tran.3D.R
##==============================================================================
## Take differences of an array
##==============================================================================
diff3D <- function(AR, along) {
dimens<-dim(AR)
if (along == 1)
return(AR[2:dimens[1],,]-AR[1:(dimens[1]-1),,])
else if (along == 2)
return(AR[,2:dimens[2],]-AR[,1:(dimens[2]-1),])
if (along == 3)
return(AR[,,2:dimens[3]]-AR[,,1:(dimens[3]-1)])
}
# function to bind two matrices to an array
mbind <- function (Mat1, Array, Mat2, along=1) {
dimens <- dim(Array) + c(0,0,2)
if (along==3)
array(dim=dimens, data=c(Mat1,Array,Mat2))
else if (along == 1)
aperm(array(dim=dimens,
data=c(Mat1,aperm(Array,c(3,2,1)),Mat2)),c(3,2,1))
else if (along == 2)
aperm(array(dim=dimens,
data=c(Mat1,aperm(Array,c(1,3,2)),Mat2)),c(1,3,2))
}
# function to bind a matrix to an array on the left
mbindl <- function (Mat1, Array, along=1) {
dimens <- dim(Array) + c(0,0,1)
if (along==3)
array(dim=dimens, data=c(Mat1,Array))
else if (along == 1)
aperm(array(dim=dimens,
data=c(Mat1,aperm(Array,c(3,2,1)))),c(3,2,1))
else if (along == 2)
aperm(array(dim=dimens,
data=c(Mat1,aperm(Array,c(1,3,2)))),c(1,3,2))
}
# function to bind a matrix to an array on the right
mbindr <- function (Array, Mat2, along=1) {
dimens <- dim(Array) + c(0,0,1)
if (along==3)
array(dim=dimens, data=c(Array,Mat2))
else if (along == 1)
aperm(array(dim=dimens,
data=c(aperm(Array,c(3,2,1)),Mat2)),c(3,2,1))
else if (along == 2)
aperm(array(dim=dimens,
data=c(aperm(Array,c(1,3,2)),Mat2)),c(1,3,2))
}
##==============================================================================
## Transport in a three-dimensional finite difference grid
##==============================================================================
tran.3D <- function(C, C.x.up=C[1,,], C.x.down=C[dim(C)[1],,],
C.y.up=C[,1,], C.y.down=C[,dim(C)[2],],
C.z.up=C[,,1], C.z.down=C[,,dim(C)[3]],
flux.x.up=NULL, flux.x.down=NULL,
flux.y.up=NULL, flux.y.down=NULL,
flux.z.up=NULL, flux.z.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,
a.bl.z.up=NULL, C.bl.z.up=NULL, a.bl.z.down=NULL, C.bl.z.down=NULL,
D.grid=NULL, D.x=NULL, D.y=D.x, D.z=D.x,
v.grid=NULL, v.x=0, v.y=0, v.z=0,
AFDW.grid=NULL, AFDW.x=1, AFDW.y=AFDW.x, AFDW.z=AFDW.x,
VF.grid=NULL, VF.x=1, VF.y=VF.x, VF.z=VF.x,
A.grid=NULL, A.x=1, A.y=1, A.z=1,
grid=NULL, dx=NULL, dy=NULL, dz=NULL,
full.check = FALSE, full.output = FALSE)
{
if (is.null(grid))
if (is.null(dx) | is.null(dy) | is.null(dz))
stop("error: either 'grid' or ('dx' and 'dy' and 'dz') should be specified ")
Nx <- dim(C)[1]
Ny <- dim(C)[2]
Nz <- dim(C)[3]
if(length (C.x.up) == 1) C.x.up <- matrix(nr=Ny,nc=Nz,C.x.up)
if(length (C.x.down) == 1) C.x.down <- matrix(nr=Ny,nc=Nz,C.x.down)
if(length (C.y.up) == 1) C.y.up <- matrix(nr=Nx,nc=Nz,C.y.up)
if(length (C.y.down) == 1) C.y.down <- matrix(nr=Nx,nc=Nz,C.y.down)
if(length (C.z.up) == 1) C.z.up <- matrix(nr=Nx,nc=Ny,C.z.up)
if(length (C.z.down) == 1) C.z.down <- matrix(nr=Nx,nc=Ny,C.z.down)
# DEFAULT INFILLING OF GRID PARAMETERS
# infilling of 3D grid
if (is.null(grid)) {
DX <- if (is.list(dx)) dx$dx else rep(dx,length.out=Nx)
DXaux <- if (is.list(dx)) dx$dx.aux else 0.5*(c(0,rep(dx,length.out=Nx))+
c(rep(dx,length.out=Nx),0))
DY <- if (is.list(dy)) dy$dx else rep(dy,length.out=Ny)
DYaux <- if (is.list(dy)) dy$dx.aux else 0.5*(c(0,rep(dy,length.out=Ny))+
c(rep(dy,length.out=Ny),0))
DZ <- if (is.list(dz)) dz$dx else rep(dz,length.out=Nz)
