tsurface.rst - pism - [fork] customized build of PISM, the parallel ice sheet model (tillflux branch)
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       ---
       tsurface.rst (21173B)
       ---
            1 .. include:: shortcuts.txt
            2 
            3 Surface mass and energy process model components
            4 ------------------------------------------------
            5 
            6 .. contents::
            7 
            8 .. _sec-surface-simple:
            9 
           10 The "invisible" model
           11 +++++++++++++++++++++
           12 
           13 :|options|: ``-surface simple``
           14 :|variables|: none
           15 :|implementation|: ``pism::surface::Simple``
           16 
           17 This is the simplest "surface model" available in PISM, enabled using ``-surface simple``.
           18 Its job is to re-interpret precipitation as climatic mass balance, and to re-interpret
           19 mean annual near-surface (2m) air temperature as the temperature of the ice at the depth
           20 at which firn processes cease to change the temperature of the ice. (I.e. the temperature
           21 *below* the firn.) This implies that there is no melt. Though primitive, this model
           22 component may be desired in cold environments (e.g. East Antarctic ice sheet) in which
           23 melt is negligible and heat from firn processes is ignored.
           24 
           25 .. _sec-surface-given:
           26 
           27 Reading top-surface boundary conditions from a file
           28 +++++++++++++++++++++++++++++++++++++++++++++++++++
           29 
           30 :|options|: ``-surface given``
           31 :|variables|: :var:`ice_surface_temp`, :var:`climatic_mass_balance` |flux|
           32 :|implementation|: ``pism::surface::Given``
           33 
           34 .. note::
           35 
           36    This is the default choice.
           37 
           38 This model component was created to force PISM with sampled (possibly periodic) climate
           39 data by reading ice upper surface boundary conditions from a file. These fields are
           40 provided directly to the ice dynamics code (see :ref:`sec-climate-inputs` for details).
           41 
           42 PISM will stop if variables :var:`ice_surface_temp` (ice temperature at the ice surface
           43 but below firn) and :var:`climatic_mass_balance` (top surface mass flux into the ice) are
           44 not present in the input file.
           45 
           46 Command-line options:
           47 
           48 - :opt:`-surface_given_file` prescribes an input file
           49 - :opt:`-surface_given_period` (*years*) makes PISM interpret data in
           50   ``-surface_given_file`` as periodic. See :ref:`sec-periodic-forcing`.
           51 - :opt:`-surface_given_reference_year` sets the reference model year; see
           52   :ref:`sec-periodic-forcing`.
           53 
           54 A file ``foo.nc`` used with ``-surface given -surface_given_file foo.nc`` should contain
           55 several records. If this file contains one record (i.e. fields corresponding to one time
           56 value only), provided forcing data is interpreted as time-independent. The :var:`time`
           57 variable should describe what model time these records correspond to; see
           58 :ref:`sec-model-time` for details.
           59 
           60 For example, to use monthly records and period of 1 year, create a file (say,
           61 "``foo.nc``") with 12 records. The :var:`time` variable may contain `0, 1, 2, 3,
           62 \dots, 11` and have the units of "month" [#]_. Then, run
           63 
           64 .. code-block:: none
           65 
           66     pismr -surface given -surface_given_file foo.nc -surface_given_period 1
           67 
           68 .. note::
           69 
           70    - This surface model *ignores* the atmosphere model selection made using the option
           71      :opt:`-atmosphere`.
           72    - PISM can handle files with virtually any number of records: it will read and store in
           73      memory at most :config:`input.forcing.buffer_size` records at any given time
           74      (default: 60, or 5 years' worth of monthly fields).
           75    - when preparing a file for use with this model, it is best to use the ``t,y,x``
           76      variable storage order: files using this order can be read in faster than ones using
           77      the ``t,x,y`` order, for reasons :ref:`explained in the User's Manual
           78      <sec-pism-io-performance>`.
           79    
           80      To change the storage order in a NetCDF file, use ``ncpdq``:
           81    
           82      .. code-block:: none
           83    
           84        ncpdq -a t,y,x input.nc output.nc
           85    
           86      will copy data from ``input.nc`` into ``output.nc``, changing the storage order to
           87      ``t,y,x`` at the same time.
