AOSS 401 Fall 2006 Lecture 10 September 28

AOSS 401, Fall 2006 Lecture 10 September 28, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich. edu 734 -647 -3530 Derek Posselt (Room 2517 D, SRB) dposselt@umich. edu 734 -936 -0502

Class News • • Homework 2 returned today Homework 3 due today (questions? ) Homework 4 posted Monday Exam 1 October 10—covers chapters 1 -3 in Holton

Weather • NCAR Research Applications Program – http: //www. rap. ucar. edu/weather/ • National Weather Service – http: //www. nws. noaa. gov/dtx/

Correction… • I made a mistake in my last set of lectures (September 19 th) • Geostrophic wind is only non-divergent if pressure is the vertical coordinate… • Corrected lecture 6 posted to ctools by Monday.

Today: Material from Chapter 3 • Natural coordinates • Balanced flow

Another Coordinate System? • We want to simplify the equations of motion • For horizontal motions on many scales, the atmosphere is in balance – Mass (p, Φ) fields in balance with wind (u) – It is easy to observe the pressure or geopotential height, much more difficult to observe the wind • Balance provides a way to infer the wind from the observed (p, Φ) • Wind is useful for prediction (remember the advection homework and in-class problems? )

The horizontal momentum equation Assume no viscosity and no vertical wind

Geostrophic balance Low Pressure High Pressure Flow initiated by pressure gradient Flow turned by Coriolis force

Geostrophic & observed wind 300 mb

Describe previous figure. What do we see? • At upper levels (where friction is negligible) the observed wind is parallel to geopotential height contours. • (On a constant pressure surface) • Wind is faster when height contours are close together. • Wind is slower when height contours are farther apart.

Geopotential (Φ) in upper troposphere ΔΦ > 0 north Φ 0+ΔΦ Φ 0+2ΔΦ Φ 0+3ΔΦ south west east

Geopotential (Φ) in upper troposphere ΔΦ > 0 north Φ 0+ΔΦ Φ 0 Δy Φ 0+2ΔΦ Φ 0+3ΔΦ south west east

Geopotential (Φ) in upper troposphere ΔΦ > 0 north δΦ = Φ 0 – (Φ 0+2ΔΦ) Φ 0+ΔΦ Φ 0 Δy Φ 0+2ΔΦ Φ 0+3ΔΦ south west east

Geopotential (Φ) in upper troposphere ΔΦ > 0 north Φ 0+ΔΦ Φ 0 Δy Φ 0+2ΔΦ Φ 0+3ΔΦ south west east

The horizontal momentum equation Assume no viscosity

Geostrophic approximation

Geopotential (Φ) in upper troposphere ΔΦ > 0 north Φ 0+ΔΦ Φ 0 Δy Φ 0+2ΔΦ Φ 0+3ΔΦ south west east

Geopotential (Φ) in upper troposphere ΔΦ > 0 north Φ 0+ΔΦ Φ 0 Δy Φ 0+2ΔΦ Φ 0+3ΔΦ south west east

Geopotential (Φ) in upper troposphere • Think about the observed wind – Flow is parallel to geopotential height lines – There is curvature in the flow

Geostrophic & observed wind 300 h. Pa

Geopotential (Φ) in upper troposphere • Think about the observed (upper level) wind – Flow is parallel to geopotential height lines – There is curvature in the flow • Geostrophic balance describes flow parallel to geopotential height lines • Geostrophic balance does not account for curvature • How to best describe balanced flow with curvature?

Another Coordinate System? • We want to simplify the equations of motion • For horizontal motions on many scales, the atmosphere is in balance – Mass (p, Φ) fields in balance with wind (u) – It is easy to observe the pressure or geopotential height, much more difficult to observe the wind • Balance provides a way to infer the wind from the observed (p, Φ) • Need to describe balance between pressure gradient, coriolis, and curvature

“Natural” Coordinate System • Follow the flow • From hydrodynamics—assumes no local changes – No local change in geopotential height – No local change in wind speed or direction • Assume – Horizontal flow only (no vertical component) – No friction

Return to Geopotential (Φ) in upper troposphere north Define one component of the horizontal wind as tangent to the direction of the wind. t Φ 0 t t t Φ 0+3ΔΦ south east west ΔΦ > 0

Return to Geopotential (Φ) in upper troposphere north Define the other component of the horizontal wind as normal to the direction of the wind. n Φ 0 n n n t t t Φ 0+3ΔΦ south east west ΔΦ > 0

“Natural” Coordinate System • Regardless of position (i, j) – t always points in the direction of flow – n always points perpendicular to the direction of the flow toward the left • Remember the “right hand rule” for vectors? Take k x t to get n • Assume – Pressure as a vertical coordinate – Flow parallel to contours of geopotential height

“Natural” Coordinate System • Advantage: We can look at a height (on a pressure surface) and pressure (on a height surface) and estimate the wind. – It is difficult to directly measure winds – We estimate winds from pressure (or hydrostatically equivalent height), a thermodynamic variable. – Natural coordinates are useful for diagnostics and interpretation.

