# gsw_internal_energy_second_derivatives

`second derivatives of internal energy (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 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 *`
```  -0.002844346969485
-0.008509204835375
-0.019897886928086
-0.042812697313995
-0.102544848040703
-0.171670276420447```
`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:

`3.05 (16th February, 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 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, P.M. Barker, 2015: Accurate
polynomial expressions for the density and specifc volume of seawater
using the TEOS-10 standard. Ocean Modelling.```
`This software is available from http://www.TEOS-10.org`