# Surface mass and energy process model components¶

## The “invisible” model¶

- Options
`-surface simple`

- Variables
none

- C++ class
`pism::surface::Simple`

This is the simplest “surface model” available in PISM, enabled using `-surface simple`

.
Its job is to re-interpret precipitation as climatic mass balance, and to re-interpret
mean annual near-surface (2m) air temperature as the temperature of the ice at the depth
at which firn processes cease to change the temperature of the ice. (I.e. the temperature
*below* the firn.) This implies that there is no melt. Though primitive, this model
component may be desired in cold environments (e.g. East Antarctic ice sheet) in which
melt is negligible and heat from firn processes is ignored.

## Reading top-surface boundary conditions from a file¶

- Options
`-surface given`

- Variables
`ice_surface_temp`

,`climatic_mass_balance`

\(kg / (m^{2} s)\)- C++ class
`pism::surface::Given`

Note

This is the default choice.

This model component was created to force PISM with sampled (possibly periodic) climate data by reading ice upper surface boundary conditions from a file. These fields are provided directly to the ice dynamics code (see Climate inputs, and their interface with ice dynamics for details).

PISM will stop if variables `ice_surface_temp`

(ice temperature at the ice surface
but below firn) and `climatic_mass_balance`

(top surface mass flux into the ice) are
not present in the input file.

A file `foo.nc`

used with `-surface given -surface_given_file foo.nc`

may contain
several records. If this file contains one record (i.e. fields corresponding to one time
value only), provided forcing data is interpreted as time-independent. Variables
`time`

and `time_bnds`

should specify model times corresponding to individual
records.

For example, to use monthly periodic forcing with a period of 1 year starting at the
beginning of \(1980\) (let’s use the \(360\)-day calendar for simplicity), create a file
(say, “`foo.nc`

”) with 12 records. The `time`

variable may contain \(15, 45, 75,
\dots, 345\) (mid-month for all \(12\) months) and have the units of “```
days since
1980-1-1
```

”. (It is best to avoid units of “months” and “years” because their meanings
depend on the calendar.) Next, add the `time_bounds`

variable for the time dimension
with the values \(0, 30, 30, 60, \dots\) specifying times corresponding to beginnings and
ends of records for each month and set the `time:bounds`

attribute accordingly. Now run

