Tidal drag and westward drift of the lithosphere - 1-s2.0-S1674987123000907-main.pdf

Research Paper

Tidal drag and westward drift of the lithosphere

Vincenzo Nesi

, Oscar Bruno

, Davide Zaccagnino

, Corrado Mascia

, Carlo Doglioni

⇑

Dipartimento di Matematica Guido Castelnuovo, Piazzale Aldo Moro, 5, Roma 00185, Italy

Istituto Nazionale di Geofisica e Vulcanologia, Via di Vigna Murata, 605, Roma 00143, Italy

Computing and Mathematical Sciences Department, Caltech, 1200 E. California Blvd., MC 305-16, Pasadena 91125, CA, USA

Dipartimento di Scienze della Terra, Piazzale Aldo Moro, 5, Roma 00185, Italy

article info

Article history:

Received 20 February 2023

Revised 12 April 2023

Accepted 21 April 2023

Available online 27 April 2023

Handling Editor: M. Santosh

Keywords:

Tidal drag

Plate motions

Polarized plate tectonics

Lithosphere-asthenosphere interaction

abstract

Tidal forces are generally neglected in the discussion about the mechanisms driving plate tectonics

despite a worldwide geodynamic asymmetry also observed at subduction and rift zones. The tidal drag

could theoretically explain the westerly shift of the lithosphere relative to the underlying mantle.

Notwithstanding, viscosity in the asthenosphere is apparently too high to allow mechanical decoupling

produced by tidal forces. Here, we propose a model for global scale geodynamics accompanied by numer-

ical simulations of the tidal interaction of the Earth with the Moon and the Sun. We provide for the first

time a theoretical proof that the tidal drag can produce a westerly motion of the lithosphere, also com-

patible with the slowing of the Earth’s rotational spin. Our results suggest a westerly rotation of the litho-

sphere with a lower bound of

/Myr in the presence of a basal effective shear viscosity

s, but it may rise to

/Myr with a viscosity of

K

s within the Low-

Velocity Zone (LVZ) atop the asthenosphere. This faster velocity would be more compatible with the

mainstream of plate motion and the global asymmetry at plate boundaries. Based on these computations,

we suggest that the super-adiabatic asthenosphere, being vigorously convecting, may further reduce the

viscous coupling within the LVZ. Therefore, the combination of solid Earth tides, ultra-low viscosity LVZ

and asthenospheric polarized small-scale convection may mechanically satisfy the large-scale decoupling

of the lithosphere relative to the underlying mantle. Relative plate motions are explained because of lat-

eral viscosity heterogeneities at the base of the lithosphere, which determine variable lithosphere-

asthenosphere decoupling and plate interactions, hence plate tectonics.

