bed_smoother compute_3d_horizontal_velocity compute_diffusive_flux compute_diffusivity compute_I compute_surface_gradient diagnostics_impl diffusivity init interglacial m_bed_smoother m_D m_delta_0 m_delta_1 m_e_factor m_e_factor_interglacial m_eemian_end m_eemian_start m_event_sia m_h_x m_h_y m_holocene_start m_seconds_per_year m_stencil_width m_work_2d_0 m_work_2d_1 m_work_3d_0 m_work_3d_1 SIAFD surface_gradient_eta surface_gradient_haseloff surface_gradient_mahaffy surface_gradient_x surface_gradient_y update ~SIAFD

compute_surface_gradient() void pism::stressbalance::SIAFD::compute_surface_gradient ( const Inputs &  inputs, array::Staggered1 &  h_x, array::Staggered1 &  h_y  ) protectedvirtual Compute the ice surface gradient for the SIA. There are three methods for computing the surface gradient. Which method is controlled by configuration parameter sia.surface_gradient_method which can have values haseloff, mahaffy, or eta. The most traditional method is to directly differentiate the surface elevation \(h\) by the Mahaffy method [Mahaffy]. The haseloff method, suggested by Marianne Haseloff, modifies the Mahaffy method only where ice-free adjacent bedrock points are above the ice surface, and in those cases the returned gradient component is zero. The alternative method, when sia.surface_gradient_method = eta, transforms the thickness to something more regular and differentiates that. We get back to the gradient of the surface by applying the chain rule. In particular, as shown in [Vazquezetal2003] for the flat bed and \(n=3\) case, if we define \[\eta = H^{(2n+2)/n}\] then \(\eta\) is more regular near the margin than \(H\). So we compute the surface gradient by \[\nabla h = \frac{n}{(2n+2)} \eta^{(-n-2)/(2n+2)} \nabla \eta + \nabla b,\] recalling that \(h = H + b\). This method is only applied when \(\eta > 0\) at a given point; otherwise \(\nabla h = \nabla b\). In all cases we are computing the gradient by finite differences onto a staggered grid. In the method with \(\eta\) we apply centered differences using (roughly) the same method for \(\eta\) and \(b\) that applies directly to the surface elevation \(h\) in the mahaffy and haseloff methods. Parameters [out] h_x the X-component of the surface gradient, on the staggered grid [out] h_y the Y-component of the surface gradient, on the staggered grid Reimplemented in pism::stressbalance::SIAFD_Regional. Definition at line 188 of file SIAFD.cc. References pism::Geometry::bed_elevation, pism::Geometry::cell_type, pism::RuntimeError::formatted(), pism::stressbalance::Inputs::geometry, pism::Geometry::ice_surface_elevation, pism::Geometry::ice_thickness, pism::Component::m_config, PISM_ERROR_LOCATION, surface_gradient_eta(), surface_gradient_haseloff(), and surface_gradient_mahaffy(). Referenced by pism::stressbalance::SIAFD_Regional::compute_surface_gradient(), and update().