Disturbance potential

The function T p) is called the disturbing potential and it is very small, T U). Now we focus our attention on the space between the geoid and the ellipsoid of rotation and assume the potential of the gravitational field of the geoid, W p), and the normal potential of the spheroid, U q), are equal ... 

At an infinite distance the disturbing potential tends to zero... 

As was shown in Chapter 1, these conditions uniquely define the function T. For determination of the disturbing potential we will make use of Poisson s integral, described in the Chapter 1, which allows one to find the harmonic function E outside the spherical surface of the radius R, Fig. 2.9b, if this function, E p), is known at points of this surface ... 

Here the point p belongs to the spherical surface A of radius R. In order to find the upper limit on the left hand side of this equality, let us recall that T is the disturbing potential. In other words, it is caused by the irregular distribution of masses whose sum is equal to zero. This means that its expansion in power series with Legendre s functions does not contain a zero term. The next term is also equal to zero, because the origin coincides with the center of mass. Therefore, the series describing the function T starts from the term, which decreases as r. This means that the product r T O if oo and... 

Making use of Equations (2.278 and 2.280) we can calculate the disturbing potential on the spherical surface and outside. In particular, at points of this surface we have... 

As in the case of Stokes problem we represent the total potential W at each point of the earth s surface as a sum of the normal and disturbing potentials ... 

The importance of Equation (2.294) is that it establishes a relationship between the height anomaly, N, and the disturbing potential T at the same point of the earth. By definition, we have for the normal height h of the point A over a quasi-geoid... 

Thus, the determination of heights of the quasi-geoid N requires knowledge of the disturbing potential T on the physical surface of the earth. As in the case of the Stokes problem, in order to calculate N we have to determine the disturbing potential, which obeys some boundary condition on the physical surface of the earth instead of the surface of a geoid. This is the main advantage of a new approach. 

Taking into account the fact that T — W—U, the disturbing potential obeys Laplace s equation outside the earth surface AT = 0 and it is a regular function at... 

Here g is the gravitational field on the physical surface of the earth, y the normal field on the surface S. At the same time, dT/dv and dy/dv have the same values along line V at both surfaces. This is the boundary condition for the disturbing potential and therefore we have to find the harmonic function regular at infinity and satisfying Equation (2.301) on the surface S. In this case, the physical surface of the earth is represented by S formed by normal heights, plotted from the reference ellipsoid. In other words, by leveling the position of the surface S becomes known. 

One can say that we have expressed the disturbing potential in terms of an unknown density. Now we demonstrate that this transition is justified because it is possible to obtain the integral equation with respect to a. In Chapter 1, it was shown that the discontinuity of the normal components of the field at both sides of the surface masses is equal to —2nka. Correspondingly, we have... 

However, in the moving frame, in which the undisturbed interface corresponds to z = 0, we see only the perturbation flow, eiij. We assume that the latter also satisfies Darcy s law and can thus be related to a disturbance potential function... 

Now, given the expression (12-100) for the disturbance shape function at the interface, an appropriate form for the solutions of (12 103) for the disturbance potential functions... 

The linearized partial differential equations for the flow and the boundary conditions having been defined, we next specify the form of the disturbance. We avoid generality and take the disturbance potential to be represented by the typical wave component... 

Another very useful relation (transformation) relates the geoid height A to the disturbing potential T T N, A T). It was first formulated by a German physicist H. Bmns (1878), and it reads... 

This is a nonhomogeneous elliptical equation of second order, known under the name of Poisson equation, that embodies all the field equations (see Section 111. A) of the earth gravity held. Stokes applied this to the disturbing potential T (see Section 111.B) outside the earth to get... 

In the mid-20th century, Russian physicist M. S. Molodenskij formulated a different scalar boundary value problem to solve for the disturbing potential outside the earth (Molodenskij, Eremeev, andYurkine 1960). His criticism of Stokes approach was that the geoid is an equipo-tential surface internal to the earth and as such requires detailed knowledge of internal (topographical) earth mass density, which we will never have. He then proceeded to replace Stokes s choice of the boundary (geoid) by the earth s surface and to solve for T(r) outside the earth. 

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