AOSS 401 Geophysical Fluid Dynamics Atmospheric Dynamics Prepared

AOSS 401 Geophysical Fluid Dynamics: Atmospheric Dynamics Prepared: 20131105 Vorticity / Physical Meaning Richard B. Rood (Room 2525, SRB) rbrood@umich. edu 734 -647 -3530 Cell: 301 -526 -8572

Class News • Ctools site (AOSS 401 001 F 13) • Second Examination on December 10, 2013 • Homework – Posted on Ctools / Due on Thursday 11/7/13

Weather • National Weather Service – Model forecasts: • Weather Underground – Model forecasts: • NCAR Research Applications Program

Outline • Vorticity and Flow

Approximated equations of motion in pressure coordinates

Geostrophic wind

Thermal wind p is an independent variable, a coordinate. Hence, x and y derivatives are taken with p constant.

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

Important mathematical and physical operators Which is the definition of vorticity.

Vorticity

Relative Vorticity

Absolute (or total) Vorticity

Physical meaning of terms in vorticity equation

Let’s go through this term by term

What are these terms?

What are these terms? Divergence • Remember that divergence is related to vertical motion • We now see that it is also related to changes in vorticity…

Why is the divergence important?

More of the answer… • Vertical wind is related to divergence of the horizontal wind. • (Which requires an ageostrophic part of the wind. ) • Changes in vorticity are related to divergence of the horizontal wind

Pure constant vorticity flow.

Pure divergent flow

Divergence influence on vorticity

Divergence influence on vorticity

What are these terms?

What are these terms?

Remember the definition of relative vorticity

What are these terms? Tilting: • Change in vertical component of vorticity by tilting horizontal vorticity into the vertical

Tilting Term Tilting: • Change in vertical component of vorticity by tilting horizontal vorticity into the vertical

Tilting Term z y x

Tilting Term z y x

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?

“Solenoidal” or “Baroclinic” Terms Something to do with horizontal gradients of thermodynamic variables

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.

“Solenoidal” or “Baroclinic” Terms Baroclinic: surfaces of constant density intersect surfaces of constant pressure z p-2Δp ρ-2Δρ p-Δp ρ-Δρ ρ p x

“Solenoidal” or “Baroclinic” Terms Barotropic: surfaces of constant density parallel to surfaces of constant pressure z p-2Δp ρ-2Δρ p-Δp ρ-Δρ p ρ x

An example in the real atmosphere… • This is a large part of what drives the development of low-level low pressure systems in middle-latitudes • It is also important in understanding circulations around fronts…

“Solenoidal” or “Baroclinic” Terms around a front Baroclinic: surfaces of constant density intersect surfaces of constant pressure z p-2Δp ρ-2Δρ p-Δp ρ-Δρ ρ p x Cold air sinking Behind front Warm air rising Ahead of front

What are these terms?

What are these terms? Advection: • Change in the vertical component of vorticity due to advection of both relative and planetary vorticity

Vorticity Equation Changes in relative vorticity are caused by: • Divergence • Tilting • Gradients in density • 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 largescale flows? Back to scale analysis…

Scale factors for “large-scale” mid-latitude

Scale of relative vorticity

Compare relative vorticity to planetary vorticity In general planetary vorticity is larger than relative vorticity.

Time rate of change of relative vorticity (Eulerian + Advection)

Remaining Terms

Scale analysis of the vorticity equation • Divergence term dominates, followed by horizontal advection and the local time rate of change of relative vorticity • Tilting term important where there is large shear and strong horizontal gradient in the vertical velocity (boundary layer, smaller scales) • Thermodynamic term important where there are strong density (temperature) gradients that intersect lines of constant pressure (sea breeze, fronts)

Consider divergence term This term scales larger than all of the other terms. But, remember that the divergence of the horizontal wind is, in actuality, small compared to the change in wind with x, y

Consider divergence term Divergence scales as the scale of the ageostrophic wind, so

Consider divergence term We saw that the relative vorticity is typically about an order of magnitude less than the planetary vorticity. So

So for quasi-nondivergent flow

Compare relative vorticity to planetary vorticity and to divergence Again we see the importance of the rotation of the Earth.

Assume balance among terms of 10 -10 s-2

A nuance on vorticity and the scaled equation: potential vorticity

A simple version of potential vorticity returned relative vorticity to equation

A simple version of potential vorticity what’s this?

A simple version of potential vorticity Assume constant density and temperature.

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 Integrate with height, z 1 z 2 over a layer of depth H. Why can we do this?

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 this is only relevant to the atmosphere in a shallow layer. But the key is… Potential vorticity is a measure of absolute vorticity relative to the depth of the vortex.

Relative vorticity with change of depth

What happens when the vortex meets the mountain? Surface with a hill.

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. • 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.

Take away points