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OBCS: Open boundary conditions for regional modeling

Authors: Alistair Adcroft, Patrick Heimbach, Samar Katiwala, Martin Losch

Introduction

The OBCS-package (:filelink:`pkg/obcs`) is fundamental to regional ocean modeling with the MITgcm, but there are so many details to be considered in regional ocean modeling that this package cannot accommodate all imaginable and possible options. Therefore, for a regional simulation with very particular details it is recommended to familiarize oneself not only with the compile-time and run-time options of this package, but also with the code itself. In many cases it will be necessary to adapt the obcs-code (in particular :filelink:`S/R OBCS_CALC <pkg/obcs/obcs_calc.F>`) to the application in question; in these cases :filelink:`pkg/obcs` (together with the :filelink:`pkg/rbcs`, see :numref:`sub_phys_pkg_rbcs`) is a very useful infrastructure for implementing special regional models.

OBCS configuration and compiling

As with all MITgcm packages, OBCS can be turned on or off at compile-time

Parts of the OBCS code can be enabled or disabled at compile-time via CPP preprocessor flags. These options are set in :filelink:`OBCS_OPTIONS.h <pkg/obcs/OBCS_OPTIONS.h>`. :numref:`tab_phys_pkg_obcs_cpp_opts` summarizes these options.

.. tabularcolumns:: |\Y{.475}|\Y{.1}|\Y{.45}|
CPP flags for the obcs package
CPP option Default Description
ALLOW_OBCS_NORTH #define enable Northern OB
ALLOW_OBCS_SOUTH #define enable Southern OB
ALLOW_OBCS_EAST #define enable Eastern OB
ALLOW_OBCS_WEST #define enable Western OB
ALLOW_OBCS_PRESCRIBE #define enable code for prescribing OB's
ALLOW_OBCS_SPONGE #undef enable sponge layer code
ALLOW_OBCS_BALANCE #define enable code for balancing transports through OB's
ALLOW_ORLANSKI #define enable Orlanski radiation conditions at OB's
ALLOW_OBCS_STEVENS #undef enable Stevens (1990) boundary conditions at OB's (currently NOT implemented for ptracers)
ALLOW_OBCS_SEAICE_SPONGE #undef Include hooks to sponge layer treatment of pkg/seaice variables
ALLOW_OBCS_TIDES #undef Add tidal contributions to normal OB flow (At the moment tidal forcing is applied only to "normal" flow)

Run-time parameters

Run-time parameters are set in files data.pkg, data.obcs, and data.exf if 'real-time' prescription is requested (i.e., :filelink:`pkg/exf` enabled). These parameter files are read in S/Rs :filelink:`PACKAGES_READPARMS <model/src/packages_readparms.F>`, :filelink:`OBCS_READPARMS <pkg/obcs/obcs_readparms.F>`, and :filelink:`EXF_READPARMS <pkg/exf/exf_readparms.F>`, respectively. Run-time parameters may be broken into three categories:

  1. switching on/off the package at runtime
  2. OBCS package flags and parameters
  3. additional timing flags in data.exf if selected.

Enabling the package

The OBCS package is switched on at runtime by setting :varlink:`useOBCS` = .TRUE. in data.pkg.

Package flags and parameters

:numref:`tab_phys_pkg_obcs_runtime_flags` summarizes the runtime flags that are set in data.obcs and their default values.

