On non-dimensional forms of basal sliding laws and flow laws for ice-sheet and glacier modelling

Journal of Glaciology, Vol. 71, e123, 2025 (doi: 10.1017/jog.2025.10100) 1 On non-dimensional forms of basal sliding laws and flow laws for ice-sheet and glacier modelling Ralf GREVE1,2 1Institute of Low Temperature Science, Hokkaido University, Sapporo, Japan 2Arctic Research Center, Hokkaido University, Sapporo, Japan Correspondence: Ralf Greve <gr[email protected]> ABSTRACT. Ice sheets and glaciers flow through basal sliding and internal deformation, each governed by physical laws commonly expressed as powerlaw relations. These formulations include coefficients – the sliding coefficient and rate factor – whose values and units depend on the respective exponents. This dependency complicates the systematic exploration of parameter space, especially in ensemble simulations. To address this, we propose dimensionless formulations of both sliding and flow laws, in which the coefficients are of order unity and decoupled from the exponents. This separation simplifies sensitivity studies and parameter variations. The dimensionless laws are straightforward to implement in existing models; we demonstrate this with the SICOPOLIS ice-sheet model using three test simulations in an idealized set-up. These simulations illustrate that independent variation of exponents and coefficients is feasible and practical, supporting the use of dimensionless laws in efforts to better constrain ice dynamics in past and future climate scenarios. 1 INTRODUCTION Ice sheets and glaciers flow due to two different processes, namely basal sliding and internal deformation. Basal sliding describes the sliding of glacier ice on the underlying substrate, which can be either hard bedrock or a deformable sediment layer between ice and bedrock. Internal deformation is governed by the – Author's Version – Greve: Non-dimensional sliding and flow laws 2 non-linear viscous properties of “hot” polycrystalline ice (that is, with a homologous temperature T{Tm near unity, where Tis the absolute temperature and Tmthe pressure melting point). In a dynamic/thermodynamic ice sheet or glacier model, both processes must be included. Basal sliding, which in reality is a complex process that depends on a multitude of factors such as the basal temperature, roughness of the bedrock, softness of the subglacial sediment layer (if existing) and hydrological conditions, is usually parameterized by a sliding law that relates the sliding velocity to the basal stresses. Internal deformation can be modelled by a non-linear viscous flow law that describes the relation between the macroscopic deformation (strain rate) and internal stresses (e.g., Hooke, 2005; Greve and Blatter, 2009; Cuffey and Paterson, 2010). Popular forms for such relations are the Weertman–Budd sliding law and the Nye–Glen flow law (see below for references). They have in common that they are expressed as power laws with some exponents, of which the optimal values are debated, and contain a factor to close the respective equation. This factor, the “sliding coefficient” in case of the sliding law and the “rate factor” in case of the flow law, may contain remaining dependencies, such as on the temperature. In a dimensional formulation, the units and numerical values of these factors depend strongly on the choice of the exponents, which makes it cumbersome to vary the exponents over their potential range of values, for instance, within an ensemble of simulations for a given scenario. To overcome this obstacle, we propose fully or partly dimensionless versions of the sliding and flow laws, which have in common that the respective factor is dimensionless and generally of order unity. These formulations decouple the value of the exponents from the value of the factor, so that the factors and exponents can be varied independently. We demonstrate this useful feature by some simple simulations with the ice-sheet model SICOPOLIS (SImulation COde for POLythermal Ice Sheets; SICOPOLIS Authors, 2025). 