A satellite orbiting at an altitude of 36000 km observes the surface in a thermal channel with a wavelength range of $13\ \mu m < \lambda < 14\ \mu m$.
Assuming a density scale height of $H_\rho=8 km$ and a mass absorption coefficient of $k_\lambda = 2 \times 10^{-4}\ m^2/kg$, find the vertical optical thickness and transmittance of the atmosphere in that channel.
If the surface is black with a temperature of 300 K, and the atmosphere has an average temperature of 260 K, find the brightness temperature observed by the satellite.
Given a pixel size 1 $km^2$, what is the maximum flux, in $W\,m^{-2}$, reaching the satellite in this situation?
In [0]:
from __future__ import print_function
from IPython.display import Image
from numpy import exp
import site
site.addsitedir('../lib')
from planck import planckInvert
klamb=2.e-4
H=8.e3
rho0=1
tau=klamb*H*rho0
trans=exp(-tau)
print("the vertical optical thickness and transmittance are {}, {:5.3f}".format(tau,trans))
#
# read the radiances off the Planck figure -- 7.9 W/m^2/sr/micron and 4.5 W/m^2/sr/micron
#
out=7.9*0.2 + 0.8*4.5
print('radiance at satellite: {:5.3f} W/m^2/micron/sr'.format(out))
print('brightness temperature at 13.5 microns is {:8.3f} K'.format(planckInvert(13.5e-6,out*1.e6)))
fov=1/36000.**2.
flux=out*fov
print("flux at satellite {:7.3g} W/m^2".format(flux))
The figure shows a two-layer atmosphere over a black surface. The shortwave insolation is 250 $W m^{-2}$ with no atmospheric absorption. What is the heating rate for layer 1, in K/hr? Assume the density in layer 1 is $\rho=1 kg m^{-3}$ and layer 1 is 1 km thick.
In [0]:
Image(filename='figures/fluxprob.jpg')
In [0]:
#define downward flux as positive
tau1=1
tau2=0.5
T1=280
T2=250
Tsfc=300
trans1=exp(-tau1*1.6666)
trans2=exp(-tau2*1.6666)
eps2=(1. - trans2)
eps1=(1. - trans1)
sigma=5.67e-8
F2d_top=eps2*sigma*T2**4.
F2d_bot=F2d_top*trans1
F1up_top=eps1*sigma*T1**4.
F1d_bot=F1up_top
FsfcTop=sigma*Tsfc**4.*trans1
FsfcBot=sigma*Tsfc**4.
balance_top=250. + F2d_top - F1up_top - FsfcTop
balance_bot=250 + F2d_bot + F1d_bot - FsfcBot
print("layer1 and layer 2 emissivities: {:5.2f} {:5.2f}".format(eps1,eps2))
print("layer1 and layer 2 transmissivities: {:5.2f} {:5.2f}".format(trans1,trans2))
print("surface flux bottom and top {:6.2f} and {:6.2f} W/m^2".format(FsfcBot,FsfcTop))
# print F2d_top,F2d_bot
# print F1up_top
out="net downward flux at top and bottom of layer 1 (W/m^2): {:6.3f} {:6.3f}"
print(out.format(balance_top,balance_bot))
heating_rate=(balance_top - balance_bot)/(1004.*1000.)*3600
print("heating rate is {:5.3f} K/hr".format(heating_rate))