           88 
           89 .. _sec-surface-elevation:
           90 
           91 Elevation-dependent temperature and mass balance
           92 ++++++++++++++++++++++++++++++++++++++++++++++++
           93 
           94 :|options|: ``-surface elevation``
           95 :|variables|: none
           96 :|implementation|: ``pism::surface::Elevation``
           97 
           98 .. include:: ../math-definitions.txt
           99 
          100 This surface model component parameterizes the ice surface temperature `T_{h}` =
          101 :var:`ice_surface_temp` and the mass balance `m` = :var:`climatic_mass_balance` as
          102 *piecewise-linear* functions of surface elevation `h`.
          103 
          104 The option :opt:`-ice_surface_temp` (*list of 4 numbers*) determines the surface
          105 temperature using the 4 parameters `\T{min}`, `\T{max}`, `\h{min}`,
          106 `\h{max}`. Let
          107 
          108 .. math::
          109 
          110   \diff{T}{h} = (\T{max} - \T{min}) / (\h{max} - \h{min})
          111 
          112 be the temperature gradient. Then
          113 
          114 .. math::
          115 
          116   T(x,y) =
          117   \begin{cases}
          118     \T{min}, & h(x,y) \le \h{min}, \\
          119     \T{min} + \diff{T}{h} \, (h(x,y) - \h{min}), & \h{min} < h(x,y) < \h{max}, \\
          120     \T{max}, & \h{max} \le h(x,y).
          121   \end{cases}
          122 
          123 The option :opt:`-climatic_mass_balance` (*list of 5 numbers*) determines the surface mass
          124 balance using the 5 parameters `\m{min}`, `\m{max}`, `\h{min}`,
          125 `\h{ELA}`, `\h{max}`. Let
          126 
          127 .. math::
          128 
          129    \diff{\m{abl}}{h} = -\m{min} / (\h{max} - \h{min})
          130 
          131 and
          132 
          133 .. math::
          134 
          135    \diff{\m{acl}}{h} = \m{max} / (\h{max} - \h{min})
          136 
          137 be the mass balance gradient in the ablation and in the accumulation area, respectively.
          138 Then
          139 
          140 .. math::
          141 
          142    m(x,y) =
          143    \begin{cases}
          144     \m{min}, & h(x,y) \le \h{min}, \\
          145     \diff{\m{abl}}{h} \, (h(x,y) - h_{\text{ELA}}), &  \h{min} < h(x,y) < \h{max}, \\
          146     \diff{\m{acl}}{h} \, (h(x,y) - h_{\text{ELA}}), & \h{min} < h(x,y) < \h{max}, \\
          147     \m{max}, & \h{max} \le h(x,y).
          148    \end{cases}
          149 
          150 The option :opt:`-climatic_mass_balance_limits` (*list of 2 numbers*) limits the mass
          151 balance below `\h{min}` to `\ms{min}` and above `\h{max}` to
          152 `\ms{max}`, thus
          153 
          154 .. math::
          155 
          156    m(x,y) =
          157    \begin{cases}
          158     m^{*}_{\text{min}}, & h(x,y) \le \h{min}, \\
          159     \diff{\m{abl}}{h} \, (h(x,y) - h_{\text{ELA}}), & \h{min} < h(x,y) < \h{max}, \\
          160     \diff{\m{acl}}{h} \, (h(x,y) - h_{\text{ELA}}), & \h{min} < h(x,y) < \h{max}, \\
          161     m^{*}_{\text{max}}, & \h{max} \le h(x,y).
          162    \end{cases}
          163 
          164 Note: this surface model *ignores* the atmosphere model selection made using the
          165 :opt:`-atmosphere` option.
          166 
          167 .. _sec-surface-pdd:
          168 
          169 Temperature-index scheme
          170 ++++++++++++++++++++++++
          171 
          172 :|options|: ``-surface pdd``
          173 :|variables|: :var:`air_temp_sd`, :var:`snow_depth`
          174 :|implementation|: ``pism::surface::TemperatureIndex``
          175                    
          176 The default PDD model used by PISM, turned on by option :opt:`-surface pdd`, is based on
          177 :cite:`CalovGreve05` and EISMINT-Greenland intercomparison (see :cite:`RitzEISMINT`).
