Bits of Vector Calculus

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Chapter 7

• Bits of Vector Calculus

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(1) Vector Magnitude and Direction

• Consider the vector to the right.
• We could determine the magnitude by determining
  the x-component and the y-component.
• Then use the equation

However, is we have a scale that shows what a
certain length means, why dont we just measure
it?
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• If the grid length shown represents 1m/s for each
  interval (0.5 inches), then the length of the
  vector (1.6 inches) represents 3.2 m/s.
• In other words, if
• 0.5 inches (a grid length) 1 m/s, then,

all we need to do is change units
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• Now consider the wind direction.
• First, you need to know which direction is
  represented by north.
• Cartesian coordinates usually are oriented with
  east to the right and north toward the top of the
  map, or along the meridian line on the map.

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• We could measure the angle with a protractor,
  remembering the wind direction is the direction
  from which the wind is blowing.
• Or, we could use the x- and y-components of the
  wind vector.

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• Wind components are usually expressed in terms of
  u, v, and w.
• Remember, that there is a difference between the
  mathematical representation of degrees of an
  angle and the azimuth degrees of a wind
  direction.

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• For our wind vector, measured with a protractor,
  the direction is about 238o.
• The components are as shown. However, this shows
  b as the u-component of the wind vector, and
  a as the y-component of the wind vector.

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• Measuring the lengths, the u-component, b, is
  2.8 grid boxes (each 0.5 inches long) for a total
  length of 1.4 inches at 1 m/s for each 0.5
  inches, or 2.8 m/s.
• The component, a, is 1.8 grid boxes long at 1
  m/s for each grid box (0.5 inches) or 1.8 m/s.

Appears to be an error on page 5. Wind speeds
are 2.8m/s and 1.8 m/s, not 1.4 and 0.9m/s
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• Determining the arctan (the angle whose
  tangent is )gives the angle from vector a to
  vector c.
• We need to add 180o to the the wind direction
  (from which the wind is blowing).

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(2) Vector Addition and Unit Vectors

• Addition
• Start the second vector at the end of the first
  vector.
• Draw a vector from the start of the first vector
  to the end of the second vector. This is the sum
  of the first two vectors.

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(3) Vector Multiplication and Components

• Unit vectors
• Vectors of 1 unit length (for whatever units you
  are using).
• The i-unit vector points in the positive
  x-direction.
• The j-unit vector points in the positive
  y-direction.
• The k-unit vector points in the positive
  z-direction (upward).

When multiplying a vector by a number, you are
simply multiplying the magnitude of the vector by
that number. The vector still points in the same
direction, unless the number is negative, in
which case the vector points in the opposite
direction as the unit vector.
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• Using unit vectors, vectors can easily be
  expressed in their components.
• Vector b could be written as 2.8 m/s i.
• Vector a could be written as 1.8 m/s j.
• Vector c could then be written as

C 2.8 m/s i 1.8 m/s j
For wind, these components have been given
special names. We say u 2.8 m/s, and v 1.8
m/s
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• Writing vectors by their components makes it much
  easier to add and subtract vectors simply add or
  subtract the i-components, then the j-components,
  then the k-components.

Vector 1 2.5 m/s i 3.5 m/s j Vector 2 1.5
m/s i - 1 m/s j
Vector 1 Vector 2 4.0 m/s i 2.5 m/s j
The vertical component, k, could also be included
to represent the air motion.
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• Vector subtraction. - Subtract b from a.
• Graphically, Method 1 add the negative of b to
  a.
• Draw a vector from the start of a to the end of
  b. This is the resultant vector.

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• Method 2 -
• Start vector b at the start of vector a.
• Then draw the resultant vector from the end of
  b to the end of a.

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• Vector subtraction plays an important roll in
  determining vertical wind shear - the difference
  in the wind at one level compared to the wind at
  another level.

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(5) Dot Product

• Assume that the u-component and v-component of
  vector a is
• ua and va.
• And for vector b they are
• ub and vb.
• Then the dot product a b is a scalar
  quantity equal to uaub vavb
• Expressed as the magnitudes of a and b and
  the vectors directions, then the dot product is
  ?a??b?cos(angle between them).

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• The dot product is a measure of the magnitude of
  two vectors and the smallness of the angle
  between them. The smallness is measured by the
  cosine of the angle. If 90 degrees to each
  other, the dot product is zero.
• The dot product can be thought of as the
  projection of one vector onto another multiplied
  by the magnitude of that second vector.

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• If one of the vectors is a unit vector i, j, or
  k, then the dot product is simply the projection
  of the other vector onto the axis in which the
  unit vector is pointing.

In this case, the result is the magnitude of the
component of vector c in the j (y-direction).
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(6) Advection as a dot product.

• Advection is greatest if
• the wind speed is a large number and
• the gradient of the thing being advected (e.g.,
  change in temperature / distance) is a large
  number and
• The wind points in the same direction as the
  gradient.

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• The advection of some scalar quantity, such as
  temperature (T) is written as
• ? is called the del operator.
• ?hT is defined as
• A vector of the gradient in space of T, where the
  subscript h refers to only the horizontal
  directions being considered.
• This vector points across the isotherms toward
  the highest values of temperature (low values
  toward high values) - so lower values are moving
  in - the reason for the - (minus) sign.

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• Remember, this could be written as
• Each component of the vector is equal to the rate
  at which temperature changes in that direction.

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• Consider this pressure analysis.
• What is the pressure gradient at the center of
  the grid?

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Method 1

• Orient the gradient vector line along the
  smallest spacing of the contours (isobars in this
  case) (perpendicular to the isopleths) at the
  point of interest. This gives the largest
  gradient
• The magnitude of the pressure gradient is simply
  the amount of pressure change along the line
  divided by the distance along the line.
• The direction of the pressure gradient is the
  direction of the vector line from lowest values
  toward higher values.

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• The component of this gradient in the x-direction
  is
• Or,
• Why (280o-90o)?
• The gradient direction (toward higher values) is
  280o.
• The x-direction is toward the east. The angle
  between the two is 280o - 90o.

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• Now for the component in the y-direction.
• This is
• Remember, the unit vector j points north.

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• Method 2
• Determine the horizontal derivatives along the
  x-axis and along the y-axis.
• Then determine the magnitude of the resultant
  vector.
• Determine the direction by adding the components
  together.

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• Method 3.
• Estimate the values (of pressure) at an equal
  distance on the x-axis and the y-axis from the
  point of interest.
• Determine the gradient in the x-direction and
  y-direction.
• Determine the magnitude as in method 2.
• Determine the direction as in method 2.

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• Suppose we are considering the horizontal
  advection of temperature.
• We could use the components of the wind and the
  components of the temperature gradient as below.
• Since v ui vj, and
• Then, the dot product of these two vectors is

The advection is then
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• Or, not using components, as in the example to
  the right,
• The magnitude of the wind is 20 knots (10m/s).
• The magnitude of the temperature gradient (?hT)
  was calculated at 0.016oC/km along the arrow.
• The angle between the vectors is (360o-45o).
• So,

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(7) Cross Product

• The magnitude of the cross product of vectors a
  and b is given by
• The result is a vector perpendicular to the plane
  on which a and b are located.
• The orientation of the vector is given as shown.

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Some terms

• Divergence the expansion or spreading out of a
  vector field. (convergence is the negative of
  divergence.)
• Horizontal
• Three dimensional
• Vorticity A vector measure of local rotation in
  a fluid flow.
• Relative vorticity the vertical component of
  vorticity, given by the curl of the horizontal
  wind.

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Questions

• Do 1, 2, 3, 4, 5, 6, 7, 8
• Show all graphs calculations.