_{1}

An overview of researches is presented, which was focused on application of a theoretical hypothesis on the turbulent vortex dynamo to the study of tropical cyclogenesis. The dynamo effect is related to the special properties of small-scale helical turbulence with the broken mirror symmetry and was hypothesized to result in large-scale vortices generation in both hydrodynamic and atmospheric turbulence. To introduce this abstract theory into tropical cyclone research, a recent discovery of vortical moist convection in the tropics is emphasized. Based on this finding, we discuss and substantiate the crucial role of rotating cumulonimbus clouds, known as vortical hot towers (VHTs), as a necessary element to provide the dynamo effect. An analogy is traced between the role of interaction “moist convection—vertical wind shear” in creating the vortex dynamo in the atmosphere and the role of the mean electromotive force providing the MHD dynamo in electrically conducting medium. Throughout the review of novel results, a pivotal role of the Russian-American collaboration on examining a helical self-organization of moist convective atmospheric turbulence under tropical cyclone formation by use of cloud-resolving numerical simulation is accented. The efforts resulted in application of the vortex dynamo theory to diagnose a time when cyclogenesis commences in a favorable tropical environment. This may help elaborate a universally accepted definition of tropical cyclogenesis that currently does not exist and contribute to practical purposes of diagnosis and forecasting.

The present review-research article is aimed at bridging the fundamentals of the theory of turbulence and the most advanced atmospheric studies based on cloud-resolving numerical simulation in order to approach unraveling the awesome natural phenomenon of tropical cyclogenesis.

Incipient research efforts towards this goal are dating back to the early 1980s, when almost simultaneously, there appeared two publications [

Following the publications [

The present work is a continuation of the efforts discussed above. In order to introduce the hypothesis on the turbulent vortex dynamo into tropical cyclone research, a recent discovery of vortical moist convection in the tropics is emphasized. Based on this finding, we discuss and substantiate the crucial role of rotating cumulonimbus clouds, known as vortical hot towers (VHTs), as a necessary element to provide the dynamo effect in conditions of tropical cyclone formation. Based on the mathematical model of the turbulent vortex dynamo in a convective system [

The paper is organized as follows. In Section 2, we discuss the mathematical model of the turbulent vortex dynamo in a rotating non-uniformly heated fluid; a numerical approach designed to study the large-scale helical-vortex instability is presented and results of numerical investigation of the model are summarized. Section 3 highlights the discovery of vortical moist convection in the tropics and its helical features. In Section 4, we describe the implementation of post-processing of atmospheric simulation data and examine the process of helicity generation on cloud- and mesoscales. Section 5 presents a mechanism of a VHT formation and helicity generation through interaction between a convective updraft and vertical wind shear. A crucial role of VHTs is emphasized in providing a special topology of a forming mesoscale vortex, which is characterized by the linkage of the primary (tangential) and secondary (transverse) circulation. Based on it, a rationale is proposed for how VHTs can participate ensuring the action of the turbulent vortex dynamo in the tropical atmosphere. Section 6 presents how the concept of vortex dynamo can be applied for practical purposes of diagnosing the commencement time of tropical cyclogenesis. In Section 7 “Conclusions”, we summarize the main findings and offer a perspective for these investigations.

A fundamental hypothesis is known in the theory of turbulence that an inverse energy transfer from small to large scales is possible in the helical turbulent medium with the broken mirror symmetry [

A mathematical model of the turbulent vortex dynamo in a convectively unstable non-uniformly heated fluid was first proposed in [

Helicity of the velocity field is a pseudoscalar quantity defined as the scalar product of velocity v ( r , t ) and vorticity ω ( r , t ) = c u r l v vectors [

H = ∫ v ⋅ ω d r , (1)

gives the helicity of vortex system, where v ⋅ ω is the helicity density of the flow. Both quantities are pseudoscalars, i.e., they change sign under change from a right-handed to a left-handed frame of reference [

A non-vanishing helicity implies the symmetry break of turbulence with respect to coordinate system reflections [

However, unlike energy the helicity can be both positive and negative. Its sign determines the predominance of the left-handed or the right-handed spiral motions in the examined flow.

