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Convergence, Shrinking and Implosion versus Divergence, Expansion and Explosion during Condensation

Some confusion apparently continues to surround the question of whether condensation lowers local pressure (and hence leads to convergence towards the condensation area) or whether it leads to a rise in local pressure and hence divergence of air from the condensation area. Three quotes below from colleagues whose diverse attitudes towards condensation-induced dynamics span the entire spectrum of possible attitudes, provide an illustration.

“Cloud formation looks like an explosion and not like an implosion. So the expansion due to the temperature rise is clearly stronger than the pressure drop due to condensation.”

“If condensation drives winds by a decreasing the number density of water vapor in the air, then as clouds form they should shrink in size due to the negative pressure. OTOH, if the release of the heat of condensation is the primary driver, they should expand. Guess what?

“Yesterday I was gazing upon a deep blue sky when a puffy white cloud came into view. It was small, but I could notice its slow growth at the diffuse edges, which I could see due to the stark contrast to the blue background. I kept watching for a time, as it grew and grew, remembering one of the classic explanations from the conventional meteorology, that is, clouds grow as a response to “latent heat” release and ensuing expansion.”

Let us try to clarify this issue.

Summary of discussion at Climate Etc.

Anastassia Makarieva | February 5, 2013 at 12:01 am | Reply
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I’d like to provide my summary of the discussion in that part that concerns our exchange with Nick Stokes and how it relates to paper content and the broader picture of our work.

I. The “0=1” argument.
It was repeated by Nick here, here, here and elsewhere.
This argument presumes that our system of equations (32)-(34) is mathematically inconsistent (contains a “mathematical fallacy”). This argument is false. The system is mathematically consistent. Those willing to consult an independent opinion please see the comment of Tomas Milanovic.

When challenged to derive the contradiction “1=0” explicitly from (32)-(34) Nick made two elementary algebraic errors (see here and here). At one place he seemingly implicitly conceded that there is no mathematical contradiction (but still made a later effort to derive one).

My take on the “mathematical fallacy” issue is here, here and here.

The argument “1=0” profoundly misinforms the reader about the actual issues that are worthy of discussion.

II. The derivation of Eq. (34).

It should be made clear that Eq. (34) in the paper was not derived from any other known equation. It was formulated based on several key physical propositions. These propositions are summarized here. They deserve a thoughtful physical consideration by anyone willing to understand the underlying physics.

1. The scenario of discussing these propositions developed as I suggested in an earlier comment here:

So, I suggest that you should instead insist that our paper contains a physical fallacy. … You should say that the assumptions that we involve to justify Eq. (34) are not convincing or anything obvious to you. To this I will respond that to us those physical assumptions do seem plausible…

In line with this, Nick once called our propositions arm-waving, another one that they are of “no relevance”. His most specific claim was

The formula is the same in all cases. Therefore it does not depend on gravity.

When asked if any formula that contains something depending on gravity must explicitly contain g, Nick changed topic. To this I responded that “to see no relevance” and “find an error” is not the same.

2. Additionally, Nick produced an alternative derivation of S (the one also presented by Dr. Held in his review with a reference to Nick). This derivation was independently reproduced by Cees de Valk. See here for a relevant thread.

Note that Eq. (34) is S = wN∂γ/∂z, where γ = pv/p is the relative pressure of water vapor and N is total air molar density. The alternative derivation of Nick et al. produces what we can call SNick = wNd∂γd/∂z, where Ndis dry air molar density and γd = pv/pd is the ratio of water vapor to dry air pressure. Since pv << p, pd and p are very close, so S (34) and SNick (Sd in our paper) differ by a small magnitude.

Now important: we do not have any objections as to from what principles SNick was derived. I accept a priori that it can be correct (which would mean that our S (34) is wrong). That would be fine. But, as everybody agrees, when put into the continuity equations (32)-(33), SNick produces a non-sensical result:
u∂N/∂x = 0. (1C)
This result is mathematically valid (no contradiction in the equations), but it implies that generally, under any possible circumstances on an isothermal surface, winds cannot have a velocity component parallel to the pressure gradient ∂p/∂x = RT∂N/∂x.
So this result is invalidated by observational evidence (e.g. radial convergence in hurricanes). By inference, whatever were the physical grounds from which SNick was derived, they were incorrect.

