Wednesday, October 29, 2014

Calculating the environmental lapse rate

I have posted over the years on the mechanisms of the lapse rate - the vertical temperature gradient in the atmosphere. It started with one of my first posts on what if there were no GHE. My basic contention has been that lapse rates are a consequence of vertical air motions in a gravity field. Wind tends to drive the gradient toward the DALR - dry adiabatic lapse rate = -9.8 °K/km. Maintaining the gradient takes kinetic energy from the wind to operate a heat pump. The pump forces heat down, to make up for the flux transported up by the gradient. The pump effect is proportional to the difference between the actual lapse rate La and the DALR L. L is the stability limit, and a steeper gradient will convert the pump into an engine, with convective instability. This also pushes La (down) toward L.

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(ρ-ρa)/dz = -(L-La)*ρ/T
I'm ignoring second order terms in dz.

The net (negative) bouyancy force is
F = V g (ρ-ρa)
dF/dz = -V g (L-La)*ρ/T

The work done against bouyancy (power) is ? F dz = 1/2 V g (L-La)*ρ/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)*ρ/T dz2
where S is entropy

Going macro

I'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)*ρ/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 entropy

If 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.
Update
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 determined

Now 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
or
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


63 comments:

  1. Nick

    A important point wrt the moist lapse rate. As a parcel of air rises that contains some amount of water vapour, eventually the relative humidity reaches 100% and some of that water vapour will condense out. Importantly however, only enough will condense to hold RH at 100% - it won't condense back to 50% RH for example. But now, if the parcel continues rising it will continuously keep bumping up against the 100% RH limit as the dew point drops and more water will condense out.

    So in principle above a certain threshold altitude condensation will be a continuous process all the way up until the parcel stops rising any further.

    So the moist ALR is potentially able to provide a near continuous adjustment to the DALR over a significant span of altitudes.

    In the real world atmosphere things are more discontinuous. Cloud formation particularly happens when one reaches the planetary boundary layer. Here a significant pressure drop due turbulence at this boundary triggers large scale cloud formation.

    Even if condensation were highly discontinuous, I think it would still significantly influence lapse rate all the way up. The simple argument about DALR and changes with altitude usually gets expressed in terms of movements of air parcels all the way from the surface. But in fact the physics of this obviously applies for any movement from any altitude to any altitude.

    So for a parcel of air with some water in it, initially its change is pure adiabatic expansion. Then at a key height condensation changes this significantly. If the parcel continues to rise above this adiabatic expansion will still apply but it isn't the same parcel as it was before condensation happened. In effect the pure adiabatic movement will now distribute any changes added by condensation.

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  2. Glenn,
    I agree that there are all sorts of things that spread the effect of condensation LH around. As you say, distributed timing, but also subsequent mixing. They all have in common that it is very hard to get a moist ALR.

    However, it seems to me that there are parts of the world where condensation is minor, and yet I don't think the ALR gets near 10.

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  3. I removed a comment that was not relevant in the form it was written. There are, however, two points related to it.

    You write

    >> The temperature inside the parcel rises at the same rate as the DALR

    As far as I understand the temperature drops rather than rises with increasing z.

    The other point is that you seem to consider both T and Ta to be functions of z. It took me some time to realize that this must be the case.

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  4. I must say that I'm confused about the basic setup that you present.

    Practically all circulation is ultimately driven by the heating of air by warm surface and cooling by IR emission at higher altitude. Thus all the driving part comes from the situations where the local lapse rate exceeds the adiabatic lapse rate that may be dry or moist depending on the relative humidity (dry, where neither condensation nor evaporation takes place). This driving atmospheric heat engine results in other parts of the atmosphere in circulation where also forced rising or subsiding motion of air parcels takes place. You seem to be concentrating solely on this forced part of the circulation, but to get average lapse rates both the driving and the forced situations must be included.

    When we have forced upwards movement of air that's not fully adiabatic the parcel cools less than it would cool in the adiabatic case. Thus forced upwards flow does, indeed, result in a lapse rate less than adiabatic. On the other hand forced subsiding flow that's not fully adiabatic results in warming that's stronger than it would be for adiabatic subsiding flow. Thus that would lead to a larger lapse rate than the adiabatic one.

    Mixing of air at the same altitude caused by large scale turbulence and by horizontal wind components might provide the explanation for the environmental lapse rate well below the dry adiabatic value even in regions, where no or little condensation occurs.

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    1. "Practically all circulation is ultimately driven by the heating of air by warm surface and cooling by IR emission at higher altitude."
      Pekka,
      I don't think that is really true. As you indicate, it can only be true where the air is convectively unstable - the lapse rate is below the DALR or moist equivalent. And that situation is not common outside the tropics. I think the main heat engine is driven by horizontal thermal inhomogeneities. The biggest is the Hadley cell, but there is land/sea etc. Once you create horizontal motion, then you have turbulence and driven vertical motion.

