grdflexure(1) GMT grdflexure(1)

## NAME

grdflexure - Compute flexural deformation of 3-D surfaces for various rheologies

## SYNOPSIS

grdflexuretopogrd-Drm/rl[/ri]/rw-ETe[u]-Goutgrid[-ANx/Ny/Nxy] [-Cppoisson] [-CyYoung] [-Fnu_a[/h_a/nu_m] ] [-Llist] [-N[f|q|s|nx/ny][+a|d|h|l][+e|n|m][+twidth][+w[suffix]][+z[p]] [-Sbeta] [-Tt0[u][/t1[u]/dt[u]|file] |n][+l] ] [-V[level] ] [-Wwd] [-Zzm] [-fg]Note:No space is allowed between the option flag and the associated arguments.

## DESCRIPTION

grdflexurecomputes the flexural response to loads using a range of user-selectable rheologies. User may select from elastic, viscoelas- tic, or firmoviscous (with one or two viscous layers). Temporal evolu- tion can also be modeled by providing incremental load grids and speci- fying a range of model output times.

## REQUIRED ARGUMENTS

topogrd2-D binary grid file with the topography of the load (in meters); See GRID FILE FORMATS below. If-Tis used,topogrdmay be a filename template with a floating point format (C syn- tax) and a different load file name will be set and loaded for each time step. The load times thus coincide with the times given via-T(but not all times need to have a corresponding file). Alternatively, givetopogrdas =flist, whereflistis an ASCII table with onetopogrdfilename and load time per record. These load times can be different from the evaluation times given via-T. For load time format, see-T.-Drm/rl[/ri]/rwSets density for mantle, load, infill (optional, otherwise it is assumed to equal the load density), and water or air. Ifridiffers fromrlthen an approximate solution will be found. Ifriis not given then it defaults torl.-ETeSets the elastic plate thickness (in meter); appendkfor km. If the elastic thickness exceeds 1e10 it will be interpreted as a flexural rigidity D (by default D is computed fromTe, Youngas modulus, and Poissonas ratio; see-Cto change these values).-GoutfileIf-Tis set thengrdfilemust be a filename template that con- tains a floating point format (C syntax). If the filename tem- plate also contains either %s (for unit name) or %c (for unit letter) then we use the corresponding time (in units specified in-T) to generate the individual file names, otherwise we use time in years with no unit.

## OPTIONAL ARGUMENTS

-ANx/Ny/NxySpecify in-plane compressional or extensional forces in the x- and y-directions, as well as any shear force [no in-plane forces]. Compression is indicated by negative values, while extensional forces are specified using positive values.-CppoissonChange the current value of Poissonas ratio [0.25].-CyYoungChange the current value of Youngas modulus [7.0e10 N/m^2].-Fnu_a[/h_a/nu_m] Specify a firmoviscous model in conjunction with an elastic plate thickness specified via-E. Just give one viscosity (nu_a) for an elastic plate over a viscous half-space, or also append the thickness of the asthenosphere (h_a) and the lower mantle viscosity (nu_m), with the first viscosity now being that of the asthenosphere. Give viscosities in Pa*s. If used, give the thickness of the asthenosphere in meter; appendkfor km.-N[a|f|m|r|s|nx/ny][+a|[+d|h|l][+e|n|m][+twidth][+v][+w[suffix]][+z[p]] Choose or inquire about suitable grid dimensions for FFT and set optional parameters. Control the FFT dimension:-Nalets the FFT select dimensions yielding the most accurate result.-Nfwill force the FFT to use the actual dimensions of the data.-Nmlets the FFT select dimensions using the least work mem- ory.-Nrlets the FFT select dimensions yielding the most rapid calculation.-Nswill present a list of optional dimensions, then exit.-Nnx/nywill do FFT on array sizenx/ny(must be >= grid file size). Default chooses dimensions >= data which optimize speed and accuracy of FFT. If FFT dimensions > grid file dimensions, data are extended and tapered to zero. Control detrending of data: Append modifiers for removing a lin- ear trend:+d: Detrend data, i.e. remove best-fitting linear trend [Default].+a: Only remove mean value.+h: Only remove mid value, i.e. 0.5 * (max + min).+l: Leave data alone. Control extension and tapering of data: Use modifiers to control how the extension and tapering are to be performed:+eextends the grid by imposing edge-point symmetry [Default],+mextends the grid by imposing edge mirror symmetry+nturns off data extension. Tapering is performed from the data edge to the FFT grid edge [100%]. Change this percentage via+twidth. When+nis in effect, the tapering is applied instead to the data margins as no extension is available [0%]. Control messages being reported:+vwill report suitable dimensions during processing. Control writing of temporary results: For detailed investigation you can write the intermediate grid being passed to the forward FFT; this is likely to have been detrended, extended by point-symmetry along all edges, and tapered. Append+w[suffix] from which output file name(s) will be created (i.e.,ingrid_prefix.ext) [tapered], whereextis your file extension. Finally, you may save the complex grid produced by the forward FFT by appending+z. By default we write the real and imaginary components toingrid_real.extandingrid_imag.ext. Appendpto save instead the polar form of magnitude and phase to filesingrid_mag.extandingrid_phase.ext.-LlistWrite the names and evaluation times of all grids that were cre- ated to the text filelist. Requires-T.-MtmSpecify a viscoelastic model in conjunction with an elastic plate thickness specified via-E. Append the Maxwell timetmfor the viscoelastic model (in ).-SbetaSpecify a starved moat fraction in the 0-1 range, where 1 means the moat is fully filled with material of densityriwhile 0 means it is only filled with material of densityrw(i.e., just water) [1].-Tt0[u][/t1[u]/dt[u]|file]|n][+l] Specifyt0,t1, and time increment (dt) for sequence of calcula- tions [Default is one step, with no time dependency]. For a single specific time, just give start timet0. The unit is years; appendkfor kyr andMfor Myr. For a logarithmic time scale, append+land specifynsteps instead ofdt. Alterna- tively, give afilewith the desired times in the first column (these times may have individual units appended, otherwise we assume year). We then write a separate model grid file for each given time step.-WwdSet reference depth to the undeformed flexed surface in m [0]. Appendkto indicate km.-ZzmSpecify reference depth to flexed surface (e.g., Moho) in m; appendkfor km. Must be positive. [0].-V[level] (morea|) Select verbosity level [c].-fgGeographic grids (dimensions of longitude, latitude) will be converted to meters via aaFlat Earthaapproximation using the current ellipsoid parameters.-^or just-Print a short message about the syntax of the command, then exits (NOTE: on Windows just use-).-+or just+Print an extensive usage (help) message, including the explana- tion of any module-specific option (but not the GMT common options), then exits.-?or no arguments Print a complete usage (help) message, including the explanation of all options, then exits.

