UDC 550.311

GEOPHYSICS

V. N. ZHARKOV, V. M. LYUBIMOV

DAMPING OF THE EARTH’S SPHEROIDAL OSCILLATIONS AT SMALL $n$

(Presented by Academician M. A. Sadovskii on 28 VII 1969)

In previous papers ($^{1,2}$) we considered the damping of torsional and radial oscillations of the Earth. In the case of torsional oscillations, comparison of experimental data with theory made it possible, in broad outline, to determine the distribution of the dissipative function $Q$ in the Earth’s mantle. Using the distribution of $Q$ thus obtained and constructing a perturbation theory for the Earth’s radial oscillations, it proved possible to give an explanation of the anomalously large values of $Q$ observed after the Chilean earthquakes for radial oscillations ($^{2}$). This gives grounds for thinking that the distribution of $Q$ obtained in ($^{1}$) for the Earth’s mantle is close to the real sought distribution of the dissipative function. As all data indicate, $Q$ in the Earth’s liquid core is at least an order of magnitude greater than $Q$ for the mantle. Therefore at the present time, because of the insufficiency of experimental data, dissipation in the core has to be neglected.

In paper ($^{3}$) a perturbation theory was constructed for spheroidal oscillations of the Earth. This made it possible to continue the investigation begun in ($^{1,2}$) and to consider the damping of spheroidal oscillations of the Earth. Below we give the results of calculations for the fundamental tone up to $n = 27$ and the first 4 overtones for $n = 1$–7. These oscillations are of interest because they depend substantially on the properties of the Earth’s core, whereas oscillations with large $n$ are already noticeably displaced from the core into the Earth’s mantle.

As the computational model of the Earth, the Gutenberg–Bullen A (GBA) model was chosen. This model is described in detail in our preceding papers. It was also used in the works of Pekeris et al. (($^{4}$), pp. 230–232). In the model the crust and mantle are divided into 34 layers with piecewise-constant parameters. The Earth’s core is liquid and is divided into 7 layers: the inner core, a transitional layer, and the outer core, consisting of 5 layers. In principle the number of layers could be increased, but for our purposes this is unimportant. The integration of the differential equations was carried out from the center of the Earth. In the first region—the inner core—exact solutions are known, which are expressed in terms of Bessel functions ($^{5}$). Then the solution was continued by the Runge–Kutta method to the boundary of the core with the mantle and through the Earth’s mantle and crust to the surface, after which, by the formulas of the perturbation theory ($^{2}$), derivatives of the frequency with respect to the parameters of the problem—the density, the compressional modulus, and the shear modulus in each layer (respectively $\chi_{\rho i}$, $\chi_{k i}$, $\chi_{\mu i}$, $i$ being the layer number)—were calculated. As usual, it was assumed that dissipation occurs only in shear processes and thus is determined only by the derivative $\chi_{\mu}$.

In the concrete calculations the GBA model is “coarsened.” As stated above, it is assumed that dissipation in the core can be neglected, and the 34 layers of the crust and mantle are combined into four enlarged layers ($j = 1, 2, 3, 4$):

1. Zone $A$ — crust ($0 \leq l \leq 38$ km).
2. Zone $B$ — subcrustal zone ($38 \leq l \leq 300$ km).

Table 1

Table 2

1. Zone $C$—transition layer $(300 \leq l \leq 1000\ \mathrm{km})$.

2. Zone $D$—lower mantle $(1000 \leq l \leq 2900\ \mathrm{km})$.

Let $\mu_{0i}$ denote the dimensionless shear modulus in the $i$-th layer and let $R_i = \chi_{\mu i}\mu_{0i}$, where the layers are counted from the center of the Earth. Then the quantities sought, calculated by us, are equal to

\[ q^1 = \frac{2}{\chi_0}\sum_{i=40}^{41} R_i,\qquad q^2 = \frac{2}{\chi_0}\sum_{i=27}^{39} R_i,\qquad q^3 = \frac{2}{\chi_0}\sum_{i=18}^{26} R_i,\qquad q^4 = \frac{2}{\chi_0}\sum_{i=8}^{17} R_i, \tag{1} \]

where $\chi_0$ is the dimensionless natural frequency.

The formula by which the damping $Q$ was determined has the form

\[ Q^{-1} = \sum_{j=1}^{4} q^j Q_j^{-1}, \]

where \(Q_j\) \((j=1,2,3,4)\) is the distribution of the dissipative function in the Earth’s mantle \((^1)\). In \((^1)\) we considered the trial distributions \(Q_j\) given in Table 1.

On the basis of the experimental data presently available, the best distribution \(Q_j\) in the mantle is given by variant 6. The results of calculations for spheroidal oscillations \({}_l Q_{sn}\) \((n=1—7,\ l=0,1,2,3,4)\) are presented in Table 2.

The most substantial result of the present calculations should be considered the large values of \(Q\) obtained for spheroidal oscillations with small \(n\). The existing uncertainties are unlikely to change the quoted results for \(Q\) by more than \(100 \div 200\). Thus, we have
\({}_0 Q_{sn}\) \((n=1—7) \sim 1100—700\), variant 6.

If we turn to the experimental data \((^4,\ \text{pp. }33—59;\ 106—114)\), it turns out that \(Q_{sn}\) \((n=2—7) \sim 400 \div 300\). These experimental values of \({}_0 Q_{sn}\) are determined with insufficient confidence. It seems to us that they are simply erroneous. This is also indicated, in particular, by Smith \((^6)\) in his report prepared for the XIV General Assembly of the International Union of Geodesy and Geophysics. Smith points out that a careful review of the old data showed that nonstationary noises (background) could have substantially affected the measured values of \(Q\) for the lower tones of spheroidal and torsional oscillations, and that the previously published values of \(Q\) were substantially underestimated in comparison with the real quantities. The results given in Table 1 are precisely what orient experimenters toward the real values of \({}_l Q_{sn}\) that should be expected from observations.

REFERENCES

\(^1\) V. N. Zharkov, V. M. Lyubimov, A. A. Movchan, A. I. Movchan, Physics of the Earth, No. 2 (1967).
\(^2\) V. N. Zharkov, V. M. Lyubimov, DAN, 177, No. 2 (1967).
\(^3\) V. N. Zharkov, V. M. Lyubimov, DAN, 180, No. 2 (1968).
\(^4\) Free Oscillations of the Earth, Moscow, 1964.
\(^5\) V. N. Zharkov, Physics of the Earth, No. 8 (1967).
\(^6\) S. W. Smith, Trans. Am. Geophys. Un., 48, No. 2, 409 (1967).