Abstract

Plate motion representing a remarkable Earth process is widely attributed to several primary forces such as ridge push and slab pull. Recently, we have presented that the ocean water pressure against the wall of continents may generate enormous force on continents. Continents are physically fixed on the top of the lithosphere that has been already broken into individual plates, this attachment enables the force to be laterally transferred to the lithospheric plates. In this study, we combine the force and the existing plate driving forces (i.e., ridge push, slab pull, collisional, and shearing) to account for plate motion. We show that the modelled movements for the South American, African, North American, Eurasian, Australian, Pacific plates are well agreement with the observed movements in both speed and azimuth, with a Root Mean Square Error (RMSE) of the modelled speed against the observed speed of 0.91, 3.76, 2.77, 2.31, 7.43, and 1.95 mm/yr, respectively.

Keywords

Ocean Water Pressure Force, Ocean-Continent Interaction, Plate Driving Force, Lithospheric Plate, Plate Motion

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1. Introduction

One of the most significant achievements in the 20^th century was the establishment of plate tectonics, which developed from the previous concept of continental drift [1] [2]. Plate tectonics mainly describes the motion of a dozen different-sized plates that connect with each other to form a giant “jigsaw puzzle” over the Earth’s surface. The evidence supporting this motion includes shape fitting of the African and American continents, a coal belt crossing from North America to Eurasia, identical directions of ice sheet movement in southern Africa and India, and Global Positioning System (GPS) speed measurements. In addition, paleomagnetic reversals in oceans [3] [4] reflect seafloor spreading, and studies of the Hawaii-Emperor seamount chain have shown that the chain is actually a trace of the lithosphere rapidly moving over relatively motionless hotspots [5] [6], which further confirms the Earth’s surface motion. During the past 50 years, our understanding of plate motion has expanded greatly. Plates were found to have been periodically dispersed and aggregated in the Mesozoic period, accompanied by 5 - 6 significant astronomical events [7]-[11]. The speed and direction of plate motion supported by paleomagnetism and deformation in the intraplate regions exhibited various styles over geological time [12].

Exploring the plate driving forces is especially important because it provides the first insights into the processes that yield plate tectonics. Throughout the history of plate tectonics, a large number of forces (i.e., centrifugal and tidal forces, ridge push, slab pull, basal drag, slab suction, mantle plume, geoid deformation, and the Coriolis force) have been presented to account for plate motion [1] [5] [13]-[31]. More than 71% of the Earth’s surface is covered by oceans, and their depths reach nearly 3700 meters [32]. Based on the principle of fluid mechanism that the water pressure against the wall of a container may generate force on the container, the ocean water pressure may generate enormous force on continents. Continents are physically fixed on the top of the lithosphere that has been already broken into individual plates, this attachment enables the force to be transferred laterally to the lithospheric plates. Recently, we have determined the distribution of this force around continents, and estimated its amplitude to be of the order of 10¹⁷ N per kilometer of continent width [33]. Our modelling suggested that the stresses yielded by this force are mostly concentrated on the upper part of the continental crust, and their magnitudes reach up to 2.0 - 6.0 MPa, which is entirely comparable to the range of earthquake stress drops (1 - 30 MPa) [34]. As a result, this force may have significantly impacted continents during a geological timescale. However, the intricate mechanism underlying how the force contributes to plate motion remains largely obscure. In this study, we discuss this issue with hope of expanding the understanding of plate motion.

2. An Ocean-Generated Force Driving Mechanism for Plate Motion

The continents are fixed on the top of the lithosphere, and the lithospheric plates connect to each other, this relationship allows the ocean-generated force (i.e., ocean water pressure force) to interact with other forces that act on the lithospheric plates. Subsequently, we list the plausible forces that act on a sample continental plate (Figure 1) and discuss the physical nature of these forces. These forces can be classified into two categories: the forces acting on the parts of the continent that connect to the oceans and those acting at both the bottom surface of the continental plate and the parts of the continental plate that connect to adjacent plates. The forces acting on the parts of the continent that connect to the ocean are treated as ocean-generated forces, denoted as F_R on the right and F_L on the left. The horizontal forces decomposed from these forces are denoted as ${F^{\prime }}_R$ on the right and ${F^{\prime }}_L$ on the left. A more detailed description of the horizontal forces generated by ocean water pressure may refer to Yang [33]. The force acting on the bottom surface of the continental plate arises from a coupling between the plate and underlying viscous asthenosphere. This force is called the basal friction force and is denoted as f_base. According to Forsyth & Uyeda [22], if there is thermal convection in the asthenosphere, f_base would be a driving force [16] [17] [20] [21]. If, instead, the asthenosphere is passive relative to plate motion, f_base would be a resistive force. Clennett et al. [35] recently treated mantle flow as a resisting force opposite to plate motions when plate reconstruction models are assessed. Here, we assume f_base to be a resistive force. Given that the continental plate moves toward the right, the forces acting on the parts of the continental plate that connect to adjacent plates include the collisional force from the plate on the right side and the push force from the ridge on the left side, they are denoted as F_C and F_ridge, respectively.

