A thorough understanding of ice thermodynamics is essential for an accurate description of glaciers, ice sheets and ice shelves. Yet there exists a significant gap in our theoretical knowledge of the time-dependent behaviour of ice temperatures due to the inevitable compromise between mathematical tractability and the accurate description of physical phenomena. In order to bridge this shortfall, we have analytically solved the 1D time-dependent advective–diffusive heat problem including additional terms due to strain heating and depth-integrated horizontal advection. Newton's law of cooling is applied as a Robin-type top boundary condition to consider potential non-equilibrium temperature states across the ice–air interface. The solution is expressed in terms of confluent hypergeometric functions following a separation of variables approach. Non-dimensionalization reduces the parameter space to four numbers that fully determine the shape of the solution at equilibrium: surface insulation, effective geothermal heat flow, the Péclet number and the Brinkman number. The initial temperature distribution exponentially converges to the stationary solution. Transient decay timescales are only dependent on the Péclet number and the surface insulation, so higher advection rates and lower insulating values imply shorter equilibration timescales, respectively. In contrast, equilibrium temperature profiles are mostly independent of the surface insulation parameter. We have extended our study to a broader range of vertical velocities by using a general power-law dependence on depth, unlike prior studies limited to linear and quadratic velocity profiles. Lastly, we present a suite of benchmark experiments to test numerical solvers. Four experiments of gradually increasing complexity capture the main physical processes for heat propagation. Analytical solutions are then compared to their numerical counterparts upon discretization over unevenly spaced coordinate systems. We find that a symmetric scheme for the advective term and a three-point asymmetric scheme for the basal boundary condition best match our analytical solutions. A further convergence study shows that

The study of ice thermodynamics is of crucial importance for understanding the behaviour of glaciers, ice sheets and ice shelves. Ice temperatures control both the rate at which ice deforms

Steady-state ice temperature distribution studies also provide analytical solutions in bounded spatial domains but fall short if the transient nature of the solution is to be captured. This is the case of the studies on the shear heating margins of West Antarctic ice streams

More recently,

Despite these simplifications, heat transfer is well-known to be a 3D process with a higher level of complexity that encompasses several mechanisms such as horizontal and vertical advection, the potential presence of liquid water within the ice, a varying ice thickness, internal heat deformation, and frictional heat production

However, numerical models require caution as their accuracy and consistency must be previously assessed. Intercomparison projects are thus fundamental since they can provide consensus in benchmark experiments that further serve as a reference solution for validation. In this context, analytical descriptions are extremely useful as they provide a control irrespective of the resolution or discretization schemes. For instance,

Traditional approaches from both numerical and analytical perspectives assume the simplest heat-flux boundary condition at the ice surface: the imposition of the air temperature at the uppermost ice layer. Knowing that glacial ice forms through snow densification, this imposition appears to be an oversimplification, given that thermal conductivity increases with density

The 1D advective–diffusive equation has been thoroughly studied in a wide range of fields, particularly in dispersion problems. In early studies, the basic approach was to reduce the advection–diffusion equation to a purely diffusive problem by eliminating the advective terms. This was achieved via a moving coordinate system

Ice temperatures are critical not only to understand the dynamics and an ice body's evolution in time but also to set the ice-sheet initialization of numerical models. Poorly known parameter fields such as the ice temperature are estimated, minimizing the mismatch between observations and model output variables. Traditional approaches compute a steady-state temperature field, incorrectly assuming that the ice is at thermal equilibrium

It is thus clear that a time-dependent analytical description would be valuable in spite of the inevitable compromise of designing a model that is both mathematically solvable and accurate. It is thus of utmost importance to carefully navigate this trade-off, deciding the appropriate level of analytical tractability and physical realism based on the specific goals of any given study. Attaining the right balance allows for meaningful insights while avoiding excessive computational demands or oversimplification that may hinder the accurate representation and understanding of the real-world system. Despite all the effort in previous works, there is still a clear gap in the understanding of the analytical nature of time-dependent ice temperatures. As a result, there are no available benchmark experiments to test numerical solvers extensively employed in ice-sheet models.

The current study presents an analytical formulation of the transient ice temperature equation and provides useful insight in two ways: first by allowing for a simplified way of studying the physics of heat transfer in ice (as demonstrated by an equilibrium timescale analysis) and second by providing a way of benchmarking numerical solvers for heat transfer. Our approach accounts for the temporal evolution of the temperature profile rather than assuming an equilibrated state, thus taking a step towards a more accurate representation of the ice thermal behaviour. The formulation of the problem is given in Sect. 2, the approach followed in this work is presented in Sect. 3, analytical solutions are shown in Sect. 4, results are presented in Sect. 5, benchmark experiments are detailed in Sect. 6, results are discussed in Sect. 7, and concluding remarks are given in Sect. 8.