          178 
          179 Our model computes the solid (snow) precipitation rate using the air temperature threshold
          180 with a linear transition. All precipitation during periods with air temperatures above
          181 :config:`surface.pdd.air_temp_all_precip_as_rain` (default of `2^\circ C`) is interpreted as
          182 rain; all precipitation during periods with air temperatures below
          183 :config:`surface.pdd.air_temp_all_precip_as_snow` (default of `0^\circ C`) is interpreted as
          184 snow.
          185 
          186 For long-term simulations, a PDD model generally uses an idealized seasonal temperature
          187 cycle. "White noise" is added to this cycle to simulate additional daily variability
          188 associated to the vagaries of weather. This additional random variation is quite
          189 significant, as the seasonal cycle may never reach the melting point but that point may be
          190 reached with some probability, in the presence of the daily variability, and thus melt may
          191 occur. Concretely, a normally-distributed, mean zero random temperature increment is added
          192 to the seasonal cycle. There is no assumed spatial correlation of daily variability. The
          193 standard deviation of the daily variability is controlled by command-line options:
          194 
          195 - :opt:`-pdd_sd_file`, which prescribes an input file containing :var:`air_temp_sd`
          196 - :opt:`-pdd_sd_period` (*years*), which interprets its data as periodic; see
          197   :ref:`sec-periodic-forcing`.
          198 - :opt:`-pdd_sd_reference_year`, which sets the reference model year; see
          199   :ref:`sec-periodic-forcing`.
          200 
          201 A file ``foo.nc`` used with ``-surface pdd -pdd_sd_file foo.nc`` should contain standard
          202 deviation of near-surface air temperature in variable :var:`air_temp_sd`, and the
          203 corresponding time coordinate in variable :var:`time`. If ``-pdd_sd_file`` is not set,
          204 PISM uses a constant value for standard deviation, which is set by the
          205 configuration parameter :config:`surface.pdd.std_dev`. The default value is `5.0` degrees
          206 :cite:`RitzEISMINT`. However, this approach is not recommended as it induces significant
          207 errors in modeled surface mass balance in both ice-covered and ice-free regions
          208 :cite:`RogozhinaRau2014`, :cite:`Seguinot2013`.
          209 
          210 Over ice-covered grid cells, daily variability can also be parameterized as a linear
          211 function of near-surface air temperature `\sigma = a \cdot T + b` using the
          212 :config:`surface.pdd.std_dev_use_param` configuration flag, and the corresponding
          213 parameters :config:`surface.pdd.std_dev_param_a` and
          214 :config:`surface.pdd.std_dev_param_b`. This parametrization replaces prescribed standard
          215 deviation values over glacierized grid cells as defined by the :var:`mask` variable (see
          216 :config:`geometry.ice_free_thickness_standard`). Default values for the slope `a` and
          217 intercept `b` were derived from the ERA-40 reanalysis over the Greenland ice sheet
          218 :cite:`SeguinotRogozhina2014`.
          219 
          220 The number of positive degree days is computed as the magnitude of the temperature
          221 excursion above `0\!\phantom{|}^\circ \text{C}` multiplied by the duration (in days)
          222 when it is above zero.
          223 
          224 In PISM there are two methods for computing the number of positive degree days. The first
          225 computes only the expected value, by the method described in :cite:`CalovGreve05`. This is
          226 the default when a PDD is chosen (i.e. option :opt:`-surface pdd`). The second is a Monte
          227 Carlo simulation of the white noise itself, chosen by adding the option :opt:`-pdd_method
          228 random_process`. This Monte Carlo simulation adds the same daily variation at every point,
          229 though the seasonal cycle is (generally) location dependent. If repeatable randomness is
          230 desired use :opt:`-pdd_method repeatable_random_process` instead.
          231 
          232 .. figure:: figures/pdd-model-flowchart.png
          233    :name: fig-pdd-model
          234 
          235    PISM's positive degree day model. `F_s` and `F_i` are PDD factors for snow
          236    and ice, respectively; `\theta_{\text{refreeze}}` is the refreeze fraction.
          237 
          238 By default, the computation summarized in :numref:`fig-pdd-model` is performed every week.
          239 (This frequency is controlled by the parameter :config:`surface.pdd.max_evals_per_year`.)
          240 To compute mass balance during each week-long time-step, PISM keeps track of the current
          241 snow depth (using units of ice-equivalent thickness). This is necessary to determine if
          242 melt should be computed using the degree day factor for snow
          243 (:config:`surface.pdd.factor_snow`) or the corresponding factor for ice
          244 (:config:`surface.pdd.factor_ice`).
