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\begin{document}

\title{\huge{Exam 2023} \\ \large{Dynamics Large Scale of Atmospheric Flow}}
\date{Time: \qty{2}{\hour}}

\maketitle

\begin{center}
    \textbf{Note}: This is a reconstruction of the exam in February 2023. Maybe questions weren't asked exactly like this, but they capture the general sense. Overall, it was very similar to older exams.
\end{center}

\vspace*{1cm}

\begin{questions}
\question State if the following are true or false
\begin{parts}
    \part Potential temperature is bigger at poles than at mid-latitudes
    \part Warm advection is associated with clockwise turning of the wind field
    \part Two potential vorticity anomalies of equal strength but opposite sign propagate parallel to each other
    \part The ageostrophic wind in the extra-tropics is typically smaller than the geostrophic wind
    \part Diabatic heating produces PV constantly
\end{parts}
\begin{center}
    \dots
\end{center}

\question The Navier-Stokes equation is given as
\begin{equation}\label{navier-stokes}
    \frac{D\vb{u}}{Dt} + (2\vb{\Omega} \wedge \vb{r}) = -\frac{1}{\rho}\nabla p + \vb{G^*} + \vb{F}
\end{equation}

\begin{parts}
    \part Explain each part of Equation~(\ref{navier-stokes}). Which are relevent on a synoptic scale?
    \part How does the coriolis force affect air parcels at the equator with a velocity of \qty{10}{\meter\per\second} (a), at \qty{45}{\degree} with \qty{60}{\meter\per\second} (b) and at the pole with a velocity of \qty{0}{\meter\per\second} (c)?
    \part How is the pressure gradient at these three points?
\end{parts}

\question A weather chart is given
\begin{parts}
    \part Cut drawn on map, draw vertical cross section with dynamic tropopause, isentropes and winds
    \part How is the Q vector, where is ascent/descent?
    \part How is the weather in norway?
\end{parts}

\question Diabatic cooling and heating (Some formulas given)
\begin{parts}
    \part Sketch a profile of $\dot{\theta}$ with a maximum of PV production at 3000m height
    \part How does $\theta$ and PV evolve along a trajectory over of an air parcel ascending through a zone of evaporation cooling and then through a zone of condensation heating? Draw a diagram
    \part How would it change, if the trajectory crosses the maximum of heating/cooling?
\end{parts}

\newpage

\question Rossby waves with given dispersion relation
\begin{parts}
    \part Given formula for $n^2$, show that it follows from dispersion relation
    \part Derive $U_\mathrm{crit}$ for $n > 0$ using the condition $0 < U_0 < U_\mathrm{crit}$
    \part Why can't synoptic Rossby waves propagate into the stratosphere? Use the condition from above to explain
\end{parts}

\question Consider the wind field $\vb{v} = (u, v)$ in the domain $x \in [0,L]$, $y \in [-d,d]$ with
\begin{align*}
    u &= U_0 \sin(\pi \frac{x}{L}) \sin(\pi \frac{y}{d}) \\
    v &= 0
\end{align*}

\begin{equation}\label{vorticity}
    \zeta = -\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} 
\end{equation}

\begin{equation}\label{wind-change}
    \nabla \vb{v} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}
\end{equation}

\begin{parts}
    \part Sketch the wind field
    \part Calculate vorticity using Equation~(\ref{vorticity}). Where are the extrema?
    \part Calulate the wind change using Equation~(\ref{wind-change}). Where are the extrema?
    \part Calculate the vorticity change over \qty{1}{\hour} with given formula using parts from b) and c)
\end{parts}

\question Eady problem
\begin{figure}[hbt]
    \centering
    \includegraphics[height=5cm]{growth-rate.png}
    \caption{Growth rate vs wave number (given without red annotations)}
    \label{growth-rate}
\end{figure}
\begin{parts}
    \part Explain the cyclonic development using less than 100 words and the terms \textit{tilted isentropes}, \textit{potential temperature anomaly}, \textit{intensification} and \textit{vertical coupling}
    \part Explain Figure~\ref{growth-rate}
    \part Calculate e-folding time for most unstable wave using given formula
\end{parts}

\end{questions}
\end{document}
