# Dynlib diag functions

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## Dynlib functions

The functions generally operate on real arrays with dimension (nz,ny,nx) where nz is number of times or levels, and ny and nx are the number of latitudes and longitudes, respectively. The function descriptions below contain detailed descriptions of arguments and returns where there is any deviation from this pattern; otherwise they may be assumed to be of the form:

Type Dim Description real (nz,ny,nx) Zonal velocity real (nz,ny,nx) Meridional velocity real (nz,ny,nx) Potential vorticity
Type Dim Description real (nz,ny,nx) output data

The ubiquitous inputs dx and dy are all of the form

Type Dim Description real (ny,nx) dx(j,i) = x(j, i+1) - x(j, i-1) (in metres) real (ny,nx) dy(j,i) = y(j+1, i) - y(j-1, i) (in metres)

Typically, the results for each level or time are computed individually in 2-D fashion, though they are returned as a 3-D array of the same size as the input.

### ddx : partial x derivative

`res=ddx(dat,dx,dy)`

Calculates the partial x derivative of dat, using centred differences. For a non-EW-cyclic grid, 0 is returned on the edges of the x domain.

### ddy : partial y derivative

`res=ddy(dat,dx,dy)`

Calculates the partial y derivative of dat, using centred differences. For a non-EW-cyclic grid, 0 is returned on all edges of the x,y domain. For an EW-cyclic grid, 0 is returned on the first and last latitudes.

`(resx,resy)=grad(dat,dx,dy)`

Calculates the 2-D gradient of dat, using centred differences in x and y. For a non-EW-cyclic grid, 0 is returned on all edges of the x,y domain. For an EW-cyclic grid, 0 is returned on the first and last latitudes.

### lap2 : 2-D laplacian

`res=lap2(dat,dx,dy)`

Calculates the 2-D laplacian of dat, using centred differences. For a non-EW-cyclic grid, 0 is returned on all edges of the x,y domain. For an EW-cyclic grid, 0 is returned on the first and last latitudes.

### vor : 2-D vorticity

`res=vor(u,v,dx,dy)`

Calculates the z component of vorticity of (u,v), using centred differences.

### div : 2-D divergence

`res=div(u,v,dx,dy)`

Calculates the 2-D divergence of (u,v), using centred differences.

### def_shear : shear deformation

`res=def_shear(u,v,dx,dy)`

Calculates the shear (antisymmetric) deformation of (u,v), using centred differences.

### def_stretch : stretch deformation

`res=def_stretch(u,v,dx,dy)`

Calculates the stretch (symmetric) deformation of (u,v), using centred differences.

### def_total : total deformation

`res=def_total(u,v,dx,dy)`

Calculates the total (rotation-independent) deformation of (u,v), using centred differences.

### def_angle : deformation angle

`res=def_angle(u,v,dx,dy)`

Calculates the angle between the x-axis and the dilatation axis of the deformation of (u,v).

### isopv_angle : iso-PV contour angle

`res=isopv_angle(pv,dx,dy)`

Calculates the angle between the x-axis and the iso-lines of PV.

### beta : angle between dilatation axis and iso-PV contours

`res=beta(u,v,pv,dx,dy)`

Calculates the angle between the dilatation axis and the iso-lines of PV.

### stretch_stir : fractional stretching rate and angular rotation rate of grad(PV)

`(stretch,stir)=stretch_stir(u,v,pv,dx,dy)`

Returns real arrays, dim (nz,ny,nx):

```stretch
= fractional PV gradient stretching rate
= gamma, 'stretching rate' (Lapeyre Klein Hua)LapKleHua1999
= -1/|gradPV| * Fn (Keyser Reeder Reed)KeyReeRee1988
=  1/|gradPV| * F (Markowski Richardson)Mar2010
```
```stir
= angular rotation rate of grad(PV) (aka stirring rate)
= d(theta)/dt (Lapeyre Klein Hua)LapKleHua1999
```
```= 1/|gradPV| * Fs  (Keyser Reeder Reed)KeyReeRee1988
```

### geop_from_montgp : geopotential

`res = geop_from_montgp(m,theta,p,dx,dy)`

Calculates geopotential (res) from montgomery potential (m), potential temperature (theta) and pressure (p)

### rev : PV gradient reversal

`(resa,resc,resai,resci,resaiy,resciy,tested) = rev(pv,highenough,latitudes,ddythres,dx,dy)`

Finds the reversals of PV y-gradient (where the negative y-gradient exceeds some threshold) and classifies them as c (cyclonic) or a (anticyclonic). Three measures of c and a reversals are returned (6 in total). Only points flagged in highenough are tested.

Type Dim Description real (nz,ny,nx) potential vorticity int*1 (nz,ny,nx) array of flags denoting whether to test the point for reversal real (ny) vector of latitudes real 0 Cutoff y-gradient for pv

Highenough is typically the output of the highenough function, which returns 1 where the surface is sufficiently above ground level and 0 elsewhere.

ddythres is the cutoff y-gradient for pv. The magnitude of (negative) d(pv)/dy must be above ddythres for reversal to be detected; this applies to revc, reva, revci,revai. Typical value: 4E-12.

