Space, Time, and Motion

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Chapter 3

• Space, Time, and Motion

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(1) Wind Observations

• Vectors have both magnitude and direction.
• Wind is a vector quantity.
• The components of wind can be expressed in the
  Cartesian coordinates x, y, z.

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• Wind Instruments
• Anemometer and wind vane
• Aerovane
• Sonic anemometer

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• Anemometer and wind vane.

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• Aerovane

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• Sonic Anemometers

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(2) Plotting Data
Each element has a standard location and format.
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o
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O
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(3) Derivatives in Time and Space

• If a graph is drawn with the values of a quantity
  on the vertical and time on the horizontal, then
  drawing a line tangent to the graph line and
  determining its value (slope) will determine the
  derivative of that quantity with respect to time.
  
• The tangent technique measures the derivative -
  how the quantity changes with respect to time.

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• Similarly, if the quantity was drawn on a graph
  with a spatial direction e.g., the x-, y-, or
  z-direction. Then, the slope of the tangent to
  the line at the location of interest determines
  the derivative of the quantity with respect to
  that direction.

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• Actually, these derivatives are estimations since
  it is very difficult to draw an accurate line
  tangent to the parameter line, and we are looking
  at the change over a rather large time interval
  or large x-direction interval.
• Calculus considers the denominator value as
  approaching zero.
  
• Secondly, these derivatives should be considered
  partial derivatives because temperature changes
  in all three directions as well as time.
  
• If we are considering only the change in the
  x-direction, we are assuming that there are no
  changes in the other directions or time.

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(4) Advection

• Advection (in meteorology) is the rate of change
  of some property of the atmosphere by the
  horizontal movement of air.
  
• The rate of change is a derivative (with respect
  to time).

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• If at 200 pm the carbon monoxide concentration
  was 80 ppb at a location 30 nm upstream from
  Savannah, Georgia, and the wind were blowing at
  15 knots toward Savannah, when would the
  concentration at Savannah reach 80 ppb with no
  sources or sinks.

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• The 80 ppb air must travel 30 nm and it is moving
  at 15 knots. Divide 30 nm by 15 knots (nm/hr)
  and you get a travel time of 2 hours. 200 pm
  2 hours 400 pm.

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• Suppose the concentration at Savannah at 200 pm
  were 60 ppb. How rapidly will the carbon
  monoxide concentration change.
  
• The rate of change (change in
  concentration)/(change in time)

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• The change in concentration of carbon monoxide
  can be written as
  
• The subscript is there to indicate we are only
  considering the change at Savannah which is not
  moving. The x-direction is toward Savannah.

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• Thus, the rate of change
• The wind speed (magnitude of the horizontal wind
  velocity vector in the direction of interest) can
  be written as
  
• Then,

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• We can then write the rate of change as
• We can get the rate of change with time
  (derivative with respect to time) from the rate
  of change with distance (derivative with respect
  to distance) if we know the velocity of the
  wind.
• Essentially, it is simply

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(5) The One-Dimensional Vector Equation

• If we consider the change in time and the change
  in x as approaching zero, we have the
  instantaneous rate of change at a point (e.g.,
  Savannah).
• Writing in calculus form (partial differential
  equation since we are only considering the change
  in the direction of the wind field (our
  x-direction), we have

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• This is a general advection equation (along the
  x-coordinate - the west to east direction). (We
  are using u the component of the wind along the
  x-coordinate
• One can write such an equation for the advection
  along the y-coordinate, or z-coordinate.

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Consider the following analysis of CO.
To get the rate of change of CO (partial
derivative of CO with respect to time) at
Savannah, we need the partial derivative of CO
with respect to x and the average wind speed.
Notice that concentrations at Savannah are less
than they are upwind, so the rate of change of
CO over time should be positive.
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Just as on a graph, to get the change in
concentration of CO, pick two points on either
side of the point of interest and get the
difference between those values. In this case,
50-70 -20.
Now, divide by the distance between those points,
30nm. This will be the slope of the graph line
at Savannah which will be the rate of change of
CO with respect to distance at Savannah.
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• The wind speed is everywhere 15 knots (nm/hr) so
  the average wind speed is 15 nm/hr.
  
