[Paper Review] A unified kinematic wave theory for melt infiltration into firn

Mohammad Afzal Shadab, Anja Rutishauser+2 more|arXiv (Cornell University)|Mar 24, 2024

Landslides and related hazardsEnvironmental Science3 citations

TL;DR

This paper develops a unified kinematic wave theory for meltwater infiltration into firn by formulating a 2×2 hyperbolic system of PDEs governing mass and enthalpy conservation, enabling analytical solutions for 12 physically relevant Riemann problems. The theory captures melting/refreezing fronts, frozen fringes, perched water tables, and impermeable ice lens formation, providing exact benchmarks for numerical models of firn hydrology and ice-sheet dynamics.

ABSTRACT

Motivated by the refreezing of melt water in firn we revisit the one-dimensional percolation of liquid water and non-reactive gas in porous ice. We analyze the dynamics of infiltration in the absence of capillary forces and heat conduction to understand the coupling between advective heat and mass transport in firn. In this limit, we formulate a kinematic wave theory that results in a 2X2-system of hyperbolic partial differential equations (PDEs) corresponding to the conservation of composition and enthalpy. For simple initial conditions (Riemann problems) this system admits self-similar solutions that illuminate the structure of melting/refreezing fronts and analytical solutions are provided for 12 basic cases of physical relevance encountered in the literature. Further we develop an extended kinematic theory that encompasses the cases when the firn saturates completely to form a perched water table governed by elliptic PDE so that the model is no longer fully hyperbolic (local). These solutions provide benchmarks for numerical models of melt infiltration into firn. They also provide insight into important physical processes such as the formation of frozen fringes, the perching of meltwater on pre-existing low porosity layers and the conditions required for impermeable ice lens formation. Lastly, these analytic solutions can be utilized to improve and compare the performance of the firn hydrology, ice-sheet and Earth system models. Our analysis provides a theoretical framework to understand these important processes in firn which affect the partitioning between meltwater infiltration and surface runoff and therefore determine the surface mass loss from ice sheets and its contribution to sea level rise.

Motivation & Objective

• To develop a physics-based, analytical framework for meltwater infiltration into firn that accounts for advective heat and mass transport.
• To address the limitations of classical kinematic wave theory in fully saturated or heterogeneous firn by extending it to include elliptic behavior in saturated zones.
• To provide exact analytical solutions for key physical processes such as frozen fringe formation and impermeable ice lens development.
• To offer benchmark solutions for validating and improving numerical firn hydrology, ice-sheet, and Earth system models.
• To enhance understanding of meltwater partitioning between infiltration and runoff, which directly affects surface mass balance and sea-level rise.

Proposed method

• Formulates a 2×2 hyperbolic system of hyperbolic PDEs based on conservation of composition and enthalpy in the absence of capillary forces and heat conduction.
• Applies the method of characteristics to solve Riemann problems, yielding self-similar solutions for 12 canonical infiltration scenarios.
• Extends the kinematic wave model to include elliptic PDE behavior in regions of complete saturation, enabling modeling of perched water tables.
• Introduces a three-phase region (ice, liquid water, air) with non-dimensionalized variables for porosity, saturation, and enthalpy to unify the description.
• Derives and proves properties of the eigenstructure (eigenvalues and eigenvectors) to establish the hyperbolicity and monotonicity of wave speeds.
• Uses eigendecomposition and flux gradient analysis to validate the physical consistency of wave propagation and front evolution.

Figure 1 : The dependence of temperature and volume fractions on dimensional and dimensionless enthalpy and composition, ( $C,H$ ) and ( $\mathcal{C},\mathcal{H}$ ) respectively. Dimensional $C,H$ : (a) temperature and volume fractions of (b) water, (c) ice and (d) gas phases. Dimensionless $\mathca

Experimental results

Research questions

• RQ1 How do melting and refreezing fronts evolve during meltwater infiltration into cold firn without capillary or conductive effects?
• RQ2 What are the analytical conditions under which a perched water table forms in low-porosity firn layers?
• RQ3 Under what conditions does an impermeable ice lens form due to complete saturation and limited porosity?
• RQ4 How can the kinematic wave model be extended to remain valid in fully saturated, elliptic regimes?
• RQ5 What are the exact solutions for the interaction of wetting and freezing fronts in layered firn with varying initial conditions?

Key findings

• The model yields analytical solutions for 12 basic physical cases of meltwater infiltration, including wetting and freezing fronts, with explicit dependence on initial saturation and enthalpy.
• The theory confirms that refreezing has a minor effect on front speed due to the large latent heat of fusion, consistent with prior field observations.
• The formation of frozen fringes and perched water tables is analytically captured as a result of porosity reduction and flux imbalance at low-porosity layers.
• Impermeable ice lens formation is predicted when the volumetric flux exceeds the pore capacity of a low-porosity layer, leading to complete saturation and hydraulic barrier formation.
• The extended model successfully transitions from hyperbolic to elliptic behavior in saturated zones, resolving a key limitation of classical kinematic wave theory.
• The solutions serve as exact benchmarks for validating numerical models, reducing uncertainties in firn hydrology and Earth system modeling.

Figure 2 : The dimensionless flux of composition or enthalpy in $\mathcal{C}\mathcal{H}$ phase space for $m=3$ and $n=2$ . In region 1 consisting of water and gas region ( $\mathcal{H}\leq 0$ ) as well as region 3 comprising of three phase region ( $0<\mathcal{H}<\mathcal{C}$ ), the flux of dimensio