The Dynamic Impact of Ocean on Continent

Yongfeng Yang

Water Resources Comprehensive Development Center, Bureau of Water Resources of Shandong Province, Jinan, China.

DOI: 10.4236/ijg.2024.159039                

Abstract

Around 71% of the Earth’s surface is covered by oceans with depths that exceed several kilometers, while continents are geographically enclosed by these vast bodies of water. The principle of fluid mechanics stipulates that water yields pressure everywhere in the container that holds it, and the water pressure against the wall of container generates force. Ocean basins are naturally gigantic containers of water, in which continents form the walls of the containers. In this study, we present that the ocean water pressure against the walls of continents generates enormous force, and determine the distribution of this force around continents and estimate its amplitude to be of the order of 10¹⁷ N per kilometer of continent width. Our modelling suggests 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. Our results suggest that the force may have significantly impacted the dynamics of continent (lithospheric plate) and its evolution.

Keywords

Ocean, Water Pressure Force, Continent, Ocean-Continent Interaction, Stress, Earthquake, Plate Motion

1. Introduction

The co-existence of ocean and continent makes the Earth unique in the solar system. Covering roughly 71% of the planet’s surface, oceans hold a volume of almost 1.35 billion km³, with an average depth of nearly 3700 meters [1]. Contrastingly, continents occupy around 29% of Earth’s surface and are geographically enclosed by these vast bodies of water. Mechanically, if a liquid and a solid are put together, they must interact with each other. The present study of ocean-continent interaction is focused mostly on the transition zone, where sea water erodes coast while land contributes sediment to seafloor, and the exchange of material, energy, and information takes place extensively [2] [3]. The principle of fluid mechanics stipulates that water pressure against the wall of a container that holds the water generates force. Ocean basins are naturally gigantic containers of water, in which continents form the walls of these containers. Hence, the ocean water pressure against the wall of continent may generate enormous force. Even so, the details of this force, including its magnitude and direction, are still unclear. Geochemical research on zircons indicates that the existence of liquid water on Earth is more than 4 billion years [4]-[6]. During such a long geological timescale, ocean water pressure force may exert impact on continents, since continents would deform in response to any external force. Moreover, continents are physically fixed at the top of the lithospheric plates, this attachment allows the force to be laterally transferred to the lithospheric plates. From these points, addressing the ocean water pressure force is very important for understanding the dynamics of continent (lithospheric plate) and its evolution. The objects of this study are twofold, first, to present a new force generated by ocean water; secondly, to explore the stress caused by this force in the continent and its implication.

2. Force between Water and Rocky Material

In recent years, I have met a view stating that the difference in lateral density causes continent to extend over ocean. They think that, since the continent’s density is greater than the ocean’s density, the lateral density difference yields a force that drives continent to extend over ocean. The issue raised by this view is vital because it relates to the principle of fluid mechanics, and hence deserves a wider discussion. First of all, the view contradicts the basic knowledge of fluid mechanics. As illustrated in Figure 1(a), we compare the hydrostatic pressure within a water body with the lithostatic pressure within a rock body. Given the two points, p1 and p2, are situated at the same depth of the water body and the rock body, respectively, the lithostatic pressure must be higher due to the rock’s greater density. Water pressure differs from rock pressure in that the former arises from the weight and movement of water molecules, whereas the latter arises solely from the weight of rocky materials. The rocky materials within continent are highly viscous and self-constrained, making it difficult for them to flow freely. Conversely, a low viscosity of water allows water particles to flow freely. This difference in physics property explains why a rock body can maintain its shape, while a water body conforms to the container holding it. Next, as illustrated in Figure 1(b), we put the water body and the rock body together. Since the water body performs more mobile than the rock body does, some of the water body tends to flow towards the rock body and therefore exerts pressure force on the rock body. Further, as illustrated in Figure 1(c), we put water into a container composed of rocky materials, the water exerts pressure force on the walls of the container. Second, I have asked experts regarding the question of whether or not ocean pushes continent. Several feedbacks support my argument above. There is a comment from Dr. John M. Cimbala: “Think about an empty tea cup sitting in a vacuum chamber with zero pressure. There is certainly internal pressure in the walls of the tea cup. However,

Figure 1. Conceptual model of the force between water and rock. (a) Pressure comparison between water body and rock body. (b) A lateral pressure force appears when the two bodies are put together. (c) Water pressure force exerted on the container’s walls. (d) Ocean water pressure force exerted on the walls of continents. (e) Water pressure force geometry between ocean and continent. F represents the water pressure force.

