CLIMATE/EARTH 401�Geophysical Fluid Dynamics�Dynamical Meteorology
Vorticity & Mountain Waves
Forced Ascent/Descent
Warming
Cooling
Barotropic Potential Vorticity
Potential vorticity is a measure of absolute vorticity relative to the depth of the vortex.
What happens if the depth (H) changes?
Relative vorticity with change of depth
What happens when the vortex meets the mountain?
Surface with a hill.
Fluid of changing depth
Application to flow on the Earth
What might cause this wave-like flow?
Flow over a mountain
Mountain
Use our simple form of potential vorticity
From scaled equation, with assumption of constant density and temperature.
Flow over a mountain�(long in the north-south)�(can’t go around the mountain)
west
east
Flow over a mountain
west
east
Depth, H
Mountain
Flow over a mountain (assume flow is adiabatic)
west
east
Depth, H
θ
θ + Δθ
Mountain
Flow over a mountain (far upstream constant zonal flow)
west
east
Depth, H
θ
θ + Δθ
ζ=0
Mountain
Use the barotropic potential vorticity equation
From scaled equation, with assumption of constant density and temperature.
What happens as air gets to mountain?
west
east
Depth, H
θ
θ + Δθ
ζ=0
Mountain
What happens as air gets to mountain?
west
east
Depth, H
θ
θ + Δθ
ζ=0
Air is lifted.
Lifting higher at ground than upper air.
(pressure gradient force spreads it out)
Mountain
What happens as air gets to mountain?
west
east
Depth, H +ΔH
θ
θ + Δθ
ζ=0
Air is lifted.
Lifting higher at ground than upper air.
(pressure gradient force spreads it out)
Mountain
What happens as air gets to mountain?
west
east
Depth, H +ΔH
θ
θ + Δθ
ζ must increase
Air is lifted.
Lifting higher at ground than upper air.
(pressure gradient force spreads it out)
Mountain
What does it mean for the relative vorticity to increase?
What happens in these waves?
Gains cyclonic vorticity
Loses cyclonic vorticity
Same as gains anticyclonic vorticity
Or schematically
Cyclonic
Anticyclonic
Rotational
Shear
What happens as air gets to mountain?
west
east
Depth, H +ΔH
θ
θ + Δθ
ζ must increase
Air turns cyclonically to increase vorticity.
In northern hemisphere turns north.
Mountain
In the (east-west, north-south) plane
MOUNTAINS
Depth, H
Depth, H +ΔH
west
east
s
n
What happens as air goes over mountain?
west
east
Depth, H -ΔH
θ
θ + Δθ
Air turns anti-cyclonically to decrease vorticity.
In northern hemisphere turns south.
ζ must decrease
Mountain
In the (east-west, north-south) plane
MOUNTAINS
Depth, H
Depth, H +ΔH
west
east
s
n
Depth, H -ΔH
What happens as air goes down mountain?
west
east
Depth, H +ΔH
θ
θ + Δθ
Air turns cyclonically to increase vorticity.
In northern hemisphere turns north.
ζ must increase
Mountain
In the (east-west, north-south) plane
MOUNTAINS
Depth, H
Depth, H +ΔH
west
east
s
n
Depth, H -ΔH
Depth, H +ΔH
Stretching here causes relative vorticity to increase; northward turning
Arrives here with northward momentum “Overshoots”
What is happening with planetary vorticity?�(In the (east-west, north-south) plane)
MOUNTAINS
Depth, H
Depth, H +ΔH
west
east
s
n
Depth, H -ΔH
Depth, H +ΔH
f is greater for deflections to north
f is less for deflections to south
What is happening with planetary vorticity?�(In the (east-west, north-south) plane)
MOUNTAINS
Depth, H
Depth, H +ΔH
west
east
s
n
Depth, H -ΔH
Depth, H +ΔH
Has an excess of potential vorticity; relative vorticity must decrease
f = f1
ζ = ζ1
H = H1
f = f2 > f1
ζ = ζ2 < ζ1
H = H2 = H1
f2 + ζ2 must = f1 + ζ1, because H is the same
Let’s explicitly map these ideas to the Earth
Local vertical / planetary vorticity
relative vorticity/planetary vorticity
relative vorticity
planetary vorticity
Compare relative vorticity to planetary vorticity
See scale analysis lectures
planetary vorticity
is usually larger than relative vorticity
for large-scale and middle latitudes
Relative and planetary vorticity
Compare relative vorticity & planetary vorticity & divergence
Flow is rotationally dominated, but divergence is crucial to understanding the flow.
End: Vorticity & Mountain Waves