def entropy_first_derivatives(SA, CT):
r"""Calculates the following two partial derivatives of specific entropy
(eta)
(1) eta_SA, the derivative with respect to Absolute Salinity at constant
Conservative Temperature, and
(2) eta_CT, the derivative with respect to Conservative Temperature at
constant Absolute Salinity.
Parameters
----------
SA : array_like
Absolute salinity [g kg :sup:`-1`]
CT : array_like
Conservative Temperature [:math:`^\circ` C (TEOS-10)]
Returns
-------
eta_SA : array_like
The derivative of specific entropy with respect to SA at constant
CT [J g :sup:`-1` K :sup:`-1`]
eta_CT : array_like
The derivative of specific entropy with respect to CT at constant
SA [ J (kg K :sup:`-2`) :sup:`-1` ]
See Also
--------
TODO
Notes
-----
TODO
Examples
--------
>>> import gsw
>>> SA = [34.7118, 34.8915, 35.0256, 34.8472, 34.7366, 34.7324]
>>> CT = [28.8099, 28.4392, 22.7862, 10.2262, 6.8272, 4.3236]
>>> gsw.entropy_first_derivatives(SA, CT)
array([[ -0.2632868 , -0.26397728, -0.2553675 , -0.23806659,
-0.23443826, -0.23282068],
[ 13.22103121, 13.23691119, 13.48900463, 14.08659902,
14.25772958, 14.38642995]])
References
----------
.. [1] 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. See Eqns. (A.12.8) and (P.14a,c).
Modifications:
2011-03-29. Trevor McDougall.
"""
n0, n1 = 0, 1
pt = pt_from_CT(SA, CT)
eta_SA = -(gibbs(n1, n0, n0, SA, pt, 0)) / (Kelvin + pt)
eta_CT = cp0 / (Kelvin + pt)
return eta_SA, eta_CT
def sigma0_CT_exact(SA, CT):
r"""Calculates potential density anomaly with reference pressure of 0 dbar,
this being this particular potential density minus 1000 kg/m^3. This
function has inputs of Absolute Salinity and Conservative Temperature.
Parameters
----------
SA : array_like
Absolute Salinity [g/kg]
CT : array_like
Conservative Temperature [:math:`^\circ` C (ITS-90)]
Returns
-------
sigma0_CT_exact: array_like
Potential density anomaly with [kg/m**3]
respect to a reference pressure of 0 dbar
that is, this potential density - 1000 kg/m**3.
Notes
-----
Note that this function uses the full Gibbs function. There is an
alternative to calling this function, namely gsw_sigma0_CT(SA,CT,p), which
uses the computationally efficient 48-term expression for density in terms
of SA, CT and p (McDougall et al., 2011).
Examples
--------
TODO
References
----------
.. [1] 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. See Eqn. (A.30.1).
.. [2] McDougall T.J., P.M. Barker, R. Feistel and D.R. Jackett, 2011: A
computationally efficient 48-term expression for the density of
seawater in terms of Conservative Temperature, and related properties
of seawater.
Modifications:
2011-04-03. Trevor McDougall and Paul Barker.
"""
pt0 = pt_from_CT(SA, CT)
return sigma0_pt0_exact(SA, pt0)
def enthalpy_second_derivatives(SA, CT, p):
r"""Calculates the following three second-order derivatives of specific
enthalpy (h),
(1) h_SA_SA, second-order derivative with respect to Absolute Salinity
at constant CT & p.
(2) h_SA_CT, second-order derivative with respect to SA & CT at
constant p.
(3) h_CT_CT, second-order derivative with respect to CT at constant SA
and p.
Parameters
----------
SA : array_like
Absolute salinity [g kg :sup:`-1`]
CT : array_like
Conservative Temperature [:math:`^\circ` C (TEOS-10)]
p : array_like
pressure [dbar]
Returns
-------
h_SA_SA : array_like
The second derivative of specific enthalpy with respect to
Absolute Salinity at constant CT & p. [J/(kg (g/kg)^2)]
h_SA_CT : array_like
The second derivative of specific enthalpy with respect to SA and
CT at constant p. [J/(kg K(g/kg))]
h_CT_CT : array_like
The second derivative of specific enthalpy with respect to CT at
constant SA and p. [J/(kg K^2)]
References
----------
.. [1] 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. See Eqns. (A.11.18), (A.11.15) and (A.11.12.)
.. [2] McDougall T.J., P.M. Barker, R. Feistel and D.R. Jackett, 2011: A
computationally efficient 48-term expression for the density of seawater in
terms of Conservative Temperature, and related properties of seawater. To
be submitted to Ocean Science Discussions.