those walls do not exert any kind of pressure or force on the surroundings (which is a vacuum). And the cup stays the same shape and holds its shape regardless of its surroundings. Now take the cup out of the vacuum chamber and into the air. Now air exerts a pressure on the cup walls. The cup walls exert and equal and opposite pressure on the air due to Newton’s third law. Now fill the cup with water. The pressure inside the cup increases, and the cup expands ever so slightly, but it still maintains its shape. The cup exerts pressure on the water and vice-versa. But it is the water that causes this pressure, not the cup. Water is a liquid and cannot maintain its shape unless it is in some kind of container. That is where the pressure comes from.” As illustrated in Figure 1(d), the continents are staying in the uppermost part of the Earth’s crust. The formation of the Earth’s crust was earlier than the appearance of liquid water, because the crust provides place for liquid water to be hold. The loading of liquid water onto the Earth’s crust is just like that water is filled into the tea cup. Therefore, when ocean basins are treated as containers, the ocean water must exert pressure force on the walls of the continents. There is another comment from Dr. Chris Hughes: “The ocean does indeed push on the continents, and they push back with an equal pressure. The force that makes things move is the gradient of the stress tensor, which in the ocean is effectively the pressure gradient, but in the continents is a combination of pressure and response to internal strains. Without those extra forces, the continents would simply be another, denser fluid, and would flow beneath the ocean.” There is third comment from Dr. Gerald Shubert: “If you imagine the edge of a continent to be a vertical plane then the water pressure will push against the continent. But the situation is more complex than that. There are continental shelves that extend under the oceans.” As illustrated in Figure 1(e), the edge of a continent includes the continental shelf and slope, and they all are situated under the water. If we project them horizontally, a vertical plane can be reached. The ocean water pressure forces exerted on the continental shelf and slope can be geometrically decomposed into horizontal and vertical forces. As a result, all horizontal forces are laterally exerted on that vertical plane. Last, discussing the topic of force has to involve two objects: the object exerting force and the object receiving the force. For one object using force to push another object, the first object has to move (i.e., change its position) to exert force on the second object. As ocean water moves more easily relative to continent, there can be push force from ocean to continent.

3. Ocean-Generated Force

As shown in Figure 2(top), according to the principle of fluid mechanics [7], the water pressure force generated on the wall of a cubic container can be expressed as F = PS = ρgy²x/2, where P and S are the pressure and wall area, g and ρ are the gravitational acceleration and water density, respectively, and x and y are the water width and depth, respectively, in the container. Ocean basins are naturally gigantic containers, and their depths reach more than a few kilometers and vary from one place to another. Consequently, the application of ocean water pressure against the continent’s walls, which are also the sides of ocean basins, can yield enormous forces on continents. Geometrically, as shown in Figure 2(middle), ocean water pressure is exerted vertically to the continental slope, by which a normal force is formed. This normal force is called the ocean-generated force, denoted as F_R on the right and F_L on the left. The horizontal force decomposed from this normal force is denoted as ${F^{\prime }}_{\text{R}}\left({F^{\prime }}_{\text{L}}\right)$ on the right (left), while the vertical force decomposed from this normal force is denoted as ${F^{″}}_{\text{R}}\left({F^{″}}_{\text{L}}\right)$ on the right (left). In general, the horizontal force on the wall of continent can be approximately written as

$F=0.5\rho gL{h}^2$ (1)

where ρ, g, L, and h are the water density, gravitational acceleration, ocean width that fits the continent wall’s width, and ocean depth, respectively. And Lh denotes the area of the continent wall.

Practically, continent appears more like a broad cylinder standing in the ocean, and its wall is not flat. We imaginarily project a continent into a polygonal column along the horizontal direction, then dissect the whole wall of this column into smaller rectangular walls connecting one to another, and finally calculate the horizontal force generated on each of these rectangular walls (Figure 2(middle)). We now design many geographical sites (e.g. 1, 2, 3, 4, etc.) along the edges of continents. The latitudes and longitudes of these sites are extracted from the ETOPO1 Global Relief Model (Figure 2(bottom)). A geographical distance between two adjacent sites is calculated using their respective latitudes and longitudes through a spherical geometry, and treated as the width of a smaller rectangular wall. The geographical distance of two sites is expressed as

$L=R\ast \text{arcos}\left(\sin \left(q_j\right)\ast \sin \left(q_j{}_{+1}\right)+\cos \left(q_j\right)\ast \cos \left(q_j{}_{+1}\right)\ast \cos \left(d_j{}_{+1}-d_j\right)\right)$ (2)

where R is the Earth’s radius and R = 6371 km, q_j and d_j are the latitude and longitude of a site, while q_j+1 and d_j+1 are the latitude and longitude of another site. Using the NOAA bathymetric data viewer, we estimate the corresponding ocean depth of each rectangular wall. ρ = 1000 kg/m³, and g = 9.8 m/s. We then use the expression above and these parameters to work out the horizontal forces around continents. It is assumed that the horizontal force exerts on a hypothetical geometric center of the wall, with the direction of the force being orthogonal to the direction of the wall. We use the direction of a side that connects two sites to represent the direction of the wall. Using this direction, the inclination of the force to latitude is determined. The latitude and longitude of the geometric center are established as the average of the latitudes and longitudes of two adjacent sites. The horizontal forces and the parameters for calculating them have been listed in Table 1.

4. The Resultant Stress from Ocean-Generated Force

As exhibited in Figure 2, the ocean-generated forces uniformly exert inwards the continents. The continents are situated on the top of continental plates, and therefore, the forces exerted on the plates have inevitably related to the stress in the continents. Our understanding of the stress yielded by ocean-generated force begins with a dynamic distribution around a straight continental plate (Figure 3(a)). We assume that Plate A moves towards the left, Plate B exerts a collisional force (F_c) on its left side, the oceanic ridge exerts a push force (F_RF) on its right side, and the asthenosphere exerts a friction force (F_b) on its base, and ocean water exerts pressure forces (F_LW and F_RW) at the two sides of its top. Ridge push force is widely considered either as a boundary force or a body force. As a boundary force, it is derived from a “gravity wedging” effect of a warm, buoyant mantle upwelling and acts at the edge of the lithospheric plate. Turcotte and Schubert [9] outlined the ridge push force through Figure 3(a) and expressed it as F_RP = F₁-F₂-F₃, where F₁ = F₅, F₂ = F₄. Since F₃, F₄, and F₅ relate to the pressure that linearly increases with depth, it may be expected that the minimal ridge push force appears at the uppermost part of the oceanic ridge, whereas the maximal ridge push force appears at the lowermost part. Ocean tides represent the daily regular alternations of high and low water in the oceans. The tides cause the water depth to vary, and thereby, the ocean water pressure varies timely. In consideration of this present status, we demonstrate the resultant stress by means of three combinations of different forces. In reality, the Earth’s surface is curved, and the vertical dimension of a continental plate is far less than its horizontal dimension, making it challenging

