Quantities and Symbols

Note: This page is highly mathematical.


This document offers a list of quantities I reference and symbols for these that I use on this site.

Some quantities are described by different people using different names and different symbols. Some quantities are likely unique to my own way of organizing concepts. I try to mostly use standard terms and symbols when these are well-established, but I’ve adjusted terms and symbols to try to achieve a measure of consistency.

Symbols come in two variants: long symbols and short symbols. Long symbols are often multi-letter, to support easier memory of what they are supposed to mean. Short symbols are a single letter (possibly with a subscript), to make it easier to work with them when doing serious mathematical analysis.

Some quantities are evaluated at the surface, while others are evaluated at “top of atmosphere” (TOA), at the interface between the atmosphere and space.

In what follows, I name the units in which quantities are most often presented. In most cases, SI units are specified, but there are exceptions. Any quantities in non-SI units should be converted to SI units when doing calculations.

Table of Contents

Local and global quantities

For most quantities listed below, there is both a location-dependent version of the quantity that varies by location around the globe, and a global average version of the quantity. Sometimes the global average version is a straight average over the surface area of the planet, and sometimes it is a weighted average, weighted to make the averaging method appropriate to how that quantity is likely to be used.

In any text that uses quantities that have both location-dependent and global average versions, I will often explicitly say which version is being used. If not, this will often be clear from the context.

To indicate whether a quantity X is global or location-dependent, I may use the notation \glob{X} or \locd{X} (where \loc indicates latitude and longitude).

When I am using the convention X=\locd{X} the global version of X is typically expressed as \ex{X} or \avg{X} where these notations represent unweighted and weighted global averages, respectively.

List of quantities and symbols


  • \sigmaStefan-Boltzmann constant (units: W m-2 K-4)
  • cSpeed of light in vacuum (units: m/s)
  • hPlanck’s constant (J/Hz)
  • \kbBoltzmann constant (J/K)


  • \thetaLatitude (units: radians formally, degrees informally)
  • \phiLongitude (units: radians formally, degrees informally)
  • \locLocation, an abbreviation for latitude and longitude (\theta, \phi)
  • tTime (units: seconds formally)
  • f or \nuFrequency of electromagnetic radiation (units: Hz)
  • f/c or \nu/cWavenumber of electromagnetic radiation (where c is the speed of light) (units: cm-1)
  • \lambda = c/fWavelength of electromagnetic radiation (units: microns, 𝜇m)

Shortwave Radiation Fluxes

Not normalized to Earth’s spherical surface area:

  • \TSI or \tsi ― [TOA] ― Total Solar Irradiance (units: W/m2)

Normalized to Earth’s spherical surface area (as are heat and longwave fluxes):

  • \MSI or \msi = \TSI/4 = \ex{\ISI} ― [TOA] ― Mean Solar Irradiance (units: W/m2)
  • \ISI or \isi ― [TOA] ― Incoming Solar Irradiance / Insolation (units: W/m2)
  • \Sn = (\Ss + \Sa) ― [TOA] ― Net solar irradiance absorbed (units: W/m2)
  • \Sa ― [atmosphere] ― Solar irradiance absorbed by atmosphere (units: W/m2)
  • \Ss ― [surface] ― Solar irradiance absorbed by surface (units: W/m2)

Heat Fluxes

  • \SNR or \snr ― [surface] ― Surface non-radiative heat loss (latent and sensible heat flows from the surface to the atmosphere) (units: W/m2)
  • \SRH or \srh = (\SLR-\DLR) ― [surface] ― Surface radiative heat loss (units: W/m2)
  • \SXH or \sxh = (\Ss-\SNR-\SRH) ― [surface] ― Surface excess heating (units: W/m2)
  • \TXH or \txh = (\isi -\OLR) ― [TOA] ― TOA excess heating (units: W/m2)

Longwave Radiation Fluxes

  • \OLR or \olr ― [TOA] ― Outgoing longwave radiation (units: W/m2)
  • \DLR or \dlr ― [surface] ― Downwelling longwave radiation at surface (units: W/m2)
  • \SLR or \slr = (\emis \sigma \Tsurf^4) ― [surface] ― Surface longwave radiation emissions (upwelling from surface) (units: W/m2)