.. tabularcolumns:: |\X{1}{3}|c|\X{1}{2}|
OBCS runtime parameters
Flag/parameter default Description
:varlink:`OB_Jnorth` 0 Nx-vector of J-indices (w.r.t. Ny) of Northern OB at each I-position (w.r.t. Nx)
:varlink:`OB_Jsouth` 0 Nx-vector of J-indices (w.r.t. Ny) of Southern OB at each I-position (w.r.t. Nx)
:varlink:`OB_Ieast` 0 Ny-vector of I-indices (w.r.t. Nx) of Eastern OB at each J-position (w.r.t. Ny)
:varlink:`OB_Iwest` 0 Ny-vector of I-indices (w.r.t. Nx) of Western OB at each J-position (w.r.t. Ny)
:varlink:`useOBCSprescribe` FALSE  
:varlink:`useOBCSsponge` FALSE  
:varlink:`useOBCSbalance` FALSE  
:varlink:`OBCS_balanceFacN`, :varlink:`OBCS_balanceFacS`, :varlink:`OBCS_balanceFacE`, :varlink:`OBCS_balanceFacW` 1 Factor(s) determining the details of the balancing code
:varlink:`OBCSbalanceSurf` FALSE include surface mass flux in balance
:varlink:`useOrlanskiNorth`, :varlink:`useOrlanskiSouth`, :varlink:`useOrlanskiEast`, :varlink:`useOrlanskiWest` FALSE Turn on Orlanski boundary conditions for individual boundary.
:varlink:`useStevensNorth`, :varlink:`useStevensSouth`, :varlink:`useStevensEast`, :varlink:`useStevensWest` FALSE Turn on Stevens boundary conditions for individual boundary
OBXyFile ' '

File name of OB field:

X: N(orth), S(outh), E(ast), W(est)

y: t(emperature), s(salinity), eta (sea surface height), u(-velocity), v(-velocity), w(-velocity), a (seaice area), h (sea ice thickness), sn (snow thickness), sl (sea ice salinity )
Orlanski Parameters OBCS_PARM02  
:varlink:`cvelTimeScale` 2000.0 Averaging period for phase speed (seconds)
:varlink:`CMAX` 0.45 Maximum allowable phase speed-CFL for AB-II (m/s)
:varlink:`CFIX` 0.8 Fixed boundary phase speed (m/s)
:varlink:`useFixedCEast` FALSE  
:varlink:`useFixedCWest` FALSE  
Sponge layer parameters OBCS_PARM03  
:varlink:`spongeThickness` 0 sponge layer thickness (in grid points)
:varlink:`Urelaxobcsinner` 0.0 relaxation time scale at the innermost sponge layer point of a meridional OB (s)
:varlink:`Vrelaxobcsinner` 0.0 relaxation time scale at the innermost sponge layer point of a zonal OB (s)
:varlink:`Urelaxobcsbound` 0.0 relaxation time scale at the outermost sponge layer point of a meridional OB (s)
:varlink:`Vrelaxobcsbound` 0.0 relaxation time scale at the outermost sponge layer point of a zonal OB (s)
Stevens parameters OBCS_PARM04  
:varlink:`TrelaxStevens` :varlink:`SrelaxStevens` 0 Relaxation time scale for temperature/salinity (s)
:varlink:`useStevensPhaseVel` TRUE  
:varlink:`useStevensAdvection` TRUE  

Defining open boundary positions

There are up to four open boundaries (OBs): Northern, Southern, Eastern, and Western. All OB locations are specified by their absolute meridional (Northern/Southern) or zonal (Eastern/Western) indices. Thus, for each zonal position i=1\ldots N_x a meridional index j specifies the Northern/Southern OB position, and for each meridional position j=1\ldots N_y a zonal index i specifies the Eastern/Western OB position. For Northern/Southern OB this defines an N_x-dimensional “row” array :varlink:`OB_Jnorth`(Nx) / :varlink:`OB_Jsouth`(Nx) and an N_y-dimenisonal “column” array :varlink:`OB_Ieast`(Ny) / :varlink:`OB_Iwest`(Ny). Positions determined in this way allows Northern/Southern OBs to be at variable j (or y) positions and Eastern/Western OBs at variable i (or x) positions. Here indices refer to tracer points on the C-grid. A zero (0) element in OB_I... / OB_J... means there is no corresponding OB in that column/row. By default all elements in OB_I... / OB_J... are zero. For a Northern/Southern OB, the OB V-point is to the South/North. For an Eastern/Western OB, the OB U-point is to the West/East. For example

