SpidermanTUba

I mean that each additional ppm produces a smaller forcing increment. A diminishing positive increment does not have to become negative.

For the approximate relationship in the post,

?F = a × ln(C/C0), with a > 0,

the slope is dF/dC = a/C > 0, while d²F/dC² = -a/C² < 0.

So the curve keeps rising while its slope decreases. There is no point where this formula turns downward. For example, doubling from 400 to 800 ppm and doubling from 800 to 1600 ppm give the same increment, a × ln(2), in this approximation. That is different from adding another 400 ppm each time.

If both axes are linear, forcing versus CO2 concentration is a rising, concave-down logarithmic curve. If the concentration axis is logarithmic and the forcing axis is linear, it becomes a straight line. It is not exponential growth on linear axes.

The logarithmic approximation has a limited physical range; I am not extrapolating it to arbitrary concentrations or using it as a temperature formula. The spectroscopy and radiative-transfer calculations behind it are discussed here:

Romps, Seeley and Edman: Why the forcing from carbon dioxide scales as the logarithm of its concentration

"preborn children"

lol

This is the fourth post in the series. The previous post connected radiation escaping to space with the atmosphere's temperature profile. Here is why adding CO2 still changes that radiation even when some absorption lines are already opaque.

1. What quantum mechanics actually says

A molecule absorbs infrared radiation through allowed transitions between its energy levels. The photon's energy must match the transition: energy difference = h × frequency, where h is Planck's constant. The transition must also satisfy the relevant selection rules; matching energy alone is not enough.

CO2's bending vibration produces an important absorption band near 15 micrometres. Rotation adds many closely spaced transitions. Molecular motion and collisions give the lines finite widths. These are measured spectral properties, not an assumption that CO2 absorbs every wavelength equally.

HITRAN's spectroscopy definitions and data parameters
MIT spectroscopy lecture: CO2 vibrations, lines and broadening

2. An opaque line centre is not an opaque spectrum

For an incident beam at a particular frequency, its transmitted fraction is:

transmitted fraction = exp(-optical depth)

Optical depth measures the absorbing material along the path, weighted by its absorption strength. Holding temperature, pressure and path fixed, doubling the CO2 amount doubles its contribution to optical depth.

For example, increasing optical depth from 1 to 2 reduces direct transmission from about 37% to 14%. Where optical depth is already 10, direct transmission is already tiny and adding more makes little difference to that beam. Both statements can be true at different frequencies.

The strong line centres can therefore be saturated while weaker parts of the band still respond. Adding CO2 primarily adds absorbers; it does not need to produce a large increase in total air pressure or individual line widths.

3. The atmosphere also emits

The transmission equation alone is not a complete atmosphere. Each layer emits infrared radiation as well as absorbing it, and collisions exchange energy among the gas molecules.

What reaches space includes radiation from the surface and from many atmospheric layers, attenuated along the way. More CO2 changes the layers and temperatures that contribute at each frequency. In much of the troposphere, temperature decreases with height, so shifting the contribution upward can reduce outgoing radiation at the existing temperatures.

That is not a claim that every wavelength comes from one altitude or that every atmospheric layer gets colder with height. The stratosphere matters too. The calculation integrates the whole spectrum and atmospheric profile. The surface and atmosphere then adjust until outgoing energy again balances absorbed sunlight.

Pierrehumbert: Infrared radiation and planetary temperature

4. Diminishing returns are still returns

Over a useful range around Earth's present conditions, CO2 radiative forcing is approximately logarithmic:

change in forcing = a × ln(C/C0)

C/C0 is the ratio of final to initial CO2 concentration. Equal percentage increases therefore give approximately equal forcing increments. Doubling adds a × ln(2); another doubling adds approximately the same again. That is different from adding the same number of ppm each time, and it is not a temperature formula.

The coefficient depends on the forcing definition and atmospheric conditions. A doubling produces a forcing of roughly 4 W/m². Understanding the logarithmic relationship requires the distribution of absorption strengths across the band; merely saying 'the lapse rate' does not derive it.

Romps et al.: Why the forcing from carbon dioxide scales as the logarithm of its concentration

5. There is an observational check

Feldman and colleagues analysed infrared spectra at sites in Oklahoma and Alaska over 2000-2010. Using those measurements, other atmospheric observations and radiative-transfer calculations, they isolated an increase in clear-sky downward surface radiation associated with the roughly 22 ppm CO2 increase: about 0.2 W/m² per decade at each site.

That is a local surface result, not the global top-of-atmosphere forcing or total greenhouse effect. It nevertheless checks the claim that additional CO2 has no radiative effect.

Feldman et al.: Observational determination of surface radiative forcing by CO2

So 'some lines already absorb nearly everything' is a valid observation. 'Therefore adding CO2 cannot change Earth's energy balance' does not follow. The question is what happens across the full spectrum in an atmosphere that both absorbs and emits.

