Геопотенциал Земли: определение, свойства и применение.

Energy related to Earth's gravity

Geopotential is the potential of the Earth's gravity field. 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 hypothetical normal potential and their difference, the disturbing potential, can also be defined.

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 can be added to form an effective potential called the geopotential,
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 that has as one of its equipotential surfaces this reference ellipsoid, with the same model potential as the true potential of the geoid; this model is called a normal potential. The difference 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.

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 can be added to form an effective potential called the geopotential,
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 that has as one of its equipotential surfaces this reference ellipsoid, with the same model potential as the true potential of the geoid; this model is called a normal potential. The difference 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.

To a rough approximation, the Earth is a sphere, or to a much better approximation, an ellipsoid. We can similarly approximate the gravity field of the Earth by a spherically symmetric field:

of which the equipotential surfaces—the surfaces of constant potential value—are concentric spheres. It is more accurate to approximate the geopotential by a field that has the Earth reference ellipsoid as one of its equipotential surfaces, however. The most recent Earth reference ellipsoid is GRS80, or Geodetic Reference System 1980, which the Global Positioning system uses as its reference. Its geometric parameters are: semi major axis a = 6378137.0 m, and flattening f = 1/298.257222101. A geopotential field is constructed, being the sum of a gravitational potential and the known centrifugal potential , that has the GRS80 reference ellipsoid as one of its equipotential surfaces. If we also require that the enclosed mass is equal to the known mass of the Earth (including atmosphere) GM = 3986005 × 108 m3·s−2, we obtain for the potential at the reference ellipsoid:

Obviously, this value depends on the assumption that the potential goes asymptotically to zero at infinity , as is common in physics. For practical purposes it makes more sense to choose the zero point of normal gravity to be that of the reference ellipsoid, and refer the potentials of other points to this.

To a rough approximation, the Earth is a sphere, or to a much better approximation, an ellipsoid. We can similarly approximate the gravity field of the Earth by a spherically symmetric field:

of which the equipotential surfaces—the surfaces of constant potential value—are concentric spheres. It is more accurate to approximate the geopotential by a field that has the Earth reference ellipsoid as one of its equipotential surfaces, however. The most recent Earth reference ellipsoid is GRS80, or Geodetic Reference System 1980, which the Global Positioning system uses as its reference. Its geometric parameters are: semi major axis a = 6378137.0 m, and flattening f = 1/298.257222101. A geopotential field is constructed, being the sum of a gravitational potential and the known centrifugal potential , that has the GRS80 reference ellipsoid as one of its equipotential surfaces. If we also require that the enclosed mass is equal to the known mass of the Earth (including atmosphere) GM = 3986005 × 108 m3·s−2, we obtain for the potential at the reference ellipsoid:

Obviously, this value depends on the assumption that the potential goes asymptotically to zero at infinity , as is common in physics. For practical purposes it makes more sense to choose the zero point of normal gravity to be that of the reference ellipsoid, and refer the potentials of other points to this.

Once a clean, smooth geopotential field has been constructed matching the known GRS80 reference ellipsoid with an equipotential surface (we call such a field a normal potential) we can subtract it from the true (measured) potential of the real Earth. The result is defined as T, the disturbing potential:

The disturbing potential T is numerically a great deal smaller than U or W, and captures the detailed, complex variations of the true gravity field of the actually existing Earth from point to point, as distinguished from the overall global trend captured by the smooth mathematical ellipsoid of the normal potential.

In practical terrestrial work, e. g., levelling, an alternative version of the geopotential is used called geopotential number , which are reckoned from the geoid upward:

where is the geopotential of the geoid.