In contrast, our S (34), despite different from SNick by a small relative magnitude, when fed into the system (32)-(33) produces instead of the above equation
-u∂N/∂x = S. (2C)
It is a classical case where a small detail matters (“devil in details” etc.) To understand why it matters requires a deeper insight into the underlying physics, not merely manipulating with the continuity equations.

3. There is an alternative physical derivation of S (34) as presented in the blog and here based on energy consideration. This complementary view, although explicitly present in the paper, might not have been sufficiently clearly articulated. This alternative derivation does not involve any small factors in deriving (2C) above. These considerations provide independent support for the physical arguments used in deriving (34). None of the commentators here, Nick included, has ever commented on that.

In the meantime, we now think that this second view on how (2C) (the main result) is obtained is more easy to understand from the physical viewpoint. We recommend people who just have their first look on the subject to evaluate it first.

4. The argument about S = CNv (see here, here, and here) is confusing.

Otherwise this argument is a distraction. When challenged during the ACPD discussion to better explain the physical foundations of Eq. (34), we showed that Eq. (34) can be interpreted as S = wkvNv, where kv is the degree to which water vapor partial pressure deviates from hydrostatic equilibrium. This proportionality is physically consistent with S being a first-order reaction in Nv, but certainly S = wkvNv cannot be formally derived from chemical kinetics.

From the empirical viewpoint, since C is not a constant (it does not depend on Nv, i.e. it is a constant with respect to Nv only), the formula does not presume rain from dry air. Since to appreciate this linearity argument requires an understanding of what kv is, this argument is of little help for those who want to get a first idea of the physics behind (34). Again, I recommend this account.

III. Issues of interest

1. Since S (34) is not formally derived from any pre-existing equations but formulated based on several plausible physical propositions (two independent sets of them), it cannot (and has not been) refuted mathematically. It has not been shown to be in conflict with any physical law either. The only way to falsify Eq. (34) consists in checking the result it yields
-u∂N/∂x = S. (2C)
against empirical evidence, as I outlined here. Nick made a few attempts, but they have so far been inconclusive.

I emphasize (2C) is an extraordinarily strong statement (see Eq. (4) in the post which is the same). It predicts that where condensation is absent, winds cannot blow along the pressure gradient or that the pressure gradient must be absent. It also predicts the reverse (lack of condensation where u∇p = 0), but the latter prediction is less informative as it does not specify the scale at which this lack of condensation should be manifested (it can be very narrow).

The meaning of the differential form of 2C (see Eq. 4 in the post) is that all potential energy released from condensation is locally converted into the power of the large-scale horizontal pressure gradient force u∇p. This is very strong. At what scale it is actually true remains to be seen. It will also help to discriminate between condensation-induced dynamics and other mechanisms at work in the atmosphere (e.g. forced convection can be different). As a bottom line, the integral form of (2C) has already produced meaningful results, so it can serve as an integral limitation on the dynamic power of circulation.

For us the main point is that our theory (unlike the existing models) yields empirically falsifiable predictions. It is a working theoretical concept for a moist atmosphere.

2. The last but one section in our blog is very important for future theoretical analyses.

3. How the theory can be empirically tested is outlined here and here in response to manacker (Max). It is said quite enough for anyone who got a basic physical idea to publish original papers based on observational analysis. There seem to be lots of relevant data around.

IV. Miscellaneous

1. My personal view on why the paper was accepted.

2. I would like to express our gratitude to Nick Stokes for his persistent attention to our work. My personal view: For people who like us have a clear picture of underlying physics Nick’s comments can provide additional details and angles. For those people who do not have a clear physical picture and make their first acquaintance with the idea, Nick’s comments are paralyzing and preventing any further understanding. Cannot be recommended for students.

Thank you very much for this exciting discussion.

