mom_regridding mom_regridding::regridding_cs character(len= *), parameter, public character(len=*), parameter, public mom_regridding::regriddingcoordinatemodedoc regriddingcoordinatemodedoc = " LAYER - Isopycnal or stacked shallow water layers\n"// " ZSTAR, Z* - stretched geopotential z*\n"// " SIGMA_SHELF_ZSTAR - stretched geopotential z* ignoring shelf\n"// " SIGMA - terrain following coordinates\n"// " RHO - continuous isopycnal\n"// " HYCOM1 - HyCOM-like hybrid coordinate\n"// " SLIGHT - stretched coordinates above continuous isopycnal\n"// " ADAPTIVE - optimize for smooth neutral density surfaces" Documentation for coordinate options. character(len= *), parameter, public character(len=*), parameter, public mom_regridding::regriddinginterpschemedoc regriddinginterpschemedoc = " P1M_H2 (2nd-order accurate)\n"// " P1M_H4 (2nd-order accurate)\n"// " P1M_IH4 (2nd-order accurate)\n"// " PLM (2nd-order accurate)\n"// " PPM_H4 (3rd-order accurate)\n"// " PPM_IH4 (3rd-order accurate)\n"// " P3M_IH4IH3 (4th-order accurate)\n"// " P3M_IH6IH5 (4th-order accurate)\n"// " PQM_IH4IH3 (4th-order accurate)\n"// " PQM_IH6IH5 (5th-order accurate)" Documentation for regridding interpolation schemes. initialize_regridding character(len= *), parameter, public character(len=*), parameter, public mom_regridding::regriddingdefaultinterpscheme regriddingdefaultinterpscheme = "P1M_H2" Default interpolation scheme. initialize_regridding logical, parameter, public logical, parameter, public mom_regridding::regriddingdefaultboundaryextrapolation regriddingdefaultboundaryextrapolation = .false. Default mode for boundary extrapolation. initialize_regridding real, parameter, public real, parameter, public mom_regridding::regriddingdefaultminthickness regriddingdefaultminthickness = 1.e-3 Default minimum thickness for some coordinate generation modes. initialize_regridding subroutine, public subroutine, public mom_regridding::initialize_regridding (CS, GV, US, max_depth, param_file, mdl, coord_mode, param_prefix, param_suffix) initialize_regridding CS CS GV GV US US max_depth max_depth param_file param_file mdl mdl coord_mode coord_mode param_prefix param_prefix param_suffix param_suffix Initialization and configures a regridding control structure based on customizable run-time parameters. cs Regridding control structure gv Ocean vertical grid structure us A dimensional unit scaling type max_depth The maximum depth of the ocean [Z ~> m]. param_file Parameter file mdl Name of calling module. coord_mode Coordinate mode param_prefix String to prefix to parameter names. If empty, causes main model parameters to be used. param_suffix String to append to parameter names. check_grid_def dz_function1 mom_string_functions::extract_integer mom_string_functions::extract_real mom_string_functions::extractword initcoord mom_error_handler::mom_error regrid_consts::regridding_adaptive regrid_consts::regridding_hycom1 regrid_consts::regridding_slight regriddingdefaultboundaryextrapolation regriddingdefaultinterpscheme regriddingdefaultminthickness regriddinginterpschemedoc rho_function1 set_regrid_max_depths set_regrid_max_thickness set_regrid_params set_target_densities set_target_densities_from_gv setcoordinateresolution uniformresolution subroutine subroutine mom_regridding::check_grid_def (filename, varname, expected_units, msg, ierr) check_grid_def filename filename varname varname expected_units expected_units msg msg ierr ierr Do some basic checks on the vertical grid definition file, variable. filename File name varname Variable name expected_units Expected units of variable msg Message to use for errors ierr True if an error occurs initialize_regridding subroutine, public subroutine, public mom_regridding::end_regridding (CS) end_regridding CS CS Deallocation of regridding memory. cs Regridding control structure coord_adapt::end_coord_adapt coord_hycom::end_coord_hycom coord_sigma::end_coord_sigma coord_slight::end_coord_slight coord_zlike::end_coord_zlike subroutine, public subroutine, public mom_regridding::regridding_main (remapCS, CS, G, GV, h, tv, h_new, dzInterface, frac_shelf_h, conv_adjust) regridding_main remapCS remapCS CS CS G G GV GV h h tv tv h_new h_new dzInterface dzInterface frac_shelf_h frac_shelf_h conv_adjust conv_adjust Dispatching regridding routine