“Natural” Coordinate System • For diagnostics and interpretation of flows, we need an equation…

Return to Geopotential (Φ) in upper troposphere north Low n t HIGH south n n t t Geostrophic assumption. Ø Do you notice that those n vectors point towards something out in the distance? east west ΔΦ > 0

Return to Geopotential (Φ) in upper troposphere Do you see some notion of a radius of curvature? Sort of like a circle, but NOT a circle. north Low HIGH south west n n n t t t east

Time to look at the mathematics Define velocity as: One direction: no (u, v) Definition of magnitude: First simplification: the velocity • Always positive • Always points in the positive t direction

Goal: Quantify Acceleration acceleration is: (Chain Rule) Change in speed Change in Direction

How to get as a function of V, R ?

Remember our circle geometry… this is not rotation of the Earth! It is an element of curvature in the flow. Δφ Δt t+Δt R= radius of curvature t Δs=RΔφ

Remember our circle geometry… this is not rotation of the Earth! It is an element of curvature in the flow. Δφ Δt t+Δt R= n radius of curvature n t Δs=RΔφ

Remember our circle geometry… Δs=RΔφ If Δs is very small, Δt is parallel to n. So, Δt points in the direction of n Δφ Δt t+Δt R= n radius of curvature n t Δs t

Remember, we want an expression for From circle geometry we have: Rearrange and take the limit Use the chain rule Remember the definition of velocity

Goal: Quantify Acceleration acceleration defined as: (Chain Rule) We just derived: So the total acceleration is

Acceleration in Natural Coordinates Along-flow speed change ?

Acceleration in Natural Coordinates The total acceleration is Definition of wind speed Circle geometry Plug in for Δs Centrifugal force angle of rotation angular velocity

Acceleration in Natural Coordinates Along-flow speed change Centrifugal Acceleration

We have seen that Coriolis force is normal to the velocity.

Pressure gradient (by definition)

The horizontal momentum equation

The horizontal momentum equation (in natural coordinates) Along-flow direction (t) Across-flow direction (n)

• Simplification? • Which coordinate system is easier to interpret? 0 0 • We are only looking at flow parallel to geopotential height contours

• Simplification? • Which coordinate system is easier to interpret? • We are only looking at flow parallel to geopotential height contours

One Diagnostic Equation Curved flow Coriolis (Centrifugal Force) Pressure Gradient

Uses of Natural Coordinates • Geostrophic balance – Definition: coriolis and pressure gradient in exact balance. – Parallel to contours straight line R is infinite 0

Geostrophic balance in natural coordinates

Which actually tells us the geostrophic wind can only be equal to the real wind if the height contours are straight. north Φ 0+ΔΦ Φ 0+2ΔΦ Δn Φ 0+3ΔΦ south west east

Therefore • If the contours are curved then the real wind is not geostrophic.

How does curvature affect the wind? (cyclonic flow/low pressure) Low R Δn HIGH Φ 0 -ΔΦ n Φ 0 t Φ 0+ΔΦ

1. Mathematical Perspective Equation of motion Split Coriolis into geostrophic and ageostrophic parts Use definition of geostrophic wind

1. Mathematical Perspective Total centrifugal force balances ageostrophic part of coriolis Look at sign of terms (R > 0) Total wind is sum of its parts Real wind speed is slower than geostrophic for cyclonic flow!

2. Physical Perspective Geostrophic balance PGF V Add curvature (centrifugal force) PGF COR V CEN COR > Pressure gradient force is the same in each case. With curvature less coriolis force is needed to balance the pressure gradient.

Geostrophic & observed wind 300 h. Pa

Geostrophic & observed wind 300 h. Pa Observed: 95 knots Geostrophic: 140 knots

How does curvature affect the wind? (anticyclonic flow/high pressure) n Low Δn t R HIGH Φ 0 -ΔΦ Φ 0+ΔΦ

1. Mathematical Perspective Total centrifugal force balances ageostrophic part of coriolis Look at sign of terms (R > 0) Total wind is sum of its parts Real wind speed is faster than geostrophic for anticyclonic flow!

2. Physical Perspective Geostrophic balance Add curvature CEN PGF V V COR < Pressure gradient force is the same in each case. With curvature more coriolis force is needed to balance the pressure gradient.

Geostrophic & observed wind 300 h. Pa

Geostrophic & observed wind 300 h. Pa Observed: 30 knots Geostrophic: 25 knots

What did we just do? • Found a way to describe balances between pressure gradient, coriolis, and curvature • We assumed friction was unimportant and only looked at flow at a particular level • We assumed flow was on pressure surfaces • We saw that the simplified system can be used to describe real flows in the atmosphere • Can we describe other flow patterns? (Different scales? Different regions of the Earth? )

Next Time: Finish balanced flows • Cyclostrophic flow (tornados, water spouts, dust devils) • Gradient wind (general description of curved flow anywhere on the globe—if friction is not important…)