We consider a 1D ice column with diffusive heat transport, vertical advection, strain heat and depth-integrated horizontal advection. Our domain is defined as the interval

Schematic view of the 1D ice column with vertical advection

In the simplest physical scenario, the ice surface temperature is set to the air temperature value

This refinement enables a more accurate representation of the surface heat transfer dynamics and contributes to a comprehensive understanding of the energy balance within the ice column. In this description, both the surface ice temperature

Considering diffusive heat transport, vertical advection and a potential heat source, the ice temperature

In order to solve this problem, we must first provide the particular form of the vertical velocity term. As in

Standard values for

The inhomogeneous term

The horizontal advection term

These assumptions allow us to include a potential strain heating source

We next outline our analytical approach. We first non-dimensionalize our problem and exploit the linearity of the differential operator by further decomposing the solution as a sum of stationary and transient components to deal with the inhomogeneity. Lastly, we apply a separation of variables to obtain a solution of the time-dependent problem and impose the corresponding initial and boundary conditions. Derivation details are elaborated in Appendix

It is natural to non-dimensionalize our problem by defining the following variables,

Non-dimensional definitions and characteristic ranges. Summation is implied over repeated indices.

Hence, we can express our Eq. (

The dimensionless problem clearly shows that five numbers completely determine the shape of the stationary solution:

Physical parameter values employed to determine the non-dimensional range shown in Table

NA: not available.

Given that Eq. (

The solution to the stationary component (Eq.

We now take a step further and allow for time evolution by solving Eq. (

The full solution

Before displaying the results of the full time-dependent problem, it is worth describing the temperature solutions at equilibrium.

Figure

The non-dimensionalization of our analytical model provides simplicity and further reduces the parameter dimensionality of the solutions to solely five numbers, each corresponding to one column in Fig.

Stationary temperature profiles

We now present the results of the full problem presented in Eq. (

To illustrate the full solutions, we show the explicit time evolution from an initial profile as it approaches the corresponding stationary solution (Fig.

Time-dependent solution

To examine the transient nature of the solutions closely, we present the temperature evolution of a given initial profile for a certain range of the non-dimensional parameters (Fig.

The particular parameter values were selected so that we could obtain four physically distinct scenarios: (a) high geothermal heat flow under a large advection regime, (b) high strain heat dissipation in a low-vertical-advection regime, (c) strong lateral advection of colder ice under surface insulating conditions and (d) weak geothermal heat flow under a low-vertical-advection regime. This setup allows us to separately determine the role played by each mechanism during the transient regime of the solution.

Figure

Time-dependent solution

We can also predict the behaviour of the transitory component directly from the eigenvalues of the problem. By calculating the inverse of the eigenvalues

Decay time and corresponding eigenvalues.

The analytical solutions obtained herein are valuable tools for testing numerical solvers. We thus propose a suite of benchmark experiments with gradually increasing complexity to test the representation of each physical process involved in ice temperature evolution (see Table

Benchmark experiments for numerical solvers and main physical processes considered for heat propagation. The experiments are named in increasing complexity order.

First, we simply consider the well-known purely diffusive case (Exp-1). Then, vertical advection is additionally included (Exp-2). Lastly, strain heating (Exp-3) and the vertically averaged horizontal advection (Exp-4) are considered. Given the analytical nature of our solutions, spatial and temporal resolutions can be set arbitrarily high as there are neither convergence nor stability constraints. This allows for a comparison against spatial and temporal resolutions found in numerical solvers. We must stress that the initial temperature profile and all other parameters can be set by the user to test the solution for any desired scenario. We also note that these are simply proposed benchmarks, but the solutions developed here can be used for any type of benchmark test that is desired and fits the limitations of the equations.

We develop a numerical model for testing by performing a finite-difference discretization of Eq. (

Finite-difference approximations employed in the numerical study (Fig.

As could be expected, Fig.

For all experiments tested, results are identical irrespective of the particular discretization of the diffusion term (Table

Additionally, we perform a resolution convergence test for the best discretization choice (Table

Convergence study of benchmark experiments. Steady-state analytical solutions shown with the solid black line.

The adoption of dimensionless variables results in enhanced generality and mathematical convenience, albeit at the expense of veiling the practical significance to real glaciers and ice sheets. We have consequently tabulated data for characteristic values to ease interpretation (Table

We first start by comparing our results with a previously obtained solution for a simpler case

The transient behaviour of the solution is intricate given the freedom to choose an arbitrary initial state. This issue can be overcome by direct inspection of the eigenvalues of the problem. An estimation of the decay time of the analytical solution shows that the advection and the surface insulation are the only parameters that determine the timescale to reach thermal equilibrium. This approach has some limitations, some of which we now discuss. The decay time dependency is subjected to the mathematical form of our problem (Eq.

The tractability of the analytical solution does not allow for further complexity, and hence additional numerical methods would be necessary if such a physical description is desired. Nonetheless, a constant horizontal advection term

It must be stressed that our analytical solutions are not limited to regions with negligible horizontal velocities, since the true constraining quantity is the vertical gradient of the horizontal velocity

The strain rate regime poses further limitations on the applicability of the solution. Particularly for regions where vertical shear dominates and the strain heat dissipation is concentrated near the base, a vertically averaged contribution appears to be inaccurate. Nevertheless, as already noted by

It is worth noting that phase changes are not considered herein, so temperature evolution is strictly confined to values below the pressure melting point. Unlike a numerical solver, where temperature is manually limited, these solutions must be taken with caution as we are describing a frozen ice column. Results are still compatible with a potential heat contribution due to basal frictional heat (Eq.

