Omega equation

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The omega equation is of great importance in meteorology and atmospheric physics. It is a partial differential equation for the vertical velocity, ω, which is defined as a Lagrangian rate of change of pressure with time, that is, \omega = \frac{dp}{dt}. The equation reads

\nabla^2\omega + \frac{f^2}{\sigma}\frac{\partial^2\omega}{\partial p^2} = \frac{f}{\sigma}\frac{\partial}{\partial p}\mathbf{V}_g\cdot\nabla_p (\zeta_g + f) + \frac{R}{\sigma p}\nabla^2_p(\mathbf{V}_g\cdot\nabla_p T),

where σ is a stability parameter and f is the Coriolis parameter.

Physically, the omega equation combines the effects of vorticity advection (first term on the right-hand side) and thermal advection (second term on the right-hand side) and determines the resulting vertical motion (as expressed by the dependent variable ω.

The above equation is used by meteorologists and operational weather forecasters to assess development from synoptic charts. In rather simple terms, positive vorticity advection (or PVA for short) and no thermal advection results in a negative ω, that is, ascending motion. Similarly, warm advection (or WA for short) also results in a negative ω corresponding to ascending motion. Negative vorticity advection (NVA) or cold advection (CA) both result in a positive ω corresponding to descending motion.

[edit] References

  • J.R. Holton, An Introduction to Dynamic Meteorology, 4th edition, Academic Press