Ó

2023 China University of Geosciences (Beijing) and Peking University Published by Elsevier B.V. This is

an open accessarticle underthe CCBY-NC-ND license (

http://creativecommons.org/licenses/by-nc-nd/4.0/

1. Introduction

The mechanisms controlling plate tectonics are still controver-

sial, ranging from bottom-up to top-down mantle convection

(e.g.,

Anderson, 2001

, and references therein). The current tech-

nique to measure the motion of the lithosphere is based on the

Global Navigation Satellite System (GNSS), having the theoretical

assumption that the sum of the kinematics of all plates equals zero,

i.e., the no-net-rotation (NNR) reference frame (

Teunissen and

Montenbruck, 2017; Weiss et al., 2017

). This mathematical

requirement provides very accurate data on the relative move-

ments among plates. However, the motion of the lithosphere rela-

tive to the mantle remains unsolved. In fact, the hotspot reference

frame (HSRF) rather shows that the sum of all plate velocities is not

null, recording a westerly directed residual, which is inferred to be

small (0

/Myr) in the deep HSRF, and much faster

/Myr) in the shallow intra-asthenospheric HSRF

(

Doglioni et al., 2005; Cuffaro and Doglioni, 2007; Crespi et al.,

2007

). Regardless the origin and source depth of the hotspots,

e.g., the shallow wetspots (

Bonatti, 1990

), the westward drift ori-

gin remains physically unexplained, although GNSS baselines sug-

gest a solid Earth tidal modulation of plate motions; in fact, tidal

harmonics seem to modulate plate motions and their speed

(

Zaccagnino et al., 2020

). Moreover, plate boundaries show a geo-

graphically tuned asymmetry that is consistent with the westerly

drift of the whole lithosphere relative to the underlying mantle,

and the mainstream of plate motions supports a shallow source

of the hotspots (

Doglioni, 1993; Doglioni and Panza, 2015

). The

westward drift of the lithosphere was postulated by

Le Pichon

(1968)

and invoked as the effect of the tidal drag (

Bostrom,

1971; Nelson and Temple, 1972

), being responsible for the asym-

metry of the Pacific slabs (

Uyeda and Kanamori, 1979; Doglioni,

1992; Riguzzi et al., 2010; Ficini et al., 2017

), whereas Doglioni

(

Doglioni, 1990

) suggested the existence of a mainstream along

which plates move, having a primary rotation and sub-rotations

https://doi.org/10.1016/j.gsf.2023.101623

1674-9871/

Ó

2023 China University of Geosciences (Beijing) and Peking University Published by Elsevier B.V.

This is an open access article under the CC BY-NC-ND license (

http://creativecommons.org/licenses/by-nc-nd/4.0/

⇑

Corresponding author.

E-mail address:

carlo.doglioni@uniroma1.it

(C. Doglioni).

Geoscience Frontiers 14 (2023) 101623

Contents lists available at

ScienceDirect

Geoscience Frontiers

journal homepage: www.else

vier.com/locate/gsf

(

Cuffaro et al., 2008

). Plate tectonics is controlled by the influence

of lateral variations of viscosity at the interface between litho-

sphere, asthenosphere and upper mantle itself. These gradients

allow variable amounts of relative motion between Earth’s layers,

hence, plate velocity. If there were a perfectly uniform global

mechanical coupling between lithosphere and mantle, then, ide-

ally, the lithosphere would behave as a single shell with no internal

relative velocities, so, there would be no surface plate interaction,

i.e., plate tectonics. Based on the velocities of the GNSS stations and

the hinge behavior of the subduction zones, nowadays it is possible

to calculate the volumes of lithosphere wedged into the Earth’s

mantle: they are about 300

30 km

/yr (

Doglioni and Anderson,

2015

;

Ficini et al., 2020

). Even in this case, there is an asymmetry.

In fact, the volumes subducting towards the west are more than

214

21 km

/yr; the difference is for the easterly-oriented subduc-

tions that provide about three times smaller mantle volume contri-

bution. Therefore, a difference in recycling between the two

systems implies an unbalance of mass which should be compen-

sated, so the Earth’s mantle must relatively migrate from west to

east to fill the mass unbalance. Asymmetries also affect geometries,

dynamics of geological structures and seismic activity. Subduction

zones are paradigmatic by this viewpoint: they are steeper if west-

erly directed (mean angle of dip about 65

) than those with

opposite polarity (27

;

Riguzzi et al., 2010

). Moreover, west-

erly oriented subduction zones are associated with back-arc

spreading, passive margins, and volcanic arcs with low topogra-

phy; conversely, east-directed subductions have trenches located

besides large orogens (

Lenci and Doglioni, 2007

). The orogens asso-

ciated with easterly-directed subduction zones are about 5–10

times larger than the accretionary prisms related to westerly-

directed subduction zones. This implies that the decollements are

deeper and exhume larger volumes along easterly-oriented sub-

duction zones than those with opposite polarity. Not only subduc-

tions and accretionary prisms: asymmetries affect back-arc basins

with concave geometry at the western margin opposed to the con-

vexity of the external arc. Despite this multifaceted evidence, the

astronomical contribution to plate tectonics remains unsolved

and mostly neglected, even though the role of tidal perturbations

has been highlighted to play a role in crustal stability, e.g.,

Kossobokov and Panza (2020)

