2. Simulación y simulacro 44.
2.2 Simulación e iconofilia 45.
When the dropwindsonde is treated as a point object falling through the atmosphere, its motions are driven by the combination of the weight force and aerodynamic forces. Furthermore, aerodynamic forces are reduced to pure drag in the point object motion
model. If the drag coefficient can be modelled as a constant regardless of angles of attack, as indicated by Hock and Franklin (1999) in their derivation of the wind finding equations, the aerodynamic drag has a simple expression. Following the derivation of the motion governing equations of vertically moving wind sensors presented by Nastrom and Vanzandt (1982), this simple expression of the aerodynamic drag, when the vertical variation of the air density is neglected, is
FDx = 1 2ρACD(u− x˙)( p (u− x˙)2+ (v− y˙)2+ (w− z˙)2) FDy = 1 2ρACD(v− y˙)( p (u− x˙)2 + (v− y˙)2+ (w− z˙)2) FDz = 1 2ρACD(w− z˙)( p (u− x˙)2+ (v− y˙)2+ (w− z˙)2) (3.1)
where FD is the drag force in which subscripts stand for the force direction, CD is the
drag coefficient, A gives the equivalent cross-section area, and ρ gives the air density. If (x, y, z), as a vector, is used to represent the dropwindsonde position, ( ˙x,y,˙ z˙) indicates its velocity at the position (x, y, z). The vertical velocity of the dropwindsonde ˙z is positive when the dropwindsonde goes upwards. Similarly to ( ˙x,y,˙ z˙), (u, v, w) is employed as the wind velocity vector. This point object model and its driving forces are illustrated in Fig. 3.1.
Figure 3.1: Sketch illustrating the point object model simulating the dropsonde motion and driving forces, including the weight force Gand the aerodynamic drag FD.
If the dropwindsonde weight force is modelled as G = mg, where m is the mass of the dropwindsonde system and g is the gravity, motions of the dropwindsonde are driven by the combination of the aerodynamic drag expressed in equation (3.1) and the weight force as,
m~a=XF~ (3.2)
Substituting the explicit expression of the aerodynamic drag and weight force into equa- tion (3.2) gives the dropwindsonde motion governing equations, which show,
mx¨ = 1 2ρACD(u− x˙)( p (u− x˙)2+ (v− y˙)2+ (w− z˙)2) my¨ = 1 2ρACD(v− y˙)( p (u− x˙)2+ (v− y˙)2+ (w− z˙)2) mz¨ = 1 2ρACD(w− z˙)( p (u− x˙)2+ (v− y˙)2+ (w− z˙)2) +mg (3.3)
where (¨x,y,¨ z¨) indicates the dropwindsonde accelerations.
Equation (3.3) is a set of nonlinear Ordinary Differential Equation (ODE)s, mainly because they contain a nonlinear termp(u− x˙)2+ (v− y˙)2+ (w− z˙)2. If the horizontal
velocity difference between the wind and dropwindsonde is far smaller than the vertical velocity difference, i.e. ( ˙z− w) >> ( ˙x− u) and ( ˙z− w) >> ( ˙y− v), and therefore is neglected in the square root expression, equation (3.3) can be simplified as,
mx¨ = 1 2ρACD|z˙− w|(u− x˙) my¨ = 1 2ρACD|z˙− w|(v− y˙) mz¨ = 1 2ρACD|z˙− w|(w− z˙) +mg (3.4)
Furthermore, when the assumption is taken that the vertical wind velocity is much smaller than the dropwindsonde falling speed and is then neglected in the term |z˙ − w|, this
simplified equation, equation (3.4), can be further simplified as, mx¨ = 1 2ρACD| − z˙|(u− x˙) (3.5) my¨ = 1 2ρACD| − z˙|(v− y˙) (3.6) mz¨ = 1 2ρACD| − z˙|(w− z˙) +mg (3.7)
Equations (3.5), (3.6) and (3.7) are the simplified equations governing the dropwind- sonde motion which are equivalent to the wind finding equations introduced by Hock and Franklin (1999). Their apparent difference is due to that the wind finding equa- tions include the drag coefficient implicitly in the falling rate term rather than explicitly as in equations (3.5), (3.6) and (3.7). To eliminate this explicit expression of the drag coefficient, the equation governing the dropwindsonde vertical motion is solved. More specifically, if the vertical wind is neglected in equation (3.7), the dropwindsonde falling speed is related to the drag coefficient as,
˙
z2 = 2m(g − z¨)
ρACD (3.8)
Noticing that the quantity (g− z¨) should be positive for expression (3.8) to be valid, this suggests that the dropwindsonde vertical acceleration, ¨z, should be less than the gravity acceleration, g. In the case ¨z > g, the instantaneous dropwindsonde vertical moving speed would be expressed as,
˙
z2 = 2m(¨z− g)
ρACD (3.9)
which is considered very unlikely.
