9.5.1 General formulation

The general formulation of biharmonic friction is a straightforward extension of the Laplacian friction given in the previous sections. What is done is to basically iterate twice on the Laplacian approach. More precisely, the components F_Bⁱ of the biharmonic friction vector are derived from the covariant divergence

$$\rho \, F_{B}^{m} \Theta^{mn}_{; n},$$ = (9.135)

where

$$\Theta^{mn} -\rho \, B \, (2 \, E^{mn} - g^{mn} \, E^{p}_{p}),$$ = (9.136)

B>0 has units of L²/t^1/2, and $\rho$ is set to $\rho_{o}$ when making the Boussinesq approximation. As shown in the next subsection, use of the ``square-root'' biharmonic viscosity in prompted by the desire to dissipate kinetic energy. This detail only matters for cases with a non-constant viscosity. Note that all labels in this section run over m,n,p=1,2. $\Theta^{mn}$ has the same form as the stress tensor used for second order friction discussed in Section 9.4.4, except with a minus sign. The components of the symmetric ``strain'' tensor are given by

$$E_{mn} = \frac{1}{2} \, (F_{m;n} + F_{n;m}).$$ (9.137)

The vector F^m is the friction vector determined through the second order frictional stress tensor

$$\rho \, F^{m} \tau^{mn}_{;n}$$ =

$$[B \, \rho \, ( 2 \, e^{mn} - g^{mn} \, e^{p}_{p}) ]_{; n}$$ = (9.138)

as derived in the previous sections, where the only difference is that the viscosity used for computing the stress tensor $\tau^{mn}$ is now set to B, and the dimensions on F^m are $L \, t^{-3/2}$. This approach ensures that the biharmonic friction is derived from the divergence of a symmetric tensor $\Theta^{mn}$, hence ensuring a proper angular momentum budget. Additionally, the computational work necessary to compute the Laplacian friction is directly employed for the biharmonic friction. Finally, as shown in the next subsection, this form for biharmonic friction also dissipates kinetic energy.