AOSS 401, Fall 2006 Lecture 18 October 24, 2007 Richard B. Rood (Room 2525, SRB) Derek Posselt (Room 2517D, SRB)

Class News Final exam will be last day of class Derek and I decided to think about good homework problems for another day. –No homework posted today.

Material from Chapter 4 Vorticity, Vorticity, Vorticity –Relative and planetary vorticity –Mid-latitude disturbances –Vorticity, divergence, in 3-D

Weather National Weather Service – –Model forecasts: 7loop.html 7loop.html Weather Underground – bin/findweather/getForecast?query=ann+arborhttp:// bin/findweather/getForecast?query=ann+arbor –Model forecasts: ?model=NAM&domain=US ?model=NAM&domain=US

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.

Absolute (or total) Vorticity

Relative and planetary vorticity Planetary vorticity is cyclonic is positive vorticity Planetary vorticity, in middle latitudes, is usually larger than relative vorticity

We derived the vorticity equation TERMS DIVERGENCE TILTING SOLENOIDAL or BAROCLINIC

Comments on the terms There are important dynamical features in the atmosphere where all of these terms are important. Baroclinic terms are due to there being gradients of temperature on pressure surfaces. (Are they explicitly there in pressure coordinates?) Like a thermodynamic “source” of rotation.

Tilting Term rotation in, say, (y, z) plane, “vorticity” in x plane as the wheel is turned there is a component of “vorticity” in the z plane

Divergence influence on vorticity

Scale factors for “large-scale” mid-latitude

Assume balance among terms of s -2

A nuance on vorticity and the scaled equation: potential vorticity

A simple version of potential vorticity Integrate with height,z 1  z 2 over a layer of depth H.

A simple version of potential vorticity This is the potential vorticity under the set of assumptions that we used to derive the equation. Constant density, constant temperature  so only in a shallow layer might this be relevant to the atmosphere. Potential vorticity is a measure of absolute vorticity relative to the depth of the vortex.

Relative vorticity with change of depth

Vorticity and depth We can see that 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. This is the play between relative and planetary vorticity.

Scaled vorticity equation

An observation The vorticity is dominated by the geostrophic component of the wind. The divergence requires the wind to be away from geostrophic balance. Generally v g /v a >= 10

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 planetary vorticity is usually larger than relative vorticity for large- scale and middle latitudes

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 intense A growing anticyclone “opposes” the planetary vorticity. –Highs less intense

Compare relative vorticity to planetary vorticity and to divergence Flow is rotationally dominated, but divergence is crucial to understanding flow.

Consider our simple form of potential vorticity From scaled equation, with assumption of constant density and temperature.

Fluid of changing depth

Two things that we have learned about vorticity. Convergence and divergence in a column of fluid, impacts the vorticity throughout the column. –Specifically, divergence above causes low pressure at the surface. Stretching and shrinking of a column of vorticity will change the relative vorticity.

Possible development of a surface low. Earth’s surface pressure surfaces

Lets return to our simple problem Earth’s surface pressure surfaces warming cooling

Lets return to our simple problem Earth’s surface pressure / height surfaces rise warming cooling pressure / height surfaces sink

Lets return to our simple problem Earth’s surface warming cooling PGF H L

Lets return to our simple problem Earth’s surface warming cooling PGF H L mass leaves column / low forms at ground L

Lets return to our simple problem Earth’s surface warming cooling PGF H L mass leaves column / low forms at ground L mass enters column / high forms at ground H

Lets return to our simple problem Earth’s surface warming cooling PGF H L mass leaves column / low forms at ground L mass enters column / high forms at ground H PGF

Mass continuity? What are the implications of mass continuity? What is your law, your equation, your tool to answer that question?

Temperature Assuming the air moves isentropically, what happens to the temperature? What is your law, your equation, your tool to answer that question?

Lets return to our simple problem Earth’s surface warming cooling PGF H L L H

Simple Thermal Circulation There is the sense of the air moves to counter the heating. If the heating ended, then the circulation would end, acting to bring back the original equilibrium situation. This sort of low is cause by heating, is called a “thermal” low, warm core. It tends to damp out.

Lets return to our simple problem Earth’s surface warm core cold core PGF H L L H

Simple Thermal Circulation This sort of low is cause by heating, is called a “thermal” low, warm core. It tends to damp out. –Remember the question about the hurricane being warm core. What about the divergence and convergence?

Lets return to our simple problem Earth’s surface warm core cold core PGF H L L H DIVERGENCE CONVERGENCE DIVERGENCE

Simple Thermal Circulation What about the divergence and convergence? –Convergence and Divergence are aligning over top of each other in the vertical. –Again, in this case there is a tendency for the circulation to damp out.

Back to the earth again

Still in the atmosphere

Flow over a hill HILL

Derived a simple form of potential vorticity From scaled equation, with assumption of constant density and temperature.

Flow over a hill (long in the north-south) (can’t go around the hill) west east

Flow over a hill HILL west east Depth, H

Flow over a hill (assume flow is adiabatic) HILL west east Depth, H θ θ + Δθ

Flow over a hill (far upstream constant zonal flow) HILL west east Depth, H θ θ + Δθ ζ=0

Derived a simple form of potential vorticity From scaled equation, with assumption of constant density and temperature.

What happens as air gets to hill? HILL west east Depth, H θ θ + Δθ ζ=0

What happens as air gets to hill? HILL 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)

What happens as air gets to hill? HILL 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)

What happens as air gets to hill? HILL 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)

How does vorticity increase?

What happens in these waves? Gains cyclonic vorticity Loses cyclonic vorticity Same as gains anticyclonic vorticity

Or schematically CyclonicAnticyclonic Rotational Shear

What happens as air gets to hill? HILL west east Depth, H + ΔH θ θ + Δθ ζ must increase Air turns cyclonically to increase vorticity. In northern hemisphere turns north.

In the (east-west, north-south) plane Depth, H Depth, H + ΔH west east s n

What happens as air goes over hill? HILL west east Depth, H - ΔH θ θ + Δθ Air turns anti-cyclonically to decrease vorticity. In northern hemisphere turns south. ζ must decrease

In the (east-west, north-south) plane Depth, H Depth, H + ΔH west east s n Depth, H - ΔH

What happens as air goes down hill? HILL west east Depth, H + ΔH θ θ + Δθ Air turns cyclonically to increase vorticity. In northern hemisphere turns north. ζ must increase

In the (east-west, north-south) plane Depth, H Depth, H + ΔH west east s n Depth, H - ΔH Depth, H + ΔH

What is happening with planetary vorticity? (In the (east-west, north-south) plane) 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) Depth, H Depth, H + ΔH west east s n Depth, H - ΔH Depth, H + ΔH f + ζ is less than earth’s vorticity and wants to turn north. Arrives here wanting vorticity. “Overshoots”

What is happening with planetary vorticity? (In the (east-west, north-south) plane) Depth, H Depth, H + ΔH west east s n Depth, H - ΔH Depth, H + ΔH

What happens if wind is from east? HILL west east θ θ + Δθ

What is happening with planetary vorticity? (In the (east-west, north-south) plane) Depth, H Depth, H + ΔH west east s n Depth, H - ΔH Depth, H + ΔH Flow from east planetary and relative vorticity interact together, no overshoot or undershoot.

Excursion into the atmosphere

Middle latitude cyclones

Weather National Weather Service – –Model forecasts: 7loop.html 7loop.html Weather Underground – bin/findweather/getForecast?query=ann+arborhttp:// bin/findweather/getForecast?query=ann+arbor –Model forecasts: ?model=NAM&domain=US ?model=NAM&domain=US