Instability and abrupt changes in marine ice sheet behaviour K. - PowerPoint PPT Presentation

Critics Workshop on Critical Transitions in Complex Systems Instability and abrupt changes in marine ice sheet behaviour K. Bulthuis 1 , 2 , M. Arnst 1 , F. Pattyn 2 , L. Favier 2 1 University of Liège, Liège, Belgium 2 Free University of Brussels, Brussels, Belgium Monday 5 September 2016 Critics Workshop, Kulhuse, Denmark Ice streams dynamics 1 / 34

Motivation � Ice streams (narrow corridors of fast-flowing ice) drain over 90% of the Antarctic mass flux. Ice stream dynamics and stability are key factors for Antarctic mass balance and future contribution to sea-level rise. Pine Island and Thwaites glaciers Siple Coast Figure: Full map of Antarctic ice flow deduced from satellite data [ NASA/JPL-Caltech/UCI]. Critics Workshop, Kulhuse, Denmark Ice streams dynamics 2 / 34

Motivation � In this presentation, I will focus on two physical models that have been proposed to explain current and past behaviours of ice streams: � Marine ice sheet instability: Marine ice streams resting on a retrograde bedrock could exhibit a rapid retreat leading to a sudden and important loss of ice (Pine Island and Thwaites glaciers) [Schoof, 2007, 2012]. � Thermally induced oscillations: Ice streams can show decadal to multi-millennial variability through a thermal feedback between ice mass and bedrock sediments (Siple Coast glaciers) [Robel et al., 2013, 2014]. � It is possible to develop a coupled model of marine ice sheet instability and thermally induced oscillations [Robel et al., 2016]. Critics Workshop, Kulhuse, Denmark Ice streams dynamics 3 / 34

Outline � Motivation. � Marine ice sheet instability. � Thermally induced oscillations. � Coupled model of marine ice sheet instability and thermally induced oscillations. � Conclusion and outlook. � References. Critics Workshop, Kulhuse, Denmark Ice streams dynamics 4 / 34

Marine ice sheet instability (MISI) Critics Workshop, Kulhuse, Denmark Ice streams dynamics 5 / 34

Marine ice sheet instability mechanism � Step 1: Steady state on an upward sloping bed ( q in = q out ). ∗ ∗ ∗ q in ∗ q out Critics Workshop, Kulhuse, Denmark Ice streams dynamics 6 / 34

Marine ice sheet instability mechanism � Step 2: Initiation of grounding line retreat ( q in < q out ). ∗ ∗ ∗ q in ∗ q out Critics Workshop, Kulhuse, Denmark Ice streams dynamics 6 / 34

Marine ice sheet instability mechanism � Step 3: Self-sustained grounding line retreat ( q in ≪ q out ). ∗ ∗ ∗ q in ∗ q out Critics Workshop, Kulhuse, Denmark Ice streams dynamics 6 / 34

A simple geometrical model for MISI � We consider an ice stream sliding on an overdeepened bed. Ice flow is described as a gravity-driven viscous flow subject to basal friction. Viscous stresses can be neglected in the ice sheet except in a narrow transition zone near the grounding line. ∗ ∗ ∗ ∗ ice sheet h ( x ) ice shelf x x g b ( x ) Critics Workshop, Kulhuse, Denmark Ice streams dynamics 7 / 34

A mathematical model for MISI � Continuity equation (nonlinear diffusion equation): �� ρ i g � m − 1 ∂ ( h − b ) 1 � � ∂ h ∂ t − ∂ � 1 / m ∂ ( h − b ) h 1+ 1 � � = a . � � m ∂ x C ∂ x ∂ x � � � Symmetry condition at the ice divide: � ∂ ( h − b ) � = 0 . � ∂ x � x =0 � Flotation condition at the grounding line: ρ i h ( x g ) = ρ w b ( x g ) . � Stress continuity at the grounding line (from boundary layer theory): 1 � � A ( ρ i g ) n +1 (1 − ρ i ρ w ) n m +1 m + n +3 m +1 . q ( x g ) = h ( x g ) 4 n C Critics Workshop, Kulhuse, Denmark Ice streams dynamics 8 / 34

Steady grounding line positions: graphical approach � Steady grounding line positions are given by 1 � � A ( ρ i g ) n +1 (1 − ρ i � m + n +3 ρ w ) n m +1 � ρ w m +1 b ( x g ) = ax g 4 n C ρ i 1 ax g , q ( x g ) (km 2 a − 1 ) 1 0 . 75 0 . 75 0 . 5 0 . 5 0 . 25 0 . 25 0 0 0 500 1000 1500 0 500 1000 1500 x g (km) x g (km) Critics Workshop, Kulhuse, Denmark Ice streams dynamics 9 / 34

Stability analysis of steady states � Graphical analysis: ax g , q B ( x g ) (km 2 a − 1 ) 1 0 . 75 0 . 5 0 . 25 0 0 500 1000 1500 0 500 1000 1500 x g (km) x (km) � Linear stability analysis: Schoof [Schoof, 2012] has shown that marine ice sheets are unstable if a ( x g ) − q ′ ( x g ) > 0 . Critics Workshop, Kulhuse, Denmark Ice streams dynamics 10 / 34

