def pt_from_entropy(SA, entropy):
r"""Calculates potential temperature with reference pressure p_ref = 0 dbar
and with entropy as an input variable.
Parameters
----------
SA : array_like
Absolute salinity [g kg :sup:`-1`]
entropy : array_like
specific entropy [J kg :sup:`-1` K :sup:`-1`]
Returns
-------
pt : array_like
potential temperature [:math:`^\circ` C (ITS-90)]
with reference sea pressure (p_ref) = 0 dbar.
See Also
--------
TODO
Notes
-----
TODO
Examples
--------
>>> import seawater.gibbs as gsw
>>> SA = [34.7118, 34.8915, 35.0256, 34.8472, 34.7366, 34.7324]
>>> entropy = [400.3892, 395.4378, 319.8668, 146.7910, 98.6471, 62.7919]
>>> gsw.pt_from_entropy(SA, entropy)
array([ 28.78317983, 28.42095483, 22.78495274, 10.23053207,
6.82921333, 4.32453778])
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 appendix A.10.
Modifications:
2011-04-03. Trevor McDougall and Paul Barker.
"""
SA = np.maximum(SA, 0)
part1 = 1 - SA / SSO
part2 = 1 - 0.05 * part1
ent_SA = (cp0 / Kelvin) * part1 * (1 - 1.01 * part1)
c = (entropy - ent_SA) * part2 / cp0
pt = Kelvin * (np.exp(c) - 1)
dentropy_dt = cp0 / ((Kelvin + pt) * part2) # Initial dentropy_dt.
for Number_of_iterations in range(0, 3):
pt_old = pt
dentropy = entropy_from_pt(SA, pt_old) - entropy
# This is half way through the modified method
# (McDougall and Wotherspoon, 2012)
pt = pt_old - dentropy / dentropy_dt
ptm = 0.5 * (pt + pt_old)
dentropy_dt = -gibbs_pt0_pt0(SA, ptm)
pt = pt_old - dentropy / dentropy_dt
"""maximum error of 2.2x10^-6 degrees C for one iteration. maximum error is
1.4x10^-14 degrees C for two iterations (two iterations is the default,
"for Number_of_iterations = 1:2")."""
return pt
def pt_from_CT(SA, CT):
r"""Calculates potential temperature (with a reference sea pressure of zero
dbar) from Conservative Temperature.
Parameters
----------
SA : array_like
Absolute salinity [g kg :sup:`-1`]
CT : array_like
Conservative Temperature [:math:`^\circ` C (ITS-90)]
Returns
-------
pt : array_like
potential temperature referenced to a sea pressure of zero dbar
[:math:`^\circ` C (ITS-90)]
See Also
--------
specvol_anom
Notes
-----
This function uses 1.5 iterations through a modified Newton-Raphson (N-R)
iterative solution procedure, starting from a rational-function-based
initial condition for both pt and dCT_dpt.
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.pt_from_CT(SA, CT)
array([ 28.78317705, 28.4209556 , 22.78495347, 10.23053439,
6.82921659, 4.32453484])
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 sections 3.1 and 3.3.
.. [2] McDougall T. J., D. R. Jackett, P. M. Barker, C. Roberts-Thomson,
R. Feistel and R. W. Hallberg, 2010: A computationally efficient 25-term
expression for the density of seawater in terms of Conservative
Temperature, and related properties of seawater.
Modifications:
2011-03-29. Trevor McDougall, David Jackett, Claire Roberts-Thomson and
Paul Barker.
"""
SA, CT, mask = strip_mask(SA, CT)
SA = np.maximum(SA, 0)
s1 = SA * 35. / SSO
a0 = -1.446013646344788e-2
a1 = -3.305308995852924e-3
a2 = 1.062415929128982e-4
a3 = 9.477566673794488e-1
a4 = 2.166591947736613e-3
a5 = 3.828842955039902e-3
b0 = 1.000000000000000e+0
b1 = 6.506097115635800e-4
b2 = 3.830289486850898e-3
b3 = 1.247811760368034e-6
a5CT = a5 * CT
b3CT = b3 * CT
CT_factor = (a3 + a4 * s1 + a5CT)
pt_num = a0 + s1 * (a1 + a2 * s1) + CT * CT_factor
pt_den = b0 + b1 * s1 + CT * (b2 + b3CT)
pt = pt_num / pt_den
dCT_dpt = pt_den / (CT_factor + a5CT - (b2 + b3CT + b3CT) * pt)
# 1.5 iterations through the modified Newton-Rapshon iterative method
CT_diff = CT_from_pt(SA, pt) - CT
pt_old = pt
pt = pt_old - CT_diff / dCT_dpt # 1/2-way through the 1st modified N-R.
