Coordinate systems in modelling balloon ascent dynamic
I coded the dynamics of a latex balloon throught standard ODE integration, set in a 3-dimensional cartesian space (z = altitude, x = North direction, y = East direction).
z-cooordinate accelleration depends, as usual, on gravity, buoyancy and ascent drag. x and y-coordinates accelleration depends on winds and lateral drag. Winds were simply modelled by a 3 columns look-up table with altitude-wind azimuth (angle from North)-wind speed: at each altitude layer, winds vector in that range were mapped onto x and y component of the ODE system. No problem in going that far.
After that, I wanted to obtain the geodetic latitude-longitude-altitude ballon path relative to Earth (WSG84). Thinking to the coordinate system where I integrated the dynamic, I decided it was considerable as a local cartesian NED system, with origin set at balloon lanch point ('z' has opposite direction, but it doesn't change the concepts). Given that, I simply transformed the (x,y,z) from this NED system (never changed along the integration) into the geodetic system (WSG84 ellipsoid).
Now, considering for example a balloon that reach a certain fixed floating altitude (z constant after a certain integration time) and after that moves only by winds (so in the x,y directions), when you convert the dynamic into geodetic, you ends with an altitude that is increasing (while 'z' doesn't move) !
Actually, if you consider the x,y 'surface' as a plane (tangent to the NED system origin), it is logical that, when the ballon displacement gets bigger, the (x,y) plane gets further and further from the Earth surface. 229
It follows then that geodetical altitude increases as the integration 'z' coordinates remains fixed. If this is the concept, you are using a FIXED 'flat Earth' system, and so when you move 'too much' from the origin, the 'flat Earth' approximation fails.
If the reasoning is correct, how to deal with this issue ? Should I have to 'update' the local system in order to take the Earth curvature into account ? And, if this is the case, how to do ?
But I propose you another point of view. The (x,y) balloon displacement depends (pratically) only on winds vector (with 'winds' I mean, from now on, the azimuth and speed at a certain altitude). Winds blows in an Earth PARALLEL fashion, so their components are naturally given (forecast or actual) for a curved surface. So, when you apply the x and y winds components to you ODE integration, you are implicitly considering a vector that stays, at any point, tangent to the Earth surface. Given that, the ODE NED system is already geodetically curved, and so the 'z' component is really the ballon altitude from Earth (so no need to translate coordinates ... it would be wrong actually).
Can you help me understand how things are ?
Thanks
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