```
pismr -surface given -surface_given_file foo.nc -surface.given.periodic
```

See Using time-dependent forcing for more information.

Note

This surface model

*ignores*the atmosphere model selection made using the option`-atmosphere`

.PISM can handle files with virtually any number of records: it will read and store in memory at most

`input`

`.forcing`

`.buffer_size`

records at any given time (default: 60, or 5 years’ worth of monthly fields).when preparing a file for use with this model, it is best to use the

`t,y,x`

variable storage order: files using this order can be read in faster than ones using the`t,x,y`

order, for reasons explained in the User’s Manual.To change the storage order in a NetCDF file, use

`ncpdq`

:ncpdq -a t,y,x input.nc output.nc

will copy data from

`input.nc`

into`output.nc`

, changing the storage order to`t,y,x`

at the same time.

Parameters

Prefix: `surface.given.`

## Elevation-dependent temperature and mass balance¶

- Options
`-surface elevation`

- Variables
none

- C++ class
`pism::surface::Elevation`

This surface model component parameterizes the ice surface temperature \(T_{h}\) =
`ice_surface_temp`

and the mass balance \(m\) = `climatic_mass_balance`

as
*piecewise-linear* functions of surface elevation \(h\).

The option `-ice_surface_temp`

(*list of 4 numbers*) determines the surface
temperature using the 4 parameters \(\T{min}\), \(\T{max}\), \(\h{min}\),
\(\h{max}\). Let

be the temperature gradient. Then

The option `-climatic_mass_balance`

(*list of 5 numbers*) determines the surface mass
balance using the 5 parameters \(\m{min}\), \(\m{max}\), \(\h{min}\),
\(\h{ELA}\), \(\h{max}\). Let

and

be the mass balance gradient in the ablation and in the accumulation area, respectively. Then

The option `-climatic_mass_balance_limits`

(*list of 2 numbers*) limits the mass
balance below \(\h{min}\) to \(\ms{min}\) and above \(\h{max}\) to
\(\ms{max}\), thus

Note: this surface model *ignores* the atmosphere model selection made using the
`-atmosphere`

option.

## Temperature-index scheme¶

- Options
`-surface pdd`

- Variables
`air_temp_sd`

,`snow_depth`

- C++ class
`pism::surface::TemperatureIndex`

The default PDD model used by PISM, turned on by option `-surface pdd`

, is based on
[18] and EISMINT-Greenland intercomparison (see [7]).

Our model computes the solid (snow) precipitation rate using the air temperature threshold
with a linear transition. All precipitation during periods with air temperatures above
`surface`

`.pdd`

`.air_temp_all_precip_as_rain`

(default of \(2^\circ C\)) is interpreted as
rain; all precipitation during periods with air temperatures below
`surface`

`.pdd`

`.air_temp_all_precip_as_snow`

(default of \(0^\circ C\)) is interpreted as
snow.

For long-term simulations, a PDD model generally uses an idealized seasonal temperature
cycle. “White noise” is added to this cycle to simulate additional daily variability
associated to the vagaries of weather. This additional random variation is quite
significant, as the seasonal cycle may never reach the melting point but that point may be
reached with some probability, in the presence of the daily variability, and thus melt may
occur. Concretely, a normally-distributed, mean zero random temperature increment is added
to the seasonal cycle. There is no assumed spatial correlation of daily variability. The
standard deviation of the daily variability is controlled by configuration parameters with
the prefix `surface.pdd.std_dev.`

:

`file`

The name of the file to read`air_temp_sd`

(standard deviation of air temperature) from.`lapse_lat_base`

(72*degree_north*) standard deviationis is a function of latitude, with value`surface`

`.pdd`

`.std_dev`

`.value`

at this latitude; this value is only active if`surface`

`.pdd`

`.std_dev`

`.lapse_lat_rate`

is nonzero`lapse_lat_rate`

(0*Kelvin / degree_north*) standard deviation is a function of latitude, with rate of change with respect to latitude given by this constant`param_a`

(-0.15) Parameter \(a\) in \(\Sigma = aT + b\), with \(T\) in degrees C. Used only if`surface`

`.pdd`

`.std_dev`

`.use_param`

is set to yes.`param_b`

(0.66*Kelvin*) Parameter \(b\) in \(\Sigma = aT + b\), with \(T\) in degrees C. Used only if`surface`

`.pdd`

`.std_dev`

`.use_param`

is set to yes.`periodic`

(no) If true, interpret`air_temp_sd`

read from`surface`

`.pdd`

`.std_dev`

`.file`

as periodic in time`use_param`

(no) Parameterize standard deviation as a linear function of air temperature over ice-covered grid cells. The region of application is controlled by`geometry`

`.ice_free_thickness_standard`

.`value`

(5*Kelvin*) standard deviation of daily temp variation; = EISMINT-Greenland value [7]

A file `foo.nc`

used with `-surface pdd -pdd_sd_file foo.nc`

should contain standard
deviation of near-surface air temperature in variable `air_temp_sd`

, and the
corresponding time coordinate in variable `time`

. If `-pdd_sd_file`

is not set,
PISM uses a constant value for standard deviation, which is set by the
configuration parameter `surface`

`.pdd`

`.std_dev`

`.value`

. The default value is \(5.0\) degrees
[7]. However, this approach is not recommended as it induces significant
errors in modeled surface mass balance in both ice-covered and ice-free regions
[19], [20].

Over ice-covered grid cells, daily variability can also be parameterized as a linear
function of near-surface air temperature \(\sigma = a \cdot T + b\) using the
`surface`

`.pdd`

`.std_dev`

`.use_param`

configuration flag, and the corresponding
parameters `surface`

`.pdd`

`.std_dev`

`.param_a`

and
`surface`

`.pdd`

`.std_dev`

`.param_b`

. This parametrization replaces prescribed standard
deviation values over glacierized grid cells as defined by the `mask`

variable (see
`geometry`

`.ice_free_thickness_standard`

). Default values for the slope \(a\) and
intercept \(b\) were derived from the ERA-40 reanalysis over the Greenland ice sheet
[21].

The number of positive degree days is computed as the magnitude of the temperature excursion above \(0\!\phantom{|}^\circ \text{C}\) multiplied by the duration (in days) when it is above zero.

In PISM there are two methods for computing the number of positive degree days. The first
computes only the expected value, by the method described in [18]. This is
the default when a PDD is chosen (i.e. option `-surface pdd`

). The second is a Monte
Carlo simulation of the white noise itself, chosen by adding the option ```
-pdd_method
random_process
```

