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PHYS-575/CSI-655PHYS-575/CSI-655Introduction to Atmospheric Physics and ChemistryIntroduction to Atmospheric Physics and Chemistry

Lecture Notes 7: Lecture Notes 7: Atmospheric DynamicsAtmospheric Dynamics

1. Kinematics of Large-Scale Horizontal Flow2. Dynamics of Horizontal Flow3. Primitive Equations4. The Atmospheric General Circulation

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Announcements: April 4, 2010Announcements: April 4, 2010

April 4: Homework #4 DueNo more homework assignments!

April 4 – Finish Clouds (Chapter 6), Begin Dynamics (7)April 11 – Dynamics (Chapter 7), Intro to Climate (10)April 18 – Climate (10); Review for exam

April 25 – Exam #2May 2 – Climate – continued; Last Day of Classes

May 16 (4:30-7:20pm)– Term Paper Presentations

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Term Paper FormatTerm Paper Format

The term paper must follow standard guides for research papers, The term paper must follow standard guides for research papers, and have the following sections:and have the following sections:

TitleTitle AbstractAbstract Introduction & backgroundIntroduction & background Body of paperBody of paper - with a significant number (10-15) references - with a significant number (10-15) references

to primary literature and/or review articles. This may include to primary literature and/or review articles. This may include discussion of scientific theories, observations, and/or methods.discussion of scientific theories, observations, and/or methods.

ConclusionsConclusions Figures Figures are important in the body of the paper.are important in the body of the paper. References to sources References to sources

The paper must be typed, double spaced, and have ~ 15-25 The paper must be typed, double spaced, and have ~ 15-25 pages of text, not including figures, and at least 3 figures (may pages of text, not including figures, and at least 3 figures (may have more, include captions). Please number all pages.have more, include captions). Please number all pages.

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Term Paper PresentationsTerm Paper PresentationsWednesday, May 16, 2010 (4:30-7:20pm)Wednesday, May 16, 2010 (4:30-7:20pm)

Overview & Summary of Term PaperOverview & Summary of Term Paper 5 minutes time limit5 minutes time limit No more than 5 slides in PowerPoint file.No more than 5 slides in PowerPoint file. No math derivations, 1-2 key equations okNo math derivations, 1-2 key equations ok Summary figuresSummary figures OutlineOutline MotivationMotivation SummarySummary Please email presentation to me by 10pm on Please email presentation to me by 10pm on Sunday, 5/15Sunday, 5/15

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Exam #2, Monday April 25Exam #2, Monday April 25Note Change of Date!!!!!!!Note Change of Date!!!!!!!

Closed Book/Notes Covers Text Chapter 5 through Chapter 7. Last approximately 1-1.5 hours Total of 100 points ~30-35 questions You are responsible for all the material from the text, lecture notes, and lecture discussion. Some sketches required. You need to know key equations and physical significance. Always define all terms.

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Hadley CirculationHadley Circulation

http://www.ux1.eiu.edu/~cfjps/1400/FIG07_014B.jpg

Spatial and Time Scales of Atmospheric Flow

Hurricane Katrina

Tornado

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The General Circulation: DetailsThe General Circulation: Details Effects of rotationEffects of rotation GeostrophyGeostrophy Atmospheric wavesAtmospheric waves Boundary layers, Boundary layers,

friction and stressesfriction and stresses Turbulence and Turbulence and

mixingmixing The planetary The planetary

boundary layerboundary layer Instabilities and wave Instabilities and wave

breakingbreaking

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Scales of Atmospheric MotionsScales of Atmospheric Motions

Type of Motion Horiz. Spatial Scale Time ScaleMolecular motion 0.1 micron 10-9 sMicro-scale flow 1 cm 0.1 sSurface Layer 10 cm 0.1 sSmall scale eddies 1 m 1 sDust devils 10 m 10 sTornado 100 m 10 sSmall clouds 1 km 1 minThunderstorm 10-100 km 10-100 minHurricane 1000 km 1 hourWeather front 1000 km 10 hoursPlanetary wave 10,000 km 3-5 days

Small and Mesoscale atmospheric flow Synoptic scale motions

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Synoptic Scale MotionsSynoptic Scale Motions Horizontal Scale ~ 100Horizontal Scale ~ 100’’s of kms of km Vertical Scale ~ depth of troposphereVertical Scale ~ depth of troposphere Timescales ~ hours to daysTimescales ~ hours to days

Motions on these scales are directly and strongly influenced Motions on these scales are directly and strongly influenced by the Earthby the Earth’’s rotation.s rotation.

