pycontrails.models.extended_k15

Support for volatile particulate matter (vPM) modeling via the extended K15 model.

See the droplet_apparent_emission_index() function for the main entry point.

A preprint is available [Ponsonby et al., 2025].

Functions

S(D, Dd, kappa, A)

Compute the supersaturation ratio at diameter D.

activation_radius(S_w, kappa, temperature[, ...])

Calculate activation radius for a given supersaturation ratio and temperature.

critical_supersaturation(Dd, kappa, T[, ...])

Compute the critical supersaturation ratio for a given particle size.

droplet_activation(n_available_all, P_w, ...)

Calculate available particles that activate to form water droplets.

droplet_apparent_emission_index(...[, ...])

Calculate the droplet apparent emissions index from nvPM, vPM and ambient particles.

droplet_number_concentration_at_saturation(T_plume)

Calculate water vapour concentration at saturation.

dynamical_regime_parameter(n_available_all, ...)

Calculate dynamical regime parameter.

emissions_index_to_number_concentration(...)

Convert particle number emissions index to number concentration.

entrained_ambient_droplet_number_concentration(...)

Calculate ambient droplet number concentration entrained in the contrail plume.

fraction_of_water_activated_particles(gmd, ...)

Calculate the fraction of particles that activate to form water droplets.

number_concentration_to_emissions_index(...)

Convert particle number concentration to apparent emissions index.

particle_growth_coefficients(T_plume, ...)

Calculate particle growth coefficients, b_1 and b_2 in Karcher et al. (2015).

plume_dilution_factor(T_plume, T_exhaust, ...)

Calculate the plume dilution factor.

plume_water_saturation_ratio_no_condensation(...)

Calculate water saturation ratio in the exhaust plume without droplet condensation.

supersaturation_loss_rate_per_droplet(...)

Calculate supersaturation loss rate per droplet.

water_droplet_activation(particles, T_plume, ...)

Calculate statistics on the water droplet activation for different particle types.

water_droplet_activation_across_all_particles(...)

Calculate the total and weighted water droplet activation outputs across all particle types.

water_supersaturation_production_rate(...)

Calculate water supersaturation production rate.

Classes

DropletActivation(particle, r_act, phi, ...)

Store the computed statistics on the water droplet activation for each particle.

Particle(type, kappa, gmd, gsd, n_ambient)

Representation of a particle with hygroscopic and size distribution properties.

ParticleType(value[, names, module, ...])

Enumeration of particle types.
class pycontrails.models.extended_k15.DropletActivation(particle, r_act, phi, n_total, n_available)

Bases: object

Store the computed statistics on the water droplet activation for each particle.

Parameters:

  • particle (Particle | None) – Source particle type, or None if this is the aggregate result.

  • r_act (npt.NDArray[np.floating]) – Activation radius for a given water saturation ratio and temperature, [\(m\)].

  • phi (npt.NDArray[np.floating]) – Fraction of particles that activate to form water droplets, between 0 and 1.

  • n_total (npt.NDArray[np.floating]) – Total particle number concentration, [\(m^{-3}\)].

  • n_available (npt.NDArray[np.floating]) – Particle number concentration available for activation, [\(m^{-3}\)].

n_available
n_total
particle
phi
r_act
class pycontrails.models.extended_k15.Particle(type, kappa, gmd, gsd, n_ambient)

Bases: object

Representation of a particle with hygroscopic and size distribution properties.

Parameters:

  • type (ParticleType) – One of ParticleType.NVPM, ParticleType.VPM, or ParticleType.AMBIENT.

  • kappa (float) – Hygroscopicity parameter, dimensionless.

  • gmd (float) – Geometric mean diameter of the lognormal size distribution, [\(m\)].

  • gsd (float) – Geometric standard deviation of the lognormal size distribution, dimensionless.

