210
S. S. Krylov et al.
distribution and the total coefficient k(θ ) in the sector after integration over longitude
is. Therefore, the choice of the sector’s angle (basically, the choice of the integration
region of the coefficient k(θ ) in longitude) is a compromise between the maximum
sensitivity of centrifugal moments of inertia to the particles motion along a surface
that is limited by the sector and the minimum area of this surface in order to reduce the
uncertainty error. Since the share of the surface area limited by a sector is 2θ 0 /π, θ 0
is determined from the condition that the function π sin θ 0 − 2 θ 0 is maximum on
the 0 < θ 0 ≤ π/2 interval. Under the condition θ 0 ≈ 0.9, the motion of the particles
in this sector determines approximately 80% variations of the tesseral harmonic of
the geopotential. However, let us choose a slightly larger value of the angle and, in
the following formulas, put for illustration purposes θ 0 = π/3, although this will not
fundamentally affect the estimates of average values.
Let us determine the share of the ocean surface area limited by one selected sector.
Since the Earth’s surface area limited by a sector is a constant and does not depend
on the Earth rotation, the share of the ocean’s surface area in the sector with an vertex
angle as (θ − π/3, θ + π/3) is proportional to the average coefficient k(θ ):
k(θ ) = =k(θ ) 2π/3 =
3
2π
θ+π/3
θ−π/3
k(θ )dθ.
(15.9)
The centrifugal moments of inertia J x z , J y z are most sensitive to the motion of
the moving medium if the ocean distributions in the selected sector and in the sector
symmetrical to it are significantly different. This condition can be replaced in a nonstrict sense by the integral condition k(θ ) − k(θ + π) = 0. In the strict sense, this
condition does not appear directly from Eq. 15.9 and thus is taken as an assumption.
If the location of the axes x
, y
meets this condition, then the assumption will be
valid.
In order to establish a correspondence, a function is defined
f (θ ) =
k(θ ) − k(θ + π)
2
(15.10)
that when k(θ ) = k(θ + π) takes the minimum value, i.e., with equal share of the
ocean surface in two opposite sectors, and the maxima corresponds to the extrema
of the function k(θ ) − k(θ + π), when the share of the ocean surface in two opposite
sectors are most different.
The correspondence between the location of the axes x
, y
and the distribution
of the ocean over the Earth’s surface is shown in Fig. 15.5. It can be seen that the
directions of the axes approximately correspond to the extrema of the function f (θ ).
That is, in the approximately orthogonal direction to the axis x
one can assume
a minimum of the amplitude of high-frequency perturbations due to less asymmetry
in the ocean distribution, which, according to the results of processing the pole
motion data, leads to high-frequency oscillations along the coordinate x
with lower
intensity. Similarly, with respect to the coordinate y
, the oscillations with a higher
S. S. Krylov et al.
distribution and the total coefficient k(θ ) in the sector after integration over longitude
is. Therefore, the choice of the sector’s angle (basically, the choice of the integration
region of the coefficient k(θ ) in longitude) is a compromise between the maximum
sensitivity of centrifugal moments of inertia to the particles motion along a surface
that is limited by the sector and the minimum area of this surface in order to reduce the
uncertainty error. Since the share of the surface area limited by a sector is 2θ 0 /π, θ 0
is determined from the condition that the function π sin θ 0 − 2 θ 0 is maximum on
the 0 < θ 0 ≤ π/2 interval. Under the condition θ 0 ≈ 0.9, the motion of the particles
in this sector determines approximately 80% variations of the tesseral harmonic of
the geopotential. However, let us choose a slightly larger value of the angle and, in
the following formulas, put for illustration purposes θ 0 = π/3, although this will not
fundamentally affect the estimates of average values.
Let us determine the share of the ocean surface area limited by one selected sector.
Since the Earth’s surface area limited by a sector is a constant and does not depend
on the Earth rotation, the share of the ocean’s surface area in the sector with an vertex
angle as (θ − π/3, θ + π/3) is proportional to the average coefficient k(θ ):
k(θ ) = =k(θ ) 2π/3 =
3
2π
θ+π/3
θ−π/3
k(θ )dθ.
(15.9)
The centrifugal moments of inertia J x z , J y z are most sensitive to the motion of
the moving medium if the ocean distributions in the selected sector and in the sector
symmetrical to it are significantly different. This condition can be replaced in a nonstrict sense by the integral condition k(θ ) − k(θ + π) = 0. In the strict sense, this
condition does not appear directly from Eq. 15.9 and thus is taken as an assumption.
If the location of the axes x
, y
meets this condition, then the assumption will be
valid.
In order to establish a correspondence, a function is defined
f (θ ) =
k(θ ) − k(θ + π)
2
(15.10)
that when k(θ ) = k(θ + π) takes the minimum value, i.e., with equal share of the
ocean surface in two opposite sectors, and the maxima corresponds to the extrema
of the function k(θ ) − k(θ + π), when the share of the ocean surface in two opposite
sectors are most different.
The correspondence between the location of the axes x
, y
and the distribution
of the ocean over the Earth’s surface is shown in Fig. 15.5. It can be seen that the
directions of the axes approximately correspond to the extrema of the function f (θ ).
That is, in the approximately orthogonal direction to the axis x
one can assume
a minimum of the amplitude of high-frequency perturbations due to less asymmetry
in the ocean distribution, which, according to the results of processing the pole
motion data, leads to high-frequency oscillations along the coordinate x
with lower
intensity. Similarly, with respect to the coordinate y
, the oscillations with a higher