Helicity is one of the most important characteristics for describing the structure of vortex fields. This quantity is a topological invariant, which measures the degree of linkage of the vortex lines [

The sources of helical turbulence are known to be the force fields of a pseudovector nature, such as magnetic or Coriolis force fields.

Let us analyze and discuss the mathematical dynamo-model for the convective

system in more detail.

The most demonstrative physical interpretation of the obtained dynamo-effect can be given in terms of toroidal and poloidal component of the vector velocity field, i.e. in the form of representation that is frequently used in magnetohydrodynamics [

V = V T + V P , V T = curl ( e ψ ) , V P = curl curl ( e φ ) , e = { 0 , 0 , 1 } . (2)

Bearing in mind the formation of tropical cyclones in the atmosphere as well as vortex flows in rotating non-uniformly heated fluids, we can also interpret this representation in common terms of tangential and transverse (or overturning) circulation, respectively, in order to apply them in our further discussion.

In terms of expressions (2), a mathematical model of the turbulent vortex dynamo in a rotating horizontal layer of incompressible non-uniformly heated fluid can be written in the dimensionless form [

( P r ∂ ∂ t − Δ ) T = − Δ ⊥ φ , ( ∂ ∂ t − Δ ) Δ φ = R a T + C [ ( e ⋅ ∇ ) 2 − Δ ⊥ ] ψ − T a 1 / 2 ∂ ψ ∂ z , ( ∂ ∂ t − Δ ) ψ = − C ( e ⋅ ∇ ) 2 φ + T a 1 / 2 ∂ φ ∂ z , P r = ν χ , R a = g β A h 4 ν χ , C ∝ Ω Λ , T a = 4 Ω 2 h 4 ν 2 . (3)

Here, T is the temperature, j and y are the poloidal and toroidal potentials of the velocity field, and Δ ⊥ = ∂ 2 / ∂ x 2 + ∂ 2 / ∂ y 2 is the two-dimensional Laplace operator. Pr, Ra, and Ta are the dimensionless Prandtl, Rayleigh, and Taylor numbers, e the unit vector directed vertically upward, A the uniform temperature gradient between the horizontal boundaries of the layer, g the gravity acceleration, b the coefficient of thermal expansion, h the layer height. The dimensionless parameter C characterizing the small-scale turbulence is related in a rather complicated manner to the turbulence characteristics such as the most energetic scale l and characteristic time t of the turbulent velocity correlation [

The system of dynamic linear Equations (3) for three large-scale fields includes two different positive feedbacks. One of them acts between the poloidal j-component of the velocity field and the field of temperature disturbance T. It links the first and the second equations in system (3) and leads to natural convective instability. The other directly links the components j and y of the velocity field, i.e. the second and the third equations from system (3). Here it is important to note that, as applied to tropical cyclones, this would imply an immediate positive feedback between the tangential and overturning circulation. This feedback, being generated by special properties of small-scale helical turbulence and, therefore, named the helical feedback [

As we can see in model (3), in the absence of volumetric heat release ( Λ = 0 ), the coefficient C = 0 and, accordingly, the corresponding terms in the equations vanish, i.e. the effect of the vortex dynamo is impossible. As applied to the atmosphere, this means that heating the layer from below owing solar radiation received by the underlying surface is insufficient for the occurrence of large-scale instability.