Bifurcation diagram � The system is bistable for some values of the parameters. The appearance or disappearance of two steady state solution branches is associated with a saddle-node bifurcation. The system can undergo hysteresis under variations of parameters. 1500 1250 x g (km) 1000 750 500 0 5 10 15 20 − 100 − 50 0 50 100 a ( C / A ) 1 / ( m +1) ( 10 15 Pa 3 m 3 / 4 ) ∆ h w (m) Critics Workshop, Kulhuse, Denmark Ice streams dynamics 11 / 34

Conclusions about MISI � Marine ice sheets have a discrete number of equilibrium profiles. � Marine ice sheets are inherently unstable on upward-sloping bed. � Marine ice sheets can undergo hysteresis under variations of physical parameters (sea level, accumulation rate, basal slipperiness and ice viscosity). � MISI mechanism has been presented for a 2D model. For 3D models, buttressing effects could stabilise marine ice sheets. Critics Workshop, Kulhuse, Denmark Ice streams dynamics 12 / 34

Thermally induced oscillations Critics Workshop, Kulhuse, Denmark Ice streams dynamics 13 / 34

Heinrich events: a thermally oscillating event � Heinrich events are quasi-periodic episodes of massive ice discharges during the last glacial period. These episodes led to a climatic cooling and high ice-rafted detritus concentrations in the North Atlantic Ocean. 100 H 1 H 2 H 3 H 4 H 5 % IRD 50 0 50 100 150 200 250 300 350 DSDP609 core depth (cm) (data from [Bond, 1996]) δ 18 O % � (NGRIP) − 35 H 1 H 2 H 3 H 4 H 5 H 6 − 40 − 45 0 10 20 30 40 50 60 70 ky before 2000 AD (data from [Andersen et al., 1996]) Critics Workshop, Kulhuse, Denmark Ice streams dynamics 14 / 34

Thermal induced oscillations mechanism � Step 1: Ice sheet build-up on a frozen bed (binge phase). ∗ ∗ ∗ ∗ Critics Workshop, Kulhuse, Denmark Ice streams dynamics 15 / 34

Thermal induced oscillations mechanism � Step 2: Binge/Purge transition. ∗ ∗ ∗ ∗ Critics Workshop, Kulhuse, Denmark Ice streams dynamics 15 / 34

Thermal induced oscillations mechanism � Step 3: Rapid basal motion (purge phase). ∗ ∗ ∗ ∗ Critics Workshop, Kulhuse, Denmark Ice streams dynamics 15 / 34

Thermal induced oscillations mechanism � Step 4: Purge/Binge transition. ∗ ∗ ∗ ∗ Critics Workshop, Kulhuse, Denmark Ice streams dynamics 15 / 34

A simple model for thermal oscillations � The system is described by a set of four dynamical variables: 1. h : Ice thickness; 2. w : Water content of the till ( 0 � w � w s ); 3. Z s : Thickness of unfrozen till with zero porosity ( 0 � Z s � Z 0 ); 4. T b : Basal temperature ( T b � T m ). w et Z s are related trough w = eZ s where e is till void ratio ( e � e c ). � The system has three main configurations: T b = T m , e > e c T b = T m , e = e c T b < T m u b u b u b = 0 h Z 0 Z s Critics Workshop, Kulhuse, Denmark Ice streams dynamics 16 / 34

A mathematical model for thermal oscillations (1) � Equation for h : dh dt = a c − u b h L ( continuity equation ) . � Equation for w ( T b = T m ): dw dt = m − Q w ( melt water budget ) with k i ( T s − T b ) ρ i L f m = + + τ b u b , G h ���� � �� � ���� geothermal flux vertical heat conduction frictional heating � 0 if w < w s or m < 0 Q w = . m otherwise Critics Workshop, Kulhuse, Denmark Ice streams dynamics 17 / 34

A mathematical model for thermal oscillations (2) � Equation for Z s ( T b = T m ):  m if e = e c and 0 < Z s < Z 0 ( ice fringe )    if e = e c and Z s = Z 0 and m < 0 ( ice fringe )  m e dZ s dt = . m if e = e c and Z s = 0 and m > 0 ( ice fringe )     0 otherwise � Equation for T b :  0 if w > 0 or ( T b = T m , w = 0 and m > 0)  dT b dt = . ρ i L f m otherwise ( basal cooling )  C i h b Critics Workshop, Kulhuse, Denmark Ice streams dynamics 18 / 34

A mathematical model for thermal oscillations (3) � Equation for u b : A g W n +1 4 n ( n +1) h n max[ τ d − τ b , 0] n u b = ( from force balance ) where τ d = ρ i g h 2 L , � a ′ exp( − b ( e − e c )) if w > 0 τ b = otherwise . ∞ Critics Workshop, Kulhuse, Denmark Ice streams dynamics 19 / 34

Characteristic modes of the ice stream � Mode 1: Steady-streaming mode with drainage ( T s = − 15 ◦ C). 300 1 0 . 75 u b (m/y) 200 w (m) 0 . 5 100 0 . 25 0 100 0 0 2 2 4 4 6 6 8 8 10 kiloyears Critics Workshop, Kulhuse, Denmark Ice streams dynamics 20 / 34

Characteristic modes of the ice stream � Mode 2: Steady-streaming mode without drainage ( T s = − 20 ◦ C). 300 1 0 . 75 u b (m/y) 200 w (m) 0 . 5 100 0 . 25 0 100 0 0 2 2 4 4 6 6 8 8 10 kiloyears Critics Workshop, Kulhuse, Denmark Ice streams dynamics 20 / 34