9.4.8 Summary of second order friction

The fundamental assumptions that determine the form of the momentum friction are the following:

• Horizontal kinetic energy is dissipated by the friction.
• The friction does not introduce interior sources or sinks of angular momentum.
• The fluid motion is quasi-hydrostatic.
• The friction exhibits lateral or transverse isotropy in which gravity picks out the only special direction.

Each of these properties is satisfied by the following two components to the friction

  

$$\fbox{$\rho \, F^{x} = g_{22}^{-1} \, (g_{22} \, \rho \, A \, D_{... ...1} \, (g_{11} \, \rho \, A \, D_{S})_{,y} + (\rho \, \kappa \, v_{,z})_{,z}$ }$$ (9.126)

$$\fbox{$\rho \, F^{y} = -g_{11}^{-1} \, (g_{11} \, \rho \, A \, D_... ...} \, (g_{22} \, \rho \, A \, D_{S})_{,x} + (\rho \, \kappa \, v_{,z})_{,z} $ }$$ (9.127)

With the Boussinesq approximation, the factors of density are formally replaced by the constant $\rho_{o}$, and so cancel out from these expressions. The metric tensor is assumed to be diagonal and to define the infinitesimal distance between two points as

$$\fbox{$ds^{2} = (h_{1} \, d\xi^{1})^{2} + (h_{2} \, d\xi^{2})^{2} + dz^{2} = dx^{2} + dy^{2} + dz^{2} $ }$$ (9.128)

In this expression, the metric components g₁₁ = h₁² and g₂₂ = h₂² are functions only of the transverse coordinates, and the physical displacements

$$h_{1} \, d\xi^{1}$$ d x = (9.129)

$$h_{2} \, d\xi^{2}$$ d y = (9.130)

have dimensions of length. The corresponding physical components of the partial derivative operators

$$\partial_{x} h_{1}^{-1} \, \partial_{1}$$ = (9.131)

$$\partial_{y} h_{y}^{-1} \, \partial_{2}$$ = (9.132)

bring the horizontal tension to the form

$$\fbox{$D_{T} = h_{2} \, (u / h_{2} \, )_{,x} -h_{1} \, (v / h_{1} \, )_{,y}$ }$$ (9.133)

and the horizontal shearing strain

$$\fbox{$D_{S} = h_{1} \, (u/h_{1} \, )_{,y} +h_{2} \, (v/h_{2} \, )_{,x}$ }$$ (9.134)

D_T and D_S are generically called the deformation rates, and the each have dimensions of t^-1. In spherical coordinates, $(\xi^{a}, \xi^{2}) =(\lambda,\phi)$, $h_{1} = a \, \cos\phi$, h₂ = a, and $\partial_{x} = (a \, \cos\phi)^{-1} \, \partial_{\lambda}$, $\partial_{y} = a^{-1} \, \partial_{\phi}$. The viscosity A is generally a function of the fluid flow, and it has dimension L²/t. The generalized curvilinear coordinates x,y,z are physical components, and so each has dimension of length. Likewise, the corresponding physical velocity components (u,v,w) have dimension L/t.