# Balancing the books¶

## 2D diagnostics¶

PISM provides a number of 2D diagnostics to keep track of mass conservation.1

All of them are computed as time-averaged fluxes over requested reporting intervals. Positive values correspond to mass gain.

For ice mass, at every grid point we have

ice_mass_accounting_error = tendency_of_ice_mass -
(tendency_of_ice_mass_due_to_flow +
tendency_of_ice_mass_due_to_conservation_error +
tendency_of_ice_mass_due_to_surface_mass_flux +
tendency_of_ice_mass_due_to_basal_mass_flux +
tendency_of_ice_mass_due_to_discharge);

ice_mass_accounting_error@long_name = "ice mass accounting error";


Click here to download this ncap2 script.

All names on the right-hand side correspond to valid PISM diagnostic quantities.

To check that all changes in mass are accounted for, download the script above and run2

ncap2 -v -S ice_mass_accounting_error.txt \
pism_output.nc mass_accounting_error.nc


The variable ice_mass_accounting_error in mass_accounting_error.nc will contain ice mass accounting errors at each point. All values of this variable should be close to or equal to zero. They are not zero (in general) due to rounding errors, though.

Use a shortcut

pismr -extra_file ex.nc -extra_times N -extra_vars mass_fluxes,...


to save all fluxes needed to “balance the books” in terms of ice mass.

Alternatively, use fluxes in terms of “ice amount” (mass per unit area):

ice_amount_accounting_error = tendency_of_ice_amount -
(tendency_of_ice_amount_due_to_flow +
tendency_of_ice_amount_due_to_conservation_error +
tendency_of_ice_amount_due_to_surface_mass_flux +
tendency_of_ice_amount_due_to_basal_mass_flux +
tendency_of_ice_amount_due_to_discharge);

ice_amount_accounting_error@long_name = "ice amount accounting error";


Click here to download this ncap2 script.

To save these, use the shortcut

pismr -extra_file ex.nc -extra_times N -extra_vars amount_fluxes,...


• tendency_of_ice_mass_due_to_flow is the change in ice mass corresponding to flux divergence

• tendency_of_ice_mass_due_to_conservation_error is the artificial change in ice mass needed to “balance the books”. This includes changes needed to preserve non-negativity of ice thickness. 4

• tendency_of_ice_mass_due_to_surface_mass_balance is the change due to the surface mass balance; note that this is not the same as the provided SMB: in ablation areas this is the effective mass balance taking into account the amount of ice present

• tendency_of_ice_mass_due_to_basal_mass_balance is the effective change due to basal (grounded and sub-shelf) melting

• tendency_of_ice_mass_due_to_discharge combines changes due to calving and frontal melt

## Scalar diagnostics¶

Diagnostics listed above are also available as scalars, integrated over the whole computational domain. The “integrated” mass accounting error can be computed using the ncap2 script below.

ice_mass_accounting_error = tendency_of_ice_mass -
(tendency_of_ice_mass_due_to_flow +
tendency_of_ice_mass_due_to_conservation_error +
tendency_of_ice_mass_due_to_basal_mass_flux +
tendency_of_ice_mass_due_to_surface_mass_flux +
tendency_of_ice_mass_due_to_discharge);

ice_mass_accounting_error@long_name = "ice mass accounting error";


Click here to download this ncap2 script.

• tendency_of_ice_mass_due_to_flow is the integral of $$-\nabla \cdot Q$$ over the computational domain. This should be zero (up to the effect of rounding errors) in simulations that do not use Dirichlet boundary conditions for ice thickness. Prescribing ice thickness creates sources and sinks, and this diagnostic describes their influence.

• tendency_of_ice_mass_due_to_conservation_error should be zero (or close to zero) in most simulations

## Mass accounting in subglacial hydrology models¶

PISM’s hydrology models provide all the diagnostic fields needed to keep track of changes in subglacial water thickness. 3

At every grid point we have

water_mass_accounting_error = tendency_of_subglacial_water_mass -
(tendency_of_subglacial_water_mass_due_to_input +
tendency_of_subglacial_water_mass_due_to_flow +
tendency_of_subglacial_water_mass_due_to_conservation_error +
tendency_of_subglacial_water_mass_at_grounded_margins +
tendency_of_subglacial_water_mass_at_grounding_line +
tendency_of_subglacial_water_mass_at_domain_boundary);

water_mass_accounting_error@long_name = "subglacial water mass accounting error";


Click here to download this ncap2 script.

All names on the right-hand side correspond to valid PISM diagnostic quantities.

Use a shortcut

pismr -extra_file ex.nc -extra_times N -extra_vars hydrology_fluxes,...


to save all diagnostics mentioned above.

## Mass accounting in the PDD model¶

PISM’s PDD model provides diagnostics needed to compare computed accumulation, melt, and runoff to the effective mass balance. Use diagnostic quantities surface_accumulation_flux, surface_melt_flux, and surface_runoff_flux (units of mass per area per time) and surface_accumulation_rate, surface_melt_rate, surface_runoff_rate (units of mass per time).

To save all these, use -extra_vars shortcuts pdd_fluxes and pdd_rates.

1

See Diagnostic quantities for the full list of diagnostics.

2

ncap2 is a part of NCO.

3

We keep track of $$W_{\text{till}} + W$$, i.e. the sum of the effective thickness of subglacial water stored in till and the effective thickness of subglacial water in the transport layer (if applicable).

4

While PISM’s mass transport scheme is not proven to be mass-conserving in every case (see [122]), in most simulations the field is uniformly zero.

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