tran.2D.Rd
\name{tran.2D}
\alias{tran.2D}
\title{
}

\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
}
\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
}
\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}
}
\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 <f.meysman@nioo.knaw.nl>,
Karline Soetaert <k.soetaert@nioo.knaw.nl>
}

\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 (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