Download Coriolis Force - Atmosphere Physics

PHYS-575/CSI-655
Introduction to Atmospheric Physics and Chemistry
Lecture Notes 7: Atmospheric Dynamics
1. Kinematics of Large-Scale
Horizontal Flow
2. Dynamics of Horizontal Flow
3. Primitive Equations
4. The Atmospheric General
Circulation
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Announcements: April 4, 2010
April 4: Homework #4 Due
No 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 #2
May 2 – Climate – continued; Last Day of Classes
May 16 (4:30-7:20pm)– Term Paper Presentations
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Term Paper Format
The term paper must follow standard guides for research papers,
and have the following sections:
 Title
 Abstract
 Introduction & background
 Body of paper - with a significant number (10-15) references
to primary literature and/or review articles. This may include
discussion of scientific theories, observations, and/or methods.
 Conclusions
 Figures are important in the body of the paper.
 References to sources
The paper must be typed, double spaced, and have ~ 15-25
pages of text, not including figures, and at least 3 figures (may
have more, include captions). Please number all pages.
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Term Paper Presentations
Wednesday, May 16, 2010 (4:30-7:20pm)
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Overview & Summary of Term Paper
5 minutes time limit
No more than 5 slides in PowerPoint file.
No math derivations, 1-2 key equations ok
Summary figures
Outline
Motivation
Summary
Please email presentation to me by 10pm on
Sunday, 5/15
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Exam #2, Monday April 25
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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Spatial and Time Scales of Atmospheric Flow
Hadley Circulation
Tornado
http://www.ux1.eiu.edu/~cfjps/1400/FIG07_014B.jpg
Hurricane Katrina
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The General Circulation: Details
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Effects of rotation
Geostrophy
Atmospheric waves
Boundary layers,
friction and stresses
Turbulence and
mixing
The planetary
boundary layer
Instabilities and wave
breaking
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Scales of Atmospheric Motions
Type of Motion
Molecular motion
Micro-scale flow
Surface Layer
Small scale eddies
Dust devils
Tornado
Small clouds
Thunderstorm
Hurricane
Weather front
Planetary wave
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Horiz. Spatial Scale
0.1 micron
1 cm
10 cm
1m
10 m
100 m
1 km
10-100 km
1000 km
1000 km
10,000 km
Time Scale
10-9 s
0.1 s
0.1 s
1s
10 s
10 s
1 min
10-100 min
1 hour
10 hours
3-5 days
Small and Mesoscale atmospheric flow
Synoptic scale motions
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Synoptic Scale Motions
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Horizontal Scale ~ 100’s of km
Vertical Scale ~ depth of troposphere
Timescales ~ hours to days
Motions on these scales are directly and strongly influenced
by the Earth’s rotation.
They are in hydrostatic balance.
The vertical component of the velocity ~ 1000 times smaller
than the horizontal component.
This scale of motion is dominant in controlling the transfer of
energy and momentum in the Earth’s atmosphere.
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1. Kinematics of Large-Scale Horizontal Flow
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 everywhere
parallel to the horizontal velocity
vector V.
Isotachs: contours of constant
scaler wind speed V.
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Natural Coordinates
Natural Coordinates: a pair of axes (s, n) where s is the arc length directed
downstream along the local streamline, and n is the distance directed normal
to the streamline and toward the left. The direction of flow is denoted by the
angle ψ, which is defined relative to a reference direction.
At any point in the flow,
the scaler wind speed:
ds
V
dt
dn
0
dt
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Properties of Flows
Shear is the rate of change of
velocity in direction transverse
to the direction of flow.
Curvature is the rate of change
of direction of flow.
dV
Shear  
dn
d
Curvature  V
ds
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.
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Kinematical properties of horizontal flow that can be defined
at any point in the flow (all have units s-1).
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Vorticity and Divergence
Vorticity and divergence are scaler quantities that can be defined
not 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=2ω
where ω is the rate of spin of an imaginary object moving with
the 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 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 the sum
of the Confluence and
Stretching terms
Even 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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Frontal Zones
Deformation can sharpen preexisting
horizontal gradients creating features
known as frontal zones.
Frontal zones can be stationary, but
more often are in motion relative as
as consequence of the flow field.
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Streamlines are horizontal trajectories only if
the flow is steady
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Atmospheric Dynamics:
The Basics
1) Forces:
Pressure gradient
Friction
Gravity
Electromagnetic
- ∂P/∂x
- ν ∂2U∂z2
-ρg
-eExB
2) Reference Frames:
Inertial
Non-Inertial
Newtonian Dynamics
Apparent Forces (Coriolis)
3) Time Tendency:
Fixed observer
Observer “riding” motion
Material Deriviative
Lagrangian Deriviative
4) Conservation Laws:
Mass
Momentum (force equation)
Energy
m
linear + angular
1/2mv2 + ρCpT + ρgz
5) Scaling of the Equations of Motion:
Geostrophic Balance
Turbulence
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Rossby Number
Reynold’s Number
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2. Dynamics of Horizontal Flow
Real Forces: are the
fundamental 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 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 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?
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 rotating
with 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 Pendulum
The Focault Pendulum is an example of
simple harmonic motion in inertial space.
The Focault Pendulum swings back and
forth in inertial space while the Earth
rotates “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 Force
Local Gravity: g = g* + Ω2R
An object on the surface of the Earth will experience an outward
directed force (away from the axis of rotation) due to the Earth’s
rotation of magnitude Ω2R.
An object moving with velocity V in the plane perpendicular to
the axis of rotation experiences an additional apparent force that
is 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 unit
vector in the vertical.
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The 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 Force
The vertical force balance is known as hydrostatic equilibrium:
1 dp
 g
 dz
The vertical force per unit mass:
In general the total pressure force:
P
p

