Equations for averaged variables - Turbulence in the Atmosphere - page 64

Then neglecting molecular diffusion Eq. (3.22) says the gradient in the x 1 -direction
evolves as

D
Dt ˜

∂ u 1 ˜

g 1 −

∂x 1 ˜

g 1 . (3.24)

This is called linear or normal strain ( Kundu , 1990 ). If ∂ u 1 /∂x 1 is negative the

magnitude of the scalar gradient is amplified; if positive, it is attenuated. This is
analogous to vortex stretching (Figure 3.5) .
Deformation can also reorient a scalar gradient. In the example of Eq. (3.23) with
a scalar field initially having a gradient only in the x 1 -direction, if the

˜

u 1 velocity ˜

component has a gradient in the α 1 direction, say, then components of the scalar

gradient are induced in that direction:

=

D g α

Dt −

˜ u 1

∂x α ˜

∂ ˜

g 1 . (3.25)

This is called shear strain. It is analogous to vortex tilting (Figure 3.5) .
In summary, a blob of c in a realization of a turbulent flow follows an irregular

trajectory and is contorted by turbulent velocity gradients as it travels; this contortion
increases themagnitude of scalar gradients within the blob, enhancing themolecular
diffusion. As a result, the blob disappears more quickly than it would in the absence
of turbulence (Figure 3.4) .

˜

Figure 3.5 A schematic of the distortion of a scalar-gradient field by a strain field.
The lines are iso-concentration contours. The upper two panels depict linear or
normal strain, Eq. (3.24) ; the lower panel depicts shear strain, Eq. (3.25) .