      "When we have forced upwards movement of air that's not fully adiabatic the parcel cools less than it would cool in the adiabatic case. Thus forced upwards flow does, indeed, result in a lapse rate less than adiabatic."
      In my version, that is just vertical mixing. It gives a turbulent heat transport component to k. And yes, that does lower the lapse rate.

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    2. Nick,
      Absorption and emission of radiation are the only basic factors that add significantly free energy to the Earth system. Only free energy can act as source of work, or in other words drive the "atmospheric heat engine". That process is really dependent on the instability of the atmosphere, i.e. a finite deviation from the stable configuration into the unstable regime.

      Absorption of solar radiation adds energy at the local temperature of the point of absorption. This is the high temperature side of the heat engine, while the cold side of the heat engine is where IR emission takes place. In addition it's well known that circulation is not driven effectively unless the cold side is at a higher altitude than the warm side, meaning that the emission from the upper troposphere is an essential component. In order to get that working it's necessary to overtake in addition also the influence of cooling of the rising air, i.e. it's necessary to exceed the adiabatic lapse rate in regions, where uplift takes place.

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    3. Pekka,
      yes, indeed. I tried to quantify this once here, with an entropy budget. But there is a degree of radiative transport, with high altitude emission, that occurs without movement. GHG at high altitude receives radiation from below and emits to space. Circulation is only encouraged if the gas, after rising and cooling, is still warmer than motionless gas would have been. It's not enough to emit - it has to emit more than default. That is a variant of saying that the lapse rate has to exceed the stability limit.

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    4. Nick,
      In this kind of simplified considerations it's probably best to consider stratosphere only as a shield that reduces net IR out of the troposphere.

      At the simplest level I would say that Hadley cells are the heat engine and Ferrel cells the main heat pump. The rest of circulation causes additional dissipation.

      The heat engine of the Hadley cells leaks a lot of energy from the hot side to the cold side by radiative heat transfer. The related reduction in the lapse rate occurs, however, on the high side of the adiabatic lapse rate. Near and below the adiabatic lapse rate convection (including latent heat) adjusts so effectively to the prevailing conditions that the effects of the radiative heat transfer are not any more important. That's in part due to the small net energy flux from any level inside troposphere (the flux leads to weak net radiative cooling under normal conditions). I don't believe that radiative heat transfer has much role in lowering the lapse rate below the adiabatic value.

      Ferrel Cells are a heat pump that transfers heat near surface towards higher latitudes. The rising flow related to the Ferrel cells is forced and that reduces the related lapse rate.

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  5. On the role of radiative heat transfer in all of this. Without radiative heat transfer within the troposphere (but allowing for radiative cooling from the upper troposphere to space) the lower troposphere would be much warmer, and the atmospheric heat engine that dives the circulation much more efficient. Thus I see radiative heat transfer as an essential moderating factor for the circulation. That means also that deviations from the adiabatic expansion and compression are reduced. The adiabatic results should thus apply better than they would apply, if all heat should be transferred by convective processes (including latent heat transfer).

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  6. Nick, what is the significance of a lapse rate of 9.8 and that of 6.5 for the earth? This 2/3 factor seems to be quite common among the known planetary atmospheres.

    http://theoilconundrum.blogspot.com/2013/05/the-homework-problem-to-end-all.html

    Like you, I think there is something universal about entropy making up for the gap, while the "moist" approximation is a convenient stop-gap but that it is inadequate in practice. Consider that there is just not one classification for "moist". What is moist? 95%, 90%, 70% relative humidity ? It could be anything but people keep on insisting that there is either dry or moist lapse rate. A small pet peeve of mine.

    Like Russ indicated, a factor of 1/2 is useful as an average energy that is lost to entropy of mixing.

    Pekka as usual is critical of any thinking outside of the box and I recall he said something to the effect that I ddn't know what I was talking about when I brought this up for discussion at the Climate Etc mudpit. So I am sure he can provide a derivation for Venus where it shows that an exceedingly linear slope for the lapse rate.

    This s just one of those areas whereby the derivations are not very clear and that there are huge holes in the argument and nothing universal across planetary atmospheres.

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    1. Web, that's interesting.
      My narrative here is that a heat pump with a finite energy source (wind) is struggling to pump heat back against various leakages (k). Or, alternatively, to remove the entropy which the irreversible processes are creating. And the fraction of the energy that it can take is proportional to the difference between the lapse rate and the stability limit. So it settles to a steady value somewhere short of the stability limit.

      It may well be that the balance point, in proportional terms, is not so different on other planets.

      I'm speaking of stability limit to include the effects of possible phase change.

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    2. Web

      When RH is less than 100% condensation can't occur, So a parcel of air at any RH less than 100% can only change dry adiabatically. Once it's RH reaches 100% however, any further change rapidly forces phase change to occur. So there is a sharp state change between dry only when RH is less than 100% and moist once 100% is reached. If that state change couldn't propagate effects further then we would expect to see sharp differences in lapse rate at different locations and altitudes.