OB_Jnorth(3)=34 means that:
  • T(3,34) is a an OB point
  • U(3,34) is a an OB point
  • V(3,34) is a an OB point
OB_Jsouth(3)=1 means that:
  • T(3,1) is a an OB point
  • U(3,1) is a an OB point
  • V(3,2) is a an OB point
OB_Ieast(10)=69 means that:
  • T(69,10) is a an OB point
  • U(69,10) is a an OB point
  • V(69,10) is a an OB point
OB_Iwest(10)=1 means that:
  • T(1,10) is a an OB point
  • U(2,10) is a an OB point
  • V(1,10) is a an OB point

For convenience, negative values for :varlink:`OB_Jnorth` / :varlink:`OB_Ieast` refer to points relative to the Northern/Eastern edges of the model, e.g. OB_Jnorth(3)=-1 means that the point (3,Ny) is a northern OB and OB_Ieast(3)=-5 means that the point (3,Nx-5) is an eastern OB.

Simple examples

For a model grid with N_x \times N_y = 120 \times 144 horizontal grid points with four open boundaries along the four edges of the domain, the simplest way of specifying the boundary points:

  OB_Ieast = 144*-1,
# or OB_Ieast = 144*120,
  OB_Iwest = 144*1,
  OB_Jnorth = 120*-1,
# or OB_Jnorth = 120*144,
  OB_Jsouth = 120*1,

When the boundaries are in single rows or columns as in the above example, the same can be achieved with the convenient parameters :varlink:`OB_singleJnorth` / :varlink:`OB_singleJsouth` / :varlink:`OB_singleIeast` / :varlink:`OB_singleIwest`:

OB_singleIeast  = -1,
OB_singleIwest  =  1,
OB_singleJnorth = -1,
OB_singleJsouth =  1,

If only the first 50 grid points of the southern boundary are boundary points:

OB_Jsouth(1:50) = 50*1,

A more complex example

Open boundaries are not restricted to single rows or columns. Each OB can be distributed in different rows and columns resulting in OBs consisting of the combination of different types of open boundaries (i.e., N, S, E and W). :numref:`fig_obcsexample` displays such an OB located on the left-bottom corner of a domain. Note there are five boundary points defined by southern and western boundaries. In particular, there are five southern boundary (blue lines) and two western boundaries points (red lines). For the boundary displayed in :numref:`fig_obcsexample` and the same dimensions as in the previous example (i.e. 120 \times 144 grid points), the namelist looks like this:

OB_Iwest  = 1*0,1*5,142*0,
OB_Jsouth = 2*3,3*2,115*0,
Example boundary

Example boundary with more than one row. The dark grey, light grey, and white boxes are points outside the domain, OB points, and ocean points, respectively. The black dots mark the OB index to write into the namelist.

For an even more complicated open boundary geometry, e.g., delimiting a concave interior domain (:varlink:`OB_Ieast` \leq :varlink:`OB_Iwest`), one might need to also specify the interior domain through an additional input file :varlink:`insideOBmaskFile` for the interior mask (=1 inside, =0 outside).

Equations and key routines

:filelink:`OBCS\_READPARMS <pkg/obcs/obcs_readparms.F>`:

Set OB positions through arrays OB_Jnorth(Nx), OB_Jsouth(Nx), OB_Ieast(Ny), OB_Iwest(Ny) and runtime flags (see Table :numref:`tab_phys_pkg_obcs_runtime_flags`).

:filelink:`OBCS\_CALC <pkg/obcs/obcs_calc.F>`:

Top-level routine for filling values to be applied at OB for T,S,U,V,\eta into corresponding “slice” arrays (x,z) (y,z) for each OB: OB[N/S/E/W][t/s/u/v]; e.g. for the salinity array at the Southern OB, the array name is :varlink:`OBSs`. Values filled are either

  • constant vertical T,S profiles as specified in file data (:varlink:`tRef`(Nr), :varlink:`sRef`(Nr)) with zero velocities U,V
  • T,S,U,V values determined via Orlanski radiation conditions (see below)
  • prescribed time-constant or time-varying fields (see below).
  • prescribed boundary fields to compute Stevens boundary conditions.

