Feb 8:
Tides and tidal currents (1)
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References:
·
Shearman,
K., and S. Lentz (2004), Observations of tidal variability on the
·
Bowden,
K.F., 1983: Physical Oceanography of Coastal Waters, Ellis Horwood,
·
Simpson,
J. H. 1998: Tidal processes in shelf seas. In “The Sea”, Vol. 10, A.R. Robinson
and K.H. Brink, eds., Wiley,
·
diagrams
of tide generating forces from Defant (1961)
·
http://www.mhl.nsw.gov.au/www/tide_glossary.htmlx
Tide glossary
·
http://co-ops.nos.noaa.gov/
Tides:
Introduction
The periodic rise and fall of sea level at the coast
due to the tides is possibly the most conspicuous phenomenon of coastal ocean
circulation.
The oscillatory flows associated with the tide are
much larger in the coastal ocean than the deep sea, and tidal currents dominate
the kinetic energy of almost all coastal seas.
[Shearman and Lentz: Energy spectrum showing peaks at
tidal frequencies]
·
we
know the origin of the tides are gravitational forces associated with the
orbits of the earth, sun and moon,
·
but
coastal tides are an indirect consequence of this
·
coastal
tides are driven by the deep ocean tides, and understanding ocean dynamics is
fundamental to explaining the magnitude and phase of the coastal response
·
tide
amplification results from energy propagation features of the tidal waves that
originate in the deep ocean – and the wave motion of tides accounts for time
lags in the response
The energetic tidal motions induce stirring and mixing
of the water, and re-suspension of sediments, and the repeated ebb and flow of
the tide leads to transport and dispersal of dissolved and suspended
substances, including plankton and larvae.
·
net
transport occurs from
o
shear
dispersion
o
rectification
of oscillatory currents into steady mean flows (through nonlinear dynamics:
momentum advection and friction)
Without mixing induced by tides, coastal waters would
be much less ventilated
·
they
would have less oxygen mixed in to them and
·
generally
be much less able to cope with the environmental pressures of coastal discharges.
·
A
predisposition toward anoxic conditions is evident in seas with weak tidal
circulations because they are not flushed with well ventilated waters.
Tidal stirring also significantly affects primary
production through its influence on water column stability, nutrient cycling,
and light availability.
·
stability:
density stratification – affects strength of vertical turbulence
·
nutrients:
deep water of offshore origin is typically nutrient enriched
·
light
attenuation with depth means vigorous vertical mixing reduces the average
exposure of phytoplankton and limits growth
Other applications:
· Shipping
· Altimetry
Overview
In these lectures I will present, in some detail, the
major features of tides and tidal currents.
These include:
·
origin
of the tide generating forces
·
harmonic
frequencies of the tidal constituents
·
statistical
tidal prediction based on these
·
the
dynamical response of the coastal ocean to tide forcing
o
the
equilibrium tide
o
gravity
waves, Kelvin waves
o
co-oscillation
o
friction
·
developments
in numerical modeling
o
non-linearity
and friction: over-tides and rectification
o
interpolation
between sparse observations
o
runoff
and storm surge
o
biological
processes
Tide generating forces
The foundation of any explanation or prediction of the
tides is based on our knowledge of the tide generating forces.
Prediction of tidal variability has traditionally been
based on statistical methods (e.g. least squares fitting) applied to
observations of sea level variability, using knowledge of the periods of the
many tidal harmonic constituents.
On most coasts, the tide is dominated by a twice daily
cycle fundamentally driven by the lunar semi-diurnal constituent of the tide,
the M2 tide, with a period of 12 h 25 m.
The forces that produce this variation are associated
with the orbit of the moon and earth around their common center of gravity.
The M2 tide generating force is actually
the residual, or difference, between the two forces that keep the moon and
earth in orbit. These are:
1. gravity, that pulls earth and moon
together
2. centrifugal force, that keeps earth and
moon apart
Components of the TGF
Centrifugal force:
·
the
force that keeps two orbiting bodies apart
·
rotation
of the bodies about their own axes (poles) is not a factor
·
each
point orbits around a different
origin in an orbit of the same radius
·
centrifugal
force is the same at all points (center and surface) and in the same direction
[First figure from Defant DefantTides.pdf ]
Gravitational force:
·
where r is the
distance apart of masses M and m
·
Let
d be the distance separating the earth and moon centers of gravity, then
the forces of attraction at the center are GMm/d2
·
This
force must balance centrifugal force or the orbit would not be stable
[Center column in table below]
·
The
gravitational force at the earth surface
will differ depending on how close the moon is:
o
earth
radius a
o
at
zenith distance is (d-a)
o
at
nadir distance is (d+a)
The tide generating forces can easily be computed for
the zenith and nadir points:
[Sketch: simple earth-moon geometry illustrating
zenith/nadir forces]
|
|
Zenith |
center of earth |
nadir |
|
forces of attraction |
|
|
|
|
centrifugal force |
|
|
|
|
tide producing forces |
|
0 |
|
|
or, neglecting terms of order a2/d2 |
|
0 |
|
The tide producing forces are directed toward the moon
at zenith, away from moon at nadir.