      However as I said in my comment above, the dry rate will propagate an change caused by phase change. Additionally, if we consider parcels of air separated horizontally, one may be drier than the other and thus potentially different lapse rates. That will give rise to horizontal gradients in temperature and pressure. These will then drive horizontal air movements that will tend to equilibrate the lapse rate.

      Are these horizontal mixing factors sufficient to smooth out all horizontal differences or will a residual remain of the dry vs wet rates? Dunno. That requires more analysis than we can do with a few calcs on a blog and also needs a detailed study of the full data set on lapse rates from around the world.

      What I would say is that we don't get a full picture from only considering simple vertical air columns. That is just the simple starting point.

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    3. The reason that I first got interested in the topic of lapse rate was because I had to apply the Standard Atmosphere for a work-related project. What initially struck me as highly coincidental was that the lapse rate used in the Standard Atmosphere (which was formulated in 1930) was almost precisely 2/3 of the dry adiabatic lapse rate predicted via gravity and MW and specific heat of air. And then looking at the other planets with polytropic atmospheres (Venus, Mars, even the Sun, and a moon of Jupiter), they all shared this factor of 2/3.

      Verkely [1] had tried to apply Max Entropy to the problem and so I started to go down that path to see where it would lead to.

      Overall it is a fascinating problem,and again one that can only be evaluated outside of the lab environment. I think that is how Carl Sagan got started in his thesis work, by researching the lapse rate on Venus.

      [1]W. T. M. Verkley and T. Gerkema, “On maximum entropy profiles,” Journal of the atmospheric sciences, vol. 61, no. 8, pp. 931–936, 2004.

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  7. Calculating averages makes it often impossible to study actual physical processes. Sometimes conservation laws that are equally valid for averages than specific situations tell so much that the connection between the real physical laws and the average is tight, but often the situation is far from that.

    When we consider lapse rate we have regular variability on diurnal and seasonal level, and we have variable weather patterns. In a specific situation the lapse rate may follow closely the adiabat, but taking the average over the full year, it's obvious that being close to the maximum given by the dry adiabatic lapse rate cannot be reached.

    The variability is least in the tropics, where high moisture situations dominate. Thus it's not surprising that the annual mean is there close to the moist adiabatic lapse rate as seen from this picture taken from the book of Marshall and Plumb (2008)

    http://scienceofdoom.files.wordpress.com/2012/02/moist-potential-temperature-mp2008.png

    from the post of Science of Doom

    http://scienceofdoom.com/2012/02/12/potential-temperature/

    Elsewhere the seasonal variability is stronger and the mixture of dry and moist conditions less dominated by any single lapse rate.

    The observation that the average lapse rate seems to be close to the "environmental lapse rate" of about 6.5 C/km is interesting. Such a value seems to be prevalent enough to justify pondering, whether there's something fundamental behind that. On the other hand deviations from that behavior are also common. As shown in the graphics the tropics deviates towards lower moist adiabat, while we have in high latitude winter relatively stable situations where the troposphere is mostly stratified with a lapse rate well less than the dry adiabatic lapse rate that would correspond to the atmosphere with very low absolute moisture. The extent of these deviations makes me think that there's really nothing deeper in the prevalence of the environmental lapse rate than that averaging over variable conditions leads typically to a value midway between the largest and lowest commonly occurring values.

    As Glenn Tamblyn wrote, any realistic analysis that starts from the consideration of actual situations that occur at specific locations at specific moments leads a very complex calculation. What's needed is a full atmospheric circulation model. Detailed weather models might be more suitable for such studies than the highly aggregated climate models, where a major part of the important factors must be described by parametrizations rather than determined by the model itself.

    It's intriguing that so much success has been found in approaches like the maximum entropy production or Mandelbrot's multifractal or power law hypotheses. They provide often descriptions of reality that seem to apply over a couple of decades in some parameter values, but the justifications for this success tend to remain on the heuristic level. Perhaps nothing better should be expected as these approaches fail outside of those couple of decades, and as we cannot test whether they would become asymptotically more and more accurate in some limit lacking situations, where approaching such a limit were possible. To me all these results remain basically rules of thumb. Rules of thumb may be very useful in practice, but that does not require that something deeper can be inferred from the success.

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    1. Pekka,
      "When we consider lapse rate we have regular variability on diurnal and seasonal level"
      It's interesting to think there how fast the LR might adapt. It seems to me that there is a flux of at least 100 W/m2 passing through and interacting with the atmosphere (mostly IR), and that's enough to raise the temperature 1°C/day. The lapse rate could change quite rapidly compared with, say, seasonal.

      I imagine the best observation is from balloons.