          245 
          246 A fraction of the melt controlled by the configuration parameter :config:`surface.pdd.refreeze`
          247 (`\theta_{\text{refreeze}}` in :numref:`fig-pdd-model`, default: `0.6`)
          248 refreezes. The user can select whether melted ice should be allowed to refreeze using the
          249 configuration flag :config:`surface.pdd.refreeze_ice_melt`.
          250 
          251 Since PISM does not have a principled firn model, the snow depth is set to zero at the
          252 beginning of the balance year. See :config:`surface.pdd.balance_year_start_day`. Default is
          253 `274`, corresponding to October 1\ :superscript:`st`.
          254 
          255 Our PDD implementation is meant to be used with an atmosphere model implementing a cosine
          256 yearly cycle such as ``searise_greenland`` (section
          257 :ref:`sec-atmosphere-searise-greenland`), but it is not restricted to parameterizations
          258 like these.
          259 
          260 This code also implements latitude- and mean July temperature dependent ice and snow
          261 factors using formulas (6) and (7) in :cite:`Faustoetal2009`; set :opt:`-pdd_fausto` to enable.
          262 The default standard deviation of the daily variability (option :opt:`-pdd_std_dev`) is
          263 2.53 degrees when :opt:`-pdd_fausto` is set :cite:`Faustoetal2009`. See also configuration
          264 parameters with the ``surface.pdd.fausto`` prefix.
          265 
          266 Note that when used with periodic climate data (air temperature and precipitation) that is
          267 read from a file (see section :ref:`sec-atmosphere-given`), use of
          268 :opt:`-timestep_hit_multiplies X` is recommended. (Here `X` is the length of the climate
          269 data period in years.)
          270 
          271 This model provides the following scalar:
          272 
          273 - :var:`surface_accumulation_rate`
          274 - :var:`surface_melt_rate`
          275 - :var:`surface_runoff_rate`
          276 
          277 and these 2D diagnostic quantities (averaged over reporting intervals; positive flux
          278 corresponds to ice gain):
          279 
          280 - :var:`surface_accumulation_flux`
          281 - :var:`surface_melt_flux`
          282 - :var:`surface_runoff_flux`
          283 
          284 This makes it easy to compare the surface mass balance computed by the model to its
          285 individual components:
          286 
          287 .. code::
          288 
          289    SMB = surface_accumulation_flux - surface_runoff_flux
          290 
          291 .. _sec-surface-pik:
          292 
          293 PIK
          294 +++
          295 
          296 :|options|: ``-surface pik``
          297 :|variables|: :var:`climatic_mass_balance` |flux|,
          298               :var:`lat` (latitude), (degrees north)
          299 :|implementation|: ``pism::surface::PIK``
          300 
          301 This surface model component implements the setup used in :cite:`Martinetal2011`. The
          302 :var:`climatic_mass_balance` is read from an input (``-i``) file; the ice surface
          303 temperature is computed as a function of latitude (variable :var:`lat`) and surface
          304 elevation (dynamically updated by PISM). See equation (1) in :cite:`Martinetal2011`.
          305 
          306 .. _sec-surface-delta-t:
          307 
          308 Scalar temperature offsets
          309 ++++++++++++++++++++++++++
          310 
          311 :|options|: ``-surface ...,delta_T``
          312 :|variables|: :var:`delta_T`
          313 :|implementation|: ``pism::surface::Delta_T``
          314 
          315 Command-line options:
          316 
          317 - :opt:`-surface_delta_T_file` sets the name of the file PISM will read :var:`delta_T`
          318   from.
          319 - :opt:`-surface_delta_T_period` (*years*) sets the period of the forcing data (section
          320   :ref:`sec-periodic-forcing`)
          321 - :opt:`-surface_delta_T_reference_year` sets the reference year (section
          322   :ref:`sec-periodic-forcing`).
          323 
          324 The time-dependent scalar offsets :var:`delta_T` are added to :var:`ice_surface_temp`
          325 computed by a surface model.
          326 
          327 Please make sure that :var:`delta_T` has the units of "``Kelvin``".
          328 
          329 This modifier is identical to the corresponding atmosphere modifier, but applies offsets
          330 at a different stage in the computation of top-surface boundary conditions needed by the
          331 ice dynamics core.