, because the inferred viscosity of

the asthenosphere would be too high to allow the tidal drag to

be efficient (e.g.,

Jordan, 1974

). Nevertheless, the lithosphere is sig-

nificantly decoupled from the underlying mantle, being able to

move relatively to it up to several centimeters per year. One of

the most straightforward proofs of this phenomenon is that the

fastest-moving plates are those showing lower basal viscosity

(e.g., the Pacific plate has about 10

Pa s (

Pollitz et al., 1998;

Becker, 2017

)). The long-term trend of motion of the lithosphere

appears to be oriented along a main flow described by the so-

called ’tectonic equator’, which is approximately a circle forming

an angle of about 30

with the geographic equator (

Crespi et al.,

2007

). A possible explanation may be provided by the Maxwell

time of the lithosphere that behaves like a fluid sensitive to the

timeframe of the precession of the Earth axis, and the low viscosity

within the low velocity zone (LVZ) constraining the mechanical

decoupling between the lithosphere and the underlying mantle

(

Doglioni and Panza, 2015

). That is, lateral rheological hetero-

geneities in the plane of uncoupling control the velocity gradients

between plates and thus seismicity. All these observations are con-

sistent with a polarized plate tectonics, i.e., with the lithosphere

moving relatively westward relative to the underlying mantle.

However, no one of the forces usually advocated as drivers of glo-

bal plate tectonics can produce the worldwide asymmetric pattern

(

Zaccagnino and Doglioni, 2022

). This suggests that the effect of an

additional force may be crucial for solving the puzzle of absolute

plate motions. In this article, we provide a new mathematical com-

putation of the lunar and solar tidal drag; moreover, we select the

most likely values of the parameters which play a key role in large

scale geodynamics and compatible with observations. In view of

this astronomical frame, we then provide a conceptual model of

the asthenosphere and its upper layer, the low-velocity zone

(LVZ), combining the external and internal phenomena, which

appear generating a strongly interacting self-organized chaotic

system governing plate tectonics (

Doglioni et al., 2007; Riguzzi

et al., 2010

2. Tidal drag modeling

The tidal acceleration on a point of the Earth,

, due to the grav-

itational effect of the Moon from

is given by

;

Þ¼

;

3 cos

ÞÞ

where

;

represents the mass of the Moon,

;

stands for the position of the observation point

in the reference

frame of the Earth (see Appendix B),

;

is the distance of the

Moon from the center of Earth and

stands for the universal grav-

itational constant. We use the angular variable

in order to

describe the relative position of the Moon with respect to the

Earth in the inertial system, while

is a temporal parameter.

For a detailed description of the meaning of different symbols,

see Appendix A and B. Such an acceleration is responsible for a

force deforming the Earth as a function of the Moon position

and complex interactions between the celestial body and the solid

and fluid components of our planet. Nonetheless, the shape of the

Earth is in a quasi-equilibrium configuration, being just weakly

perturbed by the tidal potential. While great part of the deforma-

tion acts elastically, so that the mechanical response of the system

is almost instantaneous, a part of it shows a viscous behavior

(compare with the session ‘‘Discussion”) producing an angular

delay usually estimated in the order of a few degrees (0

Hence, in order to model the effect of the tidal potential on our

planet and to understand its geodynamic implications, we need

to introduce a viscoelastic force that opposes the rotational motion

of the Earth. To describe its analytical form, we consider the aver-

age angular delay using the so-called tidal lag, henceforth

. There-

fore, we assume that, with respect the unperturbed Earth’s surface,

at a certain instant, the tidal deformation has the shape of a sine

wave

with a very large wavelength. The two highest peaks

move on the Earth’s surface according to the equation

max

Þ¼

Þð

parameterizes the position of the Moon. The tidal brake

produces a damping factor of the tangential acceleration whose

maximum is located where the function

is minimum, i.e., at the

points where the physical Earth bulge is at its minimum elongation.

