Midterm review questions II

3. Midterm review questions II#

3.1. D1. Schwartzchild with changing temperature#

From The Schwartzchild Equation

  1. Equation (3.1) allows you to find the radiance at height \(z_T\) if given enough information about the atmosphere.

    (3.1)#\[ L^\uparrow (\tau_T) = L^\uparrow (0) \, t_{tot} + \int_0^{z_T} t(\tau_T, \tau^\prime) B(\tau^\prime)\, d\tau^\prime \]

    1. Draw a sketch of the layer, and label \(L^\uparrow (0)\), \(z_T\), \(t_{tot}\), \(t(\tau_T,\tau^\prime)\), \(d\tau^\prime\), \(B(\tau^\prime)\), where \(z^\prime= z_T/2\)

    2. Use the definition of the transmissivity \(t(\tau_T, \tau^\prime)\) to prove that

      (3.2)#\[ \int_0^{z_T} t(\tau_T,\tau^\prime) B(\tau^\prime)\, d\tau^\prime = \int_0^{z_T} B(t^\prime)\, dt^\prime \]

3.2. D2 Beer’s law#

From Beers and inverse squared laws

  1. A 5 km thick ozone layer absorbs 30% of the ultraviolet sunlight that passes through it when the sun is directly overhead.

    1. What is the vertical optical thickness of the layer in the ultraviolet? (UV radiation is not reflected, only absorbed/transmitted)

    2. What is the value of the absorptivity at 4pm, when the sun is \(60^\circ\) away from the zenith?

    3. Find the solid angle subtended by the sun when viewed from the earth (i.e. – what is area/r^2 for the sun?)

    4. If the UV solar flux is 200 \(W\,m^{-2}\) for overhead sun, what is the value of the radiance and the flux at 4pm?

3.3. D3 Hydrostatic equation#

From Hydrostatic balance and the Weighting functions for temperature retrieval notebook.

  1. A 10 km thick layer of an an atmosphere has constant temperature \(T_{atm}\)=280 K, a pressure/density scale height of \(H=8\ km\) and is filled with a gas with a mass absorption coefficient of \(k_\lambda\) = 0.2 at a wavelength of 10 . Underneath this layer is a black surface with a temperature of 290 K. The atmosphere is in hydrostatic equilibrium, the gas has a constant mixing ratio and a density at the surface of \(\rho_0 = 10^{-3}\ kg\,m^{-3}\). Find:

    1. The vertical optical depth of the layer

    2. The layer transmission for radiance going straight up.

    3. The radiance, in at the top of the layer due to the surface and atmosphere.

    4. The brightness temperature \(T_b\) (K) that a satellite would measure at \(\lambda\)=10 if there were no absorption/emission above 10 km. (see Planck function curves on next page).

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