I've been thinking about the issue of True, Magnetic and Grid Norths, and the impending problem of the change of Grid Magnetic Angle from West to East, and the effect that will have on the use of the mnemonic 'Grid to Mag: Add; Mag to Grid: Get Rid'. Here's my thinking...
This is a physical property of the rotation of the Earth; it's point at which the the axis of rotation of the Earth meets the surface. This changes a little, due to the changing shape of the Earth (core changes, earthquakes etc.) but not by much, and this variation can be ignored.
This is the imaginary point in the night sky that True North points to. It changes with time (with a period of about 26000 years, or 1 degree in 72 years), as the Earth's rotation precesses
. Think of a spinning top as it slows down; the top still spins on its axis quite quickly, but the axis about which it rotates also rotates slowly; this is precession. It has no impact on map and compass navigation, but does effect celestial navigation. Slowly...
The Earth's magnetic field is caused by the moving, molten, nickel-iron core of the Earth. Lines of magnetic flux come out of the surface of the Earth at different points, at different strengths, and at different angles at different positions on the Earth's surface. A compass aligns itself with the lines of magnetic flux as they occur at that point on the Earth's surface. This gives the concept of a Magnetic North, as measured with a compass. And, for a specific point on the Earth's surface, there is a Local Magnetic North, LMN.
The concept of the difference between True North, TN, and Magnetic North due to the lines of flux is called Magnetic Declination (or Variation), and, at any point on the Earth, there is a specific Local Magnetic Declination, LMD:
LMD = TN - LMN
The Earth is an oblate spheroid. A map is a flat piece of paper, which attempts to represent the curved surface of the Earth; this it cannot do; try wrapping a ball with a piece of paper without leaving any creases... Clever Mr Mercator came up with a mathematical way of drawing a map of the Earth's surface so that bearings and distances measured between two points are correct (a projection
). OSGB and UTM maps use this (Transverse) Mercator projection. The downside of this projection is that the Grid North lines on the map, whilst being a fixed distance apart, don't point to True North. Consider that, as we go North, the distance between lines of longitude gets smaller and smaller (until they meet at the pole). Thus, in a map grid that maintains a fixed spacing between the Grid North lines, as we go North, the Grid North lines must diverge out from True North lines. The only points at which Grid North and True North are aligned is on the line forming the centre of the projection space; in the OSGB projection, this occurs at 2 degrees W. The concept of this difference between True North and Grid North is called Grid Convergence, and, at any point on the Grid, there is a Local Grid Convergence, LGC:
LGC = TN - LGN
Grid Magnetic Angle
Now, if we look at a particular area of a Mercator Projection, i.e. we look at a map, there will be a difference between the Local Magnetic North and Local Grid North. This difference is due to the Local Magnetic Declination (the relative position of the local magnetic field wrt True North), and the Local Grid Convergence due to the Mercator Projection of the map at that point (the relative position of Local Grid North wrt True North). Note that Local Grid Convergence actually varies across the entire map, but this variation is relatively small, and can be ignored on large-scale (1:50k, 1:25k) printed maps*. Let's define the Local Grid Magnetic Angle, GMA, as the difference between Local Magnetic North and Local Grid North:
GMA = LMN - LGN
GMA = (TN - LMD) - (TN - LGC)
GMA = LGC - LMD
Since, by cartographic convention, positive angles (0 <= angle <180) are E, and negative angles (-180 < angle < 0) are W, GMA measures how far Local Magnetic North is East
of Local Grid North. So, a positive GMA means LMN is E of LGN, and a negative GMA means LMN is W of LGN (as is currently the case in most of Britain).
So, we must be careful with our terminology, and must use only (Local) Grid Magnetic Angle when we wish to discuss converting between Local Grid North and Local Magnetic North; if we used the term Magnetic Declination, we would, strictly, be ignoring the Local Grid Convergence, which can be just as important a correction factor as the Local Magnetic Declination. Generally, when map-reading, we aren't concerned with Magnetic Declination or True North; this has already been dealt with by the cartographers, who have given us a handy diagrammatic representation of magnetic, true and grid Norths. The only way in which Magnetic Declination affects us is the change of Magnetic Declination with time, which causes the Local Grid Magnetic Angle to change with time.
Note that I have stressed the distinction between the general concepts, and the local
value of those concepts, since it is essential to bear in mind that all these factors vary with position on the Earth, and in the particular mapping projection employed, and with time; they are not constants either geographically, cartographically, or temporally.
Now let's look at applying this.
Consider a point on an OS map, where the Local Magnetic North is (for example) 3 degrees West (or minus 3 degrees E) of Local Grid North. By convention, the Local Grid North is 0 degrees.
GMA = LMN - LGN
GMA = -3 - 0 = -3
GMA = -3 degrees East, i.e. Local Magnetic North is 3 degrees West of Local Grid North (confirming what we stated above...)
Now, we can re-arrange
GMA = LMN - LGN
LMN = LGN + GMA
LGN = LMN - GMA
We can then replace these 'Norths' with actual measured bearings: Local Magnetic Bearing, LMB, and Local Grid Bearing, LGB:
LMB = LGB + GMA Grid to Mag: Add
LGB = LMB - GMA Mag to Grid: Get Rid
Note that, if we use the definition of GMA as above, and the E positive, W negative convention, and use signed arithmetic for the above operations, the 'Grid to Mag: Add, Mag to Grid: Get Rid' mnemonic works no matter what the values are, or where we are in the world.
nb. Unfortunately, you will often find GMA being defined as follows: "The horizontal angular difference between Grid North and Magnetic North is called GRID MAGNETIC ANGLE" (taken from the OS website below). Note that the order in which the two terms are expressed in the 'difference' is the reverse of the above definition of GMA; when expressed arithmetically, this gives a change in sign, which makes the 'Grid to Mag: Add, Mag to Grid: Get Rid' mnemonic completely wrong. Sadly, they express this GMA in terms of degrees E or degrees W, rather than the more helpful, conventional +/-degrees E.
* With the rise of electronic mapping, where there is no useful North annotation in the corner of the map, some other means of finding the GMA will be required. Ideally, the electronic mapping tool will have some means of determining GMA, probably using a look-up table of baseline and time delta values (rather like the printed map annotation).
You can put in any name and email address in the Geomag site...
Of course, having said all this, as I've observed above, I don't generally like mnemonics, and deal with grid to magnetic conversions in a completely different way, thinking about the relative position of local magnetic north wrt grid north...