The University of Miami Wave Model (UMWM) Version 1.0.0
Contents
1. 25 TRUE FALSE 45 Von Karman constant Height of the input wind speed m Random wind gustiness factor should be between O and 0 2 Depth limiter m Exponent limiter 0 69 100 growth Input factor from following winds Damping factor from opposing winds Damping factor from swell overrunning wind Breaking dissipation factor Saturation spectrum power Mean square slope adjustment to Sds Wave energy downshifting factor Dissipation due to turbulence factor Bottom friction coefficient m s Bottom percolation coefficient m s Set 1 for Cartesian or 2 for spherical projection Grid spacing in x and y in meters for gridid 1 degrees for gridid 2 Latitude of bottom left point used for gridid 2 Longitude of bottom left point used for gridid 2 Set to 1 for input from file or 0 for homogenous fields input from amp INPUT list below Bathymetry Set to 1 to exclude user chosen basins lakes Winds Surface pressure Air temperature Currents Water density Water depth m Wind magnitude m s Wind direction rad Surface pressure Pa Surface air temperature K x component ocean current m s y component ocean current m s Water density kg m 3 Gridded output Spectrum output Grid cell in x for stdout screen Grid cell in y for stdout screen 11 restart_out FALSE Restart output Parameters in the DOMAIN list define domain size and start2. 0 529 Uio fe 4fpm 3 16 For all bins higher than fe the waves are assumed to be in equilibrium with the wind traveling in the wind direction and their spectral densities are established from a balance of wind input and dissipation This approach is justified by the presumption that the quasi equilibrium range is wider in higher wind conditions 4 Stress calculation Energy E and momentum M in the wave field are related by the phase speed E Mc 4 1 The form drag components of the wind on the waves Ty and Ty are calculated from the wind input source function T Emas i Ta pos f Sin cos k dk do 4 2 min C T km z H Ty pos Sin sin k dk d 4 3 min The form drag coefficient Cd is then calculated TEETE Cd LL 4 4 pa where U is the wind speed at the measured or modeled height z The form drag is in the wave direction The skin drag coefficient Cds in the absence of waves is in the wind direction 4 and is computed from Von Karman s relationship 2 k2 a 4 5 AO where u is the friction velocity U z is wind speed at height z and is the Von Karman s constant The roughness length zp is determined by the scale of the molecular sublayer for smooth flow zo 0 1322 4 6 where Va is the kinematic viscosity of air In the presence of waves Cd is reduced due to the sheltering of the surface in the lee of steep waves The degree of sheltering
3. a gas It proceeds through a numerical solution of the wave energy balance equation on a horizontal 2 dimensional grid The wave energy is a positive definite quantity in logarithmically spaced frequency f bins and uniformly spaced directional bins The energy spectrum is carried as the wavenumber directional surface elevation variance spectrum and every frequency at each location is identified with the theoretical wavenumber Thus wave energy is a 5 dimensional quantity y k t Time t evolution is done by integrating in time the energy balance equation OE O kE O kE O GE a 1 1 t Ox Ok where E x k t is the energy spectrum and x c u where cg is the group velocity and u is the current in the wave boundary layer The second third and fourth terms on the left hand side are the advection terms in geographical wavenumber and directional space respectively Advection in wavenumber space is non zero only in case of changing ocean currents or moving bottom Here currents are considered to be quasi stationary on the time scale of wave growth decay Thus this term is neglected Advection in directional space refraction is non zero in case of variable water n depth and or inhomogeneous currents 5 S are the source sink functions that act to grow decay i 1 the waves locally pw is the liquid density and g is acceleration due to gravity The energy spectrum and the variance spectrum
4. and end times of the simulation and whether the model is to be run from a restart file See next section about restarting the model Parameters in the PHYSICS list will have effects on the physical results of the wave model Height of the input wind speed z may need to be changed depending on the source of wind fields used as model forcing The user may choose to set the gustiness parameter larger than zero Since the wind is linearly interpolated in time between each input time interval dtg this option may be used to add a stochastic component to wind fields when dtg is not short enough e g 3 or 6 hours Depth limiter dmin defines the smallest depth possible inside the model Values lower than 10 meters are allowed but may lead to a shorter time step and longer model integration time explim is the exponential growth limiter used when integrating wave spectrum in time Typically it should be between 0 4 50 growth and 0 9 140 growth Higher values may lead to numerical oscillations and instability of the model The parameters in INPUT_FLAGS list determine which fields are to be input from file For fields that are not read from input files homogeneous values are specified in INPUT list If excom is set to 1 a namelist file namelists exclude nml is being read and this file contains a list of model grid points that belong to enclosed seas that should be removed from the model domain Finally OUTPUT list specifies whether there will be gridded and