## GRID FILE FORMATS

By default GMT writes out grid as single precision floats in a COARDS-complaint netCDF file format. However, GMT is able to produce grid files in many other commonly used grid file formats and also facilitates so calledapackingaof grids, writing out floating point data as 1- or 2-byte integers. (morea|)

## GRID DISTANCE UNITS

If the grid does not have meter as the horizontal unit, append+uunitto the input file name to convert from the specified unit to meter. If your grid is geographic, convert distances to meters by supplying-fginstead.

## CONSIDERATIONS

netCDF COARDS grids will automatically be recognized as geographic. For other grids geographical grids were you want to convert degrees into meters, select-fg. If the data are close to either pole, you should consider projecting the grid file onto a rectangular coordinate system using grdproject.

## PLATE FLEXURE NOTES

The FFT solution to plate flexure requires the infill density to equal the load density. This is typically only true directly beneath the load; beyond the load the infill tends to be lower-density sediments or even water (or air). Wessel [2001, 2016] proposed an approximation that allows for the specification of an infill density different from the load density while still allowing for an FFT solution. Basically, the plate flexure is solved for using the infill density as the effec- tive load density but the amplitudes are adjusted by the factorA= sqrt ((rm - ri)/(rm - rl)), which is the theoretical difference in amplitude due to a point load using the two different load densities. The approximation is very good but breaks down for large loads on weak plates, a fairy uncommon situation.

## EXAMPLES

To compute elastic plate flexure from the loadtopo.nc, for a 10 km thick plate with typical densities, try gmt grdflexure topo.nc -Gflex.nc -E10k -D2700/3300/1035 To compute the firmoviscous response to a series of incremental loads given by file name and load time in the table l.lis at the single time 1 Ma using the specified rheological values, try gmt grdflexure -T1M =l.lis -D3300/2800/2800/1000 -E5k -Gflx/smt_fv_%03.1f_%s.nc -F2e20 -Nf+a

## REFERENCES

Cathles, L. M., 1975,Theviscosityoftheearth^<i>asmantle, Princeton University Press. Wessel. P., 2001, Global distribution of seamounts inferred from grid- ded Geosat/ERS-1 altimetry, J. Geophys. Res., 106(B9), 19,431-19,441,. Wessel, P., 2016, Regionalhttp://dx.doi.org/10.1029/2000JB000083aresidual separation of bathymetry and revised estimates of Hawaii plume flux,Geophys.J.Int.,204(2), 932-947,.http://dx.doi.org/10.1093/gji/ggv472

## SEE ALSO

gmt(1),grdfft(1),gravfft(1),grdmath(1),grdproject(1),grdseamount(1)

## COPYRIGHT

2017, P. Wessel, W. H. F. Smith, R. Scharroo, J. Luis, and F. Wobbe 5.4.2 Jun 24, 2017 grdflexure(1)

gmt5 5.4.2 - Generated Wed Jun 28 18:27:14 CDT 2017