Transcript Review of thermo and dynamics, Part 3 (pptx)

A&OS C110/C227: Review of
thermodynamics and dynamics III
Robert Fovell
UCLA Atmospheric and Oceanic Sciences
[email protected]
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Notes
• Everything in this presentation should be familiar
• Please feel free to ask questions, and remember to refer to
slide numbers if/when possible
• If you have Facebook, please look for the group
“UCLA_Synoptic”. You need my permission to join. (There are
two “Robert Fovell” pages on FB. One is NOT me, even
though my picture is being used.)
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Hydrostatic and hypsometric equations
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Hydrostatic equation
g = 9.81 m s-2 at sea-level
• Represents balance (stalemate) between vertical pressure
gradient force and gravity resulting (formally) in no vertical
acceleration and (practically) in no vertical motion
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Hypsometric equation
• Derived from the hydrostatic equation, the hypsometric
(Greek, to measure height) or thickness equation is
• where Tv is the layer mean virtual temperature of the p0-p1 layer,
p0 > p1, and ∆Z is the thickness of that layer
• Based on the idea that between two pressure levels, there is a
fixed amount of mass and that mass occupies a greater depth
as it becomes warmer and/or more moist
• ∆Z is really measured in “geopotential meters”, a gravityadjusted height, but its units evaluate as meters
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How temperature influences 1000-500 mb thickness
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How temperature influences 1000-500 mb thickness
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Temperature differences make pressure differences,
which drive winds
Note the 500 mb surface tilts downward towards the colder air.
A pressure difference at constant height is directly relatable to
a height difference on a constant pressure (isobaric) surface
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Equations of motion
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Forces: real and imaginary
• Real forces: PGF, gravity, friction
• Imaginary forces: Coriolis, centrifugal, curvature
• Vectors will be expressed with bold type or overlying
arrows
• Centrifugal force owing to Earth rotation is W2R, where W
is the Earth’s angular velocity (2p radians per day) and R
is the distance vector to the Earth’s axis of rotation
• Apparent gravity g = true gravity force (g*) + centrifugal
force W2R (see next slide)
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Apparent gravity
• Picture a perfect sphere, in
quarter section
• Note that true gravity g* is
directed towards Earth’s
center
• Note centrifugal force W2R is
directed perpendicular to
rotation axis
• As a consequence, apparent
gravity g does not point
towards center of a perfect
sphere
• Is it true that objects on the
rotating Earth, influenced by
apparent gravity, do not fall
straight down?
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Inertial and rotating frames
3D vector velocity relative to the rotating Earth
3D absolute velocity in the inertial reference frame
Total derivative with respect to time in rotating Earth frame
Absolute total derivative with respect to the in inertial frame
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We will soon see that…
-1 x Coriolis
Newton’s laws
determine this
-1 x centrifugal
Due to Earth’s rotation
What we need
to make forecasts
Start with relationship for vector A
between absolute motion and
Earth-relative motion
the “tool"
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Position vector
• Let r be the position
vector of an object,
extending from Earth
center
• Object motion
represents
• (1) that due to Earth
rotation
• (2) that due to motion
relative to Earth
• Use the “tool” with r
as A
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(*)
• Use the “tool” on Ua:
• After utilizing (*) above and skipping steps, we find the expected equation
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PGF
true gravity
friction
• Plug the above into (*), solve for dU/dt, and combine true gravity and
centrifugal into apparent gravity g:
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Pressure gradient force
Coriolis force
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• Earth rotation gave rise to the
Coriolis terms
• Earth’s sphericity give rise to the
curvature terms
Acceleration relative to flat Earth + acceleration due to curvature
r=a+z
a = Earth radius
z = height above
surface
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Flat Earth
Curvature terms
Scale analysis of the equations of motion
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Synoptic-scale scale analysis
Term
value
description
W
10-4 s-1
Earth angular velocity
a
10000 km
Earth radius
f
45˚N
midlatitudes
U, V
10 m/s
Horizontal velocity
W
1 cm/s
Vertical velocity
L
106 m
Horizontal length
scale
H
104 m
Vertical length scale
r
1 kg m-3
Sea-level air density
∆p
10 mb = 1000 Pa
Horizontal pressure
difference
∆t
100000 s
Time scale = L/U
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Horizontal equations’ 3 leading
terms on synoptic scale
10-4 s-1
10-3 s-1
10-3 s-1
• You can make a horizontal time scale with L/U,
so dU/dt = U/(L/U) = U2/L = (10)2 (m/s)2/(10000 m) =10-4 s-1
• Break into x- and y-components, as below:
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Horizontal equations’ 2 leading
terms = geostrophic balance
Geostrophic = Greek, Earth turns
where
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Note Coriolis is always pared with u, v;
it is proportional to wind speed
Horizontal equations’ 3 leading
terms, rewritten
• Deviations from geostrophic balance cause accelerations
• Accelerations serve to bring atmosphere back towards geostrophy
• Note accelerations are an order of magnitude smaller than winds
(see two slides back)
• CAUTION: only applies if our synoptic scale analysis is valid
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Vertical equation of motion
(part)
10-7 s-1
10 s-1
10 s-1
• On the synoptic scale, atmosphere is very nearly in hydrostatic balance
• For synoptic scale motions, we cannot use departures from hydrostatic
balance to compute vertical accelerations or vertical velocity
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How large-scale flow becomes geostrophic
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• Start with an air parcel subjected to a PGF
