# How does vertical wind change over time?

My question is about how vertical component of wind may change over time; in particular, as I am interested in simulating the effect of vertical wind on flight mechanics, I'd like to know whether I should consider vertical wind as a deterministic function of time or a stochastic one. Furthermore, should I consider vertical wind persistent over time or having a limited duration (e.g. few seconds)? Note that I am not interested in horizontal wind.

• There are lots of different causes. Off the top of my head, there are thermals caused by differential heating of the ground, orographic effects when horizontal winds flow across mountain ridges (ridge lift, mountain waves), downslope winds where cool air at high elevations flows downwards... Apr 11 '16 at 18:46
• Please be more specific about what exactly you want to know about vertical wind and what is your current state of knowledge on it. Which resources did you use to get to know this topic already? Apr 11 '16 at 19:52
• @sax631: All the kinds of vertical air movement I mentioned are of importance in flight. I drew the examples mostly from my experience as a pilot. I'm not sure what you mean by 'signal': do you mean rate of climb as indicated by instruments? Apr 12 '16 at 5:46
• @sax631 Could you modify your question accordingly? Apr 14 '16 at 9:00
• @daniel.neumann I've rephrased my question. Apr 15 '16 at 8:44

You can refer to the vertical component of the Navier Stoke's equations. $$\frac{Dw}{Dt}=\frac{∂w}{∂t}+\vec{v}\cdot\nabla w + w\frac{∂w}{∂z} =-\frac{1}{ρ}\frac{∂P}{∂z}-g+\nu\nabla^2w$$ Additionally, it must also be constrained by the conservation of mass. $$\frac{∂P}{∂t}=-\nabla\cdot(\rho\vec{v})$$ The Quasigeostrophic Omega Equation may also provide you some insight. This equation is filtered down from the above Navier-Stokes equation, though it may not seem like it. Omega is vertical velocity in pressure coordinates. $$(\nabla^2+\frac{f_0^2∂^2}{\sigma∂\Phi^2})\omega=\frac{f∂}{\sigma∂p}(\vec{v}\cdot\nabla(\frac{1}{f_0}\nabla^2\Phi+f)) + \frac{1}{\sigma}\nabla^2[\vec{v_g}\cdot\nabla(-\frac{∂\Phi}{∂p})]$$