5. are related by E x k pug E x k 1 2 For convenience the variance spectrum is the predicted quantity 2 Source functions The source functions S are parametric descriptions of the various phenomena that increase decrease or interchange among wavenumbers the energy in the wavenumber spectrum The wavenumber spectrum is evaluated in separable magnitude and direction bins The phenomena relevant to the prediction of storm waves in water of arbitrary depth are i Input of energy and momentum from the wind and export of wave energy and momentum to the wind when the waves overrun or run against the wind ii Dissipation of wave energy at and near the surface due to viscosity ambient turbulence and breaking iii Enhanced dissipation at and near the top and bottom interfaces due to shoaling iv Movement of energy to lower wavenumbers down shifting due to nonlinear interactions includ ing breaking v Enhanced dissipation due to straining by longer waves The theoretical and experimental justifications for the source functions corresponding to these phe nomena are given in Donelan et al 2012 Here the source functions are simply listed 2 1 Wind input function Sin Jeffreys 1924 1925 sheltering hypothesis leads to Sin of the form kw pa Sin Ar Uz j2 cos 8 c ucos v sin Uy 2 cos H c ucos v sin eat 2 1 where 0 is the angle between wind direction 7 and waves of wa
6. are stored in the same file 12 Finally parameters xp1 and ypl specify the model grid location for the simple output to screen time time step significant wave height mean wave period and drag coefficient After configuring the namelists user must provide the grid definition file This file must be written in NetCDF form and placed in input umwm gridtopo The umwm gridtopo file must contain 3 two dimensional fields lon lat and z Fields lon and lat will define the domain for the simulation Field z is the topography field where the 0 value is centered on mean sea level with positive z axis upward positive on land and negative in the ocean The model will read this field define seamask discard the land values and apply the depth limiter dmin Optionally the user may provide forcing files for wind sea level pressure air temperature surface ocean currents and water density fields Input files must be written in NetCDF form and placed in input umwmin_YYYY MM DD_hh mm ss nc These files will be read if any of the wflag pflag tflag or cflag are set to 1 and the files must contain following respective two dimensional fields U10 and V10 PSFC T2 UC and VO 7 Running the model Once namelists have been configured and input files are provided the user can start the simulation by executing umwm If the model was compiled in the MPI mode refer
7. k d 2 3 where x2 k is the sum squared slope in direction of all waves longer than k and v is the kinematic viscosity of the liquid Coefficients A gt and A3 have values 42 and 120 respectively 2 3 Dissipation by turbulence source function Sj Turbulence mixing in the wave boundary layer attenuates waves The wave dissipation due to ambient turbulence is Sat daa KELL 0 2 4 where ux is the friction velocity in the water near the surface and A 0 01 2 4 The non linear wave wave interaction function Sp A quantity proportional to the energy dissipated in spilling is passed to longer waves in the next two lower frequency bins The amount transferred decays exponentially with the square of the relative frequency separation Snilk o a As b1 Ssy k a Ak o F ba Ssb k 2Ak 0 9 Ssp k 2 5 A A where As 5 b1 exp 16 42 2 b2 ap 1025 and b and bz are normalized such that f by bg 1 2 5 The bottom friction source function Sp The bottom friction source function is related to orbital velocity at the bottom and the roughness of the bed Komen et al 1994 give the following form for a sandy bed k A a E k 0 2 6 where the roughness factor Gy varies from 0 001 to 0 01 m s depending on bed roughness 2 6 The bottom percolation source function Sep On a porous bed the percolation of flow through the bed induces wave energy dissipation The dissipation rate due to percolatio