:filelink:`ORLANSKI <pkg/obcs/ORLANSKI.h>`:

Orlanski radiation conditions :cite:`orl:76` examples can be found in example configurations :filelink:`verification/dome` and :filelink:`verification/tutorial_plume_on_slope` (as described in detail in :numref:`tutorial_plume_on_slope`).

:filelink:`OBCS\_PRESCRIBE\_READ <pkg/obcs/obcs\_prescibe\_read.F>`:

When :varlink:`useOBCSprescribe` = .TRUE. the model tries to read temperature, salinity, u- and v-velocities from files specified in the runtime parameters OB[N/S/E/W][t/s/u/v]File. These files are the usual IEEE, big-endian files with dimensions of a section along an open boundary:

  • For North/South boundary files the dimensions are (N_x\times N_r\times\mbox{time levels}), for East/West boundary files the dimensions are (N_y\times N_r\times\mbox{time levels}).
  • If a non-linear free surface is used (:numref:`nonlinear-freesurface`), additional files OB[N/S/E/W]etaFile for the sea surface height \eta with dimension (N_{x/y}\times\mbox{time levels}) may be specified.
  • If non-hydrostatic dynamics are used (:numref:`non-hydrostatic`), additional files OB[N/S/E/W]wFile for the vertical velocity w with dimensions (N_{x/y}\times N_r\times\mbox{time levels}) can be specified.
  • If :varlink:`useSEAICE` = .TRUE. then additional files OB[N/S/E/W][a,h,sl,sn,uice,vice] for sea ice area, thickness (:varlink:`HEFF`), seaice salinity, snow and ice velocities (N_{x/y}\times\mbox{time levels}) can be specified.

As in :filelink:`external_fields_load.F <model/src/external_fields_load.F>` or as done in :filelink:`pkg/exf`, the code reads two time levels for each variable, e.g., :varlink:`OBNu0` and :varlink:`OBNu1`, and interpolates linearly between these time levels to obtain the value :varlink:`OBNu` at the current model time (step). When :filelink:`pkg/exf` is used, the time levels are controlled for each boundary separately in the same way as the :filelink:`pkg/exf` fields in data.exf, namelist EXF_NML_OBCS. The run-time flags follow the above naming conventions, e.g., for the western boundary the corresponding flags are :varlink:`OBCSWstartdate1`, :varlink:`OBCSWstartdate2` and :varlink:`OBCSWperiod`. Sea-ice boundary values are controlled separately with :varlink:`siobWstartdate1`, :varlink:`siobWstartdate2` and :varlink:`siobWperiod`. When :filelink:`pkg/exf` is not used the time levels are controlled by the runtime flags :varlink:`externForcingPeriod` and :varlink:`externForcingCycle` in data; see :filelink:`verification/exp4/input/data` for an example.

:filelink:`OBCS\_CALC\_STEVENS <pkg/obcs/obcs_calc_stevens.F>`:

The boundary conditions following :cite:`stevens:90` require the vertically averaged normal velocity (originally specified as a stream function along the open boundary) \bar{u}_{ob} and the tracer fields \chi_{ob} (note: passive tracers are currently not implemented and the code stops when package :ref:`ptracers <sub_phys_pkg_ptracers>` is used together with this option). Currently the code vertically averages the normal velocity as specified in OB[E,W]u or OB[N,S]v. From these prescribed values the code computes the boundary values for the next timestep n+1 as follows (as an example, we use the notation for an eastern or western boundary):

  • u^{n+1}(y,z) = \bar{u}_{ob}(y) + (u')^{n}(y,z) where (u')^{n} is the deviation from the vertically averaged velocity at timestep n on the boundary. (u')^{n} is computed in the previous time step n from the intermediate velocity u^* prior to the correction step (see :numref:`time_stepping` equation :eq:`ustar-backward-free-surface`). (This velocity is not available at the beginning of the next time step n+1, when S/Rs :filelink:`OBCS_CALC <pkg/obcs/obcs_calc.F>` and :filelink:`OBCS_CALC_STEVENS <pkg/obcs/obcs_calc_stevens.F>` are called, therefore it needs to be saved in :filelink:`S/R DYNAMICS <model/src/dynamics.F>` by calling :filelink:`S/R OBCS_SAVE_UV_N <pkg/obcs/obcs_save_uv_n.F>` and also stored in a separate restart files pickup_stevens[N/S/E/W].${iteration}.data)