Figure 1. F_L(F_R) represents the ocean-generated force on the left (right) side of Plate A, while ${F^{\prime }}_L\left({F^{\prime }}_R\right)$ and ${F^{″}}_L\left({F^{″}}_R\right)$ denote the horizontal and vertical forces decomposed from the ocean-generated force, respectively. f_base denotes the basal friction force exerted by the underlying asthenosphere, while F_C and F_ridge denote the collisional force from Plate C on the right side and the push force from the ridge on the left side, respectively. h_L and h_R are the ocean depths on the left and right, respectively. r₁, r₂, r₃, r₄, and r₅ denote the distances of these forces to the Earth’s center. Note that the ocean depth and lithospheric plate thickness are highly exaggerated.

Plate motion is conventionally understood as a rigid plate rotating about an axis that penetrates the Earth’s center, and this rotation must be a consequence of the integrated effect of all torques acting on the plate [22] [26]. Following this understanding, we use torque balance to discuss the movement caused by these forces. According to Figure 1, a combined torque for Plate A may be written as

$\tau =\left(r_1{F^{\prime }}_L-r_2{F^{\prime }}_R\right)+r_3{F}_{\text{ridge}}-\left(r_4{F}_C+r_5{f}_{base}\right)$ (1)

where the first term $\left(r_1{F^{\prime }}_L-r_2{F^{\prime }}_R\right)$ denotes the torque yielded by the final horizontal force, the second term r₃F_ridge denotes the torque yielded by the ridge push force, both of them represent the driving torque for the continental plate, and the third term (r₄F_C + r₅f_base) denotes the resisting torque, which hinders the movement of the continental plate. Taking into consideration the reality that the plate is too thin (e.g., less than few hundred kilometers) relative to the Earth’s radius (e.g., more than six thousand kilometers), we approximate r₁ = r₂ = r₃ = r₄ = r₅.

Equation (1) provides three possibilities for the continental plate. If the driving torque is greater than the resisting torque, the combined torque is greater than zero, and the continental plate is subjected to an accelerating motion. Practically, it is impossible for the continental plate to undergo such an accelerating motion. If the driving torque is equal to the resisting torque, the combined torque is zero, and the continental plate would be subjected to a steady motion. If the driving torque is less than the resisting torque, the combined torque is less than zero, and the continental plate remains motionless.

Plate A’s movement exhibited in Figure 1 is parallel to that of Plate B and Plate C, this situation is rather idealized. Practically, the movements of most plates intersect with each other. For instance, the South American Plate moves northwest, the Nazca Plate moves eastward, the African Plate moves northeast, the Eurasian Plate moves eastward. These nonparallel movements would yield additional collisional forces and shearing forces between plates. If two plates are not moving in the opposite direction, the collisional and shearing forces between them may be driving; and if the two plates are moving in the opposite direction, the collisional and shearing forces between them may be resisting. Below, we develop two semi-analytic methods (I and II) to independently resolve plate motion.

2.1. Method I

It is assumed that the Earth’s surface is covered with Plate A, Plate B, Plate C, Plate D, and others, and that Euler pole of each plate has been established (Figure 2). For Plate A (assumed to be continental), the horizontal force F_i (i = 1, 2, 3, 4, and 5) acts on the side of the continent that is fixed on top of Plate A. The horizontal force (F₁, for instance) yields a component (${F^{\prime }}_1$ , for instance) that is orthogonal to the rotation axis of the plate; this component then yields a torque (${{\tau }^{\prime }}_1$ , for instance). The torques yielded by all the components decomposed from the horizontal forces are summed into first driving torque. The ridge push force F_r-i (i = 1, 2, 3, and 4) acts on the edge of the plate, this force also yields a component that is orthogonal to the rotation axis; this component also yields a torque. The torques yielded by all the components decomposed from the ridge push forces are summed into second driving torque. Given that Plate A, Plate B, Plate C, and Plate D move eastward, southward, westward, and eastward, respectively, and that Plate D moves faster than Plate A, these make Plate A undergo a collisional driving force F_B-c from Plate B, a shearing driving force F_D-s from Plate D, a collisional resistive force F_C-c from Plate C, a shearing resistive force F_B-s from Plate B, and a basal friction force F_basal from the underlying viscous asthenosphere. The collisional driving force F_B-c and the shearing driving force F_D-s also yield two components that are orthogonal to the rotation axis of the plate; these components also yield torques. The torques yielded by these two components are summed into third driving torque. The collisional resistive force F_C-c and the shearing resistive force F_B-s also yield two components that are orthogonal to the rotation axis; these components also yield torques. The torques yielded by these two components are summed into first resistive torque. The basal friction force F_basal yields second resistive torque.