Horizontal components of the pressure gradient force:
1 p
Px  
 x
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1 p
Py  
 y
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The Horizontal Equations of Motion
East-West direction (x, u positive toward East):
du
1 p

 fv  Fx
dt
 x
North-South direction (y, v positive toward North):
dv
1 p

 fu  Fy
dt
 y
where Fx,y is the external force (e.g. friction)
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Geostrophy and the Rossby Number
du
1 p


 fv  Fx
E-W: dt
 x
dv
1 p
 fu  Fy
N-S:  
dt
 y
Outside of the Boundary Layer (above ~1km altitude) the friction
components are insignificant. If the acceleration terms are small,
then those too may also be ignored. To determine if du/dt & dv/dt can be
ignored, compare their magnitude to the Coriolis Force for a typical
velocity (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 of
the relative magnitude of the acceleration to Coriolis terms.
Ro = U/fL
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The Horizontal Equations of Motion
Above ~1km altitude (outside the boundary layer) Fx,y =0:
1 p
 fv
 x
1 p

 fu
 y
Note that this force balance is diagnostic, not prognostic,
i.e., there is no tendency or time evolution. This is called
the Geostrophic Balance. The wind velocity that exactly
solves these equations is known as the Geostrophic Wind.
1 p
ug  
f y
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1 p
vg 
f x
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Geostrophy
Above ~1km altitude (outside the boundary layer) Fx,y =0:
1 p
1 p
 fv

 fu
 x
 y
For Synoptic Scale Motions:
L ~ 1000 KM ~ 106 m
U ~ 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 accuracy
of the Geostrophic Approximation. At mid-latitudes the geostrophic
equations are generally accurate to about 10%.
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Geostrophic Equations of Motion
The smallness of the Rossby number is a measure of the accuracy
of the Geostrophic Approximation. At mid-latitudes the geostrophic
equations are accurate to about 10%
1 p
 fv
 x
1 p