[2nd figure from Defant DefantTides.pdf
]
The vertical
component of this adds/subtracts from gravity, and is negligible.
But away from the zenith/nadir line there is a horizontal component that drives the
tides.
[Bowden fig. 2.3 BowdenFig2-3.pdf]
Using Kepler’s law for planetary motion, and the
geometry of the triangle comprising the centers of the earth, moon, and an
arbitrary point on the earth’s surface, it can be shown that the TGF is:
where 
is the angle between
the line joining the centers of the earth and moon and the point being
considered; g is vertical gravitational
acceleration at the earth surface:
( because the weight of an object is F = ma = 
)
is not
latitude. (See Bowden Figure 2.1). varies continually as
the point on the earth moves (because the earth is rotating) and the planetary
bodies that give rise to the TGF orbit around each other.
How strong is this force?
Compare to gravity:
m/M = 1/81.4 ratio of Moon to Earth masses
a =
6.37 x 103 km Earth
radius
d =
3.84 x 105 km Earth-moon
separation
so… F/g = 8.4
x 10-8 or about 1 in 10
million.
If the ocean covered
the whole earth, and did not move,
these forces would produce an equilibrium
tide displacement such that the horizontal pressure gradient due to the
sloping sea surface exactly balanced the TGF.
We can see from the distribution of horizontal forces
that there will be two bulges in the sea level, at the zenith and nadir points
of the TGF.
The equilibrium tide
If the slope of the sea surface due to the equilibrium
tide is 
then the associated
horizontal pressure gradient is:
which must balance
so that
We commonly define the potential,
, of the TGF as 
The TGF are perpendicular to contours of the potential
surface.
[3rd figure from Defant]
Then, by integration of
we get that the equilibrium tidal displacement is
The tide height is raised where the potential is low.
We can integrate the TGF with respect to distance x
to get (recognizing that 
)
The elevation is highest at the sub-lunar point 
and lowest at 
As the earth rotates about its own axis, the
equilibrium tide (if such a thing could exist) adjusts itself continuously so
that the major axis of the ellipsoid is always pointing toward the moon.
In the course of one lunar day, a given point on the
earth experiences two high, and two low, tides.
In general, the moon is not in the plane of the
earth’s equator, so the two maxima in the equilibrium tide potential are
different, and this accounts for the diurnal inequality of the tides.
The difference in elevation from high to low of the
equilibrium tide is
= 54 cm for earth and moon
This is the order of tide displacement observed at
oceanic islands or deep sea tide gauges, but much smaller than measured at many
coastal locations.
We can contrast the tide generating potential of the
moon with that of the sun.
where S = mass of the sun, and D = distance
from earth to sun.
The ratio of the maximum Fsolar/F = S/M (d/D)3
= 0.46
So the equilibrium solar tide is about half as strong
as the lunar tide.
The greater mass of the sun is offset by its greater
distance away.
A similar ratio occurs for the actual response in the
ocean.
Harmonic constituents
The TGF vary with time
·
depend
on positions of earth, moon, sun relative to a given location on the earth
·
earth
orbits sun
·
moon
orbits earth
·
orbital
planes are at angles to the earth’s equatorial plane
·
orbits
are elliptical, not circular
All these motions modulate the tidal forces so that
energy shows up at many more frequencies than just M2 and S2.
Spectrum
of sea level variations (estimated from bottom pressure) at the inshore Coastal
Mixing and Optics experiment site. Individual peaks at tidal constituent
frequencies are labeled. From Figure 3 of Shearman, R. K., and S. J.
Lentz (2004), Observations of tidal variability on the
·
One
lunar day is 24.8412 hours, so the M2 period is half this (2 bulges)
at 12.4206 hours.
·
One
solar day is 24 hours, so the S2 period is 12 hours.
·
When
the moon and sun are in alignment, M2 and S2 combine
o
spring
tides (and neap tides 7.4 days later)
[ GodinTides.pdf
]
·
earth-moon
orbit is elliptical
o
perigee
–> apogee –> perigee takes 27.6 days
o
this
modulates M2, and shows up as a lower frequency in a spectrum of
tidal height variation at period N2 = 12.6583 hours
o
the
stronger perigean tide occurs when M2 and N2 come into
phase
·
when
the lunar perigee is close to full or new moon, we get perigean spring tides
·
the
moon’s orbital plane is not in the equatorial plane (declination), so the high
tides are of different height
o
diurnal
inequality
o
produces
two lunar diurnal constituents O1 and K1 with periods
of 25.8193 and 23.9345 hours O1
is wrong in Bowden table
o
cancel
out every 13.66 days (˝ the
declinational period) when the moon is over the equator
o
The
maximum angle of the plane of the moon’s orbit and earth’s equator varies from
18o to 29o over an 18.6 year period
|
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 |