curl | 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 `\partial `

` v/\partial x - \partial u/\partial y`

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,`\partial v/\partial x`

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,`\partial v/\partial x`

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 things 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)
```

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