{"title":"Laplace’s Equation in Cartesian Coordinates and Satellite Altimetry","authors":"D. Sandwell","doi":"10.1017/9781009024822.016","DOIUrl":"https://doi.org/10.1017/9781009024822.016","url":null,"abstract":"Here we are interested in anomalies due to local structure. Consider a patch on the Earth having a width and length less than about 1000 km or 1/40 of the circumference of the Earth. Within that patch we are interested in features as small as perhaps 1-km wavelength. Using a spherical harmonic representation would require 40,000 squared coefficients! To avoid this enormous computation and still achieve accurate results, we will treat the Earth as being locally flat. Here is a remove/restore approach that has worked well in our analysis of gravity and topography:","PeriodicalId":120442,"journal":{"name":"Advanced Geodynamics","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129847421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Brief Review of Elasticity","authors":"D. Sandwell","doi":"10.1017/9781009024822.007","DOIUrl":"https://doi.org/10.1017/9781009024822.007","url":null,"abstract":"This is a very brief review of the elasticity theory needed to understand the principles of stress, strain, and flexure in Geodynamics [Turcotte and Schubert, 2002]. This review assumes that you have already taken a class in continuum mechanics. One difference from T&S is that we follow the sign convention used by seismologists and engineers where extensional strain and stress is positive. Stress Stress is a force acting on an area is measured in Newtons per meter squared (N m –2) which corresponds to a Pascal unit (Pa). The following diagram shows a cube of solid material. Each face of the cube has three components of stress so there are 9 possible components of the stress tensor. We will consider only the symmetric part of the stress tensor so only 6 of these components are independent. The antisymmetric part of the tensor represents a torque. In Cartesian coordinates the stress tensor is given by σ ij = σ xx σ xy σ xz σ xy σ yy σ yz σ xz σ yz σ zz ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ where index notation is the shorthand for dealing with tensors and vectors; a variable with a single subscript is a vector a = a i , a variable with two subscripts is a tensor σ = σ ij , and a repeated index indicates summation over the spatial coordinates. For example the pressure is given by P = −σ ii / 3. In addition, a comma preceding a subscript means differentiation with respect to that variable ∇ a = a i, j or for example a x, y = ∂a x ∂y .","PeriodicalId":120442,"journal":{"name":"Advanced Geodynamics","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117098265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Cooling of the Oceanic Lithosphere","authors":"D. Sandwell","doi":"10.1017/9781009024822.006","DOIUrl":"https://doi.org/10.1017/9781009024822.006","url":null,"abstract":"","PeriodicalId":120442,"journal":{"name":"Advanced Geodynamics","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127447100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Heat Flow Paradox","authors":"C. Scholz","doi":"10.1017/9781009024822.012","DOIUrl":"https://doi.org/10.1017/9781009024822.012","url":null,"abstract":"Paradox-The seismogenic zone extends from the surface to a depth of about 10 km. According to Byerlee's law, the shear stress on the fault should be some large fraction of the hydrostatic stress. f static coefficient of friction ~ 0.60 ρ c crustal density 2600 kg m-3 ρ w water density 1000 kg m-3 g acceleration of gravity 9.8 m s-2 D depth of seismogenic zone 12 km This assumes that water percolates to 12 km depths to lower friction on the fault. We can compute the average shear stress on the fault. The observed stress drop during an earthquake ranges from 0.1 to 10 MPa with a typical value of 5 MPa which is about 10 times smaller than the average stress from Byerlee's Law. This implies that only a fraction of the total stress is released during an earthquake. The average stress during the earthquake times the earthquake displacement produces energy both as seismic radiation (small fraction) and as heat (large fraction). If this heat energy is averaged over many earthquake cycles, then this average heat/area generated on the fault plane will appear as a heat flow anomaly on the surface having a similar heat/area as along the fault. To calculate this heat anomaly for a variety of frictional heating models, first consider a line source of heat. τ ρ ρ () z f gz c w = − () (1) τ ρ ρ ρ ρ = − () − () ∫ 1 1 2 D f gzdz f gD c w o D c w = = 56 MPa (2)","PeriodicalId":120442,"journal":{"name":"Advanced Geodynamics","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130666870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Observations Related to Plate Tectonics","authors":"D. Sandwell","doi":"10.1017/9781009024822.002","DOIUrl":"https://doi.org/10.1017/9781009024822.002","url":null,"abstract":"It is useful to assess the global data sets that are most relevant to plate tectonics. Below are a series of global maps that help to confirm various aspects of plate tectonic theory. Plate boundaries are classified as ridges, transform faults, or subduction zones based on basic observations of topography (Figure 1) and seismicity (Figure 2). Remarkably, nearly all seafloor spreading ridges lie at a depth of 2500-3000 m below sea level which is the level of isostasy for hot thin lithosphere. Depths gradually increase away from the ridges because of cooling and thermal contraction so old ocean basins are commonly 4500-5000 m deep. Fracture zones and aseismic ridges also show up on these maps. Global seismicity (magnitude > 5.1 Figure 2) highlights the plate boundaries and reveals their tectonic style. Shallow normal-faulting earthquakes (< 30 km deep) are common along slow-spreading ridges but largely absent along faster-spreading ridges where the plates are too thin and weak to retain sufficient elastic energy to generate large earthquakes. Transform faults are characterized by relatively shallow (< 30 km) strike-slip earthquakes and they are common along both fast-and slow-spreading ridges. The deeper earthquakes (green and blue dots in Figure 2) occur only in subduction zones where sheets of seismicity (i.e., Benioff zones) are critical evidence that relatively cold lithosphere is subducting back into the mantle. But even convergent boundaries are characterized by shallow extensional earthquakes on the ocean side of the trenches. Some regions (e.g., Africa. Asia, western North America, Indian ocean) have distributed earthquake activity, indicating broad deformational zones. Topography and seismicity provide strong evidence for tectonic activity but little or no information on the rate of plate motion. Marine magnetic anomalies, combined with relative plate motion directions based on satellite altimeter measurements of fracture-zone trends, have been used to construct a global age map (Figure 3) of the relatively young (< 180 Myr) oceanic lithosphere. Finally the distribution of off-ridge volcanoes that have been active during the Quaternary mainly occur directly behind trenches where wet subducting slabs reach asthenospheric depths and trigger back-arc volcanism (Figure 4). A few active volcanoes occur in the interiors of the plates and in diffuse extensional plate boundaries. The geoid (Figure 5) shows little correlation–at long wavelengths– with surface tectonics and primarily reflects mass anomalies deep in the mantle. It is expected that the dynamic topography–the topography not due to crustal and near-surface variations–and the stress-state …","PeriodicalId":120442,"journal":{"name":"Advanced Geodynamics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115257838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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