Then, we divide these five sets of torques into two exerting parts: one, which includes the second driving torque and a little portion of the first and third driving torque, balances out the first resistive torque, and the other, which including the remaining portion of the first and third driving torque, balances the second resistive torque. Consequently, all these torque balances allow Plate A to be steadily rotated under the assumption that the acceleration and inertia of the plate are neglected. The remaining portion of the first and third driving torque is called the net driving torque, and the second resistive torque is called the net resistive torque. The balance between the net driving torque and the net resistive torque may be written as

${\tau }_{driving}-{\tau }_{basal}=0$ (2)

where τ_driving is the net driving torque, τ_driving = ετ, and ε is the ratio of the net driving torque to the first and third driving torque. As shown in Figure 2(a), the component decomposed from a force (the horizontal force, for instance) may be written as F_i’ = F_icosη_i, and η_i = γ_i − λ_i, where γ_i is the inclination of this force to latitude. λ_i is the azimuth of arc P_iE with respect to latitude. This component yields a torque τ_i with respect to the rotation axis, i.e., τ_i = r_iF_i’, where r_i denotes the lever arm distance of the component F_i’, r_i = R_earthsinφ_Pi, R_earth is the Earth’s radius and R_earth = 6371 km, and φ_Pi is the angle of site P_i and the Euler pole. A sum of the torques yielded by the components, which are decomposed from the horizontal forces, collisional driving forces, and shearing driving forces, forms the first and third driving torque τ. τ_basal denotes the net resistive torque yielded by the basal friction force, it can be written as τ_basal = r_KF_basal, where r_K denotes the lever arm distance of the basal friction force. According to the principle of fluid mechanics [36], the basal friction force may be expressed as F_basal = μAu/y, μ, A, u, and y are the viscosity of the asthenosphere, the plate’s area, the plate’s speed, and the thickness of the asthenosphere, respectively. Therefore, u = yτ_driving/μA, this speed represents average level of the plate’s movement. In general, the largest speed of a plate occurs at the plate’s equator, while the smallest speed occurs at the location whose angle distance to the Euler pole is minimal or maximal. As shown in Figure 2(b), we assume that the geometric center (i.e., location K) of Plate A moves at the average speed, namely, u = u_k. And then, the speed of any location S within this plate may be expressed with u_s = u_ksinφ_s/sinφ_k, where φ_s(φ_k) is the angle distance of location S(K) to the Euler pole (i.e., location E) relative to the Earth’s center. The speed u_s can be further decomposed into the longitudinal speed u_s-lo and latitudinal speed u_s-la, and u_s-lo = u_ssin(λ_s − 90˚), u_s-la = u_scos(λ_s − 90˚), where λ_s is the azimuth of arc SE with respect to latitude. The azimuth of the movement is then calculated through the longitudinal and latitudinal speeds. All these angles and distances (i.e., η_i, γ_i, λ_i, λ_s, φ_Pi, φ_k, φ_s, r_i, r_K) may be further calculated through the latitudes and longitudes of related locations.

Figure 2. Modelling the torque balances for plate motion. (a) Geometry of ocean-generated horizontal forces, ridge push forces, collisional forces, and shearing forces over a spherical frame. The white oval within Plate A represents the scope of the continent. The green line denotes the oceanic ridge. The pink lines beneath the locations (black dots) denote the latitudinal and longitudinal directions. (b) Decomposing the average movement of the plate into the movement of any location. The location K and E are the geometric center of Plate A and its Euler pole, respectively. (c) Exhibiting the geometric centers (black dots) of the six selected plates and the established Euler pole locations (yellow stars) over a planar frame.

Here, we use six plates (South American, African, Eurasian, North American, Australian, and Pacific) to demonstrate their movements. In order to simplify the following deduction, we plot globally tectonic plates into a grid of 10˚ × 10˚ and use these grid nodes, which are within plate, to obtain the geometric center of each plate. The geometric center is approximately calculated through the average of the latitudes and longitudes of these nodes. The Euler pole location of each of these plates is cited from the GSRM v.2.1 [37]. Both the geometric center of each plate and its Euler pole location are exhibited in Figure 2(c).

Besides these forces discussed above, the additional forces (i.e., collisional and shearing) for these plates need to be considered. As exhibited in Figure 3, the African, Indian, and Australian plates provide the collisional driving forces F_AF-EU-C, F_IN-EU-C, and F_AU-EU-C for the Eurasian Plate, respectively. The Nazca plate provides a collisional driving force F_Naz-SA-C for the South American Plate. The Eurasian Plate provides a shearing resistive force F_EU-NA-S for the North American Plate; vice versa, the North American Plate provides a shearing driving force F_NA-EU-S for the Eurasian Plate. The Australian, North American, and Eurasian Plates provide the collisional driving forces F_AU-PA-C, F_NA-PA-C, and F_EU-PA-C for the Pacific Plate, respectively. The Australian, North American, and Eurasian Plates provide the collisional driving forces F_AU-PA-C, F_NA-PA-C, and F_EU-PA-C for the Pacific Plate, respectively. It’s important to note that, while slab pull has widely been linked to the Pacific Plate, Doglioni and Panza [38] made a comprehensive review on this force and presented a long list of geometric, kinematic, and mechanical arguments against it as the primary driving force. This conclusion is recently strengthened by Faccincani et al. [39]. These authors found that the lithospheric mantle density