• The parcel starts to move towards L pressure
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• If the time scale of the motion is sufficiently large,
Coriolis accelerations become important
• Coriolis always acts to the right following the motion,
in the Northern Hemisphere
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• This combination of forces causes the parcel to start
deflecting away from low pressure
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• Coriolis again acts to the right. Notice the change in
the angle between the two force vectors
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• As a consequence, the air parcel is turning even more
away from L pressure
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• You can see this process will continue until the parcel
is traveling parallel to the isobars
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• At that point, PGF and Coriolis are in opposition,
and directional change ceases
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• This adjustment process has brought the wind into
geostrophic balance
• Typically, we see the already adjusted wind, but this
process can be detected in diurnally-driven circulations,
such as the sea-breeze
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Isobar spacing represents PGF
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Isobar spacing represents PGF
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Gradient, cyclostrophic and inertial winds,
and friction
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Natural coordinates
• A coordinate system that
“goes with the flow”,
parallel to isobars
• Unit vector l points in
direction of flow
• Unit vector n is
perpendicular to the
flow, positive to the left
• Wind speed V ≥ 0
• R = radius of curvature
• R > 0 CCW
• R < 0 CW
• R = ∞ when isobars
straight
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Flow equation
Centripetal
A
Coriolis
B
PGF
C
Since the geostrophic wind can be defined as:
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• when R = ∞ (straight-like flow) V = Vg
Combinations
Centripetal
A
Coriolis
B
PGF
C
Combinations
B, C = geostrophic wind (“Earth turns”)
A, B, C = gradient wind
A, C = cyclostrophic wind (“to turn in a circle”)
A, B = inertial flow
Alternate version:
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Gradient vs. geostrophic wind
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• Cyclonic flow: R > 0 so gradient wind is subgeostrophic for same PGF
• Anticyclonic flow: R < 0 so gradient wind is supergeostrophic for same PGF
A closer look…
• Consider cyclonic
(CCW) flow
• By itself, inertia takes
a parcel on a straightline path, at constant
speed
• However, this path
would cross isobars,
towards higher
pressure
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• Start with geostrophic balance, straight-line flow
parallel to isobars
• PGF is pointing to the left of the wind, towards L
pressure. Coriolis force (CF) points to the right,
following the motion, and opposite to PGF
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• Now make the isobars curve
• Inertia is carrying our air parcel towards the curving
isobars
• As it approaches, what happens to the PGF and
Coriolis forces, and why?
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• PGF always points directly towards lower pressure
• Note a component of the PGF is now pointing
against the parcel’s motion… which slows it down
• The Coriolis force is reduced, as it is proportional to
wind speed
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• PGF gains the “upper hand” over Coriolis
• As a consequence, the parcel eases into a CCW turn,
with Coriolis again to the right of the motion AND
opposing the PGF
• To accomplish this CCW turn, the parcel must slow down
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Centripetal
A
Coriolis
B
PGF
C
• In this point of view, the centripetal force is the
force imbalance that develops between PGF and CF
owing to isobar curvature
• Centripetal force does not exist until something has a
spinning/curving motion. This explanation offers
why the curving motion emerged
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Cyclostrophic flow
• For small-scale flow, Coriolis is negligible and
curving flow represents PGF and centripetal forces
• Solving for V yields two roots, but V ≥ 0
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Cyclostrophic: by itself, spin
makes low pressure
L
L
• V ≥ 0 always, n positive to left of flow
• Radicand must be positive
• Thus, R and dp/dn must have opposite signs
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How friction affects the large-scale wind
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• Start with geostrophic balance
• How would friction alter this balance?
• Friction slows the wind, thereby reducing the Coriolis
force
• However, friction has no immediate effect on the PGF
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• As a consequence, the wind slows and turns towards
lower pressure
• Note the Coriolis force is acting to the right of the wind,
and not opposite the PGF
• Now there is a wind component from H to L
• Because of surface friction, air can flow out of surface
anticyclones, and into surface cyclones
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• Gradient wind flow around synoptic-scale L and H
• Isobar spacing is the same, so the CCW flow around the
L is somewhat slower than CW flow around H
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• Flow around synoptic-scale L and H after inclusion of
surface friction
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Friction acts…
• …near the surface, generally in lowest 1 km or so, but varies
with space and time
• On hot days, the boundary layer grows vertically, mixing more
air downward to the surface
• In very stable, nocturnal boundary layers, winds closer to the
surface are far less affected by friction
• When plotting isobaric charts, keep in mind how close a given
pressure level may be from the ground!
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Question for thought
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Picture 1
• For equal isobar spacing, flow around cyclones is slower
than flow around anticyclones
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Picture 2
• In reality, PGFs and flows around cyclones are typically
much stronger than around anticyclones. Why?
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[end]
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