The earth's imperfectly elastic response to body and loading tidal forces is discussed using complex Love numbers and complex mass loading coefficients. Exact analytical expressions were derived relating the energy dissipation within an inhomogeneous, compressible solid earth to the surface values of these complex characteristic numbers, thus relating the global dissipation function Q to the phase shifts in the potential, gravity, tilt, strain and displacement tides. Integration of a global ocean tidal model shows that energy dissipated in the solid earth due to ocean loading is at least 10% of that dissipated in the body tides

Zschau, J.

English

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rTidal Friction in the Solid Earth
Jochen Zschau
Institute of Geophysics, Kiel University, Federal Republic of Germany
N79 21507
Abstract. The earth's imperfectly
elastic response to body and loading tidal
forces is discussed using complex Love
numbers and complex mass loading coeffi-
cients. Exact analytical expressions have
been derived relating the energy dissipa-
tion within an inhomogeneous, comnressible
solid earth to the surface values of these
complex characteristic numbers, thus rela-
ting the global dissipation function Q to
the phase shifts in the potential, gravity,
tilt, strain and displacement tides.
Integration of a global ocean tidal mo-
del shows that energy dissipated in the so-
lid earth due to ocean loading is at least
10 % of that dissipated in the body tide;
however both body and loading tide together
do not account for more than a few percent
of the astronomically observed dissipation.
The commonly used relation Q =—tan<£
where <p is the observed phase lag only app-
lies when selfgravitation and hydrostatic
prestress are ignored. It, therefore, is not
applicable to the earth and in fact there is
no unique relation between the global Q and
the tidal phase shifts, this relation being
very dependent on the distribution of Q with
depth. Determinations of the global Q from
satellite observations may be in error by
70 %, and calculations on the basis of seis-
mic Q-models predict phase shifts in the
gravity tide of only a few thousands of a
degree in place of the currently predicted
tenths.
Unlike in the body tide case, dissipation
in the loading tide is sensitive to proper-
ties of the asthenosphere, and phase shifts
in the M_ loading tides in displacement, gra-
vity and tilt may be as high as several de-
grees for loads near ocean ridges and sub-
duction zones.
Rate of Tidal Dissipation in the
Solid Earth
Dissipation of body and loading ti-
dal energy within the solid earth may
be determined from its complex Love
numbers and complex mass loading co-
efficients, respectively [Zschau, 1979 a,
b] . Such the expressions for the> dissi-
pated energy turn out to be fairly simple,
for the body tide, and
2xn+l ...»("_
AE. dS (1)
E =
n 4 GR - K-, 1U-
2
n )J n (2)
Proc. of the 9th GEOP Conference, An Intermitioiwl Symposium on lite Applications of
Geodesy to GeoJywiniit-x. October 2-5. 1978. Depl. of Geodetic Science Rept. No. 280, The
Ohio State Univ!, Columbus. Ohio 43210.
for the loading tide, where AE is the
energy dissipated during one cycle of har-
monic loading, K andH*, K* are the imagi-
nary parts of the surfSce Eove numbers and
mass load coefficients, respectively, fy is
the amplitude of the body force potential,
(Jj* is the amplitude of the surface load
potential, R is the earth's radius and G is
the gravitational constant. n describes
the degree of the expansion into spherical
harmonics. The integration is taken over
the surface of the earth. The expressions
above have been obtained without approxima-
ting the real earth by an incompressible
and homogeneous one as was necessary in for-
mer calculation, for instance by Munk &
Mac Donald [i960] .
Using these formulas, and assuming the
mantle Q structure LMS as obtained from
the observation of the earth's free oscilla-
tion [see Smith, 1972] , a body tide solid
dissipation rate of ,7
3.19 x 10 erg/s,
i.e. about 1 % of the astronomically obser-
ved dissipation rate, has been obtained. The
corresponding computation for the loading
tide dissipation rate within the earth's
crust and mantle, carried out on the basis
of a global M2 ocean tide model [Hender-
shott, 1972] , yields a minimum value of
roughly 1O % of the body tide solid dissipa-
tion rate, i.e. .,
3.21 x 101 erg/s.