. This Monte Carlo simulation adds the same daily variation at every point,
though the seasonal cycle is (generally) location dependent. If repeatable randomness is
desired use `-pdd_method repeatable_random_process`

instead.

By default, the computation summarized in Fig. 46 is performed every week.
(This frequency is controlled by the parameter `surface`

`.pdd`

`.max_evals_per_year`

.)
To compute mass balance during each week-long time-step, PISM keeps track of the current
snow depth (using units of ice-equivalent thickness). This is necessary to determine if
melt should be computed using the degree day factor for snow
(`surface`

`.pdd`

`.factor_snow`

) or the corresponding factor for ice
(`surface`

`.pdd`

`.factor_ice`

).

A fraction of the melt controlled by the configuration parameter `surface`

`.pdd`

`.refreeze`

(\(\theta_{\text{refreeze}}\) in Fig. 46, default: \(0.6\))
refreezes. The user can select whether melted ice should be allowed to refreeze using the
configuration flag `surface`

`.pdd`

`.refreeze_ice_melt`

.

Since PISM does not have a principled firn model, the snow depth is set to zero at the
beginning of the balance year. See `surface`

`.pdd`

`.balance_year_start_day`

. Default is
\(274\), corresponding to October 1^{st}.

Our PDD implementation is meant to be used with an atmosphere model implementing a cosine
yearly cycle such as `searise_greenland`

(section
SeaRISE-Greenland), but it is not restricted to parameterizations
like these.

This code also implements latitude- and mean July temperature dependent ice and snow
factors using formulas (6) and (7) in [1]; set `-pdd_fausto`

to enable.
The default standard deviation of the daily variability (option `-pdd_std_dev`

) is
2.53 degrees when `-pdd_fausto`

is set [1]. See also configuration
parameters with the prefix `surface.pdd.fausto.`

:

`beta_ice_c`

(0.015*meter / (Kelvin day)*) water-equivalent thickness; for formula (6) in [1]`beta_ice_w`

(0.007*meter / (Kelvin day)*) water-equivalent thickness; for formula (6) in [1]`beta_snow_c`

(0.003*meter / (Kelvin day)*) water-equivalent thickness; for formula (6) in [1]`beta_snow_w`

(0.003*meter / (Kelvin day)*) water-equivalent thickness; for formula (6) in [1]`enabled`

(false) Set PDD parameters using formulas (6) and (7) in [1]`latitude_beta_w`

(72*degree_north*) latitude below which to use warm case, in formula (6) in [1]

Note that when used with periodic climate data (air temperature and precipitation) that is
read from a file (see section Reading boundary conditions from a file), use of
`time_stepping`

`.hit_multiples`

is recommended: set it to the length of the climate
data period in years.

This model provides the following scalar:

`surface_accumulation_rate`

`surface_melt_rate`

`surface_runoff_rate`

and these 2D diagnostic quantities (averaged over reporting intervals; positive flux corresponds to ice gain):

`surface_accumulation_flux`

`surface_melt_flux`

`surface_runoff_flux`

This makes it easy to compare the surface mass balance computed by the model to its individual components:

```
SMB = surface_accumulation_flux - surface_runoff_flux
```

Parameters

Prefix: `surface.pdd.`

.

