FLUID ROTATION Circulation and Vorticity. Arbitrary blob of fluid rotating in a horizontal plane...
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![Page 1: FLUID ROTATION Circulation and Vorticity. Arbitrary blob of fluid rotating in a horizontal plane Circulation: A measure of the rotation within a finite.](https://reader035.fdocuments.net/reader035/viewer/2022062422/56649f1d5503460f94c33a09/html5/thumbnails/1.jpg)
FLUID ROTATION
Circulation and Vorticity
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ldVldVC
cosArbitrary blob of fluid
rotating in a horizontal plane
Circulation: A measure of the rotation within a finite element of a fluid
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ldVdt
d
dt
dC
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.
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yvxy
y
uuyx
x
vvxuvdyudxC
Calculate the circulation within a small fluid element with area yx
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yvxyy
uuyx
x
vvxuvdyudxC
yxy
u
x
vC
vorticityrelativey
u
x
v
yx
C
lim
0 yx
The relative vorticity is the microscopic equivalent of macroscopic circulation
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Consider an arbitrary large fluid element, and divide it into small squares.
yvxyy
uuyx
x
vvxuvdyudxCA
yvxyy
uuyx
x
vvxuvdyudxCB
Sum circulations: common side cancels
Make infinitesimal boxes: each is a point measure of vorticity and all common sides cancel
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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:
yxy
u
x
vvdyudxC
Area
The circulation within the area is the area integral of the vorticity
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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
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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 sn
VV
(negative because we are integrating counterclockwise)
CALCULATE CIRCULATION
Note that d (s) = n
snn
VVnsVldVC
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CALCULATE VORTICITY
snn
VVnsVldVC
snn
VsVnVsVC
snn
VnVC
n
V
sV
sn
sn
n
V
sn
nV
sn
C
sn
0
lim
n
V
R
V
s
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n
V
R
V
s
Shear
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n
V
R
V
s
Flow curvature
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Vorticity due to the earth’s rotation
Consider a still atmosphere:
Earth’s rotationRV
cosaU
R
BBAAe dxUdxUldUC
no motionalong thisdirection
daadaaCe coscoscoscos
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daadaaCe coscoscoscos
AdaCe sin2sincos2sin2 2
after some algebra and trigonometry……
AfCe
fA
Ce lim0A
fvorticitysEarth sin2'
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ky
u
x
vj
x
w
z
ui
z
v
y
wV ˆˆˆ
aa Vkk
ˆˆ
y
u
x
vVkk
ˆˆ
fy
u
x
v
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
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The vorticity equation in height coordinates
xFx
pfv
z
uw
y
uv
x
uu
t
u
1
yFy
pfu
z
vw
y
vv
x
vu
t
v
1
x
pfv
dt
du
1
y
pfu
dt
dv
1(1) (2)
Expand total derivative
yxTake
)1()2(
asy
u
x
vvorticityrelativewrite
y
F
x
F
x
p
yy
p
xy
fv
x
fu
z
u
y
w
z
v
x
w
y
v
x
uf
zw
yv
xu
tyx
2
1
y
F
x
F
x
p
yy
p
xz
u
y
w
z
v
x
w
y
v
x
uf
dt
fd xy
2
1
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Rate of change of relative vorticityFollowing parcel
Divergence acting onAbsolute vorticity(twirling skater effect)
Tilting of verticallysheared flow
Gradients in forceOf friction
y
F
x
F
x
p
yy
p
xz
u
y
w
z
v
x
w
y
v
x
uf
dt
fd xy
2
1
Pressure/densitysolenoids
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Rate of change of relative vorticityFollowing parcel
Divergence acting onAbsolute vorticity(twirling skater effect)
Tilting of verticallysheared flow
Gradients in forceOf friction
y
F
x
F
x
p
yy
p
xz
u
y
w
z
v
x
w
y
v
x
uf
dt
fd xy
2
1
Pressure/densitysolenoids
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Rate of change of relative vorticityFollowing parcel
Divergence acting onAbsolute vorticity(twirling skater effect)
Tilting of verticallysheared flow
Gradients in forceOf friction
y
F
x
F
x
p
yy
p
xz
u
y
w
z
v
x
w
y
v
x
uf
dt
fd xy
2
1
Pressure/densitysolenoids
maF
am
PGF
geostrophic wind
Cold advection pattern
m (or ) largeacceleration small
m (or ) smallacceleration large
Solenoid: field loop that converts potential energy to kinetic energy
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Rate of change of relative vorticityFollowing parcel
Divergence acting onAbsolute vorticity(twirling skater effect)
Tilting of verticallysheared flow
Gradients in forceOf friction
y
F
x
F
x
p
yy
p
xz
u
y
w
z
v
x
w
y
v
x
uf
dt
fd xy
2
1
Pressure/densitysolenoids
Geostrophic wind = constant
N-S wind componentdue to friction
x
Fy
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xFfvxdt
du
The vorticity equation in pressure coordinates
yFfuydt
dv
(1) (2)
Expand total derivative
xFfvxP
u
y
uv
x
uu
t
u
yFfu
yP
v
y
vv
x
vu
t
v
yxTake
)1()2(
y
F
x
F
xP
v
yP
u
y
v
x
uf
y
u
x
v
y
u
x
v
Pf
y
u
x
v
yvf
y
u
x
v
xu
y
u
x
v
t
xy
asy
u
x
vvorticityrelativewrite
y
F
x
F
xP
v
yP
u
y
v
x
uf
Pf
yvf
xu
txy
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y
F
x
F
xP
v
yP
u
y
v
x
uf
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