This value has been determined from the low
degree harmonics of the ocean tide distribu-
tion up to n=25, and, therefore, does not
represent the high amplitude -tides in the
shelf areas. The latter may contribute
significantly to the total dissipation,
because the dissipated energy is propor-
tional to the square of the marine tidal
amplitude. On the other hand, the elastic
strain energy stored in the mantle as well
as the energy dissipated in the mantle de-
creases with increasing degree n of the
spherical harmonic loading for n> 5 as
may be seen from Fig. 1. This suggests
that solid earth dissipation in the shelf
areas does not change the total dissioa-
tion rate drastically. Anyway, both body
tide - and loading tide dissipation to-
gether most probably do not account for
more than a few percent of the astronomi-
cally observed dissipation rate.
327
https://ntrs.nasa.gov/search.jsp?R=19790013336 2020-03-21T23:58:37+00:00Z
Fig. 1. Loadincr tide shear strain energy
stored and dissipated within the solid,
earth [from Zschau, 1979b]. n is the
degree of the spherical harmonic expan-
sion of iiendershott's global i"I2 ocean
tide model [iiendershott, 1972] . Results
are valid for the Gutenberg Earth with
the free oscillation Q structure LMS for
the mantle and the surface wave 0 struc-
ture for the crust.
Body Tide Phase Shifts and the Earth's
Dissipation Function Q
In Zschau [1979b] the earth's global
dissipation function Q has been related to
the phase shifts in the potential, gravi-
ty, tilt, strain and disolacement body
tides. The following expressions valid for
constant Q distribution within the earth
have been obtained:
Tangential
displacement:
Potential va- tanlp =
riation due to
tidal defor- ___.
mation : "'".'-<•
= L/l = - O.897 Q-1
= ~ O.612 Q-1
Radial dis-
placement :
Surface areal
strain :
Tilt :
gravity va-
riation
tanip = H/h = - O.555 Q
i = - 0.318
-1
H-jK
FFh^ k = - O.O508 Q
-1
(h,H), (k,K), and (1,L) complex Love
numbers; the index n has been omitted here.
OBSERVED
DELAY ANGLES
RECORDING 0.1°
GRAVIMETERS 0.2°
SATELLITE
OBSERVATIONS 0.5°
CONVENTIONAL
I/ton ah_3k
79
39
I/land
 k
60
CORRECT
Qgi Qg£
(CONSTANT Q) (LOW 0 ZONE)
28 5.5
14 <3
36 34
TABLE 1. Conventional and correct global
Q values (Q_i) corresponding to tidal
delay angle! of gravity and satellite
observations.
The above equations show that the commonly
used relation
Q-1 (3)
where I? is the observed phase shift is not.
applicable in the earth tide case. For the
tidal gravity variation this had already
been pointed out by Slichter [I960] who
instead of (3) introduced the formula
-1 (4)
where 6 is the gravimetric fa-factorf
and<P is the gravity phase shift.
With 6 = 1-16 this'yields
tan<P = - 0.138 Q-1 (5)
which is still too large .by more than a
factor of 2 as is obvious from the compa-
rison of this formula with the correspon-
ding one given above. One may show that
Slichter's formula is equivalent to
Q (6)
which does not correspond to the basic de-
finition of Q~l as the strain energy dissi-
pated during one cycle of loading over 2 It
the peak energy stored in the system.
The general expression for the earth's glo-
bal dissipation function Q in terms of its
complex Love numbers is, however,
Q-1 t'schau, 1979b] (7)
This equation is valid for an incompressib-
le body with homogeneous density. None of
the above phase shifts is equivalent to this
expression, hence in general
Q"1 * - tanlp (8)
As shown in Zschau [l979b] the reason for
this is that in the case of tidal deforma-
tions, selfgrayitation and hydrostatic pre-
stress cannot be ignored.
