I developed these ideas in posts here, here and here. But I have wondered about the role of infrared radiation (with GHGs), and why the actual gradient is usually below the DALR. The latter is often attributed to latent heat of water, and called the moist ALR. But that is only effective if there is actual phase change (usually condensation).
I now see how it works. The heat pump reduces entropy, proportionally to the energy it takes from the wind. The entropy can indeed be related to the gradient and the effective thermal conductivity; the largest component of that is a radiative mechanism. So the lapse rate rises to the maximum level that the wind energy can sustain, given the conductive leakage.
I'll write a simplified argument first. Consider a parcel of dry air, mass m, which rises vertically dist dz for a time, at ambient pressure P=Pa, starting at ambient temperature T=Ta. The motion is adiabatic, but it then comes to rest and exchanged heat with ambient.
The temperature inside the parcel drops at the same rate as the DALR, so the difference : d(T-Ta)/dz = -(L-La)
The density difference is proportional to this
d(Ï-Ïa)/dz = -(L-La)*Ï/T
I'm ignoring second order terms in dz.
The net (negative) bouyancy force is
F = V g (Ï-Ïa)
dF/dz = -V g (L-La)*Ï/T
The work done against bouyancy (power) is ? F dz = 1/2 V g (L-La)*Ï/T dz2
Note that this is independent of sign of z; the same work is done ascending as descending.
Because the temperature on arrival is different to ambient, heat has been transported. I could work out the flux, but it isn't very useful for macroscopic work. The reason is that not only is it signed, but separate motions convey heat over different segments, and there is no easy way of adding up. Instead, an appropriate scalar to compute is the entropy removed. Heat pumps do reduce entropy; that's why they require energy. Of course, entropy is created in providing that energy.
The simplest way to calculate entropy reduction is to note that the Helmholtz Free Energy U - TS (U=internal energy) is unchanged, because the motion is adiabatic. This means T dS and P dV (pressure volume work) are balanced. And P dV is from the buoyancy work. So:
T dS = -1/2 m g (L-La)*Ï/T dz2
where S is entropy
Going macroI've shown the work done and entropy generated by a single movement. I'll try to relate that to a continuum. I've used a particular artificial example to link work done with entropy removed. In fact, turbulence typically consists of eddy motions.
Assume there is a distribution of vertical velocity components v in a slice height dz. I can then re-express the work done as a power per unit volume: F v = 1/2 v.dx' g (L-La)*Ï/T
In Latex I'd use hats to indicate averages.
I've left in a dx' which was the old distance of rise, which determines the average temperature discrepancy between parcel and ambient. It's not obvious what it should now be. But I think the best estimate for now is the Prandtl mixing length. This is related to the turbulent viscosity, and in turn to the turbulent kinetic energy (per unit volume) TKE.
So now it gets a bit more handwavy, but the formula becomes
Average power/vol (taken from wind) ~ -g (L-La)/T * TKE
This follows through to the rate of entropy removal, which is
rate of entropy ~ -g (L-La)/T2 * TKE
(power divided by T)
Temperature gradient as a source of entropyIf you have a steady temperature gradient, and a consequent heat flux Q determined by Fourier's Law:
Q = -k dT/dz
where k is a conductivity,
then the volume rate of creation of entropy is
dS/dt = -Q d(1/T)/dz = -Q/T2dT/dz
= k La T-2
So what is k?. Molecular conductivity would contribute, but where GHG's are present the main part is infrared, which is transferred from warmer regions to cooler by absorption and re-emission. In the limit of high opacity, this follows a Fourier law in the Rosseland approximation
flux = 16 s G n2 T3 dT/dz
s Boltzmann's constant, G an optical parameter (see link), n refractive index. Three optical depths is often used as a rule of thumb for high opacity; we don't have that, but you can extend down by using fuzzy boundaries, where for example there is a sink region where there is transmission direct to space.
Update: I forgot to say the main thing about G which is relevant here, which is that it is inversely proportional to absorptivity (with an offset). IOW, more GHG means less conductivity.
I've made an error here. I assumed that the flow expansion was adiabatic. This is conventional, and relates to the time scale of the motion. But I've also assumed adiabatic for the entropy balance, and that is wrong. There is a through flux of energy, mainly as IR, as indicated. And that flux carries entropy with it. So the formula should be:
dS/dt = (k La - Q) T-2
where Q is the nett flow of heat. I'll correct below. It is significant, and may change the sign.
Balancing it all - lapse rate determinedNow we have an entropy source term and a sink term. In steady state entropy can't accumulate, so they balance:
k La T-2 ~ g (L-La)/T2 * TKE
La - L ~ - (k La - Q)/g/TKE
Obviously, there is an unspecified constant of proportionality (with time units), which comes from the nature of turbulence. But I don't think it should vary greatly with, say, wind speed.
So what can we say about the discrepancy between environmental lapse rate La and theoretical DALR L (=g/cp)?
- Proportional to k, the conductivity. So if GHGs transport heat in response to the temperature gradient, as they do, the lapse rate diminishes, away from L. With no GHG's, there is much less to separate L and La. Not so clear - see above correction.
- Inversely proportional to TKE (depends on wind speed). So stronger wind brings the lapse rate closer to L
- Proportional to (La -Q/k).
So what about moisture? That is what the difference between La and L is usually attributed to.
I think moisture is best accounted for within the DALR formulation itself. The DALR L is, again, L= -g/cp, where cp is the specific heat of the gas (air). But in the derivation, it is just the heat required to raise the temperature by 1 °C(OK, that is what sh means), and you could include the heat required to overcome phase change in that. That increases cp and brings down the lapse rate. The thing about the moist ALR is that water only has a big effect when it actually changes phase. That's a point in space and time. Otherwise moist air behaves much like dry. Of course, an environmental lapse rate is only measured aftre there has been much mixing