Equations of steady motion are as usual:-
where
k is a coefficient of friction.
V denotes the absolute velocity of wind, and then
u=
Vcosδcosα,
v=
Vcosδsinα,
w=
Vsinδ where δ is the slope angle of the earth.
Pressure gradient is in direction
n, and Notations α, β, δ,
n, … are shown in fig. 1. After some simple transformations, we get (β-α) is an angle between pressure gradient and wind. (If the motion is only horizontal, the equation becomes This is Gull berg and Mohn's formula.)
The equation shows that the angle (β-α) is a function of variables α, δ and φ. For example, of the wind is southerly, we get tg (β-α) 0 and so (β-α) 0 as δ tends to φ, also the slope angle δ equals to latitude φ In such a case, the wind blows perpendicularly to isobars.
Thus, we know that (β-α) varies widely with δ, the slope angle of the earth.
Numerical example:
k is 7.6×10
-5.
Two cases (uortherly and westerly winds) are calculated on the latitudes 20°, 30° and 40°. Fig. 2 is the graph.
From these discussions, we know that slope effect is not smaller than the other effec_??_s (∂/∂
t,
w∂/∂
_??_ and so on), and we may say:- The angle between pressure gra_??_ient and wind fluctuates greatly as the result of slope effect, and not of acceleration or other factors.
If we use Naver-Stokes' equations, we get almost the same results in free air, and it may be said:-
Whem there is some vertical current, wind direction fluctuates greatly. The so-called “Ablenkungs Winkel” from geostrophic wind is not always the result of acceleration in the meaning of Ertel, Brunt-Douglas and others.
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