# Quantifying the Greenhouse Effect

When I began having conversations about the atmospheric Greenhouse Effect (GHE), those conversations were often burdened by vagueness.

It was said that the GHE meant “the Earth would be colder if it weren’t for the presence of Greenhouse gasses.” Unfortunately, that formulation seemed to leave the GHE in the realm of the hypothetical.

It led to fruitless arguments about the parameters of the hypothetical scenario. After all, water vapor is a major Greenhouse gas (GHG), and if that was removed, a lot of other things about the situation would change, not just the absorption and emission of longwave radiation in the atmosphere. And, that way of talking about the GHE also made it difficult to point to anything observable, in the world as it is, that represents the GHE.

So, I was relieved when I finally (all too recently) came to understand that there is a measure of the Greenhouse effect that is entirely quantifiable and measurable in the world as it is.

Quantitatively, the GHE is the difference between the radiative flux of longwave (LW) thermal radiation emitted by the Earth’s surface (denoted ) and the radiative flux of outgoing LW thermal radiation emitted to space at the top of Earth’s atmosphere(denoted ):

(1)

For present-day Earth, the values of these quantities are around = 398 W/m2, = 239 W/m2, and = 159 W/m2.

Why does it make sense to refer to this as being the quantitative definition of the Greenhouse effect? There are two reasons:

1. If there was nothing in the atmosphere that could absorb, emit, reflect, or scatter LW radiation (the things said to be responsible for the GHE), then the amount of LW thermal radiation reaching space would have to equal the amount emitted by the surface, . That would mean the Greenhouse effect, as I’ve defined it, would be zero, . Thus, the value of is a measure of the energetic effect of atmospheric materials which interact with LW thermal radiation.
2. , as it has been defined, directly relates to the portion of a planet’s surface temperature that is attributable to the effect of atmospheric materials which interact with LW thermal radiation.

## Relationship to surface temperature

Let’s look at how and relate to temperature. The Stefan-Boltzmann law (SB law) tells us that

(2)

where is the emissivity (a value between zero and 1, typically close to 1, which is characteristic of the material emitting thermal radiation), is the Stefan-Boltzmann constant, and is the temperature of the surface.

The SB law applies at each individual location. However, one can use it to obtain the global average by averaging the equation globally. That leads to a version of the SB equation which applies globally:

(3)

In this SB equation for global averages, is the global mean surface longwave radiative flux, is the global mean surface temperature (), is what I call the temperature-variation emissions boost factor (greater than or equal to 0), where is defined as the ratio of the average to , and is the weighted global mean emissivity (weighted by ).

So, given , , and , we can calculate the average temperature.

• is estimated to be 398 W/m2.
• = 0.0187 ± 0.0003 for Earth based on my analysis of CERES data.
• One data set puts Earth’s area-weighted average surface emissivity at = 0.935. However, I don’t think there is universal agreement on this value.

These values yield = 293 K / 20℃ / 68℉.

We can think of the surface temperature, , as being equal to the sum of two parts:

(4)

Here, the “no GHE” temperature, , is the temperature we would compute from the SB law if as would necessarily be the case if there was nothing in the atmosphere that could intercept LW radiation, and is the portion of the temperature that we attribute to the existence of the Greenhouse effect.

Using the SB law for global averages, one can show that:

(5)

(6)

For Earth, = 0.1197 .

Assuming and , then = 258 K / -15℃ / 5℉ and = 35℃ / 63℉.

So, as advertised, the quantity relates to the temperature of the surface. The Earth’s surface temperature can be thought of as involving two terms:

• A term , which is the temperature Earth’s surface would necessarily have if (a) there were no materials in Earth’s atmosphere that could interact with LW thermal radiation, and (b) the outgoing longwave radiation flux, , and emissivity, , were unchanged.
• A term , which is non-zero only because atmospheric materials that interact with LW thermal radiation make it possible for to be different than .

## GHE as energy vs. GHE as temperature

The quantities and can be thought of as, respectively, the expression of the GHE in terms of radiative flux, and the expression of the GHE in terms of surface temperature.

Either of this quantities could legitimately be called “the Greenhouse effect.”

However, in technical discussions about climate, I believe it’s usually more useful to talk about the GHE expressed in terms of radiative flux, as define by the quantity I’ve denoted .

Note that in some of my work I might use the single-letter symbol, , to denote the quantity as I’ve defined it above.

## Dimensionless Greenhouse effect

It can be useful to express the GHE in terms of a dimensionless value:

(7)

The symbol denotes the normalized greenhouse effect. The tilde over the “g” helps avoid confusing it with the gravitational constant, (9.8 m/s2), but for casual work it can be okay to omit the tilde and use to denote the normalized GHE, as long as it’s clear what is being talked about. In some of my work, I’ve alternatively thought of this quantity as being the longwave effective absorptance, , since it indicates the fraction of LW surface emissions that effectively do not reach space.

is a dimensionless quantity between 0 and 1. It is zero when no atmospheric materials interact with LW thermal radiation.

For Earth at present, 0.4.

Other quantities may be expressed in terms of as follows:

(8)

(9)

(10)

or is the effective LW absorptance of the atmosphere, not the actual LW absorptance:

• The actual LW absorptance indicates what fraction of LW power sent upwards through the atmosphere is eventually absorbed.
• or indicates what fraction of LW power sent upwards through the atmosphere is eventually absorbed without be replaced by an equivalent amount of LW power emitted from within the atmosphere; the net effect after both absorption and emission are taken into account.