Because a study can estimate a defined quantity while having limits. This one fitted heat-mortality relationships to observed temperature and death records, then compared climates with and without human influence using ten CMIP6 models, bias-corrected against local weather stations. It reports uncertainty from both the risk curves and differences among models.

Missing regions limit geographic coverage; an average risk curve cannot resolve changing vulnerability year by year. That deserves scrutiny, but does not establish that the estimated added burden is zero.

Yes, I already acknowledged the roughly 54% miss for Hansen's 1988 Scenario B. That was a particular global-warming projection through 2017, with much of the mismatch traced to assumed future forcing. It is not a measured 54% error in this different study. Which model result or assumption here do you think fails, and against what evidence?

Heat-mortality methods and limitations: LINK
Hansen comparison: LINK
Scientific American explanation: LINK

quote:

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Surely everyone has at least 3-4 close friends outside their family they talk to ....

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of course.

quote:

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...daily or close to it.

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lol. it's a shame youth is wasted on the young.

I agree that a policy needs explicit objectives, costs and tradeoffs. That is separate from claiming that physics specifies one perfect global temperature.

The measurable outcomes include heat exposure, coastal flooding, food-production risks and ecosystem losses under different warming levels. The IPCC assessment compares those risks and their uncertainties; it does not require human extinction as the test of whether harm matters.
LINK

A temperature limit is a chosen risk-management objective. Whether to pursue it through a particular tax, nuclear expansion, other energy sources or adaptation is an engineering and policy debate. The physical effect of added CO2 can be tested regardless of which policy anyone prefers. Which outcome and proposed policy would you like to compare quantitatively?

Carbon is essential to life. Reducing the additional CO2 released from fossil fuels does not mean removing all carbon, all atmospheric CO2 or photosynthesis. Those are different proposals.

The claim that life has always flourished when CO2 was high also misses major counterexamples. Penn et al. examined the end-Permian extinction: greenhouse-driven ocean warming and oxygen loss reproduced a major part of the extinction pattern seen in the fossil record. They did not simply assume that warmer meant better or worse.
LINK

That ancient event is not a prediction that the same extinction happens tomorrow. It shows why 'carbon supports life' cannot establish that any added CO2 is harmless. Concentration, rate of change, other environmental conditions and the organisms involved all matter. Which specific CO2 Coalition chart or paper would you like to test?

You're right that 'major' is an intensity threshold: Category 3-5. Counting storms above that threshold measures the frequency of strong storms; it does not give the average intensity of every storm or its rainfall. I already agreed that a century-scale increase in U.S. hurricane landfalls is not established.
LINK

But temperature analyses do not all begin in the 1970s. NASA's GISTEMP runs from 1880 and uses 1951-1980 as its reference average. A reference average sets the zero on an anomaly graph; subtracting the same constant from every year cannot create a warming trend. The record's start date, the comparison interval and the reference average are three different things.
LINK

Which temperature series and exact interval are you challenging? Then I can examine that calculation, including its uncertainty.

Thanks for the specific link. I can reproduce 100 versus 79 hurricanes, and 30 versus 25 major hurricanes, by summing NOAA's 1871-1920 and 1971-2020 rows. That table covers strikes on the mainland United States, not just the Gulf or the whole world. A count of major storms also does not measure the average intensity of every storm.
LINK

I already agreed that there is no strong evidence of a century-scale increase in U.S. landfalling hurricanes. So those counts do not contradict that point. Hurricane frequency, rainfall and intensity need separate tests; one count cannot settle them all.
LINK

Cold-related mortality matters too. The study I cited estimates the additional heat burden by comparing temperatures with and without human influence. It does not calculate the combined heat-and-cold balance, and I would not use it as though it did.

The estimate depends on the temperature-mortality relationship and the climate counterfactual, with uncertainty and geographic limits. Those are fair things to examine. Here is the free author manuscript, including its methods and limitations:
LINK

I mean risks to people and infrastructure: heat-related illness and deaths, coastal flooding, and losses to crops and water supplies. These vary by place, exposure and adaptation.

One concrete example: Vicedo-Cabrera et al. used mortality and temperature data from 732 locations in 43 countries to estimate the heat-mortality contribution of human-induced warming over 1991-2018. That is a counterfactual attribution estimate with uncertainty and geographic limits.

A temperature target is a policy choice about acceptable risks and costs. Climate research can estimate how outcomes differ under specified emissions and adaptation assumptions even if no single target is agreed. Physics does not predict what governments will choose. Those comparisons are useful for choosing and evaluating policies.

Which risk would you like to examine quantitatively?

Heat study: LINK
Impact assessment: LINK