We first consider a steady-state flow tube of an ice sheet, which starts at a dome and ends at the ice sheet margin (Fig. 1). This means that we assume that factors such as the geometry of the flow tube (e.g. ice thickness) and the vertical shape function do not change in time. In this model, we consider non-steady accumulation rate and melting rate, but this will be discussed later. The notation and equations follow that of Parrenin and Hindmarsh (2007) but the assumptions are slightly relaxed since we allow the flow tube width and relative density to vary vertically. We write the equations in the (x, z) coordinates where x, the horizontal coordinate along the direction of ice flow, is the distance from the dome and z is the vertical coordinate pointing upward. We suppose that the horizontal direction of the flow does not depend on the vertical position and is time-independent, and we represent the lateral flow divergence by the varying flow tube width Y(x,z). In practice, for grounded ice and under the shallow ice approximation, the direction of the flow can be determined from the surface elevation gradient. In this case, the steady-state assumption for the flow tube means that the shape of surface elevation contours does not change with time. The direction of the flow can also be determined by surface velocity measurements, as is done for the modelling of the flow line from DC to BELDC (Chung et al., 2025). Using surface velocity measurements is more appropriate where the surface is flat because in this case, the flow might not follow the direction of the greatest local slope.

The ice-sheet geometry is given by the bedrock elevation B(x), the surface elevation S(x) and the total ice thickness H(x)=S(x)-B(x). We let snow/ice relative density (expressed in % of volume) be D(x,z) and the surface accumulation of ice and basal melting rate at the ice–bedrock interface be a(x) and m(x), respectively. We let ux(x,z) denote the horizontal velocity and uz(x,z) the vertical velocity of the ice particles. We also let χ(x,z) represent the age of the ice particle.

We now define fluxes used to derive the stream function. The partial horizontal flux qH(x,z) is defined as the horizontal flux passing below elevation z: 1qH(x,z)=∫BzY(x,z′)D(x,z′)ux(x,z′)dz′ with QH(x)=qH(x,S) being the total horizontal flux at position x. We further define the basal melting flux as: 2Qm(x)=∫0xYB(x′)m(x′)dx′ where YB(x′)=Y(x′,B(x′)) is the tube width at the bedrock.

Because of the steady-state assumption, the two components of the velocity field can be derived from one scalar variable called the stream function, denoted by q(x,z) which follow these equations: 3YDux=∂q∂zYDuz=-∂q∂x. The flux through any path linking two points A and B is independent of the chosen path and is the difference qB–qA. Therefore, q=Qm+qH. By definition, the contours of the stream function correspond to the trajectories of ice particles.

We define the total flux at position x by Q(x)=q(x,S). We define the normalized stream function Ω(x,z) such that the stream function q is given by 4q(x,z)=Q(x)Ω(x,z). This new variable can be related to the horizontal flux shape function ω through 5Ω(x,z)=ω(x,z)+μ1+μ with ω, the horizontal-flux shape function (Parrenin et al., 2006; Reeh and Paterson, 1988) defined by: 6qH(x,z)=QH(x)ω(x,z), and with μ being the ratio of the melting flux to the horizontal flux 7μ(x)=Qm(x)QH(x). In order to determine particle trajectories and the age of ice, we can now transform the (x, z) coordinate system to a new system (π, θ) defined as (Parrenin and Hindmarsh, 2007): 8π(x)=ln⁡Q(x)Qrefθ(x,z)=ln⁡(Ω(x,z)) where Qref is a reference flux; in practice, we use the total flux at the downstream boundary of the domain. Note that the change of variable from x to π requires Q(x) to be an increasing function, and we must therefore assume that the accumulation a is strictly positive all along the flow-line. Also, the change of variable from z to θ requires Ω to be an increasing function of z, which corresponds to assuming that there is no reverse flow.

We then define the horizontal and vertical velocity components in this new coordinate system: uπ=dπdt and uθ=dθdt. Parrenin (2013) showed that these velocity components can be written as 9uπ=YSaYD∂Ω∂zuθ=-YSaYD∂Ω∂z where YS(x)=Y(x,S) is the tube width at surface. This way, it can immediately be seen (Parrenin and Hindmarsh, 2007), that these velocity components are equal in magnitude but opposite in sign. Therefore, trajectories in the (π, θ) coordinate system are straight lines of slope -1 (see Fig. 2).