Figure 3. Ocean-generated horizontal forces (yellow arrows) and related collisional/shearing forces (purple arrows). The original map is from Yang (2024) [33].

structure can be affected by variations in thermal regimes and bulk composition, and their results suggest that the lithospheric mantle is not denser than the underlying asthenospheric mantle. A difference in density between the lithospheric mantle and the underlying asthenospheric mantle means that the oceanic plate, which consists of the lithospheric crust and mantle, is unlikely to sink, forming a “negative” buoyancy to drive plate motion. Taking into account this present status, we temporarily neglect slab pull in the modelling here but will discuss it latterly. The details of these collisional and shearing forces are listed in Table 1. It is worth noting that the magnitude and direction of these forces are artificially given, potentially causing discrepancies with reality. A precise determination of these forces requires an in-depth investigation of plate boundary structures, but is not feasible within the scope of this study due to individual effort constraints. The resultant torques from all related forces are listed in Table 2.

The asthenosphere viscosity is not yet exactly determined. Many numerical studies using glacial isostatic adjustment and geoid modelling have shown that asthenospheric viscosity ranges from 10¹⁷ to 10²⁰ Pas [40]-[49]. Laboratory experiments, however, suggested that the magnitude of the asthenospheric viscosity could be substantially different from those constrained by numerical studies. The viscosity is variable and likely related to the thermodynamic state, grain size, composition of the medium, and state of stress [50]. Both the melt contents of the asthenosphere and the water in the asthenosphere may greatly affect the viscosity [51] [52]. Hirth and Kohlstedt [52] reported a variable viscosity profile for a melt-free oceanic lithosphere with a mean value of ~10¹⁸ Pas. These authors [53] [54] concluded that, in consideration of the water- and melt-rich layers characterized by much lower viscosities, a strong vertical variability of viscosity may be more realistic. The asthenosphere’s effective viscosity can be greatly lowered to 10¹⁵ Pas if the water content in the case of both diffusion and dislocation creep is included

Table 1. Given collisional and shearing forces between plates.

Notes: related sites refer to Figure 3.

Table 2. (a) Parameters for the torques of six selected plates in the method I. (b) Parameters for the torques of six selected plates in the method I (continued). (c) Parameters for the torques of six selected plates in the method I (continued). (d) Parameters for the torques of six selected plates in the method I (continued).

Note: The negative symbol “-” beneath torque denotes counterclockwise with respect to the axis of rotation.

Note: the negative symbol “−” beneath torque denotes counterclockwise with respect to the axis of rotation.

Note: the negative symbol “−” beneath torque denotes counterclockwise with respect to the axis of rotation.

Note: the negative symbol “−” beneath torque denotes counterclockwise with respect to the axis of rotation.

[55]. Scoppola et al. [54] conducted a more detailed review of asthenospheric viscosity and concluded that the presently accepted values of viscosity might be reduced through a combined experiment including these parameters (i.e., melt content, water content, mechanical anisotropy, and shear localization). A “superweak”, low-viscosity asthenosphere supported by recent observations is being accepted by the geophysical community [47] [49] [56]-[61]. Jordan [62] treated the asthenospheric thickness as 300 km. Taking into account the present status of the viscosity and thickness of the asthenosphere above, we adopt y = 300 km for each of the six selected plates, μ = 10¹⁸ Pas for the South American, African, North American, and Eurasian plates, μ = 0.6 × 10¹⁸ Pas for the Australian Plate, and μ = 0.12 × 10¹⁸ Pas for the Pacific Plate. The other parameters (i.e., plate area, the ratio of the net driving torque and the first and third driving torque) and the resultant average movements of these plates are listed in Table 3.

There have been many plate motion models (i.e., GSRM, NUVEL-1, and MORVEL) that include global navigation satellite systems (GNSS) and paleomagnetic data. For instance, GSRM v.2.1 includes more than 6739 continuous GPS velocity measurements [37]. The movements reproduced by these models may approximately represent observations. Here, the movements of 450 locations (41 for the South American Plate, 70 for the African Plate, 93 for the North American Plate, 95 for the Eurasian Plate, 47 for the Australian Plate, and 104 for the Pacific Plate) are reproduced by GSRM v.2.1. The modelled and reproduced movements are then compared in Figure 4. The Root Mean Square Error (RMSE) of the modelled speed against the reproduced speed for a plate is expressed as

$\text{RMSE}=\sqrt{{\displaystyle {\sum }_0^m{\left(u_m\left(i\right)-u_o\left(i\right)\right)}^2}/m}$ (3)

where u_m(i) represents the speed of a location (i) calculated by our model, u_o(i) represents the speed reproduced by GSRM v.2.1, and m is the total number of locations in the plate. It can been found that the modelled movements for these locations are well consistent with the reproduced movements in both speed and azimuth, the RMSE of the modelled speed against the reproduced speed for the South American, African, North American, Eurasian, Australian, and Pacific plates is 0.91, 3.76, 2.77, 2.31, 7.43, and 1.95 mm/yr, respectively.