 fu
 y
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 the
time 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 Balance
To first approximation, the horizontal balance of forces in the Earth’s atmosphere
is 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 Flow
Pressure gradients, usually due to temperature differences, cause the air
to flow. Once set in motion, the Coriolis Force deflects the force to the right
in the northern hemisphere and to the left in the southern hemisphere. This
is 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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Friction
The role of friction in the atmosphere is to produce a “drag”
on atmospheric motion. However, the magnitude of the frictional
drag is generally very difficult to quantify. The reason is that the
drag force is due to a variety of physical processes, all of which
transfer momentum between the surface and the free atmosphere.
The Frictional Force per unit mass:
1 
F 
 z
where τ represents the vertical component of the shear stress
(rate of vertical exchange of horizontal momentum) due to the
presence of smaller, unresolved scales of motion. The shear
stress at the surface of the Earth is:
τs = - ρ CD Vs V
CD = drag coefficient, Vs = scaler velocity, V = vector velocity
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Effects of Friction
The Coriolis Force and Pressure Gradient Force are always
perpendicular to the direction of flow. However, the Frictional
Force is always opposite to the direction of flow.
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Effects 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 from
high pressure to low pressure. Thus friction is a primary means
of generating flow which is not parallel to the isobars.
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The Gradient Wind
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:
du
1 p
u2

 fu 
dt
 dy
R
where R is the radius of curvature of the flow streamlines.
For steady flow, du/dt = 0, and we can use the definition of
geostrophic wind to give an equation that can be solved
algebraically for the flow velocity u.
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u2
0  u g  fu 
R
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Horizontal Balanced Flow
For rapidly rotating steady flow:
du
1 p
u2

 fu 
0
dt
 dy
R
where R is the radius of curvature of the flow streamlines.
For rapid rotation the pressure gradient
force is balanced by the centripetal term.
The relative magnitude of the centripetal
to the Coriolis term is:
Rossby No. ~ (U2/R)/fU ~ U/fR
Thus the largeness of the Rossby # is a measure of the accuracy of
the Cyclostropyhic Approximation.
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The Three-Way Balance in the Gradient Wind
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Thermal Wind Equation
A horizontal gradient in temperature
produces a vertical gradient in the
horizontal velocity field.
Hydrostatic equilibrium:
dP = -ρ g dz = - (P/RT) dz
So the difference in altitude between
two 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
Geopotential
The Thermal Wind Equation implies
a diagnostic relationship between the
temperature structure and the wind
structure.
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Vorticity (Angular Momentum) Conservation
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3. Primitive Equations
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Cross-Isobar Flow
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Wave Propagation
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4. Development of the General Circulation
Non-rotating planet:
motion is mainly in
the meridional plane.
Rotating planet:
Motion is highly
3-dimensional
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Steady Circulation
Heating:
Tropical latent heat release
IR heating from ground
Cooling:
Adiabatic expansion
IR cooling
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The Atmospheric General Circulation
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The Atmosphere as a Heat Engine
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Numerical Weather Prediction
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Ensemble Forecasts
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Weather Models
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Surface flow due to the Coriolis Effect
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Vertical Flow Consequences of the Coriolis Effect
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The Hadley Cell
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Jupiter’s Winds
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Jupiter’s Great Red Spot
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Condensation Flows: Saturation Vapor Pressure is
the Driving Force
Sulfur Dioxide flows from
the day side to night side.
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Seasonal sublimation of the polar
caps produce pressure gradients
and global scale winds.
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Questions for Discussion
(1) Is Geostrophic Flow a universal property of a planetary
atmosphere?
(2) How large can atmospheric cyclones (e.g. hurricanes)
become in a planetary atmosphere?
(3) In what way would the general circulation be different
without surface friction?
(4) If the Coriolis Force does no work on a parcel of air, then
how does the air accelerate?
(5) Which is more important for driving atmospheric winds:
Solar heating (direct and latent) and Earth’s rotation?
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Rotation Experiment
In a rotating reference frame, the only force is the Coriolis Force
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“Rotating”
Trajectories
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