We also observe that the normal component of the tidal brake is

irrelevant since it acts symmetrically in its up and down compo-

nents. In

Fig. 1

, we show the tidal vector field due to the Moon

(not to scale) and its tangential component with (

Fig. 1

b) or without

(

Fig. 1

d) the effect of the tidal brake. These considerations lead to

conjecture that the tangential component of the tidal brake acceler-

ation has the form (See

Fig. 1

tang

;

lag

cos

Þð

where, for physical reasons,

lag

;

2ð

;

Þð

Fig. 2

illustrates the physical meaning of the decoupling param-

eter, henceforth

. The blue curve represents the damping effect

due to the vertical load produced by the normal component of

V. Nesi, O. Bruno, D. Zaccagnino et al.

Geoscience Frontiers 14 (2023) 101623

the gravitational force. If such a component points inward with

respect to the surface of the Earth, the motion becomes slower.

We are now ready to write the equation for our model including

the effect of a mechanical decoupling between the inner and outer

layers due to rheological weakness in the asthenosphere (compare

with the introduction and discussions for the geological meaning

and physical motivation). We introduce the subscript

to indi-

cate we are treating the case in which only the Moon is considered.

Later we will include Sun and Moon simultaneously. We have

;

Þþ

;

Þð

and the tangential acceleration is

tang

;

Þ¼

;

sin

;

Þ½

cos

;

cos

;

The changes that result from taking into account the moment pro-

duced by the tidal torque lead to the following formula:

€

sin

Þ½

cos

Þ¼

where

;

with

the period of the diurnal

tidal harmonics and

the period of the monthly lunar harmonics.

The variable is measured in the reference system in which the Earth

is at rest (see Appendix B). The convention is that the westward

direction corresponds to increasing

. We also assume that at the

initial configuration its position is represented by

0. See Appen-

dix A for the meaning of parameters. The last ingredient needed to

complete the mathematical framework is a viscous term due to fric-

tion acting between contiguous layers inside the Earth. Here, we

assume a linear frictional term as a first hypothesis, even though

such choice is likely to be too severe. In addition, we eventually con-

sider the simultaneous effect of Moon and Sun, so that our model

equation eventually reads

;

sin

ÞÞ½

cos

ÞÞ

cos

Þ¼

;

Þ¼

where

;

represents the mass of the Moon (

m) or the

Sun (

s),

;

is the distance of the celestial body from the center of

Earth (see Appendix A). The apparent motions of the Moon and the

Sun around the Earth are described by

and

respectively.

;

depend on the gravitational constant

and

on the masses of the Moon and Sun respectively.

Now, we are interested in the motion of plates, i.e., large por-

tions of the surface of the Earth. We can represent them, as a first

approximation, as arcs of circumference, say

, of semi-opening

Fig. 1.

Tidal potential and its tangential component in the case of perfectly elastic response of the Earth to tides (a,b) and in the case of a viscous component

(c,d).

V. Nesi, O. Bruno, D. Zaccagnino et al.

Geoscience Frontiers 14 (2023) 101623

2ð

;

. The center of mass of

moves on a smaller circumfer-

ence, i.e., at a fixed distance from the center of the Earth. Therefore,

the motion is completely determined by just one angle that we

choose to be the average one,

, so that

is described by

2ð

;

. Then, assuming the plate be almost rigid,

the first cardinal equation implies that the law

relative

to the center of mass of the arc

can be considered representative

of the motion of the whole plate. Using some trigonometric identi-

ties and introducing the new definitions

Þ¼

;

sin

we get that

Eq. (8)

is equivalent to

þþ

;

cos

sin

ÞÞ

;

sin

cos

ÞÞ¼

Þ¼

;

Þ¼

with

Þ¼

þð

By construction, the solution

to Eq.

