$\begin{split}\newcommand{\Diff}{ \frac{\mathrm{d}#1}{\mathrm{d}#2} } \newcommand{\diff}{ \frac{\partial #1}{\partial #2} } \newcommand{\var}{ {#1}_{\text{#2}} } \newcommand{\h}{ \var{h}{#1} } \newcommand{\T}{ \var{T}{#1} } \newcommand{\m}{ \var{m}{#1} } \newcommand{\ms}{ \var{m^{*}}{#1} } \newcommand{\psw}{p_{\text{ocean}}} \newcommand{\pice}{p_{\text{ice}}} \newcommand{\pmelange}{p_{\text{melange}}} \newcommand{\n}{\mathbf{n}} \newcommand{\nx}{\n_{x}} \newcommand{\ny}{\n_{y}} \newcommand{\phimin}{\phi_{\mathrm{min}}} \newcommand{\phimax}{\phi_{\mathrm{max}}} \newcommand{\bmin}{b_{\mathrm{min}}} \newcommand{\bmax}{b_{\mathrm{max}}} \newcommand{\bq}{\mathbf{q}} \newcommand{\Up}{\operatorname{Up}\left(#1\big|#2\right)} \newcommand{\uppair}{\left\{\begin{matrix} #1 \\ #2 \end{matrix}\right\}} \newcommand{\div}{\nabla \cdot} \newcommand{\V}{\mathbf{V}} \newcommand{\Wtill}{W_{\mathrm{till}}}\end{split}$

Subglacial hydrology¶

At the present time, two simple subglacial hydrology models are implemented and documented in PISM, namely -hydrology null (and its modification steady) and -hydrology routing; see Table 19 and . In both models, some of the water in the subglacial layer is stored locally in a layer of subglacial till by the hydrology model. In the routing model water is conserved by horizontally-transporting the excess water (namely bwat) according to the gradient of the modeled hydraulic potential. In both hydrology models a state variable tillwat is the effective thickness of the layer of liquid water in the till; it is used to compute the effective pressure on the till (see Controlling basal strength). The pressure of the transportable water bwat in the routing model does not relate directly to the effective pressure on the till.

Note

Both null and routing described here provide all diagnostic quantities needed for mass accounting, even though null is not mass-conserving. See Mass accounting in subglacial hydrology models for details.

Table 19 Command-line options to choose the hydrology model

Option

Description

-hydrology null

The default model with only a layer of water stored in till. Not mass conserving in the map-plane but much faster than -hydrology routing. Based on “undrained plastic bed” model of . The only state variable is tillwat.

A version of the null model that computes an approximation of the steady state subglacial water flux that can be used as an input for a frontal melt parameterization such as the one described in Frontal melt parameterization.

-hydrology routing

A mass-conserving horizontal transport model in which the pressure of transportable water is equal to overburden pressure. The till layer remains in the model, so this is a “drained and conserved plastic bed” model. The state variables are bwat and tillwat.

See Table 20 for options which apply to all hydrology models. Note that the primary water source for these models is the energy conservation model which computes the basal melt rate basal_melt_rate_grounded. There is, however, also option -hydrology_input_to_bed_file which allows the user to add water directly into the subglacial layer, in addition to the computed basal_melt_rate_grounded values. Thus -hydrology_input_to_bed_file allows the user to model drainage directly to the bed from surface runoff, for example. Also option -hydrology_bmelt_file allows the user to replace the computed basal_melt_rate_grounded rate by values read from a file, thereby effectively decoupling the hydrology model from the ice dynamics (esp. conservation of energy).

To use water runoff computed by a PISM surface model, set hydrology.surface_input_from_runoff. (The Temperature-index scheme computes runoff using computed melt and the assumption that a fraction of this melt re-freezes, all other models assume that all melt turns into runoff.)

Table 20 Subglacial hydrology command-line options which apply to all hydrology models

Option

Description

-hydrology.surface_input.file

Specifies a NetCDF file which contains a time-dependent field water_input_rate which has units of water thickness per time. This rate is added to the basal melt rate computed by the energy conservation code.

-hydrology_tillwat_max (m)

Maximum effective thickness for water stored in till.

-hydrology_tillwat_decay_rate (m/a)

Water accumulates in the till at the basal melt rate basal_melt_rate_grounded, minus this rate.

-hydrology_use_const_bmelt

Replace the conservation-of-energy basal melt rate basal_melt_rate_grounded with a constant.

The default model: -hydrology null¶

In this model the water is not conserved but it is stored locally in the till up to a specified amount; the configuration parameter hydrology.tillwat_max sets that amount. The water is not conserved in the sense that water above the hydrology.tillwat_max level is lost permanently. This model is based on the “undrained plastic bed” concept of ; see also .

In particular, denoting tillwat by $$W_{till}$$, the till-stored water layer effective thickness evolves by the simple equation

(15)$\frac{\partial W_{till}}{\partial t} = \frac{m}{\rho_w} - C$

where $$m=$$ basal_melt_rate_grounded (kg $$\text{m}^{-2}\,\text{s}^{-1}$$), $$\rho_w$$ is the density of fresh water, and $$C$$ hydrology_tillwat_decay_rate. At all times bounds $$0 \le W_{till} \le W_{till}^{max}$$ are satisfied.

This -hydrology null model has been extensively tested in combination with the Mohr-Coulomb till (section Controlling basal strength) for modelling ice streaming (see  and , among others).