8. spectrum output and or whether output of restart files is enabled If restart_out is set to TRUE restart files will be written in the restart directory If any of outgrid or outspec in is set to TRUE desired output with the frequency of value of dtg will appear in the output directory Gridded output will appear in the form umwmout_YYYY MM DD_hh mm ss nc where YYYY MM DD_hh mm ss denotes model time of the output If outspec is set to TRUE a namelist file namelists spectrum nml is being read An example spectrum namelist file may look like this LL 89 66 25 89 NDBC42001 93 67 25 79 NDBC42002 96 66 25 96 NDBC42020 94 41 29 23 NDBC42035 88 80 30 09 NDBC42007 The first line of the file must contain either XY or LL It describes whether the coordinates of spectrum output location are given in model grid indices or in longitude and latitude respectively In case of XY spectrum at exact specified grid cell will be output In case of LL spectrum at grid cell nearest to specified location will be output The following lines each contain 3 elements x and y coordinate or longitude and latitude and location identifier character string up to 40 characters no spaces Spectrum output will appear in the form umwmspc_ID_YYYY MM DD_hh mm ss nc where ID is the location identifier specified by the user in namelists spectrum nml and YYYY MM DD_hh mm ss is the initial time of the simulation All output times during the simulation
9. The University of Miami Wave Model UMWM Version 1 0 0 Description and User s Manual M A Donelan and M Curcic April 2012 Rosenstiel School of Marine and Atmospheric Science University of Miami Contents 1 Model description 2 2 Source functions 2 21 Wind input function al os aa a ee a a eee 3 2 2 Wave dissipation function a oe oe eo e a a a 3 2 3 Dissipation by turbulence source function Sds e e 4 2 4 The non linear wave wave interaction function Sn A Gaba ee ah ad a Be os 4 2 5 The bottom friction source function RL I 4 2 6 The bottom percolation source function Sbp o e ee 4 3_ Numerical approaches 5 3 1 Spatial discretization oo 5 3 2 Tim discretization gt s g 4 p hoen e a ee Ge a eS 6 4 Stress calculation 7 5 Software implementation 8 6 Installation and setup 9 6 1 Obtaining the software package 2 a a a 9 6 2 Compiling the source code 2 a 9 6 3 Setting up the simulation 0 20 2 0 00000 eee ee 10 7 Running the model 13 7 1 Restarting the model 2 ee 13 8 Reading model output datal 13 9 References 14 1 Model description The University of Miami Wave Model UMWM is a prediction model for wave energy and wind stress on the interface between a liquid and
10. e y is the Courant number Depending on the choice of number of directional bins the stability criterion is more permissive At T Ad tae ag G i 3 7 where Ag is the directional bin size To ensure that 3 7 holds the number of directions must be divisible by 8 For ocean grid cells next to the land an open boundary condition is applied energy can freely propagate into land Same is applied for domain edges except that there can be incoming wave energy from the boundaries if provided by the user In the case of global domain simulation periodic boundary conditions are applied at East and West domain edges The rotation rate db in the refraction term is evaluated as csing v O ccosp u Ox Oy b 3 8 The change due to refraction is then computed using 3 2 3 5 Positive and negative values of correspond to counter clockwise and clockwise rotation of energy respectively The stability constraint for the refraction term is the same as for one dimensional advection _ At For most domain cells the allowed refraction time step is larger than the advective step given in 3 7 In case that condition 3 9 is violated which can occur on sharp bathymetric or current gradients the rotation at these points is limited so that u 1 This affects the solution insignificantly while maintaining computational efficiency Because the domain is periodic in directional space there is n
11. etup In order to successfully convey a numerical simulation with UMWM a procedure is usually as follows 1 Obtaining the software package 2 Compiling the source code 3 Setting up the simulation 4 Running the model 5 Reading model output data In general steps 1 and 2 need to be done only once A working Fortran compiler must be provided by the user Re compiling the source code is necessary only if change has been made to any of the source files Naturally step 3 needs to be repeated if the setup of the experiment is to be changed If only input forcing files are changed one needs only to re run the model step 4 6 1 Obtaining the software package Current version of UMWM can be downloaded from this URL http rsmas miami edu groups umwm download html On a UNIX like system Linux Mac OS X or other UNIX unpack the downloaded file by typing tar xzf umwm 1 x x tgz 1 x x should be replaced with the appropriate version number This will create the directory umwm In addition UMWM requires NetCDF Fortran libraries for input and output NetCDF libraries and installation instructions can be downloaded from http www unidata ucar edu downloz Also a common build tool make or gmake is recommended for easier compilation make is included by default in most UNIX Linux distributions but may need to be installed separately on a Microsoft Windows system 6 2 Compiling the source code Before doing a simulatio