In addition, if the vertical motion of the dropwindsonde is assumed in equilibrium state, which indicates a steady fall of the dropwindsonde, its vertical acceleration is zero. In this case, equation (3.8) is reduced to
˙
z =−
r
2mg
ρACD (3.10)
Using the drag coefficient expressed in equation (3.10), the equations governing horizontal dropwindsonde motions, equations (3.5) and (3.6), can be further reduced to,
¨ x = − g ˙ z(u− x˙) (3.11) ¨ y = − g ˙ z(v− y˙) (3.12)
Equations (3.11) and (3.12), depicting the dropwindsonde motion in a partially linearized and simplified manner, are essentially the wind finding equations introduced by Hock and Franklin (1999) when some terms are rearranged,
u = − z˙
g x¨+ ˙x (3.13)
v = − z˙
g y¨+ ˙y (3.14)
In summary, the dropwindsonde motion governing equations can be partially linearized and simplified, which is also a derivation of the wind finding equations currently used, under the following conditions,
• The velocity difference between the dropwindsonde and wind in the vertical direc- tion is much larger than that in other directions, i.e. ( ˙z − w) >> ( ˙x− u) and ( ˙z− w)>> ( ˙y− v).
• The vertical wind is much smaller than the dropwindsonde falling rate and can be neglected, i.e. ˙z >> w.
• The dropwindsonde vertical motion is stable and its vertical acceleration is zero.
• The drag coefficient can be modelled as a constant, independent from the drop-
in dropwindsonde falls.
In addition, for equations (3.13) and (3.14), it is possible to derive analytical solutions since they are linear ODEs. Appendix A (page 206) details the derivation of these analytical solutions and discusses dropwindsonde response characteristics based on these analytical solutions.
From the simplification shown above, it is obvious that equations (3.11) and (3.12) can only depict the dropwindsonde motion in a steady fall. Thus, they will not be valid when the local wind driving the dropwindsonde dramatically changes its magnitude or direction. In other words, the simplified equations, and therefore the wind finding equations, may not be valid if the dropwindsonde reports dramatic vertical wind velocity changes. As indicated by Aberson et al. (2006), the dropwindsonde may encounter local extreme winds when it is released around an eye-wall. In this case, the dropwindsonde may be suspended, or even ascend, due to vertical winds pointing upwards, which means the vertical wind is not negligible in equation (3.7). The horizontal velocity difference, however, still remains small due to the strong driving force of local extreme winds. In this case, the simplified equations are not valid, but the conditions for the partial linearization are still satisfied. Moreover, the drag force equation, equation (3.8), which is derived from the partially linearized motion equation in the vertical direction, is also valid with some minor changes and can be rewritten as,
(w− z˙)2 = 2m(g− z¨)
ρACD (3.15)
When the vertical dropwindsonde acceleration in equation (3.15) is neglected, which indicates the vertical motion of the dropwindsonde is in an equilibrium state, moving either upwards or downwards, equation (3.15) can be simplified as,
|w− z˙|=
r
2mg
ρACD (3.16)
Substituting equation (3.16) into the linearized motion equations, equation (3.4), and rearranging gives,
u = − w g x¨+ ˙ z g x¨+ ˙x (3.17) v = − w g y¨+ ˙ z g y¨+ ˙y (3.18)
for the case ˙z > w, in which the dropwindsonde moves upwards in an updraft, or
u = w g x¨− ˙ z g x¨+ ˙x (3.19) v = w g y¨− ˙ z gy¨+ ˙y (3.20)
for the case w > z˙, in which the dropwindsonde moves downwards with non-negligible vertical winds.
When compared to the simplified motion governing equations (3.13) and (3.14), equa- tions (3.19) and (3.20) have one additional term, w/g, to account for the vertical wind influence on dropwindsonde motions.