# gsw_internal_energy_second_derivatives

second derivatives of internal energy (75-term equation)

## Contents

## 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 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 ]

## EXAMPLE:

SA = [34.7118; 34.8915; 35.0256; 34.8472; 34.7366; 34.7324;] CT = [28.7856; 28.4329; 22.8103; 10.2600; 6.8863; 4.4036;] p = [ 10; 50; 125; 250; 600; 1000;]

[u_SA_SA, u_SA_CT, u_CT_CT, u_SA_P, u_CT_P] = ... gsw_internal_energy_second_derivatives(SA,CT,p)

u_SA_SA =

1.0e-04 *

-0.819630879789472 -0.815991440934607 -0.829998881000640 -0.850454751660842 -0.275191185099451 0.935521724807209

u_SA_CT =

1.0e-03 *

-0.131647989755567 -0.131253260179302 -0.143764305018288 -0.175089006292492 0.044242679479252 0.582715895326373

u_CT_CT =

-0.000723349499915 -0.000720061216325 -0.000745410281876 -0.000733890626293 0.000470364582417 0.003405297338392

u_SA_P =

1.0e-07 *

-0.002353268094506 -0.007029334357723 -0.016418681747556 -0.035380276284246 -0.084822481842961 -0.142006349667287

u_CT_P =

1.0e-07 *

-0.001722190998963 -0.005214908835889 -0.015155996668111 -0.048975118096772 -0.129113528684486 -0.230925961569104

## AUTHOR:

Trevor McDougall and Paul Barker. [ help@teos-10.org ]

## VERSION NUMBER:

3.06.16 (28th September, 2022)

## 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 the TEOS-10 web site.

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 and P.M. Barker, 2015: Accurate polynomial expressions for the density and specific volume of seawater using the TEOS-10 standard.Ocean Modelling,90, pp. 29-43. http://dx.doi.org/10.1016/j.ocemod.2015.04.002

This software is available from http://www.TEOS-10.org