Pho.rs: Atmospheric Physics

A1 ^0.20 Express the average net solar power received by the Earth and atmosphere system $P_0$ in terms of $F_s,a$ and $R_E$ (the radius of the Earth).

1 Correct effective cross section area
$$A = \pi R_E^2$$

0.10

2 Correct final answer
$$P_0 = (1-a) \pi R_E^2 F_s$$

If a different cross sectional area is used, 0.1 points can be given here, provided it is the only mistake.

0.10

A2 ^0.30 Estimate the temperature of the Earth's surface $T_{g0}$ assuming that it is at a steady state. Ignore the atmosphere.

1 Energy balance	0.10
2 Correct explicit blackbody radiation formula, using the surface area of a sphere	0.10
3 Correct numerical value $$T_{g0} \approx 255~К \approx -18^\circ \mbox{C}$$	0.10

A3 ^0.70 Assuming the system is in a steady state, calculate $T_g$, the temperature of the ground. Use $t_{\text{sw}}=0.9$ and $t_{\text{lw}}=0.2$.

1 Statement on radiation balance in the region outside the atmosphere	0.10
2 Statement on radiation balance in the region between the atmosphere and Earth	0.10
3 Using $t_{sw}$ correctly	0.10
4 Using $t_{lw}$ correctly	0.10
5 Correct numerical result $$T_g = \left(\frac{1+t_{sw}}{1+t_{lw}}\right)^{1/4} T_{g0} \approx 286~К \approx 13^\circ \mbox{C}$$	0.30
6 Only analytical result	0.20

B1 ^0.50 Consider a simple diatomic molecule modelled as two point masses $m_A$ and $m_B$ connected by a spring with spring constant $k$. What is the angular frequency of vibrations $\omega_d$?

1 Writing down correct equations of motion for $A$ and $B$

Points are given automatically in case the answer is correct

2 × 0.10

2 Studying the equation of motion for $x_A - x_B$

Points are given automatically in case the answer is correct

0.10

3 Correct answer
$$\omega_d = \sqrt{k \frac{m_A + m_B}{m_A m_B}}$$

0.20

B2 ^0.20 Quantum mechanics dictates that vibrational excitations due to absorbing a photon can only raise the quantum energy level by one. What is the energy of the photon $E_p$ that can excite the vibration in B.1? Neglect recoil effects.

1 Correct answer $$E = \hbar \omega_d$$	0.20
2 $h$ is used instead of $\hbar$	0.10

B3 ^0.20 What is the shift in the spectral line $f-f_0$ if the molecule is moving with velocity $v$ towards the emitter such that $|v|\ll c$, where $c$ is the speed of light.

1 Writing down an expression for Doppler effect (even if incorrect)	0.10
2 Correct answer $$f - f_0 = \frac{v}{c} f_0 \\ \text{or} \\ f - f_0 = -\frac{v}{c} f_0 \\ \text{or} \\ f - f_0 = f_0 \left(\sqrt{\frac{1+v/c}{1-v/c}}-1\right)$$	0.10

B4 ^0.20 Find the normalisation constant $C$, assuming that the velocity $v$ could range from $-\infty$ to $\infty$.

1 Normalization condition (even if done incorrectly from 0 to $\infty$)	0.10
2 Correct answer $$C = \sqrt{\frac{m}{2 \pi k_B T}}$$	0.10

B5 ^0.30 Find the probability distribution function $p_2(f)$ to find a molecule with a spectral line $f_0$ shifted to $f$ due to thermal motion, up to a normalisation factor, in terms of $f,f_0,T,m$ and fundamental constants.

1 $v$ is expressed in terms of $f$

Points are given even if the incorrect formula for Doppler effect from B3 is used.

0.10

2 Correct answer
$$p(f) \propto \exp \left[-\frac{mc^2}{2k_B T} \left(\frac{f-f_0}{f_0}\right)^2\right]$$

Points are not given if the incorrect formula for Doppler effect from B3 is used.

0.20

B6 ^0.40 Sketch $p_2(f)$ as a function of $f-f_0$, and determine the shift $f^{\star}-f_0$ at which $p_2(f^{\star})$ is a fraction $1/e$ of its peak value, where $e$ is Euler's number.

1 The distribution is has a single peak at zero	0.10
2 The distribution is symmetric	0.10
3 The distribution decays to zero for $f-f_0 \rightarrow \pm \infty$	0.10
4 Correct answer $$f^* - f_0 = f_0 \sqrt{\frac{2 k_B T}{mc^2}}$$	0.10

C1 ^0.30 Assuming that the small air mass is at hydrostatic equilibrium, derive an expression for the rate of change of pressure with respect to height, $dp/dz$ in terms of $g$ and $\rho(z)$.

1 Sum of forces equals zero	0.10
2 Correct pressure force above and below	0.10
3 Correct answer $$\frac{dp}{dz} = -\rho(z) g$$	0.10

C2 ^0.20 Express $dp/dz$ in terms of $\mu_{\text{air}},g, p(z)$, $T(z)$ (the temperature at height $z$), and fundamental constants.

1 Ideal gas law	0.10
2 Correct answer $$\frac{dp}{dz} = -\frac{\mu_{air} p(z)}{RT(z)}g$$	0.10

C3 ^0.20 Assuming an isothermal atmosphere, $T(z)=T$, find an expression for $p(z)$ in terms of $z,\mu_{\text{air}},g,p_o,T$ and fundamental constants.