`air_temp_all_precip_as_rain`

(275.15*Kelvin*) threshold temperature above which all precipitation is rain; must exceed`surface`

`.pdd`

`.air_temp_all_precip_as_snow`

to avoid division by zero, because difference is in a denominator`air_temp_all_precip_as_snow`

(273.15*Kelvin*) threshold temperature below which all precipitation is snow`balance_year_start_day`

(274*ordinal day number*) day of year for October 1st, beginning of the balance year in northern latitudes.`factor_ice`

(0.00879121*meter / (Kelvin day)*) EISMINT-Greenland value [7]; = (8 mm liquid-water-equivalent) / (pos degree day)`factor_snow`

(0.0032967*meter / (Kelvin day)*) EISMINT-Greenland value [7]; = (3 mm liquid-water-equivalent) / (pos degree day)`fausto`

`.T_c`

(272.15*Kelvin*) = -1 + 273.15; for formula (6) in [1]`fausto`

`.T_w`

(283.15*Kelvin*) = 10 + 273.15; for formula (6) in [1]`fausto`

`.beta_ice_c`

(0.015*meter / (Kelvin day)*) water-equivalent thickness; for formula (6) in [1]`fausto`

`.beta_ice_w`

(0.007*meter / (Kelvin day)*) water-equivalent thickness; for formula (6) in [1]`fausto`

`.beta_snow_c`

(0.003*meter / (Kelvin day)*) water-equivalent thickness; for formula (6) in [1]`fausto`

`.beta_snow_w`

(0.003*meter / (Kelvin day)*) water-equivalent thickness; for formula (6) in [1]`fausto`

`.enabled`

(false) Set PDD parameters using formulas (6) and (7) in [1]`fausto`

`.latitude_beta_w`

(72*degree_north*) latitude below which to use warm case, in formula (6) in [1]`firn_compaction_to_accumulation_ratio`

(0.75) How much firn as a fraction of accumulation is turned into ice`firn_depth_file`

The name of the file to read the firn_depth from.`interpret_precip_as_snow`

(no) Interpret precipitation as snow fall.`max_evals_per_year`

(52) maximum number of times the PDD scheme will ask for air temperature and precipitation to build location-dependent time series for computing (expected) number of positive degree days and snow accumulation; the default means the PDD uses weekly samples of the annual cycle; see also`surface`

`.pdd`

`.std_dev`

`.value`

`method`

(`expectation_integral`

) PDD implementation method`positive_threshold_temp`

(273.15*Kelvin*) temperature used to determine meaning of “positive” degree day`refreeze_ice_melt`

(yes) If set to “yes”, refreeze`surface`

`.pdd`

`.refreeze`

fraction of melted ice, otherwise all of the melted ice runs off.`std_dev`

`.file`

The name of the file to read`air_temp_sd`

(standard deviation of air temperature) from.`std_dev`

`.lapse_lat_base`

(72*degree_north*) standard deviationis is a function of latitude, with value`surface`

`.pdd`

`.std_dev`

`.value`

at this latitude; this value is only active if`surface`

`.pdd`

`.std_dev`

`.lapse_lat_rate`

is nonzero`std_dev`

`.lapse_lat_rate`

(0*Kelvin / degree_north*) standard deviation is a function of latitude, with rate of change with respect to latitude given by this constant`std_dev`

`.param_a`

(-0.15) Parameter \(a\) in \(\Sigma = aT + b\), with \(T\) in degrees C. Used only if`surface`

`.pdd`

`.std_dev`

`.use_param`

is set to yes.`std_dev`

`.param_b`

(0.66*Kelvin*) Parameter \(b\) in \(\Sigma = aT + b\), with \(T\) in degrees C. Used only if`surface`