ptm = 0.5 * (pt + pt_old)
# This routine calls gibbs_pt0_pt0(SA, pt0) to get the second derivative of
# the Gibbs function with respect to temperature at zero sea pressure.
dCT_dpt = -(ptm + Kelvin) * gibbs_pt0_pt0(SA, ptm) / cp0
pt = pt_old - CT_diff / dCT_dpt # End of 1st full modified N-R iteration.
CT_diff = CT_from_pt(SA, pt) - CT
pt_old = pt
pt = pt_old - CT_diff / dCT_dpt # 1.5 iterations of the modified N-R.
# Abs max error of result is 1.42e-14 deg C.
return np.ma.array(pt, mask=mask, copy=False)
def pt0_from_t(SA, t, p):
r"""Calculates potential temperature with reference pressure, pr = 0 dbar.
The present routine is computationally faster than the more general
function "pt_from_t(SA, t, p, pr)" which can be used for any reference
pressure value.
Parameters
----------
SA : array_like
Absolute salinity [g kg :sup:`-1`]
t : array_like
in situ temperature [:math:`^\circ` C (ITS-90)]
p : array_like
pressure [dbar]
Returns
-------
pt0 : array_like
potential temperature relative to 0 dbar [:math:`^\circ` C (ITS-90)]
See Also
--------
entropy_part, gibbs_pt0_pt0, entropy_part_zerop
Notes
-----
pt_from_t has the same result (only slower)
Examples
--------
>>> import gsw
>>> SA = [34.7118, 34.8915, 35.0256, 34.8472, 34.7366, 34.7324]
>>> t = [28.7856, 28.4329, 22.8103, 10.2600, 6.8863, 4.4036]
>>> p = [10, 50, 125, 250, 600, 1000]
>>> gsw.pt0_from_t(SA, t, p)
array([ 28.78319682, 28.42098334, 22.7849304 , 10.23052366,
6.82923022, 4.32451057])
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 section 3.1.
.. [2] McDougall T. J., D. R. Jackett, P. M. Barker, C. Roberts-Thomson,
R. Feistel and R. W. Hallberg, 2010: A computationally efficient 25-term
expression for the density of seawater in terms of Conservative
Temperature, and related properties of seawater.
Modifications:
2011-03-29. Trevor McDougall, David Jackett, Claire Roberts-Thomson and
Paul Barker.
"""
SA = np.maximum(SA, 0)
s1 = SA * (35. / SSO)
pt0 = t + p * (8.65483913395442e-6 -
s1 * 1.41636299744881e-6 -
p * 7.38286467135737e-9 +
t * (-8.38241357039698e-6 +
s1 * 2.83933368585534e-8 +
t * 1.77803965218656e-8 +
p * 1.71155619208233e-10))
dentropy_dt = cp0 / ((Kelvin + pt0) * (1 - 0.05 * (1 - SA / SSO)))
true_entropy_part = entropy_part(SA, t, p)
for Number_of_iterations in range(0, 2, 1):
pt0_old = pt0
dentropy = entropy_part_zerop(SA, pt0_old) - true_entropy_part
# Half way the mod. method (McDougall and Wotherspoon, 2012).
pt0 = pt0_old - dentropy / dentropy_dt
pt0m = 0.5 * (pt0 + pt0_old)
dentropy_dt = -gibbs_pt0_pt0(SA, pt0m)
pt0 = pt0_old - dentropy / dentropy_dt
"""maximum error of 6.3x10^-9 degrees C for one iteration. maximum error is
1.8x10^-14 degrees C for two iterations (two iterations is the default,
"for Number_of_iterations = 1:2"). These errors are over the full
"oceanographic funnel" of McDougall et al. (2010), which reaches down to
p = 8000 dbar."""
return pt0
def pt_from_CT(SA, CT):
r"""Calculates potential temperature (with a reference sea pressure of zero
dbar) from Conservative Temperature.