Besides the fact that (3) is not appli-
cable in the tidal case, there, further-
more, is no unique relation between the glo-
bal Q and the tidal phase shifts, this re-
lation being very dependent on the distri-
328
bution of Q with depth. For instance, a
low Q zone in the upper mantle such as given
in model LMS [see Smith, 1972] nay alter the
ratio between observed tidal phase shifts and
the global (average) Q up to a factor of 5.
Table 1 gives some examples for the
errors involved when'not taking account
of the above aspects: Let the phase delay
of the body tide gravity be O.I with re-
spect to the external forces. Proceedincr
in the conventional manner, i.e. usinq
Slichter's formula with 6 = 1.16 gives
the wrong global body tide Q = 79. In
the case of constant Q within the
mantle, we find the correct value to
be Q = 28. If we assume a low Q asthe-
nosphere, i.e. let the real Q distribu-
tion within the earth differ by only a
constant factor from the free oscilla-
tion Q model LMS, we find that the global
Q has to be chosen as low as 5.5 to
correspond to the gravity phase delay of
O.l°. This is less than 7 % of the value
Q = 79, obtained by the conventional me-
thod. For the same reason it turns out
that the body tide gravity phase delay
due to friction within the solid earth
amounts more likely to a few thousandth
of a degree than to a few tenth of a
degree as expected so far. We, therefore,
suggest that the average Ol gravity phase
shift of - O.2 as observed for Europe
may not be attributed to imperfect
elasticity in the mantle like it is pro-
posed by Melchior et al. [1976], but
rather to the indirect effect of the 0^
tide in the oceans. There seems to be
no chance at all at the moment to get in-
formation on the mantle Q from body tide
gravity investigations.
The delay angle of the potential bulge
due to tidal deformation of the earth has
been determined from the orbits of arti-
ficial satellites to be O.5° [Lambeck
et al., 1974] . Lambeck et al. relate
this delay angle to a mantle Q of 60
which one obtains by using formula (3).
The correct Q corresponding to this delay
angle is 36, if the Gutenberg earth and
constant Q values within the mantle are
adopted. For the LMS equivalent model,
we have calculated the global Q which
corresponds to the delay angle of O.5 to
be 34. This shows that the delay angle of
the tidal potential bulge is less sensi-
tive to the geometry of the Q distribution
within the earth than the phase delay of
the tidal gravity at the deformed surface
as measured by a gravimeter. The Q va-
lues of 34 and 36 are much lower than the
lowest limit of the upper mantle seismic
Q.
The usage of the exact theoretical
relationship between the bulge of the
tidal potential and the body's global Q
may also be important for other planets
as for instance for Mars. From observa-
tions of the secular acceleration of
the Mars satellite Phobos, Smith and
Born [1976] deduced the phase angle of
the potential bulge due to the body
tide of Mars. They related this nhase
angle to the global Q of Mars by the
simple formula (3), and found a global
Q between 5O and ISO. However, as one
cannot neglect selfgravitation and the
hydrostatic prestress for Mars, equation
(3) is not applicable, and, therefore,
the Q between 50 and 150 is probably too
high for Mars, provided that the obser-
ved phase angle -of the tidal bulge is
true.
Similar considerations as above may
also be important for the determination
of the lunar global Q from observations
of its physical librations [see Yoder,
1978] .