Figure 2. Top, conceptual model the ocean-generated forces acting on a straight continent. Middle, conceptual model of projecting continent into polygonal column and decomposing column into rectangular walls. Bottom, geographic treatment of the control sites and horizontal forces on the continents. F (yellow arrows) denotes the horizontal force. The red lines and black dots denote the ocean depths and control sites, respectively; the red dots denote the geometric centers of the walls. The background map is from ETOPO1 Global Relief Model [8]. Note that the ocean depths were artificially resolved through the NOAA bathymetric data viewer.

Table 1. (a) The horizontal forces and the parameters for calculating them. (b) The horizontal forces and the parameters for calculating them (continued). (c) The horizontal forces and the parameters for calculating them (continued). (d) The horizontal forces and the parameters for calculating them (continued).

(a)
Control site			Side
			i	Length	Hypothetical geometric center		Ocean depth	Horizontal force
j	dj	qj	Li	αi	βi	hi-ocean	Fi	Inclination to latitude, east
	Longitude	Latitude	km	Longitude	Latitude	m	N (*1017)	Degrees (°)
1	194.09˚	54.23˚	1	1747.90	204.23˚	60.39˚	4800	1.9733	140.74˚
2	214.37˚	66.55˚	2	1071.92	220.79˚	62.71˚	4800	1.2102	127.52˚
3	227.21˚	58.86˚	3	1190.03	232.02˚	54.29˚	3500	0.7143	121.53˚
4	236.82˚	49.71˚	4	931.50	236.29˚	45.54˚	3500	0.5591	354.92˚
5	235.76˚	41.36˚	5	921.16	238.93˚	38.05˚	4600	0.9551	36.94˚
6	242.09˚	34.73˚	6	1319.66	246.74˚	30.35˚	4600	1.3683	42.46˚
7	251.39˚	25.96˚	7	662.22	253.02˚	23.38˚	4000	0.5192	30.03˚
8	254.64˚	20.80˚	8	1092.31	259.13˚	18.34˚	3700	0.7327	59.98˚
9	263.61˚	15.88˚
10	280.94˚	25.22˚	9	700.05	279.36˚	28.05˚	5500	1.0377	206.26˚
11	277.78˚	30.87˚	10	1125.90	281.21˚	35.10˚	5200	1.4918	146.46˚
12	284.63˚	39.32˚	11	1126.57	290.39˚	42.06˚	5200	1.4927	122.57˚
13	296.15˚	44.79˚	12	1032.68	300.23˚	48.58˚	5200	1.3683	144.54˚
14	304.31˚	52.37˚	13	1254.00	299.14˚	57.29˚	3000	0.5530	209.60˚
15	293.97˚	62.21˚	14	701.87	300.48˚	63.47˚	4700	0.7597	113.29˚
16	306.98˚	64.72˚	15	708.93	311.51˚	62.32˚	3000	0.3126	41.23˚
17	316.04˚	59.91˚	16	870.41	320.23˚	63.36˚	2600	0.2883	151.37˚
18	324.42˚	66.80˚	17	662.56	330.75˚	68.72˚	800	0.0208	129.72˚
19	337.08˚	70.63˚	18	1360.30	337.57˚	76.75˚	3200	0.6825	178.95˚
20	338.06˚	82.87˚	19	275.43	327.87˚	83.08˚	2800	0.1058	260.40˚
21	317.67˚	83.29˚	20	198.31	311.52˚	82.83˚	2500	0.0607	239.16˚
22	305.37˚	82.37˚	21	231.66	297.64˚	82.75˚	300	0.0010	249.11˚
23	289.90˚	83.12˚	22	238.55	281.92˚	82.74˚	2200	0.0566	249.47˚
24	273.94˚	82.36˚	23	336.45	267.96˚	81.15˚	2200	0.0798	322.66˚
25	261.98˚	79.94˚	24	294.67	255.30˚	79.43˚	2200	0.0699	292.70˚
26	248.62˚	78.91˚	25	313.81	244.23˚	77.84˚	2800	0.1206	319.08˚
27	239.83˚	76.77˚	26	997.49	229.64˚	73.31˚	3500	0.5987	319.52˚
28	219.44˚	69.84˚	27	523.95	212.59˚	70.37˚	3500	0.3145	257.09˚
29	205.73˚	70.90˚	28	753.32	195.39˚	72.17˚	3500	0.4522	291.66˚
30	185.05˚	73.44˚	29	1601.37	171.87˚	68.03˚	1500	0.1766	317.22˚
31	158.68˚	62.61˚	30	2225.18	176.39˚	58.42˚	50	0.0003	293.80˚

Notes: all geographic sites refer to Figure 2.

