Richard B. Rood (Room 2525, SRB) AOSS 401 Geophysical Fluid Dynamics: Atmospheric Dynamics Prepared: 20131105 Vorticity / Flow Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Cell: 301-526-8572
Class News Ctools site (AOSS 401 001 F13) Second Examination on December 10, 2013 Homework Posted on Ctools / Due on Thursday 11/7/13
Weather National Weather Service Weather Underground Model forecasts: Weather Underground NCAR Research Applications Program
Outline Vorticity and Flow
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
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…
Scale factors for “large-scale” mid-latitude
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
Assume balance among terms of 10-10s-2
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.
Barotropic Potential Vorticity We can learn a lot about the atmosphere by considering the barotropic potential vorticity
Barotropic Potential Vorticity Assume constant density Integrate with height, z1 z2 over a layer of depth H.
Remember the Thermal Wind? p is an independent variable, a coordinate. Hence, x and y derivatives are taken with p constant.
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
Barotropic Potential Vorticity
What happens if the depth (H) is constant? Conservation of potential vorticity becomes conservation of absolute vorticity…
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
The vortex went over the mountain Surface with a hill.
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.
Let’s explicitly map these ideas to the Earth
Local vertical / planetary vorticity
relative vorticity/planetary vorticity
Compare relative vorticity to planetary vorticity for large-scale and middle latitudes planetary vorticity is usually larger than relative vorticity
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
Compare relative vorticity to planetary vorticity and to divergence Flow is rotationally dominated, but divergence is crucial to understanding the flow.
Return to our simple form of potential vorticity From scaled equation, with assumption of constant density.
Fluid of changing depth Stretching and shrinking of a column will change the relative vorticity.
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 Depth, H Mountain west east
Flow over a mountain (assume flow is adiabatic) θ + Δθ Depth, H θ Mountain west east
Flow over a mountain (far upstream constant zonal flow) θ + Δθ ζ=0 Depth, H θ Mountain west east
Use the barotropic potential vorticity equation From scaled equation, with assumption of constant density and temperature.
What happens as air gets to mountain? θ + Δθ ζ=0 Depth, H θ Mountain west east
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
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
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
What does it mean for the relative vorticity to increase?
What happens in these waves? Loses cyclonic vorticity Same as gains anticyclonic vorticity Gains cyclonic vorticity
Or schematically Rotational Shear Cyclonic Anticyclonic
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
In the (east-west, north-south) plane MOUNTAINS Depth, H Depth, H +ΔH n s west east
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
In the (east-west, north-south) plane MOUNTAINS Depth, H Depth, H +ΔH Depth, H -ΔH n s west east
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
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
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
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
Excursion into the atmosphere
“Colorado Lows”
What happens if wind is from east? θ + Δθ θ Mountain west east
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
Consider the vertical structure more
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
Idealized vertical cross section
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 http://www.aos.wisc.edu/weather/)
300 mb Wind Speed
Where is there positive vorticity?
500 mb Vorticity
Thermal Wind Remember, thermal wind relates Vertical shear of geostrophic wind Horizontal temperature gradients Can we identify these on weather maps?
Where are the strongest ?
850 mb Temperature
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?
Where are surface lows/highs?
Surface Precipitation
850 mb Temperature
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
Concepts Features commonly found together Coincidence? Jet stream Upper level positive vorticity Fronts Midlatitude cyclones (low pressure systems) Coincidence? More on this later…
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?
The mid-latitude cyclone
Mid-latitude cyclones: Norwegian Cyclone Model
Fronts and Precipitation Norwegian Cyclone Model CloudSat Radar
Idealized vertical cross section
Cold and warm advection
Lifting and sinking
Increasing the pressure gradient force
Almost Weather
Mid-latitude cyclones: Norwegian Cyclone Model http://www.srh.weather.gov/jetstream/synoptic/cyclone.htm
Weather National Weather Service Weather Underground Model forecasts: Weather Underground NCAR Research Applications Program