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Pure and Applied Geophysics

, Volume 175, Issue 5, pp 1649–1657 | Cite as

Influence of Tidal Forces on the Triggering of Seismic Events

  • Péter Varga
  • Erik Grafarend
Article

Abstract

Tidal stresses are generated in any three-dimensional body influenced by an external inhomogeneous gravity field of rotating planets or moons. In this paper, as a special case, stresses caused within the solid Earth by the body tides are discussed from viewpoint of their influence on seismic activity. The earthquake triggering effects of the Moon and Sun are usually investigated by statistical comparison of tidal variations and temporal distribution of earthquake activity, or with the use of mathematical or experimental modelling of physical processes in earthquake prone structures. In this study, the magnitude of the lunisolar stress tensor in terms of its components along the latitude of the spherical surface of the Earth as well as inside the Earth (up to the core-mantle boundary) were calculated for the PREM (Dziewonski and Anderson in Phys Earth Planet Inter 25(4):297–356, 1981). Results of calculations prove that stress increases as a function of depth reaching a value around some kPa at the depth of 900–1500 km, well below the zone of deep earthquakes. At the depth of the overwhelming part of seismic energy accumulation (around 50 km) the stresses of lunisolar origin are only (0.0–1.0)·103 Pa. Despite the fact that these values are much smaller than the earthquake stress drops (1–30 MPa) (Kanamori in Annu Rev Earth Planet Sci 22:207–237, 1994) this does not exclude the possibility of an impact of tidal forces on outbreak of seismic events. Since the tidal potential and its derivatives are coordinate dependent and the zonal, tesseral and sectorial tides have different distributions from the surface down to the CMB, the lunisolar stress cannot influence the break-out of every seismological event in the same degree. The influencing lunisolar effect of the solid earth tides on earthquake occurrences is connected first of all with stress components acting parallel to the surface of the Earth. The influence of load tides is limited to the loaded area and its immediate vicinity.

Keywords

Spherical tidal stress tensor zonal, tesseral and sectorial tides oceanic tidal load 

1 Historical Overview and Current Research Results

One of the founders of modern seismology, professor of mathematics at University of Dijon Perrey realized first that the study of seismology needs international cooperation. Thanks to his efforts between 1844 and 1871 a significant (for that time) earthquake data set was collected by him in form of “Annual lists of Earthquakes”. The total number of earthquakes described and catalogued by him is over 21,000. Perrey completed also regional catalogues. This way he initiated the “seismological geography”. Although examination of the temporal distribution of earthquakes has been started already in the eighteenth century (Baldivi in 1703 and Toaldo in 1797 see Davison 1927) Perrey was the first who recognised regularities in the temporal distribution of seismic events. Based on his far from complete catalogue Perrey (1875) concluded, that the series of earthquake occurrences contains periodicities. According to Davison (1927) he has identified three regularities: earthquakes are more frequent at the syzygies than at the quadratures, they are more frequent at perigee than at apogee and the seismic events are more frequent when the moon is near to the meridian.

Since Perrey’s time the problem of tidal triggering is one of the “evergreen” problems of earthquake research. The committee appointed by the French Academy of Sciences (Élie de Beaumont, Liouville and Lamé) reported favourably on Perrey’s conclusions, while the other great French seismologist F. de Montessus de Ballore (1911) strongly criticised them. Schuster (1897) was the first who applied to the study of temporal variation of seismic events the tools of statistics and on this basis he has arrived at an optimistic conclusion: “The reality of the period (i.e. tidal periodicity of earthquake occurrence) would be thereby established beyond reasonable doubt” (Schuster 1911). In his study Cotton (1922) suggested tidal stresses in the Earth as a secondary cause of earthquakes and according to him it might be possible to predict earthquake occurrence “with sufficient accuracy” by consideration of the position of the Sun and the Moon, what allows “to provide timely warnings of disastrous shocks”. From the early twentieth century, many researchers have dealt with the relationship between tides and earthquakes, in the hope that this way a tool can be found for a more accurate estimation of earthquake hazard. In addition, study of potential effects of tides may be important for seismology because it modifies the spatial and temporal distribution of earthquakes, distorts the image of the nature of seismo-tectonics, what makes more difficult understanding of the nature of earthquakes. A comprehensive summary of research dealing with tidal triggering of earthquakes and volcanic events during the first nine decades of the twentieth century was compiled by Emter (1997).