2.2. Method II

We assume that the Earth’s surface is covered with Plate A, Plate B, Plate C, Plate D, and others (Figure 5). For Plate A (assumed it to be continental), it undergoes the ocean-generated horizontal force F_i (i = 1, 2, 3, 4, and 5), the ridge push force F_r-i (i = 1, 2, 3, and 4), the collisional driving force F_B-c, the shearing driving force F_D-s, the collisional resistive force F_C-c, the shearing resistive force F_B-s, and the basal friction force F_basal. One horizontal force (F₁, for instance) yields a torque (τ₁,

Table 3. The net driving torques and their resultant movements for these selected plates in the method I.

Figure 4. The reproduced movements from GSRM v.2.1 (black arrows) verse the calculated movements from our model (red arrows) in the method I. (a)-(f) are the South American, African, North American, Eurasian, Australian, and Pacific plates, respectively.

for instance), another horizontal force (F₂, for instance) yields another torque (τ₂, for instance), a combination of these two torques results in a new torque (τ_1-2, for instance), this new torque then combines the torque yielded by third horizontal force to form another new torque. Subsequently, the torques yielded by all the horizontal forces are combined into a final torque. The collisional driving force F_B-c yields a torque, the shearing driving force F_D-s yields a torque, the final torque combines these two torques to firm first driving torque. The collisional resistive force F_C-c and the shearing resistive force F_B-s also yield two torques, they combine to firm first resistive torque. The basal friction force F_basal yields second resistive torque. The ridge push force F_r-i (i = 1, 2, 3, and 4) also yields a torque, the torques yielded by all the ridge push forces are combined into second driving torque.

Then, we divide these four sets of torques into two exerting parts: one, which includes the second driving torque and a portion of the first driving torque, balances out first the resistive torque, and the other, which including the remaining portion of the first driving torque, balances the second resistive torque. Consequently, all these torque balances allow Plate A to be steadily rotated under the assumption that the acceleration and inertia of the plate are neglected. The remaining portion of the first driving torque is called the net driving torque, and second resistive torque is called the net resistive torque. We assume that the net driving torque exerts on the geometric center (i.e., location K) of Plate A, this makes the plate move along a big circle that represents the equator of this Plate. And then, the balance between the net driving torque and the net resistive torque may be expressed with Equation (2). According to Figure 5(a), a force F_i yields a torque τ_i with respect to the Earth’s center, i.e., τ_i = R_earthF_i, where R_earth is the Earth’s radius and R_earth = 6371 km. The combination of two torques follows the trigonometric principle and may be written as

Figure 5. Modelling the torque balances for plate motion. (a) Geometry of ocean-generated horizontal forces, ridge push forces, collisional forces, and shearing forces over a spherical frame. The large light blue and yellow circles denote the orientations of the torques yielded by the horizontal force F₁ and F₂, the large green circle denotes the orientation of the combined torque of these two torques, and the large white circle denotes the possible orientation of the first driving torque. (b) Decomposing the average movement of the plate into the movement of any location. The location K and E are the geometric center of the plate and its Euler pole, respectively. (c) Exhibiting the torque balance of the selected plates over a planar frame. The calculated Euler pole locations (red dots) are compared to the established Euler pole locations (yellow stars).

${\tau }_j^2={\tau }_i^2+{\tau }_{i+1}^2+2{\tau }_i{\tau }_{i+}{}_1\cos \left({\gamma }_i-{\gamma }_{i+}{}_1\right)$ (4)

where τ_j is the combined torque, τ_i and τ_i+1 are the torque yielded by the force F_i and F_i+1, respectively. γ_i and γ_i+1 denote the inclination of the forces F_i and F_i+1 to latitude, respectively. τ_basal denotes the net resistive torque yielded by the basal friction force, it can be written as τ_basal = R_earthF_basal. According to the principle of viscous fluid mechanics, the basal friction force may be expressed as F_basal = μAu/y, μ, A, u, and y are the viscosity of the asthenosphere, the plate’s area, the plate’s speed, and the thickness of the asthenosphere, respectively. Therefore, u = yτ_driving/μA, this speed represents average level of the plate’s movement.

