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Lattice structures of the earth magnetic field
 
Impressum Copyright Klaus Piontzik  
     
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17 - The figure of the earth

17.1 - Geodetic systems

Geodetic systems explain approximation models of the earth figure. Originate from a rotation ellipsoid such systems try to define the parametres which permit a more and more precise description of the earth in her 3-dimensional expansion.
 Origin of a rotation ellipsoid   Starting point is normally an ellipsis with the great half-axis a (major axis) and the small half-axis b (minor axis). If one lets now the ellipsis rotate around the smaller axis, there originates a so-called rotation ellipsoid which is also called spheroid.

However, with enough high exactness of the observation, turns out that an earth model based on an ellipsoid is not enough for an exact solution.

Illustration 17.1 - Origin of a rotation ellipsoid    
 
The figure of the earth is described by a physical and by a mathematical surface. One understands the physical earth surface as the limitation between hard and liquid earth masses against the atmosphere. Here a density jump takes place, so to speak, in the construction of the earth, namely from the middle density of the upper earth crust with r = 2,7 gcm-3 on the air density with r = 0,0013 gcm-3.

However, the irregularly formed surface of the hard earth masses, as for example the continents, cannot be shown by a mathematical relation.
Here helps only the point by point registration and mapping. Usually this happened in 5 x 5 or also in 1 x 1 parts.

Nevertheless, the surface of the oceans which amount at least about 70% of the earth surface shows a forming law. Under certain preconditions the seas establish a surface on which a constant gravity potential exists. Then it is a level surface of the earth gravity field.
If one imagines this surface under the continents continuously, one receives the mathematical earth figure. J.B.Listing gave the name
geoid to this level surface in 1872.

 

 Geoid

Illustration 17.2 - Geoid
 
The earth is constructed from masses of different density, however, which are not steadily called homogeneous distributed. Hence, it can seem locally that the physical (measured) plumb line direction does not agree with the ellipsis-normal. These plumb line divergences must be taken into the consideration.
These divergences are called geoid undulations. Hence, a reference ellipsoid is taken, normally, and the appearing heights above the ellipsoid surface are applied.
 
With the geoid determinations effected till this day has appeared that the divergences of the geoid from a middle rotation ellipsoid amount smaller than 100 metres by height.

 

 Three-axle ellipsoid   To receive one, still better adaptation to the geoid than a biaxial rotation ellipsoid, would be seen geometrical, the next step to try with a three-axle ellipsoid. In the illustration 17.3 such an ellipsoid is shown.

Referred to the earth then is
a1, besides, the great equator half-axis, a2 the small equator half-axis and b the pole axis.
Illustration 17.3 - Three-axle ellipsoid    
     
Attempts have been started several times to determin the parametres for a three-axle ellipsoid as an earth model.
Nevertheless, the results are of each other different, namely because of the different distribution of the observations on the earth surface and the different methods by the reduction to the ellipsoid.

 

With the help of the satellite geodesy, in 1966 by C.A.Lundquist and G. Veis, the following parametres have been determined:
 
a1-a2 = 69 Meter
λo = -14 45` (longitude West)
 
Besides, a1 is the great equator half-axis, a2 the small equator half-axis and λo the longitude of the great half-axis.
If one takes now the value of Lundquist and Veis, and compares this to the zero point (-13,5 degrees longitude position -
chapter 4 - angle approach) so can be ascertained, seen on a global scale, a good correspondence.

 

Because the divergences of the biaxial rotation ellipsoid to the geoid, lie normally, in the same scale like the axis difference of 69 metres, a substantially better adaptation to the geoid is not really reached by a three-axle ellipsoid.

On the other hand, however geodetic calculations are quite complicated by a more complex geometry of the ellipsoid. Also as a geophysical normal figure the three-axle ellipsoid is worse suitable, because with the values of the earth mass and the angular speed of the earth an unnatural form arises. Hence, the three-axle ellipsoid has not asserted as a relation body in the today's geodesy.

 
Strictly speaking rank the middle rotation ellipsoid or the geoid and the three-axleellipsoid, with regard to the divergences of the actual earth figure, in the same scale.
They are according to also equivalent models.

 

 

17.2 - The average points of the earth

If one evaluates all averages which are possible in an ellipsis, with regard to the axes, five cases arise.
These are listed in the following table. Besides, the values are rounded, because all averages deviate merely a few arc seconds or arc minutes from the given numerical values.
Besides, the kind of the average forming is arbitrary. Hence, it is not distinguished here any more between the single averages, but the averages are called in general with
M.
 