  • n_ambient (float) – Ambient particle number concentration, [\(m^{-3}\)]. For ambient or background particles, this specifies the number concentration entrained in the contrail plume. For emission particles, this should be set to 0.0.

Notes

The parameters gmd and gsd define a lognormal size distribution. The hygroscopicity parameter kappa follows [Petters and Kreidenweis, 2007].

gmd
gsd
kappa
n_ambient
type
class pycontrails.models.extended_k15.ParticleType(value, names=<not given>, *values, module=None, qualname=None, type=None, start=1, boundary=None)

Bases: StrEnum

Enumeration of particle types.

AMBIENT = 'ambient'
NVPM = 'nvpm'
VPM = 'vpm'
capitalize()

Return a capitalized version of the string.

More specifically, make the first character have upper case and the rest lower case.

casefold()

Return a version of the string suitable for caseless comparisons.

center(width, fillchar=' ', /)

Return a centered string of length width.

Padding is done using the specified fill character (default is a space).

count()

Return the number of non-overlapping occurrences of substring sub in string S[start:end].

Optional arguments start and end are interpreted as in slice notation.

encode(encoding='utf-8', errors='strict')

Encode the string using the codec registered for encoding.

encoding

The encoding in which to encode the string.

errors

The error handling scheme to use for encoding errors. The default is ‘strict’ meaning that encoding errors raise a UnicodeEncodeError. Other possible values are ‘ignore’, ‘replace’ and ‘xmlcharrefreplace’ as well as any other name registered with codecs.register_error that can handle UnicodeEncodeErrors.

endswith()

Return True if the string ends with the specified suffix, False otherwise.

suffix

A string or a tuple of strings to try.

start

Optional start position. Default: start of the string.

end

Optional stop position. Default: end of the string.

expandtabs(tabsize=8)

Return a copy where all tab characters are expanded using spaces.

If tabsize is not given, a tab size of 8 characters is assumed.

find()

Return the lowest index in S where substring sub is found, such that sub is contained within S[start:end].

Optional arguments start and end are interpreted as in slice notation. Return -1 on failure.

format(*args, **kwargs)

Return a formatted version of the string, using substitutions from args and kwargs. The substitutions are identified by braces (‘{’ and ‘}’).

format_map(mapping, /)

Return a formatted version of the string, using substitutions from mapping. The substitutions are identified by braces (‘{’ and ‘}’).

index()

Return the lowest index in S where substring sub is found, such that sub is contained within S[start:end].

Optional arguments start and end are interpreted as in slice notation. Raises ValueError when the substring is not found.

isalnum()

Return True if the string is an alpha-numeric string, False otherwise.

A string is alpha-numeric if all characters in the string are alpha-numeric and there is at least one character in the string.

isalpha()

Return True if the string is an alphabetic string, False otherwise.

A string is alphabetic if all characters in the string are alphabetic and there is at least one character in the string.

isascii()

Return True if all characters in the string are ASCII, False otherwise.

ASCII characters have code points in the range U+0000-U+007F. Empty string is ASCII too.

isdecimal()

Return True if the string is a decimal string, False otherwise.

A string is a decimal string if all characters in the string are decimal and there is at least one character in the string.

isdigit()

Return True if the string is a digit string, False otherwise.

A string is a digit string if all characters in the string are digits and there is at least one character in the string.

isidentifier()

Return True if the string is a valid Python identifier, False otherwise.

Call keyword.iskeyword(s) to test whether string s is a reserved identifier, such as “def” or “class”.

islower()

Return True if the string is a lowercase string, False otherwise.

A string is lowercase if all cased characters in the string are lowercase and there is at least one cased character in the string.

isnumeric()

Return True if the string is a numeric string, False otherwise.

A string is numeric if all characters in the string are numeric and there is at least one character in the string.

isprintable()

Return True if all characters in the string are printable, False otherwise.

A character is printable if repr() may use it in its output.

isspace()

Return True if the string is a whitespace string, False otherwise.

