# Definition of Curl

1. Jul 7, 2009

### jeff1evesque

1. The problem statement, all variables and given/known data
Can someone explain the following to me,
$$\nabla x \vec{V} = -k \frac{\partial{\Phi}}{\partial{t}} \hat{a}_n$$
where $$\vec{V}, \Phi$$ are the wind velocity and pressure respectively.

2. Relevant equations
Take the cross product- thus in the matrix we have the unit vectors in the first row, the partial derivatives on the second row, and the Force (relative to each unit) on the third row.

3. The attempt at a solution
I am not sure, it was just a definition I saw in my notes. Could someone explain the purpose of $$-k, \partial{\Phi}$$. I know $$a_n$$ is the unit normal.

thanks,

JL

Last edited: Jul 7, 2009
2. Jul 8, 2009

### Redbelly98

Staff Emeritus
That doesn't look like anything from an introductory physics course. What class is this for? What was the topic of discussion?

k appears to be some constant in the equation, with units of 1/pressure.

Φ/∂t is simply the rate of change of pressure.

3. Jul 8, 2009

### cipher42

Would this happen to be for an atmospheric science class? My brother was taking one of those classes and he would occasionally ask me for help with the vector calculus that he ran into. I had the worst time trying to help him out because of the discrepancies between the notations that I had learned in physics and those that he was learning in atmospheric sciences.

4. Jul 8, 2009

### jeff1evesque

I am going over some basic math concepts like curls and divergence. In doing so, I hope I can understand concepts of basic electromagnetism.

thanks so much,

JL

5. Jul 8, 2009

### Redbelly98

Staff Emeritus
Curl gives the amount by which a vector field is "rotating"; think of the magnetic field around a straight wire for example. Or think of whirlpool eddies in a stream.

Or, taking the equation from post #1:

https://www.physicsforums.com/latex_images/22/2263762-0.png [Broken]​
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It says that, for a wind velocity pattern that circulates in the clockwise direction (curl directed downward), the pressure will increase with time. Counter-clockwise winds will result in a pressure drop over time.

Last edited by a moderator: May 4, 2017
6. Jul 8, 2009

### jeff1evesque

Thanks so much. Is there a way you could explain the above relative to the derivation of the equation- that is with respect to the determinant form of the cross product?

JL

Last edited by a moderator: May 4, 2017
7. Jul 8, 2009

### Redbelly98

Staff Emeritus
Hi,

Sorry, I have no idea how this equation is derived.

8. Jul 8, 2009

### minger

This equation looks vaguely familiar to a piece of the generalized Navier-Stokes equations (undoubtedly where it was derived from). If the generalized equations are decomposed into eigenvectors and eigenvalues, you get a similar term.

The eigenvalues represent the speed at which the waves propagate, and the eigenvectors describe what they "look" like. In one dimension, you end up with two (left and right-running) acoustic waves, an entropy wave, and two vortical waves. The vortical waves appear in the form of:
$$\nabla \times \vec{V}$$

As mentioned before, the waves can be described as a combination of the divergence and the vorticity, basically a translational and a rotational part.

Your equation essentially looks to me that the time-rate of change in the pressure field is proportional to the rotational "energy" of the air (times some constants).

p.s. You guys can use \times for a cross-product, helps to differentiate between a variable x

9. Jul 8, 2009

### jeff1evesque

Could you help me apply this concept, or perhaps direct me to a source containing this information?

Thanks so much,

JL

10. Jul 9, 2009

### minger

I think I got it. In a paper titled "Atmosphere and Earth's Rotation" by Hans Volland; Surveys in Geophysics he says that:
Then, looking up the Laplace tidal equations we get this:
http://en.wikipedia.org/wiki/Laplace's_tidal_equations#cite_note-0

With the note that
Your equation is essentially a conservation of vorticity. I would safely assume that this is derived (and possibly simplified) from the original Laplace tidal equations, which are used for atmospheric dynamics.