9.8.1 Spherical form of second order friction

In spherical coordinates, the metric takes the form

$$\mbox{diag} \, (g_{\lambda \lambda}, g_{\phi \phi}, g_{rr})$$ g_ij =

$$\mbox{diag} \, (a^{2} \cos^{2}\phi, a^{2}, 1)$$ = (9.177)

and

$$(\delta x, \delta y, \delta z) (a \cos\phi \, \delta \lambda, a \, \delta \phi, \delta r).$$ = (9.178)

Consequently, it is only $g_{11} = g_{\lambda \lambda}$ which has nonzero spatial dependence. The friction then can be written

  

$$\frac{\partial (A \, D_{T})}{\partial x} + \frac{1}{\cos^{2}\phi}... ...ac{\partial (A \, \cos^{2}\phi \, D_{S})}{\partial y} + (\kappa \, u_{,z})_{,z}$$ F^x = (9.179)

$$\frac{\partial (A \, D_{S})}{\partial x} - \frac{1}{\cos^{2}\phi}... ...c{\partial (A \, \cos^{2}\phi \, D_{T})}{\partial y} + (\kappa \, v_{,z})_{,z},$$ F^y = (9.180)

where

$$u_{,x} - v_{,y} - (v/a) \, \tan\phi$$ D_T =

$$(a \cos\phi)^{-1} \, (u_{,\lambda} - v_{,\phi} \, \cos\phi - v \, \sin\phi)$$ =

$$v_{,x} + \cos\phi \, (u/\cos\phi)_{,y}$$ D_S =

$$(a \cos\phi)^{-1} \, (v_{,\lambda} + u_{,\phi} \, \cos\phi + u \, \sin\phi).$$ = (9.181)

The terms

  

$$\frac{1}{a \, \cos\phi} \, \frac{\partial (A \, D_{T})}{\partial ... ...{a \, \cos^{2}\phi} \frac{\partial (A \, \cos^{2}\phi \, D_{S})}{\partial \phi}$$ F^u = (9.182)

$$\frac{1}{a \, \cos\phi} \, \frac{\partial (A \, D_{S})}{\partial ... ...{a \, \cos^{2}\phi} \frac{\partial (A \, \cos^{2}\phi \, D_{T})}{\partial \phi}$$ F^v = (9.183)

can be massaged into the form presented by Bryan (1969) and Wajsowicz (1993); that is the purpose of the remainder of this section.