9.4.5 Horizontal tension and shearing rate of strain

Given the form for the stress tensor in equation (9.100), it is useful to follow Smagorinsky (1993) by introducing the horizontal tension D_T

 

D_T = e¹₁ - e²₂

$$u^{1}_{\; ; 1} - u^{2}_{\; ; 2}$$ =

$$u^{1}_{\; , 1} - u^{2}_{\; , 2} + u^{m} \, ( \Gamma^{1}_{1m} - \Gamma^{2}_{2m})$$ =

$$u^{1}_{\; , 1} - u^{2}_{\; , 2} + \frac{1}{2} \, u^{m} \, \partial_{m} \, \ln (g_{11}/g_{22})$$ =

$$\sqrt{\frac{g_{22}}{g_{11}}} \, \left( u^{1} \, \sqrt{\frac{g_{11... ...{11}}} \, \right)_{,2} + u^{3} \, \partial_{3} \ln \sqrt{\frac{g_{11}}{g_{22}}}$$ =

$$\sqrt{g_{22}} \, (u / \sqrt{g_{22}} \, )_{,x} - \sqrt{g_{11}} \, (v / \sqrt{g_{11}} \, )_{,y},$$ = (9.103)

where the last step wrote D_T in terms of the physical velocity and differential components, and the $\partial_{z}$ term was dropped based on assuming the metric components are independent of depth as implied by the quasi-hydrostatic approximation (Section 9.3.7). In Cartesian coordinates, D_T = u_,x - v_,y. It is also useful to introduce the horizontal shearing strain D_S which is given by

$$2 \, \sqrt{\mathcal{G}} \, e^{12}$$ D_S =

$$2 \, \sqrt{\mathcal{G}} \, g^{11} \, e^{2}_{1}.$$ = (9.104)

A bit of work yields

$$2 \, e^{2}_{1} u^{2}_{\; ;1} + g^{22} \, g_{11} \, u^{1}_{\; ; 2}$$ =

$$u^{2}_{\; ,1} + \Gamma^{2}_{1m} \, u^{m} + g^{22} \, g_{11} \, ( u^{1}_{\; , 2} + \Gamma^{1}_{2m} \, u^{m})$$ =

$$u^{2}_{\; ,1} + \frac{1}{2} \, g^{2d}\, (g_{1d,m} + g_{md,1} - g_{1m,d}) \, u^{m} + g^{22} \, g_{11} \, u^{1}_{\; , 2}$$ =

$$\frac{1}{2} \, g^{22} \, g_{11} \, g^{1d} \, (g_{2d,m} + g_{md,2} - g_{2m,d}) \, u^{m}$$ +

$$g^{22} \, ( g_{11} \, u^{1}_{ \; ,2} + g_{22} \, u^{2}_{\; , 1}),$$ = (9.105)

which brings the horizontal shearing strain to

$$\mathcal{G}^{-1/2} \, (g_{11} \, u^{1}_{,2} + g_{22} \, u^{2}_{,1})$$ D_S =

$$\sqrt{g_{11}} \, (u/\sqrt{g_{11}} \, )_{,y} + \sqrt{g_{22}} \, (v/\sqrt{g_{22}} \, )_{,x},$$ = (9.106)

where the last step introduced the physical components. In Cartesian coordinates, D_S = u_,y + v_,x. A similar calculation yields the strain component e³₁

$$2 \, e^{3}_{1} u^{3}_{\; ;1} + g^{33} \, g_{11} \, u^{1}_{\; ; 3}$$ =

$$g^{33} \, ( g_{11} \, u^{1}_{ \; ,3} + g_{33} \, u^{3}_{\; , 1})$$ =

$$g_{11} \, u^{1}_{ \; ,3} + u^{3}_{\; , 1}$$ =

$$\approx g_{11} \, u^{1}_{ \; ,3}$$

$$\sqrt{g_{11}} \, u_{,z},$$ = (9.107)

where the last two steps follow from the quasi-hydrostatic approximation. In summary, these results bring the quasi-hydrostatic stress tensor to the form

$$\tau^{ij} \rho \, \left( \begin{array}{ccc} A \, g^{11} \, D_{T} & A \, D_{... .../\sqrt{g_{11}})_{,z} & \kappa \, (v/\sqrt{g_{22}})_{,z} & 0 \end{array}\right),$$ = (9.108)

where again g_11,z = g_22,z = 0 was used.