12 Eyl�l 1998

2. Introduction

Coastal water levels are influenced by a variety of astronomical, meteorological/ oceanographical and tectonic factors, the most readily apparent being the tides. At times, these factors interact in a complex way to elevate water levels significantly above normal tide level. Storms, which develop low atmospheric pressure, strong onshore winds and large waves, are the most common cause of elevated water levels.

Elevated water levels are of concern because they intensify damage to the coastline and to coastal developments. Elevated water levels allow larger waves to cross offshore bars and break closer to the beach, which in turn increases beach erosion and the threat to coastal developments. Elevated water levels can inundate low lying areas of the coastline and around estuaries.

� Astronomical Tides

The astronomical tide is caused by the gravitational effect of the moon, and to a lesser extent, the sun and other planets on the water mass of the oceans. Along the Ala�at� coast, tides are semi-diurnal, i.e. two high tides and two low tides per day.

� Meteorological/Oceonographical Processes
Three different meteorological processes can affect coastal water levels:

� storms

� meteorological oscillations

� climate change

Storms are local meteorological disturbances. The other two processes are semi-global or global in nature. Climate change, including the Greenhouse Effect.

�
Storms
The elevation of water level associated with a storm depends primarily on the following factors:

� the intensity, scale and direction and speed of movement of the storm;

� the bathymetry of the coastal area, including the presence or otherwise of offshore reefs and islands;

� the shape of the coastline, including the topography of the nearshore areas which may be inundated; and

� the prevailing astronomical tide.

A storm increases coastal water levels in four distinct ways: by "setup" due to barometric, wind and wave effects and by wave "runup". Figure 1.1 illustrates these components of elevated water levels. The four components are all additive and their sum represents the superelevation of storm water level above prevailing astronomical tide level.

Figure 1.1.

Barometric setup

The reduced barometric pressures that generate storm winds also cause a local rise in sea level (the inverse barometer effect). Providing low pressures persist for a sufficient length of time, the increase in water level amounts to about 0.10m for each 10hPax drop below normal barometric pressure (1,013hPa). In a severe storm with a central pressure of 980hPa, this amounts to about 0.3m.

Wind setup

Wind blowing onshore over the sea's surface drives the surface waters before it and against the coastline. This results in elevated water levels in coastal areas, the degree of elevation being higher for extensive shallow areas and semi-enclosed bays.

Wind setup is very important in Ala�at� Bay. Our mathematical analyses and practical observations show that the deviation from astronomical tide increases sea-level in stormy days.

Storm Surge

The sum of barometric and wind setup is often referred to as storm surge.

Wave setup

The breaking action of waves results in an increase in water levels in the surf zone known as "wave setup". Wave setup is associated with the conversion of the wave's kinetic energy into potential energy (Battjes, 1974). The degree of setup depends upon the type, size and period of the waves at breaking and the slope of the beach.

Wave runup

Wave runup is an oscillatory phenomenon and refers to the vertical distance the uprush of water from a breaking wave reaches above the combined level of the tide, storm surge and wave setup. A wave runup of more than 6m can occur. The magnitude of runup depends upon a variety of factors, particularly the slope and roughness of the runup surface. Runup on flat beaches is generally less than on steeper beaches; runup on smooth vertical sea walls is generally greater than on protective works with rough sloping faces.

Wave runup can result in the intermittent discharge of seawater into backbeach areas that may appear to be protected by beach barriers, such as sand dunes or seawalls.

Rainfall Runoff

Surface runoff from any rainfall accompanying a storm may cause an increase in water levels within estuaries and tidal inlets. Rainfall and runoff have no significant effect on coastal water levels.

Oceanographic Effects

Distant meteorological disturbances that are characterised by a sharp pressure gradient can generate a long low wave with a period of up to 10 days and a height of up to 0.2m. As this wave travels along the continental shelf, it becomes a "shelf wave" that is "trapped" by the shelf which acts as a wave guide. Shelf waves also modify coastal water levels.

Other effects which can result in tidal anomalies include variations in sea temperature and salinity and the influence of strong currents.

These type of effects can not be seen in our seas, especially in Aeagen Sea.

�

Climate Change

A "eustatic" sea level change refers to a change in the mean water level of the oceans around the globe. A eustatic rise can occur through two mechanisms: the expansion of the surface waters of the ocean caused by a global warming and by the melting of land-based glacier ice that accompanies any such warming. In the initial period of any global warming, i.e. the first 50 to 100 years, the first effect will be the more significant.

The term "greenhouse effect" is presently being used to describe a postulated warming of the earth due to the accumulation of certain gases in the atmosphere. In particular, the increase in levels of carbon dioxide (CO2) resulting from the burning of fossil fuels is of interest.

The current consensus of scientific opinion is that such changes could result in a global warming of 1.5� to 4.5�C over the next 30 to 50 years. Such a warming could lead to a number of changes in climate, weather and sea levels. These in turn could cause significant changes to coastal processes, e.g. increased severity and frequency of storms resulting in increased wave heights.

The atmosphere plays a crucial moderating role in the heat balance of the Earth. The principal gases of the atmosphere are nitrogen (78%) and oxygen (21%). However, their ability to absorb heat is low and they play little part in the heat balance. In contrast, carbon dioxide, nitrous oxide, methane and water vapour, which in total amount to less than 1% of the atmosphere, have high heat capacities and play a major role in the heat balance.

Relatively small changes in the concentrations of these gases may result in significant changes to the heat balance and to atmospheric temperatures. Hence the concern over CO₂ levels.

Radiocarbon dating and the analysis of small air bubbles trapped deep in antarctic ice has made it possible to reconstruct some of the past history of the world's climate. Figure 1.2 shows the variation of CO₂ levels, atmospheric temperature and sea level over the past 160,000 years. CO₂ levels were determined from air in the ice; temperatures at the time of ice formation were estimated from the relative concentrations of the isotopes oxygen-16 and deuterium (Fifield, 1988; Barnola et. al., 1987). The sea level changes shown in Figure 1.2. are taken from Chapman et. al., 1982. The variations of Figure 1.2. indicate two ice ages (150,000 and 20,000 years Before Present) and two warm periods (125,000 years ago and present time).

Figure 1.2: Atmospheric C0₂ Concentrations, Atmospheric Temperature and Sea Level over the Past 160,000 years. (After Gordon, 1989).

The data shows a good correlation between the variation of CO₂ levels in the atmosphere and change in surface temperature. The correlations between sea level and surface temperature and between sea level and CO₂ concentration appear reasonably good. These correlations are not as distinct as that between surface temperature and CO₂ concentration because of the relatively inferior accuracy and density of the sea level data set.