On the whole, the largest speed of a plate occurs at the plate’s equator, while the smallest speed occurs at the location whose angle distance to the Euler pole is minimal or maximal. According to Figure 5(b), we assume that the geometric center (i.e., location K) of Plate A moves at a speed of u_k = ζu, where ζ is an amplification coefficient of the average speed. And then, the speed of any location S within this plate may be expressed with u_s = u_ksinφ_s/sinφ_k, where φ_s(φ_k) is the angle distance of location S(K) to the Euler pole (i.e., location E) relative to the Earth’s center. The Euler pole is calculated through the location K and the orientation of the first driving torque τ. The speed u_s can be further decomposed into the longitudinal speed u_s-lo and latitudinal speed u_s-la, and u_s-lo = u_ssin(λ_s-90˚), u_s-la = u_scos(λ_s-90˚), where λ_s is the azimuth of arc SE with respect to latitude. The azimuth of the movement is then calculated through the longitudinal and latitudinal speeds. All these angles and distances (i.e., γ_i, λ_s, φ_k, φ_s) may be calculated through the latitudes and longitudes of related locations.

We here use three plates (South American, African, and Pacific) to demonstrate the resultant movements. The geometric centers of these plates, the calculated Euler pole location, and the established Euler pole location cited from GSRM v.2.1 [37] are exhibited in Figure 5(c). A few other possible forces (i.e., collisional and shearing) are considered for these plates. For example, the Eurasian Plate provides a collisional driving force F_EU-AF-C and a shearing driving force F_EU-AF-S for the African Plate. The Nazca plate provides a collisional driving force F_Naz-SA-C for the South American Plate. The Australian, North American, and Eurasian plates provide the collisional driving force F_AU-PA-C, F_NA-PA-C, and F_EU-PA-C for the Pacific Plate, respectively. Again, we temporarily neglect slab pull in the modelling here but will include it in the discussion of this study.

The details of these collisional and shearing forces have been exhibited in Figure 3 and listed in Table 1. The resultant torques from all related forces are listed in Table 4. The viscosity and thickness of the asthenosphere for these three plates are the same as that described in the method I. The other parameters (i.e., plate area, the ratio of the net driving torque to the first driving torque, and the amplification coefficient) and the resultant average movements are listed in Table 5. The movements of 215 locations (41 for the South American Plate, 70 for the African Plate, and 104 for the Pacific Plate) are reproduced by GSRM v.2.1. The calculated and reproduced movements are compared in Figure 6. The Root Mean Square Error (RMSE) of the calculated speed against the reproduced speed for these three plates is calculated through Equation (3). We find that the calculated movements for these locations are basically consistent with the reproduced movements in both speed and azimuth, the RMSE of the calculated speed against the reproduced speed for the South American, African, and Pacific plates is 0.98, 3.18, and 6.51 mm/yr, respectively. This result is not entirely as good as that demonstrated in the method I. One major cause for this is because the geometric center of a plate is strictly not calculated through the average of the latitudes and longitudes of those nodes. The less accurate first driving torque adds to the less accurate geometric center, naturally, the calculated Euler pole location and the resultant movement of the plate cannot be accurate. Even so, our goal has realized that a combination of the ocean-generated, ridge push, collisional force, and shearing forces may account for plate motion.

Table 4. (a) Parameters for the torques of three selected continental plates in the method II. (b) Parameters for the torques of three selected plates in the method II (continued).

Note: *denotes this force is changed from 1.0 × 10¹⁷ N listed in Table 1 to 2.2 × 10¹⁷ N.

Table 5. The net driving torques and their resultant movements for three selected plates in the method II.

3. Discussion

3.1. Why May Ocean-Generated Force Contribute to Plate Motion?

Based on the principle of fluid mechanics [36], a plate moving over a fluid may be expressed with a force balance equation F = μAu/y, where F, μ, A, u, and y

Figure 6. The reproduced movements from GSRM v.2.1 (black arrows) verse the calculated movements from our model (red arrows) in the method II. (a), (b), and (c) are the South American, African, and Pacific plates, respectively.

represent the driving force, the fluid’s viscosity, the plate’s area, the plate’s speed, and the thickness of the fluid, respectively. This relationship is illustrated in Figure 7(a). Practically, the lithospheric plates consist of the crust and the upper mantle and extend about 100 km below Earth’s surface, most of them have a horizontal dimension of several thousands of kilometers. Beneath the plates is the more fluid asthenosphere, which extends from roughly 100 km to 700 km below Earth’s surface. These features allow us to apply the equation above to estimate the force that is necessary to maintain the movement of the lithosphere over the asthenosphere, where F, μ, A, u, and y now denote the driving force, the asthenosphere’s viscosity, the lithosphere’s area, the lithosphere’s speed, and the asthenosphere’s thickness, respectively (Figure 7(b)). The lithosphere’s area and the asthenosphere’s thickness have been well established, while the viscosity of the asthenosphere remains a high uncertainty, which has been extensively discussed in Section 5. We here assume the lithosphere to move at a speed u = 3 cm/yr, although the fact is not so because it has been fractured into individual plates whose movements are various in both speed and direction. Given y = 300 km (Jordan, 1974), A = 510,000,000 km² [32], and μ = 10¹⁵ - 10²⁰ Pas, the driving force calculated through the equation above ranges from 1.6172 × 10¹⁵ to 1.6172 × 10²⁰ N. This result indicates that, if we want the lithosphere to move at a speed of 3.0 cm per year over the asthenosphere, a driving force of 10¹⁵ - 10²⁰ N is necessary, and that this force is simultaneously countered by a resistive force from the asthenosphere, creating a force balance that enables the movement to be realized steadily. And now, we ideally divide the lithosphere into individual plates, and exert a driving force to one of these plates, the plate encounters resistance from adjacent plates and the underlying viscous asthenosphere. Since all of plates are physically attached to the underlying asthenosphere, the upper limit of the total resistance force undergone by this plate would be 10¹⁵ - 10²⁰ N. Thus, for a lithospheric plate to move over the underlying asthenosphere, a driving force whose magnitude matches this upper limit of the total resistance force is required.

