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Richard B. Rood (Room 2525, SRB)
AOSS 401 Geophysical Fluid Dynamics: Atmospheric Dynamics Prepared: Vorticity / Flow Richard B. Rood (Room 2525, SRB) Cell:
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Class News Ctools site (AOSS 401 001 F13)
Second Examination on December 10, 2013 Homework Posted on Ctools / Due on Thursday 11/7/13
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Weather National Weather Service Weather Underground
Model forecasts: Weather Underground NCAR Research Applications Program
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Outline Vorticity and Flow
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Vorticity Equation Changes in relative vorticity are caused by:
DIVERGENCE TILTING SOLENOIDAL or BAROCLINIC Changes in relative vorticity are caused by: Divergence Tilting Gradients in density on a pressure surface Advection
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Scale Analysis of the Vorticity Equation
Changes in relative vorticity are caused by: Divergence Tilting Gradients in density Advection Which of these are most important for large-scale flows? Back to scale analysis…
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Scale factors for “large-scale” mid-latitude
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Terms in Vorticity Equation
Time rate of change Horizontal advection Divergence Planetary vorticity advection Time rate of change Horizontal advection Vertical advection Divergence Tilting Planetary vorticity advection Solenoidal term
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Assume balance among terms of 10-10s-2
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Two important definitions
barotropic – density depends only on pressure. And by the ideal gas equation, surfaces of constant pressure, are surfaces of constant density, are surfaces of constant temperature. baroclinic – density depends on pressure and temperature.
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Barotropic Potential Vorticity
We can learn a lot about the atmosphere by considering the barotropic potential vorticity
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Barotropic Potential Vorticity
Assume constant density Integrate with height, z1 z2 over a layer of depth H.
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Remember the Thermal Wind?
p is an independent variable, a coordinate. Hence, x and y derivatives are taken with p constant.
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Implications of Thermal Wind for a Barotropic Fluid…
Barotropic: temperature is constant on a pressure surface This means Geostrophic wind is constant with height in pressure coordinates in a barotropic fluid
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Barotropic Potential Vorticity
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What happens if the depth (H) is constant?
Conservation of potential vorticity becomes conservation of absolute vorticity…
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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?
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Relative vorticity with change of depth
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The vortex went over the mountain
Surface with a hill.
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Vorticity and depth There is a relationship between depth and vorticity. As the depth of the vortex changes, the relative vorticity has to change in order to conserve the potential vorticity. We have now linked the rotational and irrotational components of the wind. divergence and curl vorticity and divergence Potential vorticity indicates an interplay between relative and planetary vorticity through conservation of absolute angular momentum.
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Let’s explicitly map these ideas to the Earth
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Local vertical / planetary vorticity
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relative vorticity/planetary vorticity
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Compare relative vorticity to planetary vorticity
for large-scale and middle latitudes planetary vorticity is usually larger than relative vorticity
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Relative and planetary vorticity
Planetary vorticity is cyclonic is positive vorticity Planetary vorticity, in middle latitudes, is usually larger than relative vorticity A growing cyclone “adds to” the planetary vorticity. Lows are intense A growing anticyclone “opposes” the planetary vorticity. Highs are less intense
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Compare relative vorticity to planetary vorticity and to divergence
Flow is rotationally dominated, but divergence is crucial to understanding the flow.
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Return to our simple form of potential vorticity
From scaled equation, with assumption of constant density.
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Fluid of changing depth
Stretching and shrinking of a column will change the relative vorticity.
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Application to flow on the Earth
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What might cause this wave-like flow?
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Flow over a mountain Mountain
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Use our simple form of potential vorticity
From scaled equation, with assumption of constant density and temperature.
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Flow over a mountain (long in the north-south) (can’t go around the mountain)
west east
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Flow over a mountain Depth, H Mountain west east
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Flow over a mountain (assume flow is adiabatic)
θ + Δθ Depth, H θ Mountain west east
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Flow over a mountain (far upstream constant zonal flow)
θ + Δθ ζ=0 Depth, H θ Mountain west east
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Use the barotropic potential vorticity equation
From scaled equation, with assumption of constant density and temperature.
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What happens as air gets to mountain?
θ + Δθ ζ=0 Depth, H θ Mountain west east
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What happens as air gets to mountain?
Air is lifted. Lifting higher at ground than upper air. (pressure gradient force spreads it out) θ + Δθ ζ=0 Depth, H θ Mountain west east
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What happens as air gets to mountain?
Air is lifted. Lifting higher at ground than upper air. (pressure gradient force spreads it out) θ + Δθ ζ=0 Depth, H +ΔH θ Mountain west east
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What happens as air gets to mountain?
