104 lines
4.2 KiB
TeX
104 lines
4.2 KiB
TeX
\documentclass{exam}
|
|
\usepackage{physics}
|
|
\usepackage{siunitx}
|
|
\usepackage{graphicx}
|
|
|
|
\RenewCommandCopy\qty\SI
|
|
|
|
\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 how an upper positive PV and a lower potential temperature anomaly develop over the next day(s) 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} using fastest growth mode and shortwave cut-off
|
|
\part Calculate e-folding time for most unstable wave using given formula
|
|
\end{parts}
|
|
|
|
\end{questions}
|
|
\end{document}
|