drag drag_with_derivative IceBasalResistanceRegularizedLaw m_q m_sliding_scale_factor_reduces_tauc m_u_threshold print_info ~IceBasalResistanceRegularizedLaw

drag_with_derivative() void pism::IceBasalResistanceRegularizedLaw::drag_with_derivative ( double  tauc, double  vx, double  vy, double *  beta, double *  dbeta  ) const virtual Compute the drag coefficient and its derivative with respect to \( \alpha = \frac 1 2 (u_x^2 + u_y^2) \). \begin{align*} \beta &= \frac{\tau_{c}}{ \left( (|u|^{2})^{\frac12} + u_{\text{threshold}} \right)^{q} }\cdot (|\mathbf{u}|^{2})^{\frac{q-1}{2}} \\ \diff{\beta}{\frac12 |\mathbf{u}|^{2}} &= \frac{\tau_{c}}{ \left( (|\mathbf{u}|^{2})^{\frac12} + u_{\text{threshold}}\right)^{q} }\cdot \frac{q-1}{2}\cdot (|\mathbf{u}|^{2})^{\frac{q-1}{2} - 1}\cdot 2 - \frac{\tau_{c}}{ \left( (|\mathbf{u}|^{2})^{\frac12} + u_{\text{threshold}}\right)^{q+1} }\cdot \frac{2 \cdot q }{(|\mathbf{u}|^{2})^{\frac12}} \cdot (|\mathbf{u}|^{2})^{\frac{q-1}{2} } \\ &= \left( \frac{q-1}{|\mathbf{u}|^{2}} - \frac{q}{|\mathbf{u}| \cdot ( |\mathbf{u}| + u_{\text{threshold}}) } \right) \cdot \beta(\mathbf{u})\\ \end{align*} Same parameters are used as in the pseudo-plastic case, namely \( q, u_{\text{threshold}}, A_q \). It should be noted, that in the original equation (3) in Zoet et al, 2020 the exponent \( q=1/p \) is used. Reimplemented from pism::IceBasalResistancePlasticLaw. Definition at line 233 of file basal_resistance.cc. References pism::IceBasalResistancePlasticLaw::m_plastic_regularize, m_q, m_sliding_scale_factor_reduces_tauc, m_u_threshold, and pism::square().