Compute the gradient of a scalar function and return grids of the two horizontal components that conform with Driscoll and Healy’s (1994) sampling theorem.
Usage
call MakeGradientDH (cilm, lmax, theta, phi, n, sampling, lmax_calc, extend, radius, exitstatus)
Parameters
cilm: input, real(dp), dimension (2,lmax+1,lmax+1)- The real 4-pi normalized spherical harmonic coefficients of a scalar function. The coefficients c1lm and c2lm refer to the cosine and sine coefficients, respectively, with
c1lm=cilm(1,l+1,m+1)andc2lm=cilm(2,l+1,m+1). lmax: input, integer(int32)- The maximum spherical harmonic degree of the coefficients
cilm. This determines the number of samples of the output grids,n=2lmax+2, and the latitudinal sampling interval,90/(lmax+1). theta: output, real(dp), dimension (nlat, nlong)- A 2D map of theta component of the gradient that conforms to the sampling theorem of Driscoll and Healy (1994). If
samplingis 1, the grid is equally sampled and is dimensioned as (nbyn), wherenis2lmax+2. If sampling is 2, the grid is equally spaced and is dimensioned as (nby 2n). The first latitudinal band of the grid corresponds to 90 N, the latitudinal sampling interval is 180/ndegrees, and the default behavior is to exclude the latitudinal band for 90 S. The first longitudinal band of the grid is 0 E, by default the longitudinal band for 360 E is not included, and the longitudinal sampling interval is 360/nfor an equally sampled and 180/nfor an equally spaced grid, respectively. Ifextendis 1, the longitudinal band for 360 E and the latitudinal band for 90 S will be included, which increases each of the dimensions of the grid by 1. phi: output, real(dp), dimension (nlat, nlong)- A 2D equally sampled or equally spaced grid of the phi component of the horizontal gradient.
n: output, integer(int32)- The number of samples in latitude of the output grids. This is equal to
2lmax+2. sampling: optional, input, integer(int32), default = 1- If 1 (default) the output grids are equally sampled (
nbyn). If 2, the grids are equally spaced (nby 2n). lmax_calc: optional, input, integer(int32)- The maximum spherical harmonic degree used in evaluating the functions. This must be less than or equal to
lmax. extend: input, optional, integer(int32), default = 0- If 1, compute the longitudinal band for 360 E and the latitudinal band for 90 S. This increases each of the dimensions of the grids by 1.
radius: input, optional, real(dp), default = 1.0- The radius of the sphere used when computing the gradient of the function.
exitstatus: output, optional, integer(int32)- If present, instead of executing a STOP when an error is encountered, the variable exitstatus will be returned describing the error. 0 = No errors; 1 = Improper dimensions of input array; 2 = Improper bounds for input variable; 3 = Error allocating memory; 4 = File IO error.
Description
MakeGradientDH will compute the horizontal gradient of a scalar function on a sphere defined by the spherical harmonic coefficients cilm. The output grids of the theta and phi components of the gradient are either equally sampled (n by n) or equally spaced (n by 2n) in latitude and longitude. The gradient is given by the formula
Grad F = 1/r dF/theta theta-hat + 1/(r sin theta) dF/dphi phi-hat.
where theta is colatitude and phi is longitude. The radius r is by default set to 1, but this can be modified by use of the optional parameter radius.
The default is to use an input grid that is equally sampled (n by n), but this can be changed to use an equally spaced grid (n by 2n) by the optional argument sampling. The redundant longitudinal band for 360 E and the latitudinal band for 90 S are excluded by default, but these can be computed by specifying the optional argument extend.
Reference
Driscoll, J.R. and D.M. Healy, Computing Fourier transforms and convolutions on the 2-sphere, Adv. Appl. Math., 15, 202-250, 1994.