Bulk and dynamic viscosity in the atmosphere

I'm studying the physics of the atmosphere but I'm struggling with the matter of viscosity (Navier-Stokes equation) for gravito-acoustic waves. From Landau-Lifschitz :

$$(T)_{ij} = -p\delta_{ij} + \eta(\partial_j\mathbf{v}_i + \partial_i\mathbf{v}_{j} - \frac{2}{3}\delta_{ij}\nabla\cdot\mathbf{v}) + \zeta\delta_{ij}\nabla\cdot\mathbf{v}$$

where $\eta$ is the dynamic viscosity, $\zeta$ is the second viscosity, $\mathbf{v}$ is the velocity, $p$ is the pressure, and $\delta$ the Kronecker delta.

My question is, in a windy viscous stratified atmosphere (up to 500km) how should I treat the viscous stress tensor $T$? Should I take into account both the dynamic viscosity (normal viscosity) and the second viscosity? Can I consider the viscosity coefficients $\eta, \zeta$ to be constant? What about the Stokes assumption $\nabla \cdot \mathbf{v} = 0$?

Considering viscosity can I still use the adiabatic assumption to settle the Navier-Stokes equation?

More specifically, for gravito-acoustic propagation should I consider shear effects as well ?