9.8.2 Zonal friction

The lateral friction acting on the zonal velocity takes the expanded form

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

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

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

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

$${A \over a \, \cos\phi}( u_{, \lambda \lambda} \, \sec\phi + u_{, \phi \phi}\cos\phi + u_{, \phi}\, \sin\phi + u \, \sec\phi)$$ +

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

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

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

$${A \over a} \left( u_{, \lambda \lambda} \, \sec^{2}\phi + \sec\p... ...)_{, \phi} + u(1-\tan^{2}\phi) - 2 v_{, \lambda} \sec^{2}\phi \sin\phi \right),$$ + (9.184)

which renders

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

$${ 1 \over a \, \cos\phi} \; (D_{T} \, \partial_{\lambda}A + D_{S} \, \partial_{\phi}A \, \cos\phi).$$ + (9.185)

The second bracketed term in this expression arises from the spatial dependence of the viscosity coefficient, and should be present for any non-constant viscosity coefficient model. These non-constant viscosity coefficient terms amount to those identified by Wajsowicz (1993).

It is useful to perform one final step in the formulation in order to bring the non-constant viscosity coefficient inside the Laplacian. For this purpose, the Laplacian and non-constant viscosity coefficient terms are expanded to yield

$${A \over a^{2} \cos^{2}\phi} u_{,\lambda \lambda} + {A \over a^{2... ...i) \over a^{2}} - {2 \, A \, v_{,\lambda} \, \sin\phi \over a^{2} \cos^{2}\phi}$$ F^u =

$${A_{\lambda} \over a^{2} \cos^{2}\phi} (u_{,\lambda} - v_{,\phi} ... ...} \over a^{2} \cos \phi} (v_{,\lambda} + u_{,\phi} \, \cos\phi + u \, \sin\phi)$$ +

$${1 \over a^{2} \cos^{2}\phi} ( A \, u_{,\lambda \lambda} + A_{,\l... ...phi} ( A \, u_{,\phi} \cos\phi)_{,\phi} - A_{,\phi} \, u_{,\phi} a^{-2} \right)$$ =

$${A \, u \, (1-\tan^{2}\phi) \over a^{2}} - {2 \, A \, v_{,\lambda} \, \sin\phi \over a^{2} \cos^{2}\phi}$$ +

$${A_{,\lambda} \over a^{2} \cos^{2}\phi} (v_{,\phi} \, \cos\phi + ... ...\phi) + a^{-2} \, A_{\phi} ( v_{,\lambda} \sec\phi + u_{,\phi} + u \, \tan\phi)$$ -

$$\nabla_{h} \cdot (A \, \nabla_{h} u) + {A \, u \, (1-\tan^{2}\phi) \over a^{2}} - {2 \, A \, v_{,\lambda} \, \sin\phi \over a^{2} \cos^{2}\phi}$$ =

$${A_{,\lambda} \over a^{2} \cos\phi} (v_{,\phi} + v \, \tan\phi) + {A_{,\phi} \over a^{2} \cos\phi} ( v_{,\lambda} + u \, \sin\phi).$$ - (9.186)

This expression can be written as

 

$$\fbox{$ F^{u} = \nabla_{h} \cdot (A \, \nabla_{h} u) + old\_metric^{u} + new\_metric^{u}, $ }$$ (9.187)

where

$$\nabla_{h} \cdot (A \, \nabla_{h} u) = {1 \over a^{2} \cos^{2}\ph... ...ambda})_{,\lambda} + {1 \over a^{2} \cos\phi} (A \, u_{,\phi} \cos\phi)_{,\phi}$$ (9.188)

is the horizontal Laplacian with the generally non-constant viscosity coefficient inserted. Note that this Laplacian is acting on the zonal velocity as if it was a scalar field. The term

$$old\_metric^{u} = {A \, u \, (1-\tan^{2}\phi) \over a^{2}} - {2 \, A \, v_{,\lambda} \, \sin\phi \over a^{2} \cos^{2}\phi}$$ (9.189)

is the metric term employed for constant horizontal viscosity coefficient (Bryan 1969), and

$$new\_metric^{u} - {\partial_{\lambda}A \over a^{2} \cos\phi} (v_{,\phi} + v \, \tan\phi) + {\partial_{\phi}A \over a^{2} \cos\phi} ( v_{,\lambda} + u \, \sin\phi)$$ = (9.190)

is the metric term arising from spatial dependence in the viscosity coefficient (Wajsowicz 1993).