system ~1,700 km from Earth’s surface(because Earth is 81 times heavier than Moon) – Centrifugal = Gravitational

EQUILIBRIUM TIDE

Moon’s Gravitational Force (changes from one side of the earth to the other)F = GmM/R2

Tide Generating Force (Difference between centrifugal and gravitational)

How strong is the Tide-Generating Force?

PAB S

SP 60Since

Tide-generating Force at A: 3

2

P

SMmGFF cA

2SP

MmGFA

Gravitational Force at A:

2P

MmGFc

Centrifugal Force at A:

22 P

MmG

SP

MmGFF cA

Imbalance (Tide-generating force at A):

Tide-generating Force at B: 3

2

P

SMmGFF cB

2234

2

2

2

SPSPP

SPSMGmFF cA

22

2

SPP

SPSMGmFF cA

The mass of the sun is 2x1027 metric tons while that of the moon is only 7.3x1019 metric tons.

The sun is 390 times farther away from the earth than is the moon.

The relative Tide Generating Force on Earth = [(2x1027/7.3x1019)]/(3903)

or = 2.7x107/5.9x107 = 0.46 or 46%

How strong is the Tide-Generating Force?

PAB S

Tide-generating Force at A: 3

2

P

SMmGFF cA

Tide-generating Force at B: 3

2

P

SMmGFF cB

Equatorial Tides

Image from Hubble Telescope

Tropic Tides

Image from Hubble Telescope

What alters the range and phase of tides produced by Equilibrium Theory?

Non-astronomical factors:

coastline configurationbathymetryatmospheric forcing (wind velocity and barometric pressure)hydrography

may alter speed, produce resonance effects and seiching, storm surges

In the open ocean, tidally induced variations of sea level are a few cm.

When the tidal wave moves to the continental shelf and into confining channels, the variations may become greater.

Keep in mind that tidal waves travel as shallow (long) waves

How so?

Typical wavelengths = 4500 km (semidiurnal wave traveling over 1000 m of water)

Ratio of depth / wavelength = 1 / 4500

Then, their phase speed is: C = [ gH ]0.5

The tide observed at any location is the superposition of several constituents that arise from diverse tidal forcing mechanisms.

Main constituents: Principal Lunar Semidiurnal M2 12.42 h

Principal Solar Semidiurnal S2 12.00 h

Larger Lunar Elliptic Semidiurnal N2 12.66 h

Lunisolar diurnal K1 23.93 h

Lunar Diurnal O1 25.82 h

...)sin()sin()sin( 222222222 NNNSSSMMM tAtAtA

The Form factor F = [ K1 + O1 ] / [ M2 + S2 ] is customarily used to characterize the tide.

When 0.25 < F < 1.25 the tide is mixed - mainly semidiurnal

F > 3 the tide is diurnal F < 0.25 the tide is semidiurnal

When 1.25 < F < 3.00 the tide is mixed - mainly diurnal

F > 3 the tide is diurnal F < 0.25 the tide is semidiurnal

When 1.25 < F < 3.00 the tide is mixed - mainly diurnalWhen 0.25 < F < 1.25 the tide is mixed - mainly semidiurnal

Superposition of constituents generates modulation - e.g. fortnightly, monthly

This applies for both sea level and velocity

Subtidal modulation by two tidal constituents

In Ponce de León Inlet: M2 = 0.41 m; N2 = 0.09 m; O1: 0.06 m; S2: 0.06 m; K1= 0.08 m

F = [K1 + 01] / [S2 + M2 ] = 0.30

GNVGNV

Panama Panama CityCity

In Panama City, FL: M2 = 0.085 m; N2 = 0.017 m; O1: 0.442 m; S2: 0.035 m; K1= 0.461 m

F = [K1 + 01] / [S2 + M2 ] = 7.52

Co-oscillation

Independent tide - caused by gravitational and centrifugal forces directly on the waters of the estuary -- usually small for typical dimensions of estuaries

Co-oscillating tide - caused by the ocean tide at the entrance to the estuary as driving force

The wave propagates into the basin and may be subject to RESONANCE and RECTIFICATION -- alters tidal flows and produces subtidal motions

Let’s study what happens to the wave as it propagates into the estuary...

Progressive wave

Assume linear, frictionless motion in the x direction only, under homogeneous conditions.

The momentum balance is then:x

gtu

And the continuity equation is:tHx

u

1

The solution is d’Alembert’s solution, which can be studied with the sinusoidal wave form:

txatT

xL

a

sin

22sin

xu

txaHtH

cos

11

txaHC

txaH

u sinsin

1

-1 .5

-1

-0 .5

0

0 .5

1

1 .5

-0 .7 1 6 8 0 .2 8 3 2 1 .2 8 3 2 2 .2 8 3 2 3 .2 8 3 2 4 .2 8 3 2 5 .2 8 3 2 6 .2 8 3 2

a

time

2

22

2

2

2

2

xC

xgH

t

linear, partial differential equation (hyperbolic)

txa sin

txa sin

txaHC

u sin

CH

u

This indicates that the flow is in phasewith the elevation

-1 .5

-1

-0 .5

0

0 .5

1

1 .5

-0 .7 1 6 8 0 .2 8 3 2 1 .2 8 3 2 2 .2 8 3 2 3 .2 8 3 2 4 .2 8 3 2 5 .2 8 3 2 6 .2 8 3 2

a

time

-1 .5

-1

-0 .5

0

0 .5

1

1 .5

-0 .7 1 6 8 0 .2 8 3 2 1 .2 8 3 2 2 .2 8 3 2 3 .2 8 3 2 4 .2 8 3 2 5 .2 8 3 2 6 .2 8 3 2

