## 1. Introduction

There have been two approaches in flow dynamics: one is the deterministic Newtonian approach and the other is probabilistic (variational), stated in this article as *variational field Lagrangian formalism,* or in short, *variational formalism*. The governing equations, conventionally used in meteorology and oceanography, are deterministic, which can be derived from the variational formalism (sections 2, 3, and 4 and the appendix).

The fundamental methodology of the classical variational formalism, the Euler–Lagrange (E–L) procedure with Hamilton’s least-action principle coincides with the methodology that has been broadly used, not only in classical mechanics (Lamb 1932; Bateman 1932; Oden and Reddy 1976; Sasaki 1955), but also in modern advanced physics, with Lagrangian density designed in a physically appropriate form for each problem (Feynman and Hibbs 1965; Kaku 1993; Weinberg 1995; Nair 2005; Mandl and Shaw 2010). The research cited here represents a great number of available references with respect to field Lagrangian formalism used to describe essential mechanisms in both classical and modern physics.

Feynman and Hibbs (1965) and section 5 of this manuscript clearly show the classical limit of the quantum theory to describe the classical systems. The order of magnitudes of the dimensions, mass, times, and so on of the classical system is so large that the action *L* (of the notation used in this article) is enormous, which is estimated 0.5 × 10^{−2} mksa or *ρ* = 10^{−3} g^{−1} and |**v**| = 10 m s^{−1}. Since the Plank’s constant *ħ* = *h*/(2*π*) = 1.054 × 10^{−34} mksa rationalized system (J, s), the magnitude of *L*/*ħ* is on the order of 10^{32}, so that the quantum field formalism has not been considered normally in the classical flow formalism. We will reconsider it, showing a possible need for both systems, especially their interactions between classical and quantum systems.

The entropic balance theory was originally developed on the same basis as the variational field Lagrangian formalism to derive the same equation of Clebsch transformation (Lamb 1932). The Clebsch transformation equation was developed in the variational formalism but has not been used directly for actual phenomena because of the unobservable, nonmeteorological, but mathematical Lagrange multiplier. The Lagrange multipliers for the constraints of mass and entropy that appeared in the Clebsch transformation are analogous to the mathematical vector potential (and gauge field) of the theoretical process found known as the “Aharonov–Bohm effect” (or “A-B effect” or “A effect”) (Feynman et al. 1964; Feynman and Hibbs 1965; Landsman 1998; section 5 herein). Its experimental verification has been difficult and has not been made until two decades later (Tonomura 1997).

The entropic balance theory, changing the unobservable nonmeteorological Lagrange multiplier to observable meteorological rotational flow velocity with entropy, is now applicable to tornadic phenomena and is made applicable to solve mysteries of tornadogenesis. Furthermore, the balance is a newly found one, because it is different from the other known balance conditions, such as hydrostatic, (quasi-) geostrophic, cyclostrophic, Boussinesq, and anelastic balance.

The variational formalism is similar to the one observed by Euler in 1736 and referred to as the Euler equation (Oden and Reddy 1976; Lanczos 1970) and later as the Gateaux derivative (Gateaux 1913) in linear approximation with respect to the hypothetical variation that has two parameters, one infinitesimal and the other arbitrary, and the Frechet differential in normed space that deals with a variety of broader problems. A similar mathematical manipulation leads to a full set of dynamical and thermodynamical nonlinear equations of the ideal flow (Lamb 1932; Bateman 1932; Sasaki 1955). The E–L equations are all prognostic except for one that is diagnostic, so-called by Lamb as the Clebsch’s transformation (Clebsch 1857) of flow velocity, although the terminology was not commonly used by authors who derived the same or a similar equation.

In the action Lagrangian of the variational formalism by Bateman (1932) and Lamb (1932), the thermodynamic property is assumed to be barotropic, that is, the flow pressure is a function of density only, in order to derive the equation of motion. Baroclinicity is included by Sasaki (1955, 1999, 2009, 2010) and Dutton (1976). Salmon (1988, 1998)’s Hamiltonian formalism leads to the Clebsch transformation equation. Note that Hamiltonian formalism with a Poisson bracket is a powerful tool for quantization. Dutton compared two types of variations, Lagrangian (material) variation and Eulerian variation, and obtained similar results to those of variational formalism. Sasaki further modified the Clebsch transformation and developed the entropic balance equation that is made applicable to real meteorological phenomena (section 4). The entropic balance theory was developed by introducing the two hypotheses in the Lagrangian density: hypothesis 1 of discontinuous sudden cloud physical phase changes, compared with longer time scales of a tornado, a supercell, and daily synoptic weather, and hypothesis 2, concerned with the ensemble property of such weather systems. Of course, it also includes the continuous change of entropy flux through the boundary of the domain.

