Lecture Notes: Ice Sheets and Ice Shelves

I. Terrestrial Ice Sheets

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Formation and Flow:

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Glaciers and ice sheets originate from the persistent accumulation of snow, which over time compacts into ice1. This process requires specific climatic conditions, primarily low temperatures, allowing snow to persist through the melt season1.

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When the accumulated ice reaches a sufficient thickness, typically tens to hundreds of meters depending on the ice temperature and deformation rate, the solid ice deforms under its own weight. This deformation can be described as viscous flow, where the ice deforms continuously under stress1. Additionally, the ice may also slide at the base, particularly where it contacts the underlying bedrock or sediment1.

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Glaciers tend to form at higher elevations due to colder temperatures that allow snow to persist from winter into summer1. The topography of these glaciers is typically dictated by the mountainous terrain upon which they form1.

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In contrast, the major ice sheets in Greenland and Antarctica are massive enough that their surface topography is largely independent of the underlying land2. The sheer volume of ice in these sheets results in them generating their own high elevations2. Their melting and discharge into the ocean occur mainly around their peripheries2.

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Snow Compaction and Firn Layer:

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The formation of a glacier or ice sheet begins with the net accumulation of snow over an annual cycle3. This means that more snow must fall and persist than melts or sublimates over the course of a year.

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The accumulated snow compacts under its own weight to form glacial ice3. This process occurs in the firn layer, which is the transitional layer between snow and solid ice3. The firn layer typically extends from the surface down to depths of 10-100 meters3.

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The density of firn (ρf) is related to the density of pure ice (ρi ≈ 917 kg/m³) and the solid fraction (ϕ), which represents the proportion of the volume occupied by ice: ρf = ϕρi4. Initially, the firn contains a significant amount of air, so ϕ is less than 1, and thus ρf < ρi4.

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To deform the firn, a stress (σ) must be applied, which is analogous to packing a snowball4. This stress is related to the weight of the overlying material and the mechanical properties of the firn4.

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For a simplified elastic rheology (the study of how materials deform under stress) of the firn, the stress is considered to be linearly proportional to the strain (deformation): σ ≃ E (ϕ - ϕ0)/ϕ04, where E is the elasticity (or Young's modulus) of the firn (which represents the material's stiffness), and ϕ0 is the reference, undeformed solid fraction of firn at the surface4. This relationship describes how much the firn deforms under a given stress, assuming an elastic behavior.

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The weight of the overlying snow increases stress with depth, resulting in a hydrostatic pressure gradient. The relationship between stress and depth is ∂σ/∂z = ϕρig5, where g is the acceleration due to gravity. This equation relates the change in stress with depth to the density of the firn, the solid fraction, and gravity5.

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Integrating this relationship allows us to find that the solid fraction increases exponentially with depth: ϕ = ϕ0e^(z/δ)5, where δ is the compaction length defined as δ = E/ϕ0ρig. The compaction length (δ) parameterizes the depth over which the firn density increases towards that of solid ice5.

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In reality, firn rheology is more complex than a simple elastic model, particularly at greater depths where viscous deformation becomes significant5. Additionally, processes such as melting of the surface firn, and the percolation and refreezing of meltwater at depth also influence the density profile5.

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Model Validation: The Regional Atmospheric Climate Model (RACMO2/ANT) is used to simulate near-surface climate and mass balance in Antarctica6.... This model has been validated against observational data, showing good agreement in representing the Antarctic climate6....

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The model's surface temperature (Ts), 10-meter wind speed (V10m), and accumulation (A) are validated using data from 50 Antarctic firn sampling sites6....

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A cold bias of 2K in the modeled Ts was removed prior to analysis7.

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The model's accuracy was assessed by comparing it against measured data with assigned error bars of 1K for Ts, 1 m/s for V10m and 30% for A7.

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After removing the cold bias, the model temperature deviations follow a 1:1 line, indicating good accuracy9. Similarly, modeled and measured 10 m wind speeds show good agreement although RACMO2/ANT slightly overestimates low wind speeds and underestimates high wind speeds9.

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RACMO2/ANT captures the strong accumulation gradients in Antarctica10. Despite the complexity of accumulation processes, the model is considered reliable to drive a firn densification model8....