A significant number of recent studies found relationship between the lunisolar effect and temporal distribution of seismicity. Most of them show this relationship on a regional scale. A number of such previous studies were mentioned in Métivier et al. (2009). Nowadays many papers using significant database and advanced statistical methods discuss relationship between different seismological and tidal phenomena. Vergos et al. (2015) detected evidence for tidal triggering of earthquakes of the Hellenic arc, Arabelos et al. (2016) describe a lunisolar influence on seismic activity in central Greece. Chen et al. (2012a) suggest that in case of earthquakes which occurred in Taiwan between 1973 and 2008 “the lunar tidal force is likely a factor in the triggering of earthquakes”. According to Cochran et al. (2004) the earth tides can trigger shallow thrust fault seismic events. Wilcock (2009) found a weak correlation between tidal influence and earthquake occurrences in continental areas which can be increased in settings influenced by loading effect of oceanic tides. Stroup et al. (2007) conclude that the lunisolar periodic stress changes are dominated by oceanic or by solid earth tide in dependence of the position of seismic source zone relative to the distribution of area loaded by oceanic tidal load. At the same time Métivier et al. (2009) on basis of a study of global earthquake activity come to the conclusion that “it is highly probable that the observed triggering is caused by the solid Earth tide rather than by loading from the ocean or atmospheric tides”. Tanaka (2010, 2012) and Ide et al. (2016) found significant lunisolar triggering in case of foreshocks prior to the Sumatra megathrust earthquakes of 26 December 2004 (M w 9.0), 28 March 2005 (M w 8.6), 12 September 2007 (M w 8.5), and the Tohoku-Oki earthquake (2011, M w = 9.1). Li and Xu (2013) have found that seismicities of some provinces of China show higher correlation with the lunisolar effect during several years preceding a large (destructive) earthquake than during other times. Chen et al. (2012b) in the case of Christchurch earthquake (2011, M w = 6.3) found a correlation between the aftershock sequence and diurnal tide. At the same time these authors found also a global correlation in case of seismic events M S ≥ 7 since 1900 and the semidiurnal tidal wave M 2. Examinations carried out by different authors show slight dependence of tidal triggering effect and seismological activity on focal mechanisms. Earlier Heaton (1975) concluded that first of all shallow oblique-slip and dip-slip earthquakes M ≥ 5 correlate with tidally generated shear stresses (Heaton 1982 retracted this statement after using an extended earthquake catalogue). Tsuruoka et al. (1995) have found a slightly higher probability of earthquake occurrence for normal fault earthquakes. Métivier et al. (2009) found that more tidal triggering is exhibited in case of normal strike-slip faulting than in other cases. It was concluded in a recent paper by Ide et al. (2016), that probability of occurrence of great seismic events is higher “during periods of high tidal stresses” (i.e. during spring tides). Houston (2015) combined seismic data and calculated tidal stress and found rising sensitivity to lunisolar stresses during six large slow-slip events of tremors in Cascadia.

Summarising the results of the studies detecting significant tidal triggering it can be concluded.
  • the conclusions are almost exclusively based on statistical investigations, only sometimes supported by schematic seismic source models.

  • correlation was found in case of shallow-dipping thrust events in case of earthquakes shallower than 50 km depth.

  • the tides can act only in case of faults that are already close to rupture.

  • the normal and shear stress are essential factors of tidal triggering.

On the other hand, many researchers could not detect any significant interdependence between time distribution of tidal force and seismic activity. Young and Zürn (1979), Vidale et al. (1998), Stein (2004), Tanaka et al. (2006) could not find statistically significant relationship between tidal force and earthquake activity. Based on their investigations Tormann et al. (2015) concluded that large earthquakes “may not have a characteristic location, size or recurrence interval, and might therefore occur more randomly distributed in time”.

To contribute to the interpretation of the differences of research results the study of equations of lunisolar tidal elastic stress tensor components can be probably helpful because it could explain these differences between the research results. In the present investigation for calculations carried out the mathematical tools of author’s former study (Varga and Grafarend 1996) have been used. Our previously used mathematical procedure and—as will be seen in the following sections—the interpretation of the calculation results we have developed in the following ways:
  • magnitudes of stresses caused by the lunisolar effect and by surface load are expressed in SI units instead of the relative units used in our previous paper

  • a new interpretation was given for the new numerical results obtained both in case of tidal and load models.

