Richard B. Rood (Room 2525, SRB) AOSS 401 Geophysical Fluid Dynamics: Atmospheric Dynamics Prepared: 20131001 Balanced Flows Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Cell: 301-526-8572

Class News Ctools site (AOSS 401 001 F13) First Examination on October 22, 2013 Second Examination on December 10, 2013 Homework posted: Ctools Assignments tab Due Thursday October 10, 2013 Derivations (using notes)

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

Outline Geostrophic Balance and the Real Wind Natural Coordinates Balanced flows Geostrophic Cyclostrophic Gradient

Atmosphere in Balance Hydrostatic balance (no vertical acceleration) Geostrophic balance (no horizontal acceleration or divergence) Adiabatic lapse rate (no clouds or precipitation) Vertical motion is weak for large-scale flow. A good approximation is that flow horizontal. In the upper atmosphere the flow is often adiabatic.

Scale Analysis Scale analysis is reliant on observations of the preferred motion of the fluid. What are the size, spatial scale, of the motions? What are the time scales of the motions? How did we define large? For our flow we compared f(rotation) to U/L

I present these equations: Assume no viscosity and no vertical motion What is vertical coordinate? Pressure What conservation law? Momentum

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. There is curvature in the flow Hence acceleration

Consider this simple “map” Contours of geopotential Upper troposphere Note coordinate system What is the direction of the geostrophic wind? Where is it weaker or stronger?

Geopotential (Φ) in upper troposphere ΔΦ > 0 north Φ0+ΔΦ Φ0 Φ0+2ΔΦ Φ0+3ΔΦ south west east

Geopotential (Φ) in upper troposphere ΔΦ > 0 north Φ0+ΔΦ Φ0 Δy Φ0+2ΔΦ Φ0+3ΔΦ south west east

Geopotential (Φ) in upper troposphere ΔΦ > 0 north Φ0+ΔΦ δΦ = Φ0 – (Φ0+2ΔΦ) Φ0 Δy Φ0+2ΔΦ Φ0+3ΔΦ south west east

Geopotential (Φ) in upper troposphere ΔΦ > 0 north Φ0+ΔΦ Φ0 Δy Φ0+2ΔΦ Φ0+3ΔΦ south west east

The horizontal momentum equation Assume no viscosity

Geostrophic approximation

Geopotential (Φ) in upper troposphere ΔΦ > 0 north Φ0+ΔΦ Φ0 Δy Φ0+2ΔΦ Φ0+3ΔΦ south west east

Geopotential (Φ) in upper troposphere ΔΦ > 0 north Φ0+ΔΦ Φ0 Δy Φ0+2ΔΦ Φ0+3ΔΦ south west east

Think about this for a minute … Geopotential (Φ) in upper troposphere ΔΦ > 0 north T gradient Φ0+3ΔΦ W cf Φ0+2ΔΦ Δy Φ0+ΔΦ pgf Φ0 C south west east SOUTHERN HEMISPHERE

How did we get the wind? 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.

north Φ0+ΔΦ Φ0 Φ0+2ΔΦ Φ0+3ΔΦ south west east The geostrophic wind can only be equal to the real wind if the height contours are straight. north Φ0+ΔΦ Φ0 Φ0+2ΔΦ Φ0+3ΔΦ south west east

Geostrophic & observed wind 300 mb

Geopotential (Φ) in upper troposphere Think about the observed wind Flow is parallel to geopotential height lines There is curvature in the flow Where is the effect of curvature here?

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 What’s one way to think about this? Curvature means there is acceleration 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 Need to describe balance between pressure gradient, Coriolis and curvature

“Natural” Coordinate System Follow the flow From hydrodynamics—assumes no local changes (“steady state”) No local change in geopotential height No local change in wind speed or direction Assume Horizontal flow only (no vertical component) No friction This is like a Lagrangian parcel approach

Return to Geopotential (Φ) in upper troposphere Define one component of the horizontal wind as tangent to the direction of the wind. t north Φ0 t t t Φ0+3ΔΦ south west east ΔΦ > 0

How do these natural coordinates relate to the tangential coordinates? Ω They are still tangential, but the unit vectors do not point west to east and south to north. The coordinate system turns with the wind. And if it turns with the wind, what do we expect to happen to the forces? a Φ = latitude Earth

Looking down from above

Looking down from above

Looking down from above

Looking down from above

Looking down from above

Return to Geopotential (Φ) in upper troposphere Define the other component of the horizontal wind as normal to the direction of the wind. n north Φ0 n n n t t t Φ0+3ΔΦ south west east 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 n n t t t Geostrophic assumption. Do you notice that those n vectors point towards something out in the distance? HIGH south west east 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 n n n t t HIGH t south west east

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

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

How to get as a function of V, R ?

