Feb 19: Tides and tidal currents (4)

Wave equation with linear friction on :

Assume solution where the wave amplitude decays as the wave propagates:

Differentiating h and substituting into the damped wave equation and independently satisfying the real and imaginary parts of the equation we obtain:

showing how the wave speed is reduced, and

assuming c/c_o ~ 1 (weak friction).

The corresponding velocity follows from continuity

upon assuming a solution of the form

This incorporates the possibility of a phase shift with respect to the sea level displacement. We obtain:

Notice that the current:

· is no longer in phase with the amplitude

o the current peaks a time a/w before the sea level crest

· the magnitude is reduced compared to the same frictionless wave

The key parameter is m/ k: the ratio of the wavelength (k^-1) to the damping length scale (m^-1)

In a study of Mid-Atlantic Bight tides, Redfield (1950) found that in order to obtain satisfactory agreement with observations the product mL, where L is the wavelength, needed to be about 0.5. (He estimated values of 1 for the Bay of Fundy and 1.5 to 2 for the Strait of Juan de Fuca).

[ loss of amplitude is e^-mL = e^-0.5 = 0.6]

The phase shift of the velocity is then

Tides on the continental shelf

Tides generated in the deep ocean due to the TGF propagate as waves toward the coast.

The decreasing depth alters their dynamics.

An order of magnitude estimate of the change in wave height can be calculated by considering conservation of wave energy.

Energy flux

Energy flux in a shallow water wave:

Power = Work/time = Force*distance/time = Force*velocity

For a wave train

· Force is pressure * Area

· so wave energy flux (or power of the incoming wave field) is:

· energy flux in Dy width of wave crest and over depth Dz (the area) is

o E_flux =

The pressure perturbation and density do not alter with depth (homogeneous fluid and hydrostatic pressure balance), so integrate over depth dz and divide by width to get E_flux per unit width wave crest

Energy flux / =

Averaging over one period (or wave length) the cos² term is ½

Energy flux / E_flux =

= energy density * group velocity ( )

[ I will say more about group velocity – speed at which energy propagates through a medium – at the end of the lecture. ]

Consider a wave traveling from ocean of depth h₁ into shallow water on the water of depth h₂.

Energy conservation demands that

For, e.g., h₁ = 4000 m and h₂ = 100 m, the amplitude of the tidal wave increases by a factor of

A₂/A₁ = (4000/100)^1/4 = 2.51

A deep ocean tidal amplitude of, say, 0.35 meters will increase to 0.88 meters.

This is close to the observed amplification of the M₂ tide in the South Atlantic Bight (see Bowden fig 2.9)

The M₂ tide approaches the US Atlantic coast almost at right angles, with the phase varying little from Cape Cod to the Florida Strait.

(We can update Bowden’s figure with results from the ADCIRC numerical tidal model – R. Luettich http://adcirc.org/)

[M₂ tide amplitude and phase from ADCIRC model (AtlanticBightM2.pdf)]

M₂ and S₂ tidal elevation in the northeast US:

Velocity:

For the depth ratio 4000/100 this is a factor of 16. Velocities are amplified considerably more than sea levels.

A deep ocean tidal velocity of, say, 2.5 cm/s will increase to 40 cm/s.

In reality

· there will be some frictional loss of energy

· some energy will be reflected back to the deep ocean

· changes occurring on a sloping bottom need to be accounted for

o here we balanced energy in two locations of uniform depth without considering how the waves alter through the depth transition

Energy input is in the deep sea, and the coastal response is to this incoming energy is locally amplified.

Direct input of momentum from the TGF in coastal waters is generally not significant.

In models, coastal tidal response is frequently treated by assuming the tides are remotely forced at the deep ocean edge by imposing the amplitude and phase of the sea level and velocity variability, i.e. and ignoring the influence of the TGF within the costal region.

Solving the governing equations in coastal waters, subject to these perimeter open boundary conditions, generally does a good job of simulating the coastal tide.

The majority of energy coming onto the shelf is dissipated in the shelf seas which form the major sink of tidal energy. The stresses exert a braking effect on the earth’s rotation, and estimates of the net effect of tidal friction can be deduced from the acceleration of the moon in its orbit which is required to conserve angular momentum as the earth slows down (and the length of day increases very slightly).

Dissipation is estimated at around 3.4 TeraWatts. About 1.7 TW of this is thought to be due to the barotropic (depth-average) tide, with speculation presently that the remainder is dissipated in a few isolated locations by the baroclinic tide.

Group velocity and energy propagation

The class of waves we have considered here, shallow water gravity waves, have the property that they are non-dispersive, which denotes that their speed is independent of wavelength. Consequently, a packet of waves made up of different wavelengths/wave-numbers (in a Fourier decomposition sense) would not disperse into separate wavelengths traveling at different speeds.

This is not generally true of waves in fluids.

The general case is that wave speed is a function of wave-number, which has significant implications for the speed of propagation of energy.

