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gsw_internal_energy_second_derivatives.m
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function [u_SA_SA, u_SA_CT, u_CT_CT, u_SA_P, u_CT_P] = gsw_internal_energy_second_derivatives(SA,CT,p)
% gsw_internal_energy_second_derivatives second derivatives of
% specific interal energy of seawater
% (75-term equation)
%==========================================================================
%
% USAGE:
% [u_SA_SA, u_SA_CT, u_CT_CT, u_SA_P, u_CT_P] = ...
% gsw_internal_energy_second_derivatives(SA,CT,p)
%
% DESCRIPTION:
% Calculates the following five second-order derivatives of
% internal energy,
% (1) u_SA_SA, second order derivative with respect to Absolute Salinity
% at constant CT & p.
% (2) u_SA_CT, second order derivative with respect to SA & CT at
% constant p.
% (3) u_CT_CT, second order derivative with respect to CT at constant
% SA & p.
% (4) u_SA_P, second-order derivative with respect to SA & P at
% constant CT.
% (5) u_CT_P, second-order derivative with respect to CT & P at
% constant SA.
%
% Note that this function uses the using the computationally-efficient
% 75-term expression for specific volume (Roquet et al., 2015). There is
% an alternative to calling this function, namely
% gsw_internal_energy_second_derivatives_CT_exact(SA,CT,p) which uses the
% full Gibbs function (IOC et al., 2010).
%
% Note that the 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 [ 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:
% u_SA_SA = The second derivative of internal energy with respect to
% Absolute Salinity at constant CT & p. [ (J/kg)(g/kg)^-2 ]
% u_SA_CT = The second derivative of internal energy with respect to
% SA & CT at constant p. [ (J/kg)(g/kg)^-1 K^-1]
% u_CT_CT = The second derivative of internal energy with respect to
% CT at constant SA and p. [ (J/kg) K^-2 ]
% u_SA_P = The second derivative of internal energy with respect to
% SA & P at constant CT. [ (J/kg)(g/kg)^-1 Pa^-1 ]
% u_CT_P = The second derivative of internal energy with respect to
% CT & P at constant SA. [ (J/kg) K^-1 Pa^-1 ]
%
% AUTHOR:
% Trevor McDougall and Paul Barker. [ 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
%
% 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_internal_energy_second_derivatives: requires three inputs')
end
if ~(nargout == 5)
error('gsw_internal_energy_second_derivatives: requires five outputs')
end
[ms,ns] = size(SA);
[mt,nt] = size(CT);
[mp,np] = size(p);
if (mt ~= ms | nt ~= ns)
error('gsw_internal_energy_second_derivatives: 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_internal_energy_second_derivatives: 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;
db2Pa = 1e4; % dbar to Pa conversion factor
P = (db2Pa.*p + gsw_P0);
[h_SA_SA, h_SA_CT, h_CT_CT] = gsw_enthalpy_second_derivatives(SA,CT,p);
[v_SA_SA, v_SA_CT, v_CT_CT, v_SA_P, v_CT_P] = gsw_specvol_second_derivatives(SA,CT,p);
u_SA_SA = h_SA_SA - P.*v_SA_SA;
u_SA_CT = h_SA_CT - P.*v_SA_CT;
u_CT_CT = h_CT_CT - P.*v_CT_CT;
u_SA_P = -P.*v_SA_P;
u_CT_P = -P.*v_CT_P;
if transposed
u_SA_SA = u_SA_SA.';
u_SA_CT = u_SA_CT.';
u_CT_CT = u_CT_CT.';
u_SA_P = u_SA_P.';
u_CT_P = u_CT_P.';
end
end