Modifications:
2011-03-29. Trevor McDougall.
"""
# NOTE: The Matlab version 3.0 mentions that this function is unchanged,
# but that's not true!
pt0 = pt_from_CT(SA, CT)
abs_pt0 = Kelvin + pt0
t = pt_from_t(SA, pt0, 0, p)
temp_ratio = (Kelvin + t) / abs_pt0
rec_gTT_pt0 = 1 / gibbs(n0, n2, n0, SA, pt0, 0)
rec_gTT_t = 1 / gibbs(n0, n2, n0, SA, t, p)
gST_pt0 = gibbs(n1, n1, n0, SA, pt0, 0)
gST_t = gibbs(n1, n1, n0, SA, t, p)
gS_pt0 = gibbs(n1, n0, n0, SA, pt0, 0)
part = ((temp_ratio * gST_pt0 * rec_gTT_pt0 - gST_t * rec_gTT_t) /
(abs_pt0))
factor = gS_pt0 / cp0
# h_CT_CT is naturally well-behaved as SA approaches zero.
def enthalpy_derivative_CT_CT(SA, CT, p):
return (cp0 ** 2 * ((temp_ratio * rec_gTT_pt0 - rec_gTT_t) /
(abs_pt0 * abs_pt0)))
# h_SA_SA has a singularity at SA = 0, and blows up as SA approaches zero.
def enthalpy_derivative_SA_SA(SA, CT, p):
SA[SA < 1e-100] = 1e-100 # NOTE: Here is the changes from 2.0 to 3.0.
h_CT_CT = enthalpy_derivative_CT_CT(SA, CT, p)
return (gibbs(n2, n0, n0, SA, t, p) -
temp_ratio * gibbs(n2, n0, n0, SA, pt0, 0) +
temp_ratio * gST_pt0 ** 2 * rec_gTT_pt0 -
gST_t ** 2 * rec_gTT_t - 2.0 * gS_pt0 * part +
factor ** 2 * h_CT_CT)
"""h_SA_CT should not blow up as SA approaches zero. The following lines of
code ensure that the h_SA_CT output of this function does not blow up in
this limit. That is, when SA < 1e-100 g/kg, we force the h_SA_CT output to
be the same as if SA = 1e-100 g/kg."""
def enthalpy_derivative_SA_CT(SA, CT, p):
h_CT_CT = enthalpy_derivative_CT_CT(SA, CT, p)
return cp0 * part - factor * h_CT_CT
return (enthalpy_derivative_SA_SA(SA, CT, p),
enthalpy_derivative_SA_CT(SA, CT, p),
enthalpy_derivative_CT_CT(SA, CT, p))
def enthalpy_first_derivatives(SA, CT, p):
r"""Calculates the following three derivatives of specific enthalpy (h)
(1) h_SA, the derivative with respect to Absolute Salinity at
constant CT and p, and
(2) h_CT, derivative with respect to CT at constant SA and p.
(3) h_P, derivative with respect to pressure (in Pa) at constant SA and CT.
Parameters
----------
SA : array_like
Absolute salinity [g kg :sup:`-1`]
CT : array_like
Conservative Temperature [:math:`^\circ` C (TEOS-10)]
p : array_like
pressure [dbar]
Returns
-------
h_SA : array_like
The first derivative of specific enthalpy with respect to Absolute
Salinity at constant CT and p. [J/(kg (g/kg))] i.e. [J/g]
h_CT : array_like
The first derivative of specific enthalpy with respect to CT at
constant SA and p. [J/(kg K)]
h_P : array_like
The first partial derivative of specific enthalpy with respect to
pressure (in Pa) at fixed SA and CT. Note that h_P is specific
volume (1/rho.)
See Also
--------
TODO
Notes
-----
TODO
References
----------
.. [1] 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. See Eqns. (A.11.18), (A.11.15) and (A.11.12.)
Modifications:
2010-09-24. Trevor McDougall.
"""
# FIXME: The gsw 3.0 has the gibbs derivatives "copy-and-pasted" here
# instead of the calls to the library! Why?
pt0 = pt_from_CT(SA, CT)
t = pt_from_t(SA, pt0, 0, p)
temp_ratio = (Kelvin + t) / (Kelvin + pt0)
def enthalpy_derivative_SA(SA, CT, p):
return (gibbs(n1, n0, n0, SA, t, p) -
temp_ratio * gibbs(n1, n0, n0, SA, pt0, 0))
def enthalpy_derivative_CT(SA, CT, p):
return cp0 * temp_ratio
def enthalpy_derivative_p(SA, CT, p):
return gibbs(n0, n0, n1, SA, t, p)
return (enthalpy_derivative_SA(SA, CT, p),
enthalpy_derivative_CT(SA, CT, p),
enthalpy_derivative_p(SA, CT, p),)
def entropy_second_derivatives(SA, CT):
r"""Calculates the following three second-order partial derivatives of
specific entropy (eta)
(1) eta_SA_SA, the second derivative with respect to Absolute Salinity at
constant Conservative Temperature, and
(2) eta_SA_CT, the derivative with respect to Absolute Salinity and
Conservative Temperature.
