AOSS 401, Fall 2006 Lecture 16 October 19, 2007 Richard B. Rood (Room 2525, SRB) Derek Posselt (Room 2517D, SRB)

Material from Chapter 4 Welcome past the “midterm” –Any questions, comments. –How was your radiative transfer exam? Vorticity, Vorticity, Vorticity –Today an introduction to concepts and definitions and how things are related to the atmosphere.

Class News Decide on Final Exam Day –Can we agree on last day of class, but with an extra half hour at the end.

Let’s start with the atmosphere. What happens when we heat a spot in the atmosphere? What happens at the surface? –What have we derived that might help use think about the surface?

Let’s think about growing and decaying disturbances. Convergence (divergence) of mass into (from) column above the surface will increase (decrease) surface pressure.

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

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

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

Possible development of a surface low. Earth’s surface mass diverges up here warming warming increases thickness

Possible development of a surface low. Earth’s surface mass diverges up here warming warming increases thickness LOW generates low here

Possible development of a surface low. Earth’s surface mass diverges up here warming LOW generates low here HH and highs here

Possible development of a surface low. Earth’s surface mass diverges up here warming LOW pressure gradient initiates convergence down here HH

What else happens to the converging and diverging air?

The air turns because of the Coriolis force. –(If it is large scale.)

And a little more realistically

Wind around a system. L This is the sort of system which is often called a “vortex.”

Are you familiar with this figure?

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

What is a characteristic of these flows that is intuitively descriptive? Remember the gradient wind. Remember the difference between highs and lows.

The answer. The rotation of the wind. Which is due to the rotation of the Earth.

Less intuitive: Why is the divergence important?

The answer. Vertical wind requires divergence of the horizontal wind. Which requires an ageostrophic part of the wind.

Want to formalize the representation of the role of rotation and divergence

Suppose we have some flow

Imagine at the point flow decomposed into two “components” A “component” that flows into or away from the point.

Imagine at the point flow decomposed into two “components” A “component” that flows around the point.

In general, can imagine a “circulation” around a point. Circulation based on the component of velocity tangent to some curve summed around the curve.

Circulation Circulation is a measure of rotational part of the flow. It is, formally, an analogue to angular momentum and we can write a conservation equation for it. We define a “direction” of circulation by which direction we go around the circuit. cyclonic – counterclockwise anticyclonic -- clockwise

Lets consider a spinning skater

Motion is in the (x,y) plane Axis of rotation is in the vertical plane

Important mathematical and physical operators We have seen the dot product or divergence.

Important mathematical and physical operators Which is known as the vorticity.

A pause with circulation Vorticity is related to “circulation.” –Vorticity is a measure of rotation linked to a point in a fluid. –Circulation is a measure of rotation measured around a closed circuit... an area average. Vorticity is as fundamental a description of the fluid motion as, for example, wind. Flow can be completely described by vorticity + divergence. –rotation + vertical velocity Circulation is an important theoretical construct, but we will not study it much in this course.

An abrupt change in the lecture

How do you solve this set of equations?

The answer Multiply each of the equations by something. –Do something to the equation. Add (or subtract) them.

Back to vorticity

Vorticity

Vertical Component of Vorticity In what plane is the motion? In what direction is the vorticity?

Relative Vorticity

Vorticity

Planetary Vorticity Vorticity due to the rotation of the Earth

Picture of Earth Ω k k k Ω Ω Maximum rotation of vertical column. No rotation of vertical column.

Planetary Vorticity

Picture of Earth f=2Ωsin(Φ) 1.4X10 -4 s X10 -4 s s -1

Absolute (or total) Vorticity

Does this flow have vorticity? east westsouth north ∂x ~ Δx ∂y ~ Δy Draw a picture! lines of constant geopotential or pressure

Does this flow have vorticity? east westsouth north ∂x ~ Δx ∂y ~ Δy Draw a picture! lines of constant geopotential or pressure

Does this flow have vorticity? east westsouth north ∂x ~ Δx ∂y ~ Δy Draw a picture! What happens to this stick?

Does this flow have vorticity? east westsouth north ∂x ~ Δx ∂y ~ Δy Draw a picture! What happens to this stick?

Does this flow have vorticity? east westsouth north ∂x ~ Δx ∂y ~ Δy Draw a picture! What happens to this stick?

Does this flow have vorticity? east westsouth north ∂x ~ Δx ∂y ~ Δy Draw a picture! What happens to this stick? ∂v/∂x-∂u/∂y

A question Does the previous example tell you something about the ground? Does the flow in the previous example have divergence?

Wind around a system. L This is the sort of system which is often called a “vortex.” It is dominated by rotation. Do all curved flows have vorticity?

Vorticity Related to shear of the velocity field. ∂v/∂x-∂u/∂y

Divergence Related to stretching of the velocity field. ∂u/∂x+∂v/∂y

Full equations of motion Remember these? How would you calculate of the time rate change of the vertical component of vorticity?

The scaled horizontal momentum equation in z coordinates no viscosity

Relative Vorticity

Take derivatives

Go Back and Think about that!

Take derivatives

Pay attention to details of calculus here

Subtract these equations Conservation of vorticity

Go back and think about that!

What are these terms?

Divergence

Pure constant vorticity flow.

Pure divergent flow

Divergence influence on vorticity

Vorticity and divergence As a mid-latitude large scale flow diverges (converges) it causes a change in absolute vorticity, primarily, acting on planetary vorticity. This is the play between relative and planetary vorticity.

What are these terms?

Relative Vorticity

What are these terms? Tilting

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

What are these terms?

Something to do with horizontal gradients of thermodynamic variables

What are these terms?

And a little more realistically

Our hurricane What is the sign of vorticity at the top and the bottom? What is the sign of the divergence and convergence in the hurricane? Can you intuitively scale the horizontal and vertical components of vorticity based on this figure? Does this look like a GREAT homework problem?

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