1 1 Instability time-scales of Antarctic outlet glaciers from similitude analysis 2 3 A. Levermann1,2,3* & J. Feldmann1,3 4 5 1Potsdam Institute for Climate Impact Research, Potsdam, Germany. 6 2LDEO, Columbia University, New York, USA. 7 3Institute of Physics, Potsdam University, Potsdam, Germany. 8 9 *Correspondence to: Anders Levermann ([email protected]). 10 11 Abstract 12 Instabilities in Antarctica have the potential to significantly raise sea level globally. Recent 13 observations and ice-dynamic modeling suggest that such an instability might have been 14 triggered in West Antarctica. Although other regions have similar topographic predispositions 15 it is so far unclear how fast these instabilities would unfold if they were initiated. The relative 16 initial response time of different ice streams after destabilization can be estimated from their 17 current force balance that reflects their relation of surface-mass balance, basal friction and 18 topographic geometry. Combining observations with a similarity analysis of the ice-dynamic 19 equation we show here that the already-triggered instabilities are likely to be the fastest in 20 Antarctica. Consistent with earlier studies the response time of Thwaites Glacier is found to 21 be about twice as fast as that of Pine Island Glacier while the main outlet of East Antarctica’s 22 Wilkes Basin would be about twice as slow if destabilized. 23 2 24 Main text 25 Sea-level rise poses a future challenge for coastal regions worldwide (1). The contribution 26 from mass loss of the West Antarctic Ice Sheet has been increasing over the last two decades 27 (2). Large parts of the Antarctic Ice Sheet rest on a retrograde marine bed, i.e. on a bed 28 topography that lies below the current sea level and is down-sloping inland (3, 4). This 29 topographic situation makes these parts of the ice sheet prone to a so-called marine ice-sheet 30 instability (MISI) (5, 6). Currently these instabilities represent the largest uncertainty in 31 projections of future sea-level rise (7, 8): If the grounding line (that separates grounded from 32 floating ice) enters a region of retrograde bed slope, a positive ice-loss feedback can be 33 initiated (9–12). If it is not hindered by the buttressing effect of ice shelves and topographic 34 features (13–15) or strong basal friction (8, 16), it unfolds and yields a self-sustained ice 35 discharge from the region. 36 The ongoing retreat of two major outlets of West Antarctica, Pine Island Glacier (PIG) and 37 Thwaites Glacier (TG), initiated by warm-water entrainment into their ice-shelf cavities and 38 resulting basal melting (17), is pointing toward a developing marine ice-sheet instability (9, 39 10, 18, 19) with the potential of raising global sea level by more than 3 m (12, 20). Similar 40 warm-water exposition might apply to other Antarctic ice shelves in the future (8, 21, 41 22) bearing the potential to trigger unstable grounding-line retreat. It has been demonstrated 42 that East Antarctica's Wilkes Subglacial Basin (WSB), would, once destabilized, contribute 43 by 3 – 4 m to global sea level (11). The marine part of the adjacent Aurora Subglacial Basin 44 stores ice of around 3.5 m sea-level equivalent (22). Other studies investigated the response 45 of the Filchner-Ronne tributaries to enhanced melting (15, 23, 24) and suggest that instability 46 might not unfold there, possibly due to the stabilizing buttressing effect of the large ice shelf 47 and narrow bed troughs (13, 14). In total, the marine catchment basins of Antarctica, that are 48 connected to the ocean store ice masses of about 20 m of sea-level equivalent (Figure S1). 49 Besides the question of whether local instabilities are already triggered or will be triggered in 50 the future the question remains on which time scales a potentially unstable retreat would 51 evolve if they had been initiated. It has been argued that ice-sheet models are not suitable for 52 an analysis of these time scales because they lack necessary detail in certain physical 53 processes such as basal friction (7). 54 Here we try to contribute information from a different approach. The currently observed 55 shape of the outlet glacier represents the balance of the different forces that act on the ice. 3 56 That is the balance of the forces caused by the bed and ice topography, the surface mass 57 balance and the frictional forces from the ice interaction with itself and its surrounding. This 58 static situation carries the information for the initial ice-sheet response after destabilization. 59 We combine observations with properties of the fundamental ice-dynamic equations. Eleven 60 Antarctic outlets with similar topography are analyzed to estimate their relative response 61 times to instability. 