Neil has just come in from a walk around his garden. The grass is covered with a heavy frost everywhere except under the trees.
Was the sky clear or overcast? How do you know?
(In reply to
Solution, alternate explanation by Kenny M)
While yes, the rate of cooling of an object, even a blade of grass, is proportional to the temperature difference of the object and its environment (Newton's Law of Cooling, Fourier's Law), this applies to conductive cooling, not radiative cooling. Radiative emission does not depend on the surroundings, it obeys the Stefan-Boltzmann Law, with power output proportional to T^4. In other words, the grass radiates as per its temperature only. While it "sees" the sky and the tree, it does not care what their temperatures are for the purposes of radiative cooling.
if the grass is around 0 degrees C, it cools radiatively at the Mid-IR wavelengths centered at 10.61 um as per Wien's Displacement Law. In the range between -5 C and +5 C the grass radiation emission peak hardly alters, staying at wavelengths from 10.81 to 10.41 um. This radiation goes skyward, and is generally not absorbed by the atmosphere. The atmosphere has
near unity transmittance in this band, except for some narrow H2O and CO2 absorption lines. In reality, the thermal radiation emission spectrum of the grass blade is broad -
spanning 3 to 50um, but the peak of the distribution, located at 10um(+/-)5um, along with most of its energy, reaches space.
So then, how does the grass remain above freezing? Its temperature is a balance between its radiative cooling and conductive heating from the ground and atmosphere, and, before the Sun rises, radiative heating from the lower atmosphere, and colder upper atmosphere, the clouds, and finally yes, the tree. The radiative heating from the tree aided perhaps a little from clouds (see last post) tip it to be above freezing.
So, it's not the blocking of the upper colder sky by the tree that is keeping the grass beneath the tree from forming frost (that weak longer wavelength sky component of Mid-IR radiation is scattered-in, regardless) but rather the IR radiation downwelling from the warmer tree above. In the absence of convection (a breeze) the tree may also be warming the air just above the grass a bit as well.
Edited on October 25, 2022, 5:04 am
re: Solution, alternate explanation