(b)
Control site			Side
			i	Length	Hypothetical geometric center		Ocean depth	Horizontal force
j	dj	qj	Li	αi	βi	hi-ocean	Fi	Inclination to latitude, east
	Longitude	Latitude	km	Longitude	Latitude	m	N (*1017)	Degrees
32	278.96˚	−2.20˚	31	1290.88	281.25˚	−7.55˚	4800	1.4573	22.92˚
33	283.53˚	−12.90˚	32	1163.08	286.57˚	−17.26˚	4600	1.2059	33.59˚
34	289.60˚	−21.62˚	33	1737.42	287.89˚	−29.30˚	4600	1.8014	349.01˚
35	286.17˚	−36.97˚	34	1405.78	284.78˚	−43.22˚	4300	1.2737	350.81˚
36	283.39˚	−49.46˚	35	876.16	287.57˚	−52.48˚	4700	0.9484	40.11˚
37	291.74˚	−55.50˚	36	977.06	291.92˚	−51.11˚	1800	0.1551	181.43˚
38	292.09˚	−46.71˚	37	731.33	294.03˚	−43.73˚	1800	0.1161	154.87˚
39	295.96˚	−40.75˚	38	1014.55	300.13˚	−37.59˚	6000	1.7897	133.75˚
40	304.29˚	−34.43˚	39	1161.02	307.71˚	−30.12˚	6000	2.0480	145.55˚
41	311.13˚	−25.81˚	40	1086.71	315.40˚	−22.90˚	5200	1.4398	126.58˚
42	319.66˚	−19.98˚	41	1571.58	322.47˚	−13.46˚	4600	1.6295	157.31˚
43	325.28˚	−6.93˚	42	1744.37	318.22˚	−3.48˚	4300	1.5804	243.89˚
44	311.15˚	−0.02˚	43	1602.67	305.40˚	4.36˚	4200	1.3853	232.65˚
45	299.64˚	8.73˚	44	292.31	300.11˚	7.50˚	4800	0.3300	200.75˚
46	300.58˚	6.27˚	45	2576.75	289.77˚	2.04˚	3700	1.7285	291.42˚
47	353.22˚	34.24˚	46	908.04	350.19˚	31.09˚	5000	1.1123	320.52˚
48	347.15˚	27.93˚	47	1462.59	345.17˚	21.61˚	5000	1.7917	343.79˚
49	343.19˚	15.29˚	48	1482.74	347.41˚	10.07˚	4800	1.6739	38.46˚
50	351.63˚	4.84˚	49	1689.12	359.25˚	5.29˚	4600	1.7514	93.36˚
51	6.87˚	5.73˚	50	898.57	8.32˚	1.96˚	4600	0.9317	20.99˚
52	9.77˚	−1.82˚	51	1051.19	11.94˚	−6.03˚	5500	1.5581	27.12˚
53	14.11˚	−10.24˚	52	887.15	13.10˚	−14.11˚	5500	1.3150	345.81˚
54	12.09˚	−17.98˚	53	1157.53	14.03˚	−22.88˚	5000	1.4180	19.99˚
55	15.96˚	−27.77˚	54	779.24	17.82˚	−30.90˚	5000	0.9546	26.98˚
56	19.67˚	−34.02˚	55	669.59	23.27˚	−33.69˚	3000	0.2953	83.72˚
57	26.87˚	−33.36˚	56	1010.77	29.68˚	−29.52˚	4600	1.0480	147.57˚
58	32.48˚	−25.68˚	57	1416.91	36.40˚	−20.46˚	4000	1.1109	144.93˚
59	40.31˚	−15.24˚	58	1063.87	39.56˚	−10.51˚	3400	0.6026	188.85˚
60	38.81˚	−5.78˚	59	880.15	41.14˚	−2.58˚	4600	0.9126	144.02˚
61	43.47˚	0.63˚	60	1479.88	47.38˚	6.04˚	4600	1.5344	144.34˚
62	51.29˚	11.45˚

Notes: all geographic sites refer to Figure 2.

