Ice-sheet age computations are formulated using an Eulerian advection equation, and there are many schemes that can be used to solve them numerically. Typically, these differ in numerical characteristics such as stability, accuracy, and diffusivity. Furthermore, although various methods have been presented for ice-sheet age computations, the constrained interpolation profile method and its variants have not been examined in this context. The present study introduces one of its variants, a rational function-based constrained interpolation profile (RCIP) scheme, to one-dimensional ice age computation and demonstrates its performance levels via comparisons with those obtained from first- and second-order upwind schemes. Our results show that the RCIP scheme preserves the pattern of input surface mass balance histories in terms of the vertical profile of internal annual layer thickness better than the other schemes.

Core samples extracted from ice sheets can provide an archive of past
climate history data, and a major issue for researchers attempting to
utilize ice-core properties is defining the age of ice along the depth
of the ice sheet. This process is often called dating.
Dating with numerical ice-flow models is an important
approach because it allows researchers
to estimate age profiles before actual drilling of ice
cores. For example, in

Various methods for use in ice-sheet model dating have been adopted and compared.

To date, various methods have been presented and demonstrated for use
in ice-sheet age computations. However, there is still a
variety of numerical schemes that have not been examined within this context.
These include the constrained interpolation
profile (CIP) method

This section describes a standard algorithm of the CIP scheme family
that is used to
solve a 1-D advection equation with a non-advection term as follows:

As introduced in the previous section, there are three major approaches to solving an advection equation: Eulerian, Lagrangian, and semi-Lagrangian.
The CIP scheme family corresponds to a semi-Lagrangian method variation.
The basics of the semi-Lagrangian approach,
within the context of its comparison with the Lagrangian and Eulerian approaches, have already been presented in a number of past studies.
For example,

In CIP schemes, Eq. (

Appendix

The primary characteristic of this CIP scheme is the introduction of
an additional equation to solve the spatial derivatives of

In semi-Lagrangian approaches, a particle at

Schematic illustration of advection and
semi-Lagrangian scheme.
The new state computation for a target point

The CIP method constructs an interpolation function

The interpolation method used for the field variables, which characterize each scheme, is one of the most important topics in semi-Lagrangian schemes. Another major topic common to the semi-Lagrangian schemes is the method used to compute the departure point.

Equation (

The computation used to determine the age of the ice, i.e., the elapsed time since the ice
deposit, is performed with the pure advection equation

Some models adopt

In order to solve the time evolution of age and its gradient
(Eqs.

The age derivative,

In the present study, equivalent but different
coefficient representations (Eqs.

The spatial discretization of Eqs. (

One way to introduce a non-uniform discretization is to apply
a non-smooth grid

Another way to introduce a non-uniform discretization is to apply
a smooth grid

In the present paper, two numerical schemes, the first- and
second-order upwind schemes, are examined in comparison with the RCIP
schemes. While there are other numerical schemes suitable for such
comparisons, including Lagrangian, other
semi-Lagrangian, or even higher-order upwind schemes,
these have already been reported in past
studies

The “first-order” upwind scheme in the present paper evaluates the
advection term using the velocity at staggered grid points
as follows:

For the “second-order” upwind scheme, the derivative of the age term is
replaced by the second-order upwind difference formulation as

Following some modeling studies on the dating of deep drilling sites
that used simplified 1-D vertical ice flow models

In addition to the vertical velocity, the time evolutions of the surface and basal mass balances and the ice thickness are required for the age computations. These will be presented in each of the following sections.

The initial conditions for the

It is worth mentioning that formulations like
Eq. (

All the computations in our present study were performed on
a personal computer (PC) equipped with an Intel Xeon
E5-2609 central processing unit (CPU) and compiled with GNU Fortran.
Each surface/basal mass balance, ice thickness,
and vertical resolution configuration is repeated using four numerical schemes:
the RCIP with departure correction (

Before performing an experiment under a typical ice-sheet configuration,
verification of the numerical model used in the present study is
presented under further simplified conditions, namely the constant velocity
case.
This is easily performed using Eq. (

Experimental results obtained using a uniform velocity of

Figure

Experimental results obtained using a uniform velocity
of

Experimental results obtained under steady vertical velocity profiles with

Same as Fig.

