Download Coriolis effect - public.asu.edu

Coriolis effect
Turntable
L~1m
Ω ~ 0.1 Hz
Ω
Earth rotation: Ω = 1 cycle/day = 0.000011 Hz
HIGH
View from
above
L~1m
T ~ 1 min
-∇p
V
HIGH
LOW
L ~ 1000 km T > 1 day
Large-scale flow :
v perpendicular to ∇p
Rotation dominates (vorticity >> divergence) even if flow is not turbulent
We will see that the transformation of Navier-Stokes equations
to a rotating frame is equivalent to adding a "Coriolis force"
(and a "centrifugal force", which is however very small) to the
momentum equation.
Navier-Stokes equations in a rotating coordinate system
Coriolis & centrifugal forces
Basic setup: Consider a vector, r , that rotates
with angular velocity Ω and with its axis of
 (where
rotation pointing at the direction of 
 is a unit vector)



∆θ
A

S ≡ r t t − r t
B
 is the rotation vector
 ≡ 

S
 =  t
z
∣
S∣ =∣AB∣   (as   0 )
⇒ ∣
S∣ =∣AB∣   t
 r t
×




,
S ⊥ r t  and S ⊥  ⇒ S = 
∣×r t∣
where S is the unit vector in S direction
r t t
r t
O
y
x
 r t 
OA ⊥ AB ⇒ ∣OA∣= ⋅
 2∣r t∣2 − ⋅
 r t2 = ∣×
 r t ∣2
 r t 2 = ∣∣
⇒ ∣AB∣2 =∣OB∣2 −∣OA∣2 =∣r t∣2 −  ⋅
(See Appendix A at end of slide set)
 r t ∣ t =∣×
 r t ∣ ⇒ ∣
 r t ∣ t
⇒ ∣AB∣ =∣×
S∣ =∣×
r t  t − r t  




= × r t
Therefore, S ≡ ∣S∣S = ×r t   t , or
t
d r
 r
= ×
dt
This is the rate of change for the r-vector as measured in the inertial frame.
More precisely (the subscript "in" indicates "inertial" frame),
As  t  0 , we have

d r
 r .
 = ×
d t in
(1)
 will see the
An observer that rotates at the angular velocity and direction of 
r-vector as steady, not moving at all, i.e., (the subscript "rot" indicates rotating frame),

d r
 =0 .
d t r ot
(2)
Comparing (1) and (2), we see that the difference between 

d r
d r
 r .
 =
  ×
d t in
d t r ot
Note that the r here can be any vector.
d r
d r
 r ,
 and 
 is ×
d t in
d t r ot
(3)
1. Velocity
If is the vector of the location of an air parcel, would be its velocity so we have
 r .
v i n = v r o t  ×
2. Acceleration
Now, we need to apply the relation, 
d
d

 = [   ×]
, to v i n :
d t in
d t rot
d vi n
d vi n
 vin

 =
  ×
d t in
d t rot
d v
 d r   ×
 vrot  ×
 ×
 r
=  rot   ×
d t r ot
d t r ot
d v
 vrot  ×
 ×
 r
=  rot   2 ×
d t r ot
3. Equation of motion

d vi n
F
 =
Since Newton's law is 
, in the rotating frame we have
dt in m

d v
F
 vrot − ×
 ×
 r
 rot  = − 2 ×
d t rot m

F
1
= − ∇ p − g z for the fluid. The equation of motion (or
Previously, we have shown that
m

momentum equation) for the air parcel under the rotating frame can now be written as

d v
1
 v − ×
 ×
 r .
 = − ∇ p − g z − 
2 ×

dt

 A
 B
The momentum equation in the rotating coordinate system has two extra terms:
Term (A): Coriolis force Term (B): Centrifugal force
They are "fictitious forces" that arises from the coordinate transformation. They are
perpendicular to the velocity vector so can only act to change the "direction" of motion
but not the net kinetic energy of the flow.
Expanding 
d v
 into its Eulerian form, we have Navier-Stokes equations in the rotating frame:
dt
∂ v
1
 v − ×
 ×
 r .
= − v⋅∇ v − ∇ p − g z − 2
×

∂t

 A
 B
Appendix A
We will show that ∣A∣2∣
B∣2 −  A⋅
B2 = ∣A× 
B∣2 .
 =∣

By definition, ∣A× B∣
B vectors,
A∣∣B∣sin
 , where θ is the angle between 
A and 

while  A⋅
B =∣
A∣∣B∣cos
 . Thus, we have
 2  
 2 =∣
 2 cos 2 sin 2  =∣A∣2∣
∣A× B∣
A⋅B
A∣2∣B∣
B∣2 .