Zero-dimensional energy balance model of the climate system

(There are many types of models of the climate system, whose complexity is various.)

The simplest physics-based model of the climate system.
Governing equation: (only) conservation of energy.
Spatial dimension: spatial variation in east-west, north-south, up-down dimensions are all ignored. (Any quantity is assumed to be uniform in east-west and north-south directions, and to have a constant profile except a single-value offset in the vertical direction.)
Energy stock is assumed to be (constant) heat capacity multiplied by temperature.
Energy input is absorption of solar radiation (mainly visible and near infra-red).
Energy output is emission of terrestrial radiation (thermal infra-red).
For convenience, quantities are expressed per unit area of the earth's surface.
C dT/dt = (1/4) (1 - α_p) S - F_out
- C: (effective) heat capacity of the climate system per unit area [J/(m² K)]
- T: representative temperature of the climate system [K]
- 1/4: ratio between the cross-sectional area and the surface area of a sphere
- α_p: reflectance of solar radiation by the earth as a whole, or "planetary albedo". [non-dimensional]
- S: solar irradiance (energy flow per unit time per unit area) at a plane perpendicular to the sun-earth line. So-called "solar constant". [W/m²]
- F_out: radiance (energy flow per unit time per unit area) of terrestrial radiation emitted to space [W/m²]
Steady state
- T does not change in time. (S and α_p are assumed to be constant as well.)
- 0 = (1/4) (1 - α_p) S - F_out

Steady state, Assumption 1: F_out is blackbody radiation at temperature T.
- 0 = (1/4) (1 - α_p) S - σ_B T⁴
- Assume that S = 1.36 × 10³ W/m² [Corrected after lecture], and α_p = 0.30 (according to average values of modern observations).
- Then, T = 254 K = approx. -19 ℃
- This T is the representative temperature of the terrestrial radiation. (Note: Terrestrial radiation in the real world is a mixture of radiation emitted by the land and sea surface, by cloud top, and by atmospheric gases. It is not uniform in space.) This T is called "effective radiative temperature of the earth" T_e.
Steady state, Assumption 2: F_out is {blackbody radiation at temperature T} multiplied by ε_p.
- 0 = (1/4) (1 - α_p) S - ε_p σ_B T⁴
- ε_p may be (temporarily) called "apparent planetary emissivity".
- ε_p T⁴ = T_e⁴
- ε_p^(1/4) T = T_e
- T = T_e / ε_p^(1/4) = ε_p^(-1/4) T_e
- If ε_p < 1, T > T_e.
- Smaller ε_p ... "stronger greenhouse effect"
- If T = 287 K (14 ℃, actual global average surface air temperture), how much is ε_p ?

A stock-and-flow system as well as a feedback system
Useful for explaining gradual change between quasi-steady states
Governing equation: conservation of mass of water
Stock: mass of water, proportional to water level
Inflow: mass flow of water, independent of water level
Outflow: mass flow of water, increasing function of water level
For a given constant inflow and a given outflow function, a steady state exists. From any non-steady state, the system will approach the steady state.
What will happen if the inflow changes suddenly, and then kept at the new constant value?
What will happen if the outflow function changes suddenly, and then kept at the new constant value?

Note: There may be causes of climate change that cannot be represented by 0-dim. models.
Change of inflow of energy
- Change of energy flow emitted from the sun (Change of S)
- Change of matter in the space between the sun and the earth that interacts with radiation (unknown, but perhaps not important)
- Change of geometric configuration between the sun and the earth (important in 10000-year time scale)
- Change of absorption and reflection of solar radiation by the climate system (Change of α_p)
Change of outflow of energy
- Change of "apparent planetary emissivity" ε_p, in other words, change of "greenhouse effect"

Surface of land and ocean
- Snow and ice (feedback [to temperature])
- Vegetation, land cover (external forcing [as a 1st approx.])
Suspended liquid and solid particles in the atmosphere
- Clouds [H₂O] (feedback)
- Aerosols (external forcing)

Gaseous components of the atmosphere ("greenhouse gases")
- Water vapor [H₂O] (feedback)
- Carbon dioxide [CO₂] etc. (external forcing)
Suspended liquid and solid particles in the atmosphere (same as above)

(Note that this is not the dominant effect of the actual greenhouse made of plate glass.)
Zero-dimensional energy balance model, steady state.
Sum of 2 black arrows corresponds to (1/4) (1 - α_p) S .
The ground is assumed to be blackbody.
(a): the same as "assumption 1" above. Ground temperature T = T_e.
The atmosphere is assumed to be concentrated in a plate whose vertical dimension is negligibly small. Also, it is assumed to be transparent for solar radiation and opaque (same as blackbody) for terrestrial radiation.
(b): one plate
- Temperature of the atmospheric plate = T_e
- Ground temperature / T_e = 2^(1/4) = sqrt(sqrt(2)) = approx. 1.2
(c): 2 plates ... Ground temperature = 3^(1/4) T_e.
More plates, higher ground temperature.
The surface temperature of Venus (ca. 740 K) can be explained this way.
This model can produce indefinitely high temperature, but that is not real, because the separation of wavelength between solar and terrestrial radiation will not hold at very high ground temperature.

2019-May-16
MASUDA Kooiti