\name{tran.2D} \alias{tran.2D} \title{ General Two-Dimensional Advective-Diffusive Transport } \description{ Estimates the transport term (i.e. the rate of change of a concentration due to diffusion and advection) in a two-dimensional model domain. } \usage{ tran.2D ( C, C.x.up=C[1,], C.x.down=C[nrow(C),], C.y.up=C[,1], C.y.down=C[,ncol(C)], flux.x.up=NULL, flux.x.down=NULL, flux.y.up=NULL, flux.y.down=NULL, a.bl.x.up=NULL, a.bl.x.down=NULL, a.bl.y.up=NULL, a.bl.y.down=NULL, D.grid=NULL, D.x=NULL, D.y=D.x, v.grid=NULL, v.x=0, v.y=0, AFDW.grid=NULL, AFDW.x=1, AFDW.y=AFDW.x, VF.grid=NULL,VF.x=1, VF.y=VF.x, A.grid=NULL, A.x=1, A.y=1, grid=NULL, dx=NULL, dy=NULL, full.check = FALSE, full.output = FALSE) } \arguments{ \item{C }{concentration, expressed per unit volume, defined at the centre of each grid cell; Nx*Ny matrix [M/L3]. } \item{C.x.up }{concentration at upstream boundary in x-direction; vector of length Ny [M/L3]. } \item{C.x.down }{concentration at downstream boundary in x-direction; vector of length Ny [M/L3]. } \item{C.y.up }{concentration at upstream boundary in y-direction; vector of length Nx [M/L3]. } \item{C.y.down }{concentration at downstream boundary in y-direction; vector of length Nx [M/L3]. } \item{flux.x.up }{flux across the upstream boundary in x-direction, positive = INTO model domain; vector of length Ny [M/L2/T]. } \item{flux.x.down }{flux across the downstream boundary in x-direction, positive = OUT of model domain; vector of length Ny [M/L2/T]. } \item{flux.y.up }{flux across the upstream boundary in y-direction, positive = INTO model domain; vector of length Nx [M/L2/T]. } \item{flux.y.down }{flux across the downstream boundary in y-direction, positive = OUT of model domain; vector of length Nx [M/L2/T]. } \item{a.bl.x.up }{transfer coefficient across the upstream boundary layer. in x-direction; \code{Flux=a.bl.x.up*(C.x.up-C[1,])}. One value [L/T]. } \item{a.bl.x.down }{transfer coefficient across the downstream boundary layer in x-direction; \code{Flux=a.bl.x.down*(C[Nx,]-C.x.down)}. One value [L/T]. } \item{a.bl.y.up }{transfer coefficient across the upstream boundary layer. in y-direction; \code{Flux=a.bl.y.up*(C.y.up-C[,1])}. One value [L/T]. } \item{a.bl.y.down }{transfer coefficient across the downstream boundary layer in y-direction; \code{Flux=a.bl.y.down*(C[,Ny]-C.y.down)}. One value [L/T]. } \item{D.grid }{diffusion coefficient defined on all grid cell interfaces. A \code{prop.2D} list created by \code{\link{setup.prop.2D}} [L2/T]. } \item{D.x }{diffusion coefficient in x-direction, defined on grid cell interfaces. One value, a vector of length (Nx+1), a \code{prop.1D} list created by \code{\link{setup.prop.1D}}, or a (Nx+1)* Ny matrix [L2/T]. } \item{D.y }{diffusion coefficient in y-direction, defined on grid cell interfaces. One value, a vector of length (Ny+1), a \code{prop.1D} list created by \code{\link{setup.prop.1D}}, or a Nx*(Ny+1) matrix [L2/T]. } \item{v.grid }{advective velocity defined on all grid cell interfaces. Can be positive (downstream flow) or negative (upstream flow). A \code{prop.2D} list created by \code{\link{setup.prop.2D}} [L/T]. } \item{v.x }{advective velocity in the x-direction, defined on grid cell interfaces. Can be positive (downstream flow) or negative (upstream flow). One value, a vector of length (Nx+1), a \code{prop.1D} list created by \code{\link{setup.prop.1D}}, or a (Nx+1)*Ny matrix [L/T]. } \item{v.y }{advective velocity in the y-direction, defined on grid cell interfaces. Can be positive (downstream flow) or negative (upstream flow). One value, a vector of length (Ny+1), a \code{prop.1D} list created by \code{\link{setup.prop.1D}}, or a Nx*(Ny+1) matrix [L/T]. } \item{AFDW.grid }{weight used in the finite difference scheme for advection in the x-direction, defined on grid cell interfaces; backward = 1, centred = 0.5, forward = 0; default is backward. A \code{prop.2D} list created by \code{\link{setup.prop.2D}} [-]. } \item{AFDW.x }{weight used in the finite difference scheme for advection in the x-direction, defined on grid cell interfaces; backward = 1, centred = 0.5, forward = 0; default is backward. One value, a vector of length (Nx+1), a \code{prop.1D} list created by \code{\link{setup.prop.1D}}, or a (Nx+1)*Ny matrix [-]. } \item{AFDW.y }{weight used in the finite difference scheme for advection in the y-direction, defined on grid cell interfaces; backward = 1, centred = 0.5, forward = 0; default is backward. One value, a vector of length (Ny+1), a \code{prop.1D} list created by \code{\link{setup.prop.1D}}, or a Nx*(Ny+1) matrix [-]. } \item{VF.grid }{Volume fraction. A \code{prop.2D} list created by \code{\link{setup.prop.2D}} [-]. } \item{VF.x }{Volume fraction at the grid cell interfaces in the x-direction. One value, a vector of length (Nx+1), a \code{prop.1D} list created by \code{\link{setup.prop.1D}}, or a (Nx+1)*Ny matrix [-]. } \item{VF.y }{Volume fraction at the grid cell interfaces in the y-direction. One value, a vector of length (Ny+1), a \code{prop.1D} list created by \code{\link{setup.prop.1D}}, or a Nx*(Ny+1) matrix [-]. } \item{A.grid }{Interface area. A \code{prop.2D} list created by \code{\link{setup.prop.2D}} [L2]. } \item{A.x }{Interface area defined at the grid cell interfaces in the x-direction. One value, a vector of length (Nx+1), a \code{prop.1D} list created by \code{\link{setup.prop.1D}}, or a (Nx+1)*Ny matrix [L2]. } \item{A.y }{Interface area defined at the grid cell interfaces in the y-direction. One value, a vector of length (Ny+1), a \code{prop.1D} list created by \code{\link{setup.prop.1D}}, or a Nx*(Ny+1) matrix [L2]. } \item{dx }{distance between adjacent cell interfaces in the x-direction (thickness of grid cells). One value or vector of length Nx [L]. } \item{dy }{distance between adjacent cell interfaces in the y-direction (thickness of grid cells). One value or vector of length Ny [L]. } \item{grid }{discretization grid, a list containing at least elements \code{dx}, \code{dx.aux}, \code{dy}, \code{dy.aux} (see \code{\link{setup.grid.2D}}) [L]. } \item{full.check }{logical flag enabling a full check of the consistency of the arguments (default = \code{FALSE}; \code{TRUE} slows down execution by 50 percent). } \item{full.output }{logical flag enabling a full return of the output (default = \code{FALSE}; \code{TRUE} slows down execution by 20 percent). } } \value{ a list containing: \item{dC }{the rate of change of the concentration C due to transport, defined in the centre of each grid cell, a Nx*Ny matrix. [M/L3/T]. } \item{C.x.up }{concentration at the upstream interface in x-direction. A vector of length Ny [M/L3]. Only when \code{full.output = TRUE}. } \item{C.x.down }{concentration at the downstream interface in x-direction. A vector of length Ny [M/L3]. Only when \code{full.output = TRUE}. } \item{C.y.up }{concentration at the the upstream interface in y-direction. A vector of length Nx [M/L3]. Only when \code{full.output = TRUE}. } \item{C.y.down }{concentration at the downstream interface in y-direction. A vector of length Nx [M/L3]. Only when \code{full.output = TRUE}. } \item{x.flux }{flux across the interfaces in x-direction of the grid cells. A (Nx+1)*Ny matrix [M/L2/T]. Only when \code{full.output = TRUE}. } \item{y.flux }{flux across the interfaces in y-direction of the grid cells. A Nx*(Ny+1) matrix [M/L2/T]. Only when \code{full.output = TRUE}. } \item{flux.x.up }{flux across the upstream boundary in x-direction, positive = INTO model domain. A vector of length Ny [M/L2/T]. } \item{flux.x.down }{flux across the downstream boundary in x-direction, positive = OUT of model domain. A vector of length Ny [M/L2/T]. } \item{flux.y.up }{flux across the upstream boundary in y-direction, positive = INTO model domain. A vector of length Nx [M/L2/T]. } \item{flux.y.down }{flux across the downstream boundary in y-direction, positive = OUT of model domain. A vector of length Nx [M/L2/T]. } } \author{ Filip Meysman , Karline Soetaert } \note{ It is much more efficient to use the \emph{grid} input rather than vectors or single numbers. Thus: to optimise the code, use \link{setup.grid.2D} to create the \code{grid}, and use \link{setup.prop.2D} to create \code{D.grid}, \code{v.grid}, \code{AFDW.grid}, \code{VF.grid}, and \code{A.grid}, even if the values are 1 or remain constant. There is no provision (yet) to deal with \emph{cross-diffusion}. Set \code{D.x} and \code{D.y} different only if cross-diffusion effects are unimportant. } \examples{ ## ============================================================================= ## Testing the functions ## ============================================================================= # Parameters F <- 100 # input flux [micromol cm-2 yr-1] por <- 0.8 # constant porosity D <- 400 # mixing coefficient [cm2 yr-1] v <- 1 # advective velocity [cm yr-1] # Grid definition x.N <- 4 # number of cells in x-direction y.N <- 6 # number of cells in y-direction x.L <- 8 # domain size x-direction [cm] y.L <- 24 # domain size y-direction [cm] dx <- x.L/x.N # cell size x-direction [cm] dy <- y.L/y.N # cell size y-direction [cm] # Intial conditions C <- matrix(nrow=x.N, ncol=y.N, data=0, byrow=FALSE) # Boundary conditions: fixed concentration C.x.up <- rep(1, times=y.N) C.x.down <- rep(0, times=y.N) C.y.up <- rep(1, times=x.N) C.y.down <- rep(0, times=x.N) # Only diffusion tran.2D(full.output=TRUE, C=C, D.x=D, D.y=D, v.x=0, v.y=0, VF.x=por, VF.y=por, dx=dx, dy=dy, C.x.up=C.x.up, C.x.down=C.x.down, C.y.up=C.y.up,C.y.down=C.y.down) # Strong advection, backward (default), central and forward #finite difference schemes tran.2D(C=C, D.x=D, v.x=100*v, VF.x=por, dx=dx, dy=dy, C.x.up=C.x.up, C.x.down=C.x.down, C.y.up=C.y.up, C.y.down=C.y.down) tran.2D(AFDW.x=0.5, C=C, D.x=D, v.x=100*v, VF.x=por, dx=dx, dy=dy, C.x.up=C.x.up, C.x.down=C.x.down, C.y.up=C.y.up, C.y.down=C.y.down) tran.2D(AFDW.x=0, C=C, D.x=D, v.x=100*v, VF.x=por, dx=dx, dy=dy, C.x.up=C.x.up, C.x.down=C.x.down, C.y.up=C.y.up, C.y.down=C.y.down) # Boundary conditions: fixed fluxes flux.x.up <- rep(200, times=y.N) flux.x.down <- rep(-200, times=y.N) flux.y.up <- rep(200, times=x.N) flux.y.down <- rep(-200, times=x.N) tran.2D(C=C, D.x=D, v.x=0, VF.x=por, dx=dx, dy=dy, flux.x.up=flux.x.up, flux.x.down=flux.x.down, flux.y.up=flux.y.up, flux.y.down=flux.y.down) # Boundary conditions: convective boundary layer on all sides a.bl <- 800 # transfer coefficient C.x.up <- rep(1, times=(y.N)) # fixed conc at boundary layer C.y.up <- rep(1, times=(x.N)) # fixed conc at boundary layer tran.2D(full.output=TRUE, C=C, D.x=D, v.x=0, VF.x=por, dx=dx, dy=dy, C.x.up=C.x.up, a.bl.x.up=a.bl, C.x.down=C.x.up, a.bl.x.down=a.bl, C.y.up=C.y.up, a.bl.y.up=a.bl, C.y.down=C.y.up, a.bl.y.down=a.bl) # Runtime test with and without argument checking n.iterate <-1000 test1 <- function() { for (i in 1:n.iterate ) ST<-tran.2D(full.check=TRUE,C=C,D.x=D,v.x=0,VF.x=por, dx=dx,dy=dy,C.x.up=C.x.up,a.bl.x.up=a.bl,C.x.down=C.x.down) } system.time(test1()) test2 <- function() { for (i in 1:n.iterate ) ST<-tran.2D(full.output=TRUE,C=C,D.x=D,v.x=0,VF.x=por, dx=dx,dy=dy,C.x.up=C.x.up,a.bl.x.up=a.bl,C.x.down=C.x.down) } system.time(test2()) test3 <- function() { for (i in 1:n.iterate ) ST<-tran.2D(full.output=TRUE,full.check=TRUE,C=C,D.x=D,v.x=0, VF.x=por,dx=dx,dy=dy,C.x.up=C.x.up,a.bl.x.up=a.bl,C.x.down=C.x.down) } system.time(test3()) ## ============================================================================= ## A 2-D model with diffusion in x- and y direction and first-order ## consumption - unefficient implementation ## ============================================================================= N <- 51 # number of grid cells XX <- 10 # total size dy <- dx <- XX/N # grid size Dy <- Dx <- 0.1 # diffusion coeff, X- and Y-direction r <- 0.005 # consumption rate ini <- 1 # initial value at x=0 N2 <- ceiling(N/2) X <- seq (dx,by=dx,len=(N2-1)) X <- c(-rev(X),0,X) # The model equations Diff2D <- function (t,y,parms) { CONC <- matrix(nr=N,nc=N,y) dCONC <- tran.2D(CONC, D.x=Dx, D.y=Dy, dx=dx, dy=dy)$dC + r * CONC return (list(as.vector(dCONC))) } # initial condition: 0 everywhere, except in central point y <- matrix(nr=N,nc=N,data=0) y[N2,N2] <- ini # initial concentration in the central point... # solve for 10 time units times <- 0:10 out <- ode.2D (y=y, func=Diff2D, t=times, parms=NULL, dim = c(N,N), lrw = 160000) pm <- par (mfrow=c(2,2)) # Compare solution with analytical solution... for (i in seq(2,11,by=3)) { tt <- times[i] mat <- matrix(nr=N,nc=N,out[i,-1]) plot(X,mat[N2,],type="l",main=paste("time=",times[i]), ylab="Conc",col="red") ana <- ini*dx^2/(4*pi*Dx*tt)*exp(r*tt-X^2/(4*Dx*tt)) points(X,ana,pch="+") } legend ("bottom", col=c("red","black"), lty=c(1,NA), pch=c(NA,"+"), c("tran.2D","exact")) par("mfrow"=pm ) ## ============================================================================= ## A 2-D model with diffusion in x- and y direction and first-order ## consumption - more efficient implementation, specifying ALL 2-D grids ## ============================================================================= N <- 51 # number of grid cells Dy <- Dx <- 0.1 # diffusion coeff, X- and Y-direction r <- 0.005 # consumption rate ini <- 1 # initial value at x=0 x.grid <- setup.grid.1D(x.up=-5,x.down=5,N=N) y.grid <- setup.grid.1D(x.up=-5,x.down=5,N=N) grid2D <- setup.grid.2D(x.grid,y.grid) D.grid <- setup.prop.2D(value = Dx, y.value = Dy, grid=grid2D) v.grid <- setup.prop.2D(value = 0, grid=grid2D) A.grid <- setup.prop.2D(value = 1, grid=grid2D) AFDW.grid <- setup.prop.2D(value = 1, grid=grid2D) VF.grid <- setup.prop.2D(value = 1, grid=grid2D) # The model equations - using the grids Diff2Db <- function (t,y,parms) { CONC <- matrix(nr=N,nc=N,y) dCONC <- tran.2D(CONC, grid = grid2D, D.grid=D.grid, A.grid=A.grid, VF.grid=VF.grid, AFDW.grid = AFDW.grid, v.grid=v.grid)$dC + r * CONC return (list(as.vector(dCONC))) } # initial condition: 0 everywhere, except in central point y <- matrix(nr=N,nc=N,data=0) y[N2,N2] <- ini # initial concentration in the central point... # solve for 10 time units times <- 0:10 outb <- ode.2D (y=y, func=Diff2Db, t=times, parms=NULL, dim = c(N,N), lrw = 160000) } \references{ Soetaert and Herman, 2009. a practical guide to ecological modelling - using R as a simulation platform. Springer } \details{ The \bold{boundary conditions} are either \itemize{ \item (1) zero-gradient \item (2) fixed concentration \item (3) convective boundary layer \item (4) fixed flux } This is also the order of priority. The zero gradient is the default, the fixed flux overrules all other. } \seealso{ \code{\link{tran.polar}} for a discretisation of 2-D transport equations in polar coordinates \code{\link{tran.1D}}, \code{\link{tran.3D}} } \keyword{utilities}