Send an excel  sample file.

Here ya go.
Wind.xlsx

Is zero vertically up?
Do the angles go clockwise?

Yes. 0 (or 360) would be up.
90 right
180 down
270 left

Will something like this work?

I think so.  Is that a symbol for each data point?  

I wondering what will happen with about 100 rows of data.

That is what I am worried about. With too many points you may lose comprehendibility.

And this is the very minimum that can be shown. Any deviation from this would end up being more congested. You really need to be clear on what you would like to see.

I think it will work.

Are you going to tell me how to do that?

I need some time to complete it. I was busy today. Will get back as soon as I complete it.

Okay.  Wasn't sure if I missed something.

Thanks!

The chart looks fine, but I have no idea how it's created.

I see a second worksheet with what looks like trig functions but I'm not sure what they are for.

And I've never seen a worksheet with just a chart before.  

So I'm kind of lost as to how the reproduce this.

To reproduce this you would have to do put your data in sheet 1 exactly the way you see it in the given sheet.

You are right that the sheet 2 has trig functions which are required to display the direction of wind.
The top of this sheet contains three cells with comments.
Excel does not have geometrical graphics and the vertical and horizontal scales are always different so a circle would show as an ellipse and a square would show as a rectangle. To overcome this I have the top two cells which play with the vertical and horizontal ratio to make a square look almost a square. You may have to adjust these whenever the data size changes. The third cell is there to show a rectangle which will help you adjust the x and y.
I just chose to move the chart to a separate sheet. You may bring it back by right clicking it and selecting move chart.

So if I have more lines of data, I just add them to sheet 1?  

Will have to add them to sheet 2 or will that happen automatically?

The top of this sheet contains three cells with comments.

I do not see any comments on any of the sheets.

Did you mean the text that pops up when you hover?

I didn't know about that.

Yes, if you have more line it is a bit tricky. There are a few lines which can accommodate more data but for more the lines on sheet2 must be copied further. These lines are in threes and would  have to be copied as threes. For this you would have to select the last three rows and drag the fill handle down as far as desired. It is important that when you start dragging you should have exactly three rows selected.

Yes the comments are the texts that appear when the cursor is brought to it.