Is this a valid semi-diagnostic equation for Omega?

Question

Beginning with the Primitive equations governing atmospheric motion for a dry gas, primarily the ideal gas law, and the conservation of mass and energy, neglecting diffusivity.

$$P=\rho R T$$ $$\omega \equiv \frac{DP}{Dt}$$ Therefore $$\omega=R(\rho\frac{DT}{Dt}+T\frac{D\rho}{Dt})$$ Since $$\frac{D\rho}{Dt}=-\rho \nabla\cdot\vec{u}$$ and $$\frac{DT}{Dt}=\frac{\omega\rho}{c_p}+\frac{Q}{c_p}$$ $$\omega=R(\frac{\rho^2\omega}{c_p}+\frac{\rho Q}{c_p}-T\rho\nabla\cdot\vec{u})$$ Distributing $R$ and applying the ideal gas law $$\omega=\frac{R\rho^2\omega}{c_p}+\frac{PQ}{Tc_p}-P\nabla\cdot\vec{u}$$ Separating $\omega$ from the right hand side of the equation yields $$\omega=(1-\frac{R\rho^2}{c_p})^{-1}(\frac{PQ}{Tc_p}-P\nabla\cdot\vec{u})$$

I call this semi-diagnostic, for $\omega$ is still a part of $\nabla\cdot\vec{u}$

Is this a valid semi-diagnostic equation for $\omega$?

Answer 1

This is not a valid diagnostic equation. The vertical velocity $\omega$ is defined in vertical pressure $p$ coordinate as $\omega = Dp/Dt$. As a result, all the equations you used should be in pressure coordinate.

Then the continuity equation $$D\rho/Dt=-\rho\nabla\cdot\vec u$$

will change to $$\nabla_p\cdot\vec u=\partial_x u+\partial_y v+\partial_p \omega=0$$

Also, density $\rho$ in pressure coordinate becomes a constant $1/g$. You should use everything in pressure coordinate for a diagnostic equation of $\omega$.