They are in hydrostatic balance.They are in hydrostatic balance. The vertical component of the velocity ~ 1000 times smaller The vertical component of the velocity ~ 1000 times smaller

than the horizontal component.than the horizontal component. This scale of motion is dominant in controlling the transfer of This scale of motion is dominant in controlling the transfer of

energy and momentum in the Earthenergy and momentum in the Earth’’s atmosphere.s atmosphere.

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1. Kinematics of Large-Scale Horizontal 1. Kinematics of Large-Scale Horizontal FlowFlow

Kinematics deals with properties of flows that can be diagnosed (but not necessarily predicted) without recourse to the equations of motion.

Streamlines: lines whose orientation is such that they are everywhereparallel to the horizontal velocity vector V.

Isotachs: contours of constant scaler wind speed V.

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Natural CoordinatesNatural Coordinates

dt

dsV

Natural Coordinates: a pair of axes (s, n) where s is the arc length directeddownstream along the local streamline, and n is the distance directed normalto the streamline and toward the left. The direction of flow is denoted by theangle ψ, which is defined relative to a reference direction.

At any point in the flow, the scaler wind speed:

0dt

dn

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Properties of FlowsProperties of Flows

ShearShear is the rate of change of is the rate of change of velocity in direction transverse velocity in direction transverse to the direction of flow.to the direction of flow.

CurvatureCurvature is the rate of change is the rate of change of direction of flow.of direction of flow.

Shear and curvature are labeled as cyclonic (anticyclonic) and have a sign in the same sense as to cause an object in the flow to rotate in the same (opposite) sense as the Earth’s rotation when looking down on the N pole.

Cyclonic means counterclockwise in the N hemisphere and clockwise in the S hemisphere.

dn

dVShear

ds

dVCurvature

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Kinematical properties of horizontal flow that can be definedKinematical properties of horizontal flow that can be definedat any point in the flow (all have units sat any point in the flow (all have units s-1).).

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Vorticity and DivergenceVorticity and Divergence

Vorticity and divergence are scaler quantities that can be definednot only in natural coordinates (s,n), but also in Cartesian Coordinates (x,y), for a horizontal wind vector V.

Vorticity is the sum of the shear and curvature.

ξ = 2 ω

where ω is the rate of spin of an imaginary object moving withthe flow.

Divergence is the sum of the diffluence and the stretching,but is more easily intuited as outflux per unit volume..

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Idealized FlowsIdealized Flows

Sheared flow without curvature

Solid body rotation with cyclonic shear and cyclonic curvature

but without divergence.

Radial flow with velocity directly proportional to radius, but without curvature or shear.

Hyperbolic flow that has both shear and curvature but no

vorticity or divergence.

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Deformation is Deformation is the sumthe sum

of the Confluence of the Confluence and Stretching and Stretching

termstermsEven simple horizontal flow can rapidly distort a field of passive tracers.

This gives rise to a variety of complex transport effects such as eddy diffusion(also known as eddy mixing).

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Deformation can sharpen preexisting horizontal gradients creating features known as frontal zones.

Frontal zones can be stationary, but more often are in motion relative asas consequence of the flow field.

