The concept of relative velocity belongs to the specific field of geophysical flows.
The general equation for the vorticty vector field (Helmholtz equation) for compressible flows can be found in the Kundu textbook.
Other details are here
https://www.weather.gov/source/zhu/ZHU_Training_Page/Miscellaneous/vorticity/vorticity.html#:~:text=Relative%20Vorticity,and%20horizontal%20wind%20speed%20differences.
What is Vorticity?
Vorticity (ω⃗) is a vector quantity that represents the local spinning motion or rotation of a fluid element. Mathematically, it is defined as the curl of the velocity field (v⃗):
ω⃗=∇×v⃗
It measures the tendency of fluid particles to rotate about their own axes.
Relative Vorticity (ζ): Relative vorticity is the rotation of the fluid relative to the Earth’s surface. It depends only on the local velocity gradients within the flow. Equation (2D): ζ=∂v/∂x−∂u/∂y
where: u and v are velocity components in the x and y directions, respectively.
Absolute Vorticity (η):Absolute vorticity accounts for both relative vorticity and the Coriolis effect due to Earth's rotation. Equation: η=ζ+f
where:
- f=2Ωsin(ϕ) is the Coriolis parameter.
- Ω is Earth's angular velocity (7.292×10−5 rad/s).
- ϕ is the latitude.
Importance in Studies
- Relative Vorticity is useful in localized flow studies, such as analyzing turbulence, eddies, and cyclonic systems.
- Absolute Vorticity is more critical in large-scale atmospheric and oceanic dynamics, where Earth's rotation significantly affects the flow.
The choice depends on whether your study involves local flows (e.g., lab experiments or small-scale turbulence) or large-scale geophysical flows (e.g., atmospheric systems or ocean currents). Let me know if you need further clarifications!
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