Carbon dioxide levels in the atmosphere are thought to have increased by about 50 ppmv since the industrial revolution. This has been attributed to the burning of fossil fuels but a variety of other factors, often surprising in their nature, make significant contributions (bovine flatulence, paddy fields, etc.). Figure 1.3. shows the increase in CO₂ levels in Hawaii between 1958 and 1980. Over this period of time, the mean monthly level increased from 315 to 340 ppmv.

Figure 1.3: Mean Monthly CO₂ Level, Mauna Loa, Hawaii (NRC, 1983)

To summarize: To date, the "hard" evidence in support of a "greenhouse" increase in temperatures is limited to the observed increase in CO₂ levels from 1958 onwards and global temperature trends over the last 100 years.. Historical evidence suggests that CO₂ levels have varied between 190 and 340 ppmv over the last 160,000 years. Again, historical evidence suggests that atmospheric temperature changes follow changes in CO₂ levels. The relationships between CO₂ levels, temperature and sea level are reasonably good.

Scientific opinion is divided regarding the timing and the likely degree of "greenhouse" warming. There is however a general consensus that warming will occur. If warming occurs it is generally agreed that sea level will rise.

Figure 1.4. Variation in Mean Sea Level over the last 250,000 Years.

Figure 1.5: Sea Level Scenarios (NAS, 1987)

This figure� illustrates sea level scenarios to the year 2100 adopted by the U.S. National Research Council after deliberations by a technical committee (NAS, 1987). These scenarios were adopted after review of information available from the scientific community. The three scenarios adopted are for a sea level rise of 0.5m, 1.0m and 1.5m by the year 2100.

Figure 1.6: Sea Level Scenarios (Commonwealth Group of Experts, 1989)

This figure illustrates sea level scenarios to the year 2050 based on box diffusion modelling of ocean warming which was undertaken at the University of East Anglia (Commonwealth Group of Experts, 1989). These projections to the year 2050 range from a sea level rise of 7 to 67 cm (best estimate range 24 to 38 cm).

Figure 1.7: Sea Level Scenarios (UNEP/IPCC, 1990)

This figure illustrates "Scenario A" of sea level scenarios developed by the United Nations Environmental Program Intergovernmental Panel on Climate Change (UNEP-IPCC, 1990). "Scenario A" is based on no limitation of greenhouse gas production which is considered the most realistic option to choose for planning purposes at this time.

Examination of Figures 1.5, 6 and 7 indicates that the sea level scenarios for say 2050 are very similar. As the IPCC Working Group report is the most recent and it accounts for the views of the international scientific community, it is considered that Figure 1.6 illustrates the currently available "best estimate" of sea level scenarios.

� Surf Beats

When swell waves from two different storm sources arrive simultaneously at a beach, the resultant waves tend to occur in consecutive groups of large and small waves (leading to the popular belief that every seventh wave is the largest). This has the effect of inducing periodic water level fluctuations in the amount of wave setup at the shoreline. Longer period water level fluctuations (2 to 3 minutes) are often referred to as "surf beat" and may have amplitudes of up to 0.5m.

� Tide Measurement

Entropy Microsystems have been measuring the tidal data along the Ala�at� coastline for 2 years. A specially designed instrument is used to collect the data which essentially is in the form of water level against time. Also this tide measurement associated with wind measurements near Ala�at� Bay (Yumrukoy, Sadl�ktepe, �e�me.)

� Design Considerations

Determination of appropriate design water levels for coastal developments requires first, an assessment of each component of elevated water level at the subject site and second, the combining of these components in a realistic and statistically meaningful way. Simple addition of the values for each element is not necessarily appropriate and will usually result in a conservative design value.

Estimation of Water Level Components

Design values for water level components can be determined from measured values (if available), from analytical formulae or by numerical simulation.

Tidal data for Port Agrilia is available since 1997. These data can be used to estimate tidal behaviour at unreferenced locations. At sites where tidal effects may be significantly modified by the local bathymetry, a "harmonic analysis" of measured tidal data may be required to better define likely tidal behaviour. This requires water level data collected at the site over a period of time, the length of which depends on the complexity of the tidal system and the accuracy sought.

Mathematical modelling is necessary to derive long-term storm statistics at specific sites. Computer simulation has been used for wind field modelling (Graham and Nunn, 1959) and for storm surge modelling (Sobey, Harper & Stark, 1977).

Extreme Value Analysis

There is some difficulty in meaningfully combining storm surge statistics with tide height statistics to determine the extreme values of elevated water levels. Methods based on the application of conditional probabilities have been applied (Dexter, 1975; Haradasa et al, 1989), but inconsistencies remain. The mathematical simulation of the occurrence of a large number of random storms with coincident tides is another method of determining the likelihood of extreme water levels (McMonagle and Fidge, 1981).

3. What Is Tide?

3.1. �Definition of Tide

The word "tides" is a generic term used to define the alternating rise and fall in sea level with respect to the land, produced by the gravitational attraction of the moon and the sun. To a much smaller extent, tides also occur in large lakes, the atmosphere, and within the solid crust of the earth, acted upon by these same gravitational forces of the moon and sun. Additional nonastronomical factors such as configuration of the coastline, local depth of the water, ocean-floor topography, and other hydrographic and meteorological influences may play an important role in altering the range, interval between high and low water, and times of arrival of the tides.

The most familiar evidence of the tides along our seashores is the observed recurrence of high and low water - usually, but not always, twice daily. The term tide correctly refers only to such a relatively short-period, astronomically induced vertical change in the height of the sea surface (exclusive of wind-actuated waves and swell); the expression tidal current relates to accompanying periodic horizontal movement of the ocean water, both near the coast and offshore (but as distinct from the continuous, stream-flow type of ocean current).

Knowledge of the times, heights, and extent of inflow and outflow of tidal waters is of importance in a wide range of practical applications such as the following: Navigation through intracoastal waterways, and within estuaries, bays, and harbors; work on harbor engineering projects, such as the construction of bridges, docks, breakwaters, and deep-water channels; the establishment of standard chart datums for hydrography and for demarcation of a base line or "legal coastline" for fixing offshore territorial limits both on the sea surface and on the submerged lands of the Continental Shelf; provision of information necessary for underwater demolition activities and other military engineering uses; and the furnishing of data indispensable to fishing, boating, surfing, and a considerable variety of related water sport activities.

