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\begin{document}
\begin{center}
{\large Atmospheric Dynamics}
\par\end{center}{\large \par}
\bigskip
\noindent \begin{center}
{\large Lecture 6: Linearization Part 1}
\par\end{center}{\large \par}
\section{General concepts}
The primitive equations are sufficiently non-linear that analytical
solutions are very difficult to obtain except in special cases. A
general approach to this has been the extensive use of numerical models.
This is clearly however not a general theoretical approach and has
rather the flavour of experimental physics. A common approach to this
difficulty has been linearization about a variety of mean states.
The solutions of the resulting multi-component equations are wave-like
disturbances which may grow, decay or remain at constant amplitude
depending on what the mean state is and what dissipation exists in
the linearized equations. When the disturbances grow we refer to the
analysis as linear instability theory while in the other case one
often refers to the study as linear wave analysis. A particularly
simple and revealing starting point for linearization analysis is
the case that the mean state is at rest.
\section{Linearization about a state of rest}
\noindent Particularly useful approximate solutions of the primitive
equations can be obtained by linearizing them about a state of rest
and assuming that the background (mean) vertical density structure
is horizontally but not vertically uniform. In order to simplify the
presentation we shall assume incompressibility and hydrostatic equilibrium
for the present but later consider how the general case pans out.
The momentum equations for small perturbations about the state of
rest are\begin{equation}
\begin{array}{ccc}
\frac{\partial u}{\partial t}-fv & = & -\frac{1}{\rho_{0}}\frac{\partial p}{\partial x}\\
\frac{\partial v}{\partial t}+fu & = & -\frac{1}{\rho_{0}}\frac{\partial p}{\partial y}\\
g\rho & = & -\frac{\partial p}{\partial z}\end{array}\label{momentum}\end{equation}
where we are using the abbreviations $x$ to indicate distance along
circles of latitude (called the zonal coordinate), $y$ to indicate
distance along longitude circles (called the meridional coordinate)
and $z$ is the vertical coordinate which is perpendicular to geopotential
surfaces. The Coriolis parameter $f$ is $2\Omega\sin\varphi$ and
$\rho_{0}=\rho_{0}(z)$ is the density profile. The equation of continuity
in the incompressible case is\begin{equation}
\frac{\partial w}{\partial z}+\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0\label{continuity}\end{equation}
If we assume that there is no sources of heat and moisture then the
incompressible version of the equation of state implies that $\frac{d\rho}{dt}=0$
which when linearized becomes\begin{equation}
\frac{\partial\rho}{\partial t}+w\frac{\partial\rho_{0}}{\partial z}=0\label{density}\end{equation}
\section{Separation of Variables and the vertical Sturm-Liouville equations}
\noindent These five equations have the five unknowns $(u,v,w,p,\rho)$
and general solutions are possible. Those of interest to us are separable
in the vertical direction. We choose the following separation for
the five variables\[
\begin{array}{ccc}
\overrightarrow{u} & = & \widehat{u}(z)\overrightarrow{U}(x,y,t)\\
p & = & \widehat{p}(z)\eta(x,y,t)\\
\rho & = & \widehat{\rho}(z)\upsilon(x,y,t)\\
w & = & \widehat{h}(z)\widetilde{w}(x,y,t)\end{array}\]
where for reasons that will become clearer later we assume that $\widehat{u}$
and $\widehat{h}$ have the dimensions of length whereas $\widehat{p}$
and $\widehat{\rho}$ have their ordinary units (implying, of course,
that the horizontal dependent components are dimensionless). The vertical
equations can now be deduced up to a separation constant from the
linearized set above. We obtain the four equations for the vertical
part of the solutions:\begin{equation}
\begin{array}{ccc}
\rho_{0}\widehat{u} & = & \widehat{p}/g\\
g\widehat{\rho} & = & -\widehat{p}_{z}\\
\widehat{h}_{z} & = & \widehat{u}/H_{0}\\
\widehat{\rho} & = & \left(\rho_{0}\right)_{z}\widehat{h}\end{array}\label{vertical}\end{equation}
where the factor $g$ has been included in the first equation for
dimensional consistency and $H_{0}$ is the separation constant which
has dimension of length. Combining the first and third equations and
the second and fourth reduces this to\begin{equation}
\begin{array}{ccc}
gH_{0}\widehat{h}_{z} & = & \widehat{p}/\rho_{0}\\
\widehat{p}_{z} & = & -g\left(\rho_{0}\right)_{z}\widehat{h}\end{array}\label{reduced}\end{equation}
and this is easily condensed further to\begin{equation}
\frac{1}{\rho_{0}}(\rho_{0}\widehat{h}_{z})_{z}+\frac{N^{2}}{c_{e}^{2}}\widehat{h}=0\label{Sturm}\end{equation}
where $N\equiv\left(-g\rho_{0}^{-1}\left(\rho_{0}\right)_{z}\right)^{1/2}$
is termed the Brunt-Vaisala frequency and measures the stability of
the background stratification. Note that is always real if the density
profile is stable. Parcels of fluid displaced within a density stratification
will tend to oscillate at this frequency due to buoyancy effects.
