Helicity (dynamic of the fluids)

See also: Hélicité

In Dynamic of the fluids, the helicity is a measurement of the domino effect which a local Rotation will have on a piece of Fluide. It is a quantity used to deduce the Turbulence from the fluid and is particularly used in Météorologie to estimate the potential tornadic. The helicity is a preserved size if the fluid obeys the Navier-Stokes equations for the incompressible fluids.

It is calculated by making the summation, in a piece of fluid, relative swirl (or Rotationnel speed) with the scalar Produit local speed in the fluid:

H= \ int {\ vec V} \ cdot \ left (\ nabla \ times {\ vec V^'} \ right) \, d^3 {\ mathbf R} = \ int {\ vec V} \ cdot \ vec \ zeta \, d^3 {\ mathbf R} \ qquad \ qquad \ begin {boxes} R = dimensions \ of the \ volume \ \ V = local Speed \ \ according to \ R \ \ V^' = Speed \ of the \ particles \ in \ the \ volume \ \ \ zeta = relative swirl \ \ end {boxes}

The equation shows that such a volume in rotation around an axis in the direction of displacement, the helicity will be positive if rotation is time direction (by looking at from which volume comes) and negative if rotation is in anti-clockwise direction. Moreover, more the swirl and local speed will be parallel, more H will be large.

Meteorology

In meteorology, the helicity corresponds to the transfer of rotation of the environment towards a piece of air in convection. In this case, one simplifies the definition of the helicity to a dimension by supposing that the swirl is horizontal:

H = \ int {\ vec V_h} \ cdot \ vec \ zeta_h \, D {\ mathbf Z} = \ int {\ vec V_h} \ cdot \ nabla \ times V_h \, D {\ mathbf Z} \ qquad \ qquad \ begin {boxes} Z = altitude \ \ V_h = horizontal Speed \ \ \ \ zeta_h = relative swirl \ \ horizontal \ end {boxes}

In this formulation, if the horizontal wind does not change direction with altitude, H is null because V_h and \ nabla \ times V_h are Perpendiculaire S one to the other making their product scalar no one. It is thus obvious that H is positive if V_h turns horairement with altitude and negative in the opposite case.

The helicity thus has units of energy ( {m^2}/{s^2} ) which can be interpreted like a measurement of energy of the Cisaillement of the winds, including their change of management. One thus used the helicity to define indices of potential of tornadoes. In this case, one places oneself within the framework of reference of the storm by withdrawing the speed of this one with the ground and one also limits Z in the layer between the base of the cloud and his top since it is in this one that rotation will be generated (in general under 3 km of altitude):

HR = \ int {\ left (\ vec V_h - C \ right)} \ cdot \ nabla \ times V_h \, D {\ mathbf Z}

\ qquad \ qquad \ begin {boxes} C = Speed \ of \ D \ acute {E} placement \ of \ the storm \ end {boxes}

The breaking values of this relative helicity (HR called SRH in English) found for the storms violent one in North America are:

  • HR = 150 to 299: supercellules possible with weak Tornado according to the scale of Fujita

  • HR = 300 to 499: very favorable to the development of supercellules and strong tornadoes
  • HR > 450: violent tornadoes
  • When calculated with the helicity under 1 km, the only threshold are of 100.

However, these results are very variable according to the type of Convection and this is why an index combining the helicity and the Potential Énergie of Convection Available (EPCD) was developed. Primarily, one multiplies H by the EPCD and one divides the whole by a value threshold of the EPCD. This makes it possible to eliminate the zones from strong horizontal swirls but from low potential of convection. This index of helicity (IH or EHI in English) has the following values thresholds:

  • IH = 1: possible tornadoes

  • IH = 1 to 2: average tornadoes with strong
  • IH > 2: strong tornadoes.

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