DZaux <- if (is.list(dz)) dz$dx.aux else 0.5*(c(0,rep(dz,length.out=Nz))+
c(rep(dz,length.out=Nz),0))
grid <- list(
dx = DX, dx.aux= DXaux,
dy = DY, dy.aux= DYaux,
dz = DZ, dz.aux= DZaux
)
}
#==============================================================================
# infilling of grids with x.int, y.int, z.int, x.mid, y.mid, z.mid needed
#==============================================================================
gridFill <- function(G.x,G.y,G.z,Name) # define a function first
{
# check if G.x and G.y and G.z is not NULL
if (is.null(G.x) | is.null(G.y) | is.null(G.z))
stop( (paste("error: ",Name,"and (",Name,".x and", Name,".y and",
Name,".z) cannot be NULL at the same time", del="")))
G.grid <- list()
# infilling of x-array
if (is.array(G.x)) {
if (sum(abs(dim(G.x) - c(Nx+1,Ny,Nz)))!=0)
stop (paste("error: ",Name,".x array not of correct (Nx+1) Ny ,Nz dimensions", del=""))
G.grid$x.int <- G.x
G.grid$x.mid <- 0.5*(G.x[1:Nx,,]+G.x[2:(Nx+1),,])
} else if (length(G.x) == 1) {
G.grid$x.int <- array(data=G.x,dim=c(Nx+1,Ny,Nz))
G.grid$x.mid <- array(data=G.x,dim=c(Nx,Ny,Nz))
} else
stop (paste("error: ",Name,".x should be one element or an array", del=""))
# infilling of y-array
if (is.array(G.y)) {
if (sum(abs(dim(G.y) - c(Nx,Ny+1,Nz)))!=0)
stop (paste("error: ",Name,".x array not of correct Nx, Ny+1 ,Nz dimensions", del=""))
G.grid$y.int <- G.y
G.grid$y.mid <- 0.5*(G.y[,1:Ny,]+G.y[,2:(Ny+1),])
} else if (length(G.y) == 1) {
G.grid$y.int <- array(data=G.y,dim=c(Nx,Ny+1,Nz))
G.grid$y.mid <- array(data=G.y,dim=c(Nx,Ny,Nz))
} else
stop (paste("error: ",Name,".y should be one element or an array", del=""))
# infilling of z-array
if (is.array(G.z)) {
if (sum(abs(dim(G.z) - c(Nx,Ny,Nz+1)))!=0)
stop (paste("error: ",Name,".z array not of correct Nx, Ny ,Nz+1 dimensions", del=""))
G.grid$z.int <- G.z
G.grid$z.mid <- 0.5*(G.z[,1:Ny,]+G.z[,2:(Ny+1),])
} else if (length(G.z) == 1) {
G.grid$z.int <- array(data=G.z,dim=c(Nx,Ny,Nz+1))
G.grid$z.mid <- array(data=G.z,dim=c(Nx,Ny,Nz))
} else
stop (paste("error: ",Name,".z should be one element or an array", del=""))
G.grid
}
# Need this for VF and A (volume fraction and surface
if (is.null(VF.grid)) VF.grid <- gridFill(VF.x,VF.y,VF.z,"VF")
if (is.null(A.grid)) A.grid <- gridFill(A.x,A.y,A.z,"A")
#==============================================================================
# infilling of other grids with only x.int and y.int needed
#==============================================================================
gridInt <- function(G.x,G.y,G.z,Name) # define a function first
{
# check if G.x and G.y and G.z is not NULL
if (is.null(G.x) | is.null(G.y) | is.null(G.z))
stop( (paste("error: ",Name,"and (",Name,".x and", Name,".y and",
Name,".z) cannot be NULL at the same time", del="")))
G.grid <- list()
# infilling of x-array
if (is.array(G.x)) {
if (sum(abs(dim(G.x) - c(Nx+1,Ny,Nz)))!=0)
stop (paste("error: ",Name,".x array not of correct (Nx+1) Ny ,Nz dimensions", del=""))
G.grid$x.int <- G.x
} else if (length(G.x) == 1) {
G.grid$x.int <- array(data=G.x,dim=c(Nx+1,Ny,Nz))
} else if (length(G.x) == Nx+1) {
G.grid$x.int <- array(data=G.x,dim=c(Nx+1,Ny,Nz))
} else
stop (paste("error: ",Name,".x should be one element, a vector, or an array", del=""))
# infilling of y-array
if (is.array(G.y)) {
if (sum(abs(dim(G.y) - c(Nx,Ny+1,Nz)))!=0)
stop (paste("error: ",Name,".y array not of correct Nx, Ny+1 ,Nz dimensions", del=""))
G.grid$y.int <- G.y
} else if (length(G.y) == 1) {
G.grid$y.int <- array(data=G.y,dim=c(Nx,Ny+1,Nz))
