I was reading an article on fluid dynamics of tropical cyclones, where I came across a condition which states that,
For adiabatic friction less flow, equations (1) to (5) have a solution, v(r, z), for a steady freely spinning vortex in which u and w are identically zero and v(r, z) is an arbitrary function of r and z.
Where these 5 equations are basic primitive equations (2 horizontal and 1 vertical equation of motion,conservation of mass and thermodynamic equation) written in cylindrical coordinates.
If I try to consider the horizontal(u component only) and vertical equations of motion when they are in gradient wind and hydrostatic balance, u and w will be conserved i.e du/dt=0 and dw/dt=0. How can one conclude that they are identically zero? For the article, please open the following link https://www.google.co.in/url?sa=t&source=web&rct=j&url=https://www.meteo.physik.uni-muenchen.de/~roger/Publications/M17.pdf&ved=2ahUKEwjvzPOsmOzYAhUeSo8KHf3wC5YQFjAEegQIDRAB&usg=AOvVaw3uxV6Wyli4JcGq3qrQzR00
The primitive equations are given in section 2.1 and the lines highlighted above are taken from the first paragraph of section 2.2