# gsw_specvol_second_derivatives_CT_exact

`second derivatives of specific volume`

## USAGE:

`[v_SA_SA, v_SA_CT, v_CT_CT, v_SA_P, v_CT_P] = gsw_specvol_second_derivatives_CT_exact(SA,CT,p)`

## DESCRIPTION:

```Calculates the following three second-order derivatives of specific
volume (v),
(1) v_SA_SA, second order derivative with respect to Absolute Salinity
at constant CT & p.
(2) v_SA_CT, second order derivative with respect to SA & CT at
constant p.
(3) v_CT_CT, second order derivative with respect to CT at constant
SA & p.
(4) v_SA_P, second-order derivative with respect to SA & P at
constant CT.
(5) v_CT_P, second-order derivative with respect to CT & P at
constant SA```
```Note that this function uses the full Gibbs function.  There is an
alternative to calling this function, namely
gsw_specvol_second_derivatives(SA,CT,p), which uses the computationally
efficient 75-term expression for specific volume in terms of SA, CT
and p (Roquet et al., 2015).```

## 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:

```v_SA_SA  =  The second derivative of specific volume with respect to
Absolute Salinity at constant CT & p.  [ (m^3/kg)(g/kg)^-2 ]
v_SA_CT  =  The second derivative of specific volume with respect to
SA & CT at constant p.             [ (m^3/kg)(g/kg)^-1 K^-1]
v_CT_CT  =  The second derivative of specific volume with respect to
CT at constant SA and p.                  [ (m^3/kg) K^-2) ]
v_SA_P  =  The second derivative of specific volume with respect to
SA & P at constant CT.                    [ (m^3/kg) Pa^-1 ]
v_CT_P  =  The second derivative of specific volume with respect to
CT & P at constant SA.               [ (m^3/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;]```
`[v_SA_SA, v_SA_CT, v_CT_CT, v_SA_P, v_CT_P] = gsw_specvol_second_derivatives_CT_exact(SA,CT,p)`
`v_SA_SA =`
`   1.0e-08 *`
```   0.082747972220243
0.082798176655947
0.086916803740167
0.098324796761055
0.100275947818790
0.101230704457043```
`v_SA_CT =`
`   1.0e-08 *`
```   0.130277044003024
0.130784915228726
0.149689281804061
0.217013951069468
0.233995663194746
0.243673021659962```
`v_CT_CT =`
`   1.0e-07 *`
```   0.071415166013777
0.071591303894948
0.077547238247366
0.095261850570592
0.099967277032840
0.102907243947244```
`v_SA_P =`
`   1.0e-14 *`
```   0.116986078360622
0.116992068444784
0.121867881822378
0.136113230189008
0.139000449643749
0.140519129568244```
`v_CT_P =`
`   1.0e-14 *`
```   0.085363651161808
0.086548662105251
0.112536803643252
0.188528204436210
0.211570045408395
0.228501837692148```

## 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.
```
```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`