Conservation of energy in the "climate system"

Conservation of energy

"The first law of thermodynamics" = "The law of conservation of energy"

Conservation of X (in physics) ... No spontaneous generation of X, no spontaneous decay of X.

The everyday term "energy consumption" can be regarded as "useful energy decrease, despite total energy does not change." The meaning of "useful energy" is related to the second law of thermodynamics.

[Note added after lecture] There are many books that explain concepts of thermodynamics. One example is Atkins (2007, 2010).

Formulation of conservation of X in "isolated systems" and "open systems"

(Note: the term "closed systems" is avoided intentionally.)

• In a system where there is no exchange of X across the boundary, X does not change in time.
  • dX/dt = 0
• In a system where there are flows of X across the boundary (in a open system), X may change, but just because of net inflow of X.
  • dX/dt = F_{net in} = F_in - F_out
• If an open system is in a steady state where X does not change in time, net inflow of X is zero.
  • 0 = F_{net in} = F_in - F_out ... steady state

This concept of conservation is useful for "budget analysis", by which we can evaluate quantities which are not directly measurable.

Also, this concept of conservation is used for "prognostic" (i.e. forcasting-type) computations.

• ( X(t+Δt) - X(t) )/Δt = F_{net in}
• X(t+Δt) = X(t) + Δt * F_{net in}

Energy (as "stock")

• Unit
  • SI Unit: J [joule] = N m = kg m² / s² (conceptualized as a unit of mechanical work)
  • 1 cal (an older unit of heat. Heat required to raise temperature of 1 g of water by 1 deg.C.) = approx. 4.2 J .
  • 1 kcal = 1000 cal
• Sorts
  • mechanical energy
    • kinetic energy ... energy associated to motion ... (1/2) m v²
    • potential energy (of gravity force of the earth) ... m g z [Note: I forgot "z" when preparing for lecture. I added it later.]
  • internal energy (inexactly called "heat" or "thermal energy")
    • part related to temperature ("sensible heat")
      • approx. [internal energy] = [heat capacity] * [temperature]
        • [heat capacity] = [specific heat capacity] * [mass]
    • part related to phase (solid, liquid, gas) ... ("latent heat")
  • (potential energy of electromagnetic force)
  • (potential energy of forces between elementary particles)
• Conversion between various sorts of energy may occur, without changing the total amount.
  • Some illustrations of conversion of energy
    • ([*]=Access restricted. [ja]=Texts in Japanese.)
    • [*][ja]mechanical E = kinetic E + potential E
    • [*][ja]work-to-heat conversion (friction; Joule's experiment)

Energy exchange (as "flow")

• Dimension of physical variables; SI unit
  • Energy exchange (temporal duration and spatial magnitude unspecified): J. same as energy as stock. (e.g. 1 kWh = 1000 * 60 * 60 J)
  • Energy flow per unit time, "energy flux (rigorous meaning)" : J/s, W [watt]
  • Energy flow (arriving at, emitted from, or passing thru, a certain surface) per unit time per unit area (of the surface), "energy flux density", "energy flux (casual meaning)": W/m²; no designated name yet.
• Sorts
  • (mechanical) work ... work = force * distance. "power" = work per unit time.
  • (energy exchange other than work may be lumped as "heat transfer")
  • heat conduction. Energy moves from a material body with higher temperature to that with lower temperature, without mass exchange.
  • radiation Energy flow accompanying electromagnetic waves (in other words, photons; e.g. visible light, infra-red, radio waves). (Note: This term does not share contexts with "radioactivity", though partial overlap exists.)
  • Energy flow due to mass flow. Called "convection" in the discipline of heat transfer. In meteorology and climate science, the term "convection" has a different meaning. Energy flow due to mass flow is called advection (large-scale) or turbulent transfer (if small-scale). (Note: The term "scale" in meteorology refers to spatial magnitude of phenomena. It is different from "map scale".)

A system in quasi-steady-state

• [The river flow sustains, yet the same water never remains.]

Climate system in terms of energy conservation

• Climate system (diagram of GARP 1975)
• atmosphere, ocean, cryosphere (ice and snow), "land surface" (soil, vegetation etc.)
  • depth of "land surface" under consideration depends on time-scale in question. ca. 1 m for energy balance in a year, 10 m for 100 years, 100 m for 10000 years (Length proportional to sqrt(time)
• Almost closed with respect to mass exchange.
  • Top: incoming meteorites, escaping hydrogen gas, etc.
  • Bottom: volcanic degassing, sedimentation on the sea floor, etc.
• Open with respect to energy exchange.
  • Top: incoming sunlight (absorbed part included, reflected part excluded), outgoing infra-red radiation. Global average ca. 240 W/m² each.
  • Bottom: geothermal heat flow. Global average ca. 0.1 W/m² (ref. Pollack et al. 1993)

Technical terms about radiation

• [x] (en.wikipedia) [Electromagnetic spectrum]
• [x] (en.wikipedia) [Radiant energy]
• [x] (en.wikipedia) [Black body radiation]

Solar radiation

• The source of energy is nuclear fusion (H to He) within the sun.
• The radiation can be approximated as blackbody radiation of ca. 6000 K (temperature of the "photosphere" of the sun).
• The radiation from the sun is isotropic in a good approximation.
• The amount of energy flow emitted from the sun is constant in the first approximation.
• The space between the sun and the earth is transparent as a good approximation. The path of sunlight can be considered as straight lines.
• The amount of arrived energy flow per unit area is inversely proportional to the square of distance from the sun.
• The amount of energy flow arrived at the earth changes somewhat with sun-earth distance.
  • [ja] [Annual variation of relative insolation due to sun-earth distance]
• The amount of energy arrived at a surface perpendicular to the sun-earth line at the average sun-earth distance is S₀ = 1.36 x 10³ W/m². This quantity is conventionally called (by meteorologists) "solar constant", though it is not exactly constant.
• The amount of incoming solar radiation per unit area of the earth is 1/4 of "solar constant", because the surface area of a sphere is 4 times the area of its great circle section.
• The earth as a whole (including land, ocean, and atmosphere) reflects part of the incoming radiation, and absorbs the rest. The fraction of the reflected part is called "reflectivity" or "albedo". The albedo of the earth as a whole ("planetary albedo") is α_p = 0.30 according to observations by (artificial) satellites.
• The "energy income" of the climate system per unit time per unit area is 1/4 (1 - α_p) S_0.

The basic energy balance of the climate system

• The climate system is in a quasi-steady-state as a first approximation.
• So, its "expenditure" of energy is almost equal to the "income".
• The "expenditure" is emission of "terrestrial radiation" to space. (In this case, "terrestrial" covers not only land but also ocean, atmosphere, etc.)

References

• Peter Atkins, 2007: our Laws that Drive the Universe. Oxford University Press.
• Peter Atkins, 2010: The Laws of Thermodynamics (A Very Short Introduction). Oxford University Press. (The content is almost the same as Atkins 2007.)
• H.N. Pollack, S.J. Hurter & J.R. Johnson, 1993: Heat loss from the Earth's interior: analysis of the global data set. Reviews of Geophysics, 31: 267 - 280. doi: 10.1029/93RG01249 (abstract free, full text subscription required)