AOSS 401, Fall 2006 Lecture 9 September 26, 2007 Richard B. Rood (Room 2525, SRB) Derek Posselt (Room 2517D, SRB)

Class News Contract with class. –First exam October 10. Will make it completely through Chapter 3.

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

Outline Thermal Wind Equations of motion in pressure coordinates Maps

Full equations of motion Tangential coordinate system. z, height, as a vertical coordinate. zonal, meridional, vertical

What are the three major balances we have discussed? –What are the assumptions of those balances?

The geostrophic balance Take a vertical derivative of the equation.

The geostrophic balance Use equation of state to eliminate density. Thermal wind relationship in height (z) coordinates

moving fluid Shear? (3) Shear is a word used to describe that velocity varies in space. more slowly moving fluid z

The geostrophic balance What does this equation tell us? Zonal wind a a level is a function of average meridional temperature BELOW. Thermal wind relationship in height (z) coordinates

An estimate of the July mean zonal wind north summer south winter note the jet streams

Now we return to our march to pressure coordinates.

Pressure altitude Under virtually all conditions pressure (and density) decreases with height. ∂p/∂z < 0. That’s why it is a good vertical coordinate. If ∂p/∂z = 0, then utility as a vertical coordinate falls apart. What does this look like?

Use pressure as a vertical coordinate? What do we need. –Pressure gradient force in pressure coordinates. –Way to express derivatives in pressure coordinates. –Way to express vertical velocity in pressure coordinates.

Expressing pressure gradient force

Integrate in altitude Pressure at height z is force (weight) of air above height z.

Concept of geopotential Define a variable  such that the gradient of  is equal to g. This is called a potential function. We have assumed here that  is a function of only z.

Integrating with height

What is geopotential? Potential energy that a parcel would have if it was lifted from surface to the height z. It is analogous to the height of a pressure surface. –We seek to have an analogue for pressure on a height surface, which will be height on a pressure surface.

Implicit that this is on a constant z surface Implicit that this is on a constant p surface

Horizontal pressure gradient force in pressure coordinate is the gradient of geopotential

Our horizontal momentum equation (rotating coordinate system) Assume no viscosity Other assumptions in these equations?

2.4) Homework question Below the transformation of some scalar Ψ from the vertical coordinate z to the vertical coordinate p, pressure, is expressed symbolically. Write expressions for the horizontal (x and y) derivatives of Ψ and the vertical derivative (z) of Ψ in the pressure coordinate system. Ψ (x, y, z, t) = Ψ (x, y, p(x, y, z, t), t) What happens if ∂p/∂z = 0?

2.4) Answer Ψ (x, y, z, t) = Ψ (x, y, p(x, y, z, t), t) If ∂p/∂z=0 then the transform cannot be made, because p does not depend on z. It is not a monotonic function.

What do we do with the material derivative? By definition: Implicit that horizontal and time derivatives at constant z

What do we do with the material derivative? By definition: Implicit that horizontal and time derivatives at constant p

Continuity equation Think about this derivation!

Thermodynamic equation

Equation of state

Thermodynamic equation S p is the static stability parameter. What is static stability?

Equations of motion in pressure coordinates (plus hydrostatic and equation of state) Tangential coordinate system. p, pressure, as a vertical coordinate. zonal, meridional, vertical

Full equations of motion Tangential coordinate system. z, height, as a vertical coordinate. zonal, meridional, vertical

In the derivation Have used the conservation principle. Have relied heavily on the hydrostatic assumption. Require that the conservation principle holds in all coordinate systems. Plus we did some implicit scaling.

Let’s move this to a chart

Geopotential, Φ, in upper troposphere east west Φ0+ΔΦΦ0+ΔΦ Φ 0 +3 Δ Φ Φ0Φ0 Φ 0 +2 Δ Φ Δ Φ > 0 south north

Geopotential, Φ, in upper troposphere east west Φ0+ΔΦΦ0+ΔΦ Φ 0 +3 Δ Φ Φ0Φ0 Φ 0 +2 Δ Φ Δ Φ > 0 south north ΔyΔy

Geopotential, Φ, in upper troposphere east west Φ0+ΔΦΦ0+ΔΦ Φ 0 +3 Δ Φ Φ0Φ0 Φ 0 +2 Δ Φ Δ Φ > 0 south north ΔyΔy δΦ = Φ 0 – (Φ 0 +2ΔΦ)

Geopotential, Φ, in upper troposphere east west Φ0+ΔΦΦ0+ΔΦ Φ 0 +3 Δ Φ Φ0Φ0 Φ 0 +2 Δ Φ Δ Φ > 0 south north ΔyΔy

The horizontal momentum equation Assume no viscosity

Geostrophic approximation

Geopotential, Φ, in upper troposphere east west Φ0+ΔΦΦ0+ΔΦ Φ 0 +3 Δ Φ Φ0Φ0 Φ 0 +2 Δ Φ Δ Φ > 0 south north ΔyΔy

Think a second This is the i (east-west, x) component of the geostrophic wind. We have estimated the derivatives based on finite differences. –Does this seem like a reverse engineering of the methods we used to derive the equations? There is a consistency –The direction comes out correctly! (towards east) –The strength is proportional to the gradient.

Geopotential, Φ, in upper troposphere east west Φ0+ΔΦΦ0+ΔΦ Φ 0 +3 Δ Φ Φ0Φ0 Φ 0 +2 Δ Φ Δ Φ > 0 south north ΔyΔy

Return to Geopotential, Φ, in upper troposphere east west Φ0+ΔΦΦ0+ΔΦ Φ 0 +3 Δ Φ Φ0Φ0 Φ 0 +2 Δ Φ Δ Φ > 0 south north Do not assume geostrophic. Are the winds parallel to the height contours?

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

Return to Geopotential, Φ, in upper troposphere east west Φ 0 +3 Δ Φ Φ0Φ0 Φ 0 +2 Δ Φ Δ Φ > 0 south north Do not assume geostrophic. A qualitative velocity contour. Not the same as geopotential, but usually close.

Next Time Natural Coordinates Balanced Flows