Derivation of Coriolis Effect
3
Derivation of Coriolis Effect Cylindrical Coordinates Set v ( r,θ,z) =v r ^ r + ( rΩ +v θ ) ^ θ+v z ^ z a | ¿ | = d v dt = dv r dt ^ r +v r d ^ r dt + d ( rΩ+v θ ) dt ^ θ+ ( rΩ +v θ ) d ^ θ dt + dv z dt ^ z ¿ ¿ ( dv r dt −rΩ 2 −2 Ωv θ − v θ 2 r ) ^ r + ( dv θ dt +2 Ωv r + v r v θ r ) ^ θ+ dv z dt ^ z Note d ^ r dt = v θ r ^ θ, d ^ θ dt = −v θ r ^ r Now a frame = a | ¿ | frame + a | ¿ |¿ Where a | ¿ | frame =rΩ 2 ^ r Thus for the coordinate rotating simultaneously with the
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Transcript of Derivation of Coriolis Effect
Derivation of Coriolis EffectCylindrical CoordinatesSet
v (r ,θ , z)=vr r+( r Ω+vθ ) θ+vz za|¿|=d v
dt=d v rdtr+v r
d rdt
+d ( rΩ+v θ)
dtθ+ (rΩ+vθ ) d θdt
+d vzdtz ¿
¿( d vrdt −r Ω2−2Ωvθ−vθ2
r ) r+( d vθdt +2Ωvr+vr vθr )θ+ d vzdt z
Note
d rdt
=vθrθ , d θdt
=−vθrr
Nowa frame=a|¿|frame+ a|¿|¿
Wherea|¿|frame=r Ω
2 r
Thus for the coordinate rotating simultaneously with the disk, an object with the velocity
v frame=vr r+vθ θ+v z z
Feels an acceleration of
a frame=( d vrdt −2Ωvθ−vθ2
r ) r+( d vθdt +2Ωvr+vr vθr ) θ+ d v zdt z
d vrdtr+d vθdtθ=a frame+(2Ωvθ+ vθ2r ) r−(2Ωvr+
vr vθr
)θ
So the Coriolis Effect and the curvature effect is the additional terms on the right, i.e.
f=vθ(2Ω+vθr ) r−vr(2Ω+
vθr ) θ= v frame×( k+κ)
k=2Ω z , κ=vθrz
In Spherical Coordinates, the effect is depicted as the following:
Thus we have
f=(vθ(2Ω+vθrcosφ )cosφ+ vφ
2
r ) r+(vφ sinφ−vr cosφ )(2Ω+vθrcosφ ) θ−(v¿¿θ(2Ω+
vθrcosφ )sinφ+
vr v φr
) φ¿
Which is the Coriolis Effect we have on the surface of Earth. Note that there exist an additional angular velocity on the latitude direction, and thus we have two directions of angular momentum, the theta direction and the phi direction.
