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gsw_beta.m
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function beta = gsw_beta(SA,CT,p)
% gsw_beta saline contraction coefficient at constant
% Conservative Temperature (75-term equation)
%==========================================================================
%
% USAGE:
% beta = gsw_beta(SA,CT,p)
%
% DESCRIPTION:
% Calculates the saline (i.e. haline) contraction coefficient of seawater
% at constant Conservative Temperature using the computationally-efficient
% 75-term expression for specific volume in terms of SA, CT and p
% (Roquet et al., 2015).
%
% 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 (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:
% beta = saline contraction coefficient [ kg/g ]
% at constant Conservative Temperature
%
% AUTHOR:
% Paul Barker and Trevor McDougall [ [email protected] ]
%
% 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 Eqn. (2.19.3) 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_beta: Requires three inputs')
end %if
[ms,ns] = size(SA);
[mt,nt] = size(CT);
[mp,np] = size(p);
if (mt ~= ms | nt ~= ns)
error('gsw_beta: 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_beta: 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;
%deltaS = 24;
sfac = 0.0248826675584615; % sfac = 1/(40*(35.16504/35)).
offset = 5.971840214030754e-1; % offset = deltaS*sfac.
x2 = sfac.*SA;
xs = sqrt(x2 + offset);
ys = CT.*0.025;
z = p.*1e-4;
b000 = -3.1038981976e-4;
b001 = 2.4262468747e-5;
b002 = -5.8484432984e-7;
b003 = 3.6310188515e-7;
b004 = -1.1147125423e-7;
b010 = 3.5009599764e-5;
b011 = -9.5677088156e-6;
b012 = -5.5699154557e-6;
b013 = -2.7295696237e-7;
b020 = -3.7435842344e-5;
b021 = -2.3678308361e-7;
b022 = 3.9137387080e-7;
b030 = 2.4141479483e-5;
b031 = -3.4558773655e-6;
b032 = 7.7618888092e-9;
b040 = -8.7595873154e-6;
b041 = 1.2956717783e-6;
b050 = -3.3052758900e-7;
b100 = 1.33856134076e-3;
b101 = -6.9584921948e-5;
b102 = -9.62445031940e-6;
b103 = 3.3492607560e-8;
b110 = -8.7185357122e-5;
b111 = 2.2201669530e-5;
b112 = 1.09241497668e-5;
b120 = 7.1815645520e-5;
b121 = 5.8566692590e-6;
b122 = -1.31462208134e-6;
b130 = -2.8707266096e-5;
b131 = 6.3310612156e-7;
b140 = 8.7407361196e-6;
b200 = -2.55143801811e-3;
b201 = 1.12412331915e-4;
b202 = 1.47789320994e-5;
b210 = 1.03597385484e-4;
b211 = -2.95341353532e-5;
b212 = -4.0632556881e-6;
b220 = -5.6095752561e-5;
b221 = -1.4647841760e-6;
b230 = 6.8589973668e-6;
b300 = 2.32344279772e-3;
b301 = -6.9288874448e-5;
b302 = -7.1247898908e-6;
b310 = -4.7837639152e-5;
b311 = 1.0363690104e-5;
b320 = 1.54381356976e-5;
b400 = -1.05461852535e-3;
b401 = 1.54637136265e-5;
b410 = 6.9322972905e-6;
b500 = 1.9159474383e-4;
v000 = 1.0769995862e-3;
v001 = -6.0799143809e-5;
v002 = 9.9856169219e-6;
v003 = -1.1309361437e-6;
v004 = 1.0531153080e-7;
v005 = -1.2647261286e-8;
v006 = 1.9613503930e-9;
v010 = -1.5649734675e-5;
v011 = 1.8505765429e-5;
v012 = -1.1736386731e-6;
v013 = -3.6527006553e-7;
v014 = 3.1454099902e-7;
v020 = 2.7762106484e-5;
v021 = -1.1716606853e-5;
v022 = 2.1305028740e-6;
v023 = 2.8695905159e-7;
v030 = -1.6521159259e-5;