The separation constant has been {}``renamed'' $c_{e}^{2}\equiv gH_{0}$
where $c_{e}$ is referred to as shallow water speed for reasons that
will become clearer later. The reason for assuming that $c_{e}^{2}$
is positive is because the operator $H=\frac{-1}{\rho_{0}N^{2}}\partial_{z}\rho_{0}\partial_{z}$
may easily be shown to be positive providing that the stratification
is stable. The lower boundary condition is that the vertical velocity
vanishes and so\begin{equation}
\widehat{h}(0)=0\label{Sturmb1}\end{equation}
Equations (\ref{Sturm}) and (\ref{Sturmb1}) together with the assumption
that $N>0$ form a (semi-infinite) Sturm-Liouville eigensystem. The
mathematical literature on such systems is extensive and the eigenvalues
$c_{e}^{2}$ can be shown to be positive and the spectrum generally
has discrete and continuous parts. The discrete portion is generally
of greatest physical interest. The eigenvectors in this system are
called the \emph{normal, vertical or baroclinic/barotropic modes.}
The mode with the greatest eigenvalue and the simplest (one signed)
vertical structure for its eigenvector is the so-called \emph{barotropic}
mode which has a shallow water speed of around $200ms^{-1}.$ The
other modes are called the \emph{baroclinic} modes and have more complex
vertical structures. For the observed stratifications they have smaller
shallow water speeds (the first baroclinic mode has a typical shallow
water speed of around $50ms^{-1}$). Because the system here is Sturm-Liouville,
the vertical modes satisfy an orthogonality condition and are complete
in the sense that an arbitrary solution may be decomposed into a unique
linear combination of vertical modes. Baroclinic modes play an important
role in the understanding of tropical dynamics while barotropic modes
are of greater importance in the mid-latitudes.
\section{The linear shallow water equations}
\noindent Corresponding to the four equations in the vertical (\ref{vertical})
there are a set of equations governing the horizontal flow\begin{equation}
\begin{array}{ccc}
\frac{\partial U}{\partial t}-fV & = & -g\frac{\partial\eta}{\partial x}\\
\frac{\partial V}{\partial t}-fU & = & -g\frac{\partial\eta}{\partial y}\\
\widetilde{w} & = & -H_{0}(\frac{\partial U}{\partial x}+\frac{\partial V}{\partial y})\\
\widetilde{w} & = & \frac{\partial\upsilon}{\partial t}=\frac{\partial\eta}{\partial t}\end{array}\label{shallow1}\end{equation}
The last equality here coming from the hydrostatic equation. The third
and fourth equations here can be combined and a new variable $h\equiv g\eta$
introduced. The resulting equations\begin{equation}
\begin{array}{ccc}
U_{t}-fV & = & -h_{x}\\
V_{t}-fU & = & -h_{y}\\
h_{t}+c_{n}^{2}(U_{x}+V_{y}) & = & 0\end{array}\label{shallow}\end{equation}
are commonly referred to as the linear shallow water equations. We
shall consider their solution further in the next Lecture however
it is worth examining here the solutions which occur when the Coriolis
term is dropped as these have wide applicability in atmospheric flows
particularly and illustrate simply some important dynamical properties.
The equations in this case are easily reduced to one constant coefficient
linear PDE for $h.$ The $x$ derivative of the first equation; the
$y$ derivative of the second and the $t$ derivative of the third
are all combined to obtain\begin{equation}
h_{tt}=c_{n}^{2}(h_{xx}+h_{yy})\label{shallow2}\end{equation}
Consider now a general wave like solution of the form\[
h=h_{0}\exp(i(\omega t-kx-ly)\]
Substitution into (\ref{shallow2}) gives the wave dispersion relation\[
\omega^{2}=c_{n}^{2}(k^{2}+l^{2})\]
Waves with this dispersion relation (or approximately) are called
\emph{gravity waves} and their group velocity is given by\[
\begin{array}{ccc}
u_{g}=\frac{\partial\omega}{\partial k} & = & \frac{c_{n}^{2}k}{\sqrt{k^{2}+l^{2}}}\\
v_{g}=\frac{\partial\omega}{\partial l} & = & \frac{c_{n}^{2}l}{\sqrt{k^{2}+l^{2}}}\end{array}\]
which has magnitude equal to the shallow water speed for the particular
vertical mode they are derived from. The direction of propagation
is preferentially in the direction of greatest wavenumber or smallest
wavelength. Gravity waves are often seen in the atmosphere and are
generated by rapidly varying forcing such as convective systems and
flow over topography.
