Momentum equation
The inviscid momentum equation is \(\boxed{ \frac{D \mathbf{v}}{Dt} + 2 \boldsymbol{\Omega} \times \mathbf{v} = -\frac{1}{\rho} \nabla p - \nabla \Phi} \) where \( \Phi \) is the geopotential. In spherical coordinates, accounting for the changing directions of coordinate axes, the component expansion of the inviscid momentum equation is \[ \frac{D \mathbf{v}}{Dt} = \frac{D u}{Dt} \mathbf{i} + \frac{D v}{Dt} \mathbf{j} + \frac{D w}{Dt} \mathbf{k} + u \frac{D \mathbf{i}}{Dt} + v \frac{D \mathbf{j}}{Dt} + w \frac{D \mathbf{k}}{Dt} \text{ or } = \frac{D u}{Dt} \mathbf{i} + \frac{D v}{Dt} \mathbf{j} + \frac{D w}{Dt} \mathbf{k} + \boldsymbol{\Omega}_{\text{flow}} \times \mathbf{v} \] Using rate of change of unit vectors and the expressions for \( \boldsymbol{\Omega}_{\text{flow}} \) from the total rotation rate of a vector that moves with the flow \(\boldsymbol{\Omega}_{\text{flow}} = -\mathbf{i} \frac{v}{r} + \mathbf{j} \frac{u}{r} + \mathbf{k} \frac{u \tan \vartheta}{r}\), we find \[ \frac{D \mathbf{v}}{Dt} = \mathbf{i} \left( \frac{D u}{Dt} - \frac{u w \tan \vartheta}{r} + \frac{u w}{r} \right) + \mathbf{j} \left( \frac{D v}{Dt} + \frac{u^2 \tan \vartheta}{r} + \frac{v w}{r} \right) + \mathbf{k} \left( \frac{D w}{Dt} - \frac{u^2 + v^2}{r} \right) \] Coriolis Term Expansion
Using the vector cross product definition, the Coriolis term is \[ 2 \boldsymbol{\Omega} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0 & 2 \Omega \cos \vartheta & 2 \Omega \sin \vartheta \\ u & v & w \end{vmatrix} = \mathbf{i} (2 \Omega w \cos \vartheta - 2 \Omega v \sin \vartheta) + \mathbf{j} (2 \Omega u \sin \vartheta) - \mathbf{k} (2 \Omega u \cos \vartheta) \] Final expanded momentum equations Substituting \(\frac{D \mathbf{v}}{Dt} \) and the Coriolis term into \(\frac{D \mathbf{v}}{Dt} + 2 \boldsymbol{\Omega} \times \mathbf{v} = -\frac{1}{\rho} \nabla p - \nabla \Phi\), and using the gradient operator given by \(\nabla \phi = \mathbf{i} \, \frac{1}{r \cos \vartheta} \frac{\partial \phi}{\partial \lambda} + \mathbf{j} \, \frac{1}{r} \frac{\partial \phi}{\partial \vartheta} + \mathbf{k} \, \frac{\partial \phi}{\partial r}\)(the vector gradient of a scalar), the momentum equations become \[ \frac{D u}{Dt} - \left(2\Omega + \frac{u}{r \cos \vartheta} \right) (v \sin \vartheta - w \cos \vartheta) = -\frac{1}{\rho r \cos \vartheta} \frac{\partial p}{\partial \lambda} \] \[ \frac{D v}{Dt} + \frac{w v}{r} + \left(2 \Omega + \frac{u}{r \cos \vartheta} \right) u \sin \vartheta = -\frac{1}{\rho r} \frac{\partial p}{\partial \vartheta} \] \[ \frac{D w}{Dt} - \frac{u^2 + v^2}{r} - 2\Omega u \cos \vartheta = -\frac{1}{\rho} \frac{\partial p}{\partial r} - g \] The terms involving \( \Omega \) are Coriolis terms, and the terms involving \( 1/r \) are called metric terms