Frontal Zones

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Streamlines are horizontal trajectories only if Streamlines are horizontal trajectories only if the flow is steadythe flow is steady

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Atmospheric Atmospheric Dynamics: Dynamics: The BasicsThe Basics

1) Forces:Pressure gradient - ∂P/∂xFriction - ν ∂2U∂z2

Gravity - ρ gElectromagnetic - e E x B

2) Reference Frames:Inertial Newtonian DynamicsNon-Inertial Apparent Forces (Coriolis)

3) Time Tendency:Fixed observer Material DeriviativeObserver “riding” motion Lagrangian Deriviative

4) Conservation Laws:Mass mMomentum (force equation) linear + angularEnergy 1/2mv2 + ρCpT + ρgz

5) Scaling of the Equations of Motion:Geostrophic Balance Rossby NumberTurbulence Reynold’s Number

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2. Dynamics of Horizontal Flow2. Dynamics of Horizontal Flow

Real Forces: are thefundamental forces, e.g.- Gravitation- Electricity & Magnetism- Friction- Pressure gradient

Apparent Forces: arise due to the acceleration of the reference frame.- Centrifugal force- Corriolis force

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Real vs. Apparent ForcesReal vs. Apparent Forces

In an inertial (non-accelerating) reference frame Newton’s Laws of Motion can be directly applied to a parcel of gas in order to determine its time tendency (acceleration).

Euler’s Equation:

m Dv/Dt = - dP/dx + ρg + other forces (1-dimensional, x-direction)

Dv/Dt is the material or advective deriviative in an inertial reference frame. It can be related to the Lagrangian deriviative (“riding” the parcel) via

D/Dt = - ∂/∂t + v ∂/∂x, where v is the vector velocity.

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Real vs. Apparent ForcesReal vs. Apparent Forces

In an accelerating reference frame (planet rotating with angular velocity ω), the acceleration produces an apparent force to the fixed observer.

This is called the Coriolis Force.

The acceleration due to this “fictitious” force is given by:

Dv/Dt = -2 ω x v

The Coriolis Force is always perpendicular to the direction of motion and thus cannot do work on the fluid.

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Gravity vs. Gravitation; Or Which Way is Down?Gravity vs. Gravitation; Or Which Way is Down?

Gravitation:The force between two objects due to their mass. On the surface of the Earth gravitation is denoted by the vector g* directed toward the center of the Earth.

However, the Earth is rotatingwith angular acceleration Ω = 2π rad day-1 = 7.292 x 10-5 s-1

which produces a radial force= Ω2R, where R is the radial vector from the axis of rotation.

Gravity: g = g* + Ω2R

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Focault PendulumFocault Pendulum

The Focault Pendulum is an example ofsimple harmonic motion in inertial space.

The Focault Pendulum swings back andforth in inertial space while the Earthrotates “underneath” it.

From the point of view of someone watching the motion on the surface of the Earth, it appears that the pendulum rotates. This is inertial motion.

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The Coriolis ForceThe Coriolis Force

Local Gravity: g = g* + Ω2R

An object on the surface of the Earth will experience an outwarddirected force (away from the axis of rotation) due to the Earth’srotation of magnitude Ω2R.

An object moving with velocity V in the plane perpendicular tothe axis of rotation experiences an additional apparent force thatis known as the Coriolis Force of magnitude - 2Ω x V

In spherical coordinates the Coriolis Force = - f k x V

where f = 2Ω sinφ is the Coriolis parameter, and k is the local unitvector in the vertical.

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The Coriolis ForceThe Coriolis Force

In spherical coordinates the Coriolis Force = - f k x V

The Coriolis Force is a deflecting force, always acting perpendicular to the direction of motion.

Thus the Coriolis force cannot do work on the parcel/object.

The magnitude of the Coriolis force is = – f V

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The Pressure Gradient ForceThe Pressure Gradient Force

gdz

dp

1

p

P

x

pPx

1

The vertical force balance is known as hydrostatic equilibrium:

y

pPy

1

The vertical force per unit mass:

In general the total pressure force:

Horizontal components of the pressure gradient force:

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The Horizontal Equations of The Horizontal Equations of MotionMotion

xFfvx

p

dt

du

1

yFfuy

p

dt

dv

1

East-West direction (x, u positive toward East):

North-South direction (y, v positive toward North):

where Fx,y is the external force (e.g. friction)

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Geostrophy and the Rossby Geostrophy and the Rossby NumberNumber

xFfvx

p

dt

du

1

yFfuy

p

dt

dv

1

E-W: N-S:

Outside of the Boundary Layer (above ~1km altitude) the frictioncomponents are insignificant. If the acceleration terms are small,then those too may also be ignored. To determine if du/dt & dv/dt can beignored, compare their magnitude to the Coriolis Force for a typicalvelocity (U), lengthscale (L) and timescale (T ~ L/U)

dU/dt ~ U/T ~ U2/L; fu ~ f U

The non-dimensional Rossby Number: (dU/dt)/fu ~ U/fL is a measure ofthe relative magnitude of the acceleration to Coriolis terms.