The continents are fixed on the top of the lithosphere that is composed of individual plates, and the plates connect to each other, this frame enables the ocean-generated force to be laterally transferred to the plates. As shown by Yang [33], the ocean-generated force (i.e., the horizontal force F_i) is generally at a magnitude of 10¹⁷ N, this amount has fallen within the range of the upper limit of the total resistance force. As illustrated in Figure 7(c), three plates are totally designed in the model; along the vertical direction, the weight of each plate is balanced out by the support from the asthenosphere; thus, we only need to discuss the force balance along the horizontal direction. F_AR, F_AL, F_CL, and F_CR are the ocean-generated horizontal forces, F_RL and F_RR are the ridge push forces, F_SP is the slab pull force, F_BA, F_AB, F_CB, and F_BC are the collisional forces between three plates, f_A, f_B, and f_C are the basal friction forces exerted by the asthenosphere. According to Forsyth & Uyeda [22], if there is thermal convection in the asthenosphere, f_base would be a driving force [16] [17] [20] [21]. If, instead, the asthenosphere is passive relative to plate motion, f_base would be a resistive force. Clennett et al. [35] recently treated mantle flow as a resisting force opposite to plate motions. Here, we assume f_base to be a resistive force. Trench suction is neglected. It’s worth noting that we now include slab pull in this model, which is intended to show that this force may be incorporated into our modelling of force balance. Each of these forces can yield a torque relative to the Earth’s center, because torque is a product of force and lever arm, and therefore the lever arm may be represented with the Earth’s radius since plate is too thin relative to the Earth’s radius, the lever arm length of one plate may be approximately equal to that of another plate. This situation allows us to simplify the torque balance into the force balance for the discussion. We assume that Plate A and B rotate counterclockwise whereas Plate C rotates clockwise.

For Plate A, we set F_RL = 4.0 × 10¹² N m⁻¹, this magnitude is widely accepted by geophysical community [31]. We assume the ocean depth to be 5.00 km at the right and 3.00 km at the left, respectively. This assumption is reasonable because the ocean depths around the world are various from one place to another. These two depths correspond to F_AR = 0.1225 × 10¹² N m⁻¹ and F_AL = 0.0441 × 10¹² N m⁻¹, the final combined force of these two forces would be F_AR − F_AL = 0.0784 × 10¹² N m⁻¹. We set F_BA = 4.05 × 10¹² N m⁻¹, and use F_RL = 4.0 × 10¹² N m⁻¹ and a little portion of the final combined force, which is represented by F_ARL = 0.05 × 10¹² N m⁻¹, to balance out F_BA, and use the remaining final combined force F_RARL = 0.0284 × 10¹² N m⁻¹ to balance out the basal friction force f_A. The force balances for this plate would be F_BA − F_RL − F_ARL = 0 and F_RARL − f_A = 0.

For Plate B, which is an oceanic plate, we set F_CB = 4.0 × 10¹² N m⁻¹, due to F_BA = F_AB = 4.05 × 10¹² N m⁻¹, thus, F_AB − F_CB = 0.05 × 10¹² N m⁻¹. We use this combined force and the net slab pull to balance out the basal friction force. As exhibited in Figure 7(c), the direction of the slab pull F_SP is tilt and downwards, this means that its contribution to the horizontal plate motion is largely discounted by the geometry of force. We here term the net slab pull that contributes to the horizontal plate motion as F_NSP. Schellart [63] estimated the net slab pull F_NSP to be at a magnitude of 4.1-6.1 × 10¹² N m⁻¹. As a result, the force balance for this plate would be F_BA − F_CB + F_NSP − f_B = 0.

Figure 7. Conceptual model for the force balance of fluid mechanics. (a) A plate moving over a fluid. (b) The lithosphere moving over the asthenosphere. (c) The lithospheric plates moving over the underlying viscous asthenosphere.

