9.8.3 Meridional friction

Repeating the exercise just performed for the zonal friction yields the following lines of algebra for the meridional friction

$${ 1 \over \cos\phi} \; ( D_{S} \, \partial_{\lambda} A + A \, \partial_{\lambda} \, D_{S})$$ a F^v =

$${ 1 \over \cos^{2}\phi} \; ( D_{T} \, \partial_{\phi} A \, \cos^{... ... \partial_{\phi} D_{T} \, \cos^{2}\phi - 2 \, A \, D_{T} \, \cos\phi \sin\phi )$$ -

$${ 1 \over \cos\phi} \; (D_{S} \, \partial_{\lambda}A - D_{T} \, \partial_{\phi}A \, \cos\phi)$$ =

$${2 A \over a \, \cos\phi} (u_{, \lambda} \tan\phi - v_{,\phi} \tan\phi \cos\phi - v \tan\phi \sin\phi)$$ +

$${A \over a\, \cos\phi} \left( v_{, \lambda \lambda} \, \sec\phi + v_{, \phi \phi} \, \cos\phi + v_{, \phi} \, \sin\phi + v \, \sec\phi \right)$$ +

$${ 1 \over \cos\phi} \; (D_{S} \, \partial_{\lambda}A - D_{T} \, \partial_{\phi}A \, \cos\phi)$$ =

$${A \over a} \left( v_{, \lambda \lambda} \sec^{2}\phi + v_{, \phi... ...(\sec^{2}\phi - 2 \tan^{2}\phi) + 2 u_{, \lambda} \sec^{2}\phi \sin\phi \right)$$ +

$${ 1 \over \cos\phi} \; (D_{S} \, \partial_{\lambda}A - D_{T} \, \partial_{\phi}A \, \cos\phi)$$ =

$${A \over a} \left( v_{, \lambda \lambda} \sec^{2}\phi + \sec\phi ... ...{,\phi} + v (1 - \tan^{2}\phi) + 2 u_{, \lambda} \sec^{2}\phi \sin\phi \right),$$ + (9.191)

which renders

$$A \left( \nabla^{2}_{h} \; v + {v(1-\tan^{2}\phi) \over a^{2}} + {2 u_{, \lambda} \, \sin\phi \over a^{2} \cos^{2}\phi} \right)$$ F^v =

$${ 1 \over a \, \cos\phi} \; (D_{S} \, \partial_{\lambda}A - D_{T} \, \partial_{\phi}A \,\cos\phi).$$ + (9.192)

Again, it is useful to expand this friction one more step in order to bring the viscosity coefficient inside the Laplacian. This manipulation yields

 

$$\fbox{$ F^{v} = \nabla_{h} \cdot (A \, \nabla_{h} v) + old\_metric^{v} + new\_metric^{v}, $ }$$ (9.193)

where

$$old\_metric^{v} = {A \, v \, (1-\tan^{2}\phi) \over a^{2}} + {2 \, A \, u_{, \lambda} \, \sin\phi \over a^{2} \cos^{2}\phi}$$ (9.194)

is the metric employed for constant horizontal viscosity coefficient, and

$$new\_metric^{v} = {\partial_{\lambda}A \over a^{2} \cos\phi} (u_{... ...hi) + {\partial_{\phi}A \over a^{2} \cos\phi} (-u_{, \lambda} + v \, \sin\phi).$$ (9.195)