12. n model code needs to be compiled The source code is located in the src directory The user must edit the top section of the Makefile which contains rules about compiling the model University of Miami Wave Model Makefile Compiler flags and libraty paths FC pgf90 FCFFLAGS 03 fastsse CPPFLAGS LD FC LDFLAGS FCFFLAGS NETCDF usr local NETCDFPATH I NETCDF include L NETCDF lib lnetcdf NETCDFLIB NETCDF lib libnetcdf a Here FC is the path to the Fortran compiler to be used If the model is to be compiled in MPI mode FC should be named appropriately most commonly mpif90 FCFLAGS are the Fortran compiler flags to be used UMWM does not require any special flags so FCFLAGS are most likely to be optimization flags optional User should reference the manual pages for their compiler about the appropriate optimization flags CPPFLAGS are the C preprocessor flags macros that are optional for preprocess ing the Fortran code Only two macros are relevant for UMWM DMPI and DESMF DMPI is necessary when compiling the model in MPI mode DESMF is used when model is to be used for coupling through the Earth System Modeling Framework ESMF http www earthsystemmodeling org Most likely there is no need for the user to edit LD and LDFLAGS NETCDF is the path to the directory where NetCDF libraries are installed NETCDFPATH and NETCDFLIB should not be modified typically Once Makefile is configured ty
13. n is given by Shemdin et al 1978 k Pep Gp cosh kd E k 2 7 where the permeability factor Gp varies from 0 0006 to 0 01 m s depending on sand grain size 3 Numerical approaches 3 1 Spatial discretization The time evolution of the variance spectrum due to advection in Cartesian projection is given by OE _ 0Ol egcos u E dies sind v E SE 3 1 at Ox dy 09 l where u and v are ocean current components in x and y respectively and db is the rotation rate Both geographical propagation and refraction terms are discretized using first order upstream differencing This scheme is positive definite quantity conserving implicitly diffusive and computationally effi cient A certain amount of diffusion is desirable in order to avoid swell separation between discrete directional and frequency bins A spatial differencing operator is then discretized as OE Piyija Pi_ 1 2 N 3 2 Ox Az 22 where is a discrete index along dimension x Fluxes at cell edges 41 2 and _1 2 are defined as Tua Ti 1 21 2 Deis jot ti 17 2 i412 4 5 l Es Bea 3 3 ti 1 2 i 1 2 ti 1 2 ti 1 2l i 1 2 E BEi 1 5 E 3 4 and Li Lit Ti 1 2 gt 3 5 The above treatment of flux differencing ensures upstream definiteness For propagation in two dimensional space the stability of the scheme is ensured for _ At E 1 P min Az Ay v2 3 6 wher
14. o need for boundary conditions 3 2 Time discretization Once all the source terms in 1 1 have been evaluated E k 51 is integrated forward in time We evaluate the contribution from source and advection terms separately a a ds The contribution from source terms can be written as 5 Y se 3 11 s 1 1 where S is just S E Then by integrating 3 11 over a finite time interval At a solution is available in the form of i 1 En l Et egrp gt sia 3 12 The time increment At is dynamically computed so that the variance spectrum E can only grow by a pre determined finite factor gn l n qn exp Y 87A lt r 3 13 i l where r is usually set between 1 5 and 2 Lower values of r will draw E closer to the solution attractor Then a time splitting approach is used to achieve a more stable integration _ EE ED E 2 3 14 E is used to compute the advection and refraction terms described in the previous section Finally their contribution is evaluated by simple forward Euler differencing O c cos 0 u E Ox dlle sing v E A dE Oy Ob Be EH At 3 15 The above approach is applied to the prognostic part of the spectrum A cut off frequency fe which separates the prognostic and diagnostic parts is proportional to the peak frequency of the fully developed Pierson Moskowitz spectrum Pierson and Moskowitz 1964
15. of the skin is taken to be proportional to the ratio of total drag to skin drag and may be as large as 50 corresponding to full sheltering of the lee face of each wave The algorithm is as follows Cdol4 Cdol4 Cane 7 1 2 7 4 7 S 3 Cdd aa eo In order to obtain correct stress magnitude the high frequency limit should be equal to or greather than 2 Hz A linear wind speed dependent tail is appended to the spectrum beyond the high frequency limit Low frequency limit should be set to be equal to or lower than the expected spectrum peak frequency These are the stress components of the wind on the surface They are the mechanical couplers with the atmospheric model 5 Software implementation UMWM source code is written in standard Fortran 90 programming language It is written in read able and transparent form e g free of low level instructions and is well documented The model may be defined on any structured curvilinear grid Most common applications are on a Cartesian and spherical latitude longitude grid A choice is available between limited area regional and global simulation In the latter case a periodic boundary condition in x direction is applied auto matically The user is encouraged to look into the code for information that is not provided in this docu ment The model is open source and is licensed under a GPL General Public License Version 3 http www gnu org copyleft gpl html 6 Installation and s