} else if (length(G.y) == Ny+1) {
G.grid$y.int <- aperm(array(data=G.y,dim=c(Ny+1,Nx,Nz)),c(2,1,3))
} else
stop (paste("error: ",Name,".y should be one element, a vector or an array", del=""))
# infilling of z-array
if (is.array(G.z)) {
if (sum(abs(dim(G.z) - c(Nx,Ny,Nz+1)))!=0)
stop (paste("error: ",Name,".z array not of correct Nx, Ny ,Nz+1 dimensions", del=""))
G.grid$z.int <- G.z
} else if (length(G.z) == 1) {
G.grid$z.int <- array(data=G.z,dim=c(Nx,Ny,Nz+1))
} else if (length(G.z) == Nz+1) {
G.grid$z.int <- aperm(array(data=G.z,dim=c(Nz+1,Ny,Nx)),c(3,2,1))
} else
stop (paste("error: ",Name,".z should be one element, a vector or an array", del=""))
G.grid
}
# Need this for AFDW , D and v
if (is.null(AFDW.grid)) AFDW.grid <- gridInt(AFDW.x,AFDW.y,AFDW.z,"AFDW")
if (is.null(D.grid)) D.grid <- gridInt(D.x,D.y,D.z,"D")
if (is.null(v.grid)) v.grid <- gridInt(v.x,v.y,v.z,"v")
#==============================================================================
# INPUT CHECKS
#==============================================================================
if (full.check) {
## check dimensions of input concentrations
if (!is.null(C.x.up)) {
if (!((length(C.x.up)==1) || (sum(abs(dim(C.x.up) - c(Ny,Nz)))==0)))
stop("error: C.x.up should be of length 1 or a matrix with dim Ny, Nz")
}
if (!is.null(C.x.down)) {
if (!((length(C.x.down)==1) || (sum(abs(dim(C.x.down) - c(Ny,Nz)))==0)))
stop("error: C.x.down should be of length 1 or a matrix with dim Ny, Nz")
}
if (!is.null(C.y.up)) {
if (!((length(C.y.up)==1) || (sum(abs(dim(C.y.up) - c(Nx,Nz)))==0)))
stop("error: C.y.up should be of length 1 or a matrix with dim Nx, Nz")
}
if (!is.null(C.y.down)) {
if (!((length(C.y.down)==1) || (sum(abs(dim(C.y.down) - c(Nx,Nz)))==0)))
stop("error: C.y.down should be of length 1 or a matrix with dim Nx, Nz")
}
if (!is.null(C.z.up)) {
if (!((length(C.z.up)==1) || (sum(abs(dim(C.z.up) - c(Nx,Ny)))==0)))
stop("error: C.z.up should be of length 1 or a matrix with dim Nx, Ny")
}
if (!is.null(C.z.down)) {
if (!((length(C.z.down)==1) || (sum(abs(dim(C.z.down) - c(Nx,Ny)))==0)))
stop("error: C.z.down should be of length 1 or a matrix with dim Nx, Ny")
}
#
if (!is.null(C.bl.x.up)) {
if (!((length(C.bl.x.up)==1) || (sum(abs(dim(C.bl.x.up) - c(Ny,Nz)))==0)))
stop("error: C.bl.x.up should be of length 1 or a matrix with dim Ny, Nz")
}
if (!is.null(C.bl.x.down)) {
if (!((length(C.bl.x.down)==1) || (sum(abs(dim(C.bl.x.down) - c(Ny,Nz)))==0)))
stop("error: C.bl.x.down should be of length 1 or a matrix with dim Ny, Nz")
}
if (!is.null(C.bl.y.up)) {
if (!((length(C.bl.y.up)==1) || (sum(abs(dim(C.bl.y.up) - c(Nx,Nz)))==0)))
stop("error: C.bl.y.up should be of length 1 or a matrix with dim Nx, Nz")
}
if (!is.null(C.bl.y.down)) {
if (!((length(C.bl.y.down)==1) || (sum(abs(dim(C.bl.y.down) - c(Nx,Nz)))==0)))
stop("error: C.bl.y.down should be of length 1 or a matrix with dim Nx, Nz")
}
if (!is.null(C.bl.z.up)) {
if (!((length(C.bl.z.up)==1) || (sum(abs(dim(C.bl.z.up) - c(Nx,Ny)))==0)))
stop("error: C.bl.z.up should be of length 1 or a matrix with dim Nx, Ny")
}
if (!is.null(C.bl.z.down)) {
if (!((length(C.bl.z.down)==1) || (sum(abs(dim(C.bl.z.down) - c(Nx,Ny)))==0)))
stop("error: C.bl.z.down should be of length 1 or a matrix with dim Nx, Ny")
}
# check dimensions of input fluxes
if (!is.null(flux.x.up)) {
if (!((length(flux.x.up)==1) || (sum(abs(dim(flux.x.up) - c(Ny,Nz)))!=0)))
stop("error: flux.x.up should be of length 1 or a matrix with dim Ny, Nz")
}
