The point of this post is to show a calculation by guest, Pochas, of the decay time that should be expected from the accumulation of heat in the mixed layer of the ocean.
I realized this prediction gives another test of the accumulation theory of climate change, that potentially explains high climate sensitivity to variations in solar forcing, without recourse to feedbacks, or greenhouse gasses, in more detail here and here.
The analysis is based on the most important parameter in all dynamic systems, called the time constant, Tau. Tau quantifies two aspects of the dynamics:
1. The time taken for an impulse forcing of the system, such as a sudden spike in solar radiation, to decay to 63% of the original response.
2. The inherent gain, or amplification. That is if the Tau=10, the amplification of a step increase in forcing will be x10. This is because at Tau=10, around one tenth of an increase above the equilibrium level will be released per time period. So the new equilibrium level must be 10 times higher than the forcing, before the energy output equals the energy input.
I previously estimated Tau from global temperature series, simply from the correlation between successive temperature values, a. The Tau is then given by:
Tau = 1/(1-a)
Pochas posted the theoretical estimate of the time constant, Tau, below, that results from a reasonable assumption of the ocean mixed zone depth of 100m.
The input – output = accumulation equation is:
q sin ωt /4 – kT = nCp dT/dt
where q = input flux signal amplitude, watts/(m^2 sec). The factor 4 corrects for the disk to sphere surface geometry.
k = relates thermal flux to temperature (see below) J/(sec m^2 ºK).
T = ocean temperature,
ºKn = mass of ocean, grams.
Cp = ocean heat capacity J/(g ºK)t = time, sec or years.
Rearranging to standard form (terms with T on the left side):
nCp dT/dt + kT = q sin ωt /4
Divide by k
nCp/k dT/dt + θ = q sin ωt /(4k)
The factor nCp/k has units of time and is the time constant Tau in the solution via Laplace Transform of the above.
n = mass of water 100 m deep and 1m^2 surface area = 10E8 grams.
Cp = Joules to heat 1 gram of water by 1ºK = 4.187 J/gram.
k = thermal flux equivalent to blackbody temperature, J/(m^2 sec ºK).
Solution after inverse transform, after transients die out:
Amplitude Ratio = 1/(1+ω²T²)^½
where ω = frequency, rad/yr
Derivation of k Stefan Boltzmann equation
q = σT^4k = dq/dt
Differentiating: dq/dt = 4σT^3
Evaluating at T = blackbody temp of the earth, -18 ºC = 256 ºK
k = 4 (5.67E-8) 256^3 = 3.8 J/(sec m^2 ºK)
Calculating Time Constant Tau
Tau = nCp/k = 10E8 (4.187) / 3.8 = 1.10E8 sec
Tau = 1.10E8 / 31,557,000 sec/yr = 3.4857 yr
The figure of Tau=3.5 yrs is in good agreement with the empirical figures from the correlation of the actual global surface temperature data of 6 to 10. The effective mixed zone may be closer to 150m, and so explains the difference.
This confirms another prediction of the theory that amplification of solar forcing can be explained entirely by the accumulation of heat, without recourse to feedbacks from changing concentrations of greenhouse gases.