CASE Average condition Angle
I R=M 45
II y=M/2 30
III x=M/2 60
IV x=M 3
V y=M -
 
Remarkable is that for y=M no averages exist, because all averages are greater than the small half-axis and lie, hence, beyond the ellipsis. If one marks all existing averages in the ellipse, the following illustration 17.4 arises:
 
 Ellipsis and her averages
 
Illustration 17.4 - Ellipsis and her averages
 
If one takes as an earth model a rotation ellipsoid, the averages from the picture prove specific circles of latitude.

If one takes a three-axle ellipsoid as an earth model (values of Lundquist and Veist), single points arise. These points are called from now
average points.If one puts down all average points on a drawing of the earth, the next picture arises:
 
 Earth and average points
Illustration 17.5 - Earth and average points
 
Only in German language:
A geodetic consideration of the average points can be looked up under "
Die Gestalt der Erde-13"
A derivation of the average points can be checked under "
Die Gestalt der Erde-12"
 
A part of the average points of the earth dispose of an astonishing relation with the water course of the earth.

Water occurence can be found in the following average points:
South America - Amazon basin, North America - Ontario lake, Northern Europe - Ladoga lake,
Africa - Victoria lake, Asia - Chanka lake, Australia - two areas with several lakes.

The most important river deltas, which lie on average points are:
Po delta, Danube delta and Volga delta in Europe and directly under it the Nile delta and the Euphrates/Tigris delta. As well as in South America Rio de la Plata and Amazon delta, in North America the Hudson Bay and in Asia the Yangtse delta.

Furthermore are associated in Europe the Adriatic Sea, the Black Sea and the Caspian Sea with average points, as well as the Bottnisch bay. The position of the Mediterranean Sea between the average points is interesting also. Between two average points Baikal lake also lies in Asia.
 
The river deltas supply here virtually the answer. Because water always flows downwards, the river deltas must lie lower than the surrounding area. And this signifies:
 
The average points are, with regard to the land masses, the lowest points on the earth surface.

 

 

17.3 - Magnetic field and average points

Here the positions of the magnetic extreme values from chapter 2 still interests with regard to the average points. And this leads to the next drawing 17.6. The magnetic central areas of the extreme values for the complete field are yellow marked here and the real extreme values are bordered in red.
How to see the magnetic extreme values appear only in the edges of the average areas. And all extreme values lie on so-called specific circles of latitude (red lines) which arise from the osculating circle construction and the accompanying harmonious means of the ellipsoid axes.
The osculating circles play a special role with the ellipsis. On the one hand the ellipse can be constructed with them relatively fast. On the other hand, the radii of the osculating circles, especially the smaller radius, are used to describe the ellipsis in her properties, e.g., for the ellipse as a conic section.
Starting from a rotation model for the earth, is called from an arbitrary geodatic system, four specific circles of latitude originate from the osculating circle construction.
 
φ12 = 29 58' 19-20,5"
φ
34 = 59 58' 19-20,5"
 
Within the borders of 19-20,5 arc seconds there lie the values of all known geodetic systems.
 
 Earth, average points and magnetic extreme values
Illustration 17.6 - Earth, average points and magnetic extreme values
 
The magnetic extreme values appear only in the edges of the average areas. And all extreme values lie on the specific circles of latitude (red lines) which arise from the osculating circle construction and the harmonious means.
 
The harmonious means of the ellipsoid axes take a special position with regard to the ellipse. There exists a relation with the osculating circle construction which here, however, is not entered closer. Two other specific circles of latitude originate from the harmonious means of the ellipsoid axes:
 
φ56 = 30 08' 17,6-24,5"
 
φ56 and φ12 lie only 10 arc minutes apart and cannot be distinguished with a global representation (like in illustration 17.6) any more.
Within the borders of 17,6-24,5 arc seconds there lie the values of all known geodetic systems.
 
Only in german language:
see in addition: "
Die Gestalt der Erde-10" and "Die Gestalt der Erde-11"
 
From the drawings and the considerations the following conclusion can be drawn when one lays on a three-axle ellipsoid:
 
The positions of the magnetic extreme values of the complete field
stand in relation to the average points
and the specific circles of latitude

 

 

17.4 - Magnetic field and three-axle ellipsoid

If one puts down now all found magnetic extreme values and points, as well as the specific circles of latitude, in a whole map of the complete intensity, arises illustration 17.7 which virtually shows an extension of picture 11.6.
The blue verticals in picture 17.7 show the axes of a three-axle ellipsoid, with a 22.5 degrees partition.
The red verticals orientate themselves by the main meridian of the magnetic oscillation system, likewise with a 22.5 degrees partition.
Well to be seen is that ellipsoid system and magnetic system differ only slightly of each other, lie so, so to speak, in phase to each other. The main axis of the three-axle ellipsoid lies, with Lundquist and Veis, at -14,75 degrees of west. The main meridian of the magnetic system lies at -83,5 degrees of west.
A difference of 1.25 degrees arises from it between both systems. It is also the difference between the value of Lundquist and Veis, and the zero point from
chapter 4.
Seen on a global scale both systems agree therefore well.
 