Air is lifted. Lifting higher at ground than upper air. (pressure gradient force spreads it out) θ + Δθ ζ must increase Depth, H +ΔH θ Mountain west east
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What does it mean for the relative vorticity to increase?
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What happens in these waves?
Loses cyclonic vorticity Same as gains anticyclonic vorticity Gains cyclonic vorticity
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Or schematically Rotational Shear Cyclonic Anticyclonic
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What happens as air gets to mountain?
Air turns cyclonically to increase vorticity. In northern hemisphere turns north. θ + Δθ ζ must increase Depth, H +ΔH θ Mountain west east
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In the (east-west, north-south) plane
MOUNTAINS Depth, H Depth, H +ΔH n s west east
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What happens as air goes over mountain?
Air turns anti-cyclonically to decrease vorticity. In northern hemisphere turns south. θ + Δθ ζ must decrease Depth, H -ΔH θ Mountain west east
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In the (east-west, north-south) plane
MOUNTAINS Depth, H Depth, H +ΔH Depth, H -ΔH n s west east
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What happens as air goes down mountain?
Air turns cyclonically to increase vorticity. In northern hemisphere turns north. θ + Δθ ζ must increase Depth, H +ΔH θ Mountain west east
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In the (east-west, north-south) plane
MOUNTAINS Depth, H Depth, H +ΔH Depth, H -ΔH Depth, H +ΔH n Arrives here with northward momentum “Overshoots” Stretching here causes relative vorticity to increase; northward turning s west east
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What is happening with planetary vorticity
What is happening with planetary vorticity? (In the (east-west, north-south) plane) MOUNTAINS Depth, H Depth, H +ΔH Depth, H -ΔH Depth, H +ΔH n f is greater for deflections to north f is less for deflections to south s west east
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What is happening with planetary vorticity
What is happening with planetary vorticity? (In the (east-west, north-south) plane) MOUNTAINS Depth, H Depth, H +ΔH Depth, H -ΔH Depth, H +ΔH Has an excess of potential vorticity; relative vorticity must decrease n f = f1 ζ = ζ1 H = H1 f = f2 > f1 ζ = ζ2 = ζ1 H = H2 = H1 s west east
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Excursion into the atmosphere
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“Colorado Lows”
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What happens if wind is from east?
θ + Δθ θ Mountain west east
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What is happening with planetary vorticity
What is happening with planetary vorticity? (In the (east-west, north-south) plane) MOUNTAINS Depth, H Depth, H +ΔH Depth, H -ΔH Depth, H +ΔH n Flow from east: planetary and relative vorticity offset each other; no overshoot or undershoot. s west east
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Consider the vertical structure more
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Where is this flow more barotropic?
10 m/s 5 m/s 20 m/s 30 m/s B, cooler A, warmer - p, vertical y, north
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Idealized vertical cross section
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Vorticity on Large Scales
Remember, vorticity is caused by Wind shear Rotation in the flow Can we identify these on weather maps? (The following maps come from
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300 mb Wind Speed
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Where is there positive vorticity?
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500 mb Vorticity
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Thermal Wind Remember, thermal wind relates
Vertical shear of geostrophic wind Horizontal temperature gradients Can we identify these on weather maps?
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Where are the strongest ?
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850 mb Temperature
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Convergence/Divergence
Remember, vertical motion on large scales directly related to Convergence/divergence of ageostrophic wind Curvature in the flow Can we identify these on weather maps?
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Where are surface lows/highs?
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Surface Precipitation
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850 mb Temperature
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Concepts Vorticity: shear and curvature
Why is curvature vorticity (as opposed to shear vorticity) usually associated with developing low pressure systems? Divergence and convergence and location of surface high and low pressure systems Thermal wind—vertical shear of the horizontal wind and horizontal temperature gradients
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Concepts Features commonly found together Coincidence?
Jet stream Upper level positive vorticity Fronts Midlatitude cyclones (low pressure systems) Coincidence? More on this later…
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Mid-latitude cyclones
What we know: Low pressure systems Form through spinup of low-level positive vorticity Divergence/convergence is key This is just the beginning… Always closely associated with fronts—why? Sometimes develop rapidly, sometimes not at all—why?
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The mid-latitude cyclone
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Mid-latitude cyclones: Norwegian Cyclone Model
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Fronts and Precipitation
Norwegian Cyclone Model CloudSat Radar
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Idealized vertical cross section
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Cold and warm advection
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Lifting and sinking
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Increasing the pressure gradient force
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Almost Weather
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Mid-latitude cyclones: Norwegian Cyclone Model
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Weather National Weather Service Weather Underground
Model forecasts: Weather Underground NCAR Research Applications Program
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