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    2. Nick,
      Starting to think on details like this, I always realize, how lacking my knowledge of the actual atmosphere is. It seems that it's not enough to have some understanding of the physical basis and large scale effects, but I should know more on issues probably most familiar to meteorologists, but not discussed so much in connection of the climate.

      The lapse rate must be influenced significantly by the presence of nearby regions of rising and subsiding flows, and the mixing between these flows. All the details like that enter climate models only trough parametrizations, while some weather models might tell more, and while people who develop weather models must have significant knowledge on what's going on.

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    3. OT - in the late 1970's I was a manager at The Balloon Works, then the largest manufacturer of hot-air balloons in the world. The Chief Engineer, and part-owner, was Karl Stefan. Karl's office walls were covered with 8x11 black and whites of weather balloons and Navy fighters. At the time I did not know this, but Karl had worked at NCAR, where he did a lot of work on the balloon aspects of atmospheric research.

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    4. Look at lapse rate profiles for Mars, which fluctuate wildly with altitude and time, yet does show an average. Then look at Venus, with is constant and superlinear, and also shows that average. Mars is a thin atmosphere, Venus is a thick atmosphere. Earth is somewhere in between.

      Find a research paper that reconciles all of this information, along with the role of water vapor and any of the condensing gases that have properties like water vapor. It should make sense across the range of atmospheres.

      Thermodynamics is a science of averages and I would suggest that there is no reason that it couldn't apply here.

      The key is in the characterization of entropy. The specific heat or heat capacity terms, Cp and Cv, have the same units as entropy, and so the strategy is to figure out how the energy is mixed or dispersed. This is the "modern" way of thinking about entropy.

      http://en.wikipedia.org/wiki/Entropy(energy_dispersal)

      I realize this is hotly debated but interesting nonetheless.

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    5. Non-equlibrium thermodynamics can be considered on many scales. Fairly well established theory exists for many issues, while many other ideas have not reached a level that could be called a real theory. Much, if not all, of Maximum Entropy Production seems to be still at this stage. What I have read appears to tell that presenting well justified specifications for the conditions under which the maximum is to be determined tends to be beyond reach. Some assumptions may lead to reasonable looking results, but justifying just those assumptions may require circular argumentation.

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    6. I hope you realize that there is a distinction between Maximum Entropy Production and Maximum Entropy. The former is not as well established as the latter. In fact, the theory of MaxEnt via variational principles can be used to derive the barometric formula.


      http://arxiv.org/pdf/cond-mat/9503098.pdf
      "This is a problem falling into the realm of Jaynes’ MaxEnt [11], which is analogous
      to deriving the barometric formula, Maxwell’s velocity distribution, or Fermi- and Bose
      statistics. "


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    7. Of course I know, how entropy enters in equilibrium thermodynamics, but that part of the climate science is not problematic at all, that is determined by equilibrium thermodynamics. What's more interesting now is the behavior of the Earth system, when free energy is added and then used to drive the circulation. The Earth system may be close to stationary, but not to thermodynamic equilibrium (but many small scale subsystems are close enough to local thermodynamic equilibrium to make it applicable to them).

      Free energy and entropy are related, but I think that free energy is the more directly applicable variable for many of the considerations of the nature discussed in this thread, when we move beyond the consideration of the fully adiabatic processes. (Gibbs free energy is mostly better than Helmholtz free energy, as the subprocesses have typically an externally determined pressure rather than volume in most of the considerations of the atmosphere.)

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  8. Intriguing:
    The polytropic index and adiabatic limit: Another interpretation to the convection stability criterion

    "We propose that the adiabatic process should be the limit state of the polytropic process in one stationary system, so one inequality which can be regarded as the convection stability criterion is constructed. By applying it into the atmospheric convection of the Earth, one can find that when the vertical temperature gradient is higher than 1◦C per one hundred meters, which is called adiabatic lapse rate in aerography, convection or turbulence will take place. One important conclusion can also be drawn that a sunspot is a kind of turbulent temperature inversion phenomenon, just similar to the one in the aerography."

    Above I mentioned that the sun's "atmosphere" has the same property of the observed lapse rate being about 2/3 of that predicted from the molecular composition.

    It is possible that these turbulent sunspots are the heat pump mechanisms that Nick is hinting at. I will look at this paper more closely as it is fairly recent. It also mentions the Virial Theorem which is one of those mysterious thermodynamic principles that can reveal some important properties :)




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  9. I agree with Pekka's comments about the risks associated with averaging this type of data.

    You can split the troposphere into three basic layers, the atmospheric boundary layer (where ground interaction is important in driving the dynamics), the "cloud layer" and a "free layer" (upper troposphere, when it exists).

    I would suggest it is obvious that the sort of discussions here must be limited to the free layer. Arguments of the sort Nick is presenting obviously aren't relevant to the ABL (which can have either sign for the lapse rate, depending on time of day and weather, and varies from roughly 200-m at night to 2000-m in the daytime), or in the "cloud layer", in which changes of state are taking place.

