Theoretical Geophysics

# Theoretical Geophysics

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A1.17

comment(i) What is the polarisation $\mathbf{P}$ and slowness $\mathbf{s}$ of the time-harmonic plane elastic wave $\mathbf{u}=\mathbf{A} \exp [i(\mathbf{k} \cdot \mathbf{x}-\omega t)] ?$

Use the equation of motion for an isotropic homogenous elastic medium,

$\rho \frac{\partial^{2} \mathbf{u}}{\partial t^{2}}=(\lambda+2 \mu) \boldsymbol{\nabla}(\boldsymbol{\nabla} \cdot \mathbf{u})-\mu \boldsymbol{\nabla} \wedge(\boldsymbol{\nabla} \wedge \mathbf{u})$

to show that $\mathbf{s} \cdot \mathbf{s}$ takes one of two values and obtain the corresponding conditions on $\mathbf{P}$. If $\mathbf{s}$ is complex show that $\operatorname{Re}(\mathbf{s}) \cdot \operatorname{Im}(\mathbf{s})=0$.

(ii) A homogeneous elastic layer of uniform thickness $h, S$-wave speed $\beta_{1}$ and shear modulus $\mu_{1}$ has a stress-free surface $z=0$ and overlies a lower layer of infinite depth, $S$-wave speed $\beta_{2}\left(>\beta_{1}\right)$ and shear modulus $\mu_{2}$. Show that the horizontal phase speed $c$ of trapped Love waves satisfies $\beta_{1}<c<\beta_{2}$. Show further that

$\tan \left[\left(\frac{c^{2}}{\beta_{1}^{2}}-1\right)^{1 / 2} k h\right]=\frac{\mu_{2}}{\mu_{1}}\left(\frac{1-c^{2} / \beta_{2}^{2}}{c^{2} / \beta_{1}^{2}-1}\right)^{1 / 2}$

where $k$ is the horizontal wavenumber.

Assuming that (1) can be solved to give $c(k)$, explain how to obtain the propagation speed of a pulse of Love waves with wavenumber $k$.

A2.16

comment(i) Sketch the rays in a small region near the relevant boundary produced by reflection and refraction of a $P$-wave incident (a) from the mantle on the core-mantle boundary, (b) from the outer core on the inner-core boundary, and (c) from the mantle on the Earth's surface. [In each case, the region should be sufficiently small that the boundary appears to be planar.]

Describe the ray paths denoted by $S S, P c P, S K S$ and $P K I K P$.

Sketch the travel-time $(T-\Delta)$ curves for $P$ and $P c P$ paths from a surface source.

(ii) From the surface of a flat Earth, an explosive source emits $P$-waves downwards into a stratified sequence of homogeneous horizontal elastic layers of thicknesses $h_{1}, h_{2}, h_{3}, \ldots$ and $P$-wave speeds $\alpha_{1}<\alpha_{2}<\alpha_{3}<\ldots$. A line of seismometers on the surface records the travel times of the various arrivals as a function of the distance $x$ from the source. Calculate the travel times, $T_{d}(x)$ and $T_{r}(x)$, of the direct wave and the wave that reflects exactly once at the bottom of layer 1 .

Show that the travel time for the head wave that refracts in layer $n$ is given by

$T_{n}=\frac{x}{\alpha_{n}}+\sum_{i=1}^{n-1} \frac{2 h_{i}}{\alpha_{i}}\left(1-\frac{\alpha_{i}^{2}}{\alpha_{n}^{2}}\right)^{1 / 2}$

Sketch the travel-time curves for $T_{r}, T_{d}$ and $T_{2}$ on a single diagram and show that $T_{2}$ is tangent to $T_{r}$.

Explain how the $\alpha_{i}$ and $h_{i}$ can be constructed from the travel times of first arrivals provided that each head wave is the first arrival for some range of $x$.

A4.20

commentIn a reference frame rotating about a vertical axis with angular velocity $f / 2$, the horizontal components of the momentum equation for a shallow layer of inviscid, incompressible, fluid of uniform density $\rho$ are

$\begin{aligned} &\frac{D u}{D t}-f v=-\frac{1}{\rho} \frac{\partial p}{\partial x} \\ &\frac{D v}{D t}+f u=-\frac{1}{\rho} \frac{\partial p}{\partial y} \end{aligned}$

where $u$ and $v$ are independent of the vertical coordinate $z$, and $p$ is given by hydrostatic balance. State the nonlinear equations for conservation of mass and of potential vorticity for such a flow in a layer occupying $0<z<h(x, y, t)$. Find the pressure $p$.