In sections 2, 3, and 4 and also in the appendix, we discuss Lagrangian density and the action Lagrangian physically appropriate for the tornado mechanism and derive the Clebsch transformation and entropic balance equation.

## 2. Action Lagrangian of tornadic storm

*R*

_{e}with the molecular viscosity of the air, and a moderately high Rossby number

*R*

_{o}of Earth's rotation, which allows us to neglect Coriolis force:

*ρ*,

*U*, Φ,

*S*, and

**v**are the density of the air, internal energy, gravitational potential energy, entropy, and flow velocity, respectively; and

*α*and

*β*are the Lagrange multipliers to satisfy the constraints of conservation of mass and entropy, respectively.

*d*Ω =

*d*

^{4}

*x*

_{j}(=

*dx*

_{0},

*dx*

_{1},

*dx*

_{2},

*dx*

_{3}) represents time

*t*(=

*x*

_{0}) and three orthogonal spatial coordinates

*x*

_{1},

*x*

_{2,}and

*x*

_{3}, respectively. An ensemble of air molecules is represented by the spatial integration. The Hamilton principle of the least action imposes

*δ*represents the first variation.

**v**,

*ρ*,

*α*,

*β*, and

*S*and (2.4) may be denoted

Note that the notation *S* instead of *L* is commonly used in the variational formalism, but *S* is used for entropy and *L* is used for the action Lagrangian in this study.

## 3. Variational formalism

*L*(

*ϕ*

_{i}), where

*ϕ*

_{i}stands for

**v**,

*ρ*, and

*S*, respectively for

*i*= 1, 2, and 3 as

*δϕ*

_{i}of

*ϕ*

_{i},

**x**is the four-dimensional (time and space) vector.

*δϕ*

_{i}is an arbitrary hypothetical variation in (3.6), the steady-state condition [(3.4)] of (3.6) leads the following E–L equation:

In the application of (3.4) and (3.7) to the Lagrangian [(2.3a)] and the action Lagrangian [(2.3b)], we find that there exists a sole diagnostic E–L equation, that is, the Clebsch transformation equation and the entropic balance equation, among all other prognostic E–L equations. The Clebsch transformation has not been applied to real meteorological data in the past. The entropic balance equation is developed from the Clebsch transformation and is found to explain several mysteries of the tornadogenesis mechanism. We will discuss both the Clebsch transformation and the entropic balance equation and how the entropic balance equation plays roles to explain the tornadogenesis mechanism in the next section.

## 4. Diagnostic E–L equation: Clebsch transformation, entropic balance equation, and tornadogenesis

*δ*

_{v}of the action Lagrangian [(2.5)] with respect to

**v**;

**ω**) equation applying

**∇**

**x**to (4.2),

*Clebsch transformation*by Lamb (1932), which correspond to the solution’s diagnostic state (DS) and stationary state (SS) in Fig. 1.

*β*to couple with scalar

*S*and rotational flow velocity

**v**

_{R}, as

Equations (4.5)–(4.7) are the basic equations of the entropic balance theory, which reveal various important properties of supercells and tornadoes and furthermore the wraparound mechanism of tornadogenesis. In practice, *S* is estimated from radar reflectivity analysis, and **v**_{R} is obtained from the variational data assimilation of Doppler velocity radar observation, as planned to be reported separately, but estimated approximately from Doppler velocity observation. The approximate estimate was determined to have worked well to estimate vorticity in testing. Note that the wind under entropic balance is much more general (there is less assumption) than the thermal wind balance that is under the geostrophic or quasigeostrophic balance.