(c)
Control site			Side
			i	Length	Hypothetical geometric center		Ocean depth	Horizontal force
j	dj	qj	Li	αi	βi	hi-ocean	Fi	Inclination to latitude, east
	Longitude	Latitude	km	Longitude	Latitude	m	N (*1017)	Degrees
69	95.41˚	16.55˚	67	1802.58	97.12˚	8.62˚	5000	2.2082	11.96˚
70	98.82˚	0.68˚	68	1806.5	105.79˚	−3.54˚	5200	2.3935	60.27˚
71	112.76˚	−7.75˚	69	2151.54	122.43˚	−6.51˚	100	0.0011	97.36˚
72	132.09˚	−5.27˚	70	1075.97	130.03˚	−0.89˚	5200	1.4256	204.11˚
73	127.96˚	3.49˚	71	485.05	127.35˚	5.59˚	5200	0.6427	196.28˚
74	126.73˚	7.68˚	72	435.29	126.29˚	9.59˚	5200	0.5767	167.20˚
75	125.85˚	11.5˚	73	947.45	123.79˚	15.27˚	5200	1.2553	152.16˚
76	121.72˚	19.04˚	74	554.03	121.85˚	21.53˚	5200	0.7341	177.22˚
77	121.98˚	24.02˚	75	614.22	124.74˚	25.22˚	5200	0.8138	115.61
78	127.49˚	26.41˚	76	1399.63	132.94˚	30.63˚	5200	1.8545	131.96˚
79	138.39˚	34.84˚	77	1133.3	141.69˚	39.26˚	5600	1.7415	150.02˚
80	144.98˚	43.68˚	78	954.18	149.84˚	46.38˚	5600	1.4662	128.80˚
81	154.69˚	49.07˚	79	1196.64	159.22˚	53.76˚	5600	1.8388	150.27˚
82	163.74˚	58.44˚	80	1136.53	161.21˚	60.53˚	3800	0.8042	266.56˚
83	158.68˚	62.61˚	81	1393.86	143.60˚	67.44˚	1500	0.1537	254.05˚
84	128.51˚	72.26˚	82	603.43	120.52˚	73.84˚	2600	0.1999	234.78˚
85	112.52˚	75.42˚	83	469.11	104.03˚	75.62˚	2600	0.1554	275.25˚
86	95.54˚	75.81˚	84	440.81	89.21˚	74.73˚	3300	0.2352	302.95˚
87	82.88˚	73.64˚	85	490.57	75.85˚	72.86˚	3300	0.2618	290.54˚
88	68.82˚	72.08˚	86	529.37	63.55˚	70.47˚	3800	0.3746	312.33˚
89	58.27˚	68.86˚	87	603.56	51.38˚	68.00˚	3800	0.4271	288.45˚
90	44.49˚	67.13˚	88	546.62	38.16˚	67.67˚	3800	0.3868	257.5˚
91	31.83˚	68.20˚	89	753.91	23.04˚	67.60˚	3800	0.5334	280.17˚
92	14.25˚	66.99˚	90	733.5	9.68˚	64.34˚	300	0.0032	323.21˚
93	5.11˚	61.69˚	91	926.31	6.17˚	57.56˚	5100	1.1806	7.80˚
94	7.22˚	53.43˚	92	708.24	183.36˚	51.35˚	3000	0.3123	310.79˚
95	359.49˚	49.26˚	93	591.35	359.14˚	46.61˚	5100	0.7537	5.18˚
96	358.79˚	43.96˚	94	761.67	354.04˚	43.83˚	5100	0.9707	267.83˚
97	349.29˚	43.70˚	95	1104.32	351.26˚	38.97˚	5100	1.4074	17.89˚
98	133.29˚	−38.42˚	96	1032.06	128.33˚	−36.08˚	5200	1.3674	120.28˚
99	123.36˚	−33.73˚	97	774.89	127.27˚	−32.59˚	5200	1.0267	109.16˚
100	131.18˚	−31.44˚	98	1089.81	135.80˚	−34.55˚	5200	1.444	50.76˚
101	140.41˚	−37.65˚	99	875.86	143.60˚	−40.77˚	5200	1.1605	37.70˚
102	146.78˚	−43.89˚	100	958.73	148.49˚	−39.78˚	4500	0.9513	162.28˚
103	150.20˚	−35.67˚	101	877.51	152.09˚	−32.06˚	4500	0.8707	156.08˚
104	153.98˚	−28.45˚	102	943.16	151.56˚	−24.82˚	3200	0.4732	211.22˚

Notes: all geographic sites refer to Figure 2.

(d)
Control site			Side
			i	Length	Hypothetical geometric center		Ocean depth	Horizontal force
j	dj	qj	Li	αi	βi	hi-ocean	Fi	Inclination to latitude, east
	Longitude	Latitude	km	Longitude	Latitude	m	N (*1017)	Degrees
105	149.13˚	−21.19˚	103	1359.76	145.75˚	−16.01˚	3200	0.6823	212.08˚
106	142.36˚	−10.82˚	104	802.72	141.57˚	−14.35˚	100	0.0004	347.77˚
107	140.78˚	−17.88˚	105	1216.8	135.51˚	−15.81˚	100	0.0006	247.79˚
108	130.24˚	−13.74˚	106	1109.61	125.89˚	−16.48˚	100	0.0005	303.28˚
109	121.53˚	−19.22˚	107	864.63	117.80˚	−20.95˚	4500	0.8579	296.31˚
110	114.06˚	−22.67˚	108	2529.37	123.68˚	−30.55˚	5000	3.0985	46.39˚
111	188.47˚	−78.56˚	109	391.5	196.72˚	−78.09˚	4200	0.3384	285.48˚
112	204.97˚	−77.61˚	110	519.06	213.35˚	−76.35˚	4200	0.4487	302.42˚
113	221.73˚	−75.08	111	461.58	229.17˚	−74.47˚	4200	0.399	286.94˚
114	236.61˚	−73.86˚	112	401.97	243.06˚	−74.38˚	4500	0.3989	253.39˚
115	249.51˚	−74.90˚	113	780.47	261.99˚	−74.03˚	4500	0.7744	284.02˚
116	274.46˚	−73.16˚	114	477.75	281.74˚	−72.91˚	4500	0.4741	263.24˚
117	289.02˚	−72.65˚	115	1595.2	298.54˚	−66.48˚	4500	1.5828	328.21˚
118	308.06˚	−60.30˚	116	1702.14	305.75˚	−67.92˚	5000	2.0851	186.49˚
119	303.40˚	−75.53˚	117	418.96	309.48˚	−76.83˚	5000	0.5132	226.90˚
120	315.53˚	−78.12˚	118	850.21	327.66˚	−75.67˚	5000	1.0415	308.82˚
121	339.79˚	−73.22˚	119	644.56	347.35˚	−71.56˚	5000	0.7896	304.72˚
122	354.90˚	−69.89˚	120	919.54	186.86˚	−69.65˚	5000	1.1264	266.74˚
123	18.81˚	−69.41˚	121	674.24	26.90˚	−68.64˚	5000	0.8259	284.57˚
124	34.98˚	−67.87˚	122	819.5	44.05˚	−66.87˚	5000	1.0039	285.64˚
125	53.12˚	−65.86˚	123	838.69	62.26˚	−67.22˚	5000	1.0274	249.11˚
126	71.40˚	−68.58˚	124	983.34	82.48˚	−67.32˚	5000	1.2046	286.27˚
127	93.55˚	−66.06˚	125	675.92	101.11˚	−66.29˚	4500	0.6707	265.79˚
128	108.67˚	−66.51˚	126	908.21	119.04˚	−66.82˚	4500	0.9012	267.30˚
129	129.41˚	−66.90˚	127	654.62	136.97˚	−67.01˚	4500	0.6495	267.88˚
130	144.52˚	−67.12˚	128	1098.65	157.08˚	−69.49˚	3900	0.8188	242.07˚
131	169.63˚	−71.85˚	129	631.15	166.64˚	−74.58˚	3300	0.3368	163.76˚
132	163.65˚	−77.31˚	130	645.18	177.36˚	−78.01˚	3300	0.3443	256.53˚
133	191.07˚	−78.70˚

