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) 




semidiurnal 
principal lunar 
M_{2} 
12.421 

principal solar 
S_{2} 
12.000 

larger lunar elliptic 
N_{2} 
12.658 




diurnal 
lunisolar 
K_{1} 
23.934 

principal lunar 
O_{1} 
25.819 
http://www.math.sunysb.edu/~tony/tides/harmonic.html
The N_{2} constituent
involves the noncircularity 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 K_{1},
O_{1}, P_{1} 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.
The most common method of tidal analysis and
prediction is harmonic analysis.
This is a leastsquares 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 V_{0}+u)
·
and we compute
o
amplitude A_{i}
and phase _{}w.r.t. the equilibrium tide
·
the nodal factors
f_{o} and V_{o }(amplitude and phase) account for
the 18.6 year variability in the lunar declination, and depend only the time
datum t_{o}
Once the coefficients A_{i} 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 M_{2}, S_{2}, K_{1}, O_{1}
o
29 required to
also get N_{2} (energy from N_{2} would have gone into M_{2})
You can read about mechanical tide predicting machines at
http://coops.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.
(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
·
graduated staff
·
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
[Emery and
Thompson, fig. 1.6.3 bubbler gauge, NIWA Anawhata site]
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 M_{2}, S_{2}, K_{1} and O_{1}
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 : semidiurnal
0.25 to 1.5 : mixed, mainly semidiurnal
1.5 to 3 : mixed, mainly diurnal
greater
than3 : diurnal
Compare tidal predictions for:
Eastport,
Passamaquoddy Bay, ME
Texas
State Aquarium, Corpus Christi, TX
Apalachicola
Bay, Lower Anchorage
Cherry
Point, Strait Of Georgia, WA
Neah
Bay, Strait Of Juan De Fuca, WA
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 (F_{x}, F_{y} )
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, F_{x}, F_{y} , 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 _{}
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 C_{D} = 2 x 10^{3}
For a bottom current of 0.5 m/s this gives a stress of
0.5 N/m^{2}_{ }or