Figure 2 illustrates schematically the entropic right-hand rule. The diagnostic velocity equation [(4.5)] is universal for the ideal flow. The vorticity equation [(4.6)] derived from (4.5) is conveniently demonstrated by the mutually orthogonal vector relation, similar to the so-called Fleming’s right-hand law of electromagnetic fields (called the “entropic right-hand rule” by the author), among the orthogonal variables of the spatial three dimensions: the vorticity **ω**, the entropy gradient (1/*S*)**∇***S*, the rotational flow velocity component −*S***∇***β*, denoted by **v**_{β} or **v**_{R}, and the divergent component −**∇***α*, denoted **v**_{α} or **v**_{D}. These notations are used in the figure illustrations in this article. Based on the entropic balance theory, the wraparound mechanism is introduced by Sasaki (1999, 2009, 2010) to explain explicitly the nonlinear process of tornadogenesis. The results are consistent with tornadic storm observations and successful tornado simulations of phenomena in tornadic storms, such as overshooting hydrometeors against the upper-level westerlies. The development of a mesocyclone, a hook echo, and a wall cloud is easily explained by the entropic balance theory. The almost discontinuous transition from supercell to tornado suggested from visual observation and data analysis is also easily explained by the entropic balance theory as a transition from baroclinic to barotropic stages. Furthermore, an increase of the relative helicity to 1 (its maximum value) as a result of computer simulation is explained by the entropic balance theory. From the entropic balance theory, we can easily explain the observation of the tornado touching the ground in the perpendicular direction.

For simplicity, the vorticity in balance with the entropic gradient and rotational component of flow velocity as given by (4.6) derived from the entropic balance theory is schematically illustrated as the entropic right-hand rule (Fig. 2). Note that when the gradient of entropy is of larger magnitude, it is baroclinic (Fig. 3), while when it is of smaller magnitude, it is barotropic (Fig. 4). A schematic diagram of a supercell with a tornado is shown in Fig. 5, in which various known features of the supercell and tornado, together with a newly found mechanism, such as the overshooting of hydrometeors against the upper headwind westerlies, are explained by the entropic balance theory in the caption of the figure. Figure 6 shows a horizontal cross section of entropic source and sink that produces a baroclinic gradient in balance with the mesocyclone vortex at middle levels according to the entropic balance theory. The existence of a single vortex (Fig. 7) and that of multiple vortices (Fig. 8, for four vortices) are easily understood to exist by applying the entropic right-hand rule (Fig. 2) to each vortex.

Transition from the mesocyclone, in baroclinic state, shown in Figs. 5 and 6 to the single vortex and/or multiple vortices, in barotropic state, as shown in Figs. 7 and 8, occurs through the wraparound process. The newly found wraparound mechanism for tornadogenesis is analogous to a nonlinear process, the so-called Baker’s transformation, and the transition is discontinuous from baroclinic to barotropic stages by trapping the entropic sink core inside the vortex, like a nonlinear attractor (Figs. 1 and 9). Note also that the wraparound mechanism is two-dimensional, while Baker’s transformation is one-dimensional.

The axisymmetric single vortex is theoretically well supported by the Noether’s rotational invariant theorem (Kaku 1993) with the Lagrangian density [(2.3a)] and the application of the variational formalism. The existence of multiple vortices is also proven by the Noether’s rotational invariant theorem extending it to application for sectional rotation.

Many visual observations of cases of single-vortex and multiple-vortex tornadoes are recorded. For instance, an excellent example is shown by the case of the tornado in central Oklahoma area on 3 May 1999. Multiple-vortex funnels were observed near Chickasha, Oklahoma, and a single EF5 tornado was observed near Moore, Oklahoma, likely from the same supercell during a few hours.

## 5. Conservation constraint of the classical system: Encompassing classical flow and quantum field mechanics

^{−1}), energy and work are in joules (J = N m) and watts (W = J s

^{−1}), and electric current in is amperes (A), electric charge is in coulombs (C = A s), the charge of electron

*e*= 1.602 177 3 × 10

^{−19}C, mass of electron

*m*= 9.109 389 7 × 10

^{−31}kg, electric voltage is in volts (V), the intensity of electric filed is in volts per meter, magnetic bundle is in webers (Wb = A s), intensity of magnetic field is in amperes per meter, inductance is in henries (H = Wb A

^{−1}), and static electricity is in faradays (F = C V

^{−1}). The dielectric constant or permittivity

*€*

_{oo}of the vacuum with the speed of light

*c*and the magnetic permeability

*μ*

_{o}is

*€*

_{oo}= 1/(

*μ*

_{o}

*c*

^{2}).