3.2. �The Astronomical Tide-Producing Forces: General Considerations

At the surface of the earth, the earth's force of gravitational attraction acts in a direction inward toward its center of mass, and thus holds the ocean water confined to this surface. However, the gravitational forces of the moon and sun also act externally upon the earth's ocean waters. These external forces are exerted as tide-producing, or so-called "tractive" forces. Their effects are superimposed upon the earth's gravitational force and act to draw the ocean waters to positions on the earth's surface directly beneath these respective celestial bodies (i.e., towards the "sublunar" and "subsolar" points).

High tides are produced in the ocean waters by the "heaping" action resulting from the horizontal flow of water toward two regions of the earth representing positions of maximum attraction of combined lunar and solar gravitational forces. Low tides are created by a compensating maximum withdrawal of water from regions around the earth midway between these two humps. The alternation of high and low tides is caused by the daily (or diurnal) rotation of the earth with respect to these two tidal humps and two tidal depressions. The changing arrival time of any two successive high or low tides at any one location is the result of numerous factors later to be discussed.

3.3. �Origin of the Tide-Raising Forces

To all outward appearances, the moon revolves around the earth, but in actuality, the moon and earth revolve together around their common center of mass, or gravity. The two astronomical bodies are held together by gravitational attraction, but are simultaneously kept apart by an equal and opposite centrifugal force produced by their individual revolutions around the center-of-mass of the earth-moon system. This balance of forces in orbital revolution applies to the center-of-mass of the individual bodies only. At the earth's surface, an imbalance between these two forces results in the fact that there exists, on the hemisphere of the earth turned toward the moon, a net (or differential) tide-producing force which acts in the direction of the moon's gravitational attraction, or toward the center of the moon. On the side of the earth directly opposite the moon, the net tide-producing force is in the direction of the greater centrifugal force, or away from the moon.

Similar differential forces exist as the result of the revolution of the center-of-mass of the earth around the center-of-mass of the earth-sun system.

3.4. �Detailed Explanation of the Differential Tide Producing Forces

The tide-raising forces at the earth's surface thus result from a combination of basic forces: (1) the force of gravitation exerted by the moon (and sun) upon the earth; and (2) centrifugal forces produced by the revolutions of the earth and moon (and earth and sun) around their common centers-of-gravity (mass). The effects of those forces acting in the earth-moon system will here be discussed, with the recognition that a similar force complex exists in the earth-sun system.

With respect to this center-of-mass of the earth-moon system (known as the barycenter) the above two forces always remain in balance (i.e., equal and opposite). In consequence, the moon revolves in a closed orbit around the earth, without either escaping from, or falling into the earth - and the earth likewise does not collide with the moon. However, at local points on, above, or within the earth, these two forces are not in equilibrium, and oceanic, atmospheric, and earth tides are the result.

Figure 3.4.1: The Monthly Revolution of the Earth and Moon Around the Barycenter of the Earth-Moon System

This revolution is responsible for a centrifugal force component (Fc) necessary to the production of the tides.

a centrifugal force component (Fg) necessary to th

The center of revolution of this motion of the earth and moon around their common center-of-mass lies at a point approximately 1,068 miles beneath the earth's surface, on the side toward the moon, and along a line connecting the individual centers-of-mass of the earth and moon. (see G, Figure. 3.4.1.) The center-of-mass of the earth describes an orbit (E1, E2, E3..) around the center-of-mass of the earth-moon system (G) just as the center-of-mass of the moon describes its own monthly orbit (M1, M2, M3..) around this same point.

The Effect of Centrifugal Force.

It is this little known aspect of the moon's orbital motion which is responsible for one of the two force components creating the tides. As the earth and moon whirl around this common center-of-mass, the centrifugal force produced is always directed away from the center of revolution in the same manner that an object whirled on a string around one's head exerts a tug upon the restraining hand. All points in or on the surface of the earth acting as a coherent body acquire this component of centrifugal force. And, since the center-of-mass of the earth is always on the opposite side of this common center of revolution from the position of the moon, the centrifugal force produced at any point in or on the earth will always be directed away from the moon. This fact is indicated by the common direction of the arrows (representing the centrifugal force Fc) at points A, C, and B in Figure 3.4.1, and the thin arrows at these same points in Figure 3.4.2.

It is important to note that the centrifugal force produced by the daily rotation of the earth on it axis must be completely disregarded in tidal theory. This element plays no part in the establishment of the differential tide-producing forces.

While space does not permit here, it may be graphically demonstrated that, for such a case of revolution without accompanying rotation as above enumerated, any point on the earth will describe a circle around the earth's center-of-mass which will have the same radius as the radius of revolution of the center-of-mass of the earth around the barycenter. Thus, in Fig. 1, the magnitude of the centrifugal force produced by the revolution of the earth and moon around their common center of mass (G) is the same at point A or B or any other point on or beneath the earth's surface. Any of these values is also equal to the centrifugal force produced at the center-of-mass (C) by its revolution around the barycenter. This fact is indicated in Figure 3.4.2. by the equal lengths of the thin arrows (representing the centrifugal force Fc) at points A, C, and B, respectively.

The Effect of Gravitational Force.

While the effect of this centrifugal force is constant for all positions on the earth, the effect of the external gravitational force produced by another astronomical body may be different at different positions on the earth because the magnitude of the gravitational force exerted varies with the distance of the attracting body. According to Newton's Universal Law of Gravity, gravitational force decreases as the second power of the distance from the attracting body. As a special case, the tide-raising force varies inversely as the third power of the distance of the center-of-mass to the attracting body from the surface of the earth. Thus, in the theory of the tides, a variable influence is introduced based upon the different distances of various positions on the earth's surface from the moon's center-of-mass. The relative gravitational attraction (Fg) exerted by the moon at various positions on the earth is indicated in Figure 3.4.2. by arrows heavier than those representing the centrifugal force components.

The Net or Differential Tide-Raising Forces: Direct and Opposite Tides.

It has been emphasized above that the centrifugal force under consideration results from the revolution of the center-of-mass of the earth around the center-of-mass of the earth-moon system, and that this centrifugal force is the same anywhere on the earth. Since the individual centers-of-mass of the earth and moon remain in equilibrium at constant distances from the barycenter, the centrifugal force acting upon the center of the earth (C) as the result of their common revolutions must be equal and opposite to the gravitational force exerted by the moon on the center of the earth. This fact is indicated at point C in Figure 3.4.2. by the thin and heavy arrows of equal length, pointing in opposite directions. The net result of this circumstance is that the tide-producing force (Ft) at the earth's center is zero.