    However, averages of the vertical temperature profile of the entire troposphere clearly are mixing these various physical regimes, with the amount of "contamination" being smaller in the upper troposphere (so I'd suggest we limit any analysis to this region).

    I should also note that the "cloud layer" can extend in summer times to the top of the troposphere (and even intrude into the lower stratosphere during strong cumulus storms. Thus even averages of the upper portion of the troposphere will contain some "contamination" from weather effects, and we'd expect the amount of "contamination" to be affected by season.

    Certainly any model needs to incorporate the dynamic effects associated with the horizontal wind speed profile. Remember this can reach 100 m/s (approximately Mach 0.3!), and this "wind speed jet" is a ubiquitous feature of the upper troposphere for many latitudes.

    But's it's also worth noting that when you have a vertical gradient in wind speed, that acts as a source of turbulence, and drives vertical turbulent diffusion, which is an important mechanism for the vertical transport of heat energy.

    The time scales for this diffusive are short enough that I am pretty sure you cannot start with the assumption of adiabaticity. Rather any model you write down needs to incorporate the vertical transport of heat. I think it also probably needs to include at the least the impact of the upper tropospheric wind jet that is typically present.

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  10. Ideally any theory of the lapse rate would need to be able to match the observed lapse rate, taking in account the measured vertical profiles of the other exogenous variables. Thus, I think you should try and match the theory to specific measured temperature profiles (sticking to the "free layer", when present), rather than trying to average over profiles, then match the theory to that.

    I also think any theory needs to be predictive enough, that you can generate predictions of what the lapse rate should be for given atmospheric conditions. Otherwise, it's not really a testable theory, which is of course of limited value.

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  11. I neglected to mention it (Nick does discuss this), radiative heat energy transfer is also an important mechanism that needs to be included in any theory. I haven't looked quantitatively at the effect of water vapor on this heat energy transport, but obviously it matters, at least qualitatively.

    In principle, at least, even for a tropospheric layer not undergoing changes of state of water vapor, you might still expect a correlation between water vapor density and the observed lapse rate. (In practice a correlation is observed in the environmental lapse rate of the upper troposphere. Whether this is a real or specious correlation is something that needs to be established.)

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  12. Carrick,
    "radiative heat energy transfer is also an important mechanism that needs to be included in any theory. I haven't looked quantitatively at the effect of water vapor on this heat energy transport, but obviously it matters, at least qualitatively."

    For sure it matters. I did an interesting little test. I happened to be using an infrared 'gun' pyrometer (used to measure the temperature of surfaces like heat exchangers), and near noon on a crystal clear low humidity day (in Florida, so not REALLY low humidity ;-o ) I pointed the gun directly overhead; net radiative temperature: -11C. I did the same thing a couple of days later with about 2C higher air temperature but considerably higher humidity; net radiative temperature for clear sky: 2C. I also pointed the gun at different angles to the horizon (increasing the effective thickness of the atmosphere at lower angles). The net radiative temperature increased rapidly at lower angles, as you would expect. The bottom of a cloud (about 1 Km altitude) measured 22C. So column water vapor has to be very important for any accurate model of atmospheric heat transfer.

    Steve Fitzpatrick

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    1. Oh yeah, look at Fitz. He points his little toy at the sky and is able to deduce what is happening. Yup, that's the ticket.

      Where do they find people like Fitz and Cracker?

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    2. Said the person who thinks the polytropic approximation is valid for a layered atmosphere, with different dynamics in the different layers.

      But this is very childish behavior on your part regardless.

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    3. Oh, and Cracker thinks thermodynamics is invalid?

      This is an interesting web page, serving as an antidote to the Cracker high-school-grade quality of comments:

      http://mintaka.sdsu.edu/GF/explain/thermal/polytropes.html


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    4. The polytropic approximation is certainly not a valid approximation here.

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    5. To follow up on this, I think you might be able to assume it in the daytime ABL layer, since that is dominated by convection.

      It is implausible it would be relevant to the nocturnal stable ABL layer, to the cloud layer, or to the free layer.

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    6. Very high-school of Cracker to just simply assert that an approximation is invalid.
      You make these approximations to gain insight, not to score denialist points that the model is not detailed enough to pass muster.
      .

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  13. SteveF, I find the question of modeling the free layer (which is simple enough it should be describable with a 1-d model) an interesting intellectual question.

    But I think Nick was claiming that water vapor did not play an important role, unless there is a change of state, in the environmental lapse rate. I know factually there is a correlation observed between RH and the measured lapse rate in the free layer.

    What I don't know is whether this is a direct effect of the water vapor on the lapse rate, or just a statement of the conditional stability that occurs when you have increased water vapor concentrations.