Integrating the time spent along the trajectory from Eq. (9), the age of an ice particle can be written as 10χ=∫π0πκdπ with π0 the initial value of π when the ice particle was at surface and with the κ parameter defined as: 11κ=YDYSa∂zΩ∂Ω. We can now define the vertical thinning function, which is the ratio of the vertical thickness of a layer to its initial ice equivalent vertical thickness when it was at surface. Using this (π, θ) coordinate system, it can also be shown that the vertical thinning function can be written (Parrenin, 2013): 12τ=ΩYSaYDD0a01-1κ∫π0π∂κ∂π-1 where D0 and a0 are respectively the relative density and accumulation rate when the snow/ice particle was deposited.

We can now consider variations of surface accumulation and basal melting rates and assume they have common temporal variations (Parrenin et al., 2006; Parrenin and Hindmarsh, 2007): 13a(x,t)=a‾(x)R(t)m(x,t)=m‾(x)R(t) where a‾(x) is the temporal average accumulation rate and m‾(x) is the temporal average melting rate at position x. R(t) is a positive temporal multiplicative factor which can be determined from ice core data, e.g. using the relationship between the isotopic composition of ice and the surface accumulation rate. The implications of using the same temporal variation for basal melt rate and accumulation rate are discussed in Sect. 4.3. Using this assumption, the trajectories and isochrones are the same as those of the steady-state problem which uses a‾ and m‾, but velocities of ice particles are enhanced or reduced depending on the value of R(t). It is possible to deduce the real age of ice particles χ from the steady age χ‾ using the following change of time variable 14t‾=∫0tR(t)dt where t‾ is the steady time and t is the real time.

To simplify, we assume that snow is compressed instantaneously into ice at surface. In practice, we convert real depth into ice equivalent depth using a relative density profile informed by ice core observations or firn models in the simulated area. Moreover, we assume that the flow tube has vertical walls, that is, Y does not depend on z. This way, the equations for κ and τ become: 15κ=1a∂zΩ∂Ω16τ=Ωaa01-1κ∫π0π∂κ∂π-1.

The (π, θ) coordinate system is very useful for solving any transport equation, a class to which the age equation belongs. Indeed, if we define a regular grid in (π, θ) with the same step Δπ=Δθ=Δ, the trajectories of ice particles pass exactly through the nodes of the grid (Fig. 2). Therefore, starting from the surface where the age is χ=0, it is possible to deduce the age on the horizontal grid line below where θ=-Δ by calculating the time needed for each ice particle to cross a cell. Then the age on the horizontal grid line below (θ=-2Δ) can be calculated, etc. As particles move downstream, the conditions at the upstream boundary should be known. As the π scale is logarithmic, there is a singularity at the dome. Therefore, the horizontal π scale starts at a point slightly downstream of the dome, where an age can be prescribed, for example using a dome solution (which assumes purely vertical movement). If one is interested in a flank ice core, the upstream boundary can be chosen such that the upstream condition does not affect the age modelling of the ice core, i.e., the ice in the ice core comes from the surface boundary and not from the upstream boundary.

The time needed to cross a cell is 17Δχ=∫ππ+Δκdπ. Here, we assume that 1/a is continuous and varies linearly between two consecutive π nodes. Similarly, we assume that zΩ is continuous and varies linearly between two consecutive θ nodes along a vertical grid line (Fig. 2). It means that ∂zΩ∂Ω is a stepwise function along θ and a continuous and linear-by-parts function along π. Integrating κ along π across a cell therefore corresponds to integrating a second order polynomial.

Similarly, we can calculate the vertical thinning function using Eq. (11) by integrating the following quantity along each cell: 18∫ππ+Δ∂κ∂πdπ with our assumption that 1/a and zΩ are piecewise linear functions, ∂κ/∂π is a first order polynomial between π and π+Δ, so it can be easily integrated.

To calculate the initial horizontal positions, the total flux or the initial accumulation rate when the ice particles were at surface, we must transport these quantities from the surface along the diagonals in the (π, θ) grid (Fig. 2). The trajectories of the particles are given by the iso-contours of the stream function q.

In our age_flow_line-1.0 code, it is possible to determine the age, accumulation and thinning function on a refined vertical grid corresponding to an existing or potential ice core. We first determine the π coordinate of the ice core and calculate the age and thinning profiles by a weighted average of the two adjacent vertical profiles of the 2D grid. We then interpolate this coarse 1D vertical profile onto a refined 1D ice core profile. For the thinning function, we perform a linear interpolation while, for the age, we use a quadratic interpolation.

