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10 July 2011 [Publications]
Condensational theory of stationary tornadoes

Makarieva A.M., Gorshkov V.G., Nefiodov A.V. (2011) Physics Letters A, 375, 2259-2261.
The first-ever theoretical description of a 3D tornado circulation is presented that agrees quantitatively with observations.

As acknowledged by experts (e.g., [1]), the observed radial wind profiles in tornadoes are remarkably similar to those of hurricanes, provided a proper length scaling is adopted. Tornadoes and hurricanes have two more important features in common:

      (1) Both hurricanes and tornadoes move as a whole;
      (2) Both hurricanes and tornadoes are accompanied by intense condensation.

However, the conventional wisdom denies the two vortex types a unified explanation. A common view supported by the works of Dr. Emanuel (see, e.g., [2] for an overview) is that hurricanes are driven by oceanic evaporation that is concurrent to the vortex existence. Since surface evaporation is low on land where tornadoes form, the standard perspective has to admit that, unlike hurricanes, tornadoes must be driven by potential energy that was accumulated in the atmosphere prior to the vortex formation. As the vortex moves along having depleted the local store of PE, new stores of potential energy become available for its maintenance. (NB: Why could not one apply the same reasoning to hurricanes?)

Until recently the only candidate for this potential energy to feed tornadoes has been CAPE (convective available potential energy). CAPE is a measure of work exerted by the buoyancy force on a moist saturated air parcel rising in the ambient lapse rate. (That is, the steeper the lapse rate over a larger height, the larger the total CAPE). One might then expect intense tornadoes to occur at high CAPE soundings. However, recent comparisons of CAPE values associated with nocturnal versus diurnal tornadoes [3] show quite unequivocally that significant tornadoes form at both high and low CAPE values (as well as high and low CIN values). This adds to the suspicion that buoyancy associated with the latent heat release during moist ascent may not be the sought-for tornado driver.

In our work we have considered a different type of potential energy, the one associated with pressure gradients produced by the vapor sink -- the process of vapor removal from the gas phase that occurs during condensation. Formally, this driver of motion, condensation rate S, replaces zero in the right-hand part of the conventional continuity equation applied to a dry atmosphere. (In contrast, the differential heating driver term resides in the temperature equation).

One needs to emphasize that until recently no theory for the condensation rate S has ever been developed. This is remarkable, because other relevant rates have been investigated much better. For example, radiative heating rate is commonly represented as Newtonian relaxation to radiative equilibrium (e.g., [4, Eqs. (1)-(2)]). Evaporation rate is formulated in the various versions of the Penman equation and is proportional to the deviation of water vapor concentration from the saturated equilibrium value. But no formulations have ever been offered for condensation rate that would allow for analysis of the physical properties and dynamic importance of the vapor sink.

Ubiquitously across the literature, the account of moist properties of the atmosphere is made using the so-called microphysics schemes -- numerical techniques applied in time-dependent circulation models. However, numerical schemes are generally designed to solve an already formulated system of equations, not to define the terms for which information in the equations to be solved is missing. As we recently discussed [5], the available microphysics schemes are based on the unphysical assumption of condensation to occur at constant pressure. This assumption does not markedly affect the resulting magnitude of precipitation (which caused the procedure to become widely applied), but a priori leads to a complete neglect of the vapor sink dynamics [6, p. C10925].

In a series of papers we developed a formulation of condensation rate by considering the deviation of the vapor distribution along the temperature gradient from the equilibrium distribution that would take place in the absence of condensation. (In the case of horizontally isothermal atmosphere in hydrostatic equilibrium, condensation rate S is a product of vertical velocity and the vertical gradient of the relative partial pressure of water vapor.) This allowed one to derive an expression for the radial pressure gradient in axisymmetric steady-state vortices and obtain a general equation for their radial wind profiles.

A key difference between tornadoes and hurricanes is the smaller vertical velocity and the larger linear size of the latter. Solving the obtained equation for negligibly small vertical velocities yielded realistic radial wind profiles for intense hurricanes [7]. In our new paper (attached) we obtained a general solution for arbitrary velocities [8]. The profiles obtained for a compact vortex agree satisfactorily with the available data for 3D circulation in Mulhall tornado (world’s largest tornado on record). To our knowledge, this is the first-ever theoretical description of a 3D tornado circulation that agrees quantitatively with observations.

We thus propose a consistent theoretical approach to quantitatively explain intense atmospheric vortices on a unified physical basis. The same driver (vapor sink) as we suggest plays the dominant role in Hadley circulation [9].

We very much welcome comments from our readers and hope that our findings might ultimately generate some positive interest in the meteorological community, especially in the view of the current lack of any competitive theoretical formulations of condensation rate. Our feedback on the comments available so far is summarized in our responses to Dr. Held [10]. See also a relevant discussion in the web [11], comments by Dr. Bryan [12] and our reply to them [5].


[1] Lee W.-C., Wurman J. (2005) J. Atm. Sci., 62, 2373-2393, p. 2379.

[2] Smith, R. K. (2006) Hurricane force. Physics World, 19, 32-37.

[3] Kis, A. K. and Straka, J. M. (2010) Nocturnal tornado climatology. Wea. Forecast., 25 545-561.

[4] Held, I. M. and Hou, A. Y. (1980) Nonlinear axially symmetric circulations in a nearly inviscid atmosphere, J. Atmos. Sci., 37, 515-533.

[5] Reply to 'Erroneous comments' by Dr. George Bryan: Pinpointing where the potential energy of condensation was lost', Makarieva et al., 06 Jan 2011, ACPD

[6] 'Condensation rate and hydrostatic equilibrium of moist air', Makarieva et al., 10 Dec 2010, ACPD

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

[8] Makarieva A.M., Gorshkov V.G., Nefiodov A.V. (2011) Condensational theory of stationary tornadoes. Physics Letters A, 375, 2259-2261.

[9] Makarieva A.M., Gorshkov V.G., Sheil D., Nobre A.D., Li B.-L. (2010) Where do winds come from? A new theory on how water vapor condensation influences atmospheric pressure and dynamics. Atmospheric Chemistry and Physics Discussions, 10, 24015-24052.

[10] Reply 2 to 'Review' by Dr. Isaac Held: On publication criteria in science', Makarieva et al., 26 Apr 2011, ACPD

[11] Weight of Water and Wind, Hurricane Pro's Weigh in, discussion at Jeff Id's blog prompted by letter of Dr. Emanuel

[12] 'Erroneous comments', George Bryan, 16 Dec 2010, ACPD