At point A in Figure. 3.4.2, approximately 4,000 miles nearer to the moon than is point C, the force produced by the moon's gravitational pull is considerably larger than the gravitational force at C due to the moon (the earth's own gravity is, of course, zero at point C). The smaller lunar gravitational force at C just balances the centrifugal force at C. Since the centrifugal force at A is equal to that at C, the greater gravitational force at A must also be larger than the centrifugal force there. The net tide-producing force at A obtained by taking the difference between the gravitational and centrifugal forces is in favor of the gravitational component - or outward toward the moon. The tide-raising force at point A is indicated in Figure 3.4.2. by the double arrow extending vertically from the earth's surface toward the moon. The resulting tide produced on the side of the earth toward the moon is know as the direct tide.

Figure 3.4.2: The Combination of Forces of Lunar Origin Producing the Tides

(A similar complex of forces exists in the Earth-Sun system)

At point B, on the opposite side of the earth from the moon and about 4,000 miles farther away from the moon than is point C, the moon's gravitational force is considerably less than at point C. At point C, the centrifugal force is in balance with a gravitational force which is greater than at B. The centrifugal force at B is the same as that at C. Since gravitational force is less at B than at C, it follows that the centrifugal force exerted at B must be greater than the gravitational force exerted by the moon at B. The resultant tide-producing force at this point is, therefore, directed away from the earth's center and opposite to the position of the moon. This force is indicated by the double-shafted arrow at point B. The tide produced in this location halfway around the earth from the sublunar point, coincidentially with the direct tide, is know as the opposite tide.

The Tractive Force.

It is significant that the influence of the moon's gravitational attraction superimposes its effect upon, but does not overcome, the effect's of the earth's own gravity. Earth-gravity, although always present, plays no direct part in the tide-producing action. The tide-raising force exerted at a point on the earth's surface by the moon at its average distance from the earth (238,855 miles) is only about one 9-millionth part of the force of earth-gravity exerted toward its center (3,963 miles from the surface). The tide raising force of the moon, is, therefore, entirely insufficient to "lift" the waters of the earth physically against this far greater pull of earth's gravity. Instead, the tides are produced by that component of the tide-raising force of the moon which acts to draw the waters of the earth horizontally over its surface toward the sublunar and antipodal points. Since the horizontal component does not oppose in any way to gravity and can, therefore, act to draw particles of water freely over the earth's surface, it becomes the effective force in generating tides.

At any point on the earth's surface, the tidal force produced by the moon's gravitational attraction may be separated or "resolved" into two components of force - one in the vertical, or perpendicular to the earth's surface - the other horizontal or tangent to the earth's surface. This second component, know as the tractive ("drawing") component of force is the actual mechanism for producing the tides. The force is zero at the points on the earth's surface directly beneath and on the opposite side of the earth from the moon (since in these positions, the lunar gravitational force is exerted in the vertical - i.e., opposed to, and in the direction of the earth-gravity, respectively). Any water accumulated in these locations by tractive flow from other points on the earth's surface tends to remain in a stable configuration, or tidal "bulge".

Thus there exists an active tendency for water to be drawn from other points on the earth's surface toward the sublunar point (A, in Figure 3.4.2) and its antipodal point (B, in Figure. 3.4.2) and to be heaped at these points in two tidal bulges. Within a band around the earth at all points 90^o from the sublunar point, the horizontal or tractive force of the moon's gravitation is also zero, since the entire tide-producing force is directed vertically inward. There is, therefore, a tendency for the formation of a stable depression here. The words "tend to" and "tendency for" employed in several usages above in connection with tide-producing forces are deliberately chosen since, as will be seen below, the actual representation of the tidal forces is that of an idealized "force envelope" with which the rise and fall of the tides are influenced by many factors.

The Tidal Force Envelope.

If the ocean waters were completely to respond to the directions and magnitudes of these tractive forces at various points on the surface of the earth, a mathematical figure would be formed having the shape of a prolate spheroid. The longest (major) axis of the spheroid extended towards and directly away from the moon, and the shortest (minor) axis is center along, at right angle to, the major axis. The two tidal humps and two tidal depressions are represented in this force envelope by the directions of the major axis and rotated minor axis of the spheroid, respectively. From a purely theoretical point of view, the daily rotation of the solid earth with respect to these two tidal humps and two depressions may be conceived to be the cause of the tides.

As the earth rotates once in each 24 hours, one would ideally expect to find a high tide followed by a low tide at the same place 6 hours later; then a second high tide after 12 hours, a second low tide 18 hours later, and finally a return to high water at the expiration of 24 hours. Such would nearly be the case if a smooth, continent-free earth were covered to a uniform depth with water, if the tidal envelope of the moon alone were being considered, if the positions of the moon and sun were fixed and invariable in distance and relative orientation with respect to the earth, and if there were no other accelerating or retarding influences affecting the motions of the waters of the earth. Such, in actuality, is far from the situation which exists.

Figure 3.4.3: The Phase Inequality: Spring and Neap Tides

The gravitational attractions (and resultant tidal force envelopes) produced by the Moon and Sun reinforce each other at times of new and full moon to increase the range of the tides, and counteract each other at the first and third quarters to reduce the tidal range.

First, the tidal force envelope produced by the moon's gravitational attraction is accompanied by a tidal force envelope of considerably smaller amplitude produced by the sun. The tidal force exerted by the sun is a composite of the sun's gravitational attraction and a centrifugal force component created by the revolution of the earth's center-of-mass around the center-of-mass of the earth-sun system, in an exactly analogous manner to the earth-moon relationship. The position of this force envelope shifts with the relative orbital position of the earth in respect to the sun. Because of the great differences between the average distances of the moon (238,855 miles) and sun (92,900,000 miles) from the earth, the tide producing force of the moon is approximately 2.5 times that of the sun.

Second, there exists a wide range of astronomical variables in the production of the tides caused by the changing distances of the moon from the earth, the earth from the sun, the angle which the moon in its orbit makes with the earth's equator, the superposition of the sun's tidal envelope of forces upon that caused by the moon, the variable phase relationships of the moon, etc. Some of the principle types of tides resulting from these purely astronomical influences are describe below.

3.5. �Variations in the Range of� the Tides: Tidal Inequilities

As will be shown in Figure 3.4.6, the difference in the height, in feet, between consecutive height and low tides occurring at a given place is known as the range. The range of the tides at any location is subject to many variable factors. Those influences of astronomical origin will first be described.

