METR 2413
12 April 2004
Vorticity and Jets
Review from quiz
Review
dp
Hydrostatic balance g,
dz
Vertical pressure gradient force - 1 dp g
dz
Scale, g ~ 10 m/s2, vertical pgf ~ 10 m/s2
Horizontal pressure gradient force balanced by Coriolis force
1 p
f cv
x
f ~ 10-4 s-1, v ~ 10 m/s, Coriolis force ~ 10-3 m/s2
ρ ~ 1 kg/m3, Δp ~ 10 hPa ~ 1000 Pa, Δx ~ 1000 km ~ 106 m
Review from quiz
Thermal wind
In the presence of a horizontal temperature gradient,
the tilt of pressure surfaces increases with height.
u g g Tv
z f cTv y ug p=p2
Δz
p=p1
North
cold warm
Zonal wind increases with height in the NH because temperature decreases
towards the pole, giving increasing poleward height gradients.
Circulation and Vorticity:
Two primary measures of rotation in a fluid
By convention, both circulation and vorticity are positive in
the counterclockwise direction
(cyclonic in the Northern Hemisphere)
Circulation: Macroscopic measure of rotation for a finite
area of the fluid
= integration of the tangential component of velocity around
a closed path
Vorticity: The tendency to spin about an axis; Microscopic
measure of rotation at any point in the fluid
The tighter the spin, the larger the magnitude of the vorticity
On Earth there is:
1) Vorticity from Earth’s spin (planetary vorticity)
2) “Local” vorticity due to cyclonic/anticyclonic motion
(relative vorticity)
Absolute vorticity (total vorticity):
• measured with respect to the fixed stars
• includes Earth’s rotation (planetary vorticity) and
rotation of atmosphere relative to Earth’s surface (relative
vorticity)
• angular momentum is conserved, so absolute vorticity is
also conserved for frictionless motion
ζ a = ζ r + fc = constant
where f c is the Coriolis parameter = 2Ω sinΦ
(planetary vorticity)
Relative vorticity:
• measure of the rotation of the atmosphere about a vertical
axis relative to Earth’s surface
• units of sec-1
• Synoptic scale vorticity is analyzed and plotted on the 500
mb chart
• 500 mb vorticity may be referred to as “vertical vorticity”
(the spin is in relation to the vertical axis)
The vertical component of vorticity can be expressed as:
ζr = ∂v/∂x – ∂u/∂y
Typical magnitude of the relative vorticity for synoptic scale
flow, U ~ 10 m/s, L ~ 1000 km
ζ = ∂v/∂x - ∂u/ ∂y ≤ U/L ~ 10-5 sec-1
Typical magnitude of planetary vorticity f ~ 10-4 s-1
We’ve seen that relative vorticity is non-zero for two reasons:
1) Either streamlines of wind have curvature, or
2) The wind field has horizontal shear (or both)
Curvature vorticity: positive in troughs and negative in ridges
Positive and negative relative shear vorticity can be due to variations in
westerly wind speed with latitude
slow
Positive shear vorticity
fast
fast
slow Negative shear vorticity
Think of the above air flows as wide rivers – if you put a log oriented north-
south in the flow, the log would turn counterclockwise on the top, and
clockwise on the bottom because of the shear
Positive vorticity advection (PVA)
• found where air blows from regions of higher vorticity toward
lower vorticity
• significant because main mechanism to reduce vorticity is
divergence
• that is, in regions of PVA there tends to be divergence, which
implies upward motions beneath these areas, surface
convergence and surface pressure falls
Negative vorticity advection (NVA)
found where air blows from regions of lower vorticity towards
higher vorticity
main mechanism to increase vorticity is convergence
when there is NVA in upper levels, there tends to be
downward motion below, surface divergence and surface
pressure rises
Low Vorticity Low Vorticity
Upper Level
Convergence
High Vorticity
Anticyclonic Vorticity Advection
- Moving from low vorticity to high vorticity requires
convergence aloft.
Jet streaks
• Jet streaks are localized regions of very fast winds
embedded within the jet stream. Sometimes these local
wind maxima reach speeds in excess of 160 knots.
• Jet streaks are important as they are indicative of rising
motion/falling pressures at the surface. The figure below
represents an idealized jet streak.
Height contour
Entrance Exit
Region Region
(Rear) (Front)
As air enters from the left, it must be accelerated as the height contours
are closer together and the pressure gradient force is stronger.
The stronger pgf causes an ageostrophic flow to the north, leading to
convergence to the north and a divergence to the south. As a result, air
sinks in the northern 'quadrant', and rises in the southern quadrant of a jet
entrance.
The force to accelerate the flow to the east is supplied by the Coriolis
force as air flows from the south to the north near the jet entrance,
leading to a force to the east (the right).
In the jet exit region, the opposite happens, as air flows from
north to south to create the force necessary to decelerate
the air as it leave the jet streak. The vertical motion resulting
from this leads to rising air in the north quadrant and sinking
air in the south of the jet exit.
CONV DIV
Exit
Entrance
Region
Region
(Front)
(Rear)
However, not all jet streaks are straight. In fact, most are
curved.
The ageostrophic flow and anticyclonic vorticity advection
cause convergence in the left entrance of the jet and the
ageostrophic flow and the cyclonic vorticity advection cause
divergence in the left exit region
The divergence and ascent ahead of the upper level trough
enhance the development of surface lows
Summary
• Vorticity is a measure of the local rotation in a fluid
• Planetary vorticity due to the rotoation of the earth
• Relative vorticity due to the rotation of air relative to the
surface, curvature vorticity and shear vorticity
• Absolute vorticity = planetary vorticity + relative vorticity
is conserved in the atmosphere
• Positive (cyclonic) vorticity advection leads to upper level
divergence, and rising motion
• Jet streaks are localized regions of very strong winds,
with poleward motion at the jet entrance and
equatorward flow at the jet exit
• Enhanced divergence at the left exit region of a cyclonic
curved jet, enhances development of surface low