tran.2D.Rd

```
\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].
See last example for creating spatially-varying diffusion coefficients.
}
\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- and y- 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 <filip.meysman@nioz.nl>,
Karline Soetaert <karline.soetaert@nioz.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(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, full.output = TRUE)
# 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 <-500
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(nrow = N, ncol = N, y)
dCONC <- tran.2D(CONC, D.x = Dx, D.y = Dy, dx = dx, dy = dy)$dC + r * CONC
return (list(dCONC))
}
# initial condition: 0 everywhere, except in central point
y <- matrix(nrow = N, ncol = N, data = 0)
y[N2, N2] <- ini # initial concentration in the central point...
# solve for 10 time units
times <- 0:10
out <- ode.2D (y = y, func = Diff2D, t = times, parms = NULL,
dim = c(N,N), lrw = 160000)
pm <- par (mfrow = c(2, 2))
# Compare solution with analytical solution...
for (i in seq(2, 11, by = 3)) {
tt <- times[i]
mat <- matrix(nrow = N, ncol = N,
data = subset(out, time == tt))
plot(X, mat[N2,], type = "l", main = paste("time=", times[i]),
ylab = "Conc", col = "red")
ana <- ini*dx^2/(4*pi*Dx*tt)*exp(r*tt-X^2/(4*Dx*tt))
points(X, ana, pch = "+")
}
legend ("bottom", col = c("red","black"), lty = c(1, NA),
pch = c(NA, "+"), c("tran.2D", "exact"))
par("mfrow" = pm )
## =============================================================================
## A 2-D model with diffusion in x- and y direction and first-order
## consumption - more efficient implementation, specifying ALL 2-D grids
## =============================================================================
N <- 51 # number of grid cells
Dy <- Dx <- 0.1 # diffusion coeff, X- and Y-direction
r <- 0.005 # consumption rate
ini <- 1 # initial value at x=0
x.grid <- setup.grid.1D(x.up = -5, x.down = 5, N = N)
y.grid <- setup.grid.1D(x.up = -5, x.down = 5, N = N)
grid2D <- setup.grid.2D(x.grid, y.grid)
D.grid <- setup.prop.2D(value = Dx, y.value = Dy, grid = grid2D)
v.grid <- setup.prop.2D(value = 0, grid = grid2D)
A.grid <- setup.prop.2D(value = 1, grid = grid2D)
AFDW.grid <- setup.prop.2D(value = 1, grid = grid2D)
VF.grid <- setup.prop.2D(value = 1, grid = grid2D)
# The model equations - using the grids
Diff2Db <- function (t, y, parms) {
CONC <- matrix(nrow = N, ncol = N, data = y)
dCONC <- tran.2D(CONC, grid = grid2D, D.grid = D.grid,
A.grid = A.grid, VF.grid = VF.grid, AFDW.grid = AFDW.grid,
v.grid = v.grid)$dC + r * CONC
return (list(dCONC))
}
# initial condition: 0 everywhere, except in central point
y <- matrix(nrow = N, ncol = N, data = 0)
y[N2,N2] <- ini # initial concentration in the central point...
# solve for 8 time units
times <- 0:8
outb <- ode.2D (y = y, func = Diff2Db, t = times, parms = NULL,
dim = c(N, N), lrw = 160000)
image(outb, ask = FALSE, mfrow = c(3, 3), main = paste("time", times))
## =============================================================================
## Same 2-D model, but now with spatially-variable diffusion coefficients
## =============================================================================
N <- 51 # number of grid cells
r <- 0.005 # consumption rate
ini <- 1 # initial value at x=0
N2 <- ceiling(N/2)
D.grid <- list()
# Diffusion on x-interfaces
D.grid$x.int <- matrix(nrow = N+1, ncol = N, data = runif(N*(N+1)))
# Diffusion on y-interfaces
D.grid$y.int <- matrix(nrow = N, ncol = N+1, data = runif(N*(N+1)))
dx <- 10/N
dy <- 10/N
# The model equations
Diff2Dc <- function (t, y, parms) {
CONC <- matrix(nrow = N, ncol = N, data = y)
dCONC <- tran.2D(CONC, dx = dx, dy = dy, D.grid = D.grid)$dC + r * CONC
return (list(dCONC))
}
# initial condition: 0 everywhere, except in central point
y <- matrix(nrow = N, ncol = N, data = 0)
y[N2, N2] <- ini # initial concentration in the central point...
# solve for 8 time units
times <- 0:8
outc <- ode.2D (y = y, func = Diff2Dc, t = times, parms = NULL,
dim = c(N, N), lrw = 160000)
outtimes <- c(1, 3, 5, 7)
image(outc, ask = FALSE, mfrow = c(2, 2), main = paste("time", outtimes),
legend = TRUE, add.contour = TRUE, subset = time \%in\% outtimes)
}
\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}
```