Periodic topography
PROBLEM 4â20 A mountain range can be represented as a periodic topography with a wavelength of 100 km and an amplitude of 1.2 km. Heat flow in a valley is measured to be 46 mW mâ2. If the atmospheric gradi- ent is 6.5 K kmâ1 and k = 2.5 Wmâ1 Kâ1, determine what the heat flow would have been without topog- raphy; that is, make a topographic correction.
PROBLEM 4â29 Estimate the effects of variations in bottom water temperature on measurements of oceanic heat flow by using the model of a semi-infinite half-space subjected to periodic surface temperature fluctuations. Such water temperature variations at a specific location on the ocean floor can be due to, for example, the transport of water with variable tem- perature past the site by deep ocean currents. Find the amplitude of water temperature variations that cause surface heat flux variations of 40 mW mâ2 above and below the mean on a time scale of 1 day. As- sume that the thermal conductivity of sediments is 0.8 W mâ1 Kâ1 and the sediment thermal diffusivity is 0.2 mm2 sâ1.
PROBLEM 4â33 One way of determining the effects of erosion on subsurface temperatures is to consider the instantaneous removal of a thickness l of ground. Prior to the removal T = T0 + βy, where y is the depth, β is the geothermal gradient, and T0 is the surface temperature. After removal, the new surface is main- tained at temperature T0. Show that the subsurface temperature after the removal of the surface layer is given by
How is the surface heat flow affected by the removal of surface material?
PROBLEM 4â39 One of the estimates for the age of the Earth given by Lord Kelvin in the 1860s assumed that Earth was initially molten at a constant tem- perature Tm and that it subsequently cooled by con- duction with a constant surface temperature T0. The age of the Earth could then be determined from the present surface thermal gradient (dT/dy)0. Re- produce Kelvinâs result assuming Tm â T0 = 1700 K, c=1 kJ kgâ1 Kâ1, L =400 kJ kgâ1, κ =1 mm2 sâ1, and (dT/dy)0 = 25 K kmâ1 . In addition, determine the thickness of the solidified lithosphere. Note: Since the solidified layer is thin compared with the Earthâs radius, the curvature of the surface may be neglected.
PROBLEM4â43 Themantlerocksoftheasthenosphere from which the lithosphere forms are expected to contain a small amount of magma. If the mass frac- tion of magma is 0.05, determine the depth of the lithosphereâasthenosphere boundary for oceanic li- thosphere with an age of 60 Ma. Assume L = 400 kJ kgâ1,c=1kJkgâ1 Kâ1,Tm =1600K,T0 =275K,and κ = 1 mm2 sâ1.
PROBLEM 4â52 The ocean ridges are made up of a series of parallel segments connected by transform faults, as shown in Figure 1â13. Because of the dif- ference of age there is a vertical offset on the fracture zones. Assuming the theory just derived is applica- ble, what is the vertical offset (a) at the ridge crest and (b) 100 km from the ridge crest in Figure 4â46 (Ïm = 3300kgmâ3,κ=1mm2sâ1,αv =3Ã10â5 Kâ1,T1â T0=1300K,u=50mmyrâ1).
PROBLEM 5â7 What is the value of the acceleration of gravity at a distance b above the geoid at the equator (b ⪠a)?
PROBLEM 5â18 A volcanic plug of diameter 10 km has a gravity anomaly of 0.3 mm sâ2. Estimate the depth of the plug assuming that it can be modeled by a verti- cal cylinder whose top is at the surface. Assume that the plug has density of 3000 kg mâ3 and the rock it intrudes has a density of 2800 kg mâ3.