Ekman Surface Pumping in Polar Coordinates?

[xtamx]

Hello,

I am working on a question in a GFD textbook about tea leaves collecting in the center of the cup regardless of the direction that the tea is stirred. I have an idea of why this is the case but to prove it I need to convert the equation for Ekman pumping to polar coordinates. Its given as:

w = 1/ρf (∂τy/∂x - ∂τx/∂y)

where w is the fluid velocity in the vertical direction
ρ is the fluid density
f is the Coriolis parameter (or rotation rate in the tea cup)
τx and τy are the stresses on the fluid from the walls of the cup

The stress from the bottom of the cup is always going to be in the angular direction in polar coordinates so the net transport will always be in the radial direction according to Ekman dynamics. If the polar version of the above equation is similar then the sign on f changes to negative then the sign on τθ also changes and w remains upward.

The problem is that I have no idea how to convert that to polar coordinates. Can anyone help?

 

[mark726]

Thank you for your interesting question about tea leaves collecting in the center of a cup. Based on the information provided, it seems that the Ekman pumping equation is relevant in explaining this phenomenon. However, to fully understand and prove this, it is necessary to convert the equation to polar coordinates.

To do this, we can use the following conversion equations:

∂/∂x = cos(θ) ∂/∂r - sin(θ)/r ∂/∂θ
∂/∂y = sin(θ) ∂/∂r + cos(θ)/r ∂/∂θ

Using these equations, we can rewrite the Ekman pumping equation in polar coordinates as:

w = 1/ρf (cos(θ) ∂τy/∂r - sin(θ)/r ∂τy/∂θ - sin(θ) ∂τx/∂r - cos(θ)/r ∂τx/∂θ)

From this equation, we can see that the stress from the walls of the cup, τx and τy, will have both radial and angular components. However, the Coriolis parameter, f, will only have an angular component, as it represents the rotation rate in the tea cup. Therefore, the sign on f will not change in the polar version of the equation and the upward motion of w will still be maintained.

I hope this helps in your understanding and further exploration of this interesting phenomenon. If you have any further questions or need clarification, please do not hesitate to ask. Keep up the great work in your studies!