    Certainly though, it is implausible that any model of the free layer is going to be a valid description of how heat energy is transferred through the atmospheric boundary layer, through the cloud layer and from there to the top of the troposphere. I would imagine you would need at the least a Blackadar 1-d model, but more probably assume regional scale geostrophic forcing together with a mesoscale weather model (like WRF).

    What this says for the plausibility of the modeling of heat energy transfer by GCMs that don't have weather layer, I'll leave to others to say.

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  14. Carrick,

    I suggest you ignore any hostile lunatic who long predicted certain declining petroleum production. These sorts lack the minimum level of rational thinking needed to analyze complicated processes. They are generally rather extreme left handed wingnuts to boot. Just not worth the effort.

    I would venture a guess that non-condensing humidity does have some direct effect on the lapse rate, at least relatively near the surface. When humidity is low, less radiant heat from the surface is absorbed by air directly above (more passes to higher levels and/or directly to space). When humidity is higher, more heat is absorbed in the near surface atmosphere, which will tend to reduce the tendency toward radiational surface cooling and formation of a stable near-surface inversion at night, and so increase minimum daily temperatures relative to maximum daily temperatures. I have not checked, but I would be surprised if more of the observed warming over land during the last 50 years is not due to higher daily minimums rather than higher daily maximums. I would also be surprised if average nighttime wind velocity has not simultaneously increased (on average) due to delayed formation of surface inversions with high humidity.

    Steve Fitzpatrick

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    1. Carrick,

      Sorry, at the end that should have been 'higher humidity".

      Steve Fitzpatrick

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    2. Steve,
      You can put your user name on comments - the option is near the bottom of the list. I think once you do it, it will be the default.

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    3. Fitzy,
      Yup, The deniers like you deny everything. It's in your genes apparently.

      I suggest you go back to Lucia's Blackboard where you can continue with your pity party.





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    4. Steve,
      I include the IR effect of humidity in the conductivity k. It doesn't determine the lapse rate directly, but affects the amount of work that the heat pump has to do to counter leakage. Since more GHG means less conductivity, that actually reduces the loss and shifts LR toward DALR.

      The effect of condensation, OTOH, is effectively a boost to c_p, and brings down the "D"ALR directly.

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  15. Steve, indeed, the effect of water vapor on radiative heat transfer is what came to my mind when I saw this sentence from Nick:

    The thing about the moist ALR is that water only has a big effect when it actually changes phase.


    It's worth reiterating that the dry adiabatic lapse rate (DALR) is a condition for stability, it's not actually a predicted lapse rate. It sets an upper limit on the (magnitude) of rate of change of temperature with height, but it isn't really a theory that predicts what environmental lapse rate (ELR) you would actually observe (in general).

    But, if you look at actual profile data, what you find is you typically do see lapse rates that approach the DALR, but this is the region at the top of the atmospheric boundary layer. Presumably this is because this region is dominated by forced convection, so for this specific case, we expect to find ELRs that approach the DALR limit.

    It's probably not a surprise that the polytropic approximation works here, because isentropic is one special case of that more generalized relationship.

    At night time, as is well known, a low-altitude inversion is typically set up by cooling near the surface. Above this in a region called the "residual layer", because the daytime profile that is set up by forced convection is stable, typically what you see is a retention of this daytime lapse rate. This assumes you don't have weather or topography related processes that mix up the layer of course.

    Often, the nocturnal lapse rate is forced closer to the DALR due, I think, to radiative heat exchange. (the radiative equilibrium lapse rate is of course larger than the DALR, so radiative heat exchange will force the lapse rate to approach that the DALR stability limit.)

    In the "free layer", you almost never see anything approaching the DALR. Typically, the lapse rate shows curvature with height in this region. I believe that this is due to the increase in horizontal wind speed as you approach the tropopause.

    If horizontal wind gradients are driving the vertical temperature profile in the free layer, there is no reason to expect any particular relationship between pressure and specific volume, and I'd expect the polytropic approximation to not be valid for that case.

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    1. Living in Finland I can observe the importance of low altitude inversion in many ways. A strong inversion is common in winter but it's prevalence varies highly from time to time. One consequence of this is that monthly average temperatures vary much more in winter than in summer. The difference between the warmest and coldest January on record is 17.9C, while the difference is about 8C from April to October. Calm clear nights are the major factor in that, but in some cases inversion may persist trough the day. Variations in winter temperature seem to tell much about anything more persistent than variations in summer temperatures.

      The radiative heat exchange within the atmosphere itself is not the reason for exceeding DALR. The essential reason for that is the heating of the surface by both solar radiation and DWLR. That would result in absence of convection to the high lapse rate. The radiative processes would then be important in heating the lowest troposphere to essentially the temperature of the surface and further to the steep temperature profile through much of the troposphere. When the surface is cold this mechanism does not work and the atmosphere becomes stable against vertical convection.