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Viscous Flow of Ice:

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Ice accumulation in Greenland and Antarctica occurs over vast lateral distances of 1000 kilometers or more, with accumulated depths of 2-3 kilometers11.

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Because the lateral extent of the ice is so much greater than its vertical depth, the flow of ice from the interior to the margins of the ice sheet is primarily horizontal11.

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The vertical velocity of the ice is considered negligible11. The pressure is therefore hydrostatic. This means that the pressure at any point in the ice is determined by the weight of the ice above it: p = p0 + ρg(h-z)12, where p0 is atmospheric pressure, ρ is the density of ice, g is gravitational acceleration, h is ice thickness, and z is the local vertical coordinate (pointing upwards).

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Pressure gradients from thick to thin regions drive the flow of glacial ice12. The horizontal pressure gradient is given by ∂p/∂x = ρg(∂h/∂x)13. This equation shows that the pressure gradient depends on the density of the ice, gravity, and the slope of the ice sheet surface.

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This pressure gradient is balanced by the viscous resistance to flow of the ice13. The ice's resistance to flow is generally proportional to the shear stress acting within the ice13.

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Rheology of Ice:

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Ice is a polycrystalline material, composed of many individual ice crystals, or grains13. The deformation of ice is due to grain boundary sliding (where the grains slide past each other) and the motion of dislocations (defects in the crystal structure) within the crystalline matrix13.

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Applied stress (σ) is a tensorial quantity involving forces acting both normal and tangential to a material element. This stress can be decomposed into an isotropic (pressure p) component and a deviatoric (τ) component: σ = -pI + τ14. The isotropic component, pressure, acts equally in all directions, while the deviatoric component is responsible for viscous deformation14.

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A constitutive relationship describes the relationship between the strain rate (ϵ̇) and the applied deviatoric stress, which characterizes the deformation of the fluid. Strain rate is defined as ϵ̇ = 1/2 (∇v + ∇vᵀ)14, where v is the velocity vector.

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The simplest relationship is the Newtonian fluid model, where the strain rate is linearly proportional to the deviatoric stress: τ = ηϵ̇14, where η is the dynamic viscosity. In a Newtonian fluid, the viscosity is constant and independent of the shear rate14.

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Ice is a non-Newtonian fluid, which means that its viscosity is not constant and depends on factors such as grain size (d) and temperature (T), in addition to other thermodynamic properties such as the activation energy (Q), activation volume (V) and pressure (p)15.

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However, for many applications, where the shear is predominantly vertical, the rheology of ice can be approximated using Glenn's flow law: η = A(T) |∂u/∂z|^(1/n - 1)15, where A(T) is a temperature-dependent parameter, ∂u/∂z is the vertical shear rate (change in horizontal velocity with vertical position), and n is a power-law coefficient. The power law coefficient is typically assumed to be n=315. For a Newtonian fluid n=115. Experimental observations suggest a value of n in the range of 1.8 to 415. This equation implies that ice is a shear-thinning fluid, i.e. it's viscosity decreases with increasing shear rate15.

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Comparison Between Newtonian and Glenn’s Flow Laws: For a Newtonian fluid, the horizontal pressure gradient is balanced by the vertical gradient in the viscous shear stress: ∂p/∂x = ρg(∂h/∂x) = η(∂²u/∂z²)16. This is a direct application of Newton's law of viscosity where viscous stress is proportional to the rate of strain16. Integrating this equation (twice) with the boundary conditions of zero shear stress at the top surface and a basal sliding velocity at the bottom yields a parabolic velocity profile within the ice sheet: u = -ρg/(2η) (∂h/∂x)z(2h-z)16. This equation demonstrates that the velocity is zero at the base and increases quadratically with height until reaching a maximum at the surface (if we neglect basal sliding)16. Ice behaves as a shear-thinning fluid, where its viscosity decreases as the shear rate increases. This is modeled using Glenn's flow law17. The pressure gradient is balanced by the vertical derivative of the viscous shear stress: ∂p/∂x = ρg(∂h/∂x) = ∂/∂z(η(∂u/∂z))17. Applying Glenn’s flow law, and integrating twice, and applying the same boundary conditions (stress-free upper surface, basal sliding velocity), and assuming n=3, gives a velocity profile of u = (ρg/A)^(3) (∂h/∂x)^(3) [(h-z)^4/4 - h^4/4] + ub17. The resulting velocity profile from Glenn's law is more plug-like, with the shear concentrated near the base17. This is very different from the Newtonian case where the shear is maximized at the ice surface16....

