Feb 12: Tides and tidal currents (2)

In most locations the most significant harmonic constituents of the tide generating forces are:

 species constituent symbol period (hours) semi-diurnal principal lunar M2 12.421 principal solar S2 12.000 larger lunar elliptic N2 12.658 diurnal luni-solar K1 23.934 principal lunar O1 25.819

The N2 constituent involves the non-circularity of the moon's orbit. During the month the moon describes an ellipse, and the tides are higher when it is near its perigee (nearest the earth) and lower when it is near its apogee (farthest). The perigee itself precesses, so this pattern shifts slightly from month to month.

The diurnal constituents K1, O1, P1 take into account (among other things) the inclination of the earth's equatorial plane with respect to the plane of the moon's orbit. During a month, the moon spends about two weeks above the equator, and then two weeks below. When it is above, the high tide when the moon is visible will be higher than the next tide (when the moon is below the earth from our point of view). This effect is called the diurnal inequality. Twice a month, when the moon crosses the equator, the two tides are roughly equal.

Statistical prediction: least-squares harmonic analysis

The most common method of tidal analysis and prediction is harmonic analysis.

This is a least-squares fit of a set of tidal frequencies to observations of sea level variation made with some kind of sea level recording instrument.

This forms the basis of tide forecasts (e.g. tide tables, almanac and newspaper)

This statistical approach is required because the actual tide is not the equilibrium tide – more on this later.

We know the periods of the tidal harmonic constituents from astronomical calculations, so an observational record of sea level variability can be represented as a sum of the various constituents:

where

·        t is time reckoned from some initial epoch (such as the beginning of the year of predictions in UT)

·        i=1…N is the constituent number,

·        for each i we know

o       frequency

o       phase  of the equilibrium tide
(Bowden uses  for V0+u)

·        and we compute

o       amplitude Ai and phase w.r.t. the equilibrium tide

·        the nodal factors fo and Vo (amplitude and phase) account for the 18.6 year variability in the lunar declination, and depend only the time datum to

Once the coefficients Ai and  are computed, the tide can be predicted at any time t in the future.

More constituents

·        more accurate prediction

·        but longer time series required to separate

o       15 days of data would separate M2, S2, K1, O­1

o       29 required to also get N2 (energy from N2 would have gone into M2)

http://co-ops.nos.noaa.gov/predmach.html

[For more details on tidal constituents see Parker, B.B., A.M. Davies and J. Xing, 1999: Tidal height and current predictions, in: Coastal Ocean Prediction, Coastal and Estuarine Studies 56, C.N.K. Mooers (ed.), Amer. Geophys. Union, pp. 277-327.]

Sea-level observations

(See discussion in Emery and Thompson, Data Analysis Methods in Physical Oceanography, Elsevier Science; 2nd Rev edition (April 1, 2001), 658pp.)

Making observations of sea level has a long history, with many observational instruments and methods being developed.

Simple observations of sea level height:

·        stilling well

·        possibly automated

·        atmospheric pressure must be recorded also, and an adjustment made for the inverse barometer effect.

Modern method:

·        bubbler gauge is popular

·        slow stream of gas equilibrates its pressure with the head of water at the bubbler, so pressure in line gives the sea level.

·        the pressure transducer and instrument package (gas supply, digital recorder, transmitter, batteries) can be remote from the bubble (which might be down a precipitous cliff).

·        no moving parts, low maintenance

Types of Tides

As many as 60 harmonic constituents are generally used to make accurate tidal predictions, but the general character of the tides can be expressed by the first few constituents.

The factor F defined by:

where M2, S2, K1  and O1 are the amplitudes of the corresponding constituents may be used as an indicator of the type of tide as follows:

F:      0        to       0.25   : semi-diurnal

0.25   to       1.5     : mixed, mainly semi-diurnal

1.5     to       3        : mixed, mainly diurnal

greater than3        : diurnal

Compare tidal predictions for:

Sandy Hook, New Jersey

Eastport, Passamaquoddy Bay, ME

Texas State Aquarium, Corpus Christi, TX

Apalachicola Bay, Lower Anchorage

Neah Bay, Strait Of Juan De Fuca, WA

Coos Bay, Coos Bay

Dynamical response of the ocean

The actual response of the oceans to the TGF is a dynamical problem because of the variation of forces with time, the inertia of the water, Coriolis forces, and the confines of the coastlines.

Often, and especially in the coastal ocean, the actual response of the ocean may be nothing like the equilibrium tide.

Equations of motion in Cartesian coordinates:

where the forces on the RHS have been separated into friction and the contribution of the TGF (Fx, Fy )

By integrating the hydrostatic relation w.r.t. z and assuming constant density

the pressure gradient terms can be replaced by the sea level slope provided the horizontal gradients of the atmospheric pressure are small:

Recalling that we defined an equilibrium tide  that balanced the TGF,  Fx, Fy , we can write the momentum equations as:

This emphasizes that the ocean will respond dynamically, by generating currents u, v  if the sea surface shape differs from the equilibrium tide.

Friction

Friction arises from

·        shearing stress of wind on the surface

·        drag at the bottom

Shear stresses communicate the friction vertically:

If the stress  then momentum is added to the fluid element.

The excess stress is

Shear stress is per unit area,

so the force added to the fluid element is

The mass of the fluid element is

so the force per unit mass that would accelerate or decelerate the fluid is

with components dependent on the vector stress

Depth integrated equations

In the absence of friction, the RHS terms would be independent of z, and the response u, v would be uniform with depth.

Is this case, the velocity everywhere would simply be the depth average velocity:

,

It turns out that we can simplify the analysis by integrating the momentum equations over z and developing equations for the depth average velocity that incorporate the net influence of frictional terms on the whole water column.

This approach will give us the essence of the tidal dynamics.

It works because

·        TGF are uniform over the whole water column (body force like gravity)

·        density is assumed constant

·        pressure gradients are dominated by the sea level slope on the length scales we consider

We get

We need another equation: continuity

Integrating over depth

(Leibnitz’ rule for differentiation of an integral, cancels out the z=-h contributions even though h is h(x,y) )

To consider the tides, we can assume the surface wind stress is zero, but we still need to allow a possibly significant role for bottom friction.

For turbulent flows is it commonly observed that frictional stresses, or drag, are proportional to the velocity squared.

A widely used parameterization is

where     is the magnitude of the near bottom velocity.

A typical value for the drag coefficient CD = 2 x 10-3

For a bottom current of 0.5 m/s this gives a stress of 0.5 N/m2 or Pa.