\section{Generalization to a compressible fluid}
There is little inherent difficulty in generalizing our incompressible
derivation to the compressible case and little of great additional
qualitative interest is revealed. It is convenient in this case to
work with pressure vertical coordinates and use the equations derived
in the first section of the last Lecture. The mathematics of the derivation
of the Sturm Liouville system is essentially identical in form to
the incompressible case discussed above (exercise for interested students).
The only complication concerns boundary conditions. The vertical domain
is mapped from semi-infinite to finite by the use of pressure. The
boundary condition at $p=0$ is rather obvious ($\omega=0$) but the
boundary condition at the bottom of the atmosphere is a little less
obvious but can be shown to be\[
\omega=\rho_{0}(p_{l})\frac{\partial\Phi}{\partial t}\]
where the maximum pressure is $p_{l}$. The finite domain implies
a fully discrete spectrum for the positive eigenvalues of the problem
and they are related to the incompressible solutions in a rather clear
way albeit with modified values. The horizontal equations remain identical
i.e. they are the shallow water equations.
\section{Other linearizations of the equations}
\noindent We consider first some basic concepts in dynamical systems
theory. A general (unforced) linearized dynamical system may be written
as\begin{equation}
\frac{\partial\psi}{\partial t}=A\psi\label{dynamical}\end{equation}
where the vector $\psi$ specifies the state of the system and the
operator $A$ governs the time evolution. The eigenvectors of $A$
are often referred to as the \emph{normal modes} and the corresponding
complex eigenvalues determine the growth/decay and oscillatory frequency
of these modes. In conservative dynamical systems $A$ satisfies the
anti-hermitian property%
\footnote{One may need to transform state variables to make this apparent.%
}\[
A=-A^{\ast}\]
where the star indicates the Hermitian conjugate. This relation implies
that it has purely imaginary eigenvalues%
\footnote{As should be well known, Hermitean operators have real eigenvalues%
} which means that the normal modes oscillate with constant amplitude.
This can be shown also by defining the so-called \emph{propagator}
$U(t^{\prime},t)$ which takes a state vector at a time $t$ and transforms
it to the appropriate state vector at time $t^{\prime}.$ It is easily
demonstrated by discretizing (\ref{dynamical}) and iterating that\[
U(t^{\prime},t)=\exp((t^{\prime}-t)A)\]
from which it is easily demonstrated that if $A$ is antihermitian
then $U(t^{\prime},t)$ is orthogonal which implies that it preserves
the norm of the state vectors.
The shallow water equations control the dynamics of solutions obtained
by linearizing the primitive equations about a state of rest and we
can write these in the matrix form\[
\frac{\partial}{\partial t}\left(\begin{array}{c}
U/c\\
V/c\\
h/c^{2}\end{array}\right)=\left(\begin{array}{ccc}
0 & f & -c\frac{\partial}{\partial x}\\
-f & 0 & -c\frac{\partial}{\partial y}\\
-c\frac{\partial}{\partial x} & -c\frac{\partial}{\partial y} & 0\end{array}\right)\left(\begin{array}{c}
U/c\\
V/c\\
h/c^{2}\end{array}\right)\]
where for convenience we have {}``non-dimensionalized'' the equations
using the shallow water speed. Now as is well known (and can be shown
intuitively by discretization) the partial differential operators
$\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$
are antihermitian and so it follows easily that the operator $A$
for the shallow water equations is also. Thus these equations have
purely oscillatory solutions and the system is conservative.
If the equations are linearized about a state of non-rest then, in
general, the operator $A$ \emph{is no longer antihermitian.} An obvious
example is a linearization about a linearly varying zonal velocity
$\overline{U}.$ If we assume that this variation occurs in the meridional
direction then\[
A=\left(\begin{array}{ccc}
-\overline{U}\frac{\partial}{\partial x} & f-\overline{U}_{y} & -c\frac{\partial}{\partial x}\\
-f & -\overline{U}\frac{\partial}{\partial x} & -c\frac{\partial}{\partial y}\\
-c\frac{\partial}{\partial x} & -c\frac{\partial}{\partial y} & 0\end{array}\right)\]
In this case $A$ is obviously not antihermitian and in fact for
$\overline{U}_{y}$ large enough there exist growing normal modes.
This is commonly referred to as barotropic instability and occurs
frequently in many situations in the atmosphere.
\end{document}