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gsw_Nsquared.m
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function [N2, p_mid] = gsw_Nsquared(SA,CT,p,lat)
% gsw_Nsquared buoyancy (Brunt-Vaisala) frequency squared (N^2)
% (75-term equation)
%==========================================================================
%
% USAGE:
% [N2, p_mid] = gsw_Nsquared(SA,CT,p,{lat})
%
% DESCRIPTION:
% Calculates the buoyancy frequency squared (N^2)(i.e. the Brunt-Vaisala
% frequency squared) at the mid pressure from the equation,
%
%
% 2 2 beta.d(SA) - alpha.d(CT)
% N = g .rho_local. -------------------------
% dP
%
% The pressure increment, dP, in the above formula is in Pa, so that it is
% 10^4 times the pressure increment dp in dbar.
%
% Note. This routine uses rho from "gsw_rho", which is based on the
% computationally efficient expression for specific volume in terms of
% SA, CT and p (Roquet et al., 2015).
%
% 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 )
%
% OPTIONAL:
% lat = latitude in decimal degrees north [ -90 ... +90 ]
% Note. If lat is not supplied, a default gravitational acceleration
% of 9.7963 m/s^2 (Griffies, 2004) will be applied.
%
% SA & CT need to have the same dimensions.
% p & lat (if provided) may have dimensions 1x1 or Mx1 or 1xN or MxN,
% where SA & CT are MxN.
%
% OUTPUT:
% N2 = Brunt-Vaisala Frequency squared (M-1xN) [ 1/s^2 ]
% p_mid = Mid pressure between p grid (M-1xN) [ dbar ]
%
% AUTHOR:
% Trevor McDougall and Paul Barker [ help@teos-10.org ]
%
% VERSION NUMBER: 3.05 (27th January 2015)
%
% REFERENCES:
% Griffies, S. M., 2004: Fundamentals of Ocean Climate Models. Princeton,
% NJ: Princeton University Press, 518 pp + xxxiv.
%
% 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 section 3.10 and Eqn. (3.10.2) 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 | nargin == 4)
error('gsw_Nsquared: Requires three or four inputs')
end
if ~(nargout == 2)
error('gsw_Nsquared: Requires two outputs')
end
[ms,ns] = size(SA);
[mt,nt] = size(CT);
[mp,np] = size(p);
if (mt ~= ms | nt ~= ns)
error('gsw_Nsquared: SA and CT must have same dimensions')
end
if (ms*ns == 1)
error('gsw_Nsquared: There must be at least 2 bottles')
end
if (mp == 1) & (np == 1) % p is a scalar - must be two bottles
error('gsw_Nsquared: There must be at least 2 bottles')
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_Nsquared: Inputs array dimensions arguments do not agree')
end
if ms == 1
SA = SA.';
CT = CT.';
p = p.';
transposed = 1;
else
transposed = 0;
end
[mp,np] = size(p);
if exist('lat','var')
if transposed
lat = lat.';
end
[mL,nL] = size(lat);
[ms,ns] = size(SA);
if (mL == 1) & (nL == 1) % lat scalar - fill to size of SA
lat = lat*ones(size(SA));
elseif (ns == nL) & (mL == 1) % lat is row vector,
lat = lat(ones(1,ms), :); % copy down each column.
elseif (ms == mL) & (nL == 1) % lat is column vector,
lat = lat(:,ones(1,ns)); % copy across each row.
elseif (ms == mL) & (ns == nL)
% ok
else
error('gsw_Nsquared: Inputs array dimensions arguments do not agree')
end
grav = gsw_grav(lat,p);
else
grav = 9.7963*ones(size(p)); % (Griffies, 2004)
end
%--------------------------------------------------------------------------
% Start of the calculation
%--------------------------------------------------------------------------
db2Pa = 1e4;
Ishallow = 1:(mp-1);
Ideep = 2:mp;
grav_local = 0.5*(grav(Ishallow,:) + grav(Ideep,:));
dSA = (SA(Ideep,:) - SA(Ishallow,:));
SA_mid = 0.5*(SA(Ishallow,:) + SA(Ideep,:));
dCT = (CT(Ideep,:) - CT(Ishallow,:));
CT_mid = 0.5*(CT(Ishallow,:) + CT(Ideep,:));
dp = (p(Ideep,:) - p(Ishallow,:));
p_mid = 0.5*(p(Ishallow,:) + p(Ideep,:));
[rho_mid, alpha_mid, beta_mid] = gsw_rho_alpha_beta(SA_mid,CT_mid,p_mid);
%--------------------------------------------------------------------------
% This function calculates rho, alpha & beta using the computationally
% efficient expression for specific volume in terms of SA, CT and p. If
% one wanted to use the full TEOS-10 Gibbs function expression for specific
% volume, the following lines of code will enable this.
%
% rho_mid = gsw_rho_CT_exact(SA_mid,CT_mid,p_mid);
% alpha_mid = gsw_alpha_CT_exact(SA_mid,CT_mid,p_mid);
% beta_mid = gsw_beta_CT_exact(SA_mid,CT_mid,p_mid);
%
%--This is the end of the alternative code to evaluate rho, alpha & beta---
N2 = (grav_local.*grav_local).*(rho_mid./(db2Pa*dp)).*(beta_mid.*dSA - alpha_mid.*dCT);
if transposed
N2 = N2.';
p_mid = p_mid.';
end
end