(8)

reads

Þ¼

;

Þþ

Þð

The physical meaning of

represents the detrended modulation of

the cumulative tidal displacement of plates with respect to the deep

mantle,

, around the linear trend

defined just above. At last, let

us give the Eq.

(10)

a slightly more compact, but explicit expression

by introducing new variables (see Appendix A for details) and using

some additional transformations:

þð

sin

ÞÞ

þð

sin

ÞÞ¼

Þ¼

;

Þ¼

Since the tidal lag is only a function of the mechanical properties of

Earth without any effect due to the action of other celestial bodies,

(also compare with the discussion section for a more

detailed physical justification) and

3. Results

3.1. Tidal modulations of plate motions as a function of viscosity and

harmonic period

We now present the numerical simulations for the oscillating

component of

, i.e., the solution

Eq. (13)

and

itself in detail.

Concerning the residual displacement, we plot the diurnal,

monthly and annual solution assuming

100 s

(corresponding

s), and

. Our results reported in

Fig. 3

clearly show that high frequency tides are not able to provoke

any detectable effect on plate motions with respect to the deep

mantle, as expected. Of course, tides produce elastic displacement

up to several decimeters, but they do not affect absolute plate

velocity. Numerically, we also observe that the detrended solution

is bounded by a constant

max

that depends on the frictional

parameter

(which is related to the basal viscosity of the litho-

sphere - compare with paragraph 3.2) but also on

and

. The

bigger (i.e., the closer to one)

and

are, the smaller

max

. Con-

versely, the smaller

(i.e., the closer to zero), the bigger

max

.Bya

physical point of view, large values of

’s make the oscillatory force

smaller and smaller. Conversely, higher friction should enhance

stronger damping. The bound

max

has a physical meaning: it pro-

vides an estimation of the amplitude of tidal displacement acting

while the motion proceeds westward. Thus, we want to have

max

of the order of the observed fluctuations, i.e., compatible with

noise for high frequency tides. The regime we want to study is in

the range

2½

;

100

. Of course, the smaller

, the faster the plates

move. On the other hand, the bigger the friction

is the smaller the

high-frequency modulations are. For

2½

;

100

the maximum

oscillation is of the order of that occurring in one day. Therefore,

we propose the theoretical bound

max

Choosing

100 s

;

ffi

1 guarantees an angular velocity of

about 10

rad s

with a basal diurnal fluctuation of about 10

Fig. 2.

Illustration not to scale, for better visualization, of the physical meaning of the decoupling parameter

. (a) The green curve represents the damping effect with respect

to the acting perturbation (blue line) due to the vertical load produced by the normal component of the gravitational force. If such a component points

inward with respect to

the surface of the Earth, the motion becomes slower. (b) This panel shows the break of symmetry of the potential field obtained using the difference of th

e angle formed by the

actual Earth high tide with the position of the Moon at its Zenith (tidal lag). (c,d) Taking the absolute value of the perturbation makes more evident bo

th the angle shift

and

the asymmetry around the equilibrium position.

V. Nesi, O. Bruno, D. Zaccagnino et al.

Geoscience Frontiers 14 (2023) 101623

mm, i.e., completely negligible with respect to the usual sources of

deformation produced by other external agents such as hydrological

and thermal loading. Compare with

Fig. 4

. Now, we consider the

solution,

Eq. (8)

and make a plot of

computing

it as the sum of the linear part

Þ¼

;

, see

Eq. (12)

, and the

bounded component

, solution of

Eq. (13)

for different time

intervals. Using, for instance

25 s

(

s), we

get a westward motion of about 5 cm/yr.

3.2. On the relationship between K and

In our model, we introduce the convenient frictional term

;

however, the relevant quantity for geodynamics is the effective

shear viscosity of the low velocity zone (lower values of viscosity

in the asthenosphere), which determines the degree of mechanical

coupling between the lithosphere and the underlying layers. Even

though it is not possible to provide a exact conversion rule from

and viceversa, we are interested in writing a rough correspon-

dence between couples of values. Assuming

to be the frictional

Fig. 4.

