Total Differentiation of a Vector in a
Rotating Frame of Reference
• Before we can write Newton’s second law of motion for a
reference frame rotating with the earth, we need to
develop a relationship between the total derivative of a
vector in an inertial reference frame and the
corresponding derivative in a rotating system.
r
A
Let
be an arbitrary vector with Cartesian components
r
A = Ax iî + Ay ˆj + Az kk̂
r
A = Ax′ î ′ + A′y ˆj ′ + Az′ k̂ ′
If
i an inertial
in
i
i l frame
f
off reference,
f
and
d
in a rotating frame of reference.
r
A = Ax î + Ay ˆj + Az k̂ in an inertial frame of reference, then
r
dA ⎛ diˆ ˆ dAx ⎞ ⎛ dˆj ˆ dAy ⎞ ⎛⎜ dkˆ ˆ dAz
⎟ + ⎜ Ay + j
⎟ + Az
= ⎜ Ax + i
+k
dt ⎜⎝ dt
dt ⎟⎠ ⎜⎝ dt
dt ⎟⎠ ⎜⎝ dt
dt
⎞
⎟
⎟
⎠
Since the coordinate axes are in an inertial frame of reference,
diˆ dˆj dkˆ
= =
=0
dt dt dt
r
dA ⎛ diˆ ˆ dAx ⎞ ⎛ dˆj ˆ dAy ⎞ ⎜⎛ dkˆ ˆ dAz
⎟ + ⎜ Ay + j
⎟ + Az
+k
= ⎜ Ax + i
dt ⎜⎝ dt
dt ⎟⎠ ⎜⎝ dt
dt ⎟⎠ ⎜⎝ dt
dt
r
dA dAx ˆ dAy ˆ dAz ˆ
i+
=
j+
k
dt
dt
dt
dt
⎞
⎟
⎟
⎠
(Eq. 1)
1
r
A = Ax′ iˆ′ + A′y ˆj ′ + Az′ kˆ′ in a rotating frame of reference, then
If
r
dA ⎛ diˆ′ ˆ dAx′ ⎞ ⎛ dˆj ′ ˆ dA′y ⎞ ⎛⎜ dkˆ′ ˆ dAz′ ⎞⎟
⎟ + Az′
⎟ + ⎜ A′y
+ k′
+ j′
+i′
= ⎜ Ax′
(Eq. 2)
dt ⎜⎝
dt
dt ⎟⎠ ⎜⎝ dt
dt ⎟⎠ ⎜⎝
dt
dt ⎟⎠
Because the left hand sides of Eq. 1 and Eq. 2 are identical,
dAx ˆ dAy ˆ dAz ˆ ⎛ diˆ′ ˆ dAx′ ⎞ ⎛ dˆj ′ ˆ dA′y ⎞ ⎛⎜ dkˆ′ ˆ dAz′ ⎞⎟
⎟ + Az′
⎟ + ⎜ A′y
+ k′
+ j′
+i′
i+
j+
k = ⎜⎜ Ax′
dt
dt
dt
dt
dt ⎟⎠ ⎜⎝ dt
dt ⎟⎠ ⎜⎝
dt
dt ⎟⎠
⎝
Regrouping the terms
dAx ˆ dAy ˆ dAz ˆ dAx′ ˆ dA′y ˆ dAz′ ˆ
i+
j+
k=
i′+
j′ +
k ′ + Ax′
dt
dt
dt
dt
dt
dt
r
r
⎛ dA ⎞
⎛ dA ⎞
⎜ ⎟
⎜ ⎟
⎜ dt ⎟
⎜ dt ⎟
⎝ ⎠ rotating
⎝ ⎠inertial
diˆ′
dˆj ′
dkˆ′
+ A′y
+ Az′
dt
dt
dt
dAx ˆ dAy ˆ dAz ˆ dAx′ ˆ dA′y ˆ dAz′ ˆ
j+
k=
j′ +
k ′ + Ax′
i+
i′+
dt
dt
dt
dt
dt
dt
r
r
⎛ dA ⎞
⎛ dA ⎞
⎜ ⎟
⎜ ⎟
⎜ dt ⎟
⎜ dt ⎟
⎝ ⎠ rotating
⎝ ⎠inertial
diˆ′
dˆj ′
dkˆ′
+ Az′
+ A′y
dt
dt
dt
To interpret
effects of rotation
diˆ′ dˆj ′ dkˆ′
,
,
think of each unit vector as a position vector.
