Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 224

with ω the vector angular velocity of the rotation. Applied to the radius vector r ,
the velocity observed in the fixed frame, v fix , is related to the velocity observed in
the rotating frame, v rot ,by

v fix = v rot + ω × r . (3.73)

Taking time derivatives with respect to the fixed frame on both sides, and using
the kinematic relation (3.72) to express those on the right as time derivatives in the
rotating frame, the acceleration in the fixed frame, a fix , is related to the acceleration
in the rotating frame, a rot ,by

a fix = a rot + ω × v rot + α rot × r + ω × v rot + ω × ( ω × r )

= a rot + 2 ω × v rot + ω × ( ω × r ) + α rot × r . (3.74)

Thus, the acceleration measured in the rotating frame is augmented by the Coriolis
acceleration, 2 ω × v rot , the centripetal acceleration, ω × ( ω × r ) and the Poincare

acceleration, α rot × r , where α rot is the vector angular acceleration rate, if any,

observed in the rotating frame.
By d'Alembert's principle, the negatives of these accelerations appear as body
forces per unit mass in the equations of motion. Usually, the reference frame of the
Earth is taken to be in uniform rotation at a fixed rate, Ω , and we are left with the
Coriolis force per unit mass,

− 2 Ω × v , (3.75)

and the centrifugal force per unit mass,

− × ( Ω × r ) ,

Ω (3.76)

where v is the velocity in the Earth frame and r is the radius vector.
The centrifugal force has already been incorporated through the total geopo-
tential, described by equation (5.5), whose negative gradient is the equilibrium
gravity g 0 appearing in the equilibrium equation (3.19). For small harmonic dis-
placements at angular frequency ω, their time dependence may be represented by
the phasor e i ω t , and the body force per unit volume, arising from dynamical terms,
becomes

= ρ 0 ω u ,

2 u

F − 2 i ω Ω × (3.77)

with ρ 0 representing the equilibrium density, and u representing the vector dis-
placement field.