if (!is.null(flux.x.down)) {
if (!((length(flux.x.down)==1) || (sum(abs(dim(flux.x.down) - c(Ny,Nz)))!=0)))
stop("error: flux.x.down should be of length 1 or a matrix with dim Ny, Nz")
}
if (!is.null(flux.y.up)) {
if (!((length(flux.y.up)==1) || (sum(abs(dim(flux.y.up) - c(Nx,Nz)))!=0)))
stop("error: flux.y.up should be of length 1 or a matrix with dim Nx, Nz")
}
if (!is.null(flux.y.down)) {
if (!((length(flux.y.down)==1) || (sum(abs(dim(flux.y.down) - c(Nx,Nz)))!=0)))
stop("error: flux.y.down should be of length 1 or a matrix with dim Nx, Nz")
}
if (!is.null(flux.z.up)) {
if (!((length(flux.z.up)==1) || (sum(abs(dim(flux.z.up) - c(Nx,Ny)))!=0)))
stop("error: flux.z.up should be of length 1 or a matrix with dim Nx, Ny")
}
if (!is.null(flux.z.down)) {
if (!((length(flux.z.down)==1) || (sum(abs(dim(flux.z.down) - c(Nx,Ny)))!=0)))
stop("error: flux.z.down should be of length 1 or a matrix with dim Nx, Ny")
}
## check input of grid
if (is.null(dx) && is.null(dy) && is.null(dz) && is.null(grid))
stop("error: dx, dy, dz and grid cannot be NULL at the same time")
gn <- names(grid)
if (! "dx" %in% gn)
stop("error: grid should be a list that contains 'dx' ")
if (! "dx.aux" %in% gn)
stop("error: grid should be a list that contains 'dx.aux' ")
if (! "dy" %in% gn)
stop("error: grid should be a list that contains 'dy' ")
if (! "dy.aux" %in% gn)
stop("error: grid should be a list that contains 'dy.aux' ")
if (! "dz" %in% gn)
stop("error: grid should be a list that contains 'dz' ")
if (! "dz.aux" %in% gn)
stop("error: grid should be a list that contains 'dz.aux' ")
if (is.null(grid$dx) || is.null(grid$dx.aux))
stop("error: the grid should be a list with (numeric) values for 'dx' and 'dx.aux' ")
if (is.null(grid$dy) || is.null(grid$dy.aux))
stop("error: the grid should be a list with (numeric) values for 'dy' and 'dy.aux' ")
if (is.null(grid$dz) || is.null(grid$dz.aux))
stop("error: the grid should be a list with (numeric) values for 'dz' and 'dz.aux' ")
if (any(grid$dx <= 0) || any(grid$dx.aux <= 0) )
stop("error: the grid distances dx and dx.aux should always be positive")
if (any(grid$dy <= 0) || any(grid$dy.aux <= 0) )
stop("error: the grid distances dy and dy.aux should always be positive")
if (any(grid$dz <= 0) || any(grid$dz.aux <= 0) )
stop("error: the grid distances dz and dz.aux should always be positive")
## check input of AFDW.grid
if (is.null(AFDW.x) && is.null(AFDW.y) && is.null(AFDW.z) && is.null(AFDW.grid))
stop("error: AFDW.x, AFDW.y, AFDW.z and AFDW.grid cannot be NULL at the same time")
gn <- names(AFDW.grid)
if (! "x.int" %in% gn)
stop("error: AFDW.grid should be a list that contains 'x.int', the values at the interfaces of the grid cells in x-direction")
if (! "y.int" %in% gn)
stop("error: AFDW.grid should be a list that contains 'y.int', the values at the interfaces of the grid cells in y-direction")
if (! "z.int" %in% gn)
stop("error: AFDW.grid should be a list that contains 'z.int', the values at the interfaces of the grid cells in z-direction")
if (is.null(AFDW.grid$x.int))
stop("error: AFDW.grid$x.int should be a list with (numeric) values")
if (is.null(AFDW.grid$y.int))
stop("error: AFDW.grid$y.int should be a list with (numeric) values")
if (is.null(AFDW.grid$z.int))
stop("error: AFDW.grid$z.int should be a list with (numeric) values")
if (any (AFDW.grid$x.int < 0)||any (AFDW.grid$x.int > 1))
stop("error: the AFDW should range between 0 and 1")
if (any (AFDW.grid$y.int < 0)||any (AFDW.grid$y.int > 1))