16. pe make to compile the code The executable umwm will be generated and copied to the main model directory 6 3 Setting up the simulation Once the executable umwm has been generated the user should set up a desired experiment First step is to edit the main UMWM namelist that defines model parameters The namelist is located in namelists main nml amp DOMAIN isGlobal FALSE Global TRUE or regional FALSE mm 50 Domain size in x nm 11 Domain size in y om 37 Number of frequency bins pm 24 Number of directions 1 1 1 f1 0 0313 Lowest frequency bin Hz 1 1 1 f2 2 0 Highest frequency bin Hz startTimeStr 2008 09 08_12 00 00 Simulation start time stopTimeStr 2008 09 08_14 00 00 Simulation end time dtg 3600 Global 1 0 time step s restart FALSE Restart from file amp PHYSICS g 9 ee Gravitational acceleration m s 2 nu 0 9E Kinematic viscosity of water m 2 s sfct 0 07 Surface tension 10 kappa Z gustiness dmin explim sin_fac sin_diss1 sin_diss2 sds_fac sds_power mss_fac snl_fac sdt_fac sbf_fac sbp_fac amp GRID gridid delx dely lat_s lon_w amp INPUT_FLAGS dflag excom wflag pflag tflag cflag rflag amp INPUT depth ud wdir0 sfcpO tempO uco vco rhowO amp OUTPUT outgrid outspec xpl ypl EET E ET Oo 1000 25 0 1013 300 0 0 1025 o
17. roc Roy Soc A 110 pp 341 347 Komen G J L Cavaleri M Donelan K Hasselmann S Hasselmann and P A E M Janssen 1994 Dynamics and Modelling of Ocean Waves Cambridge University Press 532 pp Pierson W L and L Moskowitz 1964 A proposed spectral form for fully developed wind seas based on the similarity theory of S A Kitaigorodskii J Geophys Res 69 pp 5181 5190 Shemdin O H K Hasselmann S V Hsiao and K Herterich 1978 Non linear and linear bottom interaction effects in shallow water Turbulent fluzes through the sea surface wave dynamics and prediction A Favre and K Hasselmann eds Plenum New York pp 347 372 14
18. to the documentation of the MPI implementation about executing programs in parallel On a UNIX Linux system with a common modern MPI implementation MPICH2 or OpenMPT one would type mpiexec n 16 umwm This command would execute umwm on 16 parallel processes 7 1 Restarting the model If restart is set to TRUE in namelists main nml the model will read initial conditions from a restart file startTimeStr must match the time string in the restart file name For example if a user specifies 2012 09 08_18 00 00 as startTimeStr umwmrst_2008 09 08_18 00 00 nc file must be present in the restart directory 8 Reading model output data As mentioned in the previous section the output files will appear in the output directory The user can open and read output files with any program in a language that has NetCDF libraries provided Fortran MATLAB Python C C Java etc In addition convenience NetCDF viewing programs exist e g Ncview http meteora ucsd edu pierce ncview_home_page html Sample scripts 13 to read gridded and spectrum output files from UMWM will be available from the model webpage http rsmas miami edu groups umwm 9 References Donelan M A M Curcic S S Chen and A K Magnusson 2012 Modeling waves and wind stress Submitted to J Geophys Res Jeffreys H 1924 On the formation of waves by wind Proc Roy Soc A 107 pp 189 206 Jeffreys H 1925 On the formation of waves by wind II P
19. venumber k and direction A is the sheltering coefficient Sin is positive energy and momentum transferred from wind to waves when U 2cos gt c ucos vsin p and negative energy and momentum transferred from waves swell to wind when 0 lt Uy 2 cos lt c ucos p vsin or when the waves swell propagate against the wind cos lt 0 As waves approach full development Sin goes to zero i e the direct wind forcing vanishes The sheltering coefficient which describes the strength of the source sink function is different depending on whether Sin is positive wind sea or negative when the waves run before the wind or against it swell The wind velocity is that at one half wavelength above the surface up to the top of the logarithmic layer which is usually taken to be 20 m in the field 0 11 if Uy 2cos gt c ucos vsing wind sea A 40 01 if 0 lt U j2cos0 lt c ucosd vsing swell with wind 2 2 0 1 if cos lt 0 swell against wind 2 2 Wave dissipation function Sj The wave breaking dissipation source function is strongly nonlinear in the saturation spectrum B k k F k In addition the dissipation is enhanced by the straining due to the velocity field of all longer waves and by the plunging breakers that occur for small values of kd wavenumber times depth Finally viscosity acts preferentially on short waves to dissipate them 5 2 Sas k dl Ag coth kd I Asx2 k PI 4 0 wE k 4vk2E
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