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

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.

yvxy

y

uuyx

x

vvxuvdyudxC

Calculate the circulation within a small fluid element with area yx

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

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

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

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 sn

VV

(negative because we are integrating counterclockwise)

CALCULATE CIRCULATION

Note that d (s) = n

snn

VVnsVldVC

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

n

V

R

V

s

Shear

n

V

R

V

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

fA

Ce lim0A

fvorticitysEarth sin2'

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

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

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

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

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

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

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

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