Parameters
----------
SA : array_like
Absolute salinity [g kg :sup:`-1`]
CT : array_like
Conservative Temperature [:math:`^\circ` C (ITS-90)]
Returns
-------
pt : array_like
potential temperature referenced to a sea pressure of zero dbar
[:math:`^\circ` C (ITS-90)]
See Also
--------
specvol_anom
Notes
-----
This function uses 1.5 iterations through a modified Newton-Raphson (N-R)
iterative solution procedure, starting from a rational-function-based
initial condition for both pt and dCT_dpt.
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.pt_from_CT(SA, CT)
array([ 28.78317705, 28.4209556 , 22.78495347, 10.23053439,
6.82921659, 4.32453484])
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 sections 3.1 and 3.3.
.. [2] McDougall T. J., D. R. Jackett, P. M. Barker, C. Roberts-Thomson,
R. Feistel and R. W. Hallberg, 2010: A computationally efficient 25-term
expression for the density of seawater in terms of Conservative
Temperature, and related properties of seawater.
Modifications:
2011-03-29. Trevor McDougall, David Jackett, Claire Roberts-Thomson and
Paul Barker.
"""
SA, CT, mask = strip_mask(SA, CT)
s1 = SA * 35. / SSO
a0 = -1.446013646344788e-2
a1 = -3.305308995852924e-3
a2 = 1.062415929128982e-4
a3 = 9.477566673794488e-1
a4 = 2.166591947736613e-3
a5 = 3.828842955039902e-3
b0 = 1.000000000000000e+0
b1 = 6.506097115635800e-4
b2 = 3.830289486850898e-3
b3 = 1.247811760368034e-6
a5CT = a5 * CT
b3CT = b3 * CT
CT_factor = (a3 + a4 * s1 + a5CT)
pt_num = a0 + s1 * (a1 + a2 * s1) + CT * CT_factor
pt_den = b0 + b1 * s1 + CT * (b2 + b3CT)
pt = pt_num / pt_den
dCT_dpt = pt_den / (CT_factor + a5CT - (b2 + b3CT + b3CT) * pt)
# 1.5 iterations through the modified Newton-Rapshon iterative method
CT_diff = CT_from_pt(SA, pt) - CT
pt_old = pt
pt = pt_old - CT_diff / dCT_dpt # 1/2-way through the 1st modified N-R.
ptm = 0.5 * (pt + pt_old)
# This routine calls gibbs_pt0_pt0(SA,pt0) to get the second derivative of
# the Gibbs function with respect to temperature at zero sea pressure.
dCT_dpt = -(ptm + Kelvin) * gibbs_pt0_pt0(SA, ptm) / cp0
pt = pt_old - CT_diff / dCT_dpt # End of 1st full modified N-R iteration.
CT_diff = CT_from_pt(SA, pt) - CT
pt_old = pt
pt = pt_old - CT_diff / dCT_dpt # 1.5 iterations of the modified N-R.
return np.ma.array(pt, mask=mask, copy=False)
def pt_from_entropy(SA, entropy):
r"""Calculates potential temperature with reference pressure p_ref = 0 dbar
and with entropy as an input variable.
Parameters
----------
SA : array_like
Absolute salinity [g kg :sup:`-1`]
entropy : array_like
specific entropy [J kg :sup:`-1` K :sup:`-1`]
Returns
-------
pt : array_like
potential temperature [:math:`^\circ` C (ITS-90)]
with reference sea pressure (p_ref) = 0 dbar.
See Also
--------
TODO
Notes
-----
TODO
Examples
--------
>>> import seawater.gibbs as gsw
>>> SA = [34.7118, 34.8915, 35.0256, 34.8472, 34.7366, 34.7324]
>>> entropy = [400.3892, 395.4378, 319.8668, 146.7910, 98.6471, 62.7919]
>>> gsw.pt_from_entropy(SA, entropy)
array([ 28.78317983, 28.42095483, 22.78495274, 10.23053207,
6.82921333, 4.32453778])
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 appendix A.10.