If C ≠ 0 , then the mean large-scale flow V shows a special topological property ? the linkage of vortex lines of poloidal and toroidal flow component [

Thus, the positive helical feedback is responsible for a new type of instability, namely, the large-scale helical-vortex instability [

Unlike the alpha effect in an electrically conducting medium based on the interaction of two different physical fields―the magnetic field and the velocity field―in its hydrodynamic analog, the dynamo effect is to be provided only by the singularities of a single velocity field. In magnetohydrodynamics, the mean electromotive force [

The vortex dynamo should operate by linking the poloidal and toroidal component of the velocity field. In mathematical model (3), C-terms responsible for the positive feedback can also be represented as a force of some kind [

f = C { e ⋅ ( curl V ) z − ∂ ( e × V ) / ∂ z } , e = { 0 , 0 , 1 } . (4)

It seems both curious and useful to give an explicit representation of components of such hypothetical vortex-motive force

f = { ∂ v ∂ z , − ∂ u ∂ z , ∂ v ∂ x − ∂ u ∂ y } . (5)

The first two terms in Formula (5) describe the vertical shear of horizontal velocity whilst the third one is the vertical component of vorticity. As our analysis showed [

The emergence of positive helical feedback between the toroidal and poloidal component of the velocity field can be considered as a clear sign indicating the onset of helical-vortex instability. This served a basic idea for developing a numerical approach [

A key to the approach [

Numerical investigations carried out in [

・ gave a vivid example of non-zero mean helicity, 〈 H 〉 ≠ 0 , generation that implied the broken mirror symmetry,

・ demonstrated new effects in flow structure and energetics attributed to a large-scale instability,

・ confirmed a threshold type onset of this new helical instability,

・ highlighted the generation of positive helical feedback between the tangential and overturning circulation in a vortex system,

・ demonstrated how the helical feedback could be identified by examining integral kinetic energies of the tangential and overturning circulation,

・ showed a probable scenario for development of the instability by merging of helical convective cells and consequent intensification of newly forming larger-scale helical vortices,

・ emphasized the crucial role of vertical flow component in the whole scenario of new instability,

・ pointed out an increased effectiveness of heat transfer within a helical and larger vortex flow configuration.

These findings gave the authors of [

The term “helical cyclogenesis” was first introduced in [

Despite the above mentioned long standing theoretical discussions on the role of helicity in tropical cyclogenesis, helical features of the velocity field have not been highlighted in real tropical cyclone investigations until very recently.

Probably, the hypothesis on the turbulent vortex dynamo [

In particular, in the case of tropical cyclones, the hypothetical realization of the vortex dynamo would mean the generation of positive helical feedback between the tangential and transverse circulation in a forming large-scale vortex, ensuring their mutual intensification. However, in nature, no vortex-motive force is known capable of providing the feedback and it was unclear whether such a scenario would be applicable to the atmosphere at all. To substantiate a feasibility of the vortex dynamo mechanism in the atmosphere, new knowledge about atmospheric processes was strongly needed.

On the top of the third millennium, the knowledge was brought by high resolution three-dimensional numerical modeling of tropical cyclone formation. Pioneering near-cloud-resolving idealized simulations of tropical cyclogenesis under realistic meteorological conditions [

By now, the VHTs and their role in tropical cyclone formation have become a subject of significant amount of studies and reliably confirmed in observations performed by researchers over the world, see e.g. [

For the purpose of our further discussion, let us summarize briefly what has been known about these convective structures by now.

The cloud hot towers in the tropical atmosphere of the Earth (

The vortical tropical convection―VHTs―was first found nearly a half of the century later [^{−3} 1/s, what by the one-two order of magnitude exceeds the planetary rotation. The detailed evolution of the updraft observed during three successive fly-bys within approximately 40 minutes was presented in Figures 10-12 [

updrafts that rotate in helical fashion (as in rotating Rayleigh-Bénard convection), …These locally buoyant vortical plume structures amplify pre-existing cyclonic vorticity by at least an order of magnitude larger than that of the aggregate vortex.” Finally, the authors [

One of the most impressive visualizations of hot towers in a developed tropical cyclone was made by NASA (

By deepening and expanding the new knowledge gained about vortical convection [

The atmospheric scenario of self-organization described in [

The results of our collaborative research efforts are presented in publications [