In order to verify that the uncertainty due to interpolation is negligible, we also use a different method to estimate the thinning function and age at the ice core location. Once the age, thinning and the initial accumulation rate are calculated for an ice core, we derive the age to get another estimate of the thinning function, and we integrate the layer thickness to get another estimate of the age. We then check that the two estimates of age and thinning (one integrating the age and the other integrating the thinning) are consistent within numerical uncertainties.

age_flow_line-1.0 is entirely written using the Python-3 programming language with several scientific modules as dependencies (numpy, scipy, matplotlib). age_flow_line-1.0 both solves the transport equations and displays the results as figures. The figures displayed for the 2D grid are: the 2D grid in the (x, z) and (π, θ) coordinate system, the age in the x, z), (x, d) and (π, θ) coordinate systems, thinning in the (x, z) coordinate system, obtained either by direct integration of the thinning or by differentiation of the age, the ω function in the (x, z) coordinate system, the stream function q and its contours (the stream lines, which represent particle trajectories) in the (x, z) coordinate system. Along the x coordinate of the flow line, figures display the accumulation rate, the basal melting rate, the width of the flow tube and the surface velocity or the total flux Q. Along the 1D vertical grid of each ice core, one figure displays the accumulation and layer thickness as a function of the age, and another figure displays the age, spatial origin and thinning as a function of depth.

For the horizontal flux shape function ω, we use the Lliboutry profile (Lliboutry, 1979; Parrenin and Hindmarsh, 2007): 19ω(ζ)=1-p+2p+1(1-ζ)+1p+1(1-ζ)p+2 where ζ is the normalised vertical coordinate (0 at the bedrock and 1 at the surface). Therefore, only the value of p needs to be defined for each horizontal position x along the flow line. However, one could easily implement another type of horizontal flux shape function, for example the Dansgaard–Johnsen profile (Johnsen and Dansgaard, 1992).

For the upstream boundary condition on the age, we impose a dome profile: 20χ=∫θ0κdθ. The core of the code is entirely separate from the experiment directory where the results of the run are saved. The experiment directory is composed of a general parameter file in the YAML format (age_flow_line-1.0 uses the pyaml module) and text files that define the accumulation rate a, basal melting rate m, width of the flow tube Y, surface elevation S, ice sheet thickness H and exponent of the Lliboutry profile p. There is also a vertical density profile given in a text file for transforming real depths to ice equivalent depths and this profile is assumed to be the same along the whole flow line.

This age model is accurate and exhibits minimal numerical diffusion since quantities are transported from one node to the next without interpolation. This could help test the accuracy of numerical schemes of more complex 3D and transient models.

This model has similar complexity of a steady-state model but considers the variations of accumulation with time which is non-negligible along a glacial-interglacial cycle (approximately a factor of 2 on the East Antarctic plateau). For a 1000×1000 grid, the computation time is ∼0.1 s on a recent laptop computer for the numerical calculations, with a few additional seconds for the building of the figures. For this grid, the memory footprint of the simulation is ∼0.8 GB.

Due to its fast computation time, this model is appropriate for inverse simulations where the results are fitted to observations. Indeed, inverse simulations require multiple runs of the forward model to find the optimal set of parameters, which can be computationally expensive.

This model could also be appropriate for representing the advection term in a heat equation, although the diffusion term is better dealt with in a physical (x, z) coordinate system.

We have performed a simulation where both vertical and horizontal ice flows are taken into account. We show that horizontal flow has a non negligible effect on the age modelling of the BELDC ice core, since our modelled isochrones do not have the same geometry than the ones simulated by the 1D model (Chung et al., 2023) when using the same set of input parameters. Indeed, while the 1D model simulates isochrones very close to the observed ones (Chung et al., 2023), the isochrones simulated by the 2.5D model significantly deviate from them (Fig. 7), with deviations of >100 m at some places. In our forward simulation using parameters inverted by the 1D model, particles originate sometimes >15 km from their current position. Our simulation is provided as a proof of concept for the model we have developed. However, it is not appropriate to use the result of the 1D inverse model by Chung et al. (2023) to determine the parameters of the 2.5D model. For a more realistic simulation, the parameters of the model should be optimized so that modelled age fits the radar and ice core age observations. This is the purpose of the companion paper of Chung et al. (2025) who developed this inverse methodology.