Type Dim Description int*1 (nz,ny,nx) Flag =1 for cyclonic reversal (threshold test applied) int*1 (nz,ny,nx) Flag =1 for anticyclonic reversal (threshold test applied) real (nz,ny,nx) Absolute PV gradient where reversal is cyclonic (threshold test applied) real (nz,ny,nx) Absolute PV gradient where reversal is anticyclonic (threshold test applied) real (nz,ny,nx) Absolute PV y-gradient where reversal is cyclonic (no threshold test applied) real (nz,ny,nx) Absolute PV y-gradient where reversal is anticyclonic (no threshold test applied) int*1 (nz,ny,nx) flag to 1 all tested points: where highenough==1 and point not on grid edge

### prepare_fft : make data periodic in y for FFT

`res = prepare_fft(thedata,dx,dy)`

Returns the data extended along complementary meridians (for fft). For each lon, the reflected (lon+180) is attached below so that data is periodic in x and y. NOTE: Input data must be lats -90 to 90, and nx must be even.

Type Dim Description real (nz,ny,nx) input data
Type Dim Description real (nz,2*ny-2,nx) output data

### sum_kix : sum along k for flagged k-values

`(res,nres) = sum_kix(thedata,kix,dx,dy)`

Calculates sum along k dimension for k values which are flagged to 1 in kix vector (length nz).

Type Dim Description real (nz,ny,nx) input data int (nz) index flag for summation
Type Dim Description real (ny,nx) (summed) output data int 0 Number of data summed = sum(kix)

Sum_kix is typically used for calculating seasonal means. To do this, kix is set to 1 for times in the relevant season and 0 elsewhere. After (further) summing res and nres over all years, res/nres gives the mean for the season for all years.

### high_enough : flags points which are sufficiently above ground

`res = high_enough(zdata,ztest,zthres,dx,dy)`

Type Dim Description real (nz,ny,nx) geopotential of gridpoints real (1,ny,nx) geopotential of topography real 0 threshold geopotential height difference

Returns:
Type Dim Description
res int*1 (nz,ny,nx)
 Flag array set to: 1 if zdata(t,y,x) > (ztest(1,y,x) + zthres) 0 otherwise

### contour_rwb : detects RWB events, Riviere algorithm

`(beta_a_out,beta_c_out) = contour_rwb(pv_in,lonvalues,latvalues,ncon,lev,dx,dy)`

Detects the occurrence of anticyclonic and cyclonic wave-breaking events from a PV field on isentropic coordinates.

Reference: Riviere 2009Riv2009 : See the appendix C.

Type Dim Description real (nz,ny,nx) isentropic pv. Should be on a regular lat-lon grid and 180W must be the first longitude. (If 180W is not the first longitude, the outputs will have 180W as the first, so must be rearranged) real (nx) vector of longitudes real (ny) vector of latitudes int 0 number of contours to test, normally 41 or 21 real 0 potential temperature of the level
Type Dim Description int (nz,ny,nx) flag array, =1 if anticyclonic wave breaking int (nz,ny,nx) flag array, =1 if cyclonic wave breaking

### v_g : geostrophic velocity

`(resx,resy) = v_g(mont,lat,dx,dy)`

Calculates geostrophic velocity. Returns zero on equator.

### okuboweiss : Okubo-Weiss criterion

`res = okuboweiss(u,v,dx,dy)`

Calculates Okubo-Weiss criterion lambda_0=1/4 * (sigma^2-omega^2)= 1/4 W, where sigma is total deformation and omega is vorticity.

This is the square of the eigenvalues in Okubo's paperOku1969 (assumes divergence is negligible).

### laccel : Lagrangian acceleration

`(resx,resy) = laccel(u,v,mont,lat,dx,dy)`

Calculates Lagrangian acceleration on the isentropic surface, based on Montgomery potential.

Type Dim Description real (nz,ny,nx) zonal velocity real (nz,ny,nx) meridional velocity real (nz,ny,nx) Montgomery potential real (ny) vector of latitudes

`(respr,respi,resmr,resmi) = accgrad_eigs(u,v,mont,lat,dx,dy)`

Calculates eigenvalues of the lagrangian acceleration gradient tensor.

Type Dim Description real (nz,ny,nx) zonal velocity real (nz,ny,nx) meridional velocity real (nz,ny,nx) Montgomery potential real (ny) vector of latitudes
Type Dim Description real (nz,ny,nx) Real part of positive eigenvalue real (nz,ny,nx) Imaginary part of positive eigenvalue real (nz,ny,nx) Real part of negative eigenvalue real (nz,ny,nx) Imaginary part of negative eigenvalue int 0 number of contours to test, normally 41 or 21 real 0 potential temperature of the level

### dphidt : Lagrangian derivative of compression axis angle

`res = dphidt(u,v,mont,lat,dx,dy)`

Calculates Lagrangian time derivative of compression axis angle: d(phi)/dt (ref Lapeyre et. al 1999LapKleHua1999 ), from deformation and Lagrangian acceleration tensor.

Type Dim Description real (nz,ny,nx) zonal velocity real (nz,ny,nx) meridional velocity real (nz,ny,nx) Montgomery potential real (ny) vector of latitudes