stop("error: the AFDW should range between 0 and 1")
if (any (AFDW.grid$z.int < 0)||any (AFDW.grid$z.int > 1))
stop("error: the AFDW should range between 0 and 1")
## check input of D.grid
if (is.null(D.x) && is.null(D.y) && is.null(D.grid))
stop("error: D.x, D.y, and D.grid cannot be NULL at the same time")
gn <- names(D.grid)
if (! "x.int" %in% gn)
stop("error: D.grid should be a list that contains 'x.int', the D values at the interfaces of the grid cells in x-direction")
if (! "y.int" %in% gn)
stop("error: D.grid should be a list that contains 'y.int', the D values at the interfaces of the grid cells in y-direction")
if (! "z.int" %in% gn)
stop("error: D.grid should be a list that contains 'z.int', the D values at the interfaces of the grid cells in z-direction")
if (is.null(D.grid$x.int))
stop("error: D.grid$x.int should be a list with (numeric) values")
if (is.null(D.grid$y.int))
stop("error: D.grid$y.int should be a list with (numeric) values")
if (is.null(D.grid$z.int))
stop("error: D.grid$z.int should be a list with (numeric) values")
if (any (D.grid$x.int < 0)||any (D.grid$y.int < 0)||any (D.grid$z.int < 0))
stop("error: the diffusion coefficient should always be positive")
## check input of v.grid
if (is.null(v.x) && is.null(v.y) && is.null(v.grid))
stop("error: v.x, v.y, and v.grid cannot be NULL at the same time")
gn <- names(v.grid)
if (! "x.int" %in% gn)
stop("error: v.grid should be a list that contains 'x.int', the velocity values at the interfaces of the grid cells in x-direction")
if (! "y.int" %in% gn)
stop("error: v.grid should be a list that contains 'y.int', the velocity values at the interfaces of the grid cells in y-direction")
if (! "z.int" %in% gn)
stop("error: v.grid should be a list that contains 'z.int', the velocity values at the interfaces of the grid cells in z-direction")
if (is.null(v.grid$x.int))
stop("error: the advective velocity v.grid$x.int should be a list with (numeric) values")
if (is.null(v.grid$y.int))
stop("error: the advective velocity v.grid$y.int should be a list with (numeric) values")
if (is.null(v.grid$z.int))
stop("error: the advective velocity v.grid$z.int should be a list with (numeric) values")
## check input of VF.grid
gn <- names(VF.grid)
if (! "x.int" %in% gn)
stop("error: VF.grid should be a list that contains 'x.int'")
if (! "y.int" %in% gn)
stop("error: VF.grid should be a list that contains 'y.int'")
if (! "z.int" %in% gn)
stop("error: VF.grid should be a list that contains 'z.int'")
if (! "x.mid" %in% gn)
stop("error: VF.grid should be a list that contains 'x.mid'")
if (! "y.mid" %in% gn)
stop("error: VF.grid should be a list that contains 'y.mid'")
if (! "z.mid" %in% gn)
stop("error: VF.grid should be a list that contains 'z.mid'")
if (is.null(VF.grid$x.int) || is.null(VF.grid$y.int)
|| is.null(VF.grid$x.mid) || is.null(VF.grid$y.mid)
|| is.null(VF.grid$z.mid) || is.null(VF.grid$z.int))
stop("error: VF should contain (numeric) values")
if (any (VF.grid$x.int < 0) || any (VF.grid$y.int < 0)
|| any (VF.grid$x.mid < 0) || any (VF.grid$y.mid < 0)
|| any (VF.grid$z.mid < 0) || any (VF.grid$z.int < 0))
stop("error: the VF values should always be positive")
## check input of A.grid
gn <- names(A.grid)
if (! "x.int" %in% gn)
stop("error: A.grid should be a list that contains 'x.int'")
if (! "y.int" %in% gn)
stop("error: A.grid should be a list that contains 'y.int'")
if (! "z.int" %in% gn)
stop("error: A.grid should be a list that contains 'z.int'")
if (! "x.mid" %in% gn)
