Transcript Law of Motion (Physics)

Law of Motion (Physics):
F
=
m
Sum of all forces
=
time-rate of change of (mass*velocity)
� if
*
a
measured in an appropriate frame of reference
(one that does NOT rotate)
� applies
to particles (Lagrangian view)
Need to address both, but first,what forces act on the fluid?
Forces acting on geophysical fluid
• Gravity
• Pressure gradient force
• Friction (dissipation) (viscous force)
Towards center of Earth
Forces acting on geophysical fluid
• Gravity
• Pressure gradient force
• Friction (dissipation) (viscous force)
Example of
pressure gradients
daily surface
pressure map
(in the atmosphere, we
can simply measure the
pressure at the surface)
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Pressure gradient force in ocean?
Gulf Stream example
100 kilometers width
1 meter height
• Small deviations of sea surface drive all flows usually much less than 1 meter height. Pressure
gradient is very difficult to measure directly.
1 meter height
Equivalent to pressure of
1 dbar, since water
density is ~ 1000 kg/m3
(rest of water column: 5 km deep)
LOW
HIGH
Pressure gradient is directed from low to high. Calculate size.
Pressure gradient force is directed from high to low (water
pushed towards lower pressure).
Compute acceleration due to PGF
• Take the example of the Gulf Stream and compute
the velocity after 1 year of acceleration.
• You’ll find it’s ridiculously large (compared with
the observed 1 m/sec). Why can such a large
pressure gradient be maintained without large
velocities? (Earth’s rotation - Coriolis to be
discussed later)
Friction, Viscosity, and Turbulence
Turbulent: acceleration >> friction
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Laminar: acceleration ~ friction
An unsolved problem of physics
On his death-bed Physics Nobel Prize winner Werner Heisenberg is reported to have said:
"When I meet God, I am going to ask him two questions: Why relativity? And why turbulence?
I really believe he will have an answer for the first."
http://www.eng.auburn.edu/users/thurobs/Turb.html
x
Shear stress  is the force F
applied tangentially to a surface A
(think winds over the ocean),
e.g., =F/A
z
u
For water and air it is found experimentally that
the stress  is proportional to the strain u/z
(change of velocity in the direction of F perpendicular to A)
 ~ u/z
How does this accelerate the flow?
Acceleration occurs if
the stress at the upper surface
differs from
the stress at the lower surface
Eddy diffusivity and eddy viscosity
• Molecular viscosity and diffusivity are extremely small
• We know from observations that the ocean behaves as if
diffusivity and viscosity are much larger than molecular
• The ocean has lots of turbulent motion (like any fluid)
• Turbulence acts on larger scales of motion like a viscosity think of each random eddy or packet of waves acting like a
randomly moving molecule carrying its property/mean
velocity/information
Eddy diffusivity
and viscosity
Gulf Stream (top) and
Kuroshio (bottom):
meanders and makes rings
(closed eddies) that
transport properties to a new
location
Eddy diffusivity and viscosity
Example of surface drifter tracks: dominated to the eye by variability
(they can be averaged to make a very useful mean circulation)
Law of Motion (Physics):
F
=
m
Sum of all forces
=
time-rate of change of (mass*velocity)
� if
*
a
measured in an appropriate frame of reference
(one that does NOT rotate)
� applies
to particles (Lagrangian view)
What about flow fields where velocity does not
depend on tracking particles all the time?
Lagrangian View:
Observations
Salinity
Fresh riverine water turns right at the coast.
From Muenchow (1992)
Eulerian View
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Model Predictions of
Winds (pink)
Ocean Currents (black)
Surface Salinity (color)
Fresh riverine waters turn right at the coast.
From John Wilkin, Rutgers U.
Advection of velocity --> field accelerations
Particle acceleration Du/Dt
Equals
Local plus field acceleration:
Du/Dt = u/t + (u u/x + v u/y + w u/z)
local + field acceleration
zero local acceleration,
but
particles change velocity
(passing through above field)
Advection (Eulerian perspective)
•
Move “stuff” - temperature, salinity, oxygen, velocity, etc.
