# Heat and gravity.

PROBLEM 4–20 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 m−2. If the atmospheric gradi- ent is 6.5 K km−1 and *k *= 2.5 Wm−1 K−1, determine what the heat flow would have been without topog- raphy; that is, make a topographic correction.

PROBLEM 4–29 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 m−2 above and below the mean on a time scale of 1 day. As- sume that the thermal conductivity of sediments is 0.8 W m−1 K−1 and the sediment thermal diffusivity is 0.2 mm2 s−1.

PROBLEM 4–33 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 *= *T*0 + β*y*, where *y *is the depth, β is the geothermal gradient, and *T*0 is the surface temperature. After removal, the new surface is main- tained at temperature *T*0. 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 4–39 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 *T*0. The age of the Earth could then be determined from the present surface thermal gradient (*dT*/*dy*)0. Re- produce Kelvin’s result assuming *Tm *− *T*0 = 1700 K, *c*=1 kJ kg−1 K−1, *L *=400 kJ kg−1, κ =1 mm2 s−1, and (*dT*/*dy*)0 = 25 K km−1 . In addition, determine the thickness of the solidified lithosphere. Note: Since the solidified layer is thin compared with the Earth’s radius, the curvature of the surface may be neglected.

PROBLEM4–43 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 lithosphere–asthenosphere boundary for oceanic li- thosphere with an age of 60 Ma. Assume *L *= 400 kJ kg−1,*c*=1kJkg−1 K−1,*Tm *=1600K,*T*0 =275K,and κ = 1 mm2 s−1.

PROBLEM 4–52 The ocean ridges are made up of a series of parallel segments connected by transform faults, as shown in Figure 1–13. 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 4–46 (ρ*m *= 3300kgm−3,κ=1mm2s−1,α*v *=3×10−5 K−1,*T*1− *T*0=1300K,*u*=50mmyr−1).

PROBLEM 5–7 What is the value of the acceleration of gravity at a distance *b *above the geoid at the equator (*b *≪ *a*)?

PROBLEM 5–18 A volcanic plug of diameter 10 km has a gravity anomaly of 0.3 mm s−2. 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 m−3 and the rock it intrudes has a density of 2800 kg m−3.