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Basal Sliding:

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Ice may slide along its base, either over a hard bed of bare bedrock, or through the deformation of soft sediment20. This sliding is a crucial factor that affects the overall movement of ice sheets and the rate at which ice is discharged into the ocean20.

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The sliding speed (ub) is a key parameter for determining the ice export to the ocean20. This basal sliding velocity must be incorporated into the velocity profile16....

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The sliding speed is determined through a force or stress balance at the ice-bed interface. This relationship is expressed in general terms by a traction law: τb = f(ub,...)21, where τb is the basal shear stress and f represents a functional relationship depending on the sliding speed (ub), and other parameters.

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The functional relationship between basal traction (τb) and sliding speed (ub) is highly complex and depends on the properties of the bed21. This includes the bed's roughness, the water content and pressure at the bed, and the material properties (e.g., strength) of the bed21. Because the base of an ice sheet is inaccessible for direct observation, research in this area is very active21.

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Sliding over a hard bed:

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Sliding over a hard bed, such as bare bedrock, is commonly lubricated by a film of water between the ice and the bedrock22. This film is due to geothermal heat, frictional heating, and surface meltwater percolation22.

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At a macroscopic scale, the resistance to flow over hard beds is due to two main phenomena: regelation and plastic deformation22. In regelation, pressure differences cause melting on the upstream side of bed roughness elements and freezing on the downstream side, effectively allowing the ice to slide over the bumps22. Plastic deformation refers to localized flows in the ice surrounding the roughness element that result in slip at a macroscopic scale23.

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These processes lead to a basal traction law that depends on the aspect ratio of the roughness elements (a/l, where a is the amplitude and l is the length), and the basal velocity: τb = c(a/l)²ub^(2/(n+1))23. The constant c is related to the geometry of the roughness elements, and n is the power law coefficient for the viscous deformation of ice23.

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When a sufficiently large volume of water is present, smaller bumps may be effectively submerged, and the basal stress is supported only by a few larger bumps23. The proportion of these larger bumps that bear the stress is dependent on the available water and the water pressure23.

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The effective pressure (N = pi - pw), which is the difference between the ice pressure (pi) and the water pressure (pw), significantly influences the sliding speed23. When the pressure in the water equals the pressure in the ice, the basal traction becomes zero, as the ice floats entirely23.

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The Weertman sliding law is a generalized model that accounts for larger water volumes: τb = cubN^q24, where c, p and q are fitted coefficients. This is an empirical law and the coefficients must be determined through measurements.

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The maximum shear stress model, proposed by Schoof (2005), provides a more nuanced traction law: τb = µN(ub/(ub + cN^n))^(1/n)24, where µ is a friction coefficient, and c and n are fitted coefficients. This model recovers the Weertman sliding law at large effective pressures and has basal stress proportional to effective pressure at small effective pressures24.

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Sliding over a soft bed:

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When ice slides over a soft bed (such as sediment), it may deform the sediments over which it flows25. This deformation occurs at a critical yield stress25.

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Deformation of the sediment is expected only when the basal shear stress exceeds a critical value, τb > µN, where µ is the friction coefficient25.

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For a soft bed, two dominant models for soft sediment deformation are a perfect plasticity model where τb = µN25, where the bed deformation is independent of the sliding speed, and a Herschel-Bulkely or granular rheology model for the subglacial sediments (or till): τb = µN + cub26.

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These relationships suggest that deformation is either not controlled by the base, or that it is regularized by the rheology of the subglacial till26.

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It is important to note that most basal traction formulations are significant oversimplifications of reality, and are commonly applied to sparse data26. Basal traction is likely to depend on the spatially dependent properties of the bed, and also vary through time in a seasonal manner due to the dependence on water pressure26. Further research is required to refine the models of basal sliding26.