Residual displacement around the linear trend for increasing values of

compared with

. We choose

;

100 s

, corresponding to an effective

shear viscosity of about

;

Fig. 3.

Solutions of the residual of Solid Earth tidal displacements calculated

assuming

100 s

s),

1 for intervals of one day (a),

one month (b) and one year (c).

V. Nesi, O. Bruno, D. Zaccagnino et al.

Geoscience Frontiers 14 (2023) 101623

coefficient produced by a viscous force acting on the lithosphere

with surface

whose mass is represented by

(

Fig. 5

)

where

150 km is the thickness of the upper asthenosphere. If

one takes the lithospheric density

3 g/cm

;

can be written

KPa

100 km is the average thickness of the lithosphere. We propose

the following theoretical relationship

sin

By physical considerations, we also have:

ffið

Hence,

sin

5 sin

Set

rad

;

360

;

which gives a relationship for

and

(

Fig. 6

), where

represents

the westward drift velocity expressed in

/Myr.

3.3. Low frequency tidal modulations of plate motions

High frequency tidal harmonics are not able to produce a

detachment between the lithosphere and the upper mantle since

their period is much shorter than the Maxwell’s time of the

asthenosphere (

s). Therefore, high-frequency body tides

are mostly buffered by the high viscosity of the lithosphere and

the underlying mantle. Conversely, long-period tides, e.g., the lunar

nodal 18.6-years-long precession, can act effectively in modulating

plate motions. Our simulations confirm this possibility (

Fig. 7

), also

providing a computational check for the analysis reported in

Zaccagnino et al. (2020)

. The nominal and observed value of

Fig. 5.

Expected cumulative displacements produced by the action of the tidal drag

assuming

25 s

s),

1 for intervals of one day (a),

15 days (b), one month (c) and one year (d).

Fig. 6.

Westward drift velocity predicted by our model as a function of the basal

effective shear viscosity for different tidal lag angles compatible with observations.

V. Nesi, O. Bruno, D. Zaccagnino et al.

Geoscience Frontiers 14 (2023) 101623

amplitudes, about 3 mm (

Agnew, 2010

), are correctly reproduced

using a viscous frictional term

50 s

, corresponding to an

effective minimal basal shear viscosity of about

3.4. The motion of a plate

We now consider the motion of a plate specifically. From a

mathematical point of view, each portion of the lithosphere can

be represented as a rigid arc

on the surface of the Earth. A generic

point,

, on the arc has coordinates

;

Þþ

;

. There-

fore, the first of such points encountered when running counter-

clockwise (the most westward) is identified by the angle

;

, the

last one is identified by the angle

;

, where

2½

;

repre-

sents the aperture angle associated with the plate, i.e., its angular

extent. Given the angular density

, we can write the first car-

dinal equation of dynamics applied to the arc

in terms of the

unknown angle

;

, made by the center of mass

of the rigid body

around the center of mass of the Earth. We remark that our equa-

tion has an antipodal symmetry. This means that any plate,

described by the parametric form

;

tends to move

as the ‘‘opposite” arc

located at

;

. Since the crust

is divided into several plates, so that

does not coincide with the

whole surface of the Earth, the center of mass

is at constant dis-

tance form the center of the planet and, therefore, the motion of

is determined by the motion of

. Hence, we are interested in

establishing the equation describing the motion of the plate in

terms of

;

. In order to do it, we use the first cardinal equation:

the sum of the forces acting on

satisfies

;

Þ¼

;

Considering the lunar tide, the following acceleration is produced:

;

Þð

that can be written, making the calculation at first order in

;

and recalling

;

for each segment of the plate, as

;

Þ¼

;

sin

cos

;

Þð

which produces a tangential component of motion following the

equation

€

sin

ÞÞ¼

where

;