dt dt dt
linear velocity = angular velocity x position vector →
Because
δθ
r
effects of rotation
δr
Thus
r drr
V=
,
dt
diˆ′ r ˆ
= Ω × i ′,
dt
r r r
V = Ω×r
r
dr r r
= Ω×r
dt
dˆj ′ r ˆ
= Ω × j ′,
dt
dkˆ′ r ˆ
= Ω × k′
dt
2
(
r
r
r
dAx ˆ dAy ˆ dAz ˆ dAx′ ˆ dA′y ˆ dAz′ ˆ
i+
i′+
k ′ + Ax′ Ω × iˆ′ + A′y Ω × ˆj ′ + Az′ Ω × kˆ′
j+
k=
j′ +
dt
dt
dt
dt
dt
dt
(
r
⎛ dA ⎞
⎜ ⎟
⎜ dt ⎟
⎝ ⎠ rotating
r
⎛ dA ⎞
⎜ ⎟
⎜ dt ⎟
⎝ ⎠inertial
r
r
r r
⎛ dA ⎞
⎛ dA ⎞
⎜ ⎟
= ⎜⎜ ⎟⎟
+ Ω× A
⎜ dt ⎟
⎝ ⎠inertial ⎝ dt ⎠ rotating
)
(
)
)
r r
Ω× A
(effects of rotation)
This equation provides us with a
formal way of expressing the
balance of forces on a fluid parcel
in a rotating coordinate system.
We will now start with Newton
Newton’s
s second law in an
inertial reference frame and transform it to rotating
coordinates:
r
r
F
⎛ dV ⎞
∑
⎜
⎟
=
⎜ dt ⎟
m
⎝
⎠inertial
r
Substitute r, the position vector for an air parcel on the rotating earth, into
r
r
r r
⎛ dA ⎞
⎛ dA ⎞
⎜ ⎟
⎜ ⎟
=
+
Ω
× A,
⎜ dt ⎟
⎜ ⎟
⎝ ⎠inertial ⎝ dt ⎠ rotating
the transformation of the total derivative, to g
get:
r
r
r r
⎛ dr ⎞
⎛ dr ⎞
+Ω ×r
=
⎜ ⎟
⎜ ⎟
⎝ dt ⎠inertial ⎝ dt ⎠ rotating
Because velocity is the rate of change of the position vector with time, this
can be rewritten:
r drr
V=
dt
Now substitute
r
r r r
⇒ Vinertial = V + Ω × r
r
Vinertial into the transformation of the total derivative:
r
⎛ dVinertial
⎜
⎜ dt
⎝
r
⎞
dVinertial r r
⎟
=
+ Ω × Vinertial
⎟
dt
⎠inertial
3
r
r
r
⎛ dVinertial ⎞
dVinertial r r
⎟
=
+ Ω × Vinertial to get
Substitute for Vinertial in ⎜⎜
⎟
dt
⎝ dt ⎠inertial
r
r r r r
⎛ dV ⎞
d r r r
⎟
⎜
=
V +Ω ×r +Ω × V +Ω ×r
⎜ dt ⎟
⎝
⎠inertial dt
(
)
(
)
r
Using some vector identities and defining R as a vector perpendicular to the axis
of rotation with magnitude equal to the distance to the axis of rotation:
r
r
r r
r
⎛ dV ⎞
dV
⎟
⎜
=
+ 2Ω × V − Ω 2 R
⎜ dt ⎟
dt
⎝
⎠inertial
i ti l
Acceleration
following the
motion in an
inertial system
Rate of change
of relative
velocity following
the relative
motion in a
rotating
reference frame.
Coriolis
acceleration
Centrifugal
acceleration
Substituting into Newton’s second law:
r
r
F
⎛ dV ⎞
∑
⎜
⎟
=
⎜ dt ⎟
m
⎝
⎠ inertial
r
r
r r
r ∑F
dV
2
+ 2Ω × V − Ω R =
m
dt
If the real forces acting on a fluid parcel are the pressure gradient
force, gravitation and friction, then
r
r r 1
r r
dV
= −2Ω × V − ∇p + g + Fr
ρ
dt
Rate of change
of
f relative
velocity following
the relative
motion in a
rotating
reference frame.
Coriolis
acceleration
Pressure
gradient force
(per unit mass)
Gravity term
(gravitation +
centrifugal)
Friction
Vector momentum equation in rotating coordinates
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