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What goes UP must come DOWN The mathematics of tide prediction

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Title: What goes UP must come DOWN The mathematics of tide prediction


1
What goes UP must come DOWN The mathematics of
tide prediction
Rex Boggs, Glenmore State High School Kiddy
Bolger, The Rockhampton Grammar School
2
Tides are defined as The periodic rise and fall
of … water resulting from gravitational
interactions between the Sun, Moon, and Earth.
3
A model for tide height often used in the
classroom is a sine function of the form
4
But actual tides are much more complex than a
simple sine function.
5
The modern method used to generate an accurate
mathematical model for tides is called harmonic
analysis. The model generated by harmonic
analysis is a sum of cosine terms, plus a
constant.
6
The tide function for a given port typically
has upwards of twenty cosine terms. Each cosine
term is called a harmonic constituent. The
constant term Z0 is called the chart datum and is
the base from which tides are measured.
7
Until 1965, tide heights were calculated by
machines similar to this one designed by Sir
William Thomson in 1873.
8
The photo shows one such machine for calculating
tides. In the mid-1960s, these machines, were
replaced by computers.
9
Consider the tides at Mooloolaba on January 1,
2002. Here are the data from the Official Tide
Tables and Boating Safety Guide, 2002
We wish to develop a mathematical model that
accurately predicts these tide heights.
10
Our first tidal model will use only one harmonic
constituent, M2, which is the gravitational
influence of the Moon. This is the constituent
with the greatest influence on the tides.
11
From the Australian National Tide Tables for
2002, we obtain the following information for
Mooloolaba on 1 Jan 2002
Chart datum for Mooloolaba is 0.94 metres.
12
Substituting into the tide function gives us our
first model for the tides at Mooloolaba on 1
January 2002
Note that the formulas for tide prediction are
in degrees (and not radians)
13
Here is the graph of our model over a twenty-four
hour period.
This model predicts a low tide of 0.336 metres at
319 a.m., while the tide tables show a low tide
of 0.06 m at 243 a.m. Our model needs a bit of
refinement!
14
The next most important constituent is S2, the
gravitational influence of the Sun. At new moon
and full moon, the Sun and the Moon combine to
give larger tides.
15
In the first quarter and third quarter, the Sun
and the Moon are pulling at right angles to each
other, resulting in smaller tides.
16
This diagram nicely shows the effect on the tide
height of the different phases of the moon over a
thirty-day period.
17
From the Australian National Tide Tables for
2002, we obtain the following information for
Mooloolaba on 1 Jan 2002
18
This results in a more accurate model for the
tide height.
19
Here is the 24 hour graph
This model predicts a low tide of 0.219 metres
(compared to 0.06) at 309 a.m. (compared to 248
a.m.).
20
This is a better model, but still not
sufficiently accurate. We havent yet
accounted for the fact that successive high tides
usually are not equal.
21
In Mooloolaba on 1 January 2002, the high tides
differ by 0.72 metre, and the low tides differ by
0.23 metres. Such differences are a common
occurrence at mid-latitudes.
The high tides differ by 0.72 m
22
This is largely caused by the declination of the
moon (i.e. the angle of the moon above or below
the equator).
23
A person at A would experience a relatively high
tide. Slightly more than twelve hours later, the
person would be at B (due to the rotation of the
Earth) and the tide would be smaller.
24
This is called a diurnal effect, since its period
is approximately one day. There are two harmonic
constituents needed to account for this, called
K1 and O1.
25
Including these constituents gives us the
following model for tide height
26
Here is the graph of this function
This model predicts a low tide of 0.05 metres
(compared to 0.06) at 241 a.m. (compared to 248
a.m.).
27
The model also predicts a high tide of 2.02
metres (compared to 2.03) at 930 a.m. (compared
to 929 a.m.).
28
The tide model, using only four harmonic
constituents, has an error of less than 1 for
both time and height !
29
For each port in Australia, the tide models
constructed at the National Tidal Facility at
Flinders University typically include over twenty
constituents. It is interesting to learn that
tide models based solely on the four constituents
M2, S2, K1 and O1 will have a maximum error in
both height and time of no more than 5 (and
often much less than that).
30
The presenters wish to thank Mike Davis from
the National Tidal Facility at Flinders
University and Captain Arthur Diack from the
Maritime Division of Queensland Transport for
their technical assistance in the development of
this presentation.
31
Appendix A A more general model for tide
prediction
32
The tide function given earlier was a
simplified version of the more general model
so each constituent has the form
33
The amplitude of each constituent is actually the
product of two factors, H and F. H is the base
amplitude. Determining this value requires
observations to be taken over many years.
34
The amplitude of each constituent is actually the
product of two factors, H and F. F is a slowly
varying function, having a period of about
nineteen years. This function is based on the
precession of the plane of the Moons orbit,
which has a period of about 18.6 years.,
35
The phase angle has two components, g and A,
which are added together. g depends only on the
site at which the observations are made. A is
the phase of the constituent at Greenwich when t
0 (i.e. at midnight).
36
  • In summary
  • The values of Z0, H and g depend only upon the
    port. Values for each Standard Port are given
    in Table II of the National Tide Tables.
  • The values of A and F for the four main
    constituents depend only upon time. Values for
    each day of the year are given in Table III of
    the National Tide Tables.

37
As an example, here are the calculations for the
M2 constituent at Mooloolaba on 1 January 2002.
From the Australian National Tide Tables H for
Mooloolaba is 0.539 F on 1 January 2002 was
1.12 Multiplying these factors gives 0.60368,
which was the amplitude given previously.
38
From the Australian National Tide Tables g for
Mooloolaba is 234.9 A on 1 January 2002 was
041 Adding these factors gives 275.9, which was
the phase angle given previously for the M2
constituent.
39
Appendix B Further information about tides
40
Now a bit more about tides. This diagram show
why successive tides at the equator tend to be a
similar height.
41
This diagram show why there tends to be only one
high tide per day at higher latitudes. At B, the
tide is negligible.
42
Not all tides are semi-diurnal. Here are 30-day
tide charts from other parts of the world.
43
These diagrams show why typically there are two
high tides per day. One is caused by the
gravitational attraction of the moon, whilst the
other is due to centrifugal force caused by the
rotation of the Earth/Moon combination about
their centre of gravity.
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