. A graphical description of the direction and

intensity of the accelerating tidal field acting on a rigid plate is pro-

vided in

Fig. 8

. It represents a phase diagram of the effect of lunar

and solar earth tides as a function of the position of the Moon

and the Sun at time zero. While a completely symmetric force is

expected in the case of the classical field for a perfectly elastic Earth,

so that the cumulative displacement of the lithosphere with respect

to the mantle is zero, the same tidal field acting on an anelastic

Earth produces a westerly-oriented motion except for a limited

set of astronomical configurations. The point

;

corresponds to

a situation when the center of mass of the Earth lies on the same

straight line joining the center of mass of the Sun and the point

on the surface of the Earth. This is an exceptional configuration.

In the real three dimensional space, this corresponds to the Moon

crossing the ecliptic plane exactly on the segment joining

and

the Sun. In any case, the gravitational outward pull is at its peak.

What we have just observed gives an idea of the different degrees

of complication in the problem in dimension three. In that case,

viewed from Earth, the positions of the Sun and Moon have signif-

icantly richer behavior and must be parameterized on the surface of

the celestial sphere. This suggests that the measurements of

and

must accurately take into account the astronomical configuration at

the time of observation. Detailed mathematical derivations are pro-

vided in Appendix C.

3.5. Analytical results about the westward drift of the lithosphere

Given the equation of our model for both the Moon and Sun

written in explicit form

€

sin

Þþ

sin

Þ¼

;

Þ¼

(

we notice that

–

; moreover,

and

are of the same order of

magnitude and still greater that the corresponding gravitational

constants

and

. Formally, if

and

were of the same order

of magnitude, considering Moon and Sun separately, would give

Fig. 7.

Tidal modulations of plate motions around the linear long-term trend for

50 s

(corresponding to

s) produced by the 18.6-years-long

tidal harmonic. We notice an almost unfiltered signal (the nominal amplitude of the

lunar perigee is about 3 mm) with respect to the strongly buffered perturbations

produced by the daily, monthly and annual tides. This suggests that slow additional

stress, even though quite small, can act effectively in decoupling the lithosphere

from the upper mantle because its period is comparable with the Maxwell’s time of

the weakest layers of the low velocity zone (LVZ) being of the order of about 10

V. Nesi, O. Bruno, D. Zaccagnino et al.

Geoscience Frontiers 14 (2023) 101623

Fig. 8.

Phase diagram of level lines describing the direction and intensity of acceleration of the lithosphere produced by the effect of lunar and solar earth

tides as a function

of the position of the Moon and the Sun at time zero. The left panels represent the classical field for a perfectly elastic Earth, while the right panel sho

ws the same field on a

planet with an anelastic component described by a tidal lag angle of

. (a) Tangential tidal field. The figure is periodic of period one in this scale in both axes. In the lighter

zones the acceleration is westward, in the darkest in eastward. If we do not consider the brake, the maxima and minima are located at points

;

. (b) The positive

part of the tangential tidal field acting to displace the lithosphere to west with respect to the underlying mantle. Notice that almost all the phase spa

ce is covered by the a

positive tangential tidal field. The red lines are the zero level. (c) The negative part of the tangential tidal field. Notice that the part of the phase sp

ace with negative (easterly)

drag becomes quite limited in the case of a significant anelastic component in the planetary response to tidal stress. The red lines are the zero level.