*P*(

*b*,

*a*) of the trajectories is

*K*(

*b*,

*a*) is the sum of the amplitude of each trajectory to go to

*b*from

*a*,

*a*to

*b*. The amplitude

*ϕ*[

*x*(

*t*)], solution of the corresponding Schrödinger equation, appears as the phase proportional to the action

*L*:

*S*in classical and modern physics; however, in the entropic balance theory we have been using

*S*as entropy and

*L*as the action following convention [(2.3a), (2.3b), and (5.1)]. The magnitude of kinetic energy, the first term in (5.1), is estimated as 0.5 × 10

^{−2}mksa for

*ρ*= 10

^{−3}g

^{−1}and |

**v**| = 10 m s

^{−1}. The magnitudes of

*U*and Φ are assumed similar. Since the terms of the constraints vanish in classical mechanics, because

*ħ*=

*h*/(2

*π*) = 1.054 × 10

^{−34}mksa-rationalized system (J s), the magnitude of

*L*/

*ħ*is on the order of 10

^{32}, so that the quantum field formalism has not been considered in the classical flow formalism, which is on the order of magnitude of 1/10

^{32}. However, the constraints (5.5a) and (5.5b) are of the value of 0, and the value of 1/10

^{32}is extremely small compared with the classical flow terms but of the same magnitude of quantum field formalism. However, the interaction between the classical and quantum systems, like the Aharanov–Bohm effect, could be the same order of magnitude as the classical flow system. Consequently, we may modify the Lagrangian density [(5.1)] to include a quantum field formalism instead of (5.5a) and (5.5b) as

Accordingly, the inclusion of the quantum flow formalism based on Feynman and Hibbs (1965) may serve as a comprehensive way for the variational field Lagrangian approach, a new area of research. The variations in calculus of variations and the classical variational formalism are hypothetical. However, this study suggests that the variations are physical, relating to quantum variations and their interaction with classical systems. It would cast some light on the still mysterious lightning and microphysical processes of storm clouds (Petersen et al. 2008). It would justify the use of the variational formalism better than a deterministic Newtonian approach for the classical system.

## Acknowledgments

The author expresses his appreciation for the encouragement and support given to this research by Dr. Kelvin Droegemeier, Vice President for Research, Dr. Robert Palmer, Director, Advanced Radar Research Center, University of Oklahoma, and Dr. Seon Ki Park of Ewha Women’s University, and also to Weathernews, Inc., for generous support. Also, the author expresses his appreciation to faculty and scientists especially of NWC, University of Oklahoma, University of Tokyo, Kyoto University, Japan, for their useful discussion and to Vivek Mahale and Mark Laufersweiler of the University of Oklahoma for his assistance with computer graphics. I would like to acknowledge that Springer publisher gave permission to the author for the use of nine figures for this manuscript from the published book *Data Assimilation for Atmospheric, Oceanic, and Hydrostatic Applications* (Vol. 2), Chapter 18: Entropic Balance Theory and Radar Observation for Prospective Tornado Data Assimilation, Yoshi K. Sasaki, Mathew R. Kumjian, and Bradley M. Isom.

## APPENDIX

### Prognostic E–L Equations

*L*with respect to

*ρ*,

*α*,

*β*, and

*S*. The corresponding E–L equations become, respectively,

*ρ*,

*α*,

*β*,

*S*,

**∇**on (A.2) and using (A.3)–(A.5), we may prove

*R*, say the rotating Earth, relative to the fixed absolute coordinate system

*A*. A position vector

**x**that is defined on the absolute coordinate is expressed on the relative coordinates of the rotating system with the angular velocity

**Ω**measured on the absolute coordinates as follows:

**x**is measured on the absolute coordinates, becomes

**v**

_{A}of (A.10b) into (A.11), and recognizing ∂

_{t}

**v**

_{A}= ∂

_{t}

**v**

_{R}= ∂

_{t}

**v**, we get

**Ω**× (

**Ω**×

**r**) is included in the gravity field

**∇**Φ, denoted by

*g*:

*U*has the relations with density

*ρ*and entropy

*S*as

*x*,

*y*, and

*z*coordinates, cylindrical, and spherical coordinates.

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