1. Lunar Phase Effect: Spring and Neap Tides.

It has been noted above that the gravitational forces of both the moon and sun act upon the waters of the earth. It is obvious that, because of the moon's changing position with respect to the earth and sun (Figure. 3.4.3) during the monthly cycle of phases (29.53 days) the gravitational attraction of moon and sun may variously act along a common line or at changing angles relative to each other.

When the moon is at new phase and full phase (both positions being called syzygy) the gravitational attractions of the moon and sun act to reinforce each other. Since the resultant or combined tidal force is also increased, the observed high tides are higher and low tides are lower than average. This means that the tidal range is greater at all locations which display a consecutive high and low water. Such greater-than-average tides resulting at the syzygy positions of the moon are know as spring tides - a term which merely implies a "welling up" of the water and bears no relationship to the season of the year.

At first- and third-quarter phases (quadrature) of the moon, the gravitational attractions of the moon and sun upon the waters of the earth are exerted at right angles to each other. Each force tends in part to counteract the other. In the tidal force envelope representing these combined forces, both maximum and minimum forces are reduced. High tides are lower and low tides are higher than average. Such tides of diminished range are called neap tides, from a Greek word meaning "scanty".

2. Parallax Effects (Moon and Sun).

Since the moon follows an elliptical path (Figure 3.4.4), the distance between the earth and moon will vary throughout the month by about 31,000 miles. The moon's gravitational attraction for the earth's waters will change in inverse proportion to the third power of the distance between the earth and moon, in accordance with the previously mentioned variation of Newton's Law of Gravitation. Once each month, when the moon is closest to the earth (perigee), the tide-generating forces will be higher than usual, thus producing above-average ranges in the tides. Approximately two weeks later, when the moon (at apogee) is farthest from the earth, the lunar tide-raising force will be smaller, and the tidal ranges will be less than average. Similarly, in the earth-sun system, when the earth is closest to the sun (perihelion), about January 2 of each year, the tidal ranges will be enhanced, and when the earth is farthest from the sun (aphelion), around July 2, the tidal ranges will be reduced.

Figure 3.4.4: The Lunar Parallax and Solar Parallax Inequalities

Both the Moon and the Earth revolve in elliptical orbits and the distances from their centers of attraction vary. Increased gravitational influences and tide-raising forces are produced when the Moon is at position of perigee, its closest approach to the Earth (once each month) or the Earth is at perihelion, its closest approach to the Sun (once each year). This diagram also shows the possible coincidence of perigee with perihelion to produce tides of augmented range.

When perigee, perihelion, and either the new or full moon occur at approximately the same time, considerably increased tidal ranges result. When apogee, aphelion, and the first- or third-quarter moon coincide at approximately the same time, considerably reduced tidal ranges will normally occur.

3. Lunar Declination Effects: The Diurnal Inequality.

�The plane of the moon's orbit is inclined only about 5^o to the plane of the earth's orbit (the ecliptic) and thus the moon monthly revolution around the earth remains very close to the ecliptic. The ecliptic is inclined 23.5^o to the earth's equator, north and south of which the sun moves once each half year to produce the seasons. In similar fashion, the moon, in making a revolution around the earth once each month, passes from a position of maximum angular distance north of the equator to a position of maximum angular distance south of the equator during each half month. (Angular distance perpendicularly north and south of the celestial equator is termed declination.) twice each month, the moon crosses the equator. In Fig. 5, this condition is shown by the dashed position of the moon. The corresponding tidal force envelope due to the moon is depicted, in profile, by the dashed ellipse.

Figure 3.4.5: The Moon's Declination Effect (Change in Angle With Respect to the Equator) and the Diurnal Inequality; Semidiurnal, Mixed, and Diurnal Tides

A north-south cross-section through the Earth's center; the ellipse represents a meridian section through the tidal force envelope produced by the Moon.

Since the points A and A' lie along the major axis of this ellipse, the height of the high tide represented at A is the same as that which occurs as this point rotates to position A' some 12 hours later. When the moon is over the equator - or at certain other force-equalizing declinations - the two high tides and two low tides on a give day are at similar height at any location. Successive high and low tides are then also nearly equally spaced in time, and occur twice daily. (See top diagram in Figure. 3.4.6.) This is known as semidiurnal type of tides.

However, with he changing angular distance of the moon above or below the equator (represented by the position of the small solid circle in Figure 3.4.5) the tidal force envelope produced by the moon is canted, and difference between the heights of two daily tides of the same phase begin to occur. variations in the heights of the tides resulting from the changes in the declination angle of the moon and in the corresponding lines of gravitational force action give rise to a phenomenon known as the diurnal inequality.

In Figure 3.4.5, point B is beneath a bulge in the tidal envelope. One-half day later, at point B' it is again beneath the bulge, but the height of the tide is obviously not as great as at B. This situation gives rise to a twice-daily tide displaying unequal heights in successive high or low waters, or in both pairs of tides. This type of tide, exhibiting a strong diurnal inequality, is known as a mixed tide. (See the middle diagram in Figure. 3.4.6.)

Figure 3.4.6: Principal Types of Tides

Showing the Moon's declinational effect in production of semidiurnal, mixed, and diurnal tides.

�

Finally, as depicted in Figure. 3.4.5, the point C is seen to lie beneath a portion of the tidal force envelope. One-half day later, however, as this point rotates to position C', it is seen to lie above the force envelope. At this location, therefore, the tidal forces present produce only one high water and one low water each day. The resultant diurnal type of tide is shown in the bottom diagram of Figure 3.4.6.

3.6. �Factors Influencing the Local Heights and Times of Arrival of the Tides

It is noteworthy in Figure 3.4.6 that any one cycle of the tides is characterized by a definite time regularity as well as the recurrence of the cyclical pattern. However, continuing observations at coastal stations will reveal - in addition to the previously explained variations in the heights of successive tides of the same phase - noticeable differences in their successive time of occurrence. The aspects of regularity in the tidal curves are introduced by harmonic motions of the earth and moon. The variations noted both in observed heights of the tides and in their times of occurrence are the result of many factors, some of which have been discussed in the preceding section. Other influences will now be considered.

The earth rotates on its axis (from one meridian transit of the "mean" sun until the next) in 24 hours. But as the earth rotates beneath the envelope of tidal forces produced by the moon, another astronomical factor causes the time between two successive upper transits of the moon across the local meridian of the place (a period known as the lunar or "tidal" day) to exceed the 24 hours of the earth's rotation period - the mean solar day.