Consider the general superposition of two plane waves of the same amplitude but with slightly different wave-numbers and frequencies:

This can be interpreted as an approximately sinusoidal wave with phase f = kx – wt but with amplitude:

Individual crests of the modulated wave packet travel at speed c = w/k but the envelope of the packet travels at the group velocity:

In general the wave-number k is actually a vector, k = (k_x, k_y, k_z), defining the direction in 3 dimensions of an advancing wave crest. The group velocity is also a vector (whereas frequency is always scalar) and need not be in the direction of k (and can be quite counter-intuitive in its behavior).

Tidal co-oscillation in a gulf

Narrow gulf, and ignore friction

[Bowden fig 2.11]

x = 0 at the closed end, x = L at the open end

We want to consider the case of the sea level variability at the entrance to the gulf of amplitude A_L, and angular frequency

Accordingly, we denote the variation at x=L as

(A_L is set by conditions in the deep ocean)

There can be no velocity at the closed end (x=0), so the general solution is a standing wave of the form

which will satisfy the governing equations, where k is the wave-number and is the angular frequency, and U is an as yet unspecified current magnitude.

Substituting this solution for into (1), it follows that

Then integrate w.r.t. x …

This satisfies equation (2) provided

This shows the length scale of the oscillation within the bay is set by k, the wave-number of the freely propagating shallow water wave.

is the angular frequency.

Evaluating this at x=0 shows that the coefficient A is the amplitude of the sea level variation at the head of the gulf.

Since the particular solution at the mouth is:

and the general solution along the whole length of the gulf is:

then to be consistent with the imposed forcing at the mouth, the amplitude of the response in the gulf must be:

If cos kL = 0 then resonance occurs.

The first resonance occurs for , the so-called quarter-wave resonator.

The cases and are shown in [Bowden fig 2.12]

Question:

Where do the strongest velocities occur in these two situations?

·

o largest at the head, strongest U at the mouth

·

o largest at the head, strongest U at

The wavelength can be replaced by to enable an easy calculation of the resonance properties of an idealized gulf of depth h for a tidal harmonic constituent of given period T.

Resonant length

In reality, infinite resonance does not occur due to frictional dissipation and other effects that were neglected in deriving the simple equations.

Coriolis effect

The neglect of Coriolis is not a serious defect of the analysis

– the resonance property stills holds because the Kelvin wave speed does not alter with f.

Coriolis introduces an across-gulf pattern to the response because of the Rossby radius decay scale of the incoming and outgoing waves.

[Bowden fig 2.13]

Transverse oscillation:

Sea level is high on the right on flood tide, and low on ebb tide (as Kelvin wave flows back out)

The amplitude of the transverse oscillation depends on the width of the gulf compared to the Rossby radius.

For without Coriolis across entire width of gulf

For with Coriolis only in mid-point of gulf

This is an amphidromic point.

Amphidromic points will occur at all the nodes of the resonance pattern where the phase of an incoming Kelvin wave cancels the phase of the out-going wave (possibly more that one tidal period prior)

Bowden gives a simple solution (equation 2.66) for the in-going plus out-going Kelvin waves, ignoring the details of the boundary condition u=0 at the head of the gulf:

At high tide, from which we can compute the shape of the co-phase lines along which high tide occurs at the same time.

[Bowden fig 2.14]

Perpendicular to the co-phase lines are the co-range lines

In developing a resonance theory for a wider gulf in which the influence of the Coriolis cannot be neglected, the transverse component of velocity v must be included when considering the reflection process at the head of the gulf.

This introduces the other momentum equation and complicates the analysis. (Bowden presents Taylor’s solution for this Kelvin wave reflection problem).

Near the mouth of the gulf, the solution resembles two Kelvin waves with no appreciable cross-channel velocity.

At some along channel locations, the two Kelvin waves cancel out producing an amphidromic point in mid-channel (high tide occurs on one side while low tide occurs on the other).

At other locations, high occurs simultaneously on both sides, and the sea level oscillates up and down in concert.

Near the head of the gulf, to meet the boundary condition of no flow normal to the coast, other Kelvin wave-like modes are generated and the currents rotate in direction (rather than oscillating to and fro parallel to the coast).

[Bowden fig 2.14]

Frictional effects

Friction will act to damp the amplitude of the out-going Kelvin wave so that reflection is only partial.

This causes the point of cancellation to move toward the coast that is to the left looking into the gulf (in the northern hemisphere).

If the frictional dissipation is large enough, the amplitude of the reflected wave will be less than the sea level displacement due to the incoming wave and the amphidrome point will vanish.

The magnitude of the frictional dissipation that causes this will depend on the ratio of Rossby radius to channel width.

[Bowden fig 2.15]

These processes explain many of the features we observe in co-tidal diagrams for marginal seas, such as the Yellow Sea M₂ and K₁ co-tidal charts given in Simpson (fig 5.2)

[Simpson fig 5.2]