◆ drag()

double pism::IceBasalResistancePseudoPlasticLaw::drag	(	double	tauc,
		double	vx,
		double	vy
	)		const

virtual

Compute the drag coefficient for the basal shear stress.

The basal shear stress term \( \tau_b \) in the SSA stress balance for ice is minus the return value here times (vx,vy). Thus this method computes the basal shear stress as

\[ \tau_b = - \frac{\tau_c}{|\mathbf{U}|^{1-q} U_{\mathtt{th}}^q} \mathbf{U} \]

where \( \tau_b=(\tau_{(b)x},\tau_{(b)y}) \) , \( U=(u,v) \) , \( q= \) pseudo_q, and \( U_{\mathtt{th}}= \) pseudo_u_threshold. Typical values for the constants are \( q=0.25 \) and \( U_{\mathtt{th}} = 100 \) m year-1.

The linearly-viscous till case pseudo_q = 1.0 is allowed, in which case \( \beta = \tau_c/U_{\mathtt{th}} \) . The purely-plastic till case pseudo_q = 0.0 is also allowed; note that there is still a regularization with data member plastic_regularize.

One can scale tauc if desired:

A scale factor of \( A \) is intended to increase basal sliding rate by \( A \) . It would have exactly this effect if the driving stress were hypothetically completely held by the basal resistance. Thus this scale factor is used to reduce (if -sliding_scale_factor_reduces_tauc \( A \) with \( A > 1 \)) or increase (if \( A < 1 \)) the value of the (pseudo-) yield stress tauc. The concept behind this is described at the SeaRISE wiki.

Specifically, the concept behind this mechanism is to suppose equality of driving and basal shear stresses,

\[ \rho g H \nabla h = \frac{\tau_c}{|\mathbf{U}|^{1-q} U_{\mathtt{th}}^q} \mathbf{U}. \]

(For emphasis: The membrane stress held by the ice itself is missing from this incomplete stress balance.) Thus the pseudo yield stress \( \tau_c \) would be related to the sliding speed \( |\mathbf{U}| \) by

\[ |\mathbf{U}| = \frac{C}{\tau_c^{1/q}} \]

for some (geometry-dependent) constant \( C \) . Multiplying \( |\mathbf{U}| \) by \( A \) in this equation corresponds to dividing \( \tau_c \) by \( A^q \) .

Note that the mechanism has no effect whatsoever if \( q=0 \) , which is the purely plastic case. In that case there is no direct relation between the yield stress and the sliding velocity, and the difference between the driving stress and the yield stress is entirely held by the membrane stresses. (There is also no singular mathematical operation as \( A^q = A^0 = 1 \) .)

Reimplemented from pism::IceBasalResistancePlasticLaw.

Definition at line 151 of file basal_resistance.cc.

References pism::IceBasalResistancePlasticLaw::m_plastic_regularize, m_pseudo_q, m_pseudo_u_threshold, m_sliding_scale_factor_reduces_tauc, and pism::square().