Modifications:
2011-04-03. Trevor McDougall and Paul Barker.
"""
SA.clip(0, np.inf)
n0, n1 = 0, 1
part1 = 1 - SA / SSO
part2 = 1 - 0.05 * part1
ent_SA = (cp0 / Kelvin) * part1 * (1 - 1.01 * part1)
c = (entropy - ent_SA) * part2 / cp0
pt = Kelvin * (np.exp(c) - 1)
dentropy_dt = cp0 / ((Kelvin + pt) * part2) # Initial dentropy_dt.
for Number_of_iterations in range(0, 3):
pt_old = pt
dentropy = entropy_from_pt(SA, pt_old) - entropy
pt = pt_old - dentropy / dentropy_dt # Half way through mod. method.
ptm = 0.5 * (pt + pt_old)
dentropy_dt = -gibbs_pt0_pt0(SA, ptm)
pt = pt_old - dentropy / dentropy_dt
"""maximum error of 2.2x10^-6 degrees C for one iteration. maximum error is
1.4x10^-14 degrees C for two iterations (two iterations is the default,
"for Number_of_iterations = 1:2")."""
return pt
def pt0_from_t(SA, t, p):
r"""Calculates potential temperature with reference pressure, pr = 0 dbar.
The present routine is computationally faster than the more general
function "pt_from_t(SA, t, p, pr)" which can be used for any reference
pressure value.
Parameters
----------
SA : array_like
Absolute salinity [g kg :sup:`-1`]
t : array_like
in situ temperature [:math:`^\circ` C (ITS-90)]
p : array_like
pressure [dbar]
Returns
-------
pt0 : array_like
potential temperature relative to 0 dbar [:math:`^\circ` C (ITS-90)]
See Also
--------
entropy_part, gibbs_pt0_pt0, entropy_part_zerop
Notes
-----
pt_from_t has the same result (only slower)
Examples
--------
>>> import gsw
>>> SA = [34.7118, 34.8915, 35.0256, 34.8472, 34.7366, 34.7324]
>>> t = [28.7856, 28.4329, 22.8103, 10.2600, 6.8863, 4.4036]
>>> p = [10, 50, 125, 250, 600, 1000]
>>> gsw.pt0_from_t(SA, t, p)
array([ 28.78319682, 28.42098334, 22.7849304 , 10.23052366,
6.82923022, 4.32451057])
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 section 3.1.
.. [2] McDougall T. J., D. R. Jackett, P. M. Barker, C. Roberts-Thomson,
R. Feistel and R. W. Hallberg, 2010: A computationally efficient 25-term
expression for the density of seawater in terms of Conservative
Temperature, and related properties of seawater.
Modifications:
2011-03-29. Trevor McDougall, David Jackett, Claire Roberts-Thomson and
Paul Barker.
"""
s1 = SA * (35. / SSO)
pt0 = t + p * (8.65483913395442e-6 -
s1 * 1.41636299744881e-6 -
p * 7.38286467135737e-9 +
t * (-8.38241357039698e-6 +
s1 * 2.83933368585534e-8 +
t * 1.77803965218656e-8 +
p * 1.71155619208233e-10))
dentropy_dt = cp0 / ((Kelvin + pt0) * (1 - 0.05 *
(1 - SA / SSO)))
true_entropy_part = entropy_part(SA, t, p)
for Number_of_iterations in range(0, 2, 1):
pt0_old = pt0
dentropy = entropy_part_zerop(SA, pt0_old) - true_entropy_part
pt0 = pt0_old - dentropy / dentropy_dt # Half way through mod. method.
pt0m = 0.5 * (pt0 + pt0_old)
dentropy_dt = -gibbs_pt0_pt0(SA, pt0m)
pt0 = pt0_old - dentropy / dentropy_dt
"""maximum error of 6.3x10^-9 degrees C for one iteration. maximum error is
1.8x10^-14 degrees C for two iterations (two iterations is the default,
"for Number_of_iterations = 1:2"). These errors are over the full
"oceanographic funnel" of McDougall et al. (2010), which reaches down to
p = 8000 dbar."""
return pt0