Hydrodynamical helicity

this page is about helicity in fluid mechanics. For helicity of magnetic fields, see magnetic helicity. For helicity in particle physics, see helicity (particle physics).

In fluid mechanics, helicity is the extent to which helix-like motion occurs. If a parcel of fluid is moving, undergoing solid body motion rotating about an axis parallel to the direction of motion, it will have helicity. If the rotation is clockwise when viewed from ahead of the body, the helicity will be positive, if counterclockwise, it will be negative.

Formally, helicity is defined as :

\[ H=\int\mathbf{u}\cdot\left(\nabla\times\mathbf{u}\right)\,d^3{\mathbf r}. \]

Where \(\mathbf{u}\) is the three dimensional velocity vector of the fluid particles.

The concept is interesting because it is a conserved quantity\[H\] is unchanged in a fluid obeying the Euler equations (i.e. zero viscosity) for incompressible fluids. This is analogous to the conservation of magnetic helicity.

Helicity is a useful concept in theoretical descriptions of turbulence.

Meteorology

In meteorology, helicity corresponds to the transfer of vorticity from the environment to an air parcel in convective motion. Here the definition of helicity is simplified to only use the horizontal component of wind and vorticity:

\[ H = \int{ \vec V_h} \cdot \vec \zeta_h \,d{\mathbf Z} = \int{ \vec V_h} \cdot \nabla \times V_h \,d{\mathbf Z} \qquad \qquad \begin{cases} Z = Altitude \\ V_h = Horizontal\ velocity \\ \zeta_h = Horizontal\ vorticity \end{cases} \]

According to this formula, if the horizontal wind does not change direction with altitude, H will be zero as \(V_h\) and \(\nabla \times V_h\) are perpendicular one to the other making their scalar product nil. H is then positive if the wind veers (turns clockwise) with altitude and negative if it backs (turns counter-clockwise). Helicity has energy units per units of mass (\( {m^2}/{s^2}\)) and thus is interpreted as a measure of energy transfer by the wind shear with altitude, including directional.

This notion is used to predict the possibility of tornadic development in a thundercloud. In this case, the vertical integration will be limited below cloud tops (generally 3 km or 10,000 feet) and the horizontal wind will be calculated to wind relative to the storm in subtracting its motion:

\[SRH = \int{ \left ( \vec V_h - C \right )} \cdot \nabla \times V_h \,d{\mathbf Z} \qquad \qquad \begin{cases} C = Cloud\ motion\ to\ the\ ground \end{cases} \]

Critical values of SRH (Storm Relative Helicity) for tornadic development, as researched in North America, are:

• SRH = 150-299 ... supercells possible with weak tornadoes according to Fujita scale
• SRH = 300-499 ... very favourable to supercells development and strong tornadoes
• SRH > 450 ... violent tornadoes
• When calculated only below 1 km (4,000 feet), the cut-off value is 100.

Helicity in itself is not the only component of severe thunderstorms and those values are to be taken with caution. That is why the Energy Helicity Index (EHI) has been created. It is the result of SRH multiplied by the CAPE (Convective Available Potential Energy) and then divided by a threshold CAPE: EHI = (CAPE x SRH) / 160,000. This incorporates not only the helicity but the energy of the air parcel and thus tries to eliminate weak potential for thunderstorms even in strong SRH regions. The critical values of EHI:

• EHI = 1 ... possible tornadoes
• EHI = 1-2 ... moderate to strong tornadoes
• EHI > 2 ... strong tornadoes

Bibliography

• Batchelor, G. K., (1967, reprinted 2000) An Introduction to Fluid Dynamics, Cambridge Univ. Press
• Ohkitani, K., "Elementary Account Of Vorticity And Related Equations". Cambridge University Press. January 30, 2005. ISBN 0-521-81984-9
• Chorin, Alexandre J., "Vorticity and Turbulence". Applied Mathematical Sciences, Vol 103, Springer-Verlag. March 1, 1994. ISBN 0-387-94197-5
• Majda, Andrew J., Andrea L. Bertozzi, "Vorticity and Incompressible Flow". Cambridge University Press; 1st edition. December 15, 2001. ISBN 0-521-63948-4
• Tritton, D. J., "Physical Fluid Dynamics". Van Nostrand Reinhold, New York. 1977. ISBN 0-19-854493-6
• Arfken, G., "Mathematical Methods for Physicists", 3rd ed. Academic Press, Orlando, FL. 1985. ISBN 0-12-059820-5

es:Helicidad (mecánica de fluidos)