Up to the time, when our joint research with American colleagues started, no one had tried to investigate the helical features of atmospheric turbulence by direct (i.e. without introducing any parameterization) numerical simulation resolving the cloud scales. As far as we were then aware, there were close enough to this subject only papers [

The basic premise for our studies was that the inverse energy cascade and generation of large-scale structures are possible in helical turbulence characterized by the broken mirror symmetry. Such departure from the mirror symmetry in turbulence can be quantified by helicity of the velocity field. It was fairly obvious to expect helicity generation in the atmospheric turbulence under the influence of the Coriolis force. In this case, it seemed very likely to find in the atmosphere non-zero helicity fluctuations by following expectations of authors [

The foregoing determined the choice of the first step in our studies. We began by calculating the helicity of the velocity field. In papers [

To analyze the process of self-organization of moist atmospheric convection observed under conditions of tropical cyclogenesis as posed in [

The velocity fields used for post-processing in our studies [

It is important to point out that no external assumptions were imposed on the fluid motions investigated and described in [

In [

For our thorough analysis in [

In this paper, let us analyze and discuss a few helical characteristics, which have been applied in [

h = v ⋅ ω = u ( ∂ w ∂ y − ∂ v ∂ z ) + v ( ∂ u ∂ z − ∂ w ∂ x ) + w ( ∂ v ∂ x − ∂ u ∂ y ) , 〈 H 〉 , 〈 H h o r 〉 , 〈 H v e r 〉 . (6)

It is important to note that the non-zero mean helicity could be generated even in the absence of vertical flows (w = 0). However, such is only possible when the horizontal wind is changing with height, i.e., in the occurrence of vertical shear of the horizontal wind. Thus, the non-zero horizontal helicity can be considered as a sign of existing or emerging shear flow. The non-zero vertical helicity, being a product of vertical velocity and vorticity, is an indicator of the presence of vortical convection in the examined area of tropical cyclone formation and allows perfectly the identification of such flows. As our studies [

In

The color bar indicates the magnitude of h multiplied by 10^{−2}. Orange, red and dark red regions correspond to strong positive helicity. As part of our examination of the evolution of the three-dimensional helicity field (6), the vertical velocity and vertical vorticity were analyzed also (not shown). This allowed both an identification of the formation of rotating convective structures and determination of their rotational signature, i.e., cyclonic or anticyclonic.

Such rotating vertical flows observed in the field of vertical helicity h_{z} during experiment A2 were first shown in our presentation at the 31st AMS Conference on Hurricanes and Tropical Meteorology [_{z} multiplied by 10^{−4}.

The undertaken studies [

A successive development of rotating convection was traced and discussed in [

A special attention was paid to a process of merging of convective cells interpreted as a manifestation of upscale organization of atmospheric rotating moist convection. As it was shown in [

In [

Analysis carried out for experiments A2, B3, C3, and E1 from [^{12} m^{4}/s^{2}.

During next few hours, an upscale organization continued and led to formation of tropical depression vortices, which had essentially larger scales, with tens kilometers in diameter. The formation of tropical depressions occurred at t =15 h (A2―^{12}m^{4}/s^{2}. Of these vortices, a further intensification of tropical depression was observed only in experiment A2. The process of intensification up to hurricane power was accompanied by a strong increase in helicity values. Thus, at t = 62 h (^{12} m^{4}/s^{2}, respectively.

To be sure of the reliability of these new numerical results on the helicity values, an important opportunity was found that allowed their comparison with unique helicity data for intense vortical convection in Hurricane Bonnie (1998) calculated in paper [^{2}/s^{2}, was found in Hurricane Bonnie (1998) on August 24 when the maximum surface wind was about 55 m/s. We calculated the corresponding helicity for experiment A2 [^{2}/s^{2} during a time span 56 - 65 hours when the maximum surface wind was between 33.5 m/s and 42.5 m/s. Within this time interval, the helicity reached its highest local value equal to 2700 m^{2}/s^{2}, which is close to that found in [

The non-zero mean helicity within a mesoscale area of tropical cyclone formation, starting from the early hours in the vortex evolution and persistently increasing with time, signifies a break of the mirror symmetry of atmospheric turbulence. The broken mirror symmetry in turbulence is a precondition for the emergence of a large-scale alpha-like instability [

Meanwhile, the non-zero helicity does not necessarily imply that the large-scale vortex instability is underway. In fact, this only means that the existing departure of the mirror symmetry in turbulence produces an environment conducive to the onset of large-scale instability.

