tran.2D {ReacTran}R Documentation

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, C.bl.x.up=NULL, a.bl.x.down=NULL, C.bl.x.down=NULL,
  a.bl.y.up=NULL, C.bl.y.up=NULL, a.bl.y.down=NULL, C.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

C concentration, expressed per unit volume, defined at the centre of each grid cell; Nx*Ny matrix [M/L3].
C.x.up concentration at upstream boundary in x-direction; vector of length Ny [M/L3].
C.x.down concentration at downstream boundary in x-direction; vector of length Ny [M/L3].
C.y.up concentration at upstream boundary in y-direction; vector of length Nx [M/L3].
C.y.down concentration at downstream boundary in y-direction; vector of length Nx [M/L3].
flux.x.up flux across the upstream boundary in x-direction, positive = INTO model domain; vector of length Ny [M/L2/T].
flux.x.down flux across the downstream boundary in x-direction, positive = OUT of model domain; vector of length Ny [M/L2/T].
flux.y.up flux across the upstream boundary in y-direction, positive = INTO model domain; vector of length Nx [M/L2/T].
flux.y.down flux across the downstream boundary in y-direction, positive = OUT of model domain; vector of length Nx [M/L2/T].
a.bl.x.up transfer coefficient across the upstream boundary layer. in x-direction;
Flux=a.bl.x.up*(C.bl.x.up-C[1,]). One value [L/T].
C.bl.x.up concentration at the upstream boundary layer in x-direction; vector of length Ny [M/L3].
a.bl.x.down transfer coefficient across the downstream boundary layer in x-direction;
Flux=a.bl.x.down*(C[Nx,]-C.bl.x.down). One value [L/T].
C.bl.x.down concentration at the downstream boundary layer in x-direction ; vector of length Ny [M/L3].
a.bl.y.up transfer coefficient across the upstream boundary layer. in y-direction;
Flux=a.bl.y.up*(C.bl.y.up-C[,1]). One value [L/T].
C.bl.y.up concentration at the upstream boundary layer in y-direction; vector of length Nx [M/L3].
a.bl.y.down transfer coefficient across the downstream boundary layer in y-direction;
Flux=a.bl.y.down*(C[,Ny]-C.bl.y.down). One value [L/T].
C.bl.y.down concentration at the downstream boundary layer in y-direction ; vector of length Nx [M/L3].
D.grid diffusion coefficient defined on all grid cell interfaces. A prop.2D list created by setup.prop.2D [L2/T].
D.x diffusion coefficient in x-direction, defined on grid cell interfaces. One value, a vector of length (Nx+1), a prop.1D list created by setup.prop.1D, or a (Nx+1)* Ny matrix [L2/T].
D.y diffusion coefficient in y-direction, defined on grid cell interfaces. One value, a vector of length (Ny+1), a prop.1D list created by setup.prop.1D, or a Nx*(Ny+1) matrix [L2/T].
v.grid advective velocity defined on all grid cell interfaces. Can be positive (downstream flow) or negative (upstream flow). A prop.2D list created by setup.prop.2D [L/T].
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 prop.1D list created by setup.prop.1D, or a (Nx+1)*Ny matrix [L/T].
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 prop.1D list created by setup.prop.1D, or a Nx*(Ny+1) matrix [L/T].
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 prop.2D list created by setup.prop.2D [-].
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 prop.1D list created by setup.prop.1D, or a (Nx+1)*Ny matrix [-].
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 prop.1D list created by setup.prop.1D, or a Nx*(Ny+1) matrix [-].
VF.grid Volume fraction. A prop.2D list created by setup.prop.2D [-].
VF.x Volume fraction at the grid cell interfaces in the x-direction. One value, a vector of length (Nx+1), a prop.1D list created by setup.prop.1D, or a (Nx+1)*Ny matrix [-].
VF.y Volume fraction at the grid cell interfaces in the y-direction. One value, a vector of length (Ny+1), a prop.1D list created by setup.prop.1D, or a Nx*(Ny+1) matrix [-].
A.grid Interface area. A prop.2D list created by setup.prop.2D [L2].
A.x Interface area defined at the grid cell interfaces in the x-direction. One value, a vector of length (Nx+1), a prop.1D list created by setup.prop.1D, or a (Nx+1)*Ny matrix [L2].
A.y Interface area defined at the grid cell interfaces in the y-direction. One value, a vector of length (Ny+1), a prop.1D list created by setup.prop.1D, or a Nx*(Ny+1) matrix [L2].
dx distance between adjacent cell interfaces in the x-direction (thickness of grid cells). One value or vector of length Nx [L].
dy distance between adjacent cell interfaces in the y-direction (thickness of grid cells). One value or vector of length Ny [L].
grid discretization grid, a list containing at least elements dx, dx.aux, dy, dy.aux (see setup.grid.2D) [L].
full.check logical flag enabling a full check of the consistency of the arguments (default = FALSE; TRUE slows down execution by 50 percent).
full.output logical flag enabling a full return of the output (default = FALSE; TRUE slows down execution by 20 percent).

Details

The boundary conditions are either

This is also the order of priority. The zero gradient is the default, the fixed flux overrules all other.

Value

a list containing:

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].
C.x.up concentration at the upstream interface in x-direction. A vector of length Ny [M/L3]. Only when full.output = TRUE.
C.x.down concentration at the downstream interface in x-direction. A vector of length Ny [M/L3]. Only when full.output = TRUE.
C.y.up concentration at the the upstream interface in y-direction. A vector of length Nx [M/L3]. Only when full.output = TRUE.
C.y.down concentration at the downstream interface in y-direction. A vector of length Nx [M/L3]. Only when full.output = TRUE.
x.flux flux across the interfaces in x-direction of the grid cells. A (Nx+1)*Ny matrix [M/L2/T]. Only when full.output = TRUE.
y.flux flux across the interfaces in y-direction of the grid cells. A Nx*(Ny+1) matrix [M/L2/T]. Only when full.output = TRUE.
flux.x.up flux across the upstream boundary in x-direction, positive = INTO model domain. A vector of length Ny [M/L2/T].
flux.x.down flux across the downstream boundary in x-direction, positive = OUT of model domain. A vector of length Ny [M/L2/T].
flux.y.up flux across the upstream boundary in y-direction, positive = INTO model domain. A vector of length Nx [M/L2/T].
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(s)

Filip Meysman <f.meysman@nioo.knaw.nl>, Karline Soetaert <k.soetaert@nioo.knaw.nl>

References

Soetaert and Herman, 2009. a practical guide to ecological modelling - using R as a simulation platform. Springer

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.bl.x.up <- rep(1, times=(y.N)) # fixed conc at boundary layer
C.bl.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.bl.x.up=C.bl.x.up, a.bl.x.up=a.bl, C.bl.x.down=C.bl.x.up,
  a.bl.x.down=a.bl, C.bl.y.up=C.bl.y.up, a.bl.y.up=a.bl,
  C.bl.y.down=C.bl.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.bl.x.up=C.bl.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.bl.x.up=C.bl.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.bl.x.up=C.bl.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
## =============================================================================

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 )


[Package ReacTran version 1.2 Index]