Hereafter, non-uniform velocity experiments are performed
using

The ice thickness and the accumulation rate chosen in this and the following sections are

Two sets of basal melting are presented: no basal melting
and 3

Figures

The

This section presents the results of experiments conducted with non-steady velocity profiles, which were performed with the prescribed surface mass
balance time series. First, a very simple square-wave formulation is adopted for the time evolution of the surface mass balance.

Schematic figure showing the time evolution of the surface mass balance adopted in these experiments. Only the first two cycles are plotted.

Results of transient experiments with square-wave surface
mass balances of

Same as Fig.

Figures

Since there are few visible differences among the computed ages,
the computed age profiles relative to the one produced by the

Figures

Same as Fig.

Results of transient experiments with square-wave surface
mass balances of

The annual layer thickness has the following relationship in terms of
the thinning rate:

In terms of computed annual layer thickness profiles, the

The same exercises were performed using a different shape for the time
evolution of the surface mass balance.
Figure

Figure

Prescribed time evolution patterns of ice thickness adopted in
the non-steady thickness experiment.
The thickness evolution is
computed using an

The time evolution of the surface mass balance often involves the evolution
of ice thickness as a response. In this section, age
computation performance levels under non-steady mass balance and ice thickness conditions are presented.
In the present paper, the time evolution of thickness is computed as
follows:

Several experimental configuration combinations are examined.
These include square-wave or cosine-wave forcing;

A comparison with the fixed thickness experiments
(Fig.

Same as Fig.

Same as Fig.

So far, the surface mass balance values adopted in our experiments
have been positive (corresponding to the accumulation zone).
This limitation is sufficient for the usual topics relating to deep
ice-core experiments, where the interpretation of ice-core data may become too complex. However, in order to provide a complete demonstration of the
performance levels of numerical age computations for more general cases, it is worthwhile to examine other cases. Although it may be considered pointless to examine steady negative mass balance cases because they simply mirror the steady
positive cases presented above,
the surface mass balance level adopted in this section is examined with
zero or negative

The results of the transient experiments that were conducted under a square-wave surface mass balance of

Several experimental configuration combinations are examined.
In a comparison involving the experimental results of the positive mass
balance cases examined in this paper, qualitatively similar results are presented. As the prescribed surface mass balance at the lower

Vertical discretization adopted in the present study:

Results of transient experiments conducted with the square-wave surface mass balances of

Annual layer thickness becomes smaller with depth, which reflects
the vertical velocity profile. Therefore, differences in age between two neighboring levels become larger with increasing depth.
At a certain depth, the grid spacing becomes insufficient to hold the
variation of the input age cycles, which means that the preservation
of the input variation is lost below that depth.
Typically, in the experiments shown above, 100

In the same manner as computing an approximate depth–age solution under
constant surface/basal mass balance and constant thickness (e.g., the gray
benchmark lines in Fig.

Same as Fig.

Figure

Here, the same series of experiments is repeated using a higher resolution
and a uniform grid spacing of

Figure

The number of vertical layers presented above exceeds

Same as Fig.

Same as Fig.

When compared to the higher-resolution cases, the annual layer thickness patterns seem to be less preserved. The square-wave pattern in the results of

Figure

A comparison between Figs.

So far, all of the experiments were performed with uniform
discretization of either

Same as Fig.

Two non-uniform discretization types are adopted in this section.
One is a non-smooth grid and the other is a smooth grid (introduced in
Sect.

For non-uniform smooth discretization, a transformation function that
follows the reference profile is necessary between

Figures

The present study demonstrates a method for performing 1-D age computations of ice sheets under constant velocity, variable velocity responding to transient changes in surface mass balance, and/or changes in ice-sheet thickness. Herein, comparisons of the vertical profiles of computed ages, as well as annual layer thicknesses, were examined among the RCIP schemes (semi-Lagrangian) and upwind schemes (Eulerian). Although the experiments in the present study were limited to 1-D computations under summits, we believe the characteristics of the RCIP schemes have been presented sufficiently to allow evaluations of their performance levels.