The moon revolves in its orbit around the earth with an angular velocity of approximately 12.2^o per day, in the same direction in which the earth is rotating on its axis with an angular velocity of 360^o per day. In each day, therefore, a point on the rotating earth must complete a rotation of 360^o plus 12.2^o, or 372.2^o, in order to "catch up" with the moon. Since 15^o is equal to one hour of time, this extra amount of rotation equal to 12.2^o each day would require a period of time equal to 12.2^o/15^o x 60 min/hr., or 48.8 minutes - if the moon revolved in a circular orbit, and its speed of revolution did not vary. On the average it requires about 50 minutes longer each day for a sublunar point on the rotating earth to regain this position directly along the major axis of the moon's tidal force envelope, where the tide-raising influence is a maximum. In consequence, the recurrence of a tide of the same phase and similar rise (see middle diagram of Figure 3.4.6) would take place at an interval of 24 hours 50 minutes after the preceding occurrence, if this single astronomical factor known as lunar retardation were considered. This period of 24 hours 50 minutes has been established as the tidal day.

A second astronomical factor influencing the time of arrival of tides of a given phase at any location results from the interaction between the tidal force envelopes of the moon and sun. Between new moon and first-quarter phase, and between full moon and third-quarter phase, this phenomenon can cause a displacement of force components and an acceleration in tidal arrival times (known as priming the tides) resulting in the occurrence of high tides before the moon itself reaches the local meridian of the place. Between first-quarter phase and full moon, and between third-quarter phase and new moon, an opposite displacement of force components and a delaying action (known as lagging of the tides) can occur, as the result of which the arrival of high tides may take place several hours after the moon has reached the meridian.

These are the two principle astronomical causes for variation in the times of arrival of the tides. In addition to these astronomically induced variations, the tides are subject to other accelerating or retarding influences of hydraulic, hydrodynamic, hydrographic, and topographic origin - and may further be modified by meteorological conditions.

3.7. Prediction of the Tides

The first factor of consequence in this regard arises from the fact that the crests and troughs of the large-scale gravity-type traveling wave system comprising the tides strive to sweep continuously around the earth, following the position of the moon (and sun).

In the open ocean, the actual rise (see middle diagram, Figure 3.4.6) of the tidally induced wave crest is only one to a few feet. It is only when the tidal crests and troughs move into shallow water, against land masses, and into confining channels, that noticeable variations in the height of sea level can be detected.

Possessing the physical properties of a fluid, the ocean waters follow all of the hydraulic laws of fluids. This means that since the ocean waters possess a definite, although small internal viscosity, this property prevents their absolute free flow, and somewhat retards the overall movement of the tides.

Secondly, the ocean waters follow the principle of traveling waves in a fluid. As the depth of the water shallows, the speed of forward movement of a traveling wave is retarded, as deducted from dynamic considerations. In shoaling situations, therefore, the advance of tidal waters is slowed.

Thirdly, a certain relatively small amount of friction exists between the water and the ocean floor over which it moves - again slightly slowing the movement of the tides, particularly as they move inshore. Further internal friction (or viscosity) exists between tidally induced currents and contiguous current in the oceans - especially where they are flowing in opposite directions.

The presence of land masses imposes a barrier to progress of the tidal waters. Where continents interpose, tidal movements are confined to separate, nearly closed oceanic basins and the sweeps of the tides around the world is not continuous.

Topography on the ocean floor can also provide a restraint to the forward movement of tidal waters - or create sources of local-basin response to the tides. Restrictions to the advance of tidal waters imposed both by shoaling depths and the sidewalls of the channel as these waters enter confined bays, estuaries, and harbors can further considerably alter the speed of their onshore passage.

In such particularly confined bodies of water, so-called "resonance effects" between the free-period of oscillation of the traveling, tidally induced wave and that of the confining basin may cause a surging rise of the water in a phenomenon basically similar to the action of water caused to "slosh" over the sides of a wash basin by repeatedly tilting the basin and matching wave crests reflected from the opposite side of the basin.

All of the above, and other less important influences, can combine to create a considerable variety in the observed range and phase sequence of the tides - as well as variations in the times of their arrival at any location.

Of a more local and sporadic nature, important meteorological contributions to the tides know as "storm surges", caused by a continuous strong flow of winds either onshore or offshore, may superimpose their effects upon those of tidal action to cause either heightened or diminished tides, respectively. High-pressure atmospheric systems may also depress the tides, and deep low-pressure systems may cause them to increase in height.

4. Description of Data

4.1. The Measurement Method of Tidal Data

Figure 4.1: Sea level monitoring station at Port Agrilia.

The tidal data from Port Agrilia have been collected by using necessary instrumentation shown on Figure 4.1. for 2 years. This device is Entropy Microsystems� own production. It contains a microcontroller to integrate the data captured from the levelling sensor. Also this microprocessor is responsible to store gathered data to its memory. Stored data is evaluated from samples which are taken every 2 seconds in an hour.

SLMS-1

Entropy Microsystems. 1997.

The stored hourly tidal data are processed by computer program using recently developed signal processing technics. These processing technics will be mentioned later on section 5.

4.2. Port Agrilia Tidal Data

The data from Port Agrilia have been collected through the device SLMS-1 which is a measurement device from Entropy Microsystems. Internal data sample time is 2 seconds. These samples are evaluated hourly manner and stored. Data resolution is 5.7mm.� It starts from 30.03.1997 16:00 and contains 13685 records sampled every hour (60 minutes).�

4.3. Yumrukoy Wind Data

Entropy Microsystems currently measuring wind power related data in the region of interest. These measurements have begun at the end of 1994. That is Entropy microsystems has wide range of wind related data such as wind speed, wind direction and barometric pressure, etc...

Yumrukoy is about 3-4 km far from Port Agrilia. However, wind speed and wind direction is approximately same for both site.

The data from Yumrukoy is sampled every hour and sample times are synchronized. Thus, we are able to monitor both wind speed and sea height at the region.

4.4. Sadl�ktepe Wind Data

Sadl�ktepe is another site that Entropy Microsystems gathering wind data of. However this data is sampled every 10 minutes. In order to get hourly figures, the data is resampled with period of� an hour.

5. Analyses of Data

5.1. �About Analysis Methods.

Some widely used random data analysis technics was chosen to analyse the data. The data was analysed in frequency and time domain. Also some statistical processes were done in order to characterise the data.

As mentioned before tide is a periodical phenomena. Thus it must have some spectral characteristics. These characteristics were studied by using Fourier Analysis. Fourier analysis results showed that some periods have much more energy than the others.