  • If u^{n+1} is directed into the model domain, the boudary value for tracer \chi is restored to the prescribed values:

    \chi^{n+1} = \chi^{n} + \frac{\Delta{t}}{\tau_\chi} (\chi_{ob} -
  \chi^{n})

    where \tau_\chi is the relaxation time scale (either :varlink:`TrelaxStevens` or :varlink:`SrelaxStevens`). The new \chi^{n+1} is then subject to the advection by u^{n+1}.

  • If u^{n+1} is directed out of the model domain, the tracer \chi^{n+1} on the boundary at timestep n+1 is estimated from advection out of the domain with u^{n+1}+c, where c is a phase velocity estimated as \frac{1}{2} \frac{\partial\chi}{\partial{t}}/ \frac{\partial\chi}{\partial{x}}. The numerical scheme is (as an example for an eastern boundary):

    \chi_{i_{b},j,k}^{n+1} =   \chi_{i_{b},j,k}^{n} + \Delta{t}
  (u^{n+1}+c)_{i_{b},j,k}\frac{\chi_{i_{b},j,k}^{n}
    - \chi_{i_{b}-1,j,k}^{n}}{\Delta{x}_{i_{b}j}^{C}}
      \mbox{ if }u_{i_{b}jk}^{n+1}>0

    where i_{b} is the boundary index. For test purposes, the phase velocity contribution or the entire advection can be turned off by setting the corresponding parameters :varlink:`useStevensPhaseVel` and :varlink:`useStevensAdvection` to .FALSE..

See :cite:`stevens:90` for details. With this boundary condition specifying the exact net transport across the open boundary is simple, so that balancing the flow with (:filelink:`S/R OBCS_BALANCE_FLOW <pkg/obcs/obcs_balance_flow.F>` see next paragraph) is usually not necessary.

Special cases where the current implementation is not complete:

  • When you use the non-linear free surface option (parameter :varlink:`nonlinFreeSurf` > 1), the current implementation just assumes that the gradient normal to the open boundary is zero (\frac{\partial\eta}{\partial{n}} = 0). Although this is inconsistent with geostrophic dynamics and the possibility to specify a non-zero tangent velocity together with Stevens BCs for normal velocities, it seems to work. Recommendation: Always specify zero tangential velocities with Stevens BCs.
  • There is no code for passive tracers, just a commented template in :filelink:`S/R OBCS_CALC_STEVENS <pkg/obcs/obcs_calc_stevens.F>`. This means that passive tracers can be specified independently and are fluxed with the velocities that the Stevens BCs compute, but without the restoring term.
  • There are no specific Stevens BCs for sea ice, e.g., :ref:`pkg/seaice <sub_phys_pkg_seaice>`. The model uses the default boundary conditions for the sea ice packages.

:filelink:`OBCS\_BALANCE\_FLOW <pkg/obcs/obcs_balance_flow.F>`:

When turned on (CPP option :varlink:`ALLOW_OBCS_BALANCE` defined in :filelink:`OBCS_OPTIONS.h <pkg/obcs/OBCS_OPTIONS.h>` and :varlink:`useOBCSbalance` set to .TRUE. in data.obcs/OBCS_PARM01), this routine balances the net flow across the open boundaries. By default the net flow across the boundaries is computed and all normal velocities on boundaries are adjusted to obtain zero net inflow.

