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gsw_rho_second_derivatives_CT_exact.m
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function [rho_SA_SA, rho_SA_CT, rho_CT_CT, rho_SA_P, rho_CT_P] = gsw_rho_second_derivatives_CT_exact(SA,CT,p)
% gsw_rho_second_derivatives_CT_exact SA and CT partial second
% order derivatives of density
%==========================================================================
%
% USAGE:
% [rho_SA_SA, rho_SA_CT, rho_CT_CT, rho_SA_P, rho_CT_P] = ...
% gsw_rho_second_derivatives_CT_exact(SA,CT,p)
%
% DESCRIPTION:
% Calculates the following five second-order derivatives of rho,
% (1) rho_SA_SA, second-order derivative with respect to Absolute
% Salinity at constant CT & p.
% (2) rho_SA_CT, second-order derivative with respect to SA & CT at
% constant p.
% (3) rho_CT_CT, second-order derivative with respect to CT at
% constant SA & p.
% (4) rho_SA_P, second-order derivative with respect to SA & P at
% constant CT.
% (5) rho_CT_P, second-order derivative with respect to CT & P at
% constant SA.
%
% Note that this function uses the full Gibbs function. There is an
% alternative to calling this function, namely
% gsw_rho_second_derivatives(SA,CT,p), which uses the computationally
% efficient polynomial for specific volume in terms of SA, CT and p
% (Roquet et al., 2015).
%
% 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:
% rho_SA_SA = The second-order derivative of rho with respect to
% Absolute Salinity at constant CT & p. [ (kg/m^3)(g/kg)^-2 ]
% rho_SA_CT = The second-order derivative of rho with respect to
% SA and CT at constant p. [ (kg/m^3)(g/kg)^-1 K^-1 ]
% rho_CT_CT = The second-order derivative of rho with respect to CT at
% constant SA & p [ (kg/m^3) K^-2 ]
% rho_SA_P = The second-order derivative with respect to SA & P at
% constant CT. [ (kg/m^3) (g/kg)^-1 Pa^-1 ]
% rho_CT_P = The second-order derivative with respect to CT & P at
% constant SA. [ (kg/m^3) K^-1 Pa^-1 ]
%
% AUTHOR:
% Paul Barker and Trevor McDougall [ help@teos-10.org ]
%
% VERSION NUMBER: 3.05 (27th January 2015)
%
% 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 appendix A.20 and appendix K of this TEOS-10 Manual.
%
% 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_rho_second_derivatives_CT_exact: Requires three inputs')
end %if
if ~(nargout == 5)
error('gsw_rho_second_derivatives_CT_exact: Requires five outputs')
end %if
[ms,ns] = size(SA);
[mt,nt] = size(CT);
[mp,np] = size(p);
if (mt ~= ms | nt ~= ns)
error('gsw_rho_second_derivatives_CT_exact: SA and CT must have same dimensions')
end
if (mp == 1) & (np == 1) % p scalar - fill to size of SA
p = p*ones(size(SA));
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 (ns == mp) & (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_rho_second_derivatives_CT_exact: Inputs array dimensions arguments do not agree')
end %if
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;
rec_v = 1./gsw_specvol_CT_exact(SA,CT,p);
[v_SA, v_CT, v_P] = gsw_specvol_first_derivatives_CT_exact(SA,CT,p);
[v_SA_SA, v_SA_CT, v_CT_CT, v_SA_P, v_CT_P] = gsw_specvol_second_derivatives_CT_exact(SA,CT,p);
rec_v2 = rec_v.^2;
rec_v3 = rec_v2.*rec_v;
rho_CT_CT = -v_CT_CT.*rec_v2 + 2.*v_CT.^2.*rec_v3;
rho_SA_CT = -v_SA_CT.*rec_v2 + 2.*v_SA.*v_CT.*rec_v3;
rho_SA_SA = -v_SA_SA.*rec_v2 + 2.*v_SA.^2.*rec_v3;
rho_SA_P = -v_SA_P.*rec_v2 + 2.*v_SA.*v_P.*rec_v3;
rho_CT_P = -v_CT_P.*rec_v2 + 2.*v_CT.*v_P.*rec_v3;
if transposed
rho_SA_SA = rho_SA_SA.';
rho_SA_CT = rho_SA_CT.';
rho_CT_CT = rho_CT_CT.';
rho_SA_P = rho_SA_P.';
rho_CT_P = rho_CT_P.';
end
end