Category Archives: science matters

Condensation Rate: Devil in a Detail

Posted on April 2, 2011 | Leave a comment

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.    [1]

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.     [2] (If water vapor is not saturated, this assumption corresponds to a horizontally uniform surface temperature and constant relative humidity.) Combining [1] and [2] 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).    [3]

Equation [3] 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 [3] is unknown. Equation [3] 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.

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Radial profiles of velocity and pressure for condensation-induced hurricanes

Posted on March 8, 2011 | Leave a comment

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.

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