`.pdd`

`.std_dev`

`.use_param`

is set to yes.`std_dev`

`.periodic`

(no) If true, interpret`air_temp_sd`

read from`surface`

`.pdd`

`.std_dev`

`.file`

as periodic in time`std_dev`

`.use_param`

(no) Parameterize standard deviation as a linear function of air temperature over ice-covered grid cells. The region of application is controlled by`geometry`

`.ice_free_thickness_standard`

.`std_dev`

`.value`

(5*Kelvin*) standard deviation of daily temp variation; = EISMINT-Greenland value [7]

## PIK¶

- Options
`-surface pik`

- Variables
`climatic_mass_balance`

\(kg / (m^{2} s)\),`lat`

(latitude), (degrees north)- C++ class
`pism::surface::PIK`

This surface model component implements the setup used in [3]. The
`climatic_mass_balance`

is read from an input (`-i`

) file; the ice surface
temperature is computed as a function of latitude (variable `lat`

) and surface
elevation (dynamically updated by PISM). See equation (1) in [3].

## Scalar temperature offsets¶

- Options
`-surface ...,delta_T`

- Variables
`delta_T`

- C++ class
`pism::surface::Delta_T`

The time-dependent scalar offsets `delta_T`

are added to `ice_surface_temp`

computed by a surface model.

Please make sure that `delta_T`

has the units of “`Kelvin`

”.

This modifier is identical to the corresponding atmosphere modifier, but applies offsets at a different stage in the computation of top-surface boundary conditions needed by the ice dynamics core.

Parameters

Prefix: `surface.delta_T.`

## Adjustments using modeled change in surface elevation¶

- Options
`-surface ...,elevation_change`

- Variables
`surface_altitude`

(CF standard name),- C++ class
`pism::surface::LapseRates`

The `elevation_change`

modifier adjusts ice-surface temperature and surface mass balance
using modeled changes in surface elevation relative to a reference elevation read from a
file.

The surface temperature is modified using an elevation lapse rate
\(\gamma_T =\) `surface`

`.elevation_change`

`.temperature_lapse_rate`

. Here

Two methods of adjusting the SMB are available:

Scaling using an exponential factor

\[\mathrm{SMB} = \mathrm{SMB_{input}} \cdot \exp(C \cdot \Delta T),\]where \(C =\)

`surface`

`.elevation_change`

`.smb`

`.exp_factor`

and \(\Delta T\) is the temperature difference produced by applying`surface`

`.elevation_change`

`.temperature_lapse_rate`

.This mechanisms increases the SMB by \(100(\exp(C) - 1)\) percent for each degree of temperature increase.

To use this method, set

`-smb_adjustment scale`

.Elevation lapse rate for the SMB

\[\mathrm{SMB} = \mathrm{SMB_{input}} - \Delta h \cdot \gamma_M,\]where \(\gamma_M =\)

`surface`

`.elevation_change`

`.smb`

`.lapse_rate`

and \(\Delta h\) is the difference between modeled and reference surface elevations.To use this method, set

`-smb_adjustment shift`

.

Parameters

Prefix: `surface.elevation_change.`

.

`file`

Name of the file containing the reference surface elevation field (variable`usurf`

).`periodic`

(no) If true, interpret forcing data as periodic in time`smb`

`.exp_factor`

(0*Kelvin-1*) Exponential for the surface mass balance.`smb`

`.lapse_rate`

(0*(m / year) / km*) Lapse rate for the surface mass balance.`smb`

`.method`

(`shift`

) Choose the SMB adjustment method.`scale`

: use temperature-change-dependent scaling factor.`shift`

: use the SMB lapse rate.`temperature_lapse_rate`

(0*K / km*) Lapse rate for the temperature at the top of the ice.