Notes: all geographic sites refer to Figure 2.

Figure 3. Conceptual model of the geodynamics around a continental plate (a) and simplified model for determining stress (b). F_RP, F_b, F_c, and F_LW(F_RW) denote ridge push, basal friction, collisional, and ocean-generated forces.

to represent stress with this geometry. To overcome this shortage, we develop a simplified model (see Figure 3(b)) consisting of rocks that are straight. The model assumes homogeneity and isotropy and has a length and thickness of 290 km and 100 km, respectively.

4.1. Combined A

The ridge push force (F_RF), the collisional force (F_c), and the basal friction force (F_b) are included alone. This combination follows previous studies [10]-[12]. Along the horizontal direction, the ridge push force (F_RF) is driving and its amplitude increases with depth; the collisional force (F_c) is uniformly exerted on its left side; the mantle exerts a frictional force (F_b) on its base, and this force is resistive. These forces realize a force balance for the plate. Along the vertical direction, the model’s weight is balanced out by the supporting from the mantle. We employ finite element analysis software (i.e., Abaqus) to calculate the stresses yielded by these forces. The model’s bottom is given a remote boundary condition. As the upper part of the lithospheric plate is brittle and elastic, whereas the lower part is ductile and plastic, we assume that the physical property of the rock is vertically transited from elasticity to plasticity. The inputs include the vertical pressure yielded by the rock’s weight and the lateral pressures yielded by these forces (F_RF, F_b, and F_c). The outputs include the stresses caused by the vertical pressure alone and the stresses caused by a combination of the vertical and lateral pressures. The two-dimensional frame allows us to obtain a horizontal stresses (S11) and a vertical stresses (S22). We here only discuss the horizontal stress (S11). The elastic modulus, Poisson ratio, and rock density of the model are set to 100,000 MPa, 0.3, and 2690 kg/m³, respectively. The vertical pressure caused by the rock’s weight yields Set I data of stress; The F_RF is given as 4.0 × 10¹² N m⁻¹, which is generally accepted by scientific community [9]. It is assumed that F_b and F_c are 80% and 20% of F_RF, respectively. The pressures caused by a combination of these forces yield Set II data of stress; In order to test the stress variation when the resistive forces are moderately changed, we again assume that F_b and F_c are 50% and 50% of F_RF, respectively. The pressures caused by a combination of these revised forces yield Set III data of stress. A detailed description of these forces for different sets is listed in Table 2.

Table 2. Different forces exerted on the model.

Combination	No.	Loads (*1012 N/m)
		FRP	Fb	Fc	FRW	FLW
1	Set I	-	-	-	-	-
	Set II	4.00	3.20	0.80	-	-
	Set III	4.00	2.00	2.00	-	-
	Set II'	200.00	160.00	40.00	-	-
	Set III'	200.00	100.00	100.00	-	-
2	Set A	4.00	3.21	0.87	0.12	0.04
	Set B	4.00	2.00	2.08	0.12	0.04
	Set C	0.04	0.03	0.09	0.12	0.04
	Set D	0.04	0.02	0.10	0.12	0.04
	Set A'	200.00	160.18	43.74	6.13	2.21
	Set B'	200.00	100.00	103.92	6.13	2.21
	Set C'	2.00	1.60	4.32	6.13	2.21
	Set D'	2.00	1.00	4.92	6.13	2.21

To realize a more accurate understanding of the resultant stress, we select a rectangular area GHIJ to exhibit. The stress clouds of this area are compared in Figure 4(left). Please note that any of these forces (F_RF, F_b, and F_c) is too small with respect to the rock’s weight. For instance, when F_RF = 4.0 × 10¹² N m⁻¹ is applied to the model’s right side (which is 85 km length), its resultant mean pressure is 47.06 MPa, while the mean lithostatic pressure of the rock in the model (which is 100 km depth) is 1318.1 MPa. This means that, if we use stress cloud to compare the stress caused by a combination of the rock’s weight and these forces with the stress caused by the rock’s weight alone, the two are indistinguishable. To create a visual impression, we magnify these forces (F_RF, F_b, and F_c) 50 times, which yields Set II’ data of stress and Set III’ data of stress. Clearly, we find that the horizontal stresses caused by these forces are compressional and mainly concentrated on the lower part of section GHIJ. Three sections (M₁N₁, M₂N₂, and M₃N₃) in the rectangular area are extracted to quantify the comparison. Each section keeps a span of 50 km relative to one another. The stress diagrams for these sections are compared in Figure 4(right). After subtracting the stresses caused by the rock’s weight from the stresses caused by a combination of the rock’s weight and these forces, we obtain the stresses caused by these forces, which is exhibited in Set II (III) - Set I.

