Chapter 7

Bits of Vector Calculus

(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:

M x2 y 2

However, since we have a scale that shows what a certain length means, why don’t we just measure it?

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.

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.

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.

For our wind vector, measured with a protractor, the direction is about 238o.

The components are as shown. However, this shows “b” as the x-component of the wind vector (u), and “a” as the y-component of the wind

vector (v).

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.

Error on page 5. Wind speeds are approx. 2.8m/s and 1.8 m/s, not 1.4 and 0.9m/s

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).

arctanb

a

b

a

(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.

(3) Vector Multiplication and ComponentsUnit 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.

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

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 jVector 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.

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.

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”.

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

(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).

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.

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).

(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 andThe wind points in the same direction as the gradient.

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.

v hT

hT Txi

Tyj

From AMS Glossary (do you have yet?

© 2000 American Meteorological Society

This could be written as:

Each component of the vector is equal to the rate at which temperature changes in that direction.

v hT cos

hT Txi

Tyj

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

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.

Method 1

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.

Px

P i P 1 cos(angle between)

Px

0.035mb km 1cos 280o 90o

Now for the component in the y-direction.

This is

Remember, the unit vector j points north.

Py

P j P 1 cos(angle between)

Py

0.035mb km 1cos 280o 0o

Method 2Determine 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.

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.

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:

v hT uTx

vTy

hT Txi

Tyj

v hT uTx

vTy

The advection is then:

v hT uTx

vTy

u

Tx

vTy

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,

v hT v hT cos angle between 10m s 0.016 oC km cos 360o 45o = -1.1 10-4 oC s

Some termsDivergence: 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: normally, the vertical component of vorticity, given by the curl of the horizontal wind.

h v ux

vy

z

w

y

v

x

uv

y

u

x

vvk

v

(7) Cross ProductThe 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.

ab a b sin angle between