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gsw_Turner_Rsubrho.m
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function [Tu, Rsubrho, p_mid] = gsw_Turner_Rsubrho(SA,CT,p)
% gsw_Turner_Rsubrho Turner angle & Rsubrho (75-term equation)
%==========================================================================
%
% USAGE:
% [Tu, Rsubrho, p_mid] = gsw_Turner_Rsubrho(SA,CT,p)
%
% DESCRIPTION:
% Calculates the Turner angle and the Rsubrho as a function of pressure
% down a vertical water column. These quantities express the relative
% contributions of the vertical gradients of Conservative Temperature
% and Absolute Salinity to the vertical stability (the square of the
% Brunt-Vaisala Frequency squared, N^2). Tu and Rsubrho are evaluated at
% the mid pressure between the individual data points in the vertical.
% This function uses computationally-efficient 75-term expression for
% specific volume in terms of SA, CT and p (Roquet et al., 2015). Note
% that in the double-diffusive literature, papers concerned with the
% "diffusive" form of double-diffusive convection often define the
% stability ratio as the reciprocal of what is defined here as the
% stability ratio.
%
% Note that this 75-term equation has been fitted in a restricted range of
% parameter space, and is most accurate inside the "oceanographic funnel"
% described in McDougall et al. (2003). The GSW library function
% "gsw_infunnel(SA,CT,p)" is avaialble to be used if one wants to test if
% some of one's data lies outside this "funnel".
%
% INPUT:
% SA = Absolute Salinity [ g/kg ]
% CT = Conservative Temperature (ITS-90) [ deg C ]
% p = sea pressure [ dbar ]
% ( i.e. absolute pressure - 10.1325 dbar )
%
% SA & CT need to have the same dimensions,
% p may have dimensions 1x1 or Mx1 or 1xN or MxN, where SA & CT are MxN.
%
% OUTPUT:
% Tu = Turner angle, on the same (M-1)xN grid as p_mid.
% Turner angle has units of: [ degrees of rotation ]
% Rsubrho = Stability Ratio, on the same (M-1)xN grid as p_mid.
% Rsubrho is dimensionless. [ unitless ]
% p_mid = mid pressure between the indivual points of the p grid.
% That is, p_mid is on a (M-1)xN grid in the vertical.
% p_mid has units of: [ dbar ]
%
% AUTHOR:
% Trevor McDougall & Paul Barker [ [email protected] ]
%
% VERSION NUMBER: 3.04 (10th December, 2013)
%
% REFERENCES:
% IOC, SCOR and IAPSO, 2010: The international thermodynamic equation of
% seawater - 2010: Calculation and use of thermodynamic properties.
% Intergovernmental Oceanographic Commission, Manuals and Guides No. 56,
% UNESCO (English), 196 pp. Available from http://www.TEOS-10.org
% See Eqns. (3.15.1) and (3.16.1) of this TEOS-10 Manual.
%
% McDougall, T.J., D.R. Jackett, D.G. Wright and R. Feistel, 2003:
% Accurate and computationally efficient algorithms for potential
% temperature and density of seawater. J. Atmosph. Ocean. Tech., 20,
% pp. 730-741.
%
% Roquet, F., G. Madec, T.J. McDougall, P.M. Barker, 2015: Accurate
% polynomial expressions for the density and specifc volume of seawater
% using the TEOS-10 standard. Ocean Modelling.
%
% The software is available from http://www.TEOS-10.org
%
%==========================================================================
%--------------------------------------------------------------------------
% Check variables and resize if necessary
%--------------------------------------------------------------------------
if ~(nargin == 3)
error('gsw_Turner_Rsubrho: Requires three inputs')
end %if
if ~(nargout == 3 )
error('gsw_Turner_Rsubrho: Requires three outputs')
end %if
[ms,ns] = size(SA);
[mt,nt] = size(CT);
[mp,np] = size(p);
if (mt ~= ms | nt ~= ns )
error('gsw_Turner_Rsubrho: SA and CT must have same dimensions')
end
if (ms*ns == 1)
error('There must be at least 2 values')
end
if (mp == 1) & (np == 1) % p is a scalar - must be two bottles
error('There must be at least 2 pressure values')
elseif (ns == np) & (mp == 1) % p is row vector,
p = p(ones(1,ms), :); % copy down each column.
elseif (ms == mp) & (np == 1) % p is column vector,
p = p(:,ones(1,ns)); % copy across each row.
elseif (np == ms) & (np == 1) % p is a transposed row vector,
p = p.'; % transposed then
p = p(ones(1,ms), :); % copy down each column.
elseif (ms == mp) & (ns == np)
% ok
else
error('gsw_Turner_Rsubrho: Inputs array dimensions arguments do not agree')
end %if
[mp,np] = size(p);
if ms == 1
SA = SA.';
CT = CT.';
p = p.';
transposed = 1;
else
transposed = 0;
end
%--------------------------------------------------------------------------
% Start of the calculation
%--------------------------------------------------------------------------
% This line ensures that SA is non-negative.
SA(SA < 0) = 0;
Ishallow = 1:(mp-1);
Ideep = 2:mp;
p_mid = 0.5*(p(Ishallow,:) + p(Ideep,:));
SA_mid = 0.5*(SA(Ishallow,:) + SA(Ideep,:));
CT_mid = 0.5*(CT(Ishallow,:) + CT(Ideep,:));
dSA = SA(Ishallow,:) - SA(Ideep,:);
dCT = CT(Ishallow,:) - CT(Ideep,:);
[dummy, alpha, beta] = gsw_specvol_alpha_beta(SA_mid,CT_mid,p_mid);
%--------------------------------------------------------------------------
% This function evaluates Tu and Rsubrho using the computationally-efficient
% 75-term expression for specific volume in terms of SA, CT and p. If one
% wanted to compute Tu and Rsubrho using the full TEOS-10 Gibbs function
% expression, the following lines of code would do that.
%
% t_mid = gsw_t_from_CT(SA_mid,CT_mid,p_mid);
% beta = gsw_beta_const_CT_t_exact(SA_mid,t_mid,p_mid);
% alpha = gsw_alpha_wrt_CT_t_exact(SA_mid,t_mid,p_mid);
%
% --------------This is the end of the alternative code--------------------
Tu = atan2((alpha.*dCT + beta.*dSA),(alpha.*dCT - beta.*dSA));
Tu = Tu.*(180/pi);
Rsubrho = nan(size(dSA));
Rsubrho(dSA ~= 0) = (alpha(dSA ~= 0).*dCT(dSA ~= 0))./(beta(dSA ~= 0).*dSA(dSA ~= 0));
if transposed
Tu = Tu.';
Rsubrho = Rsubrho.';
p_mid = p_mid.';
end
end