# Why does the absence of diabatic heating rate and frictional terms make the zonal and vertical velocity identically zero?

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

• Please, cite your source. Context matters. For example, are you sure that those u,v,w are not formulated in a locally co-rotating frame? – AtmosphericPrisonEscape Jan 22 '18 at 17:58
• They say they're using a 'steady, freely spinning vortex' so they're already plugging in u and w as zero as initial conditions. Then by requirement u and w stay that way. Plugging that into the equations and checking, shows that there is no contradiction, so $u=w=0$ can be (part of) the solution. – AtmosphericPrisonEscape Jan 22 '18 at 18:57