Ro = U/fL

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The Horizontal Equations of The Horizontal Equations of MotionMotion

fvx

p

1

fuy

p

1

y

p

fug

1

Above ~1km altitude (outside the boundary layer) Fx,y =0:

Note that this force balance is diagnostic, not prognostic,i.e., there is no tendency or time evolution. This is calledthe Geostrophic Balance. The wind velocity that exactlysolves these equations is known as the Geostrophic Wind.

x

p

fvg

1

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GeostrophyGeostrophy

fvx

p

1

fuy

p

1

Above ~1km altitude (outside the boundary layer) Fx,y =0:

For Synoptic Scale Motions:

L ~ 1000 KM ~ 106 mU ~ 10 ms-1

f ~ 7 x 10-5 s-1

Ro = U/fL ~ 1/7 ~14%

The smallness of the Rossby number is a measure of the accuracyof the Geostrophic Approximation. At mid-latitudes the geostrophicequations are generally accurate to about 10%.

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Geostrophic Equations of MotionGeostrophic Equations of Motion

fvx

p

1 fu

y

p

1

The smallness of the Rossby number is a measure of the accuracyof the Geostrophic Approximation. At mid-latitudes the geostrophicequations are accurate to about 10%

These equations of motion are diagnostic equations, i.e., they can be used to infer the velocity field if the pressure variation is known, or they can be used to determine the pressure field if the velocity field is known. These equations cannot be used to determine thetime evolution of either the pressure or temperature fields.

In order to determine the time evolution, the du/dt term is needed.

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Schematic View of Geostrophic BalanceSchematic View of Geostrophic Balance

To first approximation, the horizontal balance of forces in the Earth’s atmosphereis between the pressure gradient force and the Coriolis force.

The approximation that they are in perfect balance is known as the Geostrophic Wind. It is typically accurate to of order 10% at mid latitudes, away from the equator, for synoptic scale motions.

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The Coriolis Force and Deflection of FlowThe Coriolis Force and Deflection of FlowPressure gradients, usually due to temperature differences, cause the airto flow. Once set in motion, the Coriolis Force deflects the force to the rightin the northern hemisphere and to the left in the southern hemisphere. Thisis an apparent force which reflects the tendency of the air to move in an inertial reference frame (“fixed to the distant stars”).

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FrictionFriction

zF

1

The role of friction in the atmosphere is to produce a “drag” on atmospheric motion. However, the magnitude of the frictionaldrag is generally very difficult to quantify. The reason is that thedrag force is due to a variety of physical processes, all of whichtransfer momentum between the surface and the free atmosphere.The Frictional Force per unit mass:

where τ represents the vertical component of the shear stress(rate of vertical exchange of horizontal momentum) due to thepresence of smaller, unresolved scales of motion. The shearstress at the surface of the Earth is:

τs = - ρ CD Vs VCD = drag coefficient, Vs = scaler velocity, V = vector velocity

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Effects of FrictionEffects of Friction

The Coriolis Force and Pressure Gradient Force are always perpendicular to the direction of flow. However, the FrictionalForce is always opposite to the direction of flow.

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Effects of Friction on Flow DirectionEffects of Friction on Flow Direction

The effect of friction is to slow down the flow. The resulting balance of forces leads to a cross-isobar drift, generally fromhigh pressure to low pressure. Thus friction is a primary meansof generating flow which is not parallel to the isobars.