Overall, the RCIP schemes show the best performance levels among the
schemes examined in the present study. In particular, the computed
vertical profiles of the annual layer thicknesses produced by RCIP
schemes follow the expected depth profiles more reasonably than the
other methods. This advantage reflects the design of the
RCIP scheme, which explicitly computes the evolution of the age
derivative, i.e., the inverse of annual layer thickness, using an
advection equation that is similar to the one used to compute the age
itself. Using the other schemes, the computed vertical profiles of
annual layer thickness either show more smoothing at shallower depths
than that were found with the RCIP scheme or the development of
oscillation at steep changes in the input surface mass balance. Such
oscillation development is shown even when the input is a smooth
cosine-wave-type pattern and the amplitude is large. Since the slope
filter adopted in this study is extremely simple, it is possible that
the results obtained by the use of a second-order upwind scheme with a
more suitable filter will change the characteristics.
The introduction of slope limiters on
general non-uniform discretization for higher-order upwind schemes is
possible

We examined two methods of computing the departure points in our RCIP
scheme experiments. Under a constant velocity case, the results
obtained by the simpler method show even less accurate
solutions than the first-order upwind scheme, while the other
“correction” method shows the best performance. The computed age
difference between the two RCIP methods is 1000 years at most
for all the configurations examined in the present study, including
the vertical resolution. As a result, the simpler method still
performs well if the expected accuracy of the application is less than
that period. Under an evolving surface mass balance, the solution of
the upwind scheme deviation is around 10

As has already been discussed in previous
studies

As long as the annual layer thickness is not a concern, we feel that the classical upwind schemes are acceptable choices for use when dating. Note that using a first-order upwind scheme causes the structural details of the surface mass balance history to disappear very rapidly, but average features will compute quite well, except for near the bottom. The second-order scheme preserves the history better than the first-order scheme, but without an effective slope limiter, strange oscillations can appear in the results, as we have demonstrated in the present paper. However, in spite of these oscillations in the annual layer thickness, the results achieved by the second-order scheme are still slightly better than those for the first-order scheme throughout most of this study's experiments.

The ice thickness and accumulation rate values used in the present
paper correspond to typical values found on the East Antarctic Plateau,
and the values used for the cycles in surface mass balance are

Although the focus of the present study is limited to 1-D age computations, implementation of the RCIP scheme for 3-D computation of the age field is also a suitable subject for future discussions.

Extension to 3-D would require the consideration of complex 3-D flow
fields and typically much lower horizontal ice age gradients.
In addition, the negative mass balance experiment demonstrated in the present
study is too simple to be compatible with the 3-D situation.
One important characteristic of the CIP scheme family is that
the spatial gradient of the field variable (age in this case) is not a
diagnostic (passive) value but is instead a prognostic field.

Furthermore, it is expected that the RCIP scheme will be applicable to
other advection problems in ice-sheet modeling. The evolutions of ice-sheet
thickness and temperature are formulated using transport or
advection equations, which are also good candidates for extending the
discussion of this study. For such cases, researchers may be
interested in mass or energy conservation in the field. Actually, a
multi-dimensional conservative formulation of CIP schemes has already
been proposed

A time-splitting technique (Eqs.

“Machine epsilon” is defined as the smallest

Although rounding up very small differences may be a possible solution for such cases, a different approach was adopted in the present study. After some
trials, the authors finally adopted the following procedure for avoiding such
oscillations, which (to the degree they used it) worked better than the rounding-up procedure. In the numerical model of the present paper,
Eq. (

All the numerical experiments in the present paper are performed with
IcIES-2/JP version 0, which is a subset package of the
Ice sheet model for Integrated Earth system Studies II (IcIES-2).
IcIES-2 is available at

The supplement related to this article is available online at:

FS developed the ice-sheet model and then implemented the RCIP and other dating schemes in the model. FS performed numerical experiments designed by all the authors. The manuscript was written by FS with contributions from TO and AAO.

The authors declare that they have no conflicts of interest.

We would like to thank Shawn Marshall and an anonymous referee for their valuable comments, which have substantially improved our manuscript. This study was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI under grant nos. 17K05664, 17H06323, and 17H06104.

This research has been supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI (grant nos. 17K05664, 17H06323, and 17H06104).

This paper was edited by Philippe Huybrechts and reviewed by Shawn Marshall and one anonymous referee.