Geophysical Fluid Dynamics

In summary, the conversation is about a homework problem involving a rotating fluid in a sphere with certain conditions. The equations are given in cylindrical coordinates and the attempt at a solution involves using the equations to find the derivative of the fluid motion and solving for four equations. The person asking for help is unsure if their solution is correct and is asking for someone to check.
  • #1
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Heya , sorry but could someone check what I've done is right ? The remainder of the question relies on these answers :S

Homework Statement



We consider a homogeneous fluid in a sphere of radius r0 = 1 in the limit of kinematic viscosity v = 0 which is rotating uniformly about its axis with a constant angular velocity.

We shall use cylindrical coordinates (s;[tex]\phi[/tex] ; z) with the corresponding unit vectors ([tex]\hat{s}[/tex]; [tex]\hat{\phi}[/tex]; [tex]\hat{z}[/tex]), with [tex]\hat{z}[/tex] being parallel to the axis of rotation.

The small-amplitude fluid motion in a rotating reference of frame is governed by the linear dimensionless vector equations

[tex]\frac{\delta \textbf{u}}{\delta t}[/tex] + 2[tex]\hat{ \textbf{z}}[/tex] x [tex]\textbf{u}[/tex] = -[tex]\nabla[/tex]p; (1)
AND

[tex]\nabla[/tex] . [tex]\textbf{u}[/tex] = 0; (2)

subject to the condition of vanishing normal flow
[tex]\hat{\textbf{r}}[/tex] . [tex]\textbf{u}[/tex] = 0 at r = 1: (3)

Let [tex]\textbf{u}[/tex]([tex]\textbf{x}[/tex], t) = [tex]\textbf{u}[/tex](s, z)[tex]e^{i(\phi + t )}[/tex]. (4)

Write down the four equations by projecting the equations (1) and (2) onto cylindrical coordinates.


Homework Equations



They are above

The Attempt at a Solution



So my attempt at this is ,
 
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  • #2
Using (1) , I find out [tex]\frac{\delta \textbf{u}}{\delta t}[/tex] which I get to be [tex]it\textbf{u}[/tex] from (4) so plugging this into (1) gives us three equations of the 4

itus + 2[tex]\hat{\textbf{z}}[/tex] x [tex]u_{s}[/tex] = - [tex]\frac{\delta p}{\delta s}[/tex]

itu[tex]\phi[/tex] + 2[tex]\hat{\textbf{z}}[/tex] x [tex]u_{\phi}[/tex] = - [tex]\frac{\delta p}{\delta \phi}[/tex]ituz + 2[tex]\hat{\textbf{z}}[/tex] x [tex]u_{z}[/tex] = - [tex]\frac{\delta p}{\delta z}[/tex]

and then using equation (2) the last equation of the 4 is

[tex]\frac{\delta u_{s}}{\delta s}[/tex] + [tex]\frac{\delta u_{\phi}}{\delta \phi}[/tex] + [tex]\frac{\delta u_{z}}{\delta z}[/tex] = 0

I think I probably have gone wrong somewhere tho , would someone please check to see what I'm doing is correct please ?
 
Last edited:
  • #3
ah nm this is wrong , seen my mistake
 

1. What is Geophysical Fluid Dynamics?

Geophysical Fluid Dynamics is a branch of geophysics that studies the movement of fluids, such as air and water, in the Earth's atmosphere and oceans. It uses mathematical models and physical principles to understand and predict the behavior of these fluids.

2. How is Geophysical Fluid Dynamics related to weather forecasting?

Geophysical Fluid Dynamics plays a crucial role in weather forecasting by providing a framework for understanding and predicting the movement of air and water in the Earth's atmosphere and oceans. This helps meteorologists to make more accurate weather predictions.

3. What are some real-world applications of Geophysical Fluid Dynamics?

Geophysical Fluid Dynamics has many practical applications, including weather and climate forecasting, ocean circulation studies, and understanding natural hazards such as hurricanes and tsunamis. It is also used in the design of ships and offshore structures, and in the study of atmospheric and oceanic pollution.

4. How does Geophysical Fluid Dynamics contribute to our understanding of climate change?

Geophysical Fluid Dynamics is an important tool for studying and predicting the Earth's climate. By analyzing the movement of fluids in the atmosphere and oceans, scientists can better understand how factors such as greenhouse gas emissions and ocean currents contribute to changes in the Earth's climate.

5. What are some challenges in studying Geophysical Fluid Dynamics?

Geophysical Fluid Dynamics is a complex and constantly evolving field of study, and there are many challenges that scientists face in understanding and predicting the behavior of fluids in the Earth's atmosphere and oceans. These challenges include the limitations of mathematical models, the vastness and complexity of the Earth's systems, and the difficulty in obtaining accurate and comprehensive data.

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