By linearising the equations about a state of rest and uniform thickness $H$, show that small disturbances $\eta=h-H$, where $\eta \ll H$, to the height of the free surface obey

$\frac{\partial^{2} \eta}{\partial t^{2}}-g H\left(\frac{\partial^{2} \eta}{\partial x^{2}}+\frac{\partial^{2} \eta}{\partial y^{2}}\right)+f^{2} \eta=f^{2} \eta_{0}-f H \zeta_{0}$

where $\eta_{0}$ and $\zeta_{0}$ are the values of $\eta$ and the vorticity $\zeta$ at $t=0$.

Obtain the dispersion relation for homogeneous solutions of the form $\eta \propto \exp [i(k x-$ $\omega t)$ ] and calculate the group velocity of these Poincaré waves. Comment on the form of these results when $a k \ll 1$ and $a k \gg 1$, where the lengthscale $a$ should be identified.

Explain what is meant by geostrophic balance. Find the long-time geostrophically balanced solution, $\eta_{\infty}$ and $\left(u_{\infty}, v_{\infty}\right)$, that results from initial conditions $\eta_{0}=A \operatorname{sgn}(x)$ and $(u, v)=0$. Explain briefly, without detailed calculation, how the evolution from the initial conditions to geostrophic balance could be found.

A1.19

comment(i) Explain the concepts of: traction on an element of surface; the stress tensor; the strain tensor in an elastic medium. Derive a relationship between the two tensors for a linear isotropic elastic medium, stating clearly any assumption you need to make.

(ii) State what is meant by an $\mathrm{SH}$ wave in a homogeneous isotropic elastic medium. An SH wave in a medium with shear modulus $\mu$ and density $\rho$ is incident at angle $\theta$ on an interface with a medium with shear modulus $\mu^{\prime}$ and density $\rho^{\prime}$. Evaluate the form and amplitude of the reflected wave and transmitted wave. Comment on the case $c^{\prime} \sin \theta / c>1$, where $c^{2}=\mu / \rho$ and $\left(c^{\prime}\right)^{2}=\mu^{\prime} / \rho^{\prime}$.

A2.16

comment(i) Explain briefly what is meant by the concepts of hydrostatic equilibrium and the buoyancy frequency. Evaluate an expression for the buoyancy frequency in an incompressible inviscid fluid with stable density profile $\rho(z)$.

(ii) Explain briefly what is meant by the Boussinesq approximation.

Write down the equations describing motions of small amplitude in an incompressible, stratified, Boussinesq fluid of constant buoyancy frequency.

Derive the resulting dispersion relationship for plane wave motion. Show that there is a maximum frequency for the waves and explain briefly why this is the case.

What would be the response to a solid body oscillating at a frequency in excess of the maximum?

A4.20

commentDefine the Rossby number. Under what conditions will a fluid flow be at (a) high and (b) low values of the Rossby number? Briefly describe both an oceanographic and a meteorological example of each type of flow.

Explain the concept of quasi-geostrophy for a thin layer of homogeneous fluid in a rapidly rotating system. Write down the quasi-geostrophic approximation for the vorticity in terms of the pressure, the fluid density and the rate of rotation. Define the potential vorticity and state the associated conservation law.

A broad current flows directly eastwards ( $+x$ direction) with uniform velocity $U$ across a flat ocean basin of depth $H$. The current encounters a low, two-dimensional ridge of width $L$ and height $H h(x)(0<x<L)$, whose axis is aligned in the north-south $(y)$ direction. Neglecting any effects of stratification and assuming a constant vertical rate of rotation $\frac{1}{2} f$, such that the Rossby number is small, determine the effect of the ridge on the current. Show that the direction of the current after it leaves the ridge is dependent on the cross-sectional area of the ridge, but not on the explicit form of $h(x)$.