(3) eta_CT_CT, the second derivative with respect to Conservative
Temperature at constant Absolute Salinity.
Parameters
----------
SA : array_like
Absolute salinity [g kg :sup:`-1`]
CT : array_like
Conservative Temperature [:math:`^\circ` C (TEOS-10)]
Returns
-------
eta_SA_SA : array_like
The second derivative of specific entropy with respect to SA at
constant CT [J (kg K (g kg :sup:`-1` ) :sup:`2`) :sup:`-1`]
eta_SA_CT : array_like
The second derivative of specific entropy with respect to
SA and CT [J (kg (g kg :sup:`-1` ) K :sup:`2`) :sup:`-1` ]
eta_CT_CT : array_like
The second derivative of specific entropy with respect to CT at
constant SA [J (kg K :sup:`3`) :sup:`-1` ]
See Also
--------
TODO
Notes
-----
TODO
Examples
--------
>>> import gsw
>>> SA = [34.7118, 34.8915, 35.0256, 34.8472, 34.7366, 34.7324]
>>> CT = [28.8099, 28.4392, 22.7862, 10.2262, 6.8272, 4.3236]
>>> gsw.entropy_second_derivatives(SA, CT)
TODO
References
----------
.. [1] 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. See Eqns. (P.14b) and (P.15a,b).
Modifications:
2011-03-29. Trevor McDougall and Paul Barker.
"""
pt = pt_from_CT(SA, CT)
abs_pt = Kelvin + pt
CT_SA = ((gibbs(n1, n0, n0, SA, pt, 0) -
(abs_pt * gibbs(n1, n1, n0, SA, pt, 0))) / cp0)
CT_pt = -(abs_pt * gibbs(n0, n2, n0, SA, pt, 0)) / cp0
eta_CT_CT = - cp0 / (CT_pt * abs_pt ** 2)
eta_SA_CT = CT_SA * eta_CT_CT
eta_SA_SA = -gibbs(n2, n0, n0, SA, pt, 0) / abs_pt - CT_SA * eta_SA_CT
return eta_SA_SA, eta_SA_CT, eta_CT_CT
def pt_first_derivatives(SA, CT):
r"""Calculates the following two partial derivatives of potential
temperature (the regular potential temperature whose reference sea
pressure is 0 dbar)
(1) pt_SA, the derivative with respect to Absolute Salinity at
constant Conservative Temperature, and
(2) pt_CT, the derivative with respect to Conservative Temperature at
constant Absolute Salinity.
Parameters
----------
SA : array_like
Absolute salinity [g kg :sup:`-1`]
CT : array_like
Conservative Temperature [:math:`^\circ` C (TEOS-10)]
Returns
-------
pt_SA : array_like
The derivative of potential temperature with respect to Absolute
Salinity at constant Conservative Temperature. [K/(g/kg)]
pt_CT : array_like
The derivative of potential temperature with respect to
Conservative Temperature at constant Absolute Salinity. [unitless]
See Also
--------
TODO
Notes
-----
TODO
Examples
--------
TODO
References
----------
.. [1] 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. See Eqns. (A.12.6), (A.12.3), (P.6) and (P.8).
.. [2] McDougall T.J., P.M. Barker, R. Feistel and D.R. Jackett, 2011: A
computationally efficient 48-term expression for the density of seawater in
terms of Conservative Temperature, and related properties of seawater. To
be submitted to Ocean Science Discussions.
Modifications:
2011-03-29. Trevor McDougall and Paul Barker.
"""
pt = pt_from_CT(SA, CT)
abs_pt = Kelvin + pt
CT_SA = ((gibbs(n1, n0, n0, SA, pt, 0) - abs_pt *
gibbs(n1, n1, n0, SA, pt, 0)) / cp0)
CT_pt = - (abs_pt * gibbs(n0, n2, n0, SA, pt, 0)) / cp0
pt_SA = - CT_SA / CT_pt
pt_CT = 1.0 / CT_pt
return pt_SA, pt_CT