Well-known environmental parameters:

  • \epsilon ― [surface] ― Emissivity (units: dimensionless, 0-1)
  • a ― [TOA] ― Albedo (units: dimensionless, 0-1)
  • \lapseLapse rate (environmental lapse rate in troposphere) (units: K/km)

Parameters characterizing the impacts of the atmosphere not being entirely transparent to longwave radiation (note: parameter values would be 1 or 0 for a LW-transparent atmosphere):

  • \GHE or \ghe = (\SLR-\OLR)Greenhouse effect (units: W/m2)
  • \nghe = (\GHE/\SLR) = \leaNormalized greenhouse effect (dimensionless, 0-1)
  • \LBTB or \lbtb =  1/\fourthroot{\ltr} = \fourthroot{(\SLR/\OLR)} = 1/\fourthroot{(1-\lea)} ― Baseline longwave temperature boost factor (units: dimensionless, \geq 1)
  • \LTR or \ltr = (\OLR/\SLR) = (1-\lea)Longwave effective transmittance (units: dimensionless, 0-1)
  • \LEA or \lea = (1-\OLR/\SLR) = \ngheLongwave effective absorptance (units: dimensionless, 0-1)
  • \LCR or \lcr = (1 - \DLR/\SLR)Longwave cooling reduction factor (units: dimensionless, 0-1)
  • \LRCF or \lrcf = (\DLR/\SLR)Longwave recirculation fraction (units: dimensionless, 0-1)

Parameters characterizing the impacts of temperature varying away from the average value (note: parameter values would be 1 or 0 for uniform temperatures):

  • \TVEB or \tveb = (\ex{\tsurf^4}/\ex{\tsurf}^4)-1 ―Baseline temperature variation emissions boost factor (units: dimensionless, \geq 1)
  • \TVTR or \tvtr = \fourthrootc{1/\tveb} ―Baseline temperature variation temperature reduction factor (units: dimensionless, \leq 1)
  • \TVCA or \tvca = (\Delta\tsurf - \ex{\Delta\tsurf}) ― Incremental temperature change anomaly (units: K or ℃)
  • \TVTS or \tvts = -\exw{\tvca}{\slr/\tsurf} ― Incremental temperature variation temperature shift (units: K or ℃)


  • \Tsurf or \tsurfSurface temperature

Effective Emission Height Model Parameters

  • \Tolr or \tolr = (\OLR/\sigma)^{\frac{1}{4}}Upwelling/Outgoing effective emission temperature (units: K)
  • \Tdlr or \tdlr = (\DLR/\sigma)^{\frac{1}{4}}Downwelling effective emission temperature (units: K)


  • \Delta F ― TOA ― Radiative forcing (units: W/m2)

Climate Response

  • \alphap = -4\bex{\OLR/\tsurf}Planck Response (units: W m-2 K-1)
  • \ECSEnvironmental Climate Sensitivity (units: W m-2 K-1)
  • \TCRTransient Climate Response (units: W m-2 K-1)


For some variables relevant to climate, it is most useful to consistently use a weighted average for that variable. How various quantities are averaged is indicated in the following sections.

Quantities Averaged in an Unweighted Manner

  • All longwave, heat flow, and shortwave fluxes (aside from \mathrm{TSI} which is neither “local” nor an average)
  • Temperature

Quantities Averaged in a Weighted Manner

  • Emissivity ― \exw{\emis}{\tsurf^4}
  • Albedo ― \exw{\albedo}{\isi}
  • Temperature Variation Change Anomaly\exw{\TVCA}{\OLR/\tsurf}

Quantities Averaged using Nonlinear Averaging

  • Upwelling/Outgoing effective emission temperature ― Global average emission temperature is computed as \fourthroot{(\ex{\OLR}/\sigma)}.
  • Downwelling effective emission temperature ― Global average emission temperature is computed as \fourthroot{(\ex{\DLR}/\sigma)}.

Averaging Not Classified

  • Lapse rate
  • Forcing
  • Climate response
  • Longwave parameters
  • Temperature distribution parameters
  • Longwave baseline temperature boost factor
  • Longwave transmission reduction
  • Longwave recirculation fraction
  • Radiant cooling reduction factor