A1.19

comment(i) In a reference frame rotating about a vertical axis with constant angular velocity $f / 2$ the horizontal components of the momentum equation for a shallow layer of inviscid, incompressible fluid of constant density $\rho$ are

$\begin{aligned} &\frac{D u}{D t}-f v=-\frac{1}{\rho} \frac{\partial P}{\partial x} \\ &\frac{D v}{D t}+f u=-\frac{1}{\rho} \frac{\partial P}{\partial y} \end{aligned}$

where $u, v$ and $P$ are independent of the vertical coordinate $z$.

Define the Rossby number $R o$ for a flow with typical velocity $U$ and lengthscale $L$. What is the approximate form of the above equations when $R o \ll 1$ ?

Show that the solution to the approximate equations is given by a streamfunction $\psi$ proportional to $P$.

Conservation of potential vorticity for such a flow is represented by

$\frac{D}{D t} \frac{\zeta+f}{h}=0$

where $\zeta$ is the vertical component of relative vorticity and $h(x, y)$ is the thickness of the layer. Explain briefly why the potential vorticity of a column of fluid should be conserved.

(ii) Suppose that the thickness of the rotating, shallow-layer flow in Part (i) is $h(y)=H_{0} \exp (-\alpha y)$ where $H_{0}$ and $\alpha$ are constants. By linearising the equation of conservation of potential vorticity about $u=v=\zeta=0$, show that the stream function for small disturbances to the state of rest obeys

$\frac{\partial}{\partial t}\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right) \psi+\beta \frac{\partial \psi}{\partial x}=0$

where $\beta$ is a constant that should be found.

Obtain the dispersion relationship for plane-wave solutions of the form $\psi \propto$ $\exp [i(k x+l y-\omega t)]$. Hence calculate the group velocity.

Show that if $\beta>0$ then the phase of these waves always propagates to the left (negative $x$ direction) but that the energy may propagate to either left or right.

A2.16

comment(i) State the equations that relate strain to displacement and stress to strain in a linear, isotropic elastic solid.

In the absence of body forces, the Euler equation for infinitesimal deformations of a solid of density $\rho$ is

$\rho \frac{\partial^{2} u_{i}}{\partial t^{2}}=\frac{\partial \sigma_{i j}}{\partial x_{j}}$

Derive an equation for $\mathbf{u}(\mathbf{x}, t)$ in a linear, isotropic, homogeneous elastic solid. Hence show that both the dilatation $\theta=\boldsymbol{\nabla} \cdot \mathbf{u}$ and the rotation $\boldsymbol{\omega}=\nabla \wedge \mathbf{u}$ satisfy wave equations and find the corresponding wave speeds $\alpha$ and $\beta$.

(ii) The ray parameter $p=r \sin i / v$ is constant along seismic rays in a spherically symmetric Earth, where $v(r)$ is the relevant wave speed $(\alpha$ or $\beta)$ and $i(r)$ is the angle between the ray and the local radial direction.

Express $\tan i$ and sec $i$ in terms of $p$ and the variable $\eta(r)=r / v$. Hence show that the angular distance and travel time between a surface source and receiver, both at radius $R$, are given by

$\Delta(p)=2 \int_{r_{m}}^{R} \frac{p}{r} \frac{d r}{\left(\eta^{2}-p^{2}\right)^{1 / 2}} \quad, \quad T(p)=2 \int_{r_{m}}^{R} \frac{\eta^{2}}{r} \frac{d r}{\left(\eta^{2}-p^{2}\right)^{1 / 2}}$

where $r_{m}$ is the minimum radius attained by the ray. What is $\eta\left(r_{m}\right)$ ?

A simple Earth model has a solid mantle in $R / 2<r<R$ and a liquid core in $r<R / 2$. If $\alpha(r)=A / r$ in the mantle, where $A$ is a constant, find $\Delta(p)$ and $T(p)$ for $\mathrm{P}$-arrivals (direct paths lying entirely in the mantle), and show that

$T=\frac{R^{2} \sin \Delta}{A}$

[You may assume that $\int \frac{d u}{u \sqrt{u-1}}=2 \cos ^{-1}\left(\frac{1}{\sqrt{u}}\right)$.]

Sketch the $T-\Delta$ curves for $\mathrm{P}$ and $\mathrm{PcP}$ arrivals on the same diagram and explain briefly why they terminate at $\Delta=\cos ^{-1} \frac{1}{4}$.