for orchestrating regridding & remapping. remapcs Remapping parameters and options cs Regridding control structure g Ocean grid structure gv Ocean vertical grid structure h Current 3D grid obtained after the last time step tv Thermodynamical variables (T, S, ...) h_new New 3D grid consistent with target coordinate dzinterface The change in position of each interface frac_shelf_h Fractional ice shelf coverage conv_adjust If true, do convective adjustment build_grid_adaptive build_grid_arbitrary build_grid_hycom1 build_grid_slight build_rho_grid build_sigma_grid build_zstar_grid calc_h_new_by_dz check_remapping_grid convective_adjustment mom_error_handler::mom_error regrid_consts::regridding_adaptive regrid_consts::regridding_hycom1 regrid_consts::regridding_slight subroutine subroutine mom_regridding::calc_h_new_by_dz (CS, G, GV, h, dzInterface, h_new) calc_h_new_by_dz CS CS G G GV GV h h dzInterface dzInterface h_new h_new Calculates h_new from h + delta_k dzInterface. cs Regridding control structure g Grid structure gv Ocean vertical grid structure h Old layer thicknesses (arbitrary units) dzinterface Change in interface positions (same as h) h_new New layer thicknesses (same as h) build_grid_hycom1 regridding_main subroutine, public subroutine, public mom_regridding::check_remapping_grid (G, GV, h, dzInterface, msg) check_remapping_grid G G GV GV h h dzInterface dzInterface msg msg Check that the total thickness of two grids match. g Grid structure gv Ocean vertical grid structure h Layer thicknesses [H ~> m or kg m-2] dzinterface Change in interface positions [H ~> m or kg m-2] msg Message to append to errors check_grid_column regridding_main subroutine, public subroutine, public mom_regridding::check_grid_column (nk, depth, h, dzInterface, msg) check_grid_column nk nk depth depth h h dzInterface dzInterface msg msg Check that the total thickness of new and old grids are consistent. nk Number of cells depth Depth of bottom [Z ~> m] or arbitrary units h Cell thicknesses [Z ~> m] or arbitrary units dzinterface Change in interface positions (same units as h) msg Message to append to errors mom_error_handler::mom_error check_remapping_grid subroutine subroutine mom_regridding::filtered_grid_motion (CS, nk, z_old, z_new, dz_g) filtered_grid_motion CS CS nk nk z_old z_old z_new z_new dz_g dz_g Returns the change in interface position motion after filtering and assuming the top and bottom interfaces do not move. The filtering is a function of depth, and is applied as the integrated average filtering over the trajectory of the interface. By design, this code can not give tangled interfaces provided that z_old and z_new are not already tangled. cs Regridding control structure nk Number of cells in source grid z_old Old grid position [H ~> m or kg m-2] z_new New grid position [H ~> m or kg m-2] dz_g Change in interface positions [H ~> m or kg m-2] mom_error_handler::mom_error build_grid_adaptive build_grid_hycom1 build_grid_slight build_rho_grid build_sigma_grid build_zstar_grid subroutine subroutine mom_regridding::build_zstar_grid (CS, G, GV, h, dzInterface, frac_shelf_h) build_zstar_grid CS CS G G GV GV h h dzInterface dzInterface frac_shelf_h frac_shelf_h Builds a z*-ccordinate grid with partial steps (Adcroft and Campin, 2004). z* is defined as z* = (z-eta)/(H+eta)*H s.t. z*=0 when z=eta and z*=-H when z=-H . cs Regridding control structure g Ocean grid structure gv ocean vertical grid structure h Layer thicknesses [H ~> m or kg m-2] dzinterface The change in interface depth [H ~> m or kg m-2]. frac_shelf_h Fractional ice shelf coverage [nondim]. adjust_interface_motion coord_zlike::build_zstar_column filtered_grid_motion mom_error_handler::mom_error regridding_main subroutine subroutine mom_regridding::build_sigma_grid (CS, G, GV, h, dzInterface) build_sigma_grid CS CS G G GV GV h h dzInterface dzInterface This routine builds a grid based on terrain-following coordinates. cs Regridding control structure g Ocean grid structure gv ocean vertical grid structure h Layer thicknesses [H ~> m or kg m-2] dzinterface The change in interface depth [H ~> m or kg m-2] coord_sigma::build_sigma_column filtered_grid_motion mom_error_handler::mom_error regridding_main subroutine subroutine mom_regridding::build_rho_grid (G, GV, US, h, tv, dzInterface, remapCS, CS) build_rho_grid G G GV GV US US h h tv tv dzInterface dzInterface remapCS remapCS CS CS This routine builds a new grid based on a given set of target interface densities. g Ocean grid structure gv Ocean vertical grid structure us A dimensional unit scaling type h Layer thicknesses [H ~> m or kg m-2] tv Thermodynamics structure dzinterface The change in interface depth [H ~> m or kg m-2] remapcs The remapping control structure cs Regridding control structure filtered_grid_motion mom_error_handler::mom_error regridding_main subroutine subroutine mom_regridding::build_grid_hycom1 (G, GV, US, h, tv, h_new, dzInterface, CS) build_grid_hycom1 G G GV GV US US h h tv tv h_new h_new dzInterface dzInterface CS CS Builds a simple HyCOM-like grid with the deepest location of potential density interpolated from the column profile and a clipping of depth for each interface to a fixed z* or p* grid. This should probably be (optionally?) changed to find the nearest location of the target density. { Based on Bleck, 2002: An oceanice general circulation model framed in hybrid isopycnic-Cartesian coordinates, Ocean Modelling 37, 55-88. http://dx.doi.org/10.1016/S1463-5003(01)00012-9 } g Grid structure gv Ocean vertical grid structure us A dimensional unit scaling type h Existing model thickness [H ~> m or kg m-2] tv Thermodynamics structure cs Regridding control structure h_new New layer thicknesses [H ~> m or kg m-2] dzinterface Changes in interface position adjust_interface_motion coord_hycom::build_hycom1_column calc_h_new_by_dz filtered_grid_motion mom_error_handler::mom_error regridding_main subroutine subroutine mom_regridding::build_grid_adaptive (G, GV, US, h, tv, dzInterface, remapCS, CS) build_grid_adaptive G G GV GV US US h h tv tv dzInterface dzInterface remapCS remapCS CS CS This subroutine builds an adaptive grid that follows density surfaces where possible, subject to constraints on the smoothness of interface heights. g The ocean's grid structure gv The ocean's vertical grid structure us A dimensional unit scaling type h Layer thicknesses [H ~> m or kg m-2] tv A structure pointing to various thermodynamic variables dzinterface The change in interface depth [H ~> m or kg m-2] remapcs The remapping control structure cs Regridding control structure adjust_interface_motion coord_adapt::build_adapt_column filtered_grid_motion regridding_main subroutine subroutine mom_regridding::build_grid_slight (G, GV, US, h, tv, dzInterface, CS) build_grid_slight G G GV GV US US h h tv tv dzInterface dzInterface CS CS Builds a grid that tracks density interfaces for water that is denser than the surface density plus an increment of some number of layers, and uses all lighter layers uniformly above this location. Note that this amounts to interpolating to find the depth of an arbitrary (non-integer) interface index which should make the results vary smoothly in space to the extent that the surface density and interior stratification vary smoothly in space. Over shallow topography, this will tend to give a uniform sigma-like coordinate. For sufficiently shallow water, a minimum grid spacing is used to avoid certain instabilities. g Grid structure gv Ocean vertical grid structure us A dimensional unit scaling type h Existing model thickness [H ~> m or kg m-2] tv Thermodynamics structure dzinterface Changes in interface position cs Regridding control structure adjust_interface_motion coord_slight::build_slight_column filtered_grid_motion mom_error_handler::mom_error regridding_main subroutine subroutine mom_regridding::adjust_interface_motion (CS, nk, h_old, dz_int) adjust_interface_motion CS CS nk nk h_old h_old dz_int dz_int Adjust dz_Interface to ensure non-negative future thicknesses. cs Regridding control structure nk Number of layers in h_old h_old Minium allowed thickness of h [H ~> m or kg m-2] dz_int Minium allowed thickness of h [H ~> m or kg m-2] mom_error_handler::mom_error build_grid_adaptive build_grid_hycom1 build_grid_slight build_zstar_grid subroutine subroutine mom_regridding::build_grid_arbitrary (G, GV, h, dzInterface, h_new, CS) build_grid_arbitrary G G GV GV h h dzInterface dzInterface h_new h_new CS CS g Ocean grid structure gv Ocean vertical grid structure h Original layer thicknesses [H ~> m or kg m-2] dzinterface