That was a critical point in our research, happened soon after our first results were published [

Having postponed for a while the problem of turbulent statistics, the work on which required the search and involvement of a highly qualified expert in both turbulence and tropical cyclone fields, it was decided to begin with a search for the large-scale helical-vortex instability.

To this end, a numerical approach [

In [

In the majority of numerical experiments in [

The described mechanism was interpreted in [

hours of those experiments were between 0.2 - 0.35 × 10^{12} m^{4}/s^{2}. Let us note, that the first updraft was initiated by a local heating at low levels, z = 2 km [

At later times in the simulations the convergence/stretching of near surface (0 < z < 2 km) vorticity by the convective plumes dominated the generation of vorticity by tilting processes [

Moreover,

Further, we briefly demonstrate, following [

The results of studies [

〈 E 〉 = 〈 E P 〉 + 〈 E S 〉 . (7)

To quantitatively diagnose an emerging feedback loop between the primary and secondary circulation, we examined the kinetic energy of both circulations calculated as squares of corresponding components of velocity in the cylindrical coordinates, integrated over the computational domain and normalized by number of grid points.

The kinetic energy evolution in experiments A2, B3, C3, and E1 is shown in

To those readers, who are interested in looking deeper into detailed hydro- and thermodynamics of tropical cyclone formation, an impressive visualization in [

In [

In numerical experiments A2, B3, C3, and E1 [^{16} m^{5}/s^{2} (

During an initial time interval, which lasts about 10 hours in A2 and E1, 15 - 16 hours in C3, and near 40 hours in B3, the tangential circulation is slightly weakening against its initial energy value (

The rotating convective cells start to generate a non-vanishing and increasing third (vertical) contribution of helicity near t = 7 - 8 h in A2 and E1, 14 - 15 h in C3, and 18 - 20 h in B3 and, thereby, a local linkage of vortex lines of horizontal and vertical flow components in a vicinity of each rotating updraft. Each rotating convective structure contributes simultaneously to both the tangential and overturning circulation, namely, by its vertical vorticity to the former and by its vertical motion to the latter. Thus, such a structure represents a natural link between the circulations on the local scale whilst the developed population should help link the primary and secondary circulations on the system scale.

A degree of such linkage is measured by helicity.

Thus, in experiment A2, an increase in the vertical contribution of helicity from zero up to approximately 0.5 × 10^{10} m^{4}/s^{2} within 6 - 9 h is associated with an amplification of horizontal and total helicity from 0.15 × 10^{12} m^{4}/s^{2} up to 0.3 × 10^{12} m^{4}/s^{2}. Just at this time, t = 6 h, a slight yet distinct increase in the kinetic energy 〈 E S 〉 starts.

Near t = 10 h in A2 and E1, 20 h in C3, and 40 h in B3, one can observe dramatic changes in the flow intensity―the kinetic energy of the transverse circulation, 〈 E S 〉 , increases sharply and soon after this, kinetic energy of the tangential circulation, 〈 E P 〉 , starts to increase as well (^{10} m^{4}/s^{2}. The helicity, 〈 H 〉 , is about 0.08 - 0.23 × 10^{12}m^{4}/s^{2}. This gives a start to the formation of a stable system-scale (hundreds of kilometers horizontally) secondary circulation during the next 1 - 2 h. The secondary circulation is sustained and linked with the primary circulation by the strong VHT and a number of smaller and less intense rotating convective cores (