•
By moving stuff, we might change the value of the stuff at the next location. We only change
the value if there is a gradient of the stuff (difference from one point to the next)
•
Proportional to velocity
•
Proportional to gradient in the same direction as the velocity, e.g.,
v T/y is the advection of temperature in the y-direction
gradient
Low temperature or velocity
High temperature or velocity
strong flow
weak flow
 to gradient
 to gradient || to gradient
Advection: --> large
--> small
strong flow
--> zero
Law of Motion (Physics):
F
=
m
Sum of all forces
=
time-rate of change of (mass*velocity)
� if
*
a
measured in an appropriate frame of reference
(one that does NOT rotate)
Lets walk the plank …
… Coriolis and Centrifugal accelerations
… ficticious forces in a rotating reference (earth)
Movie time:
1. Merry-go-round (fast and slow)
acceleration = coriolis+centrifugal
2. Idealized motion of merry-go-round
acceleration = coriolis+centrifugal
3. Coriolis simulation without friction
acceleration = coriolis
4. Coriolis simulation with friction
acceleration=coriolis+friction
Rotating coordinates
•
The Earth is rotating. We measure things relative to this “rotating reference frame”.
•
Quantity that tells how fast something is rotating:
Angular speed or angular velocity  = angle/second
360° is the whole circle, but express angle in radians (2 radians = 360°)
For Earth: 2 / 1 day = 2 / 86,400 sec = 0.707 x 10-4 /sec
Also can show  = v/R where v is the measured velocity and R is the radius to the axis
of rotation (therefore v =  R)
R
Rotating coordinates
• Vector that expresses direction of rotation and how
fast it is rotating:
vector pointing in direction of thumb using right-hand rule, curling fingers in
direction of rotation
R
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Centripetal and Centrifugal forces
(now looking straight down on the rotating plane)
Centripetal force is the actual
force that keeps the ball
“tethered” (here it is the
string, but it can be
gravitational force)
Centrifugal force is the pseudoforce (apparent force) that one
feels due to lack of awareness
that the coordinate system is
rotating or curving
centrifugal acceleration = 2R
Effect of centrifugal force on
earth and ocean
Radius:
Equatorial 6,378.135 km
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Polar 6,356.750 km
Mean 6,372.795 km
(From wikipedia entry on Earth)
The ocean is not 20 km deeper at the equator, rather the
earth itself is deformed. We bury the centrifugal force term
in the gravity term (which we call “reduced gravity”), and
ignore it henceforth.
Coriolis effect
Inertial motion:
motion in a straight line
relative to the fixed stars
Coriolis effect:
apparent deflection of that
inertially moving body just
due to the rotation of you,
the observer.
Coriolis effect deflects
bodies (water parcels, air
parcels) to the right in the
northern hemisphere and
to the left in the southern
hemisphere
Coriolis force
Additional terms in momentum
equations, at latitude 
x-momentum equation:
-sinv
= -f v
y-momentum equation:
sinu
= fu
f is the “Coriolis parameter”. It
depends on latitude.
Movie time:
1. Merry-go-round (fast and slow)
acceleration = coriolis+centrifugal
2. Idealized motion of merry-go-round
acceleration = coriolis+centrifugal
3. Coriolis simulation without friction
acceleration = coriolis
4. Coriolis simulation with friction
acceleration=coriolis+friction
Complete force balance with rotation
Three equations:
Horizontal (x) (west-east)
acceleration + advection + Coriolis =
pressure gradient force + viscous term
Horizontal (y) (south-north)
acceleration + advection + Coriolis =
pressure gradient force + viscous term
Vertical (z) (down-up)
acceleration +advection (+ neglected very small Coriolis) =
pressure gradient force + effective gravity
(including centrifugal force) + viscous term
Momentum Equation in Cartesian co-ordinates (x,y,z)
X:
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Y:
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Z:
Acceleration
= p-gradient +
friction + gravity
local acceleration
field acceleration (or advection or nonlinear advection)
Momentum Equations in Spherical co-ordinates
Radial:
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Latitude:
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Longitude:
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Acceleration = pressure gradient + friction + gravity