A string is whitespace if all characters in the string are whitespace and there is at least one character in the string.

istitle()

Return True if the string is a title-cased string, False otherwise.

In a title-cased string, upper- and title-case characters may only follow uncased characters and lowercase characters only cased ones.

isupper()

Return True if the string is an uppercase string, False otherwise.

A string is uppercase if all cased characters in the string are uppercase and there is at least one cased character in the string.

join(iterable, /)

Concatenate any number of strings.

The string whose method is called is inserted in between each given string. The result is returned as a new string.

Example: ‘.’.join([‘ab’, ‘pq’, ‘rs’]) -> ‘ab.pq.rs’

ljust(width, fillchar=' ', /)

Return a left-justified string of length width.

Padding is done using the specified fill character (default is a space).

lower()

Return a copy of the string converted to lowercase.

lstrip(chars=None, /)

Return a copy of the string with leading whitespace removed.

If chars is given and not None, remove characters in chars instead.

static maketrans()

Return a translation table usable for str.translate().

If there is only one argument, it must be a dictionary mapping Unicode ordinals (integers) or characters to Unicode ordinals, strings or None. Character keys will be then converted to ordinals. If there are two arguments, they must be strings of equal length, and in the resulting dictionary, each character in x will be mapped to the character at the same position in y. If there is a third argument, it must be a string, whose characters will be mapped to None in the result.

partition(sep, /)

Partition the string into three parts using the given separator.

This will search for the separator in the string. If the separator is found, returns a 3-tuple containing the part before the separator, the separator itself, and the part after it.

If the separator is not found, returns a 3-tuple containing the original string and two empty strings.

removeprefix(prefix, /)

Return a str with the given prefix string removed if present.

If the string starts with the prefix string, return string[len(prefix):]. Otherwise, return a copy of the original string.

removesuffix(suffix, /)

Return a str with the given suffix string removed if present.

If the string ends with the suffix string and that suffix is not empty, return string[:-len(suffix)]. Otherwise, return a copy of the original string.

replace(old, new, /, count=-1)

Return a copy with all occurrences of substring old replaced by new.

count

Maximum number of occurrences to replace. -1 (the default value) means replace all occurrences.

If the optional argument count is given, only the first count occurrences are replaced.

rfind()

Return the highest index in S where substring sub is found, such that sub is contained within S[start:end].

Optional arguments start and end are interpreted as in slice notation. Return -1 on failure.

rindex()

Return the highest index in S where substring sub is found, such that sub is contained within S[start:end].

Optional arguments start and end are interpreted as in slice notation. Raises ValueError when the substring is not found.

rjust(width, fillchar=' ', /)

Return a right-justified string of length width.

Padding is done using the specified fill character (default is a space).

rpartition(sep, /)

Partition the string into three parts using the given separator.

This will search for the separator in the string, starting at the end. If the separator is found, returns a 3-tuple containing the part before the separator, the separator itself, and the part after it.

If the separator is not found, returns a 3-tuple containing two empty strings and the original string.

rsplit(sep=None, maxsplit=-1)

Return a list of the substrings in the string, using sep as the separator string.

sep

The separator used to split the string.

When set to None (the default value), will split on any whitespace character (including n r t f and spaces) and will discard empty strings from the result.

maxsplit

Maximum number of splits. -1 (the default value) means no limit.

Splitting starts at the end of the string and works to the front.

rstrip(chars=None, /)

Return a copy of the string with trailing whitespace removed.

If chars is given and not None, remove characters in chars instead.

split(sep=None, maxsplit=-1)

Return a list of the substrings in the string, using sep as the separator string.

sep

The separator used to split the string.

When set to None (the default value), will split on any whitespace character (including n r t f and spaces) and will discard empty strings from the result.

maxsplit

Maximum number of splits. -1 (the default value) means no limit.

Splitting starts at the front of the string and works to the end.