stop("error: A.grid should be a list that contains 'x.mid'")
if (! "y.mid" %in% gn)
stop("error: A.grid should be a list that contains 'y.mid'")
if (! "z.mid" %in% gn)
stop("error: A.grid should be a list that contains 'z.mid'")
if (is.null(A.grid$x.int) || is.null(A.grid$y.int)
|| is.null(A.grid$x.mid) || is.null(A.grid$y.mid)
|| is.null(A.grid$z.mid) || is.null(A.grid$z.int))
stop("error: the VF should contain (numeric) values")
if (any (A.grid$x.int < 0) || any (A.grid$y.int < 0)
|| any (A.grid$x.mid < 0) || any (A.grid$y.mid < 0)
|| any (A.grid$z.mid < 0) || any (A.grid$z.int < 0))
stop("error: the A values should always be positive")
}
## FUNCTION BODY: CALCULATIONS
## Impose boundary flux at upstream x-boundary when needed
## Default boundary condition is no gradient
if (! is.null (flux.x.up[1])) {
nom <- flux.x.up + VF.grid$x.int[1,,]*(D.grid$x.int[1,,]/grid$dx.aux[1] +
(1-AFDW.grid$x.int[1,,])*v.grid$x.int[1,,])*C[1,,]
denom <- VF.grid$x.int[1,,]*(D.grid$x.int[1,,]/grid$dx.aux[1]+
AFDW.grid$x.int[1,,]*v.grid$x.int[1,,])
C.x.up <- nom/denom
}
## Impose boundary flux at downstream x-boundary when needed
## Default boundary condition is no gradient
if (! is.null (flux.x.down[1])) {
nom <- flux.x.down - VF.grid$x.int[(Nx+1),,]*(D.grid$x.int[(Nx+1),,]/
grid$dx.aux[Nx+1] + AFDW.grid$x.int[(Nx+1),,]*v.grid$x.int[(Nx+1),,])*C[Nx,,]
denom <- -VF.grid$x.int[(Nx+1),,]*(D.grid$x.int[(Nx+1),,]/grid$dx.aux[Nx+1]+
(1-AFDW.grid$x.int[(Nx+1),,])*v.grid$x.int[(Nx+1),,])
C.x.down <- nom/denom
}
# Impose boundary flux at upstream y-boundary when needed
# Default boundary condition is no gradient
if (! is.null (flux.y.up[1])) {
nom <- flux.y.up + VF.grid$y.int[,1,]*(D.grid$y.int[,1,]/grid$dy.aux[1] +
(1-AFDW.grid$y.int[,1,])*v.grid$y.int[,1,])*C[,1,]
denom <- VF.grid$y.int[,1,]*(D.grid$y.int[,1,]/grid$dy.aux[1]+
AFDW.grid$y.int[,1,]*v.grid$y.int[,1,])
C.y.up <- nom/denom
}
# Impose boundary flux at downstream y-boundary when needed
# Default boundary condition is no gradient
if (! is.null (flux.y.down[1])) {
nom <- flux.y.down - VF.grid$y.int[,(Ny+1),]*(D.grid$y.int[,(Ny+1),]/
grid$dy.aux[Ny+1] + AFDW.grid$y.int[,(Ny+1),]*v.grid$y.int[,(Ny+1),])*C[,Ny,]
denom <- -VF.grid$y.int[,(Ny+1),]*(D.grid$y.int[,(Ny+1),]/grid$dy.aux[Ny+1]+
(1-AFDW.grid$y.int[,(Ny+1),])*v.grid$y.int[,(Ny+1),])
C.y.down <- nom/denom
}
# Impose boundary flux at upstream z-boundary when needed
# Default boundary condition is no gradient
if (! is.null (flux.z.up[1])) {
nom <- flux.z.up + VF.grid$z.int[,,1]*(D.grid$z.int[,,1]/grid$dz.aux[1] +
(1-AFDW.grid$z.int[,,1])*v.grid$z.int[,,1])*C[,,1]
denom <- VF.grid$z.int[,,1]*(D.grid$z.int[,,1]/grid$dz.aux[1]+
AFDW.grid$z.int[,,1]*v.grid$z.int[,,1])
C.z.up <- nom/denom
}
# Impose boundary flux at downstream z-boundary when needed
# Default boundary condition is no gradient
if (! is.null (flux.z.down[1])) {
nom <- flux.z.down - VF.grid$z.int[,,(Nz+1)]*(D.grid$z.int[,,(Nz+1)]/
grid$dz.aux[Nz+1] + AFDW.grid$z.int[,,(Nz+1)]*v.grid$z.int[,,(Nz+1)])*C[,,Nz]
denom <- -VF.grid$z.int[,,(Nz+1)]*(D.grid$z.int[,,(Nz+1)]/grid$dz.aux[Nz+1]+
(1-AFDW.grid$z.int[,,(Nz+1)])*v.grid$z.int[,,(Nz+1)])
C.y.down <- nom/denom
}
## when upper boundary layer is present, calculate new C.x.up
if (!is.null(a.bl.x.up[1]) & !is.null(C.bl.x.up[1])) {
nom <- a.bl.x.up*C.bl.x.up + VF.grid$x.int[1,,]*(D.grid$x.int[1,,]/
grid$dx.aux[1] + (1-AFDW.grid$x.int[1,,])*v.grid$x.int[1,,])*C[1,,]
denom <- a.bl.x.up + VF.grid$x.int[1,,]*(D.grid$x.int[1,,]/grid$dx.aux[1]+