• Then, the rate of change of CO with time is
• Similarly, the equation can be set up for any
  spatially-varying atmospheric variable such as,
  temperature

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(6) Equations on the Brain

• To understand the equation and how it relates to
  the atmosphere
  
• Say it in words.
• See if it makes sense if each variable or
  derivative, in turn, is zero.
  
• See if the signs make sense.
• See it it makes sense if certain variables get
  larger or smaller.
  
• Make up a concrete example and work through it.

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• For the advection equation.
• The equations states that the rate of change of
  temperature with respect to time at a particular
  location is equal to the negative of the wind
  speed times the rate of variation of temperature
  in the direction toward which the wind is blowing.

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• For various terms set to zero.
• If wind speed is zero, no air is being
  transported, so the wind is not changing the
  temperature, so the advection is zero.
  
• If is zero, then the temperature
• is uniform along the x-axis, so the
• air blowing in is the same temperature as the
  air blowing out so advection is zero.

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• For this situation, remember, the equation
  relates to advection at that instant, at a
  particular location. Here the advection is zero
  because the change in temperature at that instant
  is zero.

You would have to average u and ?T/?x over a much
larger distance to get a non-zero value, not just
locally.
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• Does the sign make sense?
• If this were analyzed temperature, orient x to
  point
  
• toward where the wind is
• blowing, then u will always be positive.
• The sign of the temperature

change with time then will be determined by the
sign of the downwind variation with temperature.

• If ?T/?x is positive, (T down - T up), then if
  you graphed T versus x, the slope would be
  positive.
  
• Warmer temperatures would be downwind.

If ?T/ ?x is negative, then a slope of T vs x
would have colder temperatures downwind (slope
would be negative) as we have.
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• Check the magnitude.
• Suppose the average wind is 5 m/s toward east and
  temperatures are colder upstream, warmer
  downstream.
  
• Then, if the wind were stronger, the change would
  occur faster and temperatures would drop faster -
  greater change in temperature with time - greater
  advection.
• If the temperature change with distance is small,
  the temperature change over time would be small.

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• Check with numbers.
• Assume a wind speed of 5 m/s and temperatures are
  colder by 5oK over a distance of 100km upstream.
  
• Then,with a wind speed of 5 m/s, how long will it
  take the colder air to travel 100km?
  
• So, temperature should drop at a rate of 5oK
  every 2 x 104 seconds. This is
  

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• Check equation with numbers.
• Assume a wind speed of 5 m/s and temperatures are
  colder by 5oK over a distance of 100km upstream.
  
• Then,
• and

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(6) Space-Time Conversion

• In a constant negative wind field (wind blowing
  from the east toward the west at the same speed
  all about the point of interest, the advection
  equation becomes
• Thus, a graph of temperature vs. time would have
  the same shape as a graph of temperature vs.
  location (x-coordinate).
  
• The only difference would be the scale determined
  by the constant wind speed.
  
• And, whatever changes in temperature occur at a
  given location (?T/?t) must correspond to
  variations in the upstream temperature pattern
  (?T/?x).

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• The differences in the horizontal scale is simply
  due to the speed of the wind.

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(8) Phase Speed

• The examples have been using wind as the cause of
  the advection, but the same concept can be
  applied to anything that is moving regardless of
  the cause.
• As long as you can tell how fast it is moving
  (e.g., a cold front), the speed can be used in
  the advection equation to determine the change
  over time.
• The time record of observed meteorological
  variables at a particular space (location) can be
  converted directly into a depiction of the
  horizontal structure of the phenomena.

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• The front is moving at 20 knots and it passed
  North Platte two hours ago, so it should be 40
  miles past North Platte, at location B.
  
• If temperature at North Platte two hours ago was
  52oF and it is moving toward Dodge City at 20
  knots, the 52oF air should arrive in Dodge City
  in 5 hours after being in North Platte, or in
  another 3 hours.

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Questions

• Do 1, 2, 3, 4, 5