2 BASAL SLIDING LAWS Basal sliding laws (aka basal friction laws) relate the shear stress (drag) at the base of an ice sheet or glacier, τb, to the basal normal stress, Nband the basal sliding velocity, vb. In a general, implicit form, a basal sliding law can be expressed as fpvb, τb, Nbq “ 0,(1) Greve: Non-dimensional sliding and flow laws 3 where fis a function unspecified at this stage. The list of variables is not necessarily exhaustive as further dependencies, for instance on basal temperature or the presence of basal water, may be included. Note that, in the presence of subglacial water, the basal normal stress is often understood as the difference between the ice overburden stress, Nb,i, and the basal water pressure, pb,w, Nb“Nb,i´pb,w,(2) and then called the reduced normal stress or, alternatively, the effective pressure. A popular form of a basal sliding law is the Weertman–Budd sliding law, which results when assuming an explicit form of Eq. (1) solved for vb, with power-law dependencies on τband Nb: vb“Cb τp b Nq b ,(3) where Cbis the sliding coefficient and pp, qqare the non-negative sliding exponents (Weertman, 1957; Budd and others, 1979, 1984; Budd and Jenssen, 1987). Alternatively, Eq. (3) can be solved for the shear stress, τb“C‹ bv1{p bNq{p b,with C‹ b“C´1{p b,(4) where C‹ bis the basal friction coefficient. For the case q“0, that is, ignoring the dependence on the normal stress Nb, the above forms are often referred to as the Weertman sliding law. In principle, vband τbare vector quantities. For simplicity, we formulate the sliding laws only with the respective magnitudes. However, to interpret the results correctly, it must be kept in mind that vband τb are anti-parallel to each other due to the nature of friction. Let us now non-dimensionalize the sliding law (3) by introducing scales (typical values) for the relevant quantities (e.g., Hutter and Jöhnk, 2004). We consider a situation near the edge of an ice sheet where Greve: Non-dimensional sliding and flow laws 4 basal sliding is most relevant: rHs “ 1 km ptypical thicknessq,(5a) ε“10´2ptypical surface slopeq,(5b) rNbs “ ρgrHs “ 107Pa ptypical normal stressq,(5c) rτbs “ εrNbs “ 105Pa ptypical shear stressq,(5d) rvbs “ 100 m a´1ptypical sliding velocityq,(5e) where we have used approximate values ρ«103kg m´3for the ice density and g«10 m s´2for the acceleration due to gravity, which is sufficiently accurate for the sake of a scaling analysis. This scaling is consistent with the linear sliding law [Cb“10´3m a´1Pa´1,pp, qq“p1,0q] used for the EISMINT Phase 2 Simplified Geometry Experiments (Payne and others, 2000): 100 m a´1 loooomoooon vb “10´3m a´1Pa´1 looooooooomooooooooon Cb ˆ105Pa loomoon τb .(6) An appropriate choice for the scale of the sliding coefficient results from Eq. (3) as rCbs “ rvbsrNbsq rτbsp.(7) We now use the above scales to introduce dimensionless quantities as follows: vb“ rvbs˜vb,(8a) τb“ rτbs˜τb,(8b) Nb“ rNbs˜ Nb,(8c) Cb“ rCbs˜ Cb,(8d) where the quantities marked by the tilde symbol are the non-dimensional basal sliding velocity, shear stress, normal stress and sliding coefficient, respectively. Inserting Eqs. (8) in the sliding law (3) yields its fully non-dimensional form, ˜vb“˜ Cb ˜τp b ˜ Nq b ,(9) Greve: Non-dimensional sliding and flow laws 5 in which all quantities are supposed to be of order unity. A dimensional form of Eq. (3) can be kept by only making use of the scaling (8d) of the sliding coefficient, vb“ rCbs˜ Cb τp b Nq b ,(10) which has the advantage that its implementation in an existing model based on dimensional quantities requires only minimal adaptations. In order to obtain the fully or partly dimensionless counterparts of Eq. (4), we note the scaling and non-dimensionalization of the friction coefficient C‹ b: C‹ b“ rC‹ bs˜ C‹ b,with rC‹ bs “ rCbs´1{p“rτbs rvbs1{prNbsq{p.(11) The fully non-dimensional form of Eq. (4) results then as ˜τb“˜ C‹ b˜v1{p b˜ Nq{p b,(12) and the dimensional form in which only the scaling (11) of the friction coefficient is used reads τb“ rC‹ bs˜ C‹ bv1{p bNq{p b.