5.3.1. OBCS: Open boundary conditions for regional modeling

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

5.3.1.1. Introduction

The OBCS-package is fundamental to regional ocean modelling with the MITgcm, but there are so many details to be considered in regional ocean modelling that this package cannot accomodate 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- and runtime-options of this package, but also with the code itself. In many cases it will be necessary to adapt the obcs-code (in particular code{S/R OBCS_CALC}) to the application in question; in these cases the obcs-package (together with the rbcs-package, section ref{sec:pkg:rbcs}) is a very useful infrastructure for implementing special regional models.

5.3.1.2. OBCS configuration and compiling

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

  • using the packages.conf file by adding obcs to it,
  • or using genmake2 adding -enable=obcs or -disable=obcs switches
  • Required packages and CPP options:
    • Two alternatives are available for prescribing open boundary values, which differ in the way how OB’s are treated in time:
      • A simple time-management (e.g. constant in time, or cyclic with fixed fequency) is provided through S/R obcs_external_fields_load.
      • More sophisticated ‘real-time’ (i.e. calendar time) management is available through obcs_prescribe_read.
    • The latter case requires packages cal and exf to be enabled.

(see also Section ref{sec:buildingCode}).

Parts of the OBCS code can be enabled or disabled at compile time via CPP preprocessor flags. These options are set in OBCS_OPTIONS.h. Table 5.1 summarizes these options.

Table 5.1 OBCS CPP options
CPP option Description
ALLOW_OBCS_NORTH enable Northern OB
ALLOW_OBCS_SOUTH enable Southern OB
ALLOW_OBCS_EAST enable Eastern OB
ALLOW_OBCS_WEST enable Western OB
   
ALLOW_OBCS_PRESCRIBE enable code for prescribing OB’s
ALLOW_OBCS_SPONGE enable sponge layer code
ALLOW_OBCS_BALANCE enable code for balancing transports through OB’s
ALLOW_ORLANSKI enable Orlanski radiation conditions at OB’s
ALLOW_OBCS_STEVENS enable Stevens (1990) boundary conditions at OB’s
  (currently only implemented for eastern and
  western boundaries and NOT for ptracers)

5.3.1.3. 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. package :code:`exf enabled). These parameter files are read in S/R packages_readparms.F, obcs_readparms.F, and exf_readparms.F, respectively. Run-time parameters may be broken into 3 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 useOBCS = .TRUE. in data.pkg.

Package flags and parameters

Table 5.2 summarizes the runtime flags that are set in data.obcs, and their default values.

Table 5.2 pkg OBCS run-time parameters
Flag/parameter default Description
basic flags & parameters (OBCS_PARM01)    
OB_Jnorth 0 Nx-vector of J-indices (w.r.t. Ny) of Northern OB at each I-position (w.r.t. Nx)
OB_Jsouth 0 Nx-vector of J-indices (w.r.t. Ny) of Southern OB at each I-position (w.r.t. Nx)
OB_Ieast 0 Ny-vector of I-indices (w.r.t. Nx) of Eastern OB at each J-position (w.r.t. Ny)
OB_Iwest 0 Ny-vector of I-indices (w.r.t. Nx) of Western OB at each J-position (w.r.t. Ny)
useOBCSprescribe .FALSE.  
useOBCSsponge .FALSE.  
useOBCSbalance code{.FALSE.}  
OBCS_balanceFacN/S/E/W 1 factor(s) determining the details of the balaning code
useOrlanskiNorth/South/EastWest .FALSE. turn on Orlanski boundary conditions for individual boundary
useStevensNorth/South/EastWest .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) u(-velocity) v(-velocity)
    w(-velocity) eta (sea surface height)
    a (sea ice area) h (sea ice thickness) sn (snow thickness) sl (sea ice salinity)
     
Orlanski parameters (OBCS_PARM02)    
cvelTimeScale 2000 sec averaging period for phase speed
CMAX 0.45 m/s maximum allowable phase speed-CFL for AB-II
CFIX 0.8 m/s fixed boundary phase speed
useFixedCEast .FALSE.  
useFixedCWest .FALSE.  
     
Sponge-layer parameters (OBCS_PARM03)    
spongeThickness 0 sponge layer thickness (in grid points)
Urelaxobcsinner 0 sec relaxation time scale at the innermost sponge layer point of a meridional OB
Vrelaxobcsinner 0 sec relaxation time scale at the innermost sponge layer point of a zonal OB
Urelaxobcsbound 0 sec relaxation time scale at the outermost sponge layer point of a meridional OB
Vrelaxobcsbound 0 sec relaxation time scale at the outermost sponge layer point of a zonal OB
     
Stevens parameters (OBCS_PARM04)    
T/SrelaxStevens 0 sec relaxation time scale for temperature/salinity
useStevensPhaseVel code{.TRUE.}  
useStevensAdvection code{.TRUE.}  

5.3.1.4. Defining open boundary positions

There are four open boundaries (OBs), a 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 \(\tt OB\_Jnorth(Nx)\) / \(\tt OB\_Jsouth(Nx)\), and an \(N_y\)-dimenisonal “column” array \(\tt OB\_Ieast(Ny)\) / \(\tt OB\_Iwest(Ny)\). 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 \(\tt OB\_I\ldots\), \(\tt OB\_J\ldots\) means there is no corresponding OB in that column/row. 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 Jnorth/Ieast refer to points relative to the Northern/Eastern edges of the model eg. \(\tt OB\_Jnorth(3)=-1\) means that the point \(\tt (3,Ny)\) is a northern OB.