## Mass flux adjustment¶

- Options
`-surface ...,forcing`

- Variables
`thk`

(ice thickness),`ftt_mask`

(mask of zeros and ones; 1 where surface mass flux is adjusted and 0 elsewhere)- C++ class
`pism::surface::ForceThickness`

The `forcing`

modifier implements a surface mass balance adjustment mechanism which
forces the thickness of grounded ice to a target thickness distribution at the end of the
run. The idea behind this mechanism is that spinup of ice sheet models frequently requires
the surface elevation to come close to measured values at the end of a run. A simpler
alternative to accomplish this, namely option `-no_mass`

, represents an unmodeled,
frequently large, violation of the mass continuity equation.

In more detail, let \(H_{\text{tar}}\) be the target thickness. Let \(H\) be the time-dependent model thickness. The surface model component described here produces the term \(M\) in the mass continuity equation:

(Other details of this equation do not concern us here.) The `forcing`

modifier causes
\(M\) to be adjusted by a multiple of the difference between the target thickness and
the current thickness,

where \(\alpha>0\). We are adding mass (\(\Delta M>0\)) where \(H_{\text{tar}} > H\) and ablating where \(H_{\text{tar}} < H\).

Option `-force_to_thickness_file`

identifies the file containing the target ice
thickness field `thk`

and the mask `ftt_mask`

. A basic run modifying surface model
`given`

would look like

```
pismr -i foo.nc -surface given,forcing -force_to_thickness_file bar.nc
```

In this case `foo.nc`

contains fields `climatic_mass_balance`

and
`ice_surface_temp`

, as normal for `-surface given`

, and `bar.nc`

contains fields
`thk`

which will serve as the target thickness and `ftt_mask`

which defines the
map plane area where this adjustment is applied. Option `-force_to_thickness_alpha`

adjusts the value of \(\alpha\), which has a default value specified in the
Configuration parameters.

In addition to this one can specify a multiplicative factor \(C\) used in areas where
the target thickness field has less than
`-force_to_thickness_ice_free_thickness_threshold`

meters of ice;
\(\alpha_{\text{ice free}} = C \times \alpha\). Use the
`-force_to_thickness_ice_free_alpha_factor`

option to set \(C\).

## Using climate data anomalies¶

- Options
`-surface ...,anomaly`

- Variables
`ice_surface_temp_anomaly`

,`climatic_mass_balance_anomaly`

\(kg / (m^{2} s)\)- C++ class
`pism::surface::Anomaly`

This modifier implements a spatially-variable version of `-surface ...,delta_T`

which
also applies time-dependent climatic mass balance anomalies.

See also `-atmosphere ...,anomaly`

(section Using climate data anomalies), which is
similar but applies anomalies at the atmosphere level.

Parameters

Prefix: `surface.anomaly.`

## The caching modifier¶

- Options
`-surface ...,cache`

- C++ class
`pism::surface::Cache`

- See also

This modifier skips surface model updates, so that a surface model is called no more than
every `surface`

`.cache`

`.update_interval`

365-day “years”. A time-step of \(1\) year is
used every time a surface model is updated.

This is useful in cases when inter-annual climate variability is important, but one year differs little from the next. (Coarse-grid paleo-climate runs, for example.)

Parameters

Prefix: `surface.cache.`

`update_interval`

(10*365days*) Update interval (in 365-day years) for the`-surface cache`

modifier.

## Preventing grounding line retreat¶

- Options
`-surface ...,no_gl_retreat`

- C++ class
`pism::surface::NoGLRetreat`

This modifier adjust the surface mass balance to prevent the retreat of the grounding line. See Till friction angle optimization for an application.

Note

This modifier

*adds mass*in violation of mass conservation. Save the diagnostic`no_gl_retreat_smb_adjustment`

to get an idea about the amount added. Note, though, that this is an imperfect measure: it includes mass added to maintain non-negativity of ice thickness.We assume that the sea level and the bed elevation remain constant throughout the simulation.

This does

*not*prevent grounding line retreat caused by the thinning of the ice due to the melt at the base. Set`geometry`

`.update`

`.use_basal_melt_rate`

to “false” to ensure that basal melt has no effect on the position of the grounding line

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