A4.20

commentThe equation of motion for small displacements $\mathbf{u}$ in a homogeneous, isotropic, elastic material is

$\rho \frac{\partial^{2} \mathbf{u}}{\partial t^{2}}=(\lambda+2 \mu) \nabla(\boldsymbol{\nabla} \cdot \mathbf{u})-\mu \boldsymbol{\nabla} \wedge(\boldsymbol{\nabla} \wedge \mathbf{u})$

where $\lambda$ and $\mu$ are the Lamé constants. Derive the conditions satisfied by the polarisation $\mathbf{P}$ and (real) vector slowness s of plane-wave solutions $\mathbf{u}=\mathbf{P} f(\mathbf{s} \cdot \mathbf{x}-t)$, where $f$ is an arbitrary scalar function. Describe the division of these waves into $P$-waves, $S H$-waves and $S V$-waves.

A plane harmonic $S V$-wave of the form

$\mathbf{u}=\left(s_{3}, 0,-s_{1}\right) \exp \left[i \omega\left(s_{1} x_{1}+s_{3} x_{3}-t\right)\right]$

travelling through homogeneous elastic material of $P$-wave speed $\alpha$ and $S$-wave speed $\beta$ is incident from $x_{3}<0$ on the boundary $x_{3}=0$ of rigid material in $x_{3}>0$ in which the displacement is identically zero.

Write down the form of the reflected wavefield in $x_{3}<0$. Calculate the amplitudes of the reflected waves in terms of the components of the slowness vectors.

Derive expressions for the components of the incident and reflected slowness vectors, in terms of the wavespeeds and the angle of incidence $\theta_{0}$. Hence show that there is no reflected $S V$-wave if

$\sin ^{2} \theta_{0}=\frac{\beta^{2}}{\alpha^{2}+\beta^{2}}$

Sketch the rays produced if the region $x_{3}>0$ is fluid instead of rigid.

A1.19

comment(i) From the surface of a flat Earth, an explosive source emits P-waves downward into a horizontal homogeneous elastic layer of uniform thickness $h$ and P-wave speed $\alpha_{1}$ overlying a lower layer of infinite depth and P-wave speed $\alpha_{2}$, where $\alpha_{2}>\alpha_{1}$. A line of seismometers on the surface records the travel time $t$ as a function of distance $x$ from the source for the various arrivals along different ray paths.

Sketch the ray paths associated with the direct, reflected and head waves arriving at a given position. Calculate the travel times $t(x)$ of the direct and reflected waves, and sketch the corresponding travel-time curves. Hence explain how to estimate $\alpha_{1}$ and $h$ from the recorded arrival times. Explain briefly why head waves are only observed beyond a minimum distance $x_{c}$ from the source and why they have a travel-time curve of the form $t=t_{c}+\left(x-x_{c}\right) / \alpha_{2}$ for $x>x_{c}$.

[You need not calculate $x_{c}$ or $t_{c}$.]

(ii) A plane $\mathrm{SH}$-wave in a homogeneous elastic solid has displacement proportional to $\exp [i(k x+m z-\omega t)]$. Express the slowness vector $\mathbf{s}$ in terms of the wavevector $\mathbf{k}=(k, 0, m)$ and $\omega$. Deduce an equation for $m$ in terms of $k, \omega$ and the S-wave speed $\beta$.

A homogeneous elastic layer of uniform thickness $h$, S-wave speed $\beta_{1}$ and shear modulus $\mu_{1}$ has a stress-free surface $z=0$ and overlies a lower layer of infinite depth, S-wave speed $\beta_{2}\left(>\beta_{1}\right)$ and shear modulus $\mu_{2}$. Find the vertical structure of Love waves with displacement proportional to $\exp [i(k x-\omega t)]$, and show that the horizontal phase speed $c$ obeys

$\tan \left[\left(\frac{1}{\beta_{1}^{2}}-\frac{1}{c^{2}}\right)^{1 / 2} \omega h\right]=\frac{\mu_{2}}{\mu_{1}}\left(\frac{1 / c^{2}-1 / \beta_{2}^{2}}{1 / \beta_{1}^{2}-1 / c^{2}}\right)^{1 / 2}$

By sketching both sides of the equation as a function of $c$ in $\beta_{1} \leqslant c \leqslant \beta_{2}$ show that at least one mode exists for every value of $\omega$.