For Plate C, we set F_RR = 3.95 × 10¹² N m⁻¹, and assume the ocean depth to be 4 km at the left and 6 km at the right, respectively. These two depths correspond to F_CR = 0.0784 × 10¹² N m⁻¹ and F_CL = 0.1764 × 10¹² N m⁻¹, the final combined force of these two forces would be F_CL-F_CR = 0.098 × 10¹² N m⁻¹. Due to F_CB = F_BC = 4.0 × 10¹² N m⁻¹, we use F_RR = 3.95 × 10¹² Nm⁻¹ and a little portion of the final combined force, which is represented by F_CLR = 0.05 × 10¹² N m⁻¹, to balance out F_BC, and use the remaining final combined force F_RCLR = 0.048 × 10¹² N m⁻¹ to balance out the basal friction force f_C. The force balances for this plate would be F_BC − F_RR − F_CLR = 0 and F_RCLR − f_C = 0.

These force balances allow three plates to be steadily rotated. We find, even if the ridge push force F_RL (F_RR) is given a smaller amplitude (~10¹⁰ N m⁻¹, for example), so long as the collisional force F_BA (F_AB, F_CB, and F_BC) is properly valued, these force balances can always be realized. Nevertheless, as demonstrated by Yang [33], a ridge push force of 4.0 × 10¹² N m⁻¹ would result in a horizontal stresses that are mostly concentrated on the lower part of the lithosphere, which is not agreement with the observed stresses that are mostly concentrated on the uppermost part of the lithosphere [64]-[66]. Hence, we prefer to accept the ridge push force to be smaller than ocean-generated force. Our demonstration reaches a point that irrespective of whether slab pull is included or not included, the force balance within our model remains consistently achievable.

3.2. How Does Plate Motion Realize Mechanically?

Thus far, we have concluded that ocean-generated force is able to combine the ridge push force, the collisional force, and the shearing force to satisfy the kinematics and geometry of plate motion. Now, let us address how plate motion can be mechanically realized. As shown in Figure 7(c), it is assumed that the depth of Ocean 1 is greater than that of Ocean 2. If we use a part of Ocean 2 that connects to Plate A, which is equal in length to Ocean 1, to do comparison, the depth difference between this part of Ocean 2 and Ocean 1 creates a net gravitational potential energy relative to the asthenosphere reference level. As Plate A and Plate B move away from each other, this separation would require the Ocean 1 depth to decrease as the basin elongates horizontally, and require the Ocean 2 depth to increase as the basin shortens horizontally. Consequently, the net gravitational potential energy decreases. Therefore, if there were no external energy inputs to compensate, the net gravitational potential energy would eventually disappear, terminating plate motion. Tides may be supplying this energy. Tides represent the regular alternations of high and low water on Earth; when high water falls, the gravitational potential energy converts into kinetic energy, then, ocean water obtains movement. As all oceans are physically connected, part of the water in Ocean 2 may travel via passages to compensate the decreasing ocean depth of Ocean 1, thus sustaining the net gravitational potential energy. Given the basal friction force f_basal = 1.62 × 10¹⁸ N and the movement distance u = 3 cm/yr for the lithosphere, an energy of Q₁ = f_basal × u = 4.86 × 10¹⁶ J/yr is required to satisfy this movement distance. This energy also represents the net gravitational potential energy. The ocean water level often increases twice a day due to tides, and the resultant height is assumed to be h = 0.3 m. Given the gravitational acceleration g = 9.8 m/s, the volume v = 1.35 × 10⁹ km³ and density ρ = 1000 kg/m³ for the whole ocean, and consequently, the gravitational potential energy obtained by ocean water due to tides during a year would be Q₂ = 2 × 365 × ρvgh = 2.9 × 10²¹ J/yr. The transformation from gravitational potential energy to kinetic energy within ocean water and the energy transition between oceans must be complicated, and we believe that a small part of this tidal energy is enough to supply the net gravitational potential energy. In fact, the impact of tidal energy on plate motion has long been discussed. Rochester [67] showed that the total energy released due to tidal friction exceeds 5 × 10¹⁹ ergs/s. Several authors [68] [69] concluded that the dissipation in both shallow seas and on the solid Earth is approximately 2 × 10¹⁹ ergs/s, and this amount of energy exceeds the lower bound set by seismic energy release by 2 orders of magnitude [70] and might be driving the plate motion. Other authors (e.g., [71] [72]) reevaluated the energy budget and found that the total energy released by tidal friction may reach up to 1.2 × 10²⁰ J/yr, and approximately 0.8 × 10²⁰ J/yr is dissipated in the oceans, shallow seas, and mantle, and the remaining energy is enough to maintain the lithosphere’s rotation, estimated at approximately 1.27 × 10¹⁹ J/yr. In contrast to these studies, we provide another insight: the tidal energy obtained by ocean water may feed plate motion.

Our understanding of the impact of ocean on continent and lithospheric plate suggests that the ocean-generated force may have significantly contributed to the evolutions of continent and plate tectonics, but what’s the detail of them?

Acknowledgements

We express sincere thanks to Jinmin Chen for conducting the vector force analysis and to Bernhard Steinberger, Jeroen van Hunen, Maureen D. Long, and Thorsten Becker for their helpful comments on earlier version of this research. No funding for this research.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References