Analysis: Incoming and Stored Thermal Energy

Note: This post is highly mathematical.

I define the “climate zone” of a planet to be a region with an upper boundary above where the atmosphere becomes too thin to significantly affect the exchange of energy and with a lower boundary some distance below the surface. The analyses I offer will generally apply regardless of the precise positioning of that lower boundary. Tentatively, I suggest taking the lower boundary of the “climate zone” to be perhaps 1 km below the surface on land, and at the bottom of the ocean otherwise.

Incoming Thermal Energy

Let S be the net rate at which thermal energy enters the climate zone. This rate S may be decomposed as follows:

(1)   \begin{equation*} S = \Sn + S_x \end{equation*}

where the net absorbed solar irradiation \Sn is given by:

(2)   \begin{equation*} \Sn = (1-\albedo)\,\isi \end{equation*}

In these equations, \isi is the solar irradiation at the top of the atmosphere (TOA); the albedo, \albedo, is the fraction of incoming solar irradiation reflected back to space; and S_x reflects any non-solar sources of heat entering the climate zone, either from within it, or entering from below through the lower boundary of the climate zone.

Some planets and moons experience significant heat flux from the planetary interior. This is true, for example, for some of the moons of gas giants with eccentric orbits and associated significant tidal heating. Mainstream science does not currently believe that the term S_x has any significant impact on Earth’s climate, but this analysis aims to make no assumptions or approximations, so I will include this term.

The above quantities and the equations that relate them may be interpreted as either local (at a particular place and time on the globe) or global (averaged or summed over the entire globe, and over some period of time). It’s simple to derive the relationship between the local quantities (which will be denotes with underlines) and the global quantities (which will be denoted with overbars).

I will be making use of the notation, definitions, and identities presented in Averages and Correlations. Note that one could equally well work with sums; the results are equivalent and interchangeable.

Taking averages of the local versions of the above equations leads to:

(3)   \begin{equation*} \ex{\xloc{S}} = \ex{\xloc\Sn} + \ex{\xloc{S_x}} \end{equation*}

(4)   \begin{equation*} \ex{\xloc\Sn} = \left(1-\frac{\ex{\xloc\albedo\cdot\xloc\isi}}{\ex{\xloc\isi}}\right)\,\ex{\xloc\isi} \end{equation*}

These may be rewritten in terms of global quantities as:

(5)   \begin{equation*} \xglob{S} = \xglob\Sn + \xglob{S_x} \end{equation*}

(6)   \begin{equation*} \xglob\Sn = \left(1-\albedog\right)\,\xglob\isi \end{equation*}


(7)   \begin{equation*} \xglob{S} = \ex{\xloc\Sn} \end{equation*}

(8)   \begin{equation*} \xglob\Sn =  \ex{\xloc\Sn} \end{equation*}

(9)   \begin{equation*} \xglob{S_x} = \ex{\xloc{S_x}} \end{equation*}

(10)   \begin{equation*} \albedog = \frac{\ex{\albedol\cdot\xloc\isi}}{\ex{\xloc\isi}} = \exw{\albedol}{\xloc\isi} = \mc{\albedol}{\xloc\isi}\cdot\ex{\albedol} \end{equation*}

The complete global equation for the rate of energy entering the system is thus:

(11)   \begin{equation*} \xglob{S} = \left(1-\albedog\right)\,\xglob\isi \,+\, \xglob{S_x} \end{equation*}

We essentially knew this from the outset, but now we know how to the local and global versions of the quantities relate to one another.

This result can also be expressed in terms of sums. The total energy entering the planetary “climate zone” over the globe and over a particular period of time is given by:

(12)   \begin{equation*} \osum{\xloc{S}} = \left(1-\frac{\osum{\albedol\cdot\xloc\isi}}{\osum{\xloc\isi}}\right)\cdot\osum{\xloc\isi}\;+\; \osum{\xloc{S_x}} \end{equation*}

The global average incident solar radiation, \xglob\isi, may be expressed in terms of \TSI, the total solar irradiance at Earth’s position in space. If Earth is treated as a sphere, it can be shown that:

(13)   \begin{equation*} \xglob\isi = \frac{\TSI}{4} \end{equation*}

However, for accurate work, it is necessary to consider the oblate-spheroid shape of the Earth. This can be accounted for by replacing the 4 with 4.0034.

Stored Thermal Energy

A planet is generally a closed system, as this term is used in thermodynamics. This means that energy can enter or leave, but matter doesn’t (at least not in significant amounts). A closed system in principle contains some amount of internal energy, U. Internal energy includes both internal kinetic energy which reflects temperature and internal potential energy, which reflects things like whether matter is in a solid, liquid or gas phase. As the temperature of matter in a system increases, the internal energy U increases, and as the temperature of matter in the system decreases, the internal energy U decreases.

In other words, as matter warms, it stores energy as internal energy. The rate of energy storage may be denoted as Q_s where:

(14)   \begin{equation*} Q_s = \deriv{U}{t} \end{equation*}

The opposite of energy storage is energy release from storage. As matter cools, it releases energy from its internal energy. The rate of energy release from storage may be denoted Q_r where:

(15)   \begin{equation*} Q_r = -\deriv{U}{t} = -Q_s \end{equation*}

If we use an overbar to denote the global energy release or storage over some period, we can write:

(16)   \begin{equation*} \xglob{Q_s}= -\xglob{Q_r} \end{equation*}

When matter is warming overall, \xglob{Q_s} is positive, and when matter is cooling overall, \xglob{Q_r} is positive.

These ideas may equivalently be expressed in terms of sums over the local quantities:

(17)   \begin{equation*} \osum{\xloc{Q_s}} = -\osum{\xloc{Q_r}} \end{equation*}

Note that the term \xglob{Q_s} (and, equivalently, \xglob{Q_r}) is intended to address the storage (and release) of energy that:

  • is thermal energy (i.e., internal kinetic energy or internal energy associated with phase transitions) or which
  • is chemical energy which will be returned to being thermal energy within at most a few thousand years (as in the case of photosynthesis producing biomass which is subsequently metabolized or decomposed).

In the case where energy is converted to or from chemical energy that is stored for millions of years, as in the case of coal formation or the burning of fossil fuels, such effects should be included in the non-solar-heating term, S_x, not in the term \xglob{Q_s} (and \xglob{Q_r}). The math works out the same either way, with respect to the final temperature of the planet, but including energy flows in the right term supports improved understanding and interpretation of the meaning of the formulas.