2 The Spherical Tidal Stress Tensor

With the use of spherical harmonics \(Y_{2i} (\varPhi ,\lambda )\) (where Φ and λ denotes spherical latitude and longitude) in cases of zonal (i = 0), tesseral (i = 1) and sectorial (i = 2) tides the spherical tidal stress tensor components (Fig. 1) on the basis of Varga and Grafarend (1996) can be written in the following forms:
$$\sigma_{{rr_{i} }} (r,\lambda ,\varPhi ) = \frac{(\varLambda + 2\mu )\partial H}{\partial r} + \varLambda \left[ {\frac{2}{r}H - \frac{n(n + 1)}{{r^{2} }} T} \right] \cdot Y_{2i} (\lambda ,\varPhi ) = N(r) \cdot Y_{2i} (\lambda ,\varPhi )$$
(1)
$$\sigma_{{\varPhi \varPhi_{i} }} (r,\lambda ,\varPhi ) = \left\{ {\varLambda \left[ {\frac{\partial H}{\partial r} + \frac{2H}{r} - \frac{n(n + 1)}{{r^{2} }}T} \right] + 2\mu \frac{H}{r}} \right\} \cdot Y_{2i} (\lambda , \varPhi ) + 2\mu \frac{T}{{r^{2} }} \cdot \frac{{\partial^{2} Y_{2i} (\lambda , \varPhi )}}{{\partial \varPhi^{2} }}$$
(2)
$$\sigma_{{\lambda \lambda_{i} }} (r,\lambda ,\varPhi ) = \left\{ {\varLambda \left[ {\frac{\partial H}{\partial r} + \frac{2H}{r} - \frac{n(n + 1)}{{r^{2} }}T} \right] + 2\mu \frac{H}{r}} \right\} \cdot Y_{2i} (\lambda , \varPhi ) + 2\mu \frac{T}{{r^{2} }} \times \left[ {\frac{{\partial^{2} Y_{2i} (\lambda , \varPhi )}}{{\partial \lambda^{2} }}(\cos^{ - 2} \varPhi ) + \tan \varPhi \frac{{\partial Y_{2i} (\lambda , \varPhi )}}{\partial \varPhi }} \right]$$
(3)
$$\sigma_{{r\varPhi_{i} }} (r,\lambda ,\varPhi ) = \frac{\mu }{r}\left[ {\frac{\partial T}{\partial r} - \frac{2T}{r} + H} \right]\frac{{\partial Y_{2i} (\lambda , \varPhi )}}{\partial \varPhi }$$
(4)
$$\sigma_{{r\lambda_{i} }} (r,\lambda ,\varPhi ) = \frac{\mu }{r}\left[ {\frac{{\partial d_{ri} (r,\lambda ,\varPhi )}}{\partial \lambda }\frac{1}{\cos \varPhi } + r\frac{{\partial d_{\lambda i} (r,\lambda ,\varPhi )}}{\partial r} - d_{\lambda i} (r,\lambda ,\varPhi )} \right] = \frac{\mu }{r\cos \varPhi }\left[ {\frac{\partial T}{\partial r} - \frac{2T}{r} + H} \right]\frac{{\partial Y_{2i} (\lambda , \varPhi )}}{\partial \lambda }$$
(5)
$$\sigma_{{\varPhi \lambda_{i} }} (r,\lambda ,\varPhi ) = 2\frac{\mu T}{{r^{2} \cos \varPhi }}\left[ {\frac{{\partial^{2} (Y_{2i} (\lambda , \varPhi )}}{\partial \varPhi \partial \lambda } - \tan \varPhi \frac{{\partial Y_{2i} (\lambda , \varPhi )}}{\partial \lambda }} \right]$$
(6)
(of course \(\sigma_{{r\varPhi_{i} }} = \sigma_{{\varPhi r_{i} }} ; \sigma_{{r\lambda_{i} }} = \sigma_{{\lambda r_{i} }} ; \sigma_{{\varPhi \lambda_{i} }} = \sigma_{{\lambda \varPhi_{i} }}\)).
Fig. 1

Stress tensor components. Normal stresses: \(\sigma_{rr, } \sigma_{\varPhi \varPhi , } \sigma_{\lambda \lambda }\) and shear stresses: \(\sigma_{r\lambda } = \sigma_{\lambda r} , \;\sigma_{r\varPhi } = \sigma_{\varPhi r} ,\;\sigma_{\varPhi \lambda } = \sigma_{\lambda \varPhi }\)

In Eqs. (1)–(6) H(r) and T(r) are functions of the distance r from the centre of the Earth and describe radial and horizontal elastic deformations within the Earth and at the surface (r = a) they are the Love–Shida numbers H(a) = h and T(a) = l · μ(r), Λ(r) are the Lamé parameters and ρ(r) is density.