A2.16

comment(i) In a reference frame rotating with constant angular velocity $\boldsymbol{\Omega}$ the equations of motion for an inviscid, incompressible fluid of density $\rho$ in a gravitational field $g=-\nabla \Phi$ are

$\rho \frac{D \mathbf{u}}{D t}+2 \rho \boldsymbol{\Omega} \wedge \mathbf{u}=-\nabla p+\rho \mathbf{g}, \quad \nabla \cdot \mathbf{u}=0$

Define the Rossby number and explain what is meant by geostrophic flow.

Derive the vorticity equation

$\frac{D \boldsymbol{\omega}}{D t}=(\boldsymbol{\omega}+2 \boldsymbol{\Omega}) \cdot \nabla \mathbf{u}+\frac{\nabla \rho \wedge \nabla p}{\rho^{2}}$

$\left[\right.$ Recall that $\mathbf{u} \cdot \nabla \mathbf{u}=\nabla\left(\frac{1}{2} \mathbf{u}^{2}\right)-\mathbf{u} \wedge(\nabla \wedge \mathbf{u})$.]

Give a physical interpretation for the term $(\boldsymbol{\omega}+2 \boldsymbol{\Omega}) \cdot \nabla \mathbf{u}$.

(ii) Consider the rotating fluid of part (i), but now let $\rho$ be constant and absorb the effects of gravity into a modified pressure $P=p-\rho \mathbf{g} \cdot \mathbf{x}$. State the linearized equations of motion and the linearized vorticity equation for small-amplitude motions (inertial waves).

Use the linearized equations of motion to show that

$\nabla^{2} P=2 \rho \boldsymbol{\Omega} \cdot \boldsymbol{\omega} .$

Calculate the time derivative of the curl of the linearized vorticity equation. Hence show that

$\frac{\partial^{2}}{\partial t^{2}}\left(\nabla^{2} \mathbf{u}\right)=-(2 \mathbf{\Omega} \cdot \nabla)^{2} \mathbf{u}$

Deduce the dispersion relation for waves proportional to $\exp [i(\mathbf{k} \cdot \mathbf{x}-n t)]$. Show that $|n| \leq 2 \Omega$. Show further that if $n=2 \Omega$ then $P=0$.

A4.20

commentWrite down expressions for the phase speed $c$ and group velocity $c_{g}$ in one dimension for general waves of the form $A \exp [i(k x-\omega t)]$ with dispersion relation $\omega(k)$. Briefly indicate the physical significance of $c$ and $c_{g}$ for a wavetrain of finite size.

The dispersion relation for internal gravity waves with wavenumber $\mathbf{k}=(k, 0, m)$ in an incompressible stratified fluid with constant buoyancy frequency $N$ is

$\omega=\frac{\pm N k}{\left(k^{2}+m^{2}\right)^{1 / 2}} .$

Calculate the group velocity $\mathbf{c}_{g}$ and show that it is perpendicular to $\mathbf{k}$. Show further that the horizontal components of $\mathbf{k} / \omega$ and $\mathbf{c}_{g}$ have the same sign and that the vertical components have the opposite sign.

The vertical velocity $w$ of small-amplitude internal gravity waves is governed by

$\frac{\partial^{2}}{\partial t^{2}}\left(\nabla^{2} w\right)+N^{2} \nabla_{h}^{2} w=0$

where $\nabla_{h}^{2}$ is the horizontal part of the Laplacian and $N$ is constant.

Find separable solutions to $(*)$ of the form $w(x, z, t)=X(x-U t) Z(z)$ corresponding to waves with constant horizontal phase speed $U>0$. Comment on the nature of these solutions for $0<k<N / U$ and for $k>N / U$.

A semi-infinite stratified fluid occupies the region $z>h(x, t)$ above a moving lower boundary $z=h(x, t)$. Construct the solution to $(*)$ for the case $h=\epsilon \sin [k(x-U t)]$, where $\epsilon$ and $k$ are constants and $\epsilon \ll 1$.

Sketch the orientation of the wavecrests, the propagation direction and the group velocity for the case $0<k<N / U$.

Part II