Then tidal harmonic analysis technic was chosen to get approximate formula to the tide. Similar tidal harmonic analysis technics are also used in the USA. Some extra information about these analysis technics are included in the following sections.

The tide prediction method used today in the United States is called ``harmonic analysis.'' The ideal tide curve for any given port is represented as an average height Z0 plus a sum of terms (``constituents'') each of which in the form of f(t) = H cos(at + f). The time t is measured in hours, and f in centimeters. The numbers H, a, f are the amplitude, the speed (frequency) and the phase of the constituent. The speed is given in degrees/hour, and the phase in degrees (the cosine function that accepts input in degrees!). Different constituents have different speeds, which are the sums and differences of small integral multiples of 5 basic astronomical speeds, which are:

T: �the rotation of the Earth on its axis, with respect to the Sun, 15 degrees/hour
h:�� the rotation of the Earth about the sun, .04106864 degrees/hour
s :�� the rotation of the Moon about the Earth, .54901653 degrees/hour
p:�� the precession of the Moon's perigee, .00464183 degrees/hour
N:� the precession of the plane of the Moon's orbit, -.00220641 degrees/hour.

Thus the rotation of the Earth with respect to the fixed stars is T+h = 15.04106864 degrees/hour, and the change in the moon's longitude per hour is T+h-s= 14.49205211 degrees/hour.

The amplitude H and phase \phi for each constituent are determined from the tidal record by Fourier Analysis and Multiple Regression Method. The number of constituents needed for accurate prediction varies from port to port.

Symbol		Frequency (Degrees/Hour)	Period (Hours)
M2	2T-2s+2h	28.984	12.421
N2	2T-3s+2h+p	28.439	12.658
S2	2T	30.000	12
K1	T+h	15.041	23.934
L2	2T-s+2h-p	29.528	12.191
O1	T-2s+h	13.943	25.819
Sa	H	0.041	8780.478
\nu2	2T-3s+4h-p	28.512	12.626
K2	2T+2h	30.082	11.967
Mm	s-p	0.544	661.764
P1	T-h	14.958	24.067

�

5.2. �Frequency Domain Analysis

In frequency domain analysis, Fourier Analysis technics have been used. Spectral densities of both wind data and tide data have been analysed. Some of the results are below. In order to apply tidal harmonic analysis technics we should be sure that the tidal data is periodic with some distinct frequencies. As shown in frequency domain figures the tidal data is periodic and repeats itself with that periods. However all components in the frequeny spectrum don�t belong to tide itself. Some of the components are from wind, barometric pressure and temperature.

Mathematical Formulas

T[n] is tidal data, in centimeters.

n: is time index in hours.

�is fourier series coefficient. is complex number. is simply mean value of tidal data.

�is number of samples.

�is same as .

�is frequency index.

Then frequency is� .

��

Fourier transform transforms the data T[n] into its sinusoidal constituents as a summation of sine terms. By knowing this fact, we can see that which constituents most effects the tidal data in Port Agrilia. Following figures show that which constituents mostly effect the data.

Figure 5.2.1: Tidal Data, in time domain.

Figure 5.2.2: Spectral Power Density of Tidal Data.

Figure 4.2. shows us there are some periodicity on the tidal data at periods 12 h, 24 h, and some higher periods. In order to get accurate result we must do time domain analysis.

5.3. Time Domain Analysis

In time domain analysis multiple regression method was chosen. The analysis details are shown below.

Let �is approximate formula to T[n].

�is periodic function of n and P_ks are periods.

�is deviation from original data.

Which of the deviations are positive, negative or exactly zero depends on the choice of the parameters a_k and b_k.As a condition of optimality we minimize the sum of squares of deviations (�least squares�), that is, we determine a_k and b_kin such a way that

takes on the smallest possible value.

In order to minimize sum of squares of deviations we must take derivative of S with respect to coeficients and equalize to zero.

This systems can be solved by the following method using the computer.

A computer program can solve the linear system above� in least squared sense. Computer programs solve this kind of linear systems in least squared sense if there are many possible answers. Least squared algorithm gives most appropriate results for this kind of systems.

The results obtained here gives us the values of coefficients. Then these coefficients are rearranged with their sine and cosine components to combine together in terms of cosines. The resultant equation contains number of periods times cosine term and an average value, that is mean of T[n].

�is Tidal Constituent.

�is Period of the k_th constituent.

�is Phase of the k_th constituent.

Therefore we formed an equation which is shown below to calculate the sea level in centimeters at Port Agrilia. Note that this is astronomical tidal formula from harmonic analysis. However, other effects such as wind speed and direction (especially in stormy days), temperature and barometric pressure may change the sea height at particular time.

�(cm)

J is julian day may easily be calculated from the following equation.

The error in our approximate formula can be found in the following way.

Following constituents are found from harmonic analysis.

Symbol

Frequency

Cycle/Hour

Speed

Degrees/hour

Period

(Hours)

Amplitude

(Cm)

Phase

(Degree)

85.2

SA��

0.0001141

0.041076

8764.24

3.35

-165.9

SSA��

0.0002282

0.082152

4382.12

7.45

-0.8

MSM��

0.0013098

0.471528

763.48

0.26

56.9

MM��

0.0015122

0.544392

661.29

1.29

-104.4

MSF��

0.0028219

1.015884

354.37

1.16

-9.5

MF��

0.0030501

1.098036

327.86

0.32

-150.5

ALP1��

0.0343966

12.382776

29.07

0.21

-139.8

2Q1��

0.0357064

12.854304

28.01

0.11

34.6

SIG1��

0.0359087

12.927132

27.85

0.19

31.7

Q1��

0.0372185

13.39866

26.87

0.12

-127.5

RHO1��

0.0374209

13.471524

26.72

0.04

-130.7

O1��

0.0387307

13.943052

25.82

0.95

-148.3

TAU1��

0.0389588

14.025168

25.67

0.10

63.8

BET1��

0.0400404

14.414544

24.97

0.07

-117.7

NO1��

0.0402686

14.496696

24.83

0.05

63.8

CHI1��

0.040471

14.56956

24.71

0.23

-52.8

PI1��

0.0414385

14.91786

24.13

0.23

155.2

P1��

0.0415526

14.958936

24.07

0.56

-47.5

S1��

0.0416667

15.000012

24.00

0.52

130.4

K1��

0.0417807

15.041052

23.93

1.81

40.2

PSI1��

0.0418948

15.082128

23.87

0.35

-27.8

PHI1��

0.0420089

15.123204

23.80

0.16

-127.2

THE1��

0.0430905

15.51258

23.21

0.03

146.3

J1��

0.0432929

15.585444

23.10

0.18

53.0

Continued...

