FLUID ROTATION Circulation and Vorticity
FLUID ROTATION
Circulation and Vorticity
ldVldVC
cosArbitrary blob of fluid
rotating in a horizontal plane
Circulation: A measure of the rotation within a finite element of a fluid
ldVdtd
dtdC
In meteorology, changes in circulation are associated with changes in the intensity of weather systems. We can calculate changes in
circulation by taking the time derivative of the circulation:
Circulation is a macroscopic measure of rotation of a fluid and is a seldom used quantity in synoptic meteorology and atmospheric dynamics.
yvxy
yuuyx
xvvxuvdyudxC
Calculate the circulation within a small fluid element with area yx
yvxyyuuyx
xvvxuvdyudxC
yxyu
xvC
vorticityrelativeyu
xv
yxC
lim
0 yx
The relative vorticity is the microscopic equivalent of macroscopic circulation
Consider an arbitrary large fluid element, and divide it into small squares.
yvxyyuuyx
xvvxuvdyudxCA
yvxyyuuyx
xvvxuvdyudxCB
Sum circulations: common side cancels
Make infinitesimal boxes: each is a point measure of vorticity and all common sides cancel
Consider an arbitrary large fluid element, and divide it into small squares.
Fill area with infinitesimal boxes: each is a point measure of vorticity and all common sides cancel so that:
yxyu
xvvdyudxC
Area
The circulation within the area is the area integral of the vorticity
Understanding vorticity: A natural coordinate viewpoint
Natural coordinates: s direction is parallel to flow, positive in direction of flown direction is perpendicular to flow, positive to left of flow
Note that only the curved sides of this box will contribute to the circulation, since the wind velocity is zero on the sides in the n direction
Denote the distance along the top leg as s
Denote the distance along the bottom leg as s + d(s)
Denote the velocity along the bottom leg as V
Use Taylor series expansion and denote velocity along the top leg as snVV
(negative because we are integrating counterclockwise)
CALCULATE CIRCULATION
Note that d (s) = n
snnVVnsVldVC
CALCULATE VORTICITY
snnVVnsVldVC
snnVsVnVsVC
snnVnVC
nV
sV
snsn
nV
snnV
snC
sn
0
lim
nV
RV
s
nV
RV
s
Shear
nV
RV
s
Flow curvature
Vorticity due to the earth’s rotation
Consider a still atmosphere:
Earth’s rotationRV
cosaU
R
BBAAe dxUdxUldUC
no motionalong thisdirection
daadaaCe coscoscoscos
daadaaCe coscoscoscos
AdaCe sin2sincos2sin2 2
after some algebra and trigonometry……
AfCe
fACe lim
0A
fvorticitysEarth sin2'
kyu
xvj
xw
zui
zv
ywV ˆˆˆ
aa Vkk
ˆˆ
yu
xvVkk
ˆˆ
fyu
xv
3D relative vorticity vector
Cartesian expression for vorticity
Vertical component of vorticity vector (rotation in a horizontal plane
Absolute vorticity (flow + earth’s vorticity)
Absolute vorticity
The vorticity equation in height coordinates
xFxpfv
zuw
yuv
xuu
tu
1
yFypfu
zvw
yvv
xvu
tv
1
xpfv
dtdu
1
ypfu
dtdv
1(1) (2)
Expand total derivative
yxTake
)1()2(
asyu
xvvorticityrelativewrite
yF
xF
xp
yyp
xyfv
xfu
zu
yw
zv
xw
yv
xuf
zw
yv
xu
tyx
2
1
yF
xF
xp
yyp
xzu
yw
zv
xw
yv
xuf
dtfd xy
2
1
Rate of change of relative vorticityFollowing parcel
Divergence acting onAbsolute vorticity(twirling skater effect)
Tilting of verticallysheared flow
Gradients in forceOf friction
yF
xF
xp
yyp
xzu
yw
zv
xw
yv
xuf
dtfd xy
2
1
Pressure/densitysolenoids
Rate of change of relative vorticityFollowing parcel
Divergence acting onAbsolute vorticity(twirling skater effect)
Tilting of verticallysheared flow
Gradients in forceOf friction
yF
xF
xp
yyp
xzu
yw
zv
xw
yv
xuf
dtfd xy
2
1
Pressure/densitysolenoids
Rate of change of relative vorticityFollowing parcel
Divergence acting onAbsolute vorticity(twirling skater effect)
Tilting of verticallysheared flow
Gradients in forceOf friction
yF
xF
xp
yyp
xzu
yw
zv
xw
yv
xuf
dtfd xy
2
1
Pressure/densitysolenoids
maF
amPGF
geostrophic wind
Cold advection pattern
m (or ) largeacceleration small
m (or ) smallacceleration large
Solenoid: field loop that converts potential energy to kinetic energy
Rate of change of relative vorticityFollowing parcel
Divergence acting onAbsolute vorticity(twirling skater effect)
Tilting of verticallysheared flow
Gradients in forceOf friction
yF
xF
xp
yyp
xzu
yw
zv
xw
yv
xuf
dtfd xy
2
1
Pressure/densitysolenoids
Geostrophic wind = constant
N-S wind componentdue to friction
xFy
xFfvxdt
du
The vorticity equation in pressure coordinates
yFfuydt
dv
(1) (2)
Expand total derivative
xFfvxP
uyuv
xuu
tu
yFfu
yPv
yvv
xvu
tv
yxTake
)1()2(
yF
xF
xPv
yPu
yv
xuf
yu
xv
yu
xv
Pf
yu
xv
yvf
yu
xv
xu
yu
xv
t
xy
asyu
xvvorticityrelativewrite
yF
xF
xPv
yPu
yv
xuf
Pf
yvf
xu
txy
yF
xF
xPv
yPu
yv
xuf
Pf
yvf
xu
tyx
Local rate ofchange of relativevorticity
Horizontal advectionof absolute vorticityon a pressure surface
Vertical advectionof relative vorticity
Divergence acting onAbsolute vorticity(twirling skater effect)
Tilting of verticallysheared flow
Gradients in forceOf friction
The vorticity equation
In English: Horizontal relative vorticity is increased at a point if 1) positive vorticity is advected to the point along the pressure surface, 2) or advected vertically to the point,3) if air rotating about the point undergoes convergence (like a skater twirling up),
4) if vertically sheared wind is tilted into the horizontal due a gradient in vertical motion 5) if the force of friction varies in the horizontal.
Solenoid terms disappear in pressure coordinates: we will work in P coordinate from now on