I have the following equation for the shearing term of the deformation zone in meteorology. The reference is here.

Shearing = $\frac{dV}{dx} + \frac{dU}{dy}$

${U}$ and ${V}$ are the zonal and meridional winds, respectively; ${x}$ and ${y}$ are the longitude and latitude, respectively.

I would like to know what does it mean to have if:

a) $\frac{dV}{dx}$ < 0 and $\frac{dU}{dy}$ > 0;

b) $\frac{dU}{dy}$ < 0 and $\frac{dV}{dx}$ > 0;

c) both $\frac{dV}{dy}$ and $\frac{dU}{dx}$ are less than 0;

d) both $\frac{dV}{dy}$ and $\frac{dV}{dx}$ are greater than 0.

  • $\begingroup$ You need to clarify if if you want both a and b is less than 0 or just either one of them $\endgroup$
    – gansub
    Sep 16 '19 at 7:37
  • $\begingroup$ @gansub i want to know when both are negative and one is larger positive and the other is negative. I'll edit my post. $\endgroup$
    – Lyndz
    Sep 16 '19 at 7:40
  • $\begingroup$ I am not sure if the individual terms sign has any distinct meaning. But the the net value of shear surely h as some meaning $\endgroup$
    – gansub
    Sep 16 '19 at 8:28
  • $\begingroup$ @Lyndz The components ∂u/∂x and ∂v/∂y (c) & d)) don't occur in the shearing term... ?? $\endgroup$
    – klanomath
    Sep 18 '19 at 8:56
  • 1
    $\begingroup$ @klanomath One is shear and the other is stretch earthscience.stackexchange.com/questions/16990/… $\endgroup$
    – gansub
    Sep 22 '19 at 5:05

Did you try to sketch one of those situations? Try a square grid, define x and y, and then put $U$ and $V$ components onto one of the grid points, which you choose as starting point.

Now proceed to implement a). For example, from your starting arrows, go to the next grid point eastwards, and implement $\frac{dV}{dx}<0$, i.e. the upwards pointing arrows become smaller as you go eastwards. Do the same for U. Repeat for enough grid points until you see a pattern. Now connect $U$ and $V$ components into the final, two-dimensional velocity vector $\vec v$.

In this way you will possibly self-answer your question. I don't know if I can help you more, because I don't know what you mean with \

I would like to know what does it mean to have

The end result will be the graphical representation of a vector field. The shear tensor simply measure the amount of shear in each vector field. You will discover that some field structures surprisingly possess shear, while others don't.


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