Governing Eulerian Equations in a Rotating Frame

By use of the result obtained from Newtonian mechanics, relating the acceleration a and a, we can convert (1.78), which is written in terms of O, into a momentum balance as seen from an observer in O. 

In fluid mechanics the time rate of change of a vector tp is then written as Dtp/Dt = Dip/Dt + il xtp in analogy to the result from classical mechanics. The above relation for the generalized vector tp is applied to a fluid parcel s position r and then to its velocity v, leading to the relation  

Note that the rate of change of ft is the same in the rotating frame as in an absolute frame [6]. 

Assuming that Dw/Dt is equal to the local force acting per unit mass on a fluid parcel, the apparent forces in the rotating frame are derived. The above derivation can be called a Lagrangian approach since it exploits the concept of the fluid parcel . In this framework we define  

Recently, an Eulerian derivation of the Coriolis force has been reported by Kageyama and Hyodo [45]. They present a general procedure to derive the transformed equations in the rotating frame of reference based on the local Galilean transformation and rotational coordinate transformation of field quantities. 

The momentum equation (1.86) in O is then transformed from the flux form to the advective (non-flux) form  

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