Note, str.split() is mainly useful for data that has been intentionally delimited. With natural text that includes punctuation, consider using the regular expression module.

splitlines(keepends=False)

Return a list of the lines in the string, breaking at line boundaries.

Line breaks are not included in the resulting list unless keepends is given and true.

startswith()

Return True if the string starts with the specified prefix, False otherwise.

prefix

A string or a tuple of strings to try.

start

Optional start position. Default: start of the string.

end

Optional stop position. Default: end of the string.

strip(chars=None, /)

Return a copy of the string with leading and trailing whitespace removed.

If chars is given and not None, remove characters in chars instead.

swapcase()

Convert uppercase characters to lowercase and lowercase characters to uppercase.

title()

Return a version of the string where each word is titlecased.

More specifically, words start with uppercased characters and all remaining cased characters have lower case.

translate(table, /)

Replace each character in the string using the given translation table.

table

Translation table, which must be a mapping of Unicode ordinals to Unicode ordinals, strings, or None.

The table must implement lookup/indexing via __getitem__, for instance a dictionary or list. If this operation raises LookupError, the character is left untouched. Characters mapped to None are deleted.

upper()

Return a copy of the string converted to uppercase.

zfill(width, /)

Pad a numeric string with zeros on the left, to fill a field of the given width.

The string is never truncated.

pycontrails.models.extended_k15.S(D, Dd, kappa, A)

Compute the supersaturation ratio at diameter D.

Implements equation (6) in [Petters and Kreidenweis, 2007].

Parameters:

  • D (npt.NDArray[np.floating]) – Droplet diameter, [\(m\)]. Should be greater than Dd.

  • Dd (npt.NDArray[np.floating]) – Dry particle diameter, [\(m\)].

  • kappa (npt.NDArray[np.floating]) – Hygroscopicity parameter, dimensionless.

  • A (npt.NDArray[np.floating]) – Kelvin term coefficient, [\(m\)].

Returns:

npt.NDArray[np.floating] – Supersaturation ratio at diameter D, dimensionless.

pycontrails.models.extended_k15.activation_radius(S_w, kappa, temperature, rtol=1e-06, maxiter=30)

Calculate activation radius for a given supersaturation ratio and temperature.

The activation radius is defined as the droplet radius at which the critical supersaturation equals the ambient water supersaturation S_w. Mathematically, it is the root of the equation:

critical_supersaturation(2 * r) - S_w = 0
Parameters:

  • S_w (npt.NDArray[np.floating]) – Water saturation ratio in the aircraft plume after droplet condensation, dimensionless.

  • kappa (npt.NDArray[np.floating] | float) – Hygroscopicity parameter, dimensionless. Expected to satisfy 0 < kappa < 1.

  • temperature (npt.NDArray[np.floating]) – Temperature at which to compute the activation radius, [\(K\)].

  • rtol (float, optional) – Relative tolerance for geometric-bisection root-finding algorithm, by default 1e-6.

  • maxiter (int, optional) – Maximum number of iterations for geometric-bisection root-finding algorithm, by default 30.

Returns:

npt.NDArray[np.floating] – The activation radius, [\(m\)]. Entries where S_w <= 1.0 return nan. Only supersaturation ratios greater than 1.0 are physically meaningful for activation. The returned activation radius is the radius at which the droplet would first activate to form a water droplet in the emissions plume.

pycontrails.models.extended_k15.critical_supersaturation(Dd, kappa, T, tol=1e-12, maxiter=25)

Compute the critical supersaturation ratio for a given particle size.

The critical supersaturation ratio is the maximum of the supersaturation ratio S(D) as a function of the droplet diameter D for a given dry diameter Dd. This maximum is found by solving for the root of the derivative of log(S) with respect to D using Newton’s method.

Parameters:

  • Dd (npt.NDArray[np.floating]) – Dry diameter of the particle, [\(m\)].

  • kappa (npt.NDArray[np.floating]) – Hygroscopicity parameter, dimensionless. Expected to satisfy 0 < kappa < 1.

