Appendix 2: The equations of motion in two dimensional
orthogonal cordinates

The basic equations of motion (1) written in vector form are

$${\partial {\vec v} \over\partial t}+ {\vec v}\cdot \nabla {\vec v} = -{1\over \Sigma}\nabla \Pi - \nabla \Phi.$$ (78)

We wish to write these in component form in a general two dimensional orthogonal coordinate system $(a,\lambda)$ in the Cartesian (x,y)plane as introduced in Sect. 5. The unit vectors in these orthogonal coordinate directions are ${\vec i}_{\rm a} = \nabla a/\vert\nabla a\vert$ and ${\vec i}_{\lambda} = \nabla \lambda/\vert\nabla \lambda\vert$respectively. The orthogonal vertical coordinate z has an additional associated orthogonal unit vector ${\vec {\hat k}} ={\vec i}_{\rm a} \times {\vec i}_{\lambda}.$However, there is no dependence on z and the velocity component in that direction can be taken to be zero.

Thus we may write ${\vec v}\equiv ( v_{\rm a}, v_{\lambda})$or

$${\vec v} = v_{\rm a} {\vec i}_{\rm a} + v_{\lambda} {\vec i}_{\lambda}.$$ (79)

To deal with Eq. (.78), we first use the vector identity

$${\vec v}\cdot \nabla {\vec v}= {1\over 2} \nabla( \vert{\vec v}\vert^2) + {\vec \omega} \times {\vec v} ,$$ (80)

where ${\vec \omega} = \nabla \times {\vec v}$ is the vorticity (Arfken & Weber 2000).

Because we only wish to consider the equations in a two dimensional limit, only the z component of vorticity, $\omega_z$ is non zero so that we may write Eq. (.78) as

$${\partial {\vec v} \over\partial t}+ \omega_z {\vec {\hat k}... ...ert{\vec v}\vert^2)= -{1\over \Sigma}\nabla \Pi - \nabla \Phi.$$ (81)

We may now write Eq. (.81) in component form using the identities (see Arfken & Weber 2000)

$$\nabla \equiv \nabla a {\partial \over \partial a} + \nabla \lambda {\partial \over \partial \lambda }$$ (82)

and

$$\omega_z =\left(\nabla \times {\vec v}\right) \cdot {\vec {\h... ...la {\rm a}\vert}\right) \over \partial \lambda } \right )\cdot$$ (83)

Doing this we obtain

 

$${\partial v_{\rm a} \over\partial t}- \omega_z v _{\lambda} +{1\o... ...Pi \over \partial a} - \vert\nabla a\vert {\partial \Phi \over \partial a}\cdot$$ (84)

and

 

$${\partial v_{\lambda} \over\partial t}+ \omega_z v _{\rm a} +{1\o... ...lambda} - \vert\nabla \lambda \vert {\partial \Phi \over \partial \lambda}\cdot$$ (85)

After inserting (.83) into (.84) and (.85) it is a simple matter to obtain Eqs. (34) and (35) as given in Sect. 5.