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    2. Carrick,
      "It's worth reiterating that the dry adiabatic lapse rate (DALR) is a condition for stability, it's not actually a predicted lapse rate."

      My contention in these posts is that there is a mechanism that drives toward the DALR. It's not just a stability criterion. Vertical motion in a sub-DALR lapse pumps heat downward, building up the gradient. Or allowing the gradient to exceed the Fourier law value. In this post I've tried to predict the lapse rate. It's hand-wavy because of limited knowledge of wind turbulence, but it shows the factors that come into balance.

      "Often, the nocturnal lapse rate is forced closer to the DALR due, I think, to radiative heat exchange. (the radiative equilibrium lapse rate is of course larger than the DALR"
      Inversion takes you further from the DALR. Radiative heat exchange is less at night, even though the net E/M flux is strongly up. It's not obvious to me that the radiative equilibrium gradient is larger than the DALR. If it were, then there would have to be a very effective counter-pump.

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    3. Nick, to be clear I was referring to the lapse rate in the residual layer, not the surface boundary layer (where the nocturnal lapse rate is typically negative).

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    4. This diagram maybe helps explain it:

      http://1.bp.blogspot.com/_QVNrX4AKZAY/TPhVbPeyfLI/AAAAAAAAFAs/larwRX5QQEU/s1600/bl_evol.jpg

      Here's data from Albuquerque (nearly optimal conditions to show the effect, which is atypical for Albuquerque, due to mountain generated winds):

      https://dl.dropboxusercontent.com/u/4520911/Climate/Temperature/ABQ20140414.pdf

      Red curve is 5 PM MDT and blue curve is 5 AM MDT. Note that between 1000-2000 meters elevation, the ELR is very close to DALR, with the nocturnal lapse rate in the residual layer actual actually closer to DALR than the daytime convective layer is.

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    5. Carrick,
      My interpretation of the Albuquerque data is that there is a diurnal transient near the ground, which fades with altitude. Around 1000 m, there is plenty of wind energy, so good heat pump. Looks like at 2500 m, maybe cloud condensation?

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    6. Nick: It's not obvious to me that the radiative equilibrium gradient is larger than the DALR. If it were, then there would have to be a very effective counter-pump.

      Actually, that is typically the case near the surface, so it does act to drive convection:

      https://dl.dropboxusercontent.com/u/4520911/Climate/Temperature/RadiativeVsConvective.png

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    7. By the way, I'm pretty sure you're correct about there being cloud condensation above 2500 m.

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    8. Regarding relative magnitudes of radiative vs convective equilibrium, one of the standard papers that discusses this is:

      http://www.clidyn.ethz.ch/ese101/Papers/manabe67.pdf

      See Figure 5.

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  16. Carrick (and Pekka),

    I found this data interesting: http://www.ncdc.noaa.gov/cdo-web/datatools/records

    Note that the over the past year, there were lots of low maximum records set (no doubt reflecting unusually cold weather in the mid-west of the USA in the past winter), but far fewer low minimum records. Note also that while there were relatively fewer high maximum records set, there were many more high *minimum* records set. The early morning hours appear warmer than in the past.

    Which suggest to me that underlying GHG driven warming trend is more manifest in in the minimum temperature than in the maximum temperature, consistent with a stronger influence of higher GHG's (including water vapor) on the temperature of the nocturnal boundary layer, and weaker influence on daytime temperature, where convection is driven by solar heating. (Except at high latitudes in winter, where an inversion may be stable during the day due to very weak solar heating.)

    Boundary layer effects seem to me a good explanation for some divergence of satellite lower tropospheric temperature trends from surface temperature trends.

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  17. Carrick,

    Ignoring nocturnal boundary layer influence, it seems to me that below the altitude where clouds will form, the DALR pretty much has to be the dominant lapse rate. I mean, the DALR should apply up to where moisture starts to condense, then a lower moist lapse rate should apply... at least until some higher altitude (with lower temperature) where there just isn't enough remaining water vapor in the air to give much more latent heat.

    When I fly during the day (which is, unfortunately, too often), I often see a very clear altitude where clouds begin to form, representing the temperature where air rising air from the surface becomes saturated. 'Puffy clouds' form at this level, reflecting much solar radiation (and for sure absorbing a fair amount of solar infrared!) thus cooling the surface. But most of the time these clouds dissipate without producing any rain. The air above seems to be often dry enough that collectively formed clouds evaporate rather than generate rain. But they for sure reduce solar flux at the surface. Of course, when the sun sets, the convection and cloud formation usually stops, and the sky clears.

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    1. Steve, I agree, as long as you have strong convection --driven circulation, you'd expect to see a lapse rate close to DALR.

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  18. Pekka, to be clear, what I was arguing is that forced convection in the ABL is responsible for the ELR approaching the DALR during typical daytime conditions.