This model (-hydrology steady) adds an approximation of the subglacial water flux to the null model. It does not model the steady state distribution of the subglacial water thickness.

Here we assume that the water input from the surface read from the file specified using hydrology.surface_input.file instantaneously percolates to the base of the ice and enters the subglacial water system.

We also assume that the subglacial drainage system instantaneously reaches its steady state. Under this assumption the water flux can be approximated by using an auxiliary problem that is computationally cheaper to solve, compared to using the mass conserving routing model (at least at high grid resolutions).

We solve

(16)$\diff{u}{t} = -\div (\V u)$

on the grounded part of the domain with the initial state $$u_0 = \tau M$$, where $$\tau$$ is the scaling of the input rate (hydrology.steady.input_rate_scaling) and $$M$$ is the input rate read from an input file.

The velocity field $$\V$$ approximates the steady state flow direction. In this implementation $$\V$$ is the normalized gradient of

$\psi = \rho_i g H + \rho_w g b + \Delta \psi.$

Here $$H$$ is the ice thickness, $$b$$ is the bed elevation, $$g$$ is the acceleration due to gravity and $$\rho_i$$, $$\rho_w$$ are densities of ice and water, respectively.

Note

Note that $$\psi$$ ignores subglacial water thickness, producing flow patterns that are more concentrated than would be seen in a model that includes it. This effect is grid-dependent.

This implies that this code cannot model the distribution of the flow along the grounding line of a particular outlet glacier but it can tell us how surface runoff is distributed among the outlets.

The term $$\Delta \psi$$ is the adjustment needed to remove internal minima from the “raw” potential, filling any “lakes” it might have. This modification of $$\psi$$ is performed using an iterative process; set hydrology.steady.potential_n_iterations to control the maximum number of iterations and hydrology.steady.potential_delta to affect the amount it is adjusted by at each iteration.

The equation (16) is advanced forward in time until $$\int_{\Omega}u < \epsilon\int_{\Omega} u_0$$, where $$\epsilon$$ (hydrology.steady.volume_ratio) is a fraction controlling the accuracy of the approximation: lower values of $$\epsilon$$ would result in a better approximation and a higher computational cost (more iterations). Set hydrology.steady.n_iterations to control the maximum number of these iterations.

This model restricts the time step length in order to capture the temporal variability of the forcing: the flux is updated at least once for each time interval in the forcing file.

In simulations with a slowly-varying surface input rate the flux has to be updated every once in a while to reflect changes in the flow pattern coming from changing geometry. Use hydrology.steady.flux_update_interval (years) to set the update frequency.

See Computing steady-state subglacial water flux for technical details.

The mass-conserving model: -hydrology routing¶

In this model the water is conserved in the map-plane. Water does get put into the till, with the same maximum value hydrology.tillwat_max, but excess water is horizontally-transported. An additional state variable bwat, the effective thickness of the layer of transportable water, is used by routing. This transportable water will flow in the direction of the negative of the gradient of the modeled hydraulic potential. In the routing model this potential is calculated by assuming that the transportable subglacial water is at the overburden pressure . Ultimately the transportable water will reach the ice sheet grounding line or ice-free-land margin, at which point it will be lost. The amount that is lost this way is reported to the user.

In this model tillwat also evolves by equation (15), but several additional parameters are used in determining how the transportable water bwat flows in the model; see Table 21. Specifically, the horizontal subglacial water flux is determined by a generalized Darcy flux relation , 

(17)$\bq = - k\, W^\alpha\, |\nabla \psi|^{\beta-2} \nabla \psi$

where $$\bq$$ is the lateral water flux, $$W=$$ bwat is the effective thickness of the layer of transportable water, $$\psi$$ is the hydraulic potential, and $$k$$, $$\alpha$$, $$\beta$$ are controllable parameters (Table 21).

In the routing model the hydraulic potential $$\psi$$ is determined by

(18)$\psi = P_o + \rho_w g (b + W)$

where $$P_o=\rho_i g H$$ is the ice overburden pressure, $$g$$ is gravity, $$\rho_i$$ is ice density, $$\rho_w$$ is fresh water density, $$H$$ is ice thickness, and $$b$$ is the bedrock elevation.

For most choices of the relevant parameters and most grid spacings, the routing model is at least two orders of magnitude more expensive computationally than the null model. This follows directly from the CFL-type time-step restriction on lateral flow of a fluid with velocity on the order of centimeters to meters per second (i.e. the subglacial liquid water bwat). (By comparison, much of PISM ice dynamics time-stepping is controlled by the much slower velocity of the flowing ice.) Therefore the user should start with short runs of order a few model years. We also recommend daily or even hourly reporting for scalar and spatially-distributed time-series to see hydrology model behavior, especially on fine grids (e.g. $$< 1$$ km).

Table 21 Command-line options specific to hydrology model routing

Option

Description

-hydrology_hydraulic_conductivity $$k$$

$$=k$$ in formula (17).

-hydrology_null_strip (km)

In the boundary strip water is removed and this is reported. This option specifies the width of this strip, which should typically be one or two grid cells.

-hydrology_gradient_power_in_flux $$\beta$$

$$=\beta$$ in formula (17).

-hydrology_thickness_power_in_flux $$\alpha$$

$$=\alpha$$ in formula (17).

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