curl {oce} | R Documentation |
Calculate the z component of the curl of an x-y vector field.
curl(u, v, x, y, geographical = FALSE, method = 1)
u |
matrix containing the 'x' component of a vector field |
v |
matrix containing the 'y' component of a vector field |
x |
the x values for the matrices, a vector of length equal to the
number of rows in |
y |
the y values for the matrices, a vector of length equal to the
number of cols in |
geographical |
logical value indicating whether |
method |
A number indicating the method to be used to calculate the first-difference approximations to the derivatives. See “Details”. |
The computed component of the curl is defined by dv/dx -
du/dydv/dx - du/dy and the
estimate is made using first-difference approximations to the derivatives.
Two methods are provided, selected by the value of method
.
For method=1
, a centred-difference, 5-point stencil is used in
the interior of the domain. For example, dv/dx
is given by the ratio of v[i+1,j]-v[i-1,j] to the
x extent of the grid cell at index j. (The cell extents depend on
the value of geographical
.) Then, the edges are filled in with
nearest-neighbour values. Finally, the corners are filled in with the
adjacent value along a diagonal. If geographical=TRUE
, then x
and y
are taken to be longitude and latitude in degrees, and the
earth shape is approximated as a sphere with radius 6371km. The resultant
x
and y
are identical to the provided values, and the
resultant curl
is a matrix with dimension identical to that of
u
.
For method=2
, each interior cell in the grid is considered
individually, with derivatives calculated at the cell center. For example,
dv/dx is given by the ratio of
0.5*(v[i+1,j]+v[i+1,j+1]) - 0.5*(v[i,j]+v[i,j+1])
to the average of the x extent of the grid cell at indices j and
j+1. (The cell extents depend on the value of
geographical
.) The returned x
and y
values are the
mid-points of the supplied values. Thus, the returned x
and y
are shorter than the supplied values by 1 item, and the returned curl
matrix dimensions are similarly reduced compared with the dimensions of
u
and v
.
A list containing vectors x
and y
, along with matrix
curl
. See “Details” for the lengths and dimensions, for
various values of method
.
This function is under active development as of December 2014 and is unlikely to be stabilized until February 2015.
Dan Kelley and Chantelle Layton
Other functions relating to vector calculus: grad
library(oce) ## 1. Shear flow with uniform curl. x <- 1:4 y <- 1:10 u <- outer(x, y, function(x, y) y/2) v <- outer(x, y, function(x, y) -x/2) C <- curl(u, v, x, y, FALSE) ## 2. Rankine vortex: constant curl inside circle, zero outside rankine <- function(x, y) { r <- sqrt(x^2 + y^2) theta <- atan2(y, x) speed <- ifelse(r < 1, 0.5*r, 0.5/r) list(u=-speed*sin(theta), v=speed*cos(theta)) } x <- seq(-2, 2, length.out=100) y <- seq(-2, 2, length.out=50) u <- outer(x, y, function(x, y) rankine(x, y)$u) v <- outer(x, y, function(x, y) rankine(x, y)$v) C <- curl(u, v, x, y, FALSE) ## plot results par(mfrow=c(2, 2)) imagep(x, y, u, zlab="u", asp=1) imagep(x, y, v, zlab="v", asp=1) imagep(x, y, C$curl, zlab="curl", asp=1) hist(C$curl, breaks=100)