95 lines
3.8 KiB
TeX
95 lines
3.8 KiB
TeX
\documentclass{exam}
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\usepackage{physics}
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\usepackage{siunitx}
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\usepackage{graphicx}
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\RenewCommandCopy\qty\SI
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\begin{document}
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\title{\huge{Exam 2023} \\ \large{Dynamics Large Scale of Atmospheric Flow}}
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\date{Time: \qty{2}{\hour}}
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\maketitle
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\begin{questions}
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\question State if the following are true or false
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\begin{parts}
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\part Potential temperature is bigger at poles than at mid-latitudes
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\part Warm advection is associated with clockwise turning of the wind field
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\part Two potential vorticity anomalies of equal strength but opposite sign propagate parallel to each other
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\part The ageostrophic wind in the extra-tropics is typically smaller than the geostrophic wind
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\part Diabatic heating produces PV constantly
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\end{parts}
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\begin{center}
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\dots
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\end{center}
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\question The Navier-Stokes equation is given as
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\begin{equation}\label{navier-stokes}
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\frac{D\vb{u}}{Dt} + (2\Omega \wedge r) = -\frac{1}{\rho}\nabla p + \vb{G^*} + \vb{F}
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\end{equation}
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\begin{parts}
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\part Explain each part of Equation~(\ref{navier-stokes}). Which are relevent on a synoptic scale?
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\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)?
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\part How is the pressure gradient at these three points?
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\end{parts}
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\question A weather chart is given
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\begin{parts}
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\part Cut drawn on map, draw vertical cross section with dynamic tropopause, isentropes and winds
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\part How is the Q vector, where is ascent/descent?
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\part How is the weather in norway?
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\end{parts}
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\question Diabatic cooling and heating (Some formulas given)
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\begin{parts}
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\part Sketch a profile of $\dot{\theta}$ with a maximum of PV production at 3000m height
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\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
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\part How would it change, if the trajectory crosses the maximum of heating/cooling?
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\end{parts}
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\question Rossby waves with given dispersion relation
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\begin{parts}
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\part Given formula for $n^2$, show that it follows from dispersion relation
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\part Derive $U_\mathrm{crit}$ for $n > 0$ using the condition $0 < U_0 < U_\mathrm{crit}$
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\part Why can't synoptic Rossby waves propagate into the stratosphere? Use the condition from above to explain
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\end{parts}
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\question Consider the wind field $\vb{v} = (u, v)$ in the domain $x \in [0,L]$, $y \in [-d,d]$ with
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\begin{align*}
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u &= U_0 \sin(\pi \frac{x}{L}) \sin(\pi \frac{y}{d}) \\
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v &= 0
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\end{align*}
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\begin{equation}\label{vorticity}
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\zeta = -\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}
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\end{equation}
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\begin{equation}\label{wind-change}
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\nabla \vb{v} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}
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\end{equation}
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\begin{parts}
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\part Sketch the wind field
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\part Calculate vorticity using Equation~(\ref{vorticity}). Where are the extrema?
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\part Calulate the wind change using Equation~(\ref{wind-change}). Where are the extrema?
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\part Calculate the vorticity change over \qty{1}{\hour} with given formula using parts from b) and c)
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\end{parts}
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\question Eady problem
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\begin{figure}[hbt]
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\centering
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\includegraphics[height=5cm]{growth-rate.png}
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\caption{Growth rate vs wave number (given without red annotations)}
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\label{growth-rate}
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\end{figure}
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\begin{parts}
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\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}
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\part Explain Figure~\ref{growth-rate}
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\part Calculate e-folding time for most unstable wave using given formula
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\end{parts}
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\end{questions}
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\end{document}
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