Since the right hand side of the Eq. (1) for the radial normal stress is proportional to the equation of boundary condition at r = a \(\frac{(\varLambda + 2\mu )\partial H}{\partial r} + \varLambda \left[ {\frac{2}{a}H - \frac{n(n + 1)}{{a^{2} }} T} \right] = 0\) implies, that \(\sigma_{{rr_{i} }} (a,\lambda ,\varPhi ) = 0\).

Similarly at the surface of the Earth \(\sigma_{{r\varPhi_{i} }} (a,\lambda ,\varPhi ) = \sigma_{{r\lambda_{i} }} (a,\lambda ,\varPhi ) = 0\) whereas the terms on the right hand side of the equation are proportional to the boundary condition \(\mu \left( {\frac{\partial T}{\partial r} - \frac{2}{a}T + H} \right) = 0\) at r = a.

Therefore, it can be concluded that three out of six independent components of the stress tensor are zero at r = a and they cannot be involved in tidal triggering. It can be seen from the results of calculations (see next section) that \(\sigma_{{\varPhi \lambda_{i} }} (a,\lambda ,\varPhi ) = 0\). That is why the trigger effects at the surface of the Earth (r = a) can be associated only with two normal tensor components \(\sigma_{{\varPhi \varPhi_{i} }} (a,\lambda ,\varPhi )\) and \(\sigma_{{\lambda \lambda_{i} }} (a,\lambda ,\varPhi )\) acting parallel to the surface of the Earth.

3 Calculation of Spherical Tidal Stress Tensor Components

As a first step similarly to the procedure described in Varga and Grafarend (1996), the numerical values of H(r) and T(r) functions were determined for the elastic mantle of the Earth from the surface (r = a) to the core-mantle boundary (r = b). For this purpose the equation of motion introduced by Takeuchi (1953), Molodensky (1953), Alterman et al. (1959) should be used. For the solution Molodensky used auxiliary functions for normal (N(r)), tangential (M(r)) stresses and gravity potential (Poisson equation) (L(r)):
$$N = (\varLambda + 2\mu )\partial H/\partial r + \varLambda [2/r H - n(n + 1)/r^{2} T]$$
(7)
$$M = \mu \left( {\frac{\partial T}{\partial r} - \frac{2}{r}T + H} \right)$$
(8)
$$L = r^{2} \left( {\frac{\partial R}{\partial r} - 4\pi G\rho H} \right)$$
(9)
In Eq. (9) R(a) − 1 = k is the second Love number and G the gravitational constant. In case of calculation of functions N(r), M(r) along the radius at the surface of the Earth (r = a) the boundary conditions are equal to zero: \(N(a) = M(a) = L_{n} (a) = 0.\) The boundary conditions at the core-mantle boundary (r = b) are N(b) = −P, M(b) = L(b) = 0, where P is the hydrostatic pressure.

With the use of numerical values of H(r), T(r), N(r), and M(r) in Eqs. (1)–(6) the stress tensor components were calculated in cases of zonal (i = 0), tesseral (i = 1) and sectorial (i = 2) tides, for the latitudes 0°, 20°, 40° and 60° from the surface of the Earth till the core-mantle boundary. The numerical values of Lamé parameters μ(r), Λ(r), ρ(r) for the elastic mantle (a ≥ r ≥ b) were taken from the PREM (Dziewonski and Anderson 1981).

The results of computations (Table 1) show that
  • \(\sigma_{{r\lambda _{i = 0} }} (r,\lambda ,\varPhi )\) and \(\sigma_{{\varPhi \lambda_{i = 0} }} (r,\lambda ,\varPhi )\) are equal to zero from the surface to the core-mantle boundary in case of zonal tides for all latitudes (therefore they are missing from the table)

  • at the equator (\(\varPhi = 0^\circ\)) the zonal \(\sigma_{{r\varPhi_{i = 0} }} (r,\lambda ,0^\circ ),\) the tesseral \(\sigma_{{rr_{i = 1} }} (r,\lambda ,0^\circ ),\; \sigma_{{\varPhi \varPhi_{i = 1} }} (r,\lambda ,0^\circ )\) and \(\sigma_{{\lambda \lambda_{i = 1} }} (r,\lambda ,0^\circ )\), and the sectorial \(\sigma_{{\varPhi \lambda_{i = 2} }} (r,\lambda ,0^\circ )\), \(\sigma_{{\lambda_{i = 2} }} (r,\lambda ,0^\circ )\) tensor components have zero value through the mantle