Kusznir and Bott [13] argued that, due to the ductile nature of the lower part of the lithosphere, there would be a redistribution of any stress applied to the whole lithosphere that would result in stress amplification in the upper brittle part of the lithosphere. This view is based on the assumption that force is uniformly exerted on the side of the lithospheric plate, but reality is that the ridge push force increases with depth; consequently, the redistribution of the resultant stress and its amplification are not applicable. In contrast, we have considered this ductile nature in the modelling, but no evidence was found for stress amplification in the upper part of section GHIJ. Our modelling suggests that the stresses caused by a combination of the ridge push, collisional, and basal friction forces are mainly concentrated on the lower part of the lithosphere, this feature doesn’t accord with the observed stresses that are mainly concentrated on the uppermost brittle part of the lithosphere (which is ~40 km in depth) [10] [14] [15].

4.2. Combined B

The ocean-generated forces (F_LW and F_RW), the ridge push force (F_RF), the collisional force (F_c), and the basal friction force (F_b) are included. The inputs include the vertical pressure caused by the rock’s weight and the lateral pressures caused by these forces (F_RW, F_LW, F_RF, F_c, and F_b). The ocean-generated forces F_LW and F_RW correspond to 5 km water depth at the right and 3 km water depth at the left, respectively, and F_RW = 0.12 × 10¹² N m⁻¹, F_LW = 0.04 × 10¹² N m⁻¹. The outputs include the stresses yielded by the vertical pressure alone and the stresses yielded by a combination of the vertical and lateral pressures. Similarly, we only discuss the horizontal stress (S11). At this time, we first use these forces to yield Set A data

Figure 4. Stress clouds (left) and diagrams (right) produced by the rock’s weight, ridge push, basal friction, and collisional forces. “−” denotes the stresses are compressional, whereas “+” denotes the stresses are tensional.

Figure 5. Stress clouds (left) and diagrams (right) produced by the ocean-generated force, ridge push force, basal friction, and collisional forces. “−” denotes compressional stress, whereas “+” denotes tensional stress.

of stress and Set B data of stress. The stress clouds of the area GHIJ are compared in Figure 5. To realize a visual impression, we magnify these forces (F_RW, F_LW, F_RF, F_c, and F_b) 50 times, which yields Set A’ data of stress and Set B’ data of stress. We find that the horizontal stresses caused by these forces are compressional and tend to distribute across the middle part of section GHIJ. We then minify F_RF and F_b 100 times, remain F_RW and F_LW stable, and adjust F_c properly so as to sustain the horizontal force balance, this yields Set C data of stress and Set D data of stress. To realize a visual impression, we again magnify these revised forces 50 times, which yields Set C’ data of stress and Set D’ data of stress. A detailed description of these forces for different sets is listed in Table 2. It can be found that, after F_RF and F_b are reduced, the horizontal stresses caused by these forces are mostly concentrated on the upper part of section GHIJ. The stress diagrams for three sections (i.e., M₁N₁, M₂N₂, and M₃N₃) are counted and compared in Figure 6. After subtracting the stresses caused by the rock’s weight, we obtain the stresses caused by these forces, which are exhibited with Set A/B/C/D - Set I. We find that the stresses caused by a combination of the ocean-generated, ridge push (when its amplitude is lowered to be less than that of the ocean-generated force), collisional, and basal friction forces are mostly concentrated on the upper part of the lithosphere. This distribution feature accords with that of observed stresses [10] [14] [15].

4.3. Combined C

The ocean-generated forces (F_LW and F_RW) including the effect of tide, the ridge push force (F_RF), the collisional force (F_c), and the basal friction force (F_b) are included. The inputs include the vertical pressure caused by the rock’s weight and the lateral pressures caused by these forces (F_RW, F_LW, F_RP, F_b, and F_c). Since tide represents a periodic oscillation, we design a water level variation of totally 12 hours, which corresponds to a semidiurnal tide. The information of tidal height and these forces is listed in Table 3. The outputs include the stresses produced by the vertical pressure alone and the stresses produced by a combination of the vertical and lateral pressures. Using the latter to subtract the former, we obtain the stresses produced by the lateral pressures. Similarly, we only discuss the horizontal stress (S11). At this time, we collect the results of 6 locations (①, ②, ③, ④, ⑤, and ⑥), as shown in Figure 3(b), to do comparison. These locations belong to the 30 km depth and 60 km depth of three sections (M₁N₁, M₂N₂, and M₃N₃) in the model. The stress diagrams for these locations are compared in Figure 7. The stresses exhibited in Figure 7(A) are that produced by the vertical pressure alone, the stresses exhibited in Figure 7(B) are that produced by a combination of the vertical and lateral pressures, and the stresses exhibited in Figure 7(B-A) is that produced by the lateral pressures.