          332 
          333 .. _sec-surface-elevation-change:
          334 
          335 Adjustments using modeled change in surface elevation
          336 +++++++++++++++++++++++++++++++++++++++++++++++++++++
          337 
          338 :|options|: ``-surface ...,elevation_change``
          339 :|variables|: :var:`surface_altitude` (CF standard name),
          340 :|implementation|: ``pism::surface::LapseRates``
          341 
          342 The ``elevation_change`` modifier adjusts ice-surface temperature and surface mass balance
          343 using modeled changes in surface elevation relative to a reference elevation read from a
          344 file.
          345 
          346 The surface temperature is modified using an elevation lapse rate
          347 `\gamma_T =` :config:`surface.elevation_change.temperature_lapse_rate`. Here
          348 
          349 .. math::
          350    \gamma_T = -\frac{dT}{dz}.
          351 
          352 Two methods of adjusting the SMB are available:
          353 
          354 - Scaling using an exponential factor
          355 
          356   .. math::
          357 
          358      \mathrm{SMB} = \mathrm{SMB_{input}} \cdot \exp(C \cdot \Delta T),
          359 
          360   where `C =` :config:`surface.elevation_change.smb.exp_factor` and `\Delta T` is the
          361   temperature difference produced by applying
          362   :config:`surface.elevation_change.temperature_lapse_rate`.
          363 
          364   This mechanisms increases the SMB by `100(\exp(C) - 1)` percent for each degree of
          365   temperature increase.
          366 
          367   To use this method, set :opt:`-smb_adjustment scale`.
          368 
          369 - Elevation lapse rate for the SMB
          370 
          371   .. math::
          372 
          373      \mathrm{SMB} = \mathrm{SMB_{input}} - \Delta h \cdot \gamma_M,
          374 
          375   where `\gamma_M =` :config:`surface.elevation_change.smb.lapse_rate` and `\Delta h` is the
          376   difference between modeled and reference surface elevations.
          377 
          378   To use this method, set :opt:`-smb_adjustment shift`.
          379 
          380 It uses the following options.
          381 
          382 - :opt:`-temp_lapse_rate` gives the temperature lapse rate, in `K/km`. Note that we
          383   use the following definition of the temperature lapse rate:
          384 - :opt:`-smb_adjustment` chooses SMB lapse rate (``shift``) or SMB scaling (``scale``).
          385 - :opt:`-smb_exp_factor` specifies the exponential factor used to scale the SMB
          386 - :opt:`-smb_lapse_rate` gives the surface mass balance lapse rate, in `m/year/km`.
          387   Here, `\gamma_M=-\frac{dM}{dz}`.
          388 - :opt:`-surface_elevation_change_file` specifies the file containing the reference surface
          389   elevation field (standard name: :var:`surface_altitude`). This file can contain several
          390   surface elevation records to use lapse rate corrections relative to time-dependent
          391   surface. If one record is provided, the reference surface elevation is assumed to be
          392   time-independent.
          393 - :opt:`-surface_elevation_change_period` gives the period, in model years, to use when
          394   interpreting data in the file given with ``-surface_given_file``,
          395 - :opt:`-surface_elevation_change_reference_year` takes the time `T` in model years. The
          396   record for `t` years in ``-surface_given_file`` is interpreted as corresponding to
          397   `t` years since `T`.
          398 
          399 .. _sec-surface-forcing:
          400 
          401 Mass flux adjustment
          402 ++++++++++++++++++++
          403     
          404 :|options|: ``-surface ...,forcing``
          405 :|variables|: :var:`thk` (ice thickness), :var:`ftt_mask` (mask of zeros and ones; 1 where
          406               surface mass flux is adjusted and 0 elsewhere)
          407 :|implementation|: ``pism::surface::ForceThickness``
          408 
          409 The ``forcing`` modifier implements a surface mass balance adjustment mechanism which
          410 forces the thickness of grounded ice to a target thickness distribution at the end of the
          411 run. The idea behind this mechanism is that spinup of ice sheet models frequently requires
          412 the surface elevation to come close to measured values at the end of a run. A simpler
          413 alternative to accomplish this, namely option ``-no_mass``, represents an unmodeled,
          414 frequently large, violation of the mass continuity equation.