(13) Why do we promote using Eqs. (10) or (13) instead of Eqs. (3) or (4) in an ice sheet or glacier model? In Table 1 we have compiled some parameter combinations that were used along with Weertman or Weertman–Budd sliding laws in the literature. The impossibility of comparing the various dimensional sliding coefficients Cbfor different exponents pp, qqbecomes immediately evident. They do not even have a common unit, and the respective numerical value tells nothing about the actual strength of basal sliding. In the second case, pp, qq“p1,2q, the numerical value of Cbis greater than 109; however, the small dimensionless value means that it produces only very little basal sliding ( ˜ Cb«0.04). By contrast, in the third case, pp, qq“p3,0q, the numerical value of Cbis merely 10´12; however, it corresponds to pronounced basal sliding ( ˜ Cb“10). The dimensionless sliding coefficients ˜ Cbgive a much better idea about what the respective value means physically, and allow comparing values across different sliding laws. To further strengthen our point, suppose that we wish to test a sliding law with a new set of exponents, for instance pp, qq “ p3,1.5q. Working with the dimensional sliding coefficient Cb, we would not have any Greve: Non-dimensional sliding and flow laws 6 pp, qqCbrCbs˜ CbReference p1,0q10´3m a´1Pa´110´3m a´1Pa´11Payne and others (2000) p1,2q3.985 ˆ109m a´1Pa 1011 m a´1Pa 0.03985 Budd and others (1984): p3,0q10´12 m a´1Pa´310´13 m a´1Pa´310 Cornford and others (2020) p3,1q1.607 ˆ10´6m a´1Pa´210´6m a´1Pa´21.607 Saito and others (2016): p3,2q6.72 m a´1Pa´110 m a´1Pa´10.672 Rückamp and others (2019) Table 1. Sliding exponents pp, qq, dimensional sliding coefficients Cb, scales rCbsand dimensionless sliding coefficients ˜ Cbfor several Weertman (q“0) or Weertman–Budd (qą0) sliding laws used in the literature. :: Rather than using the normal stress Nb, these sliding laws were formulated with the pressure head Z“Nb{pρgq. We converted the sliding coefficients given in these studies accordingly, using ρ“910 kg m´3and g“9.81 m s´2. idea which order of magnitude may be suited for its numerical value, and which range of values mean strong or weak sliding. However, if the dimensionless sliding coefficient ˜ Cbis used, we can immediately start with an initial guess ˜ Cb“1and, from there on, refine the sliding law by, e.g., tuning to observed flow speeds. According to the scaling (7), (8d), the dimensional equivalent of ˜ Cb“1would be Cb“ rCbs “ 10´2.5m a´1Pa´1.5“3.162 ˆ10´3m a´1Pa´1.5. We have only discussed cases with a constant sliding parameter; however, the non-dimensionalization method is of course not limited to this. It can also be applied to a spatially variable sliding coefficient, which may arise from an inversion procedure (e.g., Morlighem and others, 2013). Alternative sliding laws, such as the Coulomb-limited rules discussed by Cornford and others (2020), allow similar non-dimensionalization, although we refrain from working out the details here. 3 FLOW LAWS A similar problem of units and hugely varying numerical values arises for the flow law of polycrystalline ice. It is a viscous flow law that relates the strain-rate (stretching) tensor dij to the stress deviator tD ij. The strain-rate tensor is defined as dij “1 2ˆBvi Bxj `Bvj Bxi˙pi, j “1,2,3q,(14) where xidenotes the Cartesian coordinates (x1“x,x2“y,x3“z), and viis the velocity vector. The stress deviator is the traceless part of the full stress tensor tij, tij “ ´p δij `tD ij ,(15) Greve: Non-dimensional sliding and flow laws 7 where p“ ´tii{3is the pressure (we assume the Einstein summation convention: summation over the twice-appearing index iimplied, thus tii is the trace of the stress tensor), and δij is the Kronecker delta symbol, in other words, the unit tensor in index notation. For the flow law, usually collinearity between the symmetric tensors dij and tD ij is assumed. We note the form given by Greve and Blatter (2009), dij “AfpτeqtD ij ,(16) where Ais the rate factor, τe“ rptD ijtD ijq{2s1{2the effective stress (summation over both iand jimplied), and fpτeqis the creep function. The rate factor depends on the temperature relative to the pressure melting point, T1, via an Arrhenius law (e.g., Cuffey and Paterson, 2010), but it is sometimes chosen as a constant parameter for simplicity. In the Nye–Glen flow law (Glen, 1955; Nye, 1957), the creep function is expressed