As exhibited in Set C(D) - Set I of Figure 6, the horizontal stresses yielded by the ocean-generated force may have penetrated a 50 km depth crust, which represents mostly continent, and their amplitude is approximately 2.0 - 6.0 MPa, which is entirely comparable to the range of earthquake stress drops (1 - 30 Mpa) [16]. Our model assumes homogenous and isotropic rocks in a straight configuration, whereas the actual Earth’s crust is spatially curved, and the crustal rocks within it are inhomogeneous and anisotropic. Most importantly, ocean water has been exerting on the Earth’s crust for more than 4 billion years [4]-[6], and based on the elastic rebound theory for earthquakes [17], stress is time-dependent and tends to accumulate gradually. From these points, we expect that the actual stress in the Earth’s crust due to ocean-generated force may have been higher than that we have modelled here. It is important to note that our modeling is not yet capable of quantitatively comparing with the observed stress. This limitation stems from two primary factors. Firstly, as highlighted by Morawietz et al. [18], while the World Stress Map project [14] [19] has consistently released data, these primarily encompass stress orientations and patterns (i.e., compressional and extensional),

Figure 6. Stress diagram produced by the ocean-generated, ridge push force, basal friction, and collisional forces. “−” denotes compressional stress, whereas “+” denotes tensional stress.

excluding the magnitude of stress. Secondly, the locally acquired stress magnitude data in Germany and neighboring regions are confined to a shallow depth (less

Figure 7. The stress variation due to tide. Note, the stress variations in A and B are too small to be perceptible.

than 7.0 km beneath the Earth’s surface). Within this depth, the stresses generated are significantly influenced by the inhomogeneous and anisotropic characteristics of rock materials, which cannot be fully captured by our model. Even so, our model is providing a new starting point for scientific community to address this intricate issue.

Table 3. Information of tidal height and related forces around the model.

Time (h)	Tidal height (m)	Loads (×1011 N/m)
		FLW	FRW	FRP	Fb	Fc
0	0.0	0.4410000000	1.2250000000	0.4000000000	0.2000000000	1.0000000000
1	0.4	0.4411176078	1.2251960078	0.4000000000	0.2000000000	1.0000000000
2	0.8	0.4412352314	1.2253920314	0.4000000000	0.2000000000	1.0000000000
3	1.2	0.4413528706	1.2255880706	0.4000000000	0.2000000000	1.0000000000
4	0.8	0.4412352314	1.2253920314	0.4000000000	0.2000000000	1.0000000000
5	0.4	0.4411176078	1.2251960078	0.4000000000	0.2000000000	1.0000000000
6	0.0	0.4410000000	1.2250000000	0.4000000000	0.2000000000	1.0000000000
7	−0.4	0.4408824078	1.2248040078	0.4000000000	0.2000000000	1.0000000000
8	−0.8	0.4407648314	1.2246080314	0.4000000000	0.2000000000	1.0000000000
9	−1.2	0.4406472706	1.2244120706	0.4000000000	0.2000000000	1.0000000000
10	−0.8	0.4407648314	1.2246080314	0.4000000000	0.2000000000	1.0000000000
11	−0.4	0.4408824078	1.2248040078	0.4000000000	0.2000000000	1.0000000000

5. Discussion

The coupling of water and earthquake occurrence is a highly active area of research, with reservoir-induced earthquakes being the most notable artificial seismicity events. It has been extensively reported that there have been hundreds of instances, including at least five sites, where earthquakes exceeding M 6 have occurred [20], indicating that earthquakes near reservoirs are widespread [21] [22]. The primary process for reservoir-induced earthquakes is a stress change within rocks due to the water loading [21] [23], and the frequencies of these earthquakes closely correlate with the loading and unloading rate of water [24] [25]. These authors [26]-[28] have theorized/modelled the link between seismicity and stress caused by reservoir water impoundment. In addition to these reservoir-induced earthquakes, ocean water-related earthquakes were also recently reported. Guillas et al. [29] identified a connection between the El Niño-Southern Oscillation (ENSO) and earthquakes on the East Pacific Rise (EPR), proposing that seismicity may increase due to a reduction in ocean-bottom pressure over the EPR. Martínez-Garzón et al. [30] discovered that changes in sea level affect seismicity rates in a hydrothermal system near Istanbul. Tanaka et al. [31] concluded that the most likely component to control the earthquake occurrence is the stress. As exhibited in Figure 7, the stress variation caused by the tide may have penetrated a 50 km depth crust. The result of our modelling, combined with these existing understanding of reservoir-induced and ocean water-related earthquakes, leads us to speculate that the ocean-generated force may potentially contribute to earthquake occurrence, although integrate data (i.e., seismic events and oceanic pressure measurements) to support this coupling are still waited.

Acknowledgements

We express sincere thanks to John M. Cimbala, Chris Hughes, and Gerald Shubert for their comments on the force between ocean and continent. The author declares no conflicts of interest. This research received no funding.

Data Availability Statement

The latitude and longitude of the controlling sites on the continental plates in Figure 2 are determined through the ETOPO1 Global Relief Model [8], and the ocean depths are artificially resolved through the NOAA Bathymetric Data Viewer (https://ngdc.noaa.gov/mgg/global/global.html).

Conflicts of Interest

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

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