Anastassia

On the results of 2012 London Olympics: Energetic records of animals and man

A. M. Makarieva, V. G. Gorshkov

• Who is the Champion of Champions: Sergey Kirdyapkin, Ivan Ukhov, Usain Bolt or Michael Phelps?
• What would have happened to a flea had it jumped to a height of 2 meters?
• In which sports is man second only to the donkey and the elephant?
• Long-term disqualification of Homo sapiens on the Olympics of Life

All animals can move. But each species has perfected in a particular type of motion. Some animals set records in running, others are particularly good at walking or crawling, yet others are outstanding arboreal acrobats. Swifts fly better than anybody else, but they are pathetic on land where they can only crawl very slowly searching for a small hill from which they could take off. Loons fly very well both in the air and underwater, but they cannot walk. It is a huge problem for a loon to approach her nest moving on the ground – because of that the loon must build the nest very close to the water edge of a lake or a river (they cannot nest on the seashore because sea water comes and goes with tides). Moles can dig tunnels under the ground at a rate of up to 10 cm per second, but they practically never show up above the ground. Brachiating monkeys that rely on their hands to move in the tree canopy make great acrobatic performances on the trees but avoid coming down to the ground.

How are all these and other species-specific skills maintained? The answer comes from considering the stability principles of life organization. Continue reading

Condensation Rate: Devil in a Detail

When gas disappears somewhere in the atmosphere, local pressure is lowered and a compensating air inflow from the surrounding areas is initiated. In our paper “Where do winds come from?” (M10) we derive the magnitude of a stationary horizontal pressure gradient $\partial p/\partial x$ that is associated with water vapor condensation — the process by which the vapor gas molecules are packed into a thousand of times smaller liquid volume and thus effectively disappear from the atmosphere.

There has been much critical discussion in the blogosphere and further on the ACPD web site of our Equation 34 for condensation rate $S$ that is key to the presented derivation. At various times and places, including Section 4.2 of our paper, it was pointed out that if one formulates $S$ in terms of water vapor mixing ratio $\gamma_d \equiv N_v/N_d = p_v/p_d$ one obtains $\partial p/\partial x = 0$. If one instead uses the relative partial pressure of water vapor, $\gamma \equiv N_v/N = p_v/p$, a horizontal pressure gradient is obtained that appears to be so significant as to substantiate the claim for a dominant role in the whole planetary dynamics. (Here $N_v$, $N_d$, $N = N_v + N_d$, $p_v$, $p_d$, $p = p_v + p_d$ are molar density and pressure of water vapor, dry air and air as a whole, respectively.)

A typical value of water vapor partial pressure $p_v$ in the lower atmosphere is around 1-3 per cent. This means that the mixing ratio and relative partial pressure $\gamma_d$ and $\gamma$ differ very insignificantly. Reckoning up the preceding criticisms, our second referee Dr. Isaac Held referred to this difference as to “a detail”. It is the purpose of this note to show that the $\gamma/\gamma_d$ dichotomy is precisely the devilish detail that, as it increasingly appears, is responsible for the fact that the condensation-induced dynamics has not received the attention it deserves.

Consider the stationary continuity (mass conservation) equations written for dry air and water vapor (Eqs. 32 and 33 in M10): $\nabla N_d \mathbf{v} = 0$, $\nabla N_v \mathbf{v} = S$.      These equations only tell us that the dry air mass is conserved, while the vapor mass may be conserved or it may be not: there can be a local source or sink of vapor $S$. Irrespective of the existence/nature/magnitude of the vapor sink/source $S$, the above equations can be combined with use of elementary algebra such that their left-hand parts take various forms. In his review Dr. Held chose $N_d \mathbf{v} \nabla (N_v/N_d) = S$, which is equivalent to $\mathbf{v} (\nabla N_v - \gamma_d \nabla N_d) = S$.    