The change in interface depth [H ~> m or kg m-2] h_new New layer thicknesses [H ~> m or kg m-2] cs Regridding control structure regridding_main subroutine, public subroutine, public mom_regridding::inflate_vanished_layers_old (CS, G, GV, h) inflate_vanished_layers_old CS CS G G GV GV h h cs Regridding control structure g The ocean's grid structure gv The ocean's vertical grid structure h Layer thicknesses [H ~> m or kg m-2] coord_rho::old_inflate_layers_1d subroutine subroutine mom_regridding::convective_adjustment (G, GV, h, tv) convective_adjustment G G GV GV h h tv tv Achieve convective adjustment by swapping layers. g The ocean's grid structure gv The ocean's vertical grid structure h Layer thicknesses [H ~> m or kg m-2] tv A structure pointing to various thermodynamic variables regridding_main real function, dimension(nk), public real function, dimension(nk), public mom_regridding::uniformresolution (nk, coordMode, maxDepth, rhoLight, rhoHeavy) uniformresolution nk nk coordMode coordMode maxDepth maxDepth rhoLight rhoLight rhoHeavy rhoHeavy Return a uniform resolution vector in the units of the coordinate. nk Number of cells in source grid coordmode A string indicating the coordinate mode. See the documenttion for regrid_consts for the recognized values. maxdepth The range of the grid values in some modes rholight The minimum value of the grid in RHO mode rhoheavy The maximum value of the grid in RHO mode The returned uniform resolution grid. mom_error_handler::mom_error regrid_consts::regridding_adaptive regrid_consts::regridding_hycom1 regrid_consts::regridding_slight initialize_regridding subroutine subroutine mom_regridding::initcoord (CS, GV, US, coord_mode) initcoord CS CS GV GV US US coord_mode coord_mode Initialize the coordinate resolutions by calling the appropriate initialization routine for the specified coordinate mode. cs Regridding control structure coord_mode A string indicating the coordinate mode. See the documentation for regrid_consts for the recognized values. gv Ocean vertical grid structure us A dimensional unit scaling type coord_adapt::init_coord_adapt coord_hycom::init_coord_hycom coord_sigma::init_coord_sigma coord_slight::init_coord_slight coord_zlike::init_coord_zlike regrid_consts::regridding_adaptive regrid_consts::regridding_hycom1 regrid_consts::regridding_slight initialize_regridding subroutine, public subroutine, public mom_regridding::setcoordinateresolution (dz, CS, scale) setcoordinateresolution dz dz CS CS scale scale Set the fixed resolution data. dz A vector of vertical grid spacings cs Regridding control structure scale A scaling factor converting dz to coordRes mom_error_handler::mom_error initialize_regridding subroutine, public subroutine, public mom_regridding::set_target_densities_from_gv (GV, US, CS) set_target_densities_from_gv GV GV US US CS CS Set target densities based on the old Rlay variable. gv Ocean vertical grid structure us A dimensional unit scaling type cs Regridding control structure initialize_regridding subroutine, public subroutine, public mom_regridding::set_target_densities (CS, rho_int) set_target_densities CS CS rho_int rho_int Set target densities based on vector of interface values. cs Regridding control structure rho_int Interface densities [R ~> kg m-3] mom_error_handler::mom_error initialize_regridding subroutine, public subroutine, public mom_regridding::set_regrid_max_depths (CS, max_depths, units_to_H) set_regrid_max_depths CS CS max_depths max_depths units_to_H units_to_H Set maximum interface depths based on a vector of input values. cs Regridding control structure max_depths Maximum interface depths, in arbitrary units units_to_h A conversion factor for max_depths into H units mom_error_handler::mom_error regrid_consts::regridding_hycom1 regrid_consts::regridding_slight coord_hycom::set_hycom_params coord_slight::set_slight_params initialize_regridding subroutine, public subroutine, public mom_regridding::set_regrid_max_thickness (CS, max_h, units_to_H) set_regrid_max_thickness CS CS max_h max_h units_to_H units_to_H Set maximum layer thicknesses based on a vector of input values. cs Regridding control structure max_h Maximum interface depths, in arbitrary units units_to_h A conversion factor for max_h into H units regrid_consts::regridding_hycom1 