  • T (npt.NDArray[np.floating]) – The temperature at which to compute the critical supersaturation, [\(K\)].

  • tol (float, optional) – Convergence tolerance for Newton’s method, by default 1e-12. Should be significantly smaller than the values in Dd.

  • maxiter (int, optional) – Maximum number of iterations for Newton’s method, by default 25.

Returns:

npt.NDArray[np.floating] – The critical supersaturation ratio, dimensionless.

pycontrails.models.extended_k15.droplet_activation(n_available_all, P_w, R_w, rho_air, dilution, nu_0)

Calculate available particles that activate to form water droplets.

Parameters:

  • n_available_all (npt.NDArray[np.floating]) – Particle number concentration entrained in the contrail plume, [\(m^{-3}\)]

  • P_w (npt.NDArray[np.floating]) – Water supersaturation production rate (P_w = dS_mw/dt), [\(s^{-1}\)]

  • R_w (npt.NDArray[np.floating]) – Supersaturation loss rate per droplet (R_w), [\(m^{3} s^{-1}\)]

  • rho_air (npt.NDArray[np.floating]) – Air density at each waypoint, [\(kg m^{-3}\)]

  • dilution (npt.NDArray[np.floating]) – Plume dilution factor, see plume_dilution_factor

  • nu_0 (float) – Initial mass-based plume mixing factor, i.e., air-to-fuel ratio, set to 60.0.

Returns:

npt.NDArray[np.floating] – Activated droplet apparent emissions index, [\(kg^{-1}\)]

References

n_2_w -> Eq. (51) of [Kärcher et al., 2015]. f -> Eq. (44) of [Kärcher et al., 2015].

pycontrails.models.extended_k15.droplet_apparent_emission_index(specific_humidity, T_ambient, T_exhaust, air_pressure, nvpm_ei_n, vpm_ei_n, G, particles=None, n_plume_points=50)

Calculate the droplet apparent emissions index from nvPM, vPM and ambient particles.

Parameters:

  • specific_humidity (npt.NDArray[np.floating]) – Specific humidity at each waypoint, [\(kg_{H_{2}O} / kg_{air}\)]

  • T_ambient (npt.NDArray[np.floating]) – Ambient temperature at each waypoint, [\(K\)]

  • T_exhaust (npt.NDArray[np.floating]) – Aircraft exhaust temperature for each waypoint, [\(K\)]

  • air_pressure (npt.NDArray[np.floating]) – Pressure altitude at each waypoint, [\(Pa\)]

  • nvpm_ei_n (npt.NDArray[np.floating]) – nvPM number emissions index, [\(kg^{-1}\)]

  • vpm_ei_n (float) – vPM number emissions index, [\(kg^{-1}\)]

  • G (npt.NDArray[np.floating]) – Slope of the mixing line in a temperature-humidity diagram.

  • particles (list[Particle] | None, optional) – List of particle types to consider. If None, defaults to a list of Particle instances representing nvPM, vPM, and ambient particles.

  • n_plume_points (int) – Number of points to evaluate the plume temperature along the mixing line. Increasing this value can improve accuracy. Values above 40 are typically sufficient. See the droplet_activation() for numerical considerations.

Returns:

npt.NDArray[np.floating] – Activated droplet apparent ice emissions index, [\(kg^{-1}\)]

Notes

All input arrays must be broadcastable to the same shape. For better performance when evaluating multiple points or grids, it is helpful to arrange the arrays so that meteorological variables (specific_humidity, T_ambient, air_pressure, G) correspond to dimension 0, while aircraft emissions (nvpm_ei_n, vpm_ei_n) correspond to dimension 1. This setup allows the plume temperature calculation to be computed once and reused for multiple emissions values.

pycontrails.models.extended_k15.droplet_number_concentration_at_saturation(T_plume)

Calculate water vapour concentration at saturation.

