20
Chapter 2
Non-dispersive Waves
We will begin this series of lectures on waves by spending some time discussing
non-dispersive waves. Non-dispersive waves have, by denition, a linear dispersion relation of the kind = k, where is constant. They have a nu
124
6.3.3
An introduction to weakly nonlinear theory
Weakly nonlinear theory is the study of the dynamics of a system that is only
weakly nonlinear, that is, a system where the amplitude of the perturbations
is just large enough for the nonlinear terms to
102
Chapter 6
Instabilities I: Convection
In the following chapters, we will study a few of the fundamental instabilities of
uid dynamics, and learn a number of tools to study their properties. The key
questions we will try to answer, each time, are:
Is
154
7.6
Interfacial shear instabilities
This section is adapted from Chapter 3 of the textbook Introduction to Hydrodynamic stability by Drazin.
In the previous Sections we have studied instabilities arising in continuously
stratied ows that have a contin
Chapter 7
Shear instabilities
In this nal Chapter, we continue our study of the stability of uid ows by
looking at another very common source of instability, shear. By denition,
shear occurs whenever two adjacent uid parcels move in parallel directions,
b
6.3. NONLINEAR STABILITY OF RAYLEIGH-BENARD CONVECTION115
6.3
Nonlinear stability of Rayleigh-Bnard cone
vection
In Chapter 1, we saw that linear stability only tells us whether a system is stable
or unstable to innitesimally-small perturbations, and tha
Chapter 5
Introduction to wave mean ow interactions
In this nal Chapter related to the study of waves, we will study what happens
to waves that travel in a uid which is not at rest, that is, in a uid that
supports a mean ow. This subject is part of the br
2.2. SOUND WAVES IN AN INHOMOGENEOUS, TIME-DEPENDENT MEDIUM37
2.2
Sound waves in an inhomogeneous, timedependent medium
So far, we have dealt with cases where c was always constant. This, however, is
usually not true in most realistic systems. In what fol
83
4.2.3
The wave packet approximation
We now derive, as we have done for pressure waves and internal gravity waves,
a wave packet approximation for surface waves. For simplicity, we will neglect
the eects of surface tension entirely, and continue to cons
Chapter 3
Incompressibility and the
Boussinesq approximation
This Chapter is based on the paper by Spiegel & Veronis, 1960. See also the
textbook Fluid Mechanics by Kundu & Cohen, pages 124-128.
In all that follows, we will be studying phenomena that occu
74
4.2
Surface waves
Surface waves are probably the most common example of waves we are visually
exposed to in everyday life from small-scale waves in the bathtub to mediumscale waves approaching a the beach, to very large-scale waves travelling on the
su
58
Chapter 4
Dispersive waves
In this Chapter we will study a much broader class of waves called dispersive
waves, for which the phase speed and group speeds are dierent. Since nondispersive wave must satisfy = k, dispersive waves include any wave whose
d
136
6.3.5
Truncated equations and the Lorentz system
As a nal foray into nonlinear theory, we now explore the idea of using truncated
systems as a tool to study the time-dependent dynamics of convection close to
onset. Indeed, as we saw previously, only v
147
7.3.2
Examples of commonly-studied shear ows
Linear shear ows
As seen in the previous section, linear shear ows of the kind u = zex are not
expected to have any growing modes. Lets see this more directly by solving
Rayleighs instability equation subje
AMS227 Final
December 7, 2013
The two questions are independent of one another. You will get a B for a substantially correct answer
to one of the two questions, and an A for substantially correct answers to both questions.
1
Richardson criterion, and Howa
Midterm
The two questions are independent of one another. You will get a B for a substantially correct answer
to one of the two questions, and an A for substantially correct answers to both questions.
1
Gravity waves in the presence of rotation
In steadil
Chapter 1
Introduction
In this series of lectures we will study waves and instabilities in uid ows, with
particular applications to geophysical and astrophysical systems, and paying
particular attention to the various mathematical methods that can be used
Homework 1
October 3, 2013
1
1.1
Steady-state solutions of the uid equations
Polytropes
As discussed in class, the equation of state usually involves 3 thermodynamic variables. However, there
are conditions under which it can be approximated fairly well b
Homework 2
October 7, 2013
1
Coursework
Question 1: Show that the Fourier solution and dAlemberts solution are indeed the same, as claimed
below equation (2.24) of the lecture notes. Hint: start with the case where q0 (x) = 0. It is a lot easier!
Then do
Homework 2
October 23, 2013
1
Coursework
Consider the governing equation for internal gravity waves, in the case where N is a slowly varying function
of X and . Assume a wave paket solution of the form
= A(X, )ei(x,t)
with k =
(1)
and = /t.
Starting fro
Homework 4
November 18, 2013
1
Local analysis
Redo the local stability analysis we did in class, but this time add the dissipation terms we neglected: the
viscous dissipation term in the momentum equation, and the thermal dissipation term in the temperatu