1. (40 pts) Seismologists report that lateral variations in seismic velocity in the deep mantle are ~0.2%. Let suppose these variations are due entirely to temperature and consider the bulk sound velocity, $V_B$.
Develop an expression for the temperature derivative of the bulk sound velocity in terms of $dK_s/dT$ and thermal expansivity, $\alpha$ by differentiating the expression defining the bulk sound velocity. Using typical values for thermodynamic properties of deep Earth minerals, estimate the size of the temperature variations necessary to produce the velocity anomalies at depths of 2471 km below the surface. (Some of the typical parameter values you may need: $dK_s/dT = -0.015$ GPa/K; $\alpha \sim 1 \times 10^{-5} K^{-1}$.) (PREM values at 2471 km depth are: $\rho = 5.35$ g/cm³ and $V_B = 10.52$ km/s). (Hint: $V_B^2 = \frac{K_s}{\rho}$)
2. (40 pts) Expressions for the Grüneisen parameter and hydrostatic equilibrium are below:
$\gamma = -\left(\frac{\partial \ln T}{\partial \ln V}\right)_s$ $\frac{dP}{dr} = -\rho g$
where $\gamma$, T, V, P, r, $\rho$ and g are the Grüneisen parameter, temperature, volume, pressure, radius, density, and gravitational acceleration.
Using these, show that, the variation of temperature with radius in an adiabatically convection region of the Earth can be written as
$\left(\frac{\partial T}{\partial r}\right)_s = \frac{-\alpha gT}{C_p}$
3. (20 pts) Prove the following relationship:
$K_s = K_T(1 + \alpha \gamma T)$