v031 = 7.9279656173e-6;
v032 = -4.6132540037e-7;
v040 = 6.9111322702e-6;
v041 = -3.4102187482e-6;
v042 = -6.3352916514e-8;
v050 = -8.0539615540e-7;
v051 = 5.0736766814e-7;
v060 = 2.0543094268e-7;
v100 = -3.1038981976e-4;
v101 = 2.4262468747e-5;
v102 = -5.8484432984e-7;
v103 = 3.6310188515e-7;
v104 = -1.1147125423e-7;
v110 = 3.5009599764e-5;
v111 = -9.5677088156e-6;
v112 = -5.5699154557e-6;
v113 = -2.7295696237e-7;
v120 = -3.7435842344e-5;
v121 = -2.3678308361e-7;
v122 = 3.9137387080e-7;
v130 = 2.4141479483e-5;
v131 = -3.4558773655e-6;
v132 = 7.7618888092e-9;
v140 = -8.7595873154e-6;
v141 = 1.2956717783e-6;
v150 = -3.3052758900e-7;
v200 = 6.6928067038e-4;
v201 = -3.4792460974e-5;
v202 = -4.8122251597e-6;
v203 = 1.6746303780e-8;
v210 = -4.3592678561e-5;
v211 = 1.1100834765e-5;
v212 = 5.4620748834e-6;
v220 = 3.5907822760e-5;
v221 = 2.9283346295e-6;
v222 = -6.5731104067e-7;
v230 = -1.4353633048e-5;
v231 = 3.1655306078e-7;
v240 = 4.3703680598e-6;
v300 = -8.5047933937e-4;
v301 = 3.7470777305e-5;
v302 = 4.9263106998e-6;
v310 = 3.4532461828e-5;
v311 = -9.8447117844e-6;
v312 = -1.3544185627e-6;
v320 = -1.8698584187e-5;
v321 = -4.8826139200e-7;
v330 = 2.2863324556e-6;
v400 = 5.8086069943e-4;
v401 = -1.7322218612e-5;
v402 = -1.7811974727e-6;
v410 = -1.1959409788e-5;
v411 = 2.5909225260e-6;
v420 = 3.8595339244e-6;
v500 = -2.1092370507e-4;
v501 = 3.0927427253e-6;
v510 = 1.3864594581e-6;
v600 = 3.1932457305e-5;
v_SA_part = b000 + xs.*(b100 + xs.*(b200 + xs.*(b300 + xs.*(b400 + b500.*xs)))) ...
+ ys.*(b010 + xs.*(b110 + xs.*(b210 + xs.*(b310 + b410.*xs))) ...
+ ys.*(b020 + xs.*(b120 + xs.*(b220 + b320.*xs)) + ys.*(b030 ...
+ xs.*(b130 + b230.*xs) + ys.*(b040 + b140.*xs + b050.*ys)))) ...
+ z.*(b001 + xs.*(b101 + xs.*(b201 + xs.*(b301 + b401.*xs))) ...
+ ys.*(b011 + xs.*(b111 + xs.*(b211 + b311.*xs)) + ys.*(b021 ...
+ xs.*(b121 + b221.*xs) + ys.*(b031 + b131.*xs + b041.*ys))) ...
+ z.*(b002 + xs.*(b102 + xs.*(b202 + b302.*xs))+ ys.*(b012 ...
+ xs.*(b112 + b212.*xs) + ys.*(b022 + b122.*xs + b032.*ys)) ...
+ z.*(b003 + b103.*xs + b013.*ys + b004.*z)));
v = v000 + xs.*(v100 + xs.*(v200 + xs.*(v300 + xs.*(v400 + xs.*(v500 ...
+ v600.*xs))))) + ys.*(v010 + xs.*(v110 + xs.*(v210 + xs.*(v310 + xs.*(v410 ...
+ v510.*xs)))) + ys.*(v020 + xs.*(v120 + xs.*(v220 + xs.*(v320 + v420.*xs))) ...
+ ys.*(v030 + xs.*(v130 + xs.*(v230 + v330.*xs)) + ys.*(v040 + xs.*(v140 ...
+ v240*xs) + ys.*(v050 + v150.*xs + v060.*ys))))) + z.*(v001 + xs.*(v101 ...
+ xs.*(v201 + xs.*(v301 + xs.*(v401 + v501.*xs)))) + ys.*(v011 + xs.*(v111 ...
+ xs.*(v211 + xs.*(v311 + v411.*xs))) + ys.*(v021 + xs.*(v121 + xs.*(v221 ...
+ v321.*xs)) + ys.*(v031 + xs.*(v131 + v231.*xs) + ys.*(v041 + v141.*xs ...
+ v051.*ys)))) + z.*(v002 + xs.*(v102 + xs.*(v202 + xs.*(v302 + v402.*xs))) ...
+ ys.*(v012 + xs.*(v112 + xs.*(v212 + v312.*xs)) + ys.*(v022 + xs.*(v122 ...
+ v222.*xs) + ys.*(v032 + v132.*xs + v042.*ys))) + z.*(v003 + xs.*(v103 ...
+ v203.*xs) + ys.*(v013 + v113.*xs + v023.*ys) + z.*(v004 + v104.*xs + v014.*ys ...
+ z.*(v005 + v006.*z)))));
beta = -v_SA_part.*0.5.*sfac./(v.*xs);
if transposed
beta = beta.';
end
end