1. Rotating frame 2. Coriolis force
3. Centrifugal force
Forces
Pressure
gradient Coriolis
Rotating Frame
The earth rotates with a period about 24 hours (23h 56m; or 366 rotation per year), although we can not feel it (human love geocentric view).
Ω = 7.292 x 10-5 rad/s (directing the North star)
Ω !
v ! = !
Ω × ! r r !
R!
Coriolis force
Newtons’ laws of motion work only a inertial frame.
However, we can still use Newtons’ law in a rotating frame in a uniform rotation, if we introduce two apparent forces:
1) Coriolis and 2) Centrifugal forces
Ω !
v ! = !
Ω × ! r r !
R! relationship working
for any vector A
Inertial frame vs. rotation frame
Da ! A
Dt = Dr ! A
Dt + !
Ω × ! A
Can be obtatined through
(1) vector calculation - Holton (2) intuition - Vallis
Relationship between total differentiations in
a inertial frame (DaA/Dt) and a rotating frame (DrA/Dt)
Ω !
v ! = !
Ω × ! r r !
R! relationship working
for any vector A
Velocity (wind)
Da! r
Dt = Dr! r
Dt + !
Ω × ! r
Relationship between total differentiations in
a inertial frame (DaA/Dt) and a rotating frame (DrA/Dt) Da !
A
Dt = Dr ! A
Dt + !
Ω × ! A
v!a = !
vr + !
Ω × ! r
Ω !
v ! = !
Ω × ! r r !
R!
Acceleration
Relationship between total differentiations in
a inertial frame (DaA/Dt) and a rotating frame (DrA/Dt)
recall
relationship working for any vector A
v!a = !
vr + !
Ω × ! r Da!
va
Dt = Dr! va
Dt + !
Ω × ! va
Da! va
Dt = Dr !
vr + !
Ω × !
(
r)
Dt + !
Ω × !
vr + !
Ω × !
(
r)
Da! va
Dt = Dr! vr
Dt + 2 !
Ω × !
vr + !
Ω × !
Ω × !
(
r)
Da ! A
Dt = Dr ! A
Dt + !
Ω × ! A
Acceleration
Coriolis force
Da! va
Dt = Dr! vr
Dt + 2 !
Ω × !
vr + !
Ω × !
Ω × !
(
r)
Centrifugal force (= −Ω2 ! R)
Da! va
Dt = − 1
ρ ∇p − kg*+ν∇2v!a Dr!
vr
Dt = −2 !
Ω × !
vr +Ω2 R!"
− kg*− 1
ρ ∇p+ν∇2v!
a
D! v
Dt = −2 !
Ω × !
v − 1
ρ ∇p − kg+ν∇2v!
Momentum equation in a rotating frame (with rotation rate Ω)
Handle as one term using the concept of "geoid"
(max of Ω2R is ~0.034 m/s2; just slightly modify g*➜g)
viscous force is local
cannot feel rotation va ➜ v
Equation set
Six equations with six known (u, v, w, T, p, ρ) Mathematically closed; and solvable
D! v
Dt = − 1
ρ ∇p − 2
Ω! × !
v − kg +ν∇2v!
cp DT
Dt − 1 ρ
Dp
Dt = J
Dρ
Dt + ρ∇ ⋅ !
v = 0 p = ρRT
(momentume eq. for u, v, w)
(thermodynamic energy eq. for T) (continuity eq. for ρ)
(equation of state for p) 3
1
1 1 6
num.
eq.