This behavior can be controlled with the runtime flags :varlink:`OBCS_balanceFacN`, :varlink:`OBCS_balanceFacS`, :varlink:`OBCS_balanceFacE`, and :varlink:`OBCS_balanceFacW`. The values of these flags determine how the net inflow is redistributed as small correction velocities between the individual sections. A value -1 balances an individual boundary, values >0 determine the relative size of the correction. For example, the values

OBCS_balanceFacE = 1.,
OBCS_balanceFacW = -1.,
OBCS_balanceFacN = 2.,
OBCS_balanceFacS = 0.,

make the model

  • correct Western :varlink:`OBWu` by substracting a uniform velocity to ensure zero net transport through the Western open boundary;
  • correct Eastern and Northern normal flow, with the Northern velocity correction two times larger than the Eastern correction, but not the Southern normal flow, to ensure that the total inflow through East, Northern, and Southern open boundary is balanced.

The old method of balancing the net flow for all sections individually can be recovered by setting all flags to -1. Then the normal velocities across each of the four boundaries are modified separately, so that the net volume transport across each boundary is zero. For example, for the western boundary at i=i_{b}, the modified velocity is:

u(y,z) - \int_{\mbox{western boundary}}u dy dz \approx OBNu(j k) - \sum_{j k}
OBNu(j k) h_{w}(i_{b} j k)\Delta{y_G(i_{b} j)}\Delta{z(k)}.

This also ensures a net total inflow of zero through all boundaries, but this combination of flags is not useful if you want to simulate, for example, a sector of the Southern Ocean with a strong ACC entering through the western and leaving through the eastern boundary, because the value of -1 for these flags will make sure that the strong inflow is removed. Clearly, global balancing with OBCS_balanceFacE/W/N/S \ge 0 is the preferred method.

Setting runtime parameter :varlink:`OBCSbalanceSurf` to TRUE., the surface mass flux contribution, say, from surface freshwater flux :varlink:`EmPmR` is included in the balancing scheme.

OBCS_APPLY_*:

:filelink:`OBCS\_SPONGE <pkg/obcs/obcs_sponge.F>`:

The sponge layer code (turned on with CPP option :varlink:`ALLOW_OBCS_SPONGE` and run-time parameter :varlink:`useOBCSsponge`) adds a relaxation term to the right-hand-side of the momentum and tracer equations. The variables are relaxed towards the boundary values with a relaxation time scale that increases linearly with distance from the boundary

G_{\chi}^{\mbox{(sponge)}} =
- \frac{\chi - [( L - \delta{L} ) \chi_{BC} + \delta{L}\chi]/L}
{[(L-\delta{L})\tau_{b}+\delta{L}\tau_{i}]/L}
= - \frac{\chi - [( 1 - l ) \chi_{BC} + l\chi]}
{[(1-l)\tau_{b}+l\tau_{i}]}

where \chi is the model variable (U/V/T/S) in the interior, \chi_{BC} the boundary value, L the thickness of the sponge layer (runtime parameter :varlink:`spongeThickness` in number of grid points), \delta{L}\in[0,L] (\frac{\delta{L}}{L}=l\in[0,1]) the distance from the boundary (also in grid points), and \tau_{b} (runtime parameters :varlink:`Urelaxobcsbound` and :varlink:`Vrelaxobcsbound`) and \tau_{i} (runtime parameters :varlink:`Urelaxobcsinner` and :varlink:`Vrelaxobcsinner`) the relaxation time scales on the boundary and at the interior termination of the sponge layer. The parameters :varlink:`Urelaxobcsbound` and :varlink:`Urelaxobcsinner` set the relaxation time scales for the Eastern and Western boundaries, :varlink:`Vrelaxobcsbound` and :varlink:`Vrelaxobcsinner` for the Northern and Southern boundaries.

OB's with nonlinear free surface

OB's with sea ice

Flow chart

C     !CALLING SEQUENCE:
c ...

OBCS diagnostics

Diagnostics output is available via the diagnostics package (see :numref:`sub_outp_pkg_diagnostics`). Available output fields are summarized below:

------------------------------------------------------
 <-Name->|Levs|grid|<--  Units   -->|<- Tile (max=80c)
------------------------------------------------------

Experiments and tutorials that use obcs

In the directory :filelink:`verification` the following experiments use :filelink:`pkg/obcs`: