Coordinate Systems, The Hypsometric Equation, and Thickness

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Chapter 8

• Coordinate Systems, The Hypsometric Equation, and
  Thickness

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(1) Conventional Coordinate Systems

• Cartesian Three axes perpendicular to each
  other.

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Plane polar coordinates are a natural way to
describe circular motion.

• Instead of describing the position of an object
  using x and y coordinates, this system uses two
  coordinates that locate the object's position
  using its distance from the origin (called "r"
  for radius) and its angle (theta) from the
  positive x-axis .
• Angles measured counterclockwise from the x-axis
  are defined to be positive angles, and angles
  measured clockwise from the x-axis are negative
  angles. The distance from the origin is ALWAYS
  positive.

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Spherical Coordinate System.

• The angle (?) defines the elevation above (or
  below) the plane of (T).
• Latitude and longitude are examples of a
  spherical coordinate system.
• r is the distance from the center of the Earth.

• T is the longitude with the Greenwich Meridian
  as the starting point.
• ? is the latitude with the equator as the
  starting point.
• Up, where the elevation angle/latitude 90o
  is the north pole.
• Looking down from the up position, longitude
  increases in a counterclockwise direction.

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Use of spherical coordinates with radar

• The elevation angle, ?, is zero along a line
  parallel to the horizontal.
• The azimuth angle, ?, begins with zero at north,
  and moves clockwise just as the azimuth angle for
  wind.
• r is the radial distance from the radar location.

• Up refers to moving outward directly from the
  center of the Earth.
• Looking down from the up direction, ? moves in
  a clockwise direction just as wind.

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• Because the Earth rotates, an object moving in a
  rotating coordinate system with the Earth
  experiences an apparent force outward from the
  axis of rotation of the Earth, the centrifugal
  force, FCN ?2R.
• That object also experiences gravity trying to
  move it toward the center of the Earth.

The resultant of these two forces can be
expressed as a single force, FEG, effective
gravity.
Since the centrifugal force is small compared to
gravity, the direction of the effective gravity
force is close to, but not exactly in the same
direction as gravity.
We, air molecules, etc. experience this and call
it the force of gravity, except it is really the
result of true gravity and the Earths motion.
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• An object moving around and experiencing the same
  effective gravitational force on it is said to
  be moving on a surface of constant geopotential.
  A surface of constant geopotential is a surface
  along which a parcel of air could move without
  undergoing any changes in its potential energy.
  Also known as equigeopotential surface or a level
  surface.

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• Because the Earth rotates, it bulges at the
  equator.
• For large scale motions, the spherical coordinate
  system works well for identifying the location of
  air parcels. (We are ignoring the bulge).
• For small scale motions, the Cartesian coordinate
  system works well enough. (We are ignoring that
  the earth isnt flat).

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• Pressure coordinate. In the atmosphere in the
  vertical, pressure is easier to measure than
  height. Often, the location of a parcel of air,
  or other characteristics of the air, is expressed
  in terms of the pressure where it is located
  rather than the height above some surface (as sea
  level).
• Isentropic coordinate. An isentrope is a surface
  along which the potential temperature is
  constant. Parcels remain on isentropic surfaces
  as long as no diabatic processes (e.g. latent
  heat release due to condensation) are occuring.
  Also, other conserved properties of the parcel
  remain intact (like mixing ratio).

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(3) Hydrostatic equation

• The hydrostatic equation represents a balance of
  the vertical pressure gradient force (acting
  upward) and the gravitational force (effective
  gravitational force) acting down.
• When these two forces are not in balance, there
  is an acceleration of a parcel of air either
  upward or downward.
• This is expressed by the equation below from
  chapter 6.

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• When they are in balance, the right side of the
  equation is zero and we have the hydrostatic
  equation.
• Using the Ideal Gas Law Equation to get an
  expression for density
• 
  (as done in chapter 5) and substitute
  that in for density gives.

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• Divide both sides by gives.
• This can be written as

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(4) Hypsometric equation

• Integrate from some low level to some high level.
• Multiply both sides by -1, reverse the limits on
  the left (gets rid of the - sign on the left) and
  perform the integration on the right. Gives

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• Graphically, integration can be expressed as
  computing the area under a curve.
• For the integration of the term on the right, the
  graph would look like the following.
• The integrand is constant and equal to 1 along
  the distance (z2-z1).

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• For the left side of the equation, there is the
  temperature term, and temperature changes as we
  move vertically in the atmosphere. However, we
  can simplify the equation by using the average
  value of temperature between the levels
  represented by ln(P1) and ln(P2).

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• Then, we can write the equation as
• The quantity is just a constant by
  which we are multiplying the integrand between
  the limits ln(P1) and ln(P2).
• Integrating the left side between those limits
  gives

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• Graphically, it would look like

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• Geopotential height.
• Remember, a surface of constant geopotential is a
  surface along which a parcel of air could move
  without undergoing any changes in its potential
  energy
• The geopotential height is the height at which an
  object (such as a volume of air) would have the
  same geopotential (assuming constant value for
  gravity) as it would in the real atmosphere
  (where gravity changes).
• It is expressed as

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• Virtual Temperature
• The real atmosphere has water vapor in it, which
  varies. Therefore, Rd is not a good value to use
  for the gas constant to express how the volume is
  going to behave. An R value should be used that
  expresses how each particular volume would behave
  based on the particular molecules (including
  water vapor of varying amounts) it has in it.
• However, that means calculating a new R value for
  every different volume of air, or coming up with
  an equation that expresses how R changes with
  different types of air.

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• We can get around that problem with water vapor
  and still use Rd by using virtual temperature.
• Virtual temperature is defined as the temperature
  a dry parcel of air would have if it had the same
  density and pressure as a moist parcel of air.
• Virtual temperature is always greater than actual
  temperature of the parcel because having moisture
  in the parcel makes it less dense.
• Thus, virtual temperature accounts for the effect
  of having water vapor in a volume of air.

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• Virtual temperature is given by
• Where w is the mixing ratio of the air.
• Sometimes it is expressed as g/kg.
• Thus, by using virtual temperature, we can use Rd
  and still account for the effects of water vapor
  being in the volume, as long as that water vapor
  amount doesnt change.

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• So, now we can use gravity as a constant as long
  as we realize we are dealing with geopotential
  heights (Z), not true height(z) - as measured
  with a ruler.
• And, we can use Rd as long as we use Tv, virtual
  temperature, to account for moisture.

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Hypsometric equation

• This integrated form of the hydrostatic equation
  shows that the vertical distance between two
  pressure surfaces in the atmosphere is
  proportional to the average temperature between
  those two surfaces.

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• By knowing pressure and temperature, the height
  between pressure surfaces can be calculated (with
  only a small error if water vapor is ignored).
• By also knowing the water vapor amount (which can
  be obtained from knowing the dew point) a more
  accurate determination of the thickness between
  pressure surface can be made.

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• By knowing the pressure at the ground, and
  temperatures up through the atmosphere, one can
  work their way up through the atmosphere
  determining the thickness of each layer between
  pressure surfaces and thus the height of pressure
  surfaces.
• Its easy to see that where temperatures are
  cold, pressure surfaces are going to be at a
  lower height in the atmosphere. And, where
  temperatures are warm, pressure surfaces will be
  at a higher height in the atmosphere.

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(8) Critical thickness

• The thickness of a layer is used for forecasting
  (e.g., snow) because it gives a good indicator of
  the average temperature of the layer.
• The smaller the thickness between the layers, the
  colder the air.
• Some use 1000-500 mb thickness.
• Some use 850-700 mb thickness.
• Some use both.
• The critical thickness varies with the elevation
  of the station and the stability of the air mass.

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• Large changes in the thickness across a short
  distance (thickness gradient) can also give an
  indication of the strength of a front.
• So, on a thickness chart, thickness contours
  close together indicate a rapid change in the
  average temperature of the air perpendicular to
  those thickness contours.

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Homework

• Do 1, 3, 4, 5, 6, 7