  • as it was already mentioned, at the surface of the Earth normal stresses \(\sigma_{{rr_{i} }} (a,\lambda ,\varPhi ) = \sigma_{{r\varPhi_{i} }} (a,\lambda ,\varPhi ) = \sigma_{{r\lambda_{i} }} (a,\lambda ,\varPhi ) = 0\) (i = 0, 1, 2). The same is valid for shear stress: \(\sigma_{{\varPhi \lambda_{i = 0} }} (a,\lambda ,\varPhi ) = 0\). These stress tensor components are negligibly small to depths of approximately 200–300 km

  • only the horizontal components of the normal stress, \(\sigma_{\varPhi \varPhi }\) and \(\sigma_{\lambda \lambda }\), have non zero values at r = a, which are ≤2 kPa.

Therefore it can be concluded that the influence of aforementioned conclusions that the influence of the earth tides on earthquake activity is a complex phenomenon. Their impact depends on the geographic location, on the depth and on the direction of stress tensor components. The strongest influence have the zonal (long-periodic) and sectorial (semi-diurnal) tides on and λλ (Fig. 2; Table 2). Their largest magnitude (~1 kPa) can be observed in the equatorial region and at high latitudes. In contrast the tesseral (diurnal) tides have probably less triggering effect as the two aforementioned ones.
Table 1

Lunisolar zonal, tesseral and sectorial stress tensor components (in kPa) within the mantle (a ≥ r ≥ 0.55—the approximate radius of the core) at latitudes 0°, 20°, 40° and 60°

 