u

a C/H

time

Standing wave

The momentum balance is also:x

gtu

And the continuity equation is:tHx

u

1

The solution is: txatxatxa sincossinsin

xu

txaHtH

coscos

11

xtaH

Cxta

Hu

sincossincos1

txa sincos

txaHC

u cossin

This indicates that the flow is out of phase with the elevation by 90 degrees

-1 .5

-1

-0 .5

0

0 .5

1

1 .5

-0 .7 1 6 8 0 .2 8 3 2 1 .2 8 3 2 2 .2 8 3 2 3 .2 8 3 2 4 .2 8 3 2 5 .2 8 3 2 6 .2 8 3 2

a

time

-1 .5

-1

-0 .5

0

0 .5

1

1 .5

-0 .7 1 6 8 0 .2 8 3 2 1 .2 8 3 2 2 .2 8 3 2 3 .2 8 3 2 4 .2 8 3 2 5 .2 8 3 2 6 .2 8 3 2

u

a C/Htime

Resonance

At the mouth x = L, ta sin0

txa sincosSubstituting into ta sin0 at x = L

L

tatLa sinsincos 0

tLxa

sin

coscos0

La

acos

0

For resonance to exist, the denominator should tend to zero, i.e.,

122

22

22

2

nL

nLL

and CTnL ;12

4

The natural period of oscillation is then:

1214

nCL

TN

1214

nCL

TN

tLxa

sin

coscos0

tLx

HCa

u

coscossin0

L

u

4

Example of seiching

For an estuary with length < λ /4, u is zero at the head and maximum at the mouth

For longer estuaries u is zero at x = 0, λ / 2, 3 λ / 2,… or where sin κx = 0

and maximum at x = λ /4, 3 λ / 4, 5 λ / 4, …, i.e., where sin κx is max

1214

nCL

TNMerion’s FormulaMode 1(n =1)

H (m) L (km) C (m/s) TN (h)

Long Island Sound 20 180 14 14

Chesapeake Bay 10 250 10 28

Bay of Fundy 70 250 26 10.7

Tidal Waves With Friction

zu

Azx

gtu

z

Integrating vertically:

01

H

udzH

U

H

uuC

xg

zU

AHx

gtU bbb

z

1

tUub cos If 0

The bottom stress becomes: ttUCb coscos20

UHUC

xg

tU

r

b

0

38

Momentum balance for a progressive wave:

With continuity:xU

Ht

These are the governing equations for progressive tidal motion with friction.

Expanding this in a Fourier cosine series, to lowest order:

tUCttUC bb

cos38

coscos 20

20

maximum U precedes maximum eta

If we letHUC

r b 0

38

and gHr

Cr

22

where

2

tanr

a

We obtain the solution:

xteH

aU x cos

22

xtae x cos

UHUC

xg

tU

r

b

0

38

xU

Ht

For resonance, we have again U =0 at the head of the estuary, i.e.,

xtexteH

aU xx coscos

22

xtaextae xx coscos

Effects of Friction on a Standing Tidal Wave in an Estuary

Effects of Rotation on a Progressive Tidal Wave in a Semi-enclosed basin

xg

tu

tHxu

1

ygfu

Solution:

0 v;txcoseH

Cau Ry

txae Ry cos

R = C / f

KELVIN WAVE

Effects of Rotation on a Standing Tidal Wave in an Estuary

Two Kelvin waves of equal amplitude progressing in opposite directions.

0

v

txcosetxcoseH

Cau RyRy

txaetxae RyRy coscos

Instead of having lines of no motion, we are now reduced to a central region -- amphidromic region-- of no motion at the origin. The interference of two geostrophically controlled simple harmonic waves produces a change from a linear standing wave to a rotary wave.

0

coscos

v

txetxeHC

aU RyRy

txaetxae RyRy coscos

0

coscos

v

txetxeHC

aU RyRy

txaetxae RyRy coscos

Pinet (2006)

Pinet (2006)

Pinet (2006)

Effects of Bottom Friction on an amphydromicsystem

Parker (1990)

Virtual Amphidromes

Parker (1990)

Virtual amphidromesin Chesapeake Bay

Fisher (1986)

Tidal Rectification

Oscillatory tidal currents can produce a mean current or residual

Noticeable in areas of sharp coastline bends and bathymetric gradients

Non-linear phenomenon

Energy is transferred from the dominant tidal frequencies (e.g. M2) to both higher harmonics (e.g. M4 and M6) and to mean flows (zero frequency).

Three mechanisms responsible for this energy transfer to mean flow. Studied from vorticity tendencies.

Conservation of Potential Vorticity

Lateral Variations of Bottom Stress

Lateral Shear in the Depth-Distributed Friction Force

Robinson (1981)

Tidal flow oscillations at a single frequency (e.g. M2) will generate tidal vorticity by the above 3 mechanisms.

Because of non-linearities (products of u times u), the vorticity generated will not only be at the fundamental tidal frequency, but also at higher harmonics and at zero frequency.

tUtUtUtU 2cos121

coscoscos 20

22000

The three mechanisms can be identified from the vorticity equation.

This is derived from the depth-averaged 2-D momentum equation appropriate to barotropic tidal motion in shallow seas

UAUUHCp

UkUUtU

hb 2ˆ2)(

Taking the curl of that equation and

yu

xv

2ˆ)(

hbb A

H

UUCU

HC

Ukft

processes edissipativ

2

bottom sloping

2

shear lateralConservPot Vort

h

bbb AHUC

HUH

UCUU

HC

dtdH

Hf

dtdwhich may

be rewritten as:

Max ebb

End ebb

Max flood

End flood

Example:

Geyer and Signell (1990, JGR, 95, 3189)

Geyer and Signell (1990, JGR, 95, 3189)