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

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

Remember our circle geometry… Δs=RΔφ If Δs is very small, Δt is parallel to n. So, Δt points in the direction of n as Δs  0 Δφ Δt t+Δt n t R= radius of curvature n 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 defined as: (Product 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 angle of rotation Circle geometry angular velocity Plug in for Δs Centrifugal force

Acceleration in Natural Coordinates Along-flow speed change Centrifugal Acceleration

Horizontal Momentum Equation? Our horizontal momentum equation will consist of three terms Acceleration Coriolis force Pressure gradient force

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

Horizontal Pressure Gradient (by definition)

The horizontal momentum equation Tangential Cartesian coordinate system / pressure in vertical Natural coordinate system for horizontal flow in 2D

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

Simplification? Which coordinate system is easier to interpret? 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 (Centrifugal Force) Coriolis Pressure Gradient

Uses of Natural Coordinates Geostrophic balance Definition: coriolis and pressure gradient in exact balance. Parallel to contours  straight line  R becomes infinite

Geostrophic balance in natural coordinates

north Φ0+ΔΦ Φ0 Φ0+2ΔΦ Δn Φ0+3ΔΦ south west east Which actually tells us the geostrophic wind can only be equal to the real wind if the height contours are straight. north Φ0+ΔΦ Φ0 Φ0+2ΔΦ Δn Φ0+3ΔΦ south west east

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

Cyclonic and Anticyclonic What do these mean? Cyclonic is flow around a low pressure system. Anticyclonic is flow around a high pressure system. These are words to describe rotational flow.

How does curvature affect the wind? (cyclonic flow/low pressure) Φ0 Φ0+ΔΦ Φ0-ΔΦ HIGH Low R t n Δn

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!

Add curvature (centrifugal force) 2. Physical Perspective Geostrophic balance Add curvature (centrifugal force) V PGF COR V PGF COR CEN > 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 hPa

Geostrophic & observed wind 300 hPa Observed: 95 knots Geostrophic: 140 knots

How does curvature affect the wind? (anticyclonic flow/high pressure) Φ0 Φ0+ΔΦ Φ0-ΔΦ HIGH Low t n Δn R

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 V PGF COR PGF 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 hPa

Geostrophic & observed wind 300 hPa 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?)

Summary:“Natural” Coordinates & Balanced Flows Another perspective on flows Within meteorological features 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) / better estimate than geostrophic

Calculation / Interpretation: Gradient Wind Balance For homework

Gradient Flow (Momentum equation in natural coordinates) Balance in the normal, as opposed to tangential, component of the momentum equation Balance between pressure gradient, coriolis, and centrifugal force

Gradient Flow (Momentum equation in natural coordinates)

Gradient Flow (Momentum equation in natural coordinates) Look for real and non-negative solutions for V

Gradient Flow Solution (V) must be real and non-negative 8 Possible Solutions Anomalous Low Regular Low Anomalous High Regular High

Gradient Flow (Solutions for Lows, remember that square root.) Pressure gradient force NORMAL ANOMALOUS Low Low V n n V Coriolis Force Centrifugal force

Gradient Flow (Solutions for Highs, remember that square root.) Pressure gradient force NORMAL ANOMALOUS High High V n n V Coriolis Force Centrifugal force

Gradient Flow: Implications Solution (V) must be real

Gradient Flow Regular High and Low Definition of normal, n, direction Low High R > 0 R < 0 n n

Gradient Flow: Implications Solution must be real Always satisfied High ∂Φ/∂n < 0 R < 0 Trouble! pressure gradient MUST go to zero faster than R goes to zero