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Coriolis Force The Cross Product 1 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… 3 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 4 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 5 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 6 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. 7