Definition of Curl - Explaining Nabla x V = -k Partial Phi/Partial t a_n

[jeff1evesque]

Homework Statement

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.

Homework 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.

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

 

[Redbelly98]

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.

 

[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.

 

[jeff1evesque]

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.

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

 

[Redbelly98]

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​

[/URL]

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.

 

[jeff1evesque]

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​

[/URL]

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.

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

 

[Redbelly98]

Hi,

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

 

[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

 

[jeff1evesque]

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

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

Thanks so much,


JL

 

[minger]

I think I got it. In a paper titled "Atmosphere and Earth's Rotation" by Hans Volland; Surveys in Geophysics he says that:

The theoretical aspects of the transfer of angular momentum between atmosphere and Earth
are treated with particular emphasis on analytical solutions. This is made possible by the consequent
usage of spherical harmonics of low degree and by the development of large-scale atmospheric
dynamics in terms of orthogonal wave modes as solutions of Laplace's tidal equations.

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

William Thomson (Lord Kelvin) rewrote Laplace's momentum terms using the curl to find an equation for vorticity. Under certain conditions this can be further rewritten as a conservation of vorticity.

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.