Parameters:

T_plume (npt.NDArray[np.floating]) – Plume temperature evolution along mixing line, [\(K\)]

Returns:

npt.NDArray[np.floating] – Water vapour concentration at water saturated conditions, [\(m^{-3}\)]

Notes

  • This is approximated based on the ideal gas law: p V = N k_b T, so (N/v) = p / (k_b * T).

pycontrails.models.extended_k15.dynamical_regime_parameter(n_available_all, S_mw, P_w, r_act_nw, b_1, b_2)

Calculate dynamical regime parameter.

Parameters:

  • n_available_all (npt.NDArray[np.floating]) – Particle number concentration that can be activated across all particles, [\(m^{-3}\)]

  • S_mw (npt.NDArray[np.floating]) – Water saturation ratio in the aircraft plume without droplet condensation

  • P_w (npt.NDArray[np.floating]) – Water supersaturation production rate (P_w = dS_mw/dt), [\(s^{-1}\)]

  • r_act_nw (npt.NDArray[np.floating]) – Number-weighted droplet activation radius, [\(m\)]

  • b_1 (npt.NDArray[np.floating]) – Particle growth coefficient, [\(m s^{-1}\)]

  • b_2 (npt.NDArray[np.floating]) – Particle growth coefficient, [\(m s^{-1}\)]

Returns:

npt.NDArray[np.floating] – Dynamical regime parameter (kappa_w)

References

Eq. (49) of [Kärcher et al., 2015].

pycontrails.models.extended_k15.emissions_index_to_number_concentration(number_ei, rho_air, dilution, nu_0)

Convert particle number emissions index to number concentration.

Parameters:

  • number_ei (npt.NDArray[np.floating] | float) – Particle number emissions index, [\(kg^{-1}\)].

  • rho_air (npt.NDArray[np.floating]) – Air density at each waypoint, [\(kg m^{-3}\)].

  • dilution (npt.NDArray[np.floating]) – Plume dilution factor.

  • nu_0 (float) – Initial mass-based plume mixing factor, i.e., air-to-fuel ratio, set to 60.0.

Returns:

npt.NDArray[np.floating] – Particle number concentration entrained in the contrail plume, [\(m^{-3}\)]

References

Eq. (37) of [Kärcher et al., 2015] without the phi term.

pycontrails.models.extended_k15.entrained_ambient_droplet_number_concentration(n_ambient, T_plume, T_ambient, dilution)

Calculate ambient droplet number concentration entrained in the contrail plume.

Parameters:

  • n_ambient (npt.NDArray[np.floating] | float) – Ambient particle number concentration, [\(m^{-3}\)].

  • T_plume (npt.NDArray[np.floating]) – Plume temperature evolution along mixing line, [\(K\)].

  • T_ambient (npt.NDArray[np.floating]) – Ambient temperature for each waypoint, [\(K\)].

  • dilution (npt.NDArray[np.floating]) – Plume dilution factor.

Returns:

npt.NDArray[np.floating] – Ambient droplet number concentration entrained in the contrail plume, [\(m^{-3}\)].

References

Eq. (37) of [Kärcher et al., 2015] without the phi term.

pycontrails.models.extended_k15.fraction_of_water_activated_particles(gmd, gsd, r_act)

Calculate the fraction of particles that activate to form water droplets.

Parameters:

  • gmd (npt.NDArray[np.floating] | float) – Geometric mean diameter, [\(m\)]

  • gsd (npt.NDArray[np.floating] | float) – Geometric standard deviation

  • r_act (npt.NDArray[np.floating] | float) – Droplet activation threshold radius for a given supersaturation (s_w), [\(m\)]

Returns:

npt.NDArray[np.floating] – Fraction of particles that activate to form water droplets (phi)

Notes

The cumulative distribution is estimated directly using the SciPy error function.

pycontrails.models.extended_k15.number_concentration_to_emissions_index(n_conc, rho_air, dilution, nu_0)

Convert particle number concentration to apparent emissions index.

