curl: Curl of 2D Vector Field

In oce: Analysis of Oceanographic Data

Curl of 2D Vector Field

Description

Calculate the z component of the curl of an x-y vector field.

Usage

curl(u, v, x, y, geographical = FALSE, method = 1)

Arguments

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 u and v.
y	the y values for the matrices, a vector of length equal to the number of cols in u and v.
geographical	logical value indicating whether x and y are longitude and latitude, in which case spherical trigonometry is used.
method	A number indicating the method to be used to calculate the first-difference approximations to the derivatives. See “Details”.

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.

Value

A list containing vectors x and y, along with matrix curl. See “Details” for the lengths and dimensions, for various values of method.

Author(s)

Dan Kelley and Chantelle Layton

See Also

Other things relating to vector calculus: grad()

Examples

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)

oce documentation built on Sept. 11, 2024, 7:09 p.m.