20°

40°

60°

Zonal tides

σ rr r/a

  1.00

0

0

0

0

  0.90

−0.38

−0.24

0.03

0.35

  0.80

−0.52

−0.35

0.07

0.76

  0.70

−0.41

−0.28

0.03

0.59

  0.60

−0.28

−0.14

0.17

0.31

  0.55

−0.69

−0.35

0.41

0.93

σ r/a

  1.00

0.00

0.00

0.00

0.00

  0.90

0.00

0.38

0.55

0.76

  0.80

0.00

0.86

1.10

1.38

  0.70

0.00

1.21

1.66

2.00

  0.60

0.00

0.86

1.14

1.62

  0.55

0.00

0.00

0.00

0.00

σ φφ r/a

  1.00

−1.83

−0.41

0.07

0.69

  0.90

−0.52

0.10

0.69

1.97

  0.80

0.69

0.79

1.38

2.52

  0.70

2.83

2.76

2.66

2.35

  0.60

9.32

7.76

3.90

0.41

  0.55

14.84

8.45

5.35

0.17

 σ λλ r/a

  1.00

0.00

0.00

0.00

1.38

  0.90

−2.66

−0.52

1.28

4.55

  0.80

−3.86

−2.07

2.83

12.42

  0.70

−4.83

−2.42

5.00

16.91

  0.60

−4.83

−1.38

7.07

17.77

  0.55

−4.14

−1.04

7.59

18.11

Tesseral tides

σ rr r/a

  1.00

0.00

0.00

0.00

0.00

  0.90

0.00

0.10

0.14

0.10

  0.80

0.00

0.31

0.55

0.38

  0.70

0.00

0.21

0.48

0.41

  0.60

0.00

0.21

0.28

0.24

  0.55

0.00

0.31

0.55

0.41

σ r/a

  1.00

0.00

0.00

0.00

0.00

  0.90

0.48

0.41

0.17

−0.28

  0.80

1.04

0.76

0.28

−0.66

  0.70

1.41

1.17

0.35

−0.83

  0.60

1.10

0.76

0.10

−0.59

  0.55

0.00

0.00

0.00

0.00

σ φφ r/a

  1.00

0.00

0.28

0.41

0.35

  0.90

0.00

0.59

0.83

0.59

  0.80

0.00

0.41

0.66

0.48

  0.70

0.00

0.07

0.10

0.14

  0.60

0.00

−1.28

−1.55

1.38

  0.55

0.00

−3.28

−3.28

−5.35

σ φλ r/a

  1.00

0.00

0.00

0.00

0.00

  0.90

0.59

0.38

−0.10

−1.38

  0.80

1.41

0.97

−0.24

−3.69

  0.70

2.76

1.62

−0.35

−6.04

  0.60

4.04

2.76

−1.38

−7.49

  0.55

5.21

3.28

−1.62

−8.18

σ λλ r/a

  1.00

0.00

0.10

0.24

0.08

  0.90

0.00

1.38

1.73

0.10

  0.80

0.00

2.42

2.76

−0.35

  0.70

0.00

3.11

3.11

−2.76

  0.60

0.00

3.45

1.73

−5.87

  0.55

0.00

3.11

1.21

−6.73

σ r/a

  1.00

0.00

0.00

0.00

0.00

  0.90

0.00

0.17

0.28

0.41

  0.80

0.00

0.38

0.62

0.93

  0.70

0.00

0.59

1.00

1.31

  0.60

0.00

0.35

0.72

1.04

  0.55

0.00

0.00

0.00

0.00

Sectorial tides

σ rr r/a

  1.00

0.00

0.00

0.00

0.00

  0.90

0.38

0.38

2.42

1.24

  0.80

0.55

0.48

0.35

1.41

  0.70

0.38

0.31

0.24

1.04

  0.60

0.35

0.28

0.21

0.10

  0.55

0.62

0.52

0.38

1.31

σ r/a

  1.00

0.00

0.00

0.00

0.00

  0.90

0.00

−0.10

−0.17

−0.14

  0.80

0.00

−0.28

−0.41

−0.41

  0.70

0.00

−0.59

−0.69

−0.55

  0.60

0.00

−0.31

−0.59

−0.45

  0.55

0.00

0.00

0.00

0.00

σ φφ r/a

  1.00

0.69

0.59

0.52

−0.10

  0.90

1.97

1.83

1.38

0.93

  0.80

2.31

2.14

2.00

1.73

  0.70

2.17

2.17

2.31

2.45

  0.60

1.10

1.66

2.42

2.90

  0.55

0.00

0.00

0.00

0.00

σ φλ r/a

  1.00

0.00

0.00

0.00

0.00

  0.90

0.00

−0.59

−1.10

−1.73

  0.80

0.00

−1.66

−2.66

−0.41

  0.70

0.00

−2.66

−4.07

−6.62

  0.60

0.00

0.00

0.00

0.00

  0.55

0.00

0.00

0.00

0.00

σ λλ r/a

  1.00

0.93

0.62

0.55

0.24

  0.90

1.38

1.14

0.41

−0.17

  0.80

1.21

0.69

−0.17

−0.86

  0.70

−0.52

−0.41

−1.21

−1.93

  0.60

−3.62

−3.69

−3.76

−3.80

  0.55

−5.87

−6.56

−7.59

−8.97

σ r/a

  1.00

0.00

0.00

0.00

0.00

  0.90

0.48

0.45

0.38

0.24

  0.80

0.97

0.90

0.79

0.55

  0.70

1.52

1.45

1.14

0.76

  0.60

1.10

1.04

0.93

0.62

  0.55

0.00

0.00

0.00

0.00

Fig. 2

Latitude dependence of normal horizontal stress tensor components along the surface of the Earth (Z zonal, T tesseral and S sectorial tides)

Table 2

Latitude dependence of the horizontal shear stress tensor components σ λλ and σ λλ at the Earth’s surface in kPa

 

σ φφ

σ λλ

Zonal

 0°

−1.83

0.00

 20°

−0.41

0.00

 40°

0.07

0.00

 60°

0.66

1.38

Tesseral

  

 0°

0.00

0.00

 20°

0.27

0.17

 40°

0.41

0.35

 60°

0.35

0.07

Sectorial

  