25

SO1��

0.0446027

16.056972

22.42

0.26

-143.1

Symbol

Frequency

Cycle/Hour

Speed

Degrees/hour

Period

(Hours)

Amplitude

(Cm)

Phase

(Degree)

OO1��

0.0448308

16.139088

22.31

0.15

90.9

UPS1��

0.046343

16.68348

21.58

0.10

-2.7

OQ2��

0.0759749

27.350964

13.16

0.12

-84.2

EPS2��

0.0761773

27.423828

13.13

0.08

8.8

2N2��

0.0774871

27.895356

12.91

0.14

53.3

MU2��

0.0776895

27.96822

12.87

0.17

157.0

N2��

0.0789993

28.439748

12.66

0.50

34.4

NU2��

0.0792016

28.512576

12.63

0.13

125.4

H1��

0.0803973

28.943028

12.44

0.17

135.6

M2��

0.0805114

28.984104

12.42

2.33

-56.9

H2��

0.0806255

29.02518

12.40

0.73

-22.2

MKS2��

0.0807396

29.066256

12.39

0.45

179.7

LDA2��

0.0818212

29.455632

12.22

0.03

-59.9

L2��

0.0820236

29.528496

12.19

0.06

-124.0

T2��

0.0832193

29.958948

12.02

0.28

-74.5

S2��

0.0833333

29.999988

12.00

1.60

10.7

R2��

0.0834474

30.041064

11.98

0.33

81.2

K2��

0.0835615

30.08214

11.97

0.53

-46.6

MSN2��

0.0848455

30.54438

11.79

0.10

-50.8

ETA2��

0.0850736

30.626496

11.75

0.26

-114.5

MO3��

0.1192421

42.927156

8.39

0.09

8.7

M3��

0.1207671

43.476156

8.28

0.11

6.5

SO3��

0.122064

43.94304

8.19

0.17

-68.9

MK3��

0.1222921

44.025156

8.18

0.20

-91.5

SK3��

0.1251141

45.041076

7.99

0.10

-65.2

MN4��

0.1595106

57.423816

6.27

0.05

-23.7

M4��

0.1610228

57.968208

6.21

0.11

-154.9

SN4��

0.1623326

58.439736

6.16

0.03

64.4

MS4��

0.1638447

58.984092

6.10

0.21

-3.3

MK4��

0.1640729

59.066244

6.09

0.08

-84.2

S4��

0.1666667

60.000012

6.00

0.10

87.3

SK4� ��

0.1668948

60.082128

5.99

0.17

11.2

2MK5��

0.2028036

73.009296

4.93

0.18

-123.9

2SK5��

0.2084474

75.041064

4.80

0.05

173.0

2MN6��

0.240022

86.40792

4.17

0.07

-94.8

M6��

0.2415342

86.952312

4.14

0.07

53.0

2MS6��

0.2443561

87.968196

4.09

0.07

28.7

2MK6��

0.2445843

88.050348

4.09

0.05

-36.3

2SM6��

0.2471781

88.984116

4.05

0.05

-98.1

MSK6��

0.2474062

89.066232

4.04

0.12

178.2

3MK7��

0.2833149

101.993364

3.53

0.08

-92.0

M8��

0.3220456

115.936416

3.11

0.12

2.6

M10��

0.402557

144.92052

2.48

0.05

-21.0

�� Table 5.3.1� - Calculated tidal constituents.

The error from our approximate formula using 68 constituents listed above is 0.096, that is nearly 1%.

7 constituents effects the tidal equation more than 1 centimeters. These are Ssa, Sa, M2, K1, S2, Mm, Msf� respectively.

Hence, for any given date, the sea height is approximately given by the subsequent equation.

Figure 5.3.1: Measured and Predicted Tide (Astronomical Tide).

The sources of deviation from astronomical tide are wind, barometric pressure and other effects. Deviation from wind is called as wind setup. Next, we are going to study wind setup.

Figure 5.3.2: Wind Speed vs. Deviation from Astronomical Tide.

One of the important issue in evaluating or predicting the sea height is the wind. Wind speed and wind direction are both important concepts. As shown above, we formulated astronomical tide. However, astronomical tide is a harmonic concept. That is, it is repeating itself in certain periods.

While we can predict annual wind power in any given region, of course there is always an uncertainty about the wind strength at any given time.

As shown in Figure 5.3.2. wind speed and deviation from the astronomical tide has a relationship. Deviation does not only depend on wind speed. Also wind direction affects the deviation. Our analyses show us that wind blowing from South directions causes an increase in sea-level. In contrast to the wind from the North which causes decreasing effect. �

5.4. �Statistical Analysis

Commonly used statistical analysis technics have been applied to the tidal data. Results from the statistical analysis listed at the table below.

	Mean	Standard Error	Median	Mode	Standard Deviation	Sample Variance	Kurtosis	Skewness	Range	Minimum	Maximum	Sum	Count	Largest(1)		Smallest(1)	Confidence Level(95.0%)
1997-1998	84.81	0.09	84.36	77.94	10.05	100.93	3.07	-0.51	111.72	3.27	114.99	1160559.75	13685.00	114.99	3.27			0.17
Apr, 1997	89.73	0.46	89.91	89.91	12.24	149.84	6.12	-1.39	105.45	8.40	113.85	64515.27	719.00	113.85	8.40			0.90
May, 1997	79.65	0.18	79.08	77.94	4.99	24.86	17.27	-1.49	64.41	31.20	95.61	59181.09	743.00	95.61	31.20			0.36
Jun, 1997	78.83	0.23	78.51	77.94	6.15	37.77	9.11	-0.84	70.11	27.21	97.32	56677.83	719.00	97.32	27.21			0.45
Jul, 1997	81.77	0.22	81.36	84.21	6.01	36.14	-0.10	-0.08	32.49	64.26	96.75	60756.57	743.00	96.75	64.26			0.43
Aug, 1997	83.76	0.20	84.21	84.21	5.42	29.42	-0.22	0.10	31.35	68.82	100.17	62230.02	743.00	100.17	68.82			0.39
Sep, 1997	89.61	0.22	90.48	91.62	5.95	35.38	42.74	-3.66	90.06	10.68	100.74	64428.69	719.00	100.74	10.68			0.44

Table of Contents

Barometric setup

Oceanographic Effects

Climate Change

Mathematical Formulas