          415 
          416 In more detail, let `H_{\text{tar}}` be the target thickness. Let `H` be the
          417 time-dependent model thickness. The surface model component described here produces the
          418 term `M` in the mass continuity equation:
          419 
          420 .. math::
          421 
          422    \frac{\partial H}{\partial t} = M - S - \nabla\cdot \mathbf{q}.
          423 
          424 (Other details of this equation do not concern us here.) The ``forcing`` modifier causes
          425 `M` to be adjusted by a multiple of the difference between the target thickness and
          426 the current thickness,
          427 
          428 .. math::
          429 
          430    \Delta M = \alpha (H_{\text{tar}} - H)
          431 
          432 where `\alpha>0`. We are adding mass (`\Delta M>0`) where
          433 `H_{\text{tar}} > H` and ablating where `H_{\text{tar}} < H`.
          434 
          435 Option :opt:`-force_to_thickness_file` identifies the file containing the target ice
          436 thickness field ``thk`` and the mask ``ftt_mask``. A basic run modifying surface model
          437 ``given`` would look like
          438 
          439 .. code-block:: none
          440 
          441     pismr -i foo.nc -surface given,forcing -force_to_thickness_file bar.nc
          442 
          443 In this case ``foo.nc`` contains fields :var:`climatic_mass_balance` and
          444 :var:`ice_surface_temp`, as normal for ``-surface given``, and ``bar.nc`` contains fields
          445 :var:`thk` which will serve as the target thickness and :var:`ftt_mask` which defines the
          446 map plane area where this adjustment is applied. Option :opt:`-force_to_thickness_alpha`
          447 adjusts the value of `\alpha`, which has a default value specified in the
          448 :ref:`sec-parameter-list`.
          449 
          450 In addition to this one can specify a multiplicative factor `C` used in areas where
          451 the target thickness field has less than
          452 :opt:`-force_to_thickness_ice_free_thickness_threshold` meters of ice;
          453 `\alpha_{\text{ice free}} = C \times \alpha`. Use the
          454 :opt:`-force_to_thickness_ice_free_alpha_factor` option to set `C`.
          455 
          456 .. _sec-surface-anomaly:
          457 
          458 Using climate data anomalies
          459 ++++++++++++++++++++++++++++
          460     
          461 :|options|: :opt:`-surface ...,anomaly`
          462 :|variables|: :var:`ice_surface_temp_anomaly`,
          463               :var:`climatic_mass_balance_anomaly` |flux|
          464 :|implementation|: ``pism::surface::Anomaly``
          465 
          466 This modifier implements a spatially-variable version of ``-surface ...,delta_T`` which
          467 also applies time-dependent climatic mass balance anomalies.
          468 
          469 It takes the following options:
          470 
          471 - :opt:`-surface_anomaly_file` specifies a file containing variables
          472   :var:`ice_surface_temp_anomaly` and :var:`climatic_mass_balance_anomaly`.
          473 - :opt:`-surface_anomaly_period` (years) specifies the period of the forcing data, in
          474   model years; see :ref:`sec-periodic-forcing`
          475 - :opt:`-surface_anomaly_reference_year` specifies the reference year; see
          476   :ref:`sec-periodic-forcing`
          477 
          478 See also to ``-atmosphere ...,anomaly`` (section :ref:`sec-atmosphere-anomaly`), which is
          479 similar, but applies anomalies at the atmosphere level.
          480 
          481 .. _sec-surface-cache:
          482 
          483 The caching modifier
          484 ++++++++++++++++++++
          485 
          486 :|options|: ``-surface ...,cache``
          487 :|implementation|: ``pism::surface::Cache``
          488 :|seealso|: :ref:`sec-ocean-cache`
          489     
          490 This modifier skips surface model updates, so that a surface model is called no more than
          491 every :opt:`-surface.cache.update_interval` years. A time-step of `1` year is used every
          492 time a surface model is updated.
          493 
          494 This is useful in cases when inter-annual climate variability is important, but one year
          495 differs little from the next. (Coarse-grid paleo-climate runs, for example.)
          496 
          497 It takes the following options:
          498 
          499 - :opt:`-surface.cache.update_interval` (*years*) Specifies the minimum interval between
          500   updates. PISM may take longer time-steps if the adaptive scheme allows it, though.
          501 
          502 .. rubric:: Footnotes
          503 
          504 .. [#] You can use other time units supported by UDUNITS_.