1 Recognizing a separable differential equation	0.10
2 Correct answer $$p(z) = p_0 \exp \left(-\frac{\mu_{air} g}{RT} z\right)$$	0.10

C4 ^0.60 For the adiabatically rising air mass, find the adiabatic lapse rate $\Gamma_a$ in terms of $c_p$ (the molar specific heat capacity at constant pressure), $\mu_{\text{air}}$ and $g$.

1 Writing the adiabatic relation in any form	0.10
2 Relating $dT/dz$ to $dP/dz$	0.30
3 Correct answer $$\Gamma_a = \frac{\mu_{air}}{c_p} g$$	0.20

C5 ^1.40 Find the equation of motion for $\delta z$, the instantaneous vertical displacement. Under what condition is the equilibrium at $z$ stable? What is the angular frequency $\omega$ of small oscillations? Express your answers in terms of $T,\Gamma, g, \mu_{\text{air}}$ and $c_p$.

1 Inclusion of gravitational force with parcel density	0.20
2 Inclusion of buoyancy force with air density	0.20
3 Correct equation of motion	0.20
4 Relating density to temperature $$\rho \propto \frac{1}{T}$$	0.20
5 Smallness of $\delta z$ is used	0.20
6 Correct stability requirements $$\mu_{air} g / c_p > \Gamma$$	0.20
7 Correct answer $$\omega = \sqrt{\frac{\mu_{air} g / c_p - \Gamma}{T}g}$$	0.20

D1 ^0.50 Express $dp_s/dT$ for the water liquid-vapour coexistence curve in terms of the water latent heat of evaporation $L, \mu_{\text{H}_2\text{O}}, p_s, T$ and fundamental constants.

1 Correct entropy change $$\Delta S = \frac{Lm}{T}$$	0.20
2 Volume of liquid is much less than volume of vapor	0.10
3 Correct answer $$\frac{dp_s}{dT} = \frac{\mu_{\mbox{H}_2\mbox{O}} L p_s}{RT^2}$$	0.20

D2 ^0.20 If for some reference temperature $T_o$, $p_s=p_{so}$, find an expression for $p_s(T)$ in terms of $p_{so},\mu_{\text{H}_2\text{O}},L,T,T_o$ and fundamental constants.

1 Recognizing a separable differential equation	0.10
2 Correct answer $$p_s(T) = p_{so} \exp \left[ -\frac{\mu_{\mathrm{H_2O}} L}{R} \left( \frac{1}{T} - \frac{1}{T_o} \right) \right]$$	0.10

D3 ^2.00 Assume that the air mass starts at $T_i=17.0 ^\circ \text{C}$ and $p_i=10^5 \text{ Pa}$. Find the temperature $T_l$ at which liquid water starts forming if $\phi=10^{-2}$. Assume that the water content in the air mass stays constant during the rise. Use $L= 2460\text{ kJ/kg}$ and $p_{so}=1.94\times10^3\text{ Pa}$ at $T_i=17.0^{\circ}\text{C}$.

1 According to Dalton's law obtained $$p_w = \phi \frac{\mu_{air}}{\mu_{\mbox{H}_2\mbox{O}}} p$$ If the formula $p_w = \phi p$ is used further in this task, the maximum score for D3 is 1.2	0.40
2 The dependence $p(T)$ for an adiabatic process	0.20
3 Understanding that partial pressure of water needs to reach saturation for condensation to start	0.40
4 An equation in a convenient form for numerical solving is obtained	0.40
5 A method for numerical solving is suggested	0.20
6 Correct numerical answer $$T \approx 286.8~К \approx 13.7^\circ \mbox{С}$$	0.40

E1 ^0.80 Consider a simple prism with an apex angle of $\varphi$ and direct a light ray into it at an incidence angle $\alpha$, as shown in Figure E.1. Let the refractive index of the prism be n. Express the angle of deviation $\delta$ of the light ray after passing through the prism in terms of $\alpha$, $n$ and $\varphi$.

1 Writing Snell’s law for the two refractions	2 × 0.20
2 A dependence between angles is used	0.10
3	0.00
4 Correct calculations for $\delta$	0.00
5 Correct answer $$\delta = \alpha + \arcsin \left\{ n \sin \left[ \varphi - \arcsin \left( \frac{\sin \alpha}{n} \right) \right] \right\} - \varphi$$	0.30

E2 ^0.60 Plot on the answer sheet how the deviation angle $\delta$ of the examined light ray depends on the incidence angle $\alpha$ within the interval $[20^\circ,70^\circ]$ in $5^\circ$ increments. The refractive index of ice is n=1.31.

1 Calculation of $\delta$ for all values of $\alpha \in [20^\circ,70^\circ]$ with a step of $5^\circ$	0.20
2 Calculation of $\delta$ for 6 values of $\alpha$	0.10
3 A graph is plotted	0.20
4 The graph has a local minimum	0.20

E3 ^0.20 Using the graph from the previous part, determine at what angle the halo appears the brightest relative to the direction of the Sun.

1 The minimal value of $\delta$ is obtained $$\delta_{min} \approx 21.8^\circ$$	0.10
2 Concluding that the angular size of halo corresponds to the minimal value of $\delta$	0.10

Atmospheric Physics

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