II. Floating Ice Shelves

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Role of Ice Shelves:

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In large ice sheets, the flow of ice ultimately transports a significant amount of ice mass to the ocean. This is where the ice begins to float as an ice shelf27.

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Ice shelves play a vital role in the dynamics of ice sheets, particularly in regions like West Antarctica27. They act to moderate the flow of ice from the continent to the ocean27.

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Flow Dynamics:

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The flow within the ice shelf is again characterized by being long and thin, similar to the ice sheet itself28.

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The pressure within the ice shelf is to leading order hydrostatic, implying that the pressure at any depth is determined by the weight of the overlying ice and water28.

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The flow within the ice shelf is predominantly driven by hydrostatic pressure gradients28.

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Since the ice shelf is floating on the ocean, there is negligible stress exerted by the ocean below and the atmosphere above28. Consequently, the velocity within the ice shelf is uniform in the vertical direction, and varies only in the lateral dimensions28.

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Flotation Condition:

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Ice floats because it is less dense than ocean water29. The density of ice is approximately ρice = 917 kg/m³29. Seawater density typically ranges from ρwater = 1020 to 1030 kg/m³ depending on salinity, although it is typically assumed to be 1025 kg/m³29.

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The Archimedes’ principle dictates that a floating body displaces its own weight in water29.

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The depth to which ice floats can be calculated using the equilibrium between the hydrostatic pressure in the ice and the ocean water. For an iceberg floating a height h above the water and a depth b submerged below the water the pressure in both ice and water must be the same30. Therefore ρwgb = ρg(h+b)30. This gives a relationship h = ((ρw - ρ)/ρ) * b ≃ 1/9 b30. This means that approximately one-ninth of the ice is above the water surface and eight-ninths of the ice is submerged.

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When applied to the transition from a grounded ice sheet to a floating ice shelf, the ice grounding line position is determined by the first point at which the base of the ice equals the depth of the ocean water, so that the ice begins to float: b(xG) = (ρ/(ρw - ρ)) h(xG) = d(xG)31, where d(xG) is the depth of the ocean at the grounding line.

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Locating the Grounding Line Through Flexure:

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The grounding line position is a critical parameter for determining the amount of ice directly contributing to sea-level rise31.

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The flexure, or bending of ice, is modeled classically as the bending of an elastic beam31. The beam has a bending stiffness given by B = EH³/12(1-ν²)31, where E is Young’s modulus, H is the thickness of the ice, and ν is Poisson’s ratio (a measure of how much a material compresses in one direction when it is stretched in another)31. Typical values for these parameters are E ≃ 9 GPa, H ≃ 200-1000 m, and ν ≃ 0.332.

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The bending of an elastic beam is modeled using Euler-Bernoulli beam theory: B(d⁴w/dx⁴) = p(x,y)32, where w is the deflection of the center-line of the ice, and p(x,y) is the loading acting on the ice sheet32. This theory balances the elastic stresses within a thin elastic beam with the forces acting on its surfaces32.

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Over the ocean, the loading force is due to the weight of the ice and the buoyancy force of the ocean32. The deflection of the ice is modeled by the equation: B(d⁴w/dx⁴) = -ρigH + ρwg(H/2 - w)33. Here, the first term is the weight of the ice and the second term is the buoyancy force of the water.

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Far from the grounding line, where the flexure is negligible and dw/dx → 0, the load and buoyancy must balance, giving the isostatic flotation condition: w = ((ρw - 2ρi)/ρw) * H33.

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The flexural length scale, which describes the distance over which significant flexure occurs, is given by l ∼ (B/ρwg)^(1/4)33. This length scale represents the typical distance over which the ice flexes near the grounding line33.

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The solutions to the flexure equation consist of a particular solution wp = H/2 - (ρi/ρw)H34 which represents the far field deflection, and a homogeneous solution which describes the flexural deformation, wh = A1e^(-x/α)cos(x/α) + A2e^(-x/α)sin(x/α)34, where the flexural length scale has been incorporated into a new parameter α, such that α⁴ = 4l⁴.

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Over land, the compression of subglacial sediments can be modeled as the deformation of an elastic medium with a spring constant k35. The deflection is described by: B(d⁴w/dx⁴) = -ρigH + k(H/2 - w + d)35, where d(x,y) is the basal topography. This equation balances the flexural forces in the ice with the load due to the weight of the ice, and the restoring force due to the compression of sediments35.