Once the linkage on the system scale is formed, this marks a critical point in a process of tropical cyclone formation when the vortex becomes self-sustaining. The time after which both 〈 E S 〉 and 〈 E P 〉 mutually increase may be considered a practical definition for the moment of tropical cyclogenesis―“G”. The simulations indicate that a positive feedback formed between the two circulations is sustained by convective instability and vortical convection in the central region of the developing circulation. The convective instability there is maintained primarily by latent heat fluxes from the underlying sea surface, which need not increase with wind speed [

We have provided an overview of the efforts undertaken to apply the fundamental ideas on self-organization in helical turbulence to study the formation of tropical cyclones in the atmosphere, and also contributed to this problem by a new suggestion concerning the analogy between the vortex dynamo in the atmosphere and MHD dynamo in electrically conducting medium. Beginning with the short excursus to the 1980s, when the hypothesis on the turbulent vortex dynamo was put forward and wide-ranging research program was arranged to test it, we have emphasized recent results of our collaborative Russian-American investigations.

1) Our first finding [

2) By adapting for post-processing of atmospheric simulation data our earlier developed approach [

3) For the first time in tropical cyclone research, we have emphasized a role of special topology of forming vortex provided by interaction of motions of different scales [

For purposes of quantitative diagnosis, we have analyzed the evolution of energetics and structure of the forming vortex. The integral kinetic energy of primary and secondary circulation was applied to diagnose the onset of large-scale vortex instability (

4) The process of helicity generation in conditions of tropical cyclogenesis was examined in [

5) Thus, due to the crucial role of topology in the new instability and role of VHTs in providing this, the helicity can be suggested as a measure to quantify the chaotic influence resulting from moist convection.

6) In the present paper, we discuss and substantiate the key role of VHTs as a necessary element to provide the dynamo effect. Based on the mathematical model for the turbulent vortex dynamo, an analogy is traced between the role of interaction “moist convection―vertical wind shear” in creating the vortex dynamo in the atmosphere and the role of the mean electromotive force providing the MHD dynamo in electrically conducting medium.

Although some successful steps have been undertaken to substantiate the contribution of the turbulent vortex dynamo to tropical cyclogenesis, further research is needed:

・ It is indispensable to study the turbulence statistics with a focus on analysis of the transport of kinetic energy and helicity over the spectrum of scales. The discovery of an inverse energy cascade would become a serious argument in order to confirm the vortex dynamo.

・ Another challenging task is connected with the search for a possible threshold of discovered instability. Bearing in mind the above analysis based on the vortex dynamo model and discussion for the vortex-motive force, it seems useful to combine helicity with Convective Available Potential Energy (CAPE), see, e.g. [

・ The discovery of the instability threshold would allow us to recall the idea on impacting on tropical cyclogenesis, which was investigated within the framework of the Soviet Program, in order to consider it using the most advanced tools of modern science.

・ To implement the above suggestions, it seems very promising to use as a basis for investigations a platform similar to the “Hurricane Nature Run” developed and applied in [

Meanwhile, it would be possible to begin without postponing and to apply our approach presented in Section 6 to a real case of observed tropical cyclone. This implies a combination of our approach with the theory of tropical cyclogenesis in an easterly wave critical layer [

This work was supported in part by the Russian Foundation for Basic Research (RFBR) under Grant 16-05-00551a. The results based on near-cloud-resolving numerical simulation could only be obtained in close collaboration with my American co-author in many works, M.T. Montgomery, and his colleagues M.E. Nicholls, and S. Barve. I am deeply grateful for discussions and recommendation about writing the review to R.Z. Sagdeev, the co-author of the vortex dynamo theory and Director of Space Research Institute in the 1970-80s, who supervised the Soviet Research Program discussed in the Introduction. I also thank the referees for their valuable comments, which have led to significant improvements in presentation.

Levina, G. (2018) On the Path from the Turbulent Vortex Dynamo Theory to Diagnosis of Tropical Cyclogenesis. Open Journal of Fluid Dynamics, 8, 86-114. https://doi.org/10.4236/ojfd.2018.81008