Vectors
A vector is a quantity that has both a
magnitude and a direction. In two
dimensions, vector A = a1i + a2j where
i and j are unit vectors in directions
perpendicular to each other. In
meteorology, we usually assign i as the
unit vector toward the east and j the
unit vector toward the north. The
scalars a1 and a2 are components in
the coordinate directions.
Vector Addition and Subtraction
• For vectors A = a1i + a2j and B = b1i + b2j,
the vector sum C = (a1+b1)i + (a2+b2)j. See
diagram on the board in class using the
diagonal of the parallelogram for A and B.
• The vector difference A - B is the vector
that must be added to B to get vector A. It
is the vector that extends from the tip of B
to the tip of A. See diagram on the board
in class notes.
• Example: Find A+B and A-B for A= -3i -2j and B
= 2i -4j. Draw a diagram to show your results.
Wind Vector
• The horizontal wind V is an example of a
two-dimensional vector. If we let V = ui +
vj, then u is the east-west component of V
and v = north-south component of V. Note
that both u and v are scalars. If dd is the
meteorological wind direction and ff is the
speed in m/s, then
• u = ff cos(270-dd), v = ff sin(270-dd).
Check if you need to convert argument of
sin and cos to radians – multiply by π/180.
Example
• For a north wind at 20 kt (~10m/s),
u = 10m/s cos(270-360) = 0 m/s
• V = 10m/s sin(270-360) = -10 m/s
• For a southeast wind at 30 kt (~15 m/s),
u = 15 m/s cos(270-135) = -10.6 m/s
• v = 15 m/s sin(270-135) = 10,6 m/s
Magnitude of a vector
• The magnitude of a 2-dimensional
vector A = a1i + a2j is sqrt(a12 + a22).
For the wind vector V, the magnitude
(speed) is |V| = sqrt(u12 + v22). In the
previous examples, the magniude of
the north wind is 10 m/s and the
magnitude of the southeast wind is 15
m/s.
Gradient of a scalar
• Recall that the gradient of a scalar such
as temperature T is the rate of change
with distance. The gradient of a scalar
is always a vector. The magnitude is
the rate of change in the direction
perpendicular to the contours
(isotherms in this case), and the
direction is perpendicular to the
isotherms toward the higher values
(warm air in this case). We use T.
Scalar Product (aka dot product)
• The dot product of two vectors A and B is
A ● B = a1b1 + a2b2 (1) where a1 and b1 are
components in the i direction and a2 and
b2 are the components in the j direction.
We can also write A ● B = |A||B|cosθ (2)
where θ is the angle between A and B.
Find the dot product of A and B if
A = 2i -5j and B = -3i + 6j. Easy way is to
multiply components to get 2(-3) + (-5)(6)
giving the answer -36. Find it using (2).
Thermal Advection
• Meteorologists define thermal advection as
-V●T where V is the wind vector and T
is the temperature gradient in some set of
units such as degrees C per 100 km.
• Example: Suppose temperature decreases
toward the north at the rate of 3oC per 100
km, compute the thermal advection (oC per
hr) if the wind is 30 kt (~15 m/s) from the:
• a) south, b) north, c) east, d) northwest
• For each case, draw a diagram.
Terrain-induced Vertical Motion
• When air is forced up (or down) a mountain
slope, the vertical motion is wo = Vo•h
where Vo is the surface wind vector and h
is the gradient of terrain elevation.
• Example: If terrain slopes upwards toward
the south at the rate of 800 m per 100 km,
compute the terrain-induced vertical motion
if the wind is from: a) southwest at 8 m/s;
b) east at 10 m/s; c) 030 degrees at 12 m/s
Vector Product (aka cross product)
• The cross product C of vectors A and B is
a vector with magnitude |C| = |A||B|sinθ
where θ is the angle between A and B.
We write the cross product C = AxB which
is a vector which is perpendicular to both
A and B (perpendicular to the plane
formed by A and B). Note AxB = -BxA.
• The direction of AxB is determined from
the right-hand rule. If you use your thumb
for A and index finger for B, then your
middle finger points in direction of AxB.
Vector representation of Geostrophic Wind
• We can express the geostrophic wind vector
as (g/f) k xz where g = 9.8 ms-2 is the
acceleration of gravity, f = 2Ω sinφ is the
Coriolis parameter, Ω = 7.292 x 10-5 s-1 is the
earth’s rotation rate, and z is the height
gradient on a constant pressure surface.