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The flexural wavelength on land is given by ll ∼ (B/k)^(1/4)35. This wavelength is typically much shorter than the flexural length over the ocean, because k is often much larger than ρwg36.

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The solution to the flexure equation over land is given by w = H/2 - (ρigH/k) + C1e^(x/αl)cos(x/αl) + C2e^(x/αl)sin(x/αl)36.

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These solutions for deflection on the ocean and on the land must be matched across the grounding zone36. The ice sheet, which is treated as an elastic beam, must remain continuous across this zone36.

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Therefore the following continuity conditions must be satisfied across the grounding line: [w] = 0, [dw/dx] = 0, [d²w/dx²] = 0, [d³w/dx³] = 037, which means that the deflection, the slope, the bending moment, and the shear stress are all continuous37.

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These four continuity conditions, in addition to the flotation constraint w(xG) = H/2 + d(xG)37, form a set of five equations which are sufficient to constrain the four coefficients A1, A2, C1, and C2, and the grounding line position xG37.

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Grounding Line Stability:

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The flux of ice within the ice sheet is a function of ice thickness, and can be expressed as: q = -(ρig/A)^(3) (∂h/∂x)^(3) (h + d)^(5)/5 + ub(h + d)38. This equation shows that the ice flux increases rapidly with increasing ice thickness38.

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The stability of grounding lines is assessed differently depending on whether the underlying bedrock slopes downwards into the ocean (prograde) or downwards towards the ice sheet (retrograde). In both cases the argument is based on the flux across the grounding line increasing with increasing ice thickness38.

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Prograde slopes: If the grounding line position increases, it leads to deeper water depths and a larger ice thickness due to the flotation condition39. The increased thickness of ice results in a larger ice flux across the grounding line. The increased mass loss leads to thinning of the ice sheet behind the grounding line. This thinning at the grounding line leads to a retreat of the grounding line due to the flotation condition39. Thus, prograde slopes have a negative feedback on grounding line migration39.

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Retrograde slopes: If the grounding line position increases it leads to shallower water depths, and therefore thinner ice according to the flotation condition40. This thinner ice leads to a reduction of ice flux across the grounding line, and an accumulation of mass in the ice sheet behind the grounding line. This accumulation drives the grounding line to advance further, leading to yet shallower water, creating a positive feedback loop40. Retrograde slopes therefore have a positive feedback on grounding line migration40.

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The grounding line position is determined by a combination of two physical processes41:

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The flotation condition (Archimedean buoyancy)41.

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A force balance at the grounding line, which determines the flux of ice across the grounding line, and hence its position41.

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Therefore, for any bedrock topography, the grounding line position occurs where both the flotation condition and the force balance are satisfied, and where both the water depth and ice flux are prescribed41.

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Back Stresses on the Ice Sheet:

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The back stress on the ice sheet is the force applied by the ice shelf on the ice sheet at the grounding line42.

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For an ice shelf that extends freely into the ocean, and makes no contact with any bedrock pinning points, the shelf cannot exert any force on the ice at the grounding line. It only transmits the hydrostatic pressure from the ocean42.

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However, if the ice shelf makes contact with pinning points, (submerged islands or high points in the ocean bathymetry or the side walls of a fjord), then the force from the ground may be transmitted back to the grounding line42.

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This force transmitted back to the grounding line is known as a back stress. It effectively thickens the flow of ice at the grounding line, analogous to how a pinning point may slow down a viscous flow and cause upstream build up42.

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The thickening of ice at the grounding line will drive the grounding line to advance on either prograde or retrograde slopes43.

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Ice shelf buttressing is when the ice shelf, through pinning points, acts to stabilize the ice sheet43.

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Melting of ice shelves, particularly by warm ocean water, can decrease this buttressing effect, thus destabilizing the grounding lines, especially in areas of retrograde slopes43.

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Examples of the importance of ice shelf buttressing include the breakup of the Larsen B ice shelf in 2002 which led to an increase in the speed of the feeder glaciers along the Antarctic peninsula, and the response of the Rutford ice sheet to ocean tides43.