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In a lecture we learned that instantaneously removing all clouds from earth would give radiation forcing of about 18W/m² [1], leading to significant warming. So clouds, in simple words, cool the earth significantly.

However, in the same lecture we learned that clouds do have a positive feedback, which means (in my opinion), that by increasing temperature we have an increased amount of net radiation to earth. How is this to be understood? Does it mean that higher temperature gives rise to less clouds and therefore more net radiation?

Somehow, this appears not very intuitive, because on the one hand clouds are "good for cooling", on the other hand they have positive feedback, which is "worse for cooling". Is there a way to understand this "tradeoff" on a pure qualitative level?

I know, that clouds are quite complex and not well understood, but maybe there is a convincing explanation for that "discrepancy".

[1] https://journals.ametsoc.org/view/journals/clim/31/2/jcli-d-17-0208.1.xml

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The cloud radiative forcing and the cloud feedback are two separate properties of the climate system that both operate through various cloud mechanisms. As that paper you cite shows in Table 5, overall cloud radiative forcing is split broadly into two opposing components: a shortwave forcing of -46 W/m2 (reflected solar radiation), and a longwave forcing of +28 W/m2 (thermal emission). That gives a net cloud radiative forcing of -18 W/m2 at the top-of-atmosphere.

We need to be mindful of the sign conventions here. You mention the removal of clouds giving a forcing of +18 W/m2, which is not wrong, but we normally frame this as the addition or presence of cloud giving a forcing of -18 W/m2. That’s because we’re thinking of it as describing how clouds contribute to the long-term balanced state of the system, such as the pre-industrial state, rather than a “what if” experiment about them disappearing.

Cloud (and other) feedbacks come into play when we think about what happens when something perturbs the radiative balance of the system from this baseline. That something could be a change in GHG or aerosol concentrations, large-scale deforestation, a large volcanic eruption, solar output, etc. This will cause changes to the system state through these feedback mechanisms that can either oppose or strengthen the original radiative perturbation.

In the case of cloud feedbacks, those two opposing components (shortwave and longwave) could each change in either direction in response to a radiative perturbation, so the net effect of cloud feedbacks could also go in either direction. Models disagree a lot about what the cloud net response to greenhouse gas increase is; the CMIP5 ensemble had -0.04±0.53 W/m2/K in response to a CO2 doubling, but individual models can go in either direction.

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In a lecture we learned that instantaneously removing all clouds from earth would give radiation forcing of about 18W/m², leading to significant warming.

First of all, this is non-physical. The Earth is about 70% ocean. The only way to eliminate all clouds from the atmosphere would be an Earth covered from pole to pole by ice. (This may have happened long ago, in the Cryogenic.) Water has an extremely low albedo, and land has a much lower albedo than clouds. Much of the Earth's roughly 30% albedo results from clouds. Magically removing all clouds (and this would require magic) would significantly reduce the Earth's albedo. That magic is the source of that increased radiative forcing. But it is not physical.

However, in the same lecture we learned that clouds do have a positive feedback, which means (in my opinion), that by increasing temperature we have an increased amount of net radiation to earth. How is this to be understood?

I suspect you may have a misunderstanding of what "positive feedback" means. If you've ever attended a concert or speech where someone goes up to the microphone and says "testing, testing, testing", they are testing whether the sound system is exhibiting positive feedback. Positive feedback is a bad thing in this regard. The "testing" turns into an awful screech if positive feedback is present. Positive feedbacks in climate science are things that magnify the effects of increased climate forcings. Negative feedbacks counteract the effects of increased climate forcings.

Whether smallish increases or decreases in cloudiness is a positive or negative feedback has long been debated. (Keep in mind that 0% cloudiness is non-physical, so no one investigates this.) There apparently is a difference between high clouds versus low clouds, but the community is leaning toward increased cloudiness being a net positive feedback (a bad thing) rather than a net negative feedback (a good thing).

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The cloud radiative feedback, as commonly referred to, is a measure of the clouds's effect on radiation change due to changes in temperature (i.e. "climate"):

Climate warming affects both cloud number and cloud properties, which in turn affect warming itself.

Clouds reflect shortwave radiation back to space, yes, but they also reflect longwave ratiation down, back to earth, warming the surface. Remember that water vapor is a natural greenhouse gas. Therefore clouds, or moisture in general, increase the greenhouse gas effect. This is illustrated in the image below taken from this website, referred to as "back radiation".

enter image description here

The net effect seems to be positive, but uncertainties are large. Dessler et al. (2010) estimate the cloud feedback magnitude at 0.54 plus/minus 0.74 (two standard deviations) (W/m^2)/K. Note that the uncertainties are larger than the mean estimate.

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