## Global and local versions of GHE

The GHE is usually talked about as a global effect related to global average radiative fluxes and global average surface temperature. All the specific values I’ve calculated for Earth have reflected this global orientation.

However, it is also valid to interpret the quantities I’ve worked with above as being local or location specific, i.e, having values which depend on the particular latitude and longitude, . In particular, , , and can be evaluated locally, provided we omit the factor which is relevant only in the context of global mean values.

Each equation I’ve presented so far remains valid if the quantities are interpreted locally rather than globally (aside from the factor being present in global equations and absent in local equations).1Note, however, that this particular description of the Greenhouse effect hasn’t included equations derived from TOA energy conservation. Those equations would be need to be modified to be valid, if interpreted locally. In particular, one would need to include a term for advection or lateral heat transport. Lateral heat transfer averages to zero globally, and so does not appear in global equations.

It may be useful to be more explicit about the relationships between local and global values. For some quantities, the global value needed to make the equations work is a simple area-weighted global average. For others, it is a weighted average. And, for still others, deriving the global value is best done by referring to another global value, rather than through working directly with the location-specific values.

I’ll use the notation to refer to the unweighted (i.e., area-weighted) global average of , and to refer to the global average of weighted by . To explicitly differentiate local and global versions of a quantity, I will write the local quantity as and the global quantity as . For many quantities, the global value will simply be the unweighted average of the local value. But, for some quantities, the global quantity will be calculated another way.

For the following quantities, the relevant global version of the quantity is simply the unweighted global average:

• Greenhouse effect:
• Surface temperature:

The global values of these quantities are calculated as the indicated weighted average:

• Emissivity: (and replace with )
• Dimensionless GHE or :

The global values of these quantities should be calculated from other global quantities, rather than trying to calculate them from their local values:

• GHE temperature component:
• No-GHE temperature:

In the global equations, if is not uniform, then wherever appears, it should be replaced by , where or is the temperature variation emissivity boost factor, , defined as:

(11)

## Small changes

For small changes in temperature and emissions, one can effectively linearize the SB law. Small changes in surface thermal emissions and surface temperature are related, to a good approximation, by:

(12)

(13)

These equations apply to both local and global quantities.

## Response to TOA radiative forcing

It is important to keep in mind that radiative forcing is not equal to change in .

The effect of GHGs is often quantified in terms of the top-of-atmosphere (TOA) radiative forcing it produces. The TOA radiative forcing, associated with a change in GHG concentration is the negation of the change in that would result if the GHG concentration change happened instantly and the surface wasn’t able to make any adjustments in response (so is unchanged), and atmosphere was only allowed to make a few minor adjustments. (There are lots of technicalities in the definition, and several variants of the definition.)

Momentarily and theoretically, given those constraints:

(14)

This implies an equivalent change in :

(15)

However, it’s important to notice that this state, with (and with ) is an initial state, in which there is a TOA energy imbalance and the system is not in thermal equilibrium.

By the time that response happens, there will be no flux or change in flux anywhere in the system that has a value . Instead, there will be an equilibrium state in which will be zero and has some nonzero value.

There is no need for to equal any more than there is a need for the distance that one moves one end of a lever (the “input” movement) to match the distance that the other end of the lever will travel (the “output” movement).

Just as with being smaller than by a factor , with being smaller than by a response factor defined so that:

(16)

(17)

Climatologists use an (arguably regrettable) standard sign convention for such that the net value of is generally negative, and a negative value of means that a positive forcing at TOA leads to a surface temperature increase and surface emissions increase . (For that reason, when using official notation, I usually write , since I expect that will be positive.)

I’ve only seen be used in the context of global equations, not location-specific equations. Although we could define a location-specific multiplier, that would appear to be outside the scope of what I’m aware of others doing. Yet, if you’re going to write location-specific equations, be aware that the actual multiplier will surely vary by location.

The value of depends on whether one puts any constraints on the system’s ability to respond, and on how long one waits for a response. As one waits for a response, different aspects of the climate system have time to respond.

The IPCC typically focuses on three different time frames, a short-term time frame, a 70-year mid-term time frame which is reported on via a metric called the Transient Climate Response, TCR, and a centuries long long-term time frame which is reported on via a metric called the Equilibrium Climate Sensitivity, ECS.

It’s critically important to understand that, even in the short term .

I’ve shown elsewhere[LINK] that for an instantaneous forcing , if the system is constrained so that nothing but surface temperature is permitted to adjust as the system comes to equilibrium, then:

(18)

(19)

(20)

I’m using location-specific versions of , , and in these equations, and am explicitly indicating how they are averaged.

For Earth at present, 3.3 W⋅m-2⋅K-1.

The IPCC defines a response called the Planck response, , which corresponds to similar constraints (though there may be nuances that I have yet to digest). They report that 3.3 W⋅m-2⋅K-1 is a “crude guess” at the value of , but estimate a most likely value of = 3.2 W⋅m-2⋅K-1. (See IPCC Climate Change 2021: The Physical Science Basis, p. 968.)

## Spectral version of GHE

One can define versions of and that characterize the GHE for specific frequencies or wavelengths of LW radiation.

Suppose and are the spectral irradiance curves that describe the spectra of LW emissions at the surface and TOA, respectively.

Then we could define the spectral GHE, , as:

(21)

Similarly, we could define the spectral longwave effective absorptance (or spectral normalized GHE), , as:

(22)