Horizontal divergence is expressed in cartesian coordinates as
D = (∂u/∂x) + (∂v/∂y)
Note there are two terms: the instantaneous rate of change of the u component of the winds in the x direction, and the instantaneous rate of change of the v component in the y direction.
We can estimate the value of the divergence by using a centered differencing approach, where:
D = (Δu/Δx) + (Δv/Δy)
In the first term, we must take the change in the u component of the wind in the x direction. In the 2nd term, we must take the change in the v component in the y direction.
So if we have our points laid out as follows:
P3
P2 P1
P4
then Δu/Δx will involve the points P1 and P2 -- the two points
that vary in the x direction. It will be convenient to have Δx be
positive, so we do "P1 minus P2" to get our delta -- P1 having a greater value
of x than P2. Similarly, for Δv/Δy we would do "P3 minus P4".
So to estimate the term ∂u/∂x we would take a centered difference as follows:
(u at P1) - (u at P2) --------------------- (x at P1) - (x at P2)
Simlarly, to estimate the term ∂v/∂y we would do:
(v at P3) - (v at P4) --------------------- (y at P3) - (y at P4)
Consider the homework problem, where:
At P1 the wind is blowing 10mph towards the north.
At P2 the wind is blowing 10mph towards the south.
At P3 the wind is blowing 10mph towards the west.
At P4 the wind is blowing 10mph towards the east.
It looks sort of like this:
<------
P3 /|\
|
|
P2 P1
|
|
\|/ ------->
P4
Well, anyway, it gives us a rough idea.
Here's the u and v components for the points:
P1: u is zero; v is 10
P2: u is zero; v is −10
P3: u is −10; v is zero
P4: u is 10; v is zero
To estimate Divergence, we do:
(u at P1) - (u at P2) (zero - zero) --------------------- = ------------- (x at P1) - (x at P2) 1
This gives us zero for this term.
Then for the 2nd term:
(v at P3) - (v at P4) (zero - zero) --------------------- = ------------- (y at P3) - (y at P4) 1
Again, we have zero. So the two terms added together give us zero.
So, D = zero
Now for vorticity, the terms are different, but the approach is the same.
ζ = (∂v/∂x) - (∂u/∂y)
The first term is the rate of change of v in the x direction. The x direction means we are using P1 and P2. We estimate the first term as follows:
(v at P1) - (v at P2) (10 - ( -10 )) --------------------- = ------------- (x at P1) - (x at P2) 1
So we get a value of 20 for the first term. Remember we are just assuming, for now, that Δx is 1.
The 2nd term is the rate of change of u in the y direction. That means we are using P3 and P4. We estimate the 2nd term as follows:
(u at P3) - (u at P4) ( ( -10 ) - 10 ) --------------------- = ---------------- (y at P3) - (y at P4) 1
So the value of the 2nd term is −20.
To estimate ζ we have to subtract the 2nd term from the 1st term: 20 − (−20) = 40
We get a value of 40, which is positive. Since we do not know what Δx and Δy actually are, we cannot calculate a value for ζ. However, the question just asked us if ζ was negative, zero, or positive. It is positive.
Consider the homework problem, where:
At P1 the wind is blowing 10mph towards the east.
At P2 the wind is blowing 10mph towards the west.
At P3 the wind is blowing 10mph towards the north.
At P4 the wind is blowing 10mph towards the south.
P1: u is 10; v is zero
P2: u is −10; v is zero
P3: u is zero; v is 10
P4: u is zero; v is −10
Looking at Divergence, we do the 1st term:
(u at P1) - (u at P2) (10 - ( -10 )) --------------------- = ------------- (x at P1) - (x at P2) 1
The value of this term is 20. Looking at the 2nd term:
(v at P3) - (v at P4) (10 - ( -10 )) --------------------- = ------------- (y at P3) - (y at P4) 1
The value of this term is also 20. To estimate the Divergence, we add the two terms, and get a positive value. So to answer the question: is D negative, zero, or positive? D is positive.
We would need to know the values of Δx or Δy to calculate the value of D.
It may be possible to solve problems of this type by inspection, if one understands what vorticity and divergence are.