62 The time-scaling laws used here are derived from the governing equations of shallow ice- 63 stream flow (25) (see Supporting Online Material). In their dimensionless form these 64 equations have three independent parameters, analogous to the Reynolds number in the 65 Navier-Stokes equation (26). If all three of theses parameters are the same for topographically 66 similar situations the solution of the ice-dynamical equation will be similar. The parameters 67 are combinations of characteristics of the ice dynamics, physical ice properties, boundary 68 conditions and geometry (concept of similitude/similarity (27), for a detailed derivation 69 compare ref. (25)). The resulting scaling laws are consistent with analytic solutions of the ice 70 equations (6). 71 Here the individual geometry and physical conditions of the examined ice streams, that are 72 obtained from observational data (4, 28) and model inversion (29), are used to calculate 73 response times after destabilization relative to the reference, PIG. The applied method 74 assumes conditions of unstable retreat in which ice-shelf buttressing is a minor constraint 75 taking into account bed friction as a possible stabilizing factor. The investigation is 76 constrained to the short initial time period after destabilization during which the grounding 77 line passes the retrograde bed slope near the coast. 78 The procedure is schematized in Fig. 1: For each of the considered tributaries a transect is 79 defined that represents a center line of the ice stream and covers its potentially unstable bed 80 section close to the grounding line (Figs. 1a and b). Fitting a linear slope to the retrograde bed 81 section (blue line), the bed elevation b at the starting point of the slope and the slope 82 magnitude s serve as characteristic geometric scales for the sloping bed. Basal friction C is 83 averaged along this section to represent local conditions whereas the surface mass balance a 84 is obtained from averaging over the catchment that feeds the ice stream (Figures S2-S12). For 85 each tributary these physical parameters are taken relative to the reference values of PIG (Fig. 86 1c). 4 87 The resulting scaling ratios (Fig. 1d, values given in Table 1) are then used to calculate 88 response-time ratios τ=T' /T via two independent scaling conditions. The first equation 89 results from the shallow-shelf approximation of the stress-balance which accounts for the fast 90 plug-like flow of ice streams (see Supporting Online Material) and involves the ratios for 91 vertical scale β=b'/b , slope σ =s'/ s and friction γ=C' /C which yields τ = σ− 2 γ . (1) 92 As a consequence, for example, more slippery bed conditions and a steeper bed slope yield a 93 faster response. An independent second relation is derived from mass conservation and 94 incorporates the ratios for the vertical scale and the surface mass balance, δ=a' /a , i.e., ~τ =βδ− 1 . (2) 95 Here an initially stable situation with stronger accumulation but initially thinner ice at the 96 grounding line, for example, results in a faster response of the instability after destabilization. 97 The two time-scaling values τ and ~τ are used as independent estimates of the response 98 time and to a certain extent give a measure for the uncertainty range for each analyzed ice 99 stream that accounts for observational uncertainty and approximations in our approach. Ice 100 streams with large uncertainty in their calculated time scaling are reported, but considered 101 unsuitable for the approach used here (see Supporting Online Material). For each of the six 102 remaining tributaries (Table 1, blue) the relative error of the two calculated response-time 103 ratios with respect to their mean value is smaller than 28 %. 104 The individual retrograde bed slopes vary in magnitude by a factor of 2 and bed elevation 105 differs by up to ~1000 m (Fig. 2a). The lengths of the slope sections indicate how far the 106 considered retrograde slopes reach inland before an (intermediate) section with vanishing or 107 up-sloping bed follows (Figs. 1a,b and S2-12). After applying a geometric scaling according 108 to the obtained ratios of vertical and horizontal scales (Table 1) the bed profiles collapse 109 towards the reference showing similar down-sloping, while still exhibiting their characteristic 110 pattern (Figs. 2b and S7). This assures that the chosen geometric measures adequately reflect 111 the characteristics of the retrograde bed slopes. 112 Our results reveal that TG has the smallest response time to destabilization of all analyzed 113 outlets (Fig. 3), being between 1.25 to almost 2 times as fast as PIG, which is the second 114 fastest in the analyzed ensemble. For the chosen outlet of WSB the two calculated ratios 115 indicate a response that would be twice as slow as PIG.