The Effect of Imperfect Mantle Elasti-
city on Loading Tides
Unlike in the body tide case, also
high degree harmonics of the load are
important in the loading tide case. It -
is found that for the free oscillation
Q model LMS, the loading tides of harmo-
nic degrees slightly less than 1OO are
strongly effected by the low Q astheno-
sphere, the global loading tide Qs being
up to nearly 7 times smaller than the
global body tide Q (see Fig."2). Corres-
pondingly, the loading tide phase shifts
due to imperfections in the elasticity
of the mantle are by more than one order
of magnitude higher than those of the
body tide. Near ocean ridges and near
subduction zones the M- loading tide
phase shifts may even Be as high as a
several degrees for the displacements as
well as for gravity and tilt, provided the
Maxwell constitutive law is valid. From
the computation of phase shift Green's
functions it is obvious that these maxi-
mum phase shifts occur at about 8O to
1OO km distance from the load, this
distance depending on the depth of the
assumed low viscous asthenosphere
(see Fig. 3). Our numerical results
suggest that loading tide investigations
could become an effective tool for
studying the upper mantle viscosity in
regions where viscosities lower than
1O*9 Poise may be expected.
Acknowledgement; A portion of this
study was completed while the author
was a visiting fellow at the Coopera-
tive Institute for Research in Envi-
ronmental Sciences, Boulder, Colorado.
References
Hendershott, M.C., The effects of solid
earth deformation on global ocean
tides, Geophys. J.R. astr. Soc. 29,
389, 1972.
329
10 10J
-750 © BODY TIDE
@ LOADING TIDE
10*
750 H
500-
Fig. 2. Body tide Q and loading tide O
for different degrees n of the spheri-
cal harmonic expansions of the tides
[ from Zschau, 1979b]. Pesults are
valid for the Gutenberg Earth with the
LMS Q structure from free oscillation
data for the mantle, and the MM 8 Q
structure from surface wave data for the
crust. The low Q values for n slightly
lower than 10O in the loading tide case,
are due to the low Q asthenosphere 'in
model LMS. The high Q values for h'igher
n and for lower n are due to the high Q
values in the crust and in the lower
mantle, respectively".
Lambeck, K., Cazenave and G. Balmino,
solid earth and ocean tides estima-
ted from satellite orbit analysis,
Rev. Geophys. Space Phys. 12, p. 421,
1974.
Melchior, P., J.T. Kuo, and B. Ducarme,
earth, tide gravity maps for Western
Europe, Phys. Earth Planet, Interiors,
13, p. 184, 1976.
Munk, W.H., .and G.J.F. MacDonald, The ro-
tation of the earth, Cambridge Univer-
sity Press, New York, I960.
Slichter, L.B., in P. Melchior: Compte
Rendu des Reunions de la Commission
Permanente des marees terrestres a
I'AssembleeGenerale d'Helsinki,
Marees Terr. Bull. Inf. 21, p. 369,
1960.
Smith, S.W., The anelasticity of the
mantle, in: The upper mantle, edited
by A.R. Ritsema, Tectonophysics'13,
p. 601, 1972. .
Smith, J.C., and G.H. Born, secular acce-
leration of Phobos and Q of Mars,
Icarus 27, p. 51, 1976.
Joder, C.F., Effects of the Spin-Spin
interaction and the inelastic tidal
deformation on the Lunar physical
librations, 1979, in print (Preprint
from JPL, Pasadena, California 911O3).
-2°
-0°
-2°
--4°
--5°
.FUNCTION
(OCEAN RIDGE MODEL, M2)
01
RADIAL DISPL.
TANGENTIAL DISPL.'
GRAVITY
A (KM)
100
Fig. 3. M2 phase Green's functions for an
ocean ridge model [ from Zschau, 1979a,b.].
They represent the phase shifts between the
viscoelastic earth's responses to a varying
point load, and the responses of the corres-
ponding elastic earth. The high phase shifts
at distances of about 100 km from the load
are due to the low viscous asthenosphere.
A : distance from the point load, PHI:
phase shift.
Zschau, J., Phase shifts of tidal sea
load deformations of the earth's sur-
face due to low viscosity layers .in
the interior, Proceed. 8th Int.
Symp. on Earth Tides held at Bonn
1977, in print (1979 O.)
Zschau, J., Tidal friction in the solid
Earth: Loading tides versus body .tides,
in:-P. Brosche and J. Silndermann, TirJal
friction and the earth's rotation,
Springer 1979b,in print.
330


Tidal friction in the solid earth

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