Simple examples: For a model grid with :math:` N_{x}times N_{y} = 120times144` horizontal grid points with four open boundaries along the four egdes of the domain, the simplest way of specifying the boundary points in is:

  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,

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

OB_Jsouth(1:50) = 50*1,

5.3.1.5. Equations and key routines

OBCS_READPARMS:

Set OB positions through arrays OB_Jnorth(Nx), OB_Jsouth(Nx), OB_Ieast(Ny), OB_Iwest(Ny), and runtime flags (see Table [tab:pkg:obcs:runtime:sub:flags]).

OBCS_CALC:

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: \(\tt OB[N/S/E/W][t/s/u/v]\); e.g. for salinity array at Southern OB, array name is \(\tt OBSt\). Values filled are either

  • constant vertical \(T,S\) profiles as specified in file data (tRef(Nr), 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).
  • use prescribed boundary fields to compute Stevens boundary conditions.

ORLANSKI:

Orlanski radiation conditions [Orl76], examples can be found in verification/dome and verification/tutorial\_plume\_on\_slope

(ref{sec:eg-gravityplume}).

OBCS_PRESCRIBE_READ:

When 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 (ref{sec: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 (ref{sec: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 useSEAICE=.TRUE. then additional files OB[N/S/E/W][a,h,sl,sn,uice,vice] for sea ice area, thickness (HEFF), seaice salinity, snow and ice velocities \((N_{x/y}\times\mbox{time levels})\) can be specified.

As in S/R external\_fields\_load or the exf-package, the code reads two time levels for each variable, e.g.OBNu0 and OBNu1, and interpolates linearly between these time levels to obtain the value OBNu at the current model time (step). When the exf-package is used, the time levels are controlled for each boundary separately in the same way as the exf-fields in data.exf, namelist EXF\_NML\_OBCS. The runtime flags follow the above naming conventions, e.g. for the western boundary the corresponding flags are OBCWstartdate1/2 and OBCWperiod. Sea-ice boundary values are controlled separately with siobWstartdate1/2 and siobWperiod. When the exf-package is not used, the time levels are controlled by the runtime flags externForcingPeriod and externForcingCycle in data, see verification/exp4 for an example.

OBCS_CALC_STEVENS:

(THE IMPLEMENTATION OF THESE BOUNDARY CONDITIONS IS NOT COMPLETE. PASSIVE TRACERS, SEA ICE AND NON-LINEAR FREE SURFACE ARE NOT SUPPORTED PROPERLY.)

The boundary conditions following [Ste90] 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 code{ptracers} is used together with this option). Currently, the code vertically averages the normal velocity as specified in code{OB[E,W]u} or code{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 section [sec:time:sub:stepping], e.g., eq.([eq:ustar-backward-free-surface])). (This velocity is not available at the beginning of the next time step \(n+1\), when S/R OBCS_CALC/OBCS_CALC_STEVENS are called, therefore it needs to be saved in S/R DYNAMICS by calling S/R OBCS_SAVE_UV_N 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:

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

    where \(\tau_\chi\) is the relaxation time scale T/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):

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

    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 useStevensPhaseVel and useStevensAdvection to .FALSE..

See [Ste90] for details. With this boundary condition specifying the exact net transport across the open boundary is simple, so that balancing the flow with (S/R~OBCS_BALANCE_FLOW, see next paragraph) is usually not necessary.

OBCS_BALANCE_FLOW:

When turned on (ALLOW\_OBCS\_BALANCE defined in OBCS\_OPTIONS.h and useOBCSbalance=.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 OBCS\_balanceFacN/S/E/W. 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 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:

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

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, say, 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, gobal balancing with OBCS_balanceFacE/W/N/S \(\ge 0\) is the preferred method.

OBCS_APPLY_*:

OBCS_SPONGE:

The sponge layer code (turned on with ALLOW\_OBCS\_SPONGE and 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

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

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 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 Urelaxobcsbound and Vrelaxobcsbound) and \(\tau_{i}\) (runtime parameters Urelaxobcsinner and Vrelaxobcsinner) the relaxation time scales on the boundary and at the interior termination of the sponge layer. The parameters Urelaxobcsbound/inner`set the relaxation time scales for the Eastern and Western boundaries, :code:`Vrelaxobcsbound/inner for the Northern and Southern boundaries.

OB’s with nonlinear free surface

5.3.1.6. Flow chart

C     !CALLING SEQUENCE:
c ...

5.3.1.7. OBCS diagnostics

Diagnostics output is available via the diagnostics package (see Section [sec:pkg:diagnostics]). Available output fields are summarized in Table [tab:pkg:obcs:diagnostics].

[tab:pkg:obcs:diagnostics]

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

5.3.1.8. Reference experiments

In the directory verifcation, the following experiments use obcs:

  • exp4: box with 4 open boundaries, simulating flow over a Gaussian bump based on , also tests Stevens-boundary conditions;
  • dome: based on the project “Dynamics of Overflow Mixing and Entrainment” (http://www.rsmas.miami.edu/personal/tamay/DOME/dome.html), uses Orlanski-BCs;
  • internal_wave: uses a heavily modified S/R~OBCS\_CALC
  • :code:seaice_obcs`: simple example who to use the sea-ice related code, based on lab_sea;
  • tutorial_plume_on_slope: uses Orlanski-BCs, see also section [sec:eg-gravityplume].

5.3.1.9. References

5.3.1.10. Experiments and tutorials that use obcs

  • tutorial\_plume\_on\_slope (section~ref{sec:eg-gravityplume})