Download Coriolis Force The Cross Product

Coriolis Force
The Cross Product
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Properties of the Vector Cross
Product
B
A
θ
Note:
A X B is orthogonal to and A and to B
When A and B are parallel A X B = 0.
Rotational Kinematics and the
Cross Product
Ω = angular speed
V
θ
R
2
Ω
y
z
r
x
R
rate of change relative
to fixed reference
point on earth
Let…
Note…
rate of change of the
fixed reference
relative to the earth’s
center
= rate of change relative to a fixed point
on earth
A fancy way of saying linear
velocity due to rotation is radius
times angular velocity
Eq 1
The velocity of a parcel of water is equal to the
Eq 2
observed velocity plus the velocity due to the rotation
of the earth
The next step is to determine the expression for observed acceleration and
the acceleration due to the rotation of the earth.
Let…
Define an operator….
Using Eq 2…
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Using the
derivative
chain rule
because steady rotation of the earth…
acceleration relative to
a fixed point on the
surface of the earth
acceleration of
the fixed point
on the surface of
the earth
Ω
j
Ignored because it is independent
of velocity and magnitude is small
k
location on earth
observed velocity
rotational velocity
φ
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horizontal terms of Coriolis
vertical term of Coriolis
Ignoring…
Define a new parameter called the Coriolis Parameter:
Acceleration of the reference point on
the surface of the earth
full acceleration
of a point moving
relative to the
center of the
earth
acceleration
observed relative
to a reference
point fixed on
the earth surface
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small compared to other terms
Ignored because length
scales over which w acts is
small
If no external forces applied
fv and -fu are the Coriolis force
components that act
perpendicular to the direction
of motion.
Navier-Stokes Equation
Adding in Coriolis Force…
u component
v component
w component
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A parcel of fluid moving in the northward direction (v) reduces its distance from the axis
of rotation (R). In order to conserve angular momentum, the parcel of fluid must rotate
faster in the eastward direction (u - directed into the page). The apparent force used to
describe the sudden eastward motion is the Coriolis force (acting to the right of the
direction of northward motion). Note that the more rapid the parcel of fluid in moved to
the north, the more rapid the parcel must increase it velocity (i.e., accelerate) to the
right meaning the stronger the Coriolis force.
A parcel of fluid with a velocity directed in the eastward direction (u - directed into the
page) produces a centrifugal force that is directed outward and perpendicular to the axis of
rotation. This centrifugal force can be broken down into separate force components. The
force component that is directed normal to the surface is extremely small relative to gravity
and can be reasonably ignored. The force component that is tangential to the surface,
and southward directed, is the Coriolis force (acting to the right of the direction of motion).
Note that a stronger motion in the u direction produces a stronger centrifugal force and a
stronger Coriolis force pushing the parcel to the right.
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