Geopotential (symbol W) is the potential of the Earth's gravity field. It has SI units of square metre per square seconds (m2/s2). For convenience it is often defined as the negative of the potential energy per unit mass, so that the gravity vector is obtained as the gradient of the geopotential, without the negation. In addition to the actual potential (the geopotential), a theoretical normal potential (symbol U) and their difference, the disturbing potential (T = W − U), can also be defined.
Concepts
For geophysical applications, gravity is distinguished from gravitation. Gravity is defined as the resultant force of gravitation and the centrifugal force caused by the Earth's rotation. Likewise, the respective scalar potentials, gravitational potential and centrifugal potential, can be added to form an effective potential called the geopotential, W {\displaystyle W} . The surfaces of constant geopotential or isosurfaces of the geopotential are called equigeopotential surfaces (sometimes abbreviated as geop), also known as geopotential level surfaces, equipotential surfaces, or simply level surfaces. Global mean sea surface is close to one equigeopotential called the geoid. How the gravitational force and the centrifugal force add up to a force orthogonal to the geoid is illustrated in the figure (not to scale). At latitude 50 deg the off-set between the gravitational force (red line in the figure) and the local vertical (green line in the figure) is in fact 0.098 deg. For a mass point (atmosphere) in motion the centrifugal force no more matches the gravitational and the vector sum is not exactly orthogonal to the Earth surface. This is the cause of the coriolis effect for atmospheric motion.
The geoid is a gently undulating surface due to the irregular mass distribution inside the Earth; it may be approximated however by an ellipsoid of revolution called the reference ellipsoid. The currently most widely used reference ellipsoid, that of the Geodetic Reference System 1980 (GRS80), approximates the geoid to within a little over ±100 m. One can construct a simple model geopotential U {\displaystyle U} that has as one of its equipotential surfaces this reference ellipsoid, with the same model potential U 0 {\displaystyle U_{0}} as the true potential W 0 {\displaystyle W_{0}} of the geoid; this model is called a normal potential. The difference T = W − U {\displaystyle T=W-U} is called the disturbing potential. Many observable quantities of the gravity field, such as gravity anomalies and deflections of the vertical (plumb-line), can be expressed in this disturbing potential.
Background
Newton's law of universal gravitation states that the gravitational force F acting between two point masses m1 and m2 with centre of mass separation r is given by
F = − G m 1 m 2 r 2 r ^ , {\displaystyle \mathbf {F} =-G{\frac {m_{1}m_{2}}{r^{2}}}\mathbf {\hat {r}} ,}
where G is the gravitational constant, and r̂ is the radial unit vector. For a non-pointlike object of continuous mass distribution, each mass element dm can be treated as mass distributed over a small volume, so the volume integral over the extent of object 2 gives
with corresponding gravitational potential
where ρ2 = ρ(x, y, z) is the mass density at the volume element and of the direction from the volume element to point mass 1. u {\displaystyle u} is the gravitational potential energy per unit mass. Earth's gravity field can be derived from a gravity potential (geopotential) field as follows:
g = ∇ W = grad W = ∂ W ∂ X i + ∂ W ∂ Y j + ∂ W ∂ Z k , {\displaystyle \mathbf {g} =\nabla W=\operatorname {grad} W={\frac {\partial W}{\partial X}}\mathbf {i} +{\frac {\partial W}{\partial Y}}\mathbf {j} +{\frac {\partial W}{\partial Z}}\mathbf {k} ,}
which expresses the gravity acceleration vector as the gradient of W {\displaystyle W} , the potential of gravity. The vector triad { i , j , k } {\displaystyle \{\mathbf {i} ,\mathbf {j} ,\mathbf {k} \}} is the orthonormal set of base vectors in space, pointing along the X , Y , Z {\displaystyle X,Y,Z} coordinate axes. Here, X {\displaystyle X} , Y {\displaystyle Y} and Z {\displaystyle Z} are geocentric coordinates.
Formulation Both gravity and its potential contain a contribution from the centrifugal pseudo-force due to the Earth's rotation. We can write
W = V + Φ , {\displaystyle W=V+\Phi ,}
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