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The Gradient WindThe Gradient Wind

R

ufu

dy

p

dt

du 21

For rapidly rotating flow outside of the surface friction layer,the angular rotation produces a centripetal acceleration.The three-way balance of forces in Gradient Flow (Pressure gradient, Coriolis, and Centripetal) leads to a modification of the force balance equations so that:

where R is the radius of curvature of the flow streamlines.For steady flow, du/dt = 0, and we can use the definition ofgeostrophic wind to give an equation that can be solvedalgebraically for the flow velocity u.

R

ufuug

2

0

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Horizontal Balanced FlowHorizontal Balanced Flow

01 2

R

ufu

dy

p

dt

du

For rapidly rotating steady flow:

where R is the radius of curvature of the flow streamlines.

For rapid rotation the pressure gradientforce is balanced by the centripetal term.The relative magnitude of the centripetalto the Coriolis term is:

Rossby No. ~ (U2/R)/fU ~ U/fR

Thus the largeness of the Rossby # is a measure of the accuracy ofthe Cyclostropyhic Approximation.

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The Three-Way Balance in the Gradient The Three-Way Balance in the Gradient WindWind

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Thermal Wind Thermal Wind EquationEquation

A horizontal gradient in temperatureproduces a vertical gradient in thehorizontal velocity field.

Hydrostatic equilibrium:

dP = -ρ g dz = - (P/RT) dz

So the difference in altitude betweentwo pressure levels differing by dP is

-dz = (RT/P) dP

So increasing T (in the horizontal) implies increasing dz in the vertical.

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Isotherms and Isotherms and GeopotentialGeopotential

The Thermal Wind Equation impliesa diagnostic relationship between thetemperature structure and the windstructure.

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Vorticity (Angular Momentum) ConservationVorticity (Angular Momentum) Conservation

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3. Primitive Equations3. Primitive Equations

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Cross-Isobar FlowCross-Isobar Flow

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Wave PropagationWave Propagation

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4. Development of the General 4. Development of the General CirculationCirculation

Non-rotating planet:motion is mainly in the meridional plane.

Rotating planet:Motion is highly3-dimensional

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Steady CirculationSteady Circulation

Heating:Tropical latent heat releaseIR heating from ground

Cooling:Adiabatic expansionIR cooling

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The Atmospheric General CirculationThe Atmospheric General Circulation

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The Atmosphere as a Heat EngineThe Atmosphere as a Heat Engine

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Numerical Weather PredictionNumerical Weather Prediction

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Ensemble ForecastsEnsemble Forecasts

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Weather ModelsWeather Models

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Surface flow due to the Coriolis EffectSurface flow due to the Coriolis Effect

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Vertical Flow Consequences of the Coriolis EffectVertical Flow Consequences of the Coriolis Effect

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The Hadley CellThe Hadley Cell

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JupiterJupiter’’s Windss Winds

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JupiterJupiter’’s Great Red Spots Great Red Spot

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Condensation Flows: Saturation Vapor Pressure is Condensation Flows: Saturation Vapor Pressure is the Driving Forcethe Driving Force

Sulfur Dioxide flows from the day side to night side.

Seasonal sublimation of the polarcaps produce pressure gradientsand global scale winds.

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Questions for DiscussionQuestions for Discussion

(1) Is Geostrophic Flow a universal property of a planetary (1) Is Geostrophic Flow a universal property of a planetary atmosphere?atmosphere?

(2) How large can atmospheric cyclones (e.g. hurricanes) (2) How large can atmospheric cyclones (e.g. hurricanes) become in a planetary atmosphere?become in a planetary atmosphere?

(3) In what way would the general circulation be different(3) In what way would the general circulation be different

without surface friction?without surface friction?

(4) If the Coriolis Force does no work on a parcel of air, then(4) If the Coriolis Force does no work on a parcel of air, then

how does the air accelerate?how does the air accelerate?

(5) Which is more important for driving atmospheric winds: (5) Which is more important for driving atmospheric winds: Solar heating (direct and latent) and EarthSolar heating (direct and latent) and Earth’’s rotation? s rotation?

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Rotation ExperimentRotation Experiment

In a rotating reference frame, the only force is the Coriolis Force

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““RotatingRotating”” TrajectoriesTrajectories