Forces and dynamics - Physical Processes in Earth and Environmental Sciences - page 59

Definition of vertical component of
vorticity due to horizontal rotation

Vertical paddle stick
with fins

Conventionally
+ve anticlockwise
-ve clockwise

z z

Anticlockwise (positive) vertical vorticity contribution 1

u 3

+ve

u 2
u 1

+ y

u 1 > u 2 > u 3 so gradient of u across direction
+ y is negative, that is, d u/ d y = -ve

Coriolis

Anticlockwise (positive) vertical vorticity contribution 2

Fig. 3.34 Vorticity sign conventions and the negative vorticity
evident from the flow of Coriolis's hair. w 3

w 2

w 1

(Fig. 3.34), we define positive cyclonic vorticity with
anticlockwise rotation viewed looking down on or into
the vortical axis; vice versa for negative or anticyclonic
vorticity . Looked at this way it is clear that vorticity,

+ x

w 1 < w 2 < w 3 so gradient of w across direction
+ x is positive, that is, d w/ d x = +ve

is a vector quantity; it has both magnitude and direc-
tion with vertical,

y ,
components. Each of these components defines rotation
in the plane orthogonal to itself, for example, stream-
wise vorticity involves rotations in the plane orthogonal
to the streamwise direction and since x is the stream-
wise component the vorticity refers to rotation in the
plane yz .
Now here is the tricky bit (Figs 3.35 and 3.36).
In order for rotation to occur there must be a gradient of
velocity acting upon a parcel of fluid; if there is no gradient
there can be no vorticity. The velocity gradient sets up
gradients of shearing stress and hence this kind of shear
vorticity (also called relative vorticity ) depends upon the
magnitude of the gradient, not the absolute velocity of the
flow itself. This is best imagined by spinning-up a small
object, like a top, with one's fingers to create vertical vor-
ticity (ignore the tendency for precession): a shear couple
is required from you to turn the object into rotation.
Better still for use in flowing fluids, you can make your
own vorticity top from a wooden stick and two orthogonal
fins (or you can just imagine the vorticity top in a thought
experiment). Now, with respect to the plane normal to the
vertical spin axis of the vorticity top, only two velocity
gradients may exist in the xy plane that, between them,

z , streamwise, x , and spanwise,

Vertical component of vorticity, z z

−∂ u / ∂ y + ∂ w / ∂ x

is overall positive

Fig. 3.35 Shear vorticity and the Taylor vorticity top.

d w/ d x negative or positive

+ y ( w )

+ x ( u )

d u/ d y
negative
or positive

Fig. 3.36 Combination of velocity gradients that might produce
overall positive vorticity.