 Magnetic field and three-axle ellipsoid
 
Illustration 17.7 - Magnetic field and three-axle ellipsoid
 
Then from the drawing 17.7 the following conclusion can be drawn:
 
The positions of all magnetic extreme values of the earth
stands in relation to a three-axle ellipsoid
 
From the drawings 17.5 to 17.7 from these chapter and the previous considerations, primarily from the chapter 4.4, chapter 9.5 and chapter 13, the following conclusion can be drawn when one lays on a three-axle ellipsoid:
 
The positions of all magnetic extreme values of the earth
stands in relation to the figure of the earth

 

 

17.5 - The C-grid

Based on the 4-structure of the sektoral part, a partition of 22.5 degrees (chapter 9) originates round the equator.
Caused by the specific circles of latitude and the 45 degrees of latitude from the tesseral system respectively the source points, a 15 degrees partition arises from north to south.
With it the existing B-grid can be refined. If one draw in all latitudes and longitudes of both partitions 15 and 22.5 degrees, the C-grid appears:
 
 The C-grid
 
Illustration 17.8 - The C-grid
 
 
The C grid contains therefore all information above the magnetic field and about the earth figure, taking into account a three-axle ellipsoid.

 

 

17.6 - The Bermuda triangle

Since the appearance of the books "the Bermuda triangle" in 1974/75 and "Without a trace" in 1977 from Charles Berlitz, numerous rumours have entwined themselves round the phenomena, which have achieved attention also in the wider public.
The Bermudas triangle lies in the Atlantic sea area to the east of the peninsula of Florida.
 
 The Bermuda triangle
Illustration 17.9 - The Bermuda triangle
 
 
The results of the magnetic field investigation of the author produced knowledge which allows a statement to one of the phenomena of the Bermuda triangle. Namely about the cases described up to now of mysterious compass behaviour like gyrating or swaying the compass needle or also static magnetic variation.
If one compares the position of the Bermuda triangle to the C-grid (illustration 17.8) so one recognises immediately that the triangle
lies directly at the main meridian (λ =-83,5 degrees of west) of the planetary magnetic oscillation system. Because round the main meridian the extreme zone expands up to 15 degrees on both sides, the Bermuda triangle lies in the magnetic maximum zone of this planet. At the same time the 30th degree of latitude also proceeds through the area, and how in chapter 17.3 was to be seen, two geometrically specific circles of latitude lie here. (see also illustration 9.2, 9.5, 9.8 in chapter 9)
If one pulls up now the map of the annual magnetic change (illustration 17.10) from the WMM 2005 ("World Magnetic Model" see also
chapter 2) and marks the main meridian (with his maximum zones) of the planetary magnetic oscillation system, an amazing correspondence arises:
 
 Map of the annual magnetic change with main meridian
Illustration 17.10 - Map of the annual magnetic change with main meridian - WMM 2005
 
 
Only two areas exist with a maximum in change of the field. An area lies to the east of the southern tip of Africa, the other area partially corresponds with the Bermuda triangle.
If one compares illustration 17.10 to illustration 17.7 or 17.8 so one recognises that the north maximum of the tesseral field and also a source point area are in immediate nearness (see also illustration 9.11in
chapter 9). Because also the main meridian of the planetary oscillation system runs here, the consequence is that the Bermuda triangle is to be looked as the only place on the earth at which the strongest fluctuations can appear in the magnetic field.
These fluctuations can appear temporarily as variations, as well as vary in the intensity and in the direction of field parts. Effects like gyrating or also swaying to a compass needle as well as static magnetic variation can be looked here as absolutely plausible. And this is exactly the kind of compass phenomena about which is reported that they appear in the Bermudas triangle.
(see double book „The Bermuda triangle“ from Berlitz and Valentine, page 19, 24, 35, 73, 76, 77, 80, 91, 92 respectively "Without a trace", page 16, 17, 24, 25, 30, 46)

 

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