Knowing that ice forms by snow densification through time

Our suite of benchmark experiments allows us to test numerical solvers and assess reliability for different discretization schemes and resolutions. The basal boundary condition is sensitive to the particular discretization scheme, as the geothermal flux is the main source of heat in the ice column and is considered via a Neumann boundary condition. The simplest two-point stencil does not correctly represent the equilibrium temperatures, yielding larger deviations at the base (Fig.

Resolution plays a fundamental role in obtaining a reliable temperature profile. A sigma coordinate system with quadratic spacing accurately (

We have determined the analytical solution to the 1D time-dependent advective–diffusive heat problem including additional terms due to strain rate deformation and depth-integrated horizontal advection. A Robin-type top boundary condition further considers potential non-equilibrium temperature states across the ice–air interface. The solution was expressed in terms of confluent hypergeometric functions following a separation of variables approach. Non-dimensionalization reduced the parameter space to five numbers that fully determine the shape of the solution at equilibrium. We further overcome the arbitrariness of the initial temperature profile by directly calculating the eigenvalues of the problem and their corresponding decay times as an estimation of the timescale of our system in different physical scenarios. The transient component exponentially converges to the stationary solution with a decay time that solely depends on vertical advection and surface insulation.

The sign of vertical advection is of utmost importance as it determines the direction along which temperature gradients are transported. We have focused in the present study on the downward advective scenario, given the implausibility of an upward advection of ice. At equilibrium, basal temperatures are particularly sensitive to four physical quantities: vertical advection, geothermal heat flow, strain heat and lateral advection. In contrast, the surface insulation yields negligible changes in the stationary solution. This is true even for highly insulating conditions at the ice surface, so long as colder ice is transported more efficiently than heat travels upwards due to diffusion.

The transient regime shows a strongly distinct behaviour. The arbitrariness of the initial state is overcome by a direct inspection of the eigenvalues of the problem. We then obtain a magnitude that represents the decay time of each Fourier mode that provides information about the equilibration time of the system. We find that the decay time of the transient component solely depends on two magnitudes: advection (

Our suite of benchmark experiments are convenient for assessing the accuracy and reliability of numerical schemes. We have employed unevenly spaced grid discretizations to obtain higher resolution near the base whilst minimizing the total number of grid points, thus reducing computational costs. A symmetric discretization of the advective term combined with a three-point basal boundary condition yields the best agreement compared to analytical solutions. In terms of convergence and grid resolution, we find that

Lastly, we note that our analytical solutions are general and can be applied to any initial boundary value problem that fulfils the conditions herein described. They can provide temperature distributions for any 1D problem at arbitrarily high spatial and temporal resolutions that consider the combined effects of diffusion, advection and strain heating without any additional numerical implementation. Furthermore, they present a reliable benchmark test for any numerical thermomechanical solver to quantify accuracy losses and necessary spatial and temporal resolutions.

Let us briefly outline the separation of variables technique before elaborating on the solutions of our general problem. Consider the following initial-boundary value problem (IBVP) on an interval

This technique looks for a solution of the form

Additionally, in order for

Thus, for each eigenfunction

We can then express the transitory solution as

Since the confluent hypergeometric functions are orthogonal, the normalized eigenfunctions form an orthonormal basis under the

For the stationary regime, we do not need to apply separation of variables because the problem reduces to a second-order ordinary differential equation in only one independent variable

In this section, we also assume thermal equilibrium, thus reducing the problem again to a second-order ordinary differential equation in only one independent variable

Unlike the general stationary solution shown in Eq. (

Our finite-difference discretization considers unevenly spaced grids, commonly used in the glaciological community where higher resolutions are desired near the base whilst minimizing the required number of points to reduce computational costs. We thus build a new coordinate system

We now present the numerical schemes necessary to account for non-homogeneous grids

All scripts to obtain the results presented herein and to further plot figures can be found at

DMP formulated the problem, derived the analytical solutions, coded the numerical solvers, analysed the results and wrote the paper. All other authors contributed to the analysis of the results and the writing of the paper.

At least one of the (co-)authors is a member of the editorial board of

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This research has been supported by the Spanish Ministry of Science and Innovation (project IceAge, grant no. PID2019-110714RA-100); the Ramón y Cajal Programme of the Spanish Ministry for Science, Innovation and Universities (grant no. RYC-2016-20587); and the European Commission, H2020 Research Infrastructures (TiPES, grant no. 820970).

This research has been supported by the Spanish Ministry of Science and Innovation (project IceAge, grant no. PID2019-550 110714RA-100); the Ramón y Cajal Programme of the Spanish Ministry for Science, Innovation and Universities (grant no. RYC-2016-20587); and the European Commission, H2020 Research Infrastructures (TiPES, grant no. 820970). The article processing charges for this open-access publication were covered by the CSIC Open Access Publication Support Initiative through its Unit of Information Resources for Research (URICI).

This paper was edited by Carlos Martin and reviewed by Ed Bueler and four anonymous referees.