regrid_consts::regridding_slight coord_hycom::set_hycom_params coord_slight::set_slight_params initialize_regridding real function, dimension(cs%nk), public real function, dimension(cs%nk), public mom_regridding::getcoordinateresolution (CS, undo_scaling) getcoordinateresolution CS CS undo_scaling undo_scaling Query the fixed resolution data. cs Regridding control structure undo_scaling If present and true, undo any internal rescaling of the resolution data. real function, dimension(cs%nk+1), public real function, dimension(cs%nk+1), public mom_regridding::getcoordinateinterfaces (CS, undo_scaling) getcoordinateinterfaces CS CS undo_scaling undo_scaling Query the target coordinate interface positions. cs Regridding control structure undo_scaling If present and true, undo any internal rescaling of the resolution data. Interface positions in target coordinate mom_error_handler::mom_error character(len=20) function, public character(len=20) function, public mom_regridding::getcoordinateunits (CS) getcoordinateunits CS CS Query the target coordinate units. cs Regridding control structure mom_error_handler::mom_error regrid_consts::regridding_adaptive regrid_consts::regridding_hycom1 regrid_consts::regridding_slight character(len=20) function, public character(len=20) function, public mom_regridding::getcoordinateshortname (CS) getcoordinateshortname CS CS Query the short name of the coordinate. cs Regridding control structure mom_error_handler::mom_error regrid_consts::regridding_adaptive regrid_consts::regridding_hycom1 regrid_consts::regridding_slight subroutine, public subroutine, public mom_regridding::set_regrid_params (CS, boundary_extrapolation, min_thickness, old_grid_weight, interp_scheme, depth_of_time_filter_shallow, depth_of_time_filter_deep, compress_fraction, ref_pressure, dz_min_surface, nz_fixed_surface, Rho_ML_avg_depth, nlay_ML_to_interior, fix_haloclines, halocline_filt_len, halocline_strat_tol, integrate_downward_for_e, remap_answers_2018, adaptTimeRatio, adaptZoom, adaptZoomCoeff, adaptBuoyCoeff, adaptAlpha, adaptDoMin, adaptDrho0) set_regrid_params CS CS boundary_extrapolation boundary_extrapolation min_thickness min_thickness old_grid_weight old_grid_weight interp_scheme interp_scheme depth_of_time_filter_shallow depth_of_time_filter_shallow depth_of_time_filter_deep depth_of_time_filter_deep compress_fraction compress_fraction ref_pressure ref_pressure dz_min_surface dz_min_surface nz_fixed_surface nz_fixed_surface Rho_ML_avg_depth Rho_ML_avg_depth nlay_ML_to_interior nlay_ML_to_interior fix_haloclines fix_haloclines halocline_filt_len halocline_filt_len halocline_strat_tol halocline_strat_tol integrate_downward_for_e integrate_downward_for_e remap_answers_2018 remap_answers_2018 adaptTimeRatio adaptTimeRatio adaptZoom adaptZoom adaptZoomCoeff adaptZoomCoeff adaptBuoyCoeff adaptBuoyCoeff adaptAlpha adaptAlpha adaptDoMin adaptDoMin adaptDrho0 adaptDrho0 Can be used to set any of the parameters for MOM_regridding. cs Regridding control structure boundary_extrapolation Extrapolate in boundary cells min_thickness Minimum thickness allowed when building the new grid [H ~> m or kg m-2] old_grid_weight Weight given to old coordinate when time-filtering grid interp_scheme Interpolation method for state-dependent coordinates depth_of_time_filter_shallow Depth to start cubic [H ~> m or kg m-2] depth_of_time_filter_deep Depth to end cubic [H ~> m or kg m-2] compress_fraction Fraction of compressibility to add to potential density [nondim] ref_pressure The reference pressure for density-dependent coordinates [R L2 T-2 ~> Pa] dz_min_surface The fixed resolution in the topmost SLight_nkml_min layers [H ~> m or kg m-2] nz_fixed_surface The number of fixed-thickness layers at the top of the model rho_ml_avg_depth Averaging depth over which to determine mixed layer potential density [H ~> m or kg m-2] nlay_ml_to_interior Number of layers to offset the mixed layer density to find resolved stratification [nondim] fix_haloclines Detect regions with much weaker stratification in the coordinate halocline_filt_len Length scale over which to filter T & S when looking for spuriously unstable water mass profiles [H ~> m or kg m-2] halocline_strat_tol Value of the stratification ratio that defines a problematic halocline region. integrate_downward_for_e