Our model relies on the pseudo-steady assumption, which states that all variables are in steady-state except for a temporal multiplicative factor which is both applied to accumulation and melting.

First, the geometry is assumed to be in steady-state. This assumption is adapted for the interior of East Antarctica, where relative variations in ice thickness were small (Ritz et al., 2001), but it still requires that the flow lines have not varied too much in the past, which is still unclear. Greenland and West Antarctica have probably encountered more important changes since they are more sensitive to climatic variations in the ocean and in the atmosphere (Quiquet et al., 2013; Wolff et al., 2025).

Second, the same temporal multiplicative factor is applied to both surface accumulation and basal melting. There are no physical reason to assume surface accumulation and basal melting have varied in concert; this is just a mathematical convenience. However, in regions where basal melt rates are low, this assumption should not introduce major errors, as long as a realistic time-averaged basal melt rate is prescribed. Moreover, this temporal factor assumes that the surface accumulation spatial pattern has remained stable in time. This assumption relies on a stable snow precipitation process and a stable snow re-deposition by wind, which might not always be the case (Cavitte et al., 2018).

Our pseudo-steady model can therefore be seen as a first approach to model the age field along flow lines at high resolution, but more complex 3D and transient models are necessary to fully investigate the age of the ice in ice sheets.

In this implementation, we have made a few simplifying assumptions that could be relaxed in the future. First we assumed that the flow tube has vertical walls, which is for example not the case along a ridge (Passalacqua et al., 2016). Second, we converted real depths into ice equivalent depth using a unique density profile along the flow line, while clearly for long flow lines the density profile might change with the distance from the dome. Third, we used a Lliboutry profile for the horizontal flux shape function ω, while other profiles could be used. In particular at a steady-state dome where Raymond arches develop (Raymond, 1983), the Lliboutry profile does not seem to be suitable (Martín and Gudmundsson, 2012). Fourth, we prescribed a dome solution on the upstream boundary of the domain, but any boundary condition could be prescribed.

Due to the numerical simplicity of the model, it may be possible to derive its Jacobian, which would make inverse simulations more efficient.

The value of this model lies in its high numerical accuracy and very fast computation time. It is appropriate for long flow lines, old age and for parameter inferences. The companion paper of Chung et al. (2025) outlines its interest, since we can optimize its parameters in a couple of minutes on a personal computer. However, this model has physical assumptions which might introduce errors. To overcome this limitation, our model could be used in conjunction with a transient model like the one developed by Buchardt and Dahl-Jensen (2007) or Gerber et al. (2021). For example, our model could provide an initial condition which could be refined using the transient model. It could also provide a “first guess” and a fast approximation of the Jacobian of the transient model, to speed up the optimization of its parameters.

This flow tube model could be applied to several nearly steady-state flow lines of the current Greenland and Antarctic ice sheets, especially those that have ice core information.

The Vostok flow line in East Antarctica was originally modelled by Ritz (1992) but only the age along the Vostok ice core was calculated by a Lagrangian tracer scheme. This work was later extended by Parrenin et al. (2001, 2004) who optimized some parameters of the model to fit the ice core age observations, but still without accounting for the isochronal information. Another flow tube model was developed and applied to the Vostok flow line by Salamatin et al. (2009), accounting for both the ice core and isochronal age observations. It would be valuable to apply the current numerical model to this Vostok flow line and compare the results with these previous modelling efforts.

The EDML ice core in East Antarctica is situated on a ridge and horizontal flow also needs to be taken into account. Huybrechts et al. (2007) applied a large scale ice sheet model of Antarctica with a local high resolution and high order model nested inside around EDML. It would be valuable to see how our numerical model, with its very high resolution and numerical accuracy and its fitting to isochronal observations, compares with this previous transient and large scale model.

The NEEM and NorthGRIP ice cores in Greenland are also situated on a ridge coming from Summit. Isochrones around NorthGRIP were simulated using a flow tube model by Buchardt and Dahl-Jensen (2007) and fitted to radar observations. The basal melting was hence deduced. Applying our numerical model to this flow line, we could compare it to the previous work and see if its high resolution and accuracy can improve the modelling scenario.

Another interesting flow line in Greenland is the one going from Summit to the EastGRIP ice core. The age of the ice along this profile was modelled by Gerber et al. (2021) using a similar model to that of the NorthGRIP flow line, based on a Dansgaard–Johnsen velocity profile (Johnsen and Dansgaard, 1992). Again, our numerical model could be compared to this previous modelling effort.