|
1
|
- Circulation
- Bjerknes Circulation Theorem
- Vorticity
- Potential Vorticity
- Conservation of Potential Vorticity
|
|
2
|
- Circulation and vorticity are the two primary measures of rotation in a
fluid.
- Circulation, which is a scalar integral quantity, is a macroscopic measure
of rotation for a finite area of the fluid.
- Vorticity, however, is a vector field that gives a microscopic measure
of the rotation at any point in the fluid.
|
|
3
|
- The circulation, C, about a closed contour in a fluid is defined as the
line integral evaluated along the contour of the component of the
velocity vector that is locally tangent to the contour.
|
|
4
|
- In this case the circulation is just 2π times the angular momentum
of the fluid ring about the axis of rotation. Alternatively, note that
C/(πR2) = 2Ω so that the circulation divided by the
area enclosed by the loop is just twice the angular speed of rotation of
the ring.
- Unlike angular momentum or angular velocity, circulation can be computed
without reference to an axis of rotation; it can thus be used to
characterize fluid rotation in situations where “angular velocity” is
not defined easily.
|
|
5
|
- In fluid mechanics, the state when no part of the fluid has motion
relative to any other part of the fluid is called 'solid body rotation'.
|
|
6
|
- Circulation can be considered as the amount of force that pushes along a
closed boundary or path.
- Circulation is the total “push” you get when going along a path, such as
a circle.
|
|
7
|
- The circulation theorem is obtained by taking the line integral of
Newton’s second law for a closed chain of fluid particles.
|
|
8
|
- For a barotropic fluid, Bjerknes circulation theorem can be integrated
following the motion from an initial state (designated by subscript 1)
to a final state (designated by subscript 2), yielding the circulation
change:
|
|
9
|
- In a barotropic fluid, the solenoid term (Term 2) vanishes.
- č The absolute
circulation (Ca) is
conserved following the parcel.
|
|
10
|
- Suppose that the air within a circular region of radius 100 km centered
at the equator is initially motionless with respect to the earth. If
this circular air mass were moved to the North Pole along an isobaric
surface preserving its area, the circulation about the circumference
would be:
- C = −2Wπr2[sin(π/2) −
sin(0)]
- Thus the mean tangential velocity at the radius r = 100 km would be:
- V = C/(2πr) = − Wr ≈ −7
m/sec
- The negative sign here indicates that the air has acquired anticyclonic
relative circulation.
|
|
11
|
|
|
12
|
- A counter-clockwise circulation (i.e., sea breeze) will develop in which
lighter fluid (the warmer land air; T2) is made to rise and
heavier fluid (the colder sea air; T1) is made to sink.
- The effect is this circulation will be to tilt the isopycnals into an
oritentation in which they are more nearly parallel with the isobars –
that is, toward the barotropic state, in which subsequent circulation
change would be zero.
- Such a circulation also lowers the center of mass of the fluid system
and thus reduces the potential energy of that system.
|
|
13
|
- Use the following value for the typical sea-land contrast:
- p0 = 1000 hPa
- p1 = 900 hPa
- T2 − T1
= 10◦ C
- L = 20 km
- h = 1 km
- We obtain an acceleration of about 7 × 10−3 ms−2
for an acceleration of sea-breeze circulation driven by the
solenoidal effect of sea-land temperature contrast.
|
|
14
|
- Vorticity is the tendency for elements of the fluid to "spin.“.
- Vorticity can be related to the amount of “circulation” or
"rotation" (or more strictly, the local angular rate of
rotation) in a fluid.
- Definition:
|
|
15
|
- In large-scale dynamic meteorology, we are in general concerned only
with the vertical components of absolute and relative vorticity, which
are designated by η and ζ
, respectively.
|
|
16
|
|
|
17
|
- Stokes’theorem states that the circulation about any closed loop is
equal to the integral of the normal component of vorticity over the area
enclosed by the contour.
- For a finite area, circulation divided by area gives the average normal
component of vorticity in the region.
- Vorticity may thus be regarded as a measure of the local angular
velocity of the fluid.
|
|
18
|
- Vorticity can be associated with only two broad types of flow
configuration.
- It is easier to demonstrate this by considering the vertical component
of vorticity in natural coordinates.
|
|
19
|
|
|
20
|
|
|
21
|
- The quantity P [units: K kg−1 m2 s−1]
is the isentropic coordinate form of Ertel’s potential vorticity.
- It is defined with a minus sign so that its value is normally positive
in the Northern Hemisphere.
- Potential vorticity is often expressed in the potential vorticity unit
(PVU), where 1 PVU = 10−6 K kg−1 m2
s−1.
- Potential vorticity is always in some sense a measure of the ratio of
the absolute vorticity to the effective depth of the vortex.
- The effective depth is just the differential distance between potential
temperature surfaces measured in pressure units (−∂θ/∂p).
|
|
22
|
|
|
23
|
|
|
24
|
- The Rossby potential vorticity conservation law indicates that in a
barotropic fluid, a change in the depth is dynamically analogous to a
change in the Coriolis parameter.
- Therefore, in a barotropic fluid, a decrease of depth with increasing
latitude has the same effect on the relative vorticity as the increase
of the Coriolis force with latitude.
|
|
25
|
|
|
26
|
- If the horizontal flow is divergent, the area enclosed by a chain of
fluid parcels will increase with time and if circulation is to be
conserved, the average absolute vorticity of the enclosed fluid must
decrease (i.e., the vorticity will be diluted).
- If, however, the flow is convergent, the area enclosed by a chain of
fluid parcels will decrease with time and the vorticity will be
concentrated.
- This mechanism for changing vorticity following the motion is very
important in synoptic-scale
disturbances.
|
|
27
|
- Convert vorticity in X and Y directions into the Z-direction by the
tilting/twisting effect produced by the vertical velocity
(əw/əx and əw/əy).
|
|
28
|
- Given appropriate horizontal configurations of p and ρ, vorticity
can be produced.
- In this example, cyclonic vorticity will rotate the iosteres until they
are parallel with the isobars in a configuration in which high pressure
corresponds to high density and vice versa.
|
|
29
|
|
|
30
|
- In intense cyclonic storms, the relative vorticity should be retained in
the divergence term.
|
|
31
|
|
|
32
|
- For horizontal motion that is non-divergent (∂u/∂x
+∂v/∂y = 0), the flow field can be represented by a streamfunction
ψ (x, y) defined so that the velocity components are given as
- u =
−∂ψ/∂y,
- v = +∂ψ/∂x.
- The vorticity is then given by
|
|
33
|
|
Notes
