A way to parameterize helical boundary layer turbulence in numerical modeling of tropical cyclogenesis

Despite many years' observation and investigation of tropical cyclones the problem of cyclogenesis is still far from being completely understood. It is well-known, for example, that not every tropical depression may develop into a hurricane. This suggests the existence of a threshold for vortex instability. That is why one of primary tasks needing the fastest solution is that on factors governing the early stages of cyclone evolution to identify those of them predetermining the hurricane formation.

As a possible mechanism, which may contribute to early stages of tropical cyclogenesis, we consider the mechanism of vortex dynamo. This effect is based on specific generating properties of small-scale helical turbulence. According to modern physics small-scale helical turbulence has a number of special features and under certain conditions is capable of intensifying and sustaining large-scale vortex disturbances by means of energy transfer from small to large scales. It is a well-defined fact that turbulent convection in a rotating medium, for example in the atmosphere, is helical and mirror-nonsymmetrical. The action of the Coriolis force on a horizontal velocity component in convective cells makes them helical. The non-zero mean helicity of convection can arise in non-homogeneous atmosphere.

In our Laboratory of Hydrodynamics we are carrying out theoretical and experimental investigations of thermal turbulent convection in rotating fluids. Our work is focused on the modeling of physical mechanisms and conditions leading to the formation of intensive large-scale vortices.

Both theoretical and experimental studies resulted in two important findings:

1. Helical boundary layer turbulence can initiate a large-scale helical–vortex instability

2. There exists a threshold of such instability.

Progress in experimental studying of rapidly rotating large-scale spiral vortex from a localized heat source has been regularly discussed at international geophysical and turbulent meetings since 2003.

Our experimental approach allows inducing such vortex due to interaction between the Coriolis force and the large-scale shear flow when there exist small helical heat carriers in a fluid. A horizontal temperature difference between a heater in the central zone of the bottom and a cold periphery results in a large-scale advective shear flow. A vertical temperature difference induces an intensive updraft formed in turbulent states by chaotically rising small-scale convective elements over the heater. In a rotating layer the Coriolis force affects these flows. After exceeding some critical intensity of heating a rotating helical vortex appears and penetrates the whole layer height. “Neutral” curves to describe the threshold of this coherent structure generation versus background rotation and the intensity of convection have been obtained. An explanation for the discovered phenomenon has been suggested. It may result from an instability caused by an interaction between the large-scale spiral disturbance and small-scale spiral heat carriers (similar to rolls in the tropical cyclone boundary layer) rising from the convective boundary layer near the heater.

The experiments are carried out using two apparatuses with rotating cuvettes of cylindrical and rectangular form. An aspect ratio of typical horizontal scale of the fluid layer to its height can be changed and was probed up to 10. Several liquids are tested so that the corresponding Prandtl numbers are varied in the range 60–180. Since 2004 these investigations have used modern techniques of Particle Image Velocimeter measurements and are assigned to become more systematical and giving quantitative results. They are aimed at both better understanding the vortex formation and searching for ways to control this process.

Our numerical approach uses as a basis recent achievements in the vortex dynamo theory. Theoretical model for the vortex dynamo in a convectively unstable rotating fluid involves a mean-field velocity equation. This suggests a parameterization of small-scale helical turbulence of thermoconvective origin induced in conditions of the combined heating (from below and by internal heat sources due to vapor condensation) and subjected the affect of the Coriolis force. It is extremely important to note that the additional volumetric heat release is a binding condition for obtaining the non-zero dynamo effect.

In accordance with the vortex dynamo theory the main averaged effect of small-scale helical turbulence and the first sign indicating the onset of helical-vortex instability is the generation of positive feedback linking the horizontal and vertical circulation in a forming vortex structure. This is described mathematically by the so-called generating alpha-term in the mean-field velocity equation.

By means of numerical simulation we are demonstrating how such feedback can work, for example, for the Rayleigh-Benard convection in an extended horizontal layer of incompressible fluid. To this purpose a forcing function, which has an identical tensor structure with the generating alpha-term has been included in the momentum equation from the Boussinesq system. We performed the linear stability analysis and showed that there existed a threshold of instability.

We obtained new effects in the flow structure and energetics, which might be of immediate relevance to the initiation of large-scale vortex instability. The most important finding is a merging of helical convective cells accompanied by an enlargement of the typical horizontal scale of the forming structures, an essential increase in the kinetic energy of flows and intensification of heat transfer. The results of modeling clearly demonstrate how the energy of the additional helical source can be effectively converted into the energy of intensive large-scale vortex flow.

We do like to draw attention of those who work with numerical meteorological models to our results. Inclusion of our forcing function into such models can give a simple parameterization of helical boundary layer turbulence. This also gives a hope to find a threshold of large-scale vortex instability in real atmospheric conditions.

This work is supported by the Russian Foundation for Basic Research under Projects NN 04-05-64315, 03-05-64593 and the International Science and Technology Center under Project #2021.