as a power law, fpτeq “ τn´1 e,(17) so that dij “Aτn´1 etD ij ,(18) where nis the stress exponent. A value of n“1would correspond to a Newtonian fluid; however, the deformability of ice differs markedly from that behaviour, and the value is frequently chosen as n“3, or within the range from 1.5 to 4.2 (Cuffey and Paterson, 2010) (while recent evidence from laboratory experiments actually supports n“1for temperate ice; Schohn and others, 2025). The Nye–Glen flow law (18) can also be inverted for the stress deviator, tD ij “A‹d´p1´1{nq edij ,with A‹“A´1{n,(19) where A‹is the associated rate factor and de“ rpdijdijq{2s1{2the effective strain rate (e.g., Greve and Blatter, 2009). Similar to the procedure in Sect. 2, we now introduce scales (typical values) for the relevant quantities, Greve: Non-dimensional sliding and flow laws 8 considered suitable for areas where rather large deformations take place: rτs “ 105Pa ptypical deviatoric stressq,(20a) rds “ 2.5ˆ10´2a´1ptypical strain rateq,(20b) where the scale rτsis deemed appropriate for both tD ij and τe. Using Eq. (18) entails the choice for the scale of the rate factor: rAs “ rds rτsn.(21) We introduce the dimensionless quantities tD ij “ rτs˜ tD ij ,(22a) τe“ rτs˜τe,(22b) dij “ rds˜ dij ,(22c) A“ rAs˜ A , (22d) where the quantities marked by the tilde symbol are the non-dimensional components of the stress deviator, effective stress, components of the strain-rate tensor and rate factor, respectively. For n“3, Eq. (21) yields rAs “ 2.5ˆ10´17 a´1Pa´3“7.922 ˆ10´25 s´1Pa´3. This value is close to the recommendation by Cuffey and Paterson (2010) for T1“ ´6˝C, which demonstrates the validity of our scaling. We obtain the fully non-dimensional form of the flow law (18) as ˜ dij “˜ A˜τn´1 e˜ tD ij ,(23) (all quantities supposed to be of order unity). A form with only Ascaled results if we only apply the scaling (22d): dij “ rAs˜ A τn´1 etD ij .(24) Analogous to the sliding law (10), this form can be implemented in a model based on dimensional quantities with only minimal changes. To obtain the fully or partly dimensionless versions of Eq. (19), we note the scaling and non-dimensio- Greve: Non-dimensional sliding and flow laws 9 Fig. 1. Dimensionless rate factor ˜ Aas a function of the temperature relative to pressure melting T1, following the recommendation by Cuffey and Paterson (2010): Arrhenius law with activation energies Q“60 kJ mol´1for T1ď ´10˝C,Q“115 kJ mol´1for T1ě ´10˝C,A“3.5ˆ10´25 s´1Pa´3for T1“ ´10˝Cand n“3. nalization of the associated rate factor A‹, A‹“ rA‹s˜ A‹,with rA‹s “ rAs´1{n“rτs rds1{n,(25) and for the effective strain rate, the scale (20b) is used, de“ rds˜ de.(26) The fully non-dimensional form of Eq. (19) is then ˜ tD ij “˜ A‹˜ d´p1´1{nq e˜ dij ,(27) and the form with only A‹scaled reads tD ij “ rA‹s˜ A‹d´p1´1{nq edij ,(28) Figure 1 shows the temperature-dependent rate factor following the recommendation by Cuffey and Paterson (2010), non-dimensionalized with the scaling (21), (22d). However, while the original recommendation in dimensional form is valid only for n“3, this dimensionless form can be used for any value of the stress exponent n. It is therefore much more flexible. Consider the case n“4, which has recently been discussed by, e.g., Bons and others (2018); Millstein Greve: Non-dimensional sliding and flow laws 16 Saito F, Abe-Ouchi A, Takahashi K and Blatter H (2016) SeaRISE experiments revisited: potential sources of spread in multi-model projections of the Greenland ice sheet. The Cryosphere,10(1), 43–63 (doi: 10.5194/tc-10-43-2016) Sayag R and Tziperman E (2008) Spontaneous generation of pure ice streams via flow instability: Role of longitudinal shear stresses and subglacial till. Journal of Geophysical Research: Solid Earth,113(B5), B05411 (doi: 10.1029/2007JB005228) Schohn CM, Iverson NR, Zoet LK, Fowler JR and Morgan-Witts N (2025) Linear-viscous flow of temperate ice. Science,387(6730), 182–185 (doi: 10.1126/science.adp7708) SICOPOLIS Authors (2025) SICOPOLIS v25. GitHub, URL https://github.com/sicopolis/sicopolis Weertman J (1957) On the sliding of glaciers. Journal of Glaciology,3(21), 33–38 (doi: 10.3189/S0022143000024709)