    The issue with the residual layer, is it is a layer of air that becomes "detached" from the surface after the establishment of a nocturnal (stable) boundary layer. When it is not disturbed (typically this is between sunset and roughly sunset + 4 hours), what you see is a further steeping of the lapse rate.

    In the daytime surface layer (roughly the first 100-m), the lapse rate typically exceeds the DALR, but again this is generally due to convective forcing, and there is a lot of low-altitude turbulence associated with it.

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  19. Carrick,
    I've realised I made a substantial error here. I assumed entropy balance as if the air were adiabatic. But as you say, there is a significant radiative flux interacting with the air, and that carries entropy with it. So it isn't the case that the heat pump has to completely counter the entropy creation.

    I'm not sure what to make of your plots showing a radiative flux much greater than the gradient would imply. They say that convection transfers heat up to straighten the gradient to the LR. The logic of my corrected equation agrees. But the LR is generally in the convectively stable region. Maybe transient instability is enough.

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    1. Nick, I think it's the case that daytime lapse rates at the top of the ABL tend to look nearly adiabatic.

      I believe point of the radiation equilibrium lapse rate being larger than the DALR near the surface, is that it implies radiative transfer will force the daytime lapse rate to be close to the DALR stability limit. So the DALR is a good zeroth order approximation to the lapse rate you will observe in the upper layer of the daytime convectively dominated ABL.

      The other important thing to see is that the DALR isn't a prediction of the observed lapse rate, rather it's an important "speed limit" on the maximum rate at which temperature can vary vertically in the atmosphere. So while it's useful for the specific application of the ABL, it has little to do with the actual observed lapse rate in the free atmosphere layer above the cloud interaction layer, other than being a not very interesting upper bound on the ELR.

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  20. Nick,
    I am puzzled. Sometimes I try to make a comment and the 'capcha' (sp?) window pops up, and the comment shows up. Sometimes the 'capcha' window does NOT pop up, my comment disappears, and is lost forever (happened twice this AM, once yesterday). Does blogger not play well with certain browsers?

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    1. Steve, this happens to me too… what I found works is to always press the preview button. I also write my comment in an editor then copy and paste it in, which helps reduce the chance of lost data (at the risk of copy and paste errors).

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    2. Steve,
      Sorry about that problem. I don't know what is happening. There is nothing in the spam bin (which is itself unusual). AFAICS, every comment of yours that I have email notification of has appeared.

      I invoke captcha during quiet commenting periods, but not in recent days.

      I have found acquiring a Google ID helpful. My wordpress ID was also good, but recently problematic. The google ID means you don't have to enter anything, and it used to mean slightly better privileges, like being able to paste into comments.

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    3. As I mentioned to Steve, I do lose comments sometimes. This happens even when I am logged into my google account.

      The only work around I've found is to press the preview button before the submit button.

      0n my browser the preview window is virtually useless, except for very short comments—this is not a complaint, it's just why I wouldn't automatically preview my comments.

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  21. Nick,

    Here is what I tried to post earlier:

    I think Carrick's graph (day and night lapse rate profiles) is very informative. The daytime curve shows that the actual lapse rate is higher than the DALR, especially below ~1 Km, which makes perfect sense to me. The DALR is a stability criteria, and there can only be active convection when the DALR is exceeded. At or below the DALR the column is stable and any mixing can only take place due to horizontal shear driven turbulence. Upward heat transfer is going on both by convection and radiation, with radiation becoming relatively more important at higher altitudes, until near the tropopause the lapse rate falls below the DALR and convection stops. Higher absolute humidity in the air (but below saturation) reduces the relative importance of radiative flux at all altitudes, and so increases the relative importance of convection. In a simplified system where there is no horizontal wind shear, the size of the discrepancy between the actual lapse rate and the DALR should dictate how much convective transfer there is, and at higher humidity this discrepancy should increase all along the air column, indicating the increased importance of convection.

    Carrick's night time curve shows that the lower troposphere becomes stable (lapse rate below the DALR), where absent shear driven turbulence only radiative transfer is important. Wind shear driven turbulent mixing, opposing a stable lapse rate (that is, below the DALR), serves to transport heat from above to below because of adiabatic heating.

    When you add enough moisture to get condensation, there must be a two-tier lapse rate: low altitude where the DALR governs convective stability, and a higher altitude above which the MALR governs convective stability. Seems to me no simple model of this very complicated process is likely to do a reasonable job, especially when the influence of wind shear and cloud formation is included.

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    1. Hey, I tinkered with the blog javascript and I think I found a bug. The comment window should now be adjustable. I'll see what else I can fix.

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  22. Err, 'stability criterion'

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  23. How does this relate to air mass variation or real time application in physical contexts other than paelogeographic and atmospheric structured arguments? Is there a unique and qualitative attribute with the parrametrics of a specific feature?

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    1. Sorry, I'm having trouble understanding the question. The total air mass doesn't vary much (just a little with humidity). The rest I can't really understand.

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