Vorticity

1
Vorticity

• Relative vorticity
• where U/R is the curvature term and -?U/?n the
  shear term
• Absolute vorticity
• where f is the Coriolis parameter. ? written
  without a subscript normally refers to ?a.
• Potential vorticity ratio of absolute vorticity
  to depth of an air column
• More formally we can derive Ertels formulation
  of PV

2
Vorticity equation

• We can derive the barotropic vorticity equation
• Which relates stretching to changes in vorticity

3
Dines compensation

• In the stratosphere vertical motion is inhibited
  - so ? reaches a maximum in the mid-troposphere
• Divergence at jet stream level
• gt Convergence at the surface
• This is how the jet stream drives the weather

Tropopause
4
Rossby waves
Minimum in ?ain ridge Maximum in ?a in trough So
d?a/dt gt0 ?a increases as air goes from ridge
to trough By conservation of PV, air columns must
stretch. Stratosphere resists vertical motion so
the columns stretch downwards descending motion
between A and B By Dines, this results in
divergence at the surface anticyclone tends to
form.
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Example

• Upper tropospheric air flows at a speed of 30
  ms-1 through a sinusoidal trough-ridge pattern at
  50oN, of peak-to-peak amplitude 500 km and
  wavelength 3000 km.
• Calculate the change in absolute vorticity
  between ridge and trough, and derive the
  fractional change in the depth of an air column
  as it traverses the pattern.
• Draw a diagram to mark areas of upward and
  downward motion in the flow, and hence describe
  the effect of the pattern on the surface weather.
• (The radius of curvature of y A sin(kx) is
  (Ak2)-1 at the crests).

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Example

• Upper tropospheric air flows at a speed of 30
  ms-1 through a sinusoidal trough-ridge pattern at
  50oN, of peak-to-peak amplitude 500 km and
  wavelength 3000 km.
• Calculate the change in absolute vorticity
  between ridge and trough, and derive the
  fractional change in the depth of an air column
  as it traverses the pattern.
• Draw a diagram to mark areas of upward and
  downward motion in the flow, and hence describe
  the effect of the pattern on the surface weather.
• (The radius of curvature of y A sin(kx) is
  (Ak2)-1 at the crests).
• First, calculate Rmax from (Ak2)-1
• A 250 km, ?3x106 m so k 2.09x10-6 m-1
• Ak2 1.09 x 10-6 m-1 so Rmax 916 km
• U/Rmax 3.28 x 10-5 s-1 (ve in trough, -ve in
  ridge)

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1. Absolute vorticity 2. Potential vorticity
  use 1, with no shear vorticity term
(uniform speed). 500 km corr. to 4.5 lat
2.25 (1 111 km) ftrough (47.75?)
1.08?10-4 s-1 so ?a 1.41?10-4 s-1 fridge
(52.25?) 1.15?10-4 s-1 so ?a 0.82?10-4
s-1   Fractional change in depth of air column is
calculated using 2. In the straight part of the
flow, ?a f 1.12?10-4 s-1. Since
is conserved .    
Divergence aloft causes pressure drop Convergence
at surface causes cyclone to spin-up
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Shear Vorticity
on left side of jet
on right side of jet
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Example

• A zonal jet streak develops in a uniform zonal
  flow of 30 m s-1 at 60N. The jet has a maximum
  speed of 80 m s-1. The cyclonic side is 200 km
  wide and the anticyclonic side 600 km wide. If
  the initial depth of a column of air which enters
  the jet is 100 mb, use the barotropic vorticity
  equation to estimate its depth at maximum
  velocity if it is positioned
• (i) poleward of the jet core
• (ii) equatorward of the jet core
• (iii) directly upstream of the jet core.
• What limits the accuracy of these estimates?

10
Curved jet streams
C, D convergence/divergence due to trough or
ridge C,D convergence/divergence due to jet
entrance/exit
Diffluent trough
Confluent trough
D D
D C
D D
D C
Black arrow denotes jet axis (location of wind
maximum)
C D
C C
C C
C D
Confluent ridge
Diffluent ridge