Introduction
The atmosphere can be divided into several layers, distinguished by different temperature profiles.
The first layer is the troposphere, from the earth’s surface to a height of 8 km at the poles, and 15
km at the equator. This layer is characterised by a linear decrease of temperature with height (6.5
K km−1 ). All the weather takes place in this layer. The second layer is the stratosphere, starting
from the top of the troposphere, called the tropopause, to a height of 50 km. In the lower stratosphere the temperature is constant and then it increases with height. Above the stratosphere, we
find the mesosphere, which reaches to a height of 85 km, and is characterised by a decrease of
temperature. And on top of this layer,we find the thermosphere, with an increase of temperature
with height. This structure of the atmosphere can also be seen in the figure below.
Figure 1: The atmosphere divided into several layers, distinguished by different temperature profiles.(Holton [3])
In this thesis I discuss a simple model of the lower atmosphere, containing the troposphere and the
stratosphere. The question is whether this simple model has realistic dynamical properties. To
find an answer to this question, we first consider the physical conservation laws that describe all
the motions in the atmosphere. These so-called primitive equations are discussed in chapter 1. In
chapter 2, the equations of motion are derived for the troposphere. For the troposphere we assume
that the potential temperature is constant. The potential temperature is the temperature that a
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parcel of dry air at pressure p and temperature T would have if it were expanded or compressed to
a reference pressure pr , without energy transport to or from the parcel. This assumption implies
the linear decrease of temperature with height and it simplifies the conservation laws of chapter
1.
We see in figure 1 that the temperature is constant in the lower part of the stratosphere, and
that the temperature increases with height, higher in the stratosphere. For the stratosphere we
make the assumption that the temperature is constant within the layer, since this will be a good
approximation. This also gives a set of simplified equations of motion. These equations are derived
in chapter 3.
In the next chapter, chapter 4, the two layers are added together into a simple model of the atmosphere. The two layers are coupled by the pressure at the tropopause. All the motions in the
two layers are governed by the equations derived in chapters 2 and 3. We make a few assumptions
for some of the variables at the boundaries and look at vertical profiles of temperature, pressure,
density and potential temperature.
One way to look at this model is to linearise the equations of motion and only look at small perturbation in the variables. Here we look at plane wave solutions with different frequencies. This
is done in chapter 5. We consider two cases, the first case is for f , the Coriolis parameter, equal
to zero. In the second case we take f constant and unequal to zero. For the second case, we can
look if dynamics are realistic.
In chapter 6 we discuss another way to look at the dynamics of the model. Here we look at an
initial value problem. We look only at waves travelling in the zonal direction, first for the troposphere only, then for both layers. In chapter 7 we also consider waves travelling in the meridional
direction. Again we look at initial value problems first for the troposphere only, and then for both
layers.
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