V. Nesi, O. Bruno, D. Zaccagnino et al.

Geoscience Frontiers 14 (2023) 101623

; therefore, if only takes just one measure, it

could give

. We do not have a physical reason to

take

–

since they only depend on the mechanical properties

of the Earth. So, we have

€

Þ¼

We can find an upper estimate of the limit angular velocity for plate

motions produced by the drag taking the absolute value of the tidal

oscillation

€

Þj

which would lead to

ffi

For a proof of this formula, see Appendix D. If plates move at the

observed velocity, then, the maximum allowed value for the effec-

tive viscous coupling between lithosphere and the mantle,

, would

satisfy:

max

where the average westward drift angular velocity is assumed in

the order of magnitude of

rad

. In the linear fric-

tion model, the value

max

cannot be achieved, since it requires

a non physical behavior for the force; nevertheless, it can be consid-

ered a benchmark. In order to provide a more accurate estimation of

friction and plate velocity, we can write

;

max

where the tidal lag is an angle

that measures the misalignment of

the Moon and Sun with respect to the maximum height of the tidal

bulge, the normal decoupling parameter

is a measure of the ease

of lithosphere detachment from asthenosphere. Since

;

ffi

max

represents the limiting case of the previous dynamics in which the

Earth’s lithosphere is modeled as a thin spherical layer with ele-

vated rigidity except for the boundary of plates in which it is seg-

mented. Therefore, no matter how small the phase delay is, the

tidal brake produces, for long times, a westward motion of the litho-

sphere. In fact, given the model equation

€

sin

cos

Þð

sin

Þ¼

;

Þ¼

it can be written in the equivalent form

€

sin

where

Þ¼

ÞÞ

; so that, using the homogeneous initial

conditions, one gets

€

sin

Þ¼

€

sin

where

remains bounded, while

, the dominant term, is linear.

Therefore, the limiting angular velocity is furnished by the

relationship

lim

!þ1

which gives

So, we have

plate

ffi

(

)

;

ffi

max

This means that our model predicts a westward motion of plates

with constant velocity depending on the basal viscosity of the litho-

sphere and the tidal lag.

4. Discussion

4.1. Mathematical and physical achievements of our model

The causes and extent of the displacement of the surface layer

of the lithosphere relative to the underlying mantle have long

been debated. One topic considered by many to be controversial

is whether gravitational forces are able to contribute appreciably.

For the most skeptical, the ‘‘westward drift” corresponds to

/Myr, inferred only as a temporary accident related to

the faster motion of the Pacific plate being related to the slab pull

(e.g.,

Ricard et al., 1991

), being the tidal component irrelevant.

Moreover, the literature is poor in estimates about the tidal lag.

In our model, we aim to find a mathematical relationship

between three potentially measurable physical quantities: the

viscosity

, the westward drift velocity with respect to the mantle

for long times

, and the tidal lag

. To keep the mathematical

treatment simple, we treat a two-dimensional model in which

Moon and Sun travel their apparent motion with respect to Earth

in a single plane. We do not want to hide the fact that the three-

dimensional treatment is more complex. But we will not delve

into such analysis in this first study. To the best of our knowl-

edge, for the first time the tidal lag

is included in the model,

in a way that appears physically quite natural. Even in this sim-

plified two-dimensional version, an independent theoretical esti-

mation of

based on the solution of an appropriate free boundary

problem does not seem to be known. Our assumptions are as fol-

lows. First, the tangential tidal acceleration is restrained in areas

where the normal component points toward the Earth’s interior

as seems unavoidable. This is a very inexpensive way of consider-

ing a disconnection parameter,

. The value of

is dictated by

physical considerations, but could in principle be measured. Sec-

ond, the peak of the decoupling does not align with the Moon’s

position, but with the maximum amplitude of the observed solid

tide. The exact values of

are not known. There are various mea-

surements, but the theoretical references to be able to assess

their accuracy appear to be uncertain. This will be the subject

of future studies. However, we obtain a law that fixes the product

and

by constraining it to that of

. Assuming that

remains in the range assumed by the most accepted measures,

we draw very interesting conclusions. We obtained the following

crucial result

sin

so that, recalling

rad

;

360

;

1 corresponds to one degree every million years and

2 would be the most skeptical viewpoint. The value

1 corresponds to a tidal lag of one degree. We have

V. Nesi, O. Bruno, D. Zaccagnino et al.

Geoscience Frontiers 14 (2023) 101623