Parameters:

  • n_conc (npt.NDArray[np.floating]) – Particle number concentration entrained in the contrail plume, [\(m^{-3}\)]

  • rho_air (npt.NDArray[np.floating]) – Air density at each waypoint, [\(kg m^{-3}\)]

  • dilution (npt.NDArray[np.floating]) – Plume dilution factor, see plume_dilution_factor

  • nu_0 (float) – Initial mass-based plume mixing factor, i.e., air-to-fuel ratio, set to 60.0.

Returns:

npt.NDArray[np.floating] – Particle apparent number emissions index, [\(kg^{-1}\)]

pycontrails.models.extended_k15.particle_growth_coefficients(T_plume, air_pressure, S_mw, n_w_sat, vol_molecule_h2o)

Calculate particle growth coefficients, b_1 and b_2 in Karcher et al. (2015).

Parameters:

  • T_plume (npt.NDArray[np.floating]) – Plume temperature evolution along mixing line, [\(K\)]

  • air_pressure (npt.NDArray[np.floating]) – Pressure altitude at each waypoint, [\(Pa\)]

  • S_mw (npt.NDArray[np.floating]) – Water saturation ratio in the aircraft plume without droplet condensation

  • n_w_sat (npt.NDArray[np.floating]) – Droplet number concentration at water saturated conditions, [\(m^{-3}\)]

  • vol_molecule_h2o (float) – Volume of a supercooled water molecule, [\(m^{3}\)]

Returns:

tuple[npt.NDArray[np.floating], npt.NDArray[np.floating]] – Particle growth coefficients b_1 and b_2, [\(m s^{-1}\)]

References

pycontrails.models.extended_k15.plume_dilution_factor(T_plume, T_exhaust, T_ambient, tau_m, beta)

Calculate the plume dilution factor.

Parameters:

  • T_plume (npt.NDArray[np.floating]) – Plume temperature evolution along mixing line, [\(K\)].

  • T_exhaust (npt.NDArray[np.floating]) – Aircraft exhaust temperature for each waypoint, [\(K\)].

  • T_ambient (npt.NDArray[np.floating]) – Ambient temperature for each waypoint, [\(K\)].

  • tau_m (float) – Mixing timescale, i.e., the time for an exhaust volume element at the center of the jet plume to remain unaffected by ambient air entrainment, [\(s\)].

  • beta (float) – Plume dilution parameter, set to 0.9.

Returns:

npt.NDArray[np.floating] – Plume dilution factor.

References

Eq. (12) of [Kärcher et al., 2015].

pycontrails.models.extended_k15.plume_water_saturation_ratio_no_condensation(T_plume, p_mw)

Calculate water saturation ratio in the exhaust plume without droplet condensation.

Parameters:

  • T_plume (npt.NDArray[np.floating]) – Plume temperature evolution along mixing line, [\(K\)]

  • p_mw (npt.NDArray[np.floating]) – PWater vapour partial pressure along mixing line, [\(Pa\)]

Returns:

npt.NDArray[np.floating] – Water saturation ratio in the aircraft plume without droplet condensation (S_mw).

References

Page 7894 of [Kärcher et al., 2015].

Notes

  • When expressed in percentage terms, S_mw is identical to relative humidity.

  • Water saturation ratio in the aircraft plume with droplet condensation (S_w)

  • In contrail-forming conditions, S_w <= S_mw because the supersaturation in the contrail plume is quenched from droplet formation and growth.

pycontrails.models.extended_k15.supersaturation_loss_rate_per_droplet(kappa_w, r_act_nw, n_w_sat, b_1, b_2, vol_molecule_h2o)

Calculate supersaturation loss rate per droplet.