 0°

0.69

0.93

 20°

0.55

0.62

 40°

0.52

0.06

 60°

−1.04

0.24

4 Spherical Stresses Produced by Oceanic Load

The calculation of the functions H′(r) and T′(r) describing the response of the Earth to external load (relative to the solid Earth). Load is similar to the case of the lunisolar effect, but the surface boundary conditions are modified: two of them remain the same as in case of earth tides: \(M'(a) = L'(a) = 0,\) but the third one expresses that the loading mass is on the surface of the Earth: \(N^{'} (a) = p = \rho^{*} gh\), where p is the surface pressure, \(\rho^{*}\) is the density of sea water (1025 kg/m3) and h is tidal height in m. Because \(N^{'} (a) \ne 0\) consequently \(\sigma_{rr} (a) \ne 0\) (see Eq. 1). The gravitational potential generated by unit load acting on a square shaped (Δφ = Δλ) spheroidal layer on the surface (r = a) and characterised by a centri angle \(\alpha = \left[ {(\Delta \phi \times \Delta \lambda )/\pi } \right]^{1/2}\), at a spherical distance θ from the centre of the loaded area, can be given in the following form (Pertzev 1976):
$$W(\varTheta ) = \mathop \sum \limits_{n = 1}^{\infty } W_{n} = \frac{2\pi Ga}{g}\mathop \sum \limits_{n = 1}^{\infty } \frac{1}{2n + 1} \left[ {P_{n - 1} (\cos \alpha ) - P_{n + 1} (\cos \alpha )} \right]P_{n} (\cos \varTheta ) \left( {\frac{r}{a}} \right)^{n}$$
(10)
Similarly to the earth tidal stress tensor, the load stress tensor can be obtained, considering that the surface load is axially symmetric, and therefore the solution does not depend on the azimuth (Varga and Grafarend 1996):
$$\sigma_{rr} = \mathop \sum \limits_{n = 1}^{\infty } N_{n}^{'} W_{n} (\varTheta )\left( {\frac{r}{a}} \right)^{n}$$
(11)
$$\sigma_{\varTheta \varTheta } = \mathop \sum \limits_{n = 1}^{\infty } \left\{ {\left[ {\varLambda \left( {\frac{{\partial H_{n}^{'} }}{\partial r} + \frac{{2H_{n}^{'} }}{r} - \frac{n(n + 1)}{{r^{2} }}T_{n}^{'} } \right) + 2\mu \frac{{H_{n}^{'} }}{r}} \right]W_{n} (\varTheta ) + 2\mu \frac{{T_{n}^{'} \frac{{\partial W_{n} }}{\partial \varTheta }}}{{r^{2} }}} \right\}\left( {\frac{r}{a}} \right)^{n}$$
(12)
$$\sigma_{\lambda \lambda } = \mathop \sum \limits_{n = 1}^{\infty } \left\{ {\left[ {\varLambda \left( {\frac{{\partial H_{n}^{'} }}{\partial r} + \frac{{2H_{n}^{'} }}{r} - \frac{n(n + 1)}{{r^{2} }}T_{n}^{'} } \right) + 2\mu \frac{{H_{n}^{'} }}{r}} \right]W_{n} (\varTheta ) + 2\mu T_{n}^{'} \tan \varTheta \frac{{\partial W_{n} }}{\partial \varTheta }} \right\}\left( {\frac{r}{a}} \right)^{n}$$
(13)
$$\sigma_{r\varTheta } = \mathop \sum \limits_{n = 1}^{\infty } \frac{{M_{n}^{'} }}{r}\frac{{\partial W_{n} (\varTheta )}}{\partial \varTheta }\left( {\frac{r}{a}} \right)^{n}$$
(14)
$$\sigma_{r\lambda } = 0$$
(15)
$$\sigma_{\varTheta \lambda } = 0$$
(16)
Equations (11)–(16) show that the load generated stresses are mainly the normal ones (\(\sigma_{rr}\), \(\sigma_{\varPhi \varPhi }\) and \(\sigma_{\lambda \lambda }\)). The shear stress \(\sigma_{r\theta } = 0\), while M′(a) = 0. The magnitude of \(\sigma_{rr}\) is significant only locally, in the middle of the loaded area it may reach 100 kPa in case of a tidal height of 1 m on an area 10° by 10° (approx. 106 km2), while the other two components \(\sigma_{\varPhi \varPhi }\) and \(\sigma_{\lambda \lambda }\) are 30 kPa (Fig. 3). If the area of the tidal load is reduced the amplitude of the load stress also decreases. According to model calculations carried out: the amplitudes of the load stresses in case of spherical segments 5° by 5° (~2.5 × 105 km2), and 2° by 2° (~4 × 104 km2) are 25 and 7% of the amplitude obtained for spherical segment 10° by 10°. In the immediate vicinity of the loaded area the magnitude of load stress is (30–40)% of the maximum obtained for the middle of the loaded area. In contrast to the tidal stress, which inside the Earth moving away from the surface significantly increases its value (Table 3), the magnitude of the load stress decreases with depth rapidly.
Fig. 3

Radial normal and lateral stress tensor components \((\sigma_{rr, } \sigma_{\varTheta \varTheta } )\) caused by 102 Pa normal load on a spherical segment 100 × 100 in function of spherical distance θ

Table 3

Distribution of vertical (σ rr ) and horizontal (σ θθ ) load stress components (in kPa) at the surface in function of angular distance from the centre of the loaded area 10° × 10°. 1 m high water column load is considered

