PISM, A Parallel Ice Sheet Model  stable v2.0.5 committed by Constantine Khrulev on 2022-10-14 09:56:26 -0800

## ◆ effective_viscosity() [1/2]

 void pism::rheology::FlowLaw::effective_viscosity ( double B, double gamma, double * nu, double * dnu ) const

Computes the regularized effective viscosity and its derivative with respect to the second invariant $$\gamma$$.

\begin{align*} \nu &= \frac{1}{2} B \left( \epsilon + \gamma \right)^{(1-n)/(2n)},\\ \diff{\nu}{\gamma} &= \frac{1}{2} B \cdot \frac{1-n}{2n} \cdot \left(\epsilon + \gamma \right)^{(1-n)/(2n) - 1}, \\ &= \frac{1-n}{2n} \cdot \frac{1}{2} B \left( \epsilon + \gamma \right)^{(1-n)/(2n)} \cdot \frac{1}{\epsilon + \gamma}, \\ &= \frac{1-n}{2n} \cdot \frac{\nu}{\epsilon + \gamma}. \end{align*}

Here $$\gamma$$ is the second invariant

\begin{align*} \gamma &= \frac{1}{2} D_{ij} D_{ij}\\ &= \frac{1}{2}\, ((u_x)^2 + (v_y)^2 + (u_x + v_y)^2 + \frac{1}{2}\, (u_y + v_x)^2) \\ \end{align*}

and

$D_{ij}(\mathbf{u}) = \frac{1}{2}\left(\diff{u_{i}}{x_{j}} + \diff{u_{j}}{x_{i}}\right).$

Either one of nu and dnu can be NULL if the corresponding output is not needed.

Parameters
 [in] B ice hardness [in] gamma the second invariant [out] nu effective viscosity [out] dnu derivative of $$\nu$$ with respect to $$\gamma$$

Definition at line 163 of file FlowLaw.cc.

References m_schoofReg.