If true, integrate for interface positions downward from the top. remap_answers_2018 If true, use the order of arithmetic and expressions that recover the remapping answers from 2018. Otherwise use more robust but mathematically equivalent expressions. adapttimeratio Ratio of the ALE timestep to the grid timescale [nondim]. adaptzoom Depth of near-surface zooming region [H ~> m or kg m-2]. adaptzoomcoeff Coefficient of near-surface zooming diffusivity [nondim]. adaptbuoycoeff Coefficient of buoyancy diffusivity [nondim]. adaptalpha Scaling factor on optimization tendency [nondim]. adaptdomin If true, make a HyCOM-like mixed layer by preventing interfaces from being shallower than the depths specified by the regridding coordinate. adaptdrho0 Reference density difference for stratification-dependent diffusion. [R ~> kg m-3] mom_error_handler::mom_error regrid_consts::regridding_adaptive regrid_consts::regridding_hycom1 regrid_consts::regridding_slight coord_adapt::set_adapt_params coord_hycom::set_hycom_params regrid_interp::set_interp_extrap regrid_interp::set_interp_scheme coord_sigma::set_sigma_params coord_slight::set_slight_params coord_zlike::set_zlike_params mom_oda_driver_mod::init_oda initialize_regridding integer function, public integer function, public mom_regridding::get_regrid_size (CS) get_regrid_size CS CS Returns the number of levels/layers in the regridding control structure. cs Regridding control structure type(zlike_cs) function, public type(zlike_cs) function, public mom_regridding::get_zlike_cs (CS) get_zlike_cs CS CS This returns a copy of the zlike_CS stored in the regridding control structure. cs Regridding control structure mom_diag_remap::diag_remap_update type(sigma_cs) function, public type(sigma_cs) function, public mom_regridding::get_sigma_cs (CS) get_sigma_cs CS CS This returns a copy of the sigma_CS stored in the regridding control structure. cs Regridding control structure mom_diag_remap::diag_remap_update type(rho_cs) function, public type(rho_cs) function, public mom_regridding::get_rho_cs (CS) get_rho_cs CS CS This returns a copy of the rho_CS stored in the regridding control structure. cs Regridding control structure mom_diag_remap::diag_remap_update real function, dimension(cs%nk), public real function, dimension(cs%nk), public mom_regridding::getstaticthickness (CS, SSH, depth) getstaticthickness CS CS SSH SSH depth depth Return coordinate-derived thicknesses for fixed coordinate systems. cs Regridding control structure ssh The sea surface height, in the same units as depth depth The maximum depth of the grid, often [Z ~> m] The returned thicknesses in the units of depth mom_error_handler::mom_error regrid_consts::regridding_adaptive regrid_consts::regridding_hycom1 regrid_consts::regridding_slight mom_ale::ale_initthicknesstocoord subroutine subroutine mom_regridding::dz_function1 (string, dz) dz_function1 string string dz dz Parses a string and generates a dz(:) profile that goes like k**power. string String with list of parameters in form dz_min, H_total, power, precision dz Profile of nominal thicknesses mom_error_handler::mom_error initialize_regridding integer function integer function mom_regridding::rho_function1 (string, rho_target) rho_function1 string string rho_target rho_target Parses a string and generates a rho_target(:) profile with refined resolution downward and returns the number of levels. string String with list of parameters in form dz_min, H_total, power, precision rho_target Profile of interface densities [kg m-3] initialize_regridding Generates vertical grids as part of the ALE algorithm. A vertical grid is defined solely by the cell thicknesses, $h$. Most calculations in this module start with the coordinate at the bottom of the column set to -depth, and use a increasing value of coordinate with decreasing k. This is consistent with the rest of MOM6 that uses position, $z$ which is a negative quantity for most of the ocean. A change in grid is define through a change in position of the interfaces: \[ z^n_{k+1/2} = z^{n-1}_{k+1/2} + \Delta z_{k+1/2} \] with the positive upward coordinate convention \[ z_{k-1/2} = z_{k+1/2} + h_k \] so that \[ h^n_k = h^{n-1}_k + ( \Delta z_{k-1/2} - \Delta z_{k+1/2} ) \] Original date of creation: 2008.06.09 by L. White