AFDW.grid$x.int[1,,]*v.grid$x.int[1,,])
C.x.up <- nom/denom
}
## when lower boundary layer is present, calculate new C.x.down
if (!is.null(a.bl.x.down[1]) & !is.null(C.bl.x.down[1])) {
nom <- a.bl.x.down*C.bl.x.down + VF.grid$x.int[(Nx+1),,]*(D.grid$x.int[(Nx+1),,]/
grid$dx.aux[(Nx+1)] + (1-AFDW.grid$x.int[(Nx+1),,])*
v.grid$x.int[(Nx+1),,])*C[Nx,,]
denom <- a.bl.x.down + VF.grid$x.int[(Nx+1),,]*(D.grid$x.int[(Nx+1),,]/
grid$dx.aux[(Nx+1)]+ AFDW.grid$x.int[(Nx+1),,]*v.grid$x.int[(Nx+1),,])
C.x.down <- nom/denom
}
## when upper y boundary layer is present, calculate new C.y.up
if (!is.null(a.bl.y.up[1]) & !is.null(C.bl.y.up[1])) {
nom <- a.bl.y.up*C.bl.y.up + VF.grid$y.int[,1,]*(D.grid$y.int[,1,]/
grid$dy.aux[1] + (1-AFDW.grid$y.int[,1,])*v.grid$y.int[,1,])*C[,1,]
denom <- a.bl.y.up + VF.grid$y.int[,1,]*(D.grid$y.int[,1,]/grid$dy.aux[1]+
AFDW.grid$y.int[,1,]*v.grid$y.int[,1,])
C.y.up <- nom/denom
}
## when lower y boundary layer is present, calculate new C.y.down
if (!is.null(a.bl.y.down[1]) & !is.null(C.bl.y.down[1])) {
nom <- a.bl.y.down*C.bl.y.down + VF.grid$y.int[,(Ny+1),]*
(D.grid$y.int[,(Ny+1),]/grid$dy.aux[(Ny+1)] +
(1-AFDW.grid$y.int[,(Ny+1),])*v.grid$y.int[,(Ny+1),])*C[,Ny,]
denom <- a.bl.y.down + VF.grid$y.int[,(Ny+1),]*(D.grid$y.int[,(Ny+1),]/
grid$dy.aux[(Ny+1)]+ AFDW.grid$y.int[,(Ny+1),]*v.grid$y.int[,(Ny+1),])
C.y.down <- nom/denom
}
## when upper z boundary layer is present, calculate new C.z.up
if (!is.null(a.bl.z.up[1]) & !is.null(C.bl.z.up[1])) {
nom <- a.bl.z.up*C.bl.z.up + VF.grid$z.int[,1,]*(D.grid$z.int[,1,]/
grid$dz.aux[1] + (1-AFDW.grid$z.int[,1,])*v.grid$z.int[,1,])*C[,1,]
denom <- a.bl.z.up + VF.grid$z.int[,1,]*(D.grid$z.int[,1,]/grid$dz.aux[1]+
AFDW.grid$z.int[,1,]*v.grid$z.int[,1,])
C.z.up <- nom/denom
}
## when lower z boundary layer is present, calculate new C.z.down
if (!is.null(a.bl.z.down[1]) & !is.null(C.bl.z.down[1])) {
nom <- a.bl.z.down*C.bl.z.down + VF.grid$z.int[,(Nz+1),]*
(D.grid$z.int[,,(Nz+1)]/grid$dz.aux[(Nz+1)] +
(1-AFDW.grid$z.int[,,(Nz+1)])*v.grid$z.int[,,(Nz+1)])*C[,,Nz]
denom <- a.bl.z.down + VF.grid$z.int[,,(Nz+1)]*(D.grid$z.int[,,(Nz+1)]/
grid$dz.aux[(Nz+1)]+ AFDW.grid$z.int[,,(Nz+1)]*v.grid$z.int[,,(Nz+1)])
C.z.down <- nom/denom
}
## Calculate diffusive part of the flux
# Nx = 10, Ny = 2, Nz = 3
# DX <- array(dim = c(Nx, Ny, Nz), dx.aux)
# DY <- aperm(array(dim = c(Ny,Nx,Nz), dy.aux),c(2,1,3))
# DZ <- aperm(array(dim=c(Nz,Ny,Nx),dz.aux),c(3,2,1))
x.Dif.flux <- -VF.grid$x.int * D.grid$x.int *
diff3D(mbind(C.x.up, C, C.x.down, along=1), along=1)/
array(dim=c(Nx+1,Ny,Nz),data=grid$dx.aux)
y.Dif.flux <- -VF.grid$y.int * D.grid$y.int *
diff3D(mbind(C.y.up, C, C.y.down, along=2), along=2)/
aperm(array(data=grid$dy.aux,dim=c(Ny+1,Nx,Nz)),c(2,1,3))
z.Dif.flux <- -VF.grid$z.int * D.grid$z.int *
diff3D(mbind(C.z.up, C, C.z.down, along=3), along=3)/
aperm(array(data=grid$dz.aux,dim=c(Nz+1,Ny,Nx)),c(3,2,1))
## Calculate advective part of the flux
x.Adv.flux <- 0
if (any(v.grid$x.int >0) ) {
vv <- v.grid$x.int
vv[vv<0]<-0
x.Adv.flux <- x.Adv.flux + VF.grid$x.int * vv * (
(1-AFDW.grid$x.int) * mbindl (C.x.up, C,along=1)
+ AFDW.grid$x.int * mbindr (C,C.x.down,along=1))
}
if (any (v.grid$x.int < 0)) {
vv <- v.grid$x.int
vv[vv>0]<-0
x.Adv.flux <- x.Adv.flux + VF.grid$x.int * vv * (
AFDW.grid$x.int * mbindl(C.x.up,C,along=1)
+ (1-AFDW.grid$x.int) * mbindr(C,C.x.down,along=1))
}
y.Adv.flux <- 0
if (any(v.grid$y.int >0) ) {
vv <- v.grid$y.int
vv[vv<0]<-0