Parameters:

  • kappa_w (npt.NDArray[np.floating]) – Dynamical regime parameter. See dynamical_regime_parameter

  • r_act_nw (npt.NDArray[np.floating]) – Number-weighted droplet activation radius, [\(m\)]

  • n_w_sat (npt.NDArray[np.floating]) – Droplet number concentration at water saturated conditions, [\(m^{-3}\)]

  • b_1 (npt.NDArray[np.floating]) – Particle growth coefficient, [\(m s^{-1}\)]

  • b_2 (npt.NDArray[np.floating]) – Particle growth coefficient, [\(m s^{-1}\)]

  • vol_molecule_h2o (float) – Volume of a supercooled water molecule, [\(m^{3}\)]

Returns:

npt.NDArray[np.floating] – Supersaturation loss rate per droplet (R_w), [\(m^{3} s^{-1}\)]

Notes

Originally calculated using Eq. (50) of [Kärcher et al., 2015], but has been updated in [Ponsonby et al., 2025] and is now calculated using Eq. (6) and Eq. (7) of [Kärcher et al., 2006].

pycontrails.models.extended_k15.water_droplet_activation(particles, T_plume, T_ambient, nvpm_ei_n, vpm_ei_n, S_mw, dilution, rho_air, nu_0)

Calculate statistics on the water droplet activation for different particle types.

Parameters:

  • particles (list[Particle]) – Properties of different particles in the contrail plume.

  • T_plume (npt.NDArray[np.floating]) – Plume temperature evolution along mixing line, [\(K\)].

  • T_ambient (npt.NDArray[np.floating]) – Ambient temperature for each waypoint, [\(K\)].

  • nvpm_ei_n (npt.NDArray[np.floating]) – nvPM number emissions index, [\(kg^{-1}\)].

  • vpm_ei_n (float) – vPM number emissions index, [\(kg^{-1}\)].

  • S_mw (npt.NDArray[np.floating]) – Water saturation ratio in the aircraft plume without droplet condensation.

  • dilution (npt.NDArray[np.floating]) – Plume dilution factor, see plume_dilution_factor().

  • rho_air (npt.NDArray[np.floating]) – Density of air, [\(kg / m^{-3}\)].

  • nu_0 (float) – Initial mass-based plume mixing factor, i.e., air-to-fuel ratio, set to 60.0.

Returns:

list[DropletActivation] – Computed statistics on the water droplet activation for each particle type.

pycontrails.models.extended_k15.water_droplet_activation_across_all_particles(particle_droplets)

Calculate the total and weighted water droplet activation outputs across all particle types.

Parameters:

particle_droplets (list[DropletActivation]) – Computed statistics on the water droplet activation for each particle type. See DropletActivation and water_droplet_activation().

Returns:

DropletActivation – Total and weighted water droplet activation outputs across all particle types.

References

Eq. (37) and Eq. (43) of [Kärcher et al., 2015].

pycontrails.models.extended_k15.water_supersaturation_production_rate(T_plume, T_exhaust, T_ambient, dilution, S_mw, tau_m, beta)

Calculate water supersaturation production rate.

Parameters:

  • T_plume (npt.NDArray[np.floating]) – Plume temperature evolution along mixing line, [\(K\)]

  • T_exhaust (npt.NDArray[np.floating]) – Aircraft exhaust temperature for each waypoint, [\(K\)]

  • T_ambient (npt.NDArray[np.floating]) – Ambient temperature for each waypoint, [\(K\)]

  • dilution (npt.NDArray[np.floating]) – Plume dilution factor, see plume_dilution_factor

  • S_mw (npt.NDArray[np.floating]) – Water saturation ratio in the aircraft plume without droplet condensation

  • tau_m (float) – Mixing timescale, i.e., the time for an exhaust volume element at the center of the jet plume to remain unaffected by ambient air entrainment, [\(s\)]

  • beta (float) – Plume dilution parameter, set to 0.9

Returns:

npt.NDArray[np.floating] – Water supersaturation production rate (P_w = dS_mw/dt), [\(s^{-1}\)]