θ

σ rr

σ θθ

 0°

1.14

0.48

 5°

0.62

0.35

 10°

−0.10

0.17

 15°

0.00

0.00

 20°

0.07

−0.03

 25°

0.03

0.00

 30°

0.03

0.00

5 What is the Impact of the Earth and Oceanic Tides on Earthquake Activity?

The results of the calculations show that in case of all tide types (zonal, tesseral, and sectorial) the stress components significantly increase with depth, but not approaching their maximum values in the depth range of interest from the viewpoint of earthquakes. The deepest earthquake focal depth data reliably determined is 68,410 km (South of Fiji, 17 June 1977). The radial and shear stress components show a monotonous increase at least until the middle of the mantle between 1000 and 2000 km depth zone (0.85–0.70) r/a. Moreover, some stress types reach their maximum value in this depth range. Results of model calculations show that distribution of tidal stress tensor components up to 1000 km depth does not depend to a substantial degree on the inner structure of the Earth. The distribution was compared in case of PREM and a model of the Earth with homogeneous mantle where the density depends only on the hydrostatic pressure. The biggest observed difference did not exceed 20%.

The depth distribution of stresses has a minor importance for the study possible tidal triggering, since 95% of the seismic energy is released within the depth interval (0–50) km (Varga et al. 2017). As it was shown in Sect. 2, the role of lunisolar effect can be significant in case of horizontal shear stresses \(\sigma_{\varPhi \varPhi }\) and \(\sigma_{\lambda \lambda }\). The likelihood of triggering effect in case of shallow seismicity in all probability can be increased by the fact, that the majority of the forces generating earthquakes are horizontal (or nearly horizontal) (e.g. normal, reverse or strike slips). If we accept that these two stress types may be the most likely earth tidal stresses, we must also take into account that the geographical distributions of \(\sigma_{\varPhi \varPhi }\) and \(\sigma_{\lambda \lambda }\) are not the same for different tides. In the equatorial region it is unlikely that tesseral tides, trigger while sectorial and zonal tides are big. At mid-latitudes the zonal tide generated stresses are small, while in polar regions only the zonal stress tensor component are significant.

Stresses produced by surface load are generated directly on the loaded area or in its immediate vicinity. Despite of the fact that the tidally generated load can be sometimes very significant, it has a local effect only (Fig. 3). If a huge area of about one million square kilometres (~10° × 10°) is loaded by 1 cm high tide (100 Pa) the generated vertical (\(\sigma_{rr }\)) and horizontal (σθΘ) stresses are only 1.13 and 0.48 kPa (Table 3; Fig. 3). Of course in case of 1 m high tide the generated stresses are of the order of 10−1 MPa, that is still very small in comparison with the accumulated energy in earthquake foci of significant seismic events (105–107 Pa; 1–10 bar). Of course the area of 10° × 10° is very large. In case of 1° × 1° the 100 Pa load generates stresses about ten times smaller. As it was mentioned already earthquakes are shear fracture processes and consequently they are controlled predominantly by maximum stresses. Therefore—as in the case of tidal stress—for triggering the horizontal load stress (θθ) is of importance.

6 Summary

Reviewing the results of the calculations it can be concluded that the magnitude of triggering effect of earth tides is different in case of zonal, tesseral, and sectorial tides and also significantly depends on the latitude.

The results of calculations carried out using theoretical models show that only the horizontal shear stresses \(\sigma_{\varPhi \varPhi }\) and \(\sigma_{\lambda \lambda }\) produced by earth tides are most likely to influence the outbreak of an earthquake.

The load caused by oceanic tides due to their local influence on solid Earth can have only a limited role in earthquake triggering by tides.

Notes

Acknowledgements

We thank our reviewers (Walter Zürn and an anonymous colleague) for their helpful comments. The research described in this paper was completed during research stay of P. Varga (01.03.2016–31.05.2016) supported by the Alexander Humboldt Foundation at the Department of Geodesy and Geoinformatics, Stuttgart University. P. Varga thanks Professor Nico Sneeuw for the excellent research conditions provided by him. Financial support from the Hungarian Scientific Research Found OTKA (Project K12508) is acknowledged.

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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Research Centre for Astronomy and Earth Sciences, Hungarian Academy of Sciences, Geodetic and Geophysical InstituteKövesligethy Seismological ObservatoryBudapestHungary
  2. 2.Department of Geodesy and GeoinformaticsStuttgart UniversityStuttgartGermany

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