y.Adv.flux <- y.Adv.flux + VF.grid$y.int * vv * (
(1-AFDW.grid$y.int) * mbindl(C.y.up,C,along=2)
+ AFDW.grid$y.int * mbindr(C,C.y.down,along=2))
}
if (any (v.grid$y.int < 0)) {
vv <- v.grid$y.int
vv[vv>0]<-0
y.Adv.flux <- y.Adv.flux + VF.grid$y.int * vv * (
AFDW.grid$y.int * mbindl(C.y.up ,C,along=2)
+ (1-AFDW.grid$y.int) * mbindr(C,C.y.down,along=2))
}
z.Adv.flux <- 0
if (any(v.grid$z.int >0) ) {
vv <- v.grid$z.int
vv[vv<0]<-0
z.Adv.flux <- z.Adv.flux + VF.grid$z.int * vv * (
(1-AFDW.grid$z.int) * mbindl(C.z.up,C,along=3)
+ AFDW.grid$z.int * mbindr(C,C.z.down,along=3))
}
if (any (v.grid$z.int < 0)) {
vv <- v.grid$z.int
vv[vv>0]<-0
z.Adv.flux <- z.Adv.flux + VF.grid$z.int * vv * (
AFDW.grid$z.int * mbindl(C.z.up,C,along=3)
+ (1-AFDW.grid$z.int) * mbindr(C,C.z.down,along=3))
}
x.flux <- x.Dif.flux + x.Adv.flux
y.flux <- y.Dif.flux + y.Adv.flux
z.flux <- z.Dif.flux + z.Adv.flux
## Impose boundary fluxes when needed
## Default boundary condition is no gradient
if (! is.null (flux.x.up[1]))
x.flux[1,,] <- flux.x.up
if (! is.null (flux.x.down[1]))
x.flux[dim(x.flux)[1],,] <- flux.x.down
if (! is.null (flux.y.up[1]))
y.flux[,1,] <- flux.y.up
if (! is.null (flux.y.down[1]))
y.flux[,dim(y.flux)[2],] <- flux.y.down
if (! is.null (flux.z.up[1]))
z.flux[,,1] <- flux.z.up
if (! is.null (flux.z.down[1]))
z.flux[,,dim(z.flux)[3]] <- flux.z.down
## Calculate rate of change = flux gradient NOG DOEN
dFdx <- - (diff3D(A.grid$x.int*x.flux,along=1)/ A.grid$x.mid/grid$dx) / VF.grid$x.mid
dFdy <- - (diff3D(A.grid$y.int*y.flux,along=2)/ A.grid$y.mid/grid$dy) / VF.grid$y.mid
dFdz <- - (diff3D(A.grid$z.int*z.flux,along=3)/ A.grid$z.mid/grid$dz) / VF.grid$z.mid
if (!full.output) {
return (list (dC = dFdx + dFdy + dFdz, # Rate of change due to advective-diffuisve transport in each grid cell
flux.x.up = x.flux[1,,], # flux across lower boundary interface; positive = IN
flux.x.down = x.flux[dim(x.flux)[1],,], # flux across lower boundary interface; positive = OUT
flux.y.up = y.flux[,1,], # flux across lower boundary interface; positive = IN
flux.y.down = y.flux[,dim(y.flux)[2],], # flux across lower boundary interface; positive = OUT
flux.z.up = z.flux[,,1], # flux across lower boundary interface; positive = IN
flux.z.down = z.flux[,,dim(z.flux)[3]])) # flux across lower boundary interface; positive = OUT
} else {
return (list (dC = dFdx + dFdy + dFdz, # Rate of change due to advective-diffuisve transport in each grid cell
C.x.up = C.x.up, # concentration at upper interface
C.x.down = C.x.down, # concentration at upper interface
C.y.up = C.y.up, # concentration at upper interface
C.y.down = C.y.down, # concentration at upper interface
C.z.up = C.z.up, # concentration at upper interface
C.z.down = C.z.down, # concentration at upper interface
x.flux = x.flux, # flux across at the interface of each grid cell
y.flux = y.flux, # flux across at the interface of each grid cell
z.flux = z.flux, # flux across at the interface of each grid cell
flux.x.up = x.flux[1,,], # flux across lower boundary interface; positive = IN
flux.x.down = x.flux[dim(x.flux)[1],,], # flux across lower boundary interface; positive = OUT
flux.y.up = y.flux[,1,], # flux across lower boundary interface; positive = IN
flux.y.down = y.flux[,dim(y.flux)[2],], # flux across lower boundary interface; positive = OUT
flux.z.up = z.flux[,,1], # flux across lower boundary interface; positive = IN
flux.z.down = z.flux[,,dim(z.flux)[3]])) # flux across lower boundary interface; positive = OUT
}
} # end tran.3D
