Vorticity

The Law of conservation of Vorticity

• Relative vorticity is defined as

• The time rate of change of vorticity can be written by taking the equations of horizontal momentum and deriving the x component with respect to y and the y component with respect to x

• By doing that, the relative vorticity and Coriolis parameter can be isolated (try to show this is true –Problem 7.32) and we can rewrite the equation in the vorticity equation in the form:

y

u

x

v

In Cartesian coordinate form

x

pfv

z

uw

y

uv

x

uu

t

u

y 1

y

pfu

z

vw

y

vv

x

vu

t

v

x 1

Momentum Equations:

dy

dfv

z

u

y

w

z

v

x

w

y

v

x

uf

zw

yv

xu

t

x

p

yy

p

x

21

dt

d V

x

p

yy

p

xz

u

y

w

z

v

x

w

y

v

x

uf

dt

fd

2

1)(

)(

Divergence Term:Very important in synoptic-scale movements

Tilting Term: important for mesocale processes (e.g. formation of tornadoes

Solenoidal term important only in particular small-scale circulations

xx

yy

z zSuppose v increasing with height δv/ δz Suppose w varying

with xδw/ δx

• For synoptic purposes, lets neglect the tilting and solenoidal terms. In Lagrangian form the vorticity variation equation is:

y

v

x

uf

dt

fd)(

)(

y

v

x

uff

t

f)()(

)( V

y

v

x

uff

t

f)()(

)( V

Local variation (Eulerian)

• f can be interpreted as the planetary vorticity (2Ωsinφ)

Divergence term

7.21a

Relative importance of planetary and relative vorticity

• High latitudes: f~10-4 and ζ~10-5s-1

• tropical latitudes: f and ζ are comparable• In fast-moving extratropical systems that are not

rapidly amplifying or decaying, the divergence term is relatively small then:

)( ft

V

0)()()(

ft

f

dt

fd V

Lagrangian form:

(7.22a)

0t

f

In conclusion:

• (ζ+f) behaves as a conservative tracer, a property whose numerical values are conserved by air parcels as they move along (are advected) by the horizontal wind field

0)(

dt

fd

What does it mean? Suppose you are a parcel moving in the wave (Lagrangian approach)

• As the particle moves northward in the NH it experiences changes in f (increases). To compensate, relative vorticity must decrease (ζ+f) = constant. (anticyclonic tendency)

ζ > 0ζ < 0

ζ > 0

• If a parcel moves southward in the NH then f decreases and ζ must increase (cyclonic tendency)

How to use the non-divergent form of the vorticity equation to predict the weather?

There is another interesting relationship between geostrophic vorticity and geopotential height that helps to predict the weather. The geostrophic vorticity can be written as:

2

2

2

21

yxfy

u

x

v ggg

Related to the second derivative (Laplacian ) of the geopotential.

When the Laplacian of the geopotential height is negative (positive), it indicates that the geopotential height is actually INCREASING (DECREASING)

How to use that?• Estimate the advection of vorticity by the wind (right

side of the eq 7.22)

)( ftg

V

• If the relative vorticity is increasing in a given region it indicates that the geopotential height is actually DECREASING – which indicates the approaching of a trough – Lower temperatures can be expected

• If the relative vorticity is decreasing in a given region it indicates that the geopotential height is INCREASING – which indicates the approaching of a ridge – Higher temperatures can be expected

Let’s try our own weather forecast for Santa Barbara using these concepts

Remarks• The relative importance of the tendencies induced by the advection of

relative vorticity and planetary vorticity depends on the strength of the flow U and on the zonal wavelength L of the waves:

• The smaller the scale of the waves, the stronger the ζ perturbations and the stronger the advection of relative vorticity

• For baroclinic waves with zonal wavelengths ~ 4000km the advection of relative vorticity is much stronger than the advection of planetary vorticity and the influence of advection of planetary vorticity is barely discernible

• For planetary waves with wavelengths roughly comparable to the radius of the Earth, the much weaker eastward advection of relative vorticity is almost entirely cancelled by the advection of planetary vorticity and the net vorticity tendencies is quite small.

• The monthly, seasonal climatologies tend to be dominated by ‘stationary waves’

The maps below give an idea about these differences

Understanding Rossby Waves• Suppose a closed chain of fluid particles initially aligned along

a circle of latitudes. The absolute vorticity is given by f

• Assume ζ = 0 at time to. If the chain of parcels is subjected to a sinusoidal wave at the time t1, then

to

t1

• ζt1 =fto – ft1 = –βδy where (β=δf/δy)• If it moves north, ζt1 <0 and if it moves south ζt1 >0

• observes a decrease in ζt1 and an increase in ζt1

It is like the entire pattern of waves move westward: conclusionsRossby waves move westward in an environment with zero zonal winds

ζt1

Barotropic vorticity

• Expand the equation:• into:

• where:

Middle latitudes

Beta effect is important for the dynamics of Rossby Waves

)( ft

V

vy

fv

t

VV

ERyy

f cos2sin2

111110~ ms

Vorticity and divergence

• High latitudes: f~10-4 and ζ~10-5s-1

VV

)()()(

fft

f

Important in amplifying waves When >= 10-6s-1

Consider that planetary vorticity dominates relative vorticity. How divergence contributes to increase/decrease the absolute vorticity? => f

V

ft

f )( 0 V (Divergence)(f>0) => contributes to the decrease of absolute vorticity

0 V(f>0) => contributes to the increase of absolute vorticity

(Convergence)

Divergence in the upper atmosphere is related to decreasing vorticity,draws air upward and provides a lifting mechanism, thus initiating

and maintaining low-pressure at the surface.

Increasing upper-level vorticity is related to convergence and sinking air,which creates high pressure at the surface.

This mechanism helps to maintain low and high pressure systems at the surface, provided the existence of friction (that causes

convergence/divergence in low levels)

Northern Hemisphere

925hPa300hPa

Potential Vorticity

• Suppose an incompressible block of fluid with area A and height H moving with velocity V.

δz=H

Vdt

dA

Adt

dH

H

11

dt

dH

H

f

dt

fd )()(

0)(

H

f

dt

d Demonstration: Exercise 7.35

V

Explanation and implications

0)(

z

f

dt

d

IF const

z

f

)(THEN

M

MMΘo+δθ

ΘoWesterly flow

Suppose a westerly zonal flow (that is ζ =0 ), NH

ζo +fo =fo

M

MMΘo+δθ

ΘoWesterly flow

y

x

const

H

f )(

φo

φo +δφ -> f>fo- ζ <0 fo

ζ <0 to compensate decrease in H

v

ζ >0 – H returns to normal and f has decreased

y

x

const

H

f )(

φofo

ζ >0 – to compensate the decrease in H upstreamf decreases

M

MMΘo+δθ

Θo

ζ <0 – to compensate the increase in H downslope (as f increases until return to the original latitude)

Lets check what happens as the trough moves from the west coast toward continental USA and crosses the Rock