[ISSUE]: Mass conservation equation integration in Level-4 data

[bpoujol]

It seems that the integration of the mass conservation equation equation in JOANNE Level-4 data does not account for vertical variations in density.
The mass conservation equation for a compressible fluid reads $\vec \nabla \cdot (\rho \vec v) = 0$, or (neglecting horizontal density variations) :
$$\delta = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = - \frac{1}{\rho} \frac{\partial (\rho w)}{\partial z} = - \frac{\partial \omega}{\partial p}$$
Pressure velocity should therefore be computed as :
$$\omega = - \int_{p_0}^p \delta \mathrm{d}p = \int_{0}^z \delta \rho g \mathrm{d}z$$
However it seems that within the dataset Level-4 data vertical velocity is computed as :
$$\omega = \rho g \int_{0}^z \delta \mathrm{d}z$$
and therefore density is out of the integral.

The difference is very small in the lower troposphere but it can reach several hPa/hr in the mid- and upper-troposphere as shown in the example below computed from one EUREC4A circling set.