In the two-dimensional circulation considered in M10 the velocity vector is the sum of the horizontal and vertical components, $\mathbf{v} = \mathbf{u} + \mathbf{w}$. We also consider a horizontally uniform surface temperature, which dictates a constant saturated pressure of water vapor, such that $\mathbf{u} \nabla N_v = 0$.      (If water vapor is not saturated, this assumption corresponds to a horizontally uniform surface temperature and constant relative humidity.) Combining  and  we obtain $\displaystyle \mathbf{u} \nabla N_d = \left(S - S_d\right)\frac{1}{\gamma_d}$,     where $S_d \equiv \mathbf{w} \left(\nabla N_v - \gamma_d \nabla N_d \right)$.    

Equation  has two important implications for any given horizontal velocity $u \ne 0$. First, it shows that when $S = S_d$, the horizontal density gradient $\partial N/\partial x = \partial N/\partial x = 0$. Second, it shows that if $S$ and $S_d$ differ by a small relative magnitude of the order of $\gamma_d \ll 1$, this magnitude is multiplied by a large relative magnitude $1/\gamma_d \gg 1$ to determine the horizontal density gradient $\partial N/\partial x$.

We should emphasize that the two above conclusions are unrelated to any ideas about what the condensation rate could look like. The value of $S$ in  is unknown. Equation  shows that any minor difference of the order of $\gamma_d$ in the theoretical formulation of $S$ — whatever the latter might be — is not a detail but is the zeroth order term in determining the horizontal density and pressure gradients associated with vapor condensation.

Radial profiles of velocity and pressure for condensation-induced hurricanes

Makarieva A.M., Gorshkov V.G. (2011) Radial profiles of velocity and pressure for condensation-induced hurricanes, Physics Letters A, 375, 1053-1058.

Abstract.

The Bernoulli integral in the form of an algebraic equation is obtained for the hurricane air ﬂow as the sum of the kinetic energy of wind and the condensational potential energy.With an account for the eye rotation energy and the decrease of angular momentum towards the hurricane center it is shown that the theoretical profiles of pressure and velocity agree well with observations for intense hurricanes. The previous order of magnitude estimates obtained in pole approximation are confirmed.

In simple words

Maximum wind speeds that arise in hurricanes and tornadoes are developed when the pressure gradient force significantly exceed friction forces. In the absence of friction energy along the streamline (the Bernoulli integral) is conserved, equal to the sum of kinetic and potential energy. Potential energy coincides with air pressure, while the pressure gradient determines the force that makes the air move.

The stream power is equal to the product of force and velocity. The characteristic change of pressure along the vertical is $\Delta p = \beta p_v$, where $p_v$ is partial pressure of water vapor at the Earth’s surface, $\beta$ is the share of water vapor that ultimately condenses in the ascending air. The mean vertical force acting per unit air volume is equal to $\Delta p/h$, where $h$ is the characteristic scale height of the condensation process.

Due to the continuity constraints the vertical power $(\Delta p/h)w$, where $w$ is vertical velocity of the rising air, should be equal to radial power $(\partial p/\partial r) u$, where $u$ is radial velocity directed towards the center of the condensation area, $r$ is radial distance from the center. On the other hand, change $d(2\pi r hu)$ of the radial air flux entering the condensation region via a circular wall of area $2 \pi r h$ is equal to $2\pi rw dr$, which is the vertical air outflow via the ring of radius $r$ and area $2\pi r dr$. This equality relates the vertical and radial velocity by the well-know relationship $w = (h/r) (\partial ur/\partial r)$. Putting this relationship into the equality of the vertical and horizontal powers, we obtain that condensation causes the air pressure to fall along an unusual non-linear law: $p(r) = p_1 + \Delta p \ln ur$.

The smallness of vertical velocity $w$ allows one to neglect it compared to radial $u$ and tangential (rotational) $v$ velocities, $v \approx a/r$, where $a$ is the (partially) conserved angular momentum. This causes the Bernoulli integral for the air flow to take the form of an algebraic equation on radial velocity $\rho/2 (u^2 + a^2/r^2) + \Delta p \ln ur = \rm const.$

The obtained transparent form of the Bernoulli integral describes the major quantitative hurricane characteristics, including profiles of pressure and velocity and the size of the eye.