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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\Omega \wedge 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} + +\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}