How can I interpret the 2D advection equation?

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Discussion Overview

The discussion revolves around the interpretation and modeling of the 2D advection equation in the context of debris transport on a glacier. Participants are exploring the physical differences between equations related to the velocity components and the behavior of the debris layer.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant seeks clarification on the physical differences between two equations related to the advection of a debris rock layer on a glacier.
  • Several participants request diagrams to clarify whether the velocity components (u and v) represent averages over the thickness of the layer or values at the top.
  • There is a question about whether the rock layer is moving as a rigid body or deforming, with one participant asserting that the rocks are not deforming and are instead transported by the ice.
  • Another participant suggests that if the rocks are not deforming, then the velocity components u and v do not vary with x and y.
  • A participant emphasizes the importance of understanding the physics behind the process and suggests that one should consider how one equation can be derived from another, referencing a previous comment by another participant.
  • There is a suggestion that understanding the divergence term related to the change of debris height over time is crucial for grasping the underlying model.

Areas of Agreement / Disagreement

Participants express differing views on the behavior of the debris layer (rigid vs. deforming) and the interpretation of velocity components, indicating that multiple competing views remain in the discussion.

Contextual Notes

Participants have not reached a consensus on the interpretation of the equations or the physical behavior of the debris layer, and there are unresolved questions regarding the definitions of the velocity components.

jones1234
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TL;DR
How can I interpret the 2D advection equation?
I want to model the advection of debris rock layer with a thickness hd on top of a glacier through ice flow with velocity components u and v. Can anybody explain the physical difference between these 2 equations and which one I should take? Thanks

advection.png
 
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Please provide a diagram. Are the v's in the equations the averages with respect to z over the thickness of the layer or the values at the top of the layer?
 
Chestermiller said:
Please provide a diagram. Are the v's in the equations the averages with respect to z over the thickness of the layer or the values at the top of the layer?
u_s = the x-component of the surface velocity,
v_s = the y-component of the surface velocity,
h_d = the thickness of the rock layer
 
jones1234 said:
u_s = the x-component of the surface velocity,
v_s = the y-component of the surface velocity,
h_d = the thickness of the rock layer
Is the rock moving as a rigid layer or is it deforming?
 
Chestermiller said:
Is the rock moving as a rigid layer or is it deforming?
The rocks are not deforming. They are on top of the ice and get transported by the ice.
 
jones1234 said:
The rocks are not deforming. They are on top of the ice and get transported by the ice.
Then u and v do not vary with x and y
 
Chestermiller said:
Then u and v do not vary with x and y
The most important part of modeling is actually understanding the physics of the process and then deriving the underlying model. Do you see how eq. (2) can be obtained from eq. (1) with the comments of @Chestermiller ?

If you haven't done so already, it will also be good for your understanding to go back even one step further and ask yourself why the change of debris height in time is given by this divergence term in the first place. The diagram that @Chestermiller asked for will be crucial for this understanding
 
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