# This file is part of OpenDrift.
#
# OpenDrift is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, version 2
#
# OpenDrift is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with OpenDrift. If not, see <https://www.gnu.org/licenses/>.
#
# Copyright 2015, 2023, Knut-Frode Dagestad, MET Norway
# Copyright 2023, Achref Othmani & 2024, Lenny Hucher, NERSC, Norway
import numpy as np
import logging; logger = logging.getLogger(__name__)
from opendrift.models.oceandrift import OceanDrift, Lagrangian3DArray
from opendrift.models.physics_methods import PhysicsMethods
from scipy.integrate import solve_ivp
# Constants
rho_water = 1027
rho_air = 1.293
rho_ice = 917
rho_iceb = 900
g = 9.81
csi = 1 # Sea ice coefficient of resistance
wave_drag_coef = 0.3
[docs]
class IcebergObj(Lagrangian3DArray):
""" Extending Lagrangian3DArray with relevant properties for an Iceberg """
variables = Lagrangian3DArray.add_variables([
('sail', {'dtype': np.float32, # Sail of iceberg (part above waterline )
'units': 'm',
'default': 10}),
('draft', {'dtype': np.float32, # Draft of iceberg (part below waterline)
'units': 'm',
'default': 90}),
('length', {'dtype': np.float32, # length of iceberg
'units': 'm',
'default': 100}),
('width', {'dtype': np.float32, # width of iceberg
'units': 'm',
'default': 30}),
('weight_coeff', {'dtype': np.float32, # Relative to the shape of iceberg (e.g. 1 for tabular; 0.3 for pinnacle)
'units': '1',
'default': 1}),
('water_drag_coeff', {'dtype': np.float32, # Ocean drag coeff.
'units': '1',
'default': 0.25}),
('wind_drag_coeff', {'dtype': np.float32, # Wind drag coeff.
'units': '1',
'default': 0.7}),
("iceb_x_velocity", {"dtype": np.float32, # Iceberg velocity in the x-direction
"units": "m/s",
"default": 0.0}),
("iceb_y_velocity", {"dtype": np.float32, # Iceberg velocity in the y-direction
"units": "m/s",
"default": 0.0}),
])
# Define the functions needed
[docs]
def water_drag(iceb_vel, water_vel, Ao, rho_water, water_drag_coef):
""" Ocean force
Args:
iceb_vel : Iceberg velocity at time t
water_vel : Ocean current velocity
Ao : Iceberg's area exposed to the ocean current(length x draft)
rho_water : Water density
water_drag_coef : Co is the drag coefficient applied on the iceberg's draft
Returns:
Ocean force
"""
vxo, vyo = water_vel[0], water_vel[1] # water velocity
x_vel, y_vel = iceb_vel[0], iceb_vel[1] # iceberg velocity
rel_water_x_vel = vxo - x_vel
rel_water_y_vel = vyo - y_vel
rel_water_norm = np.sqrt(rel_water_x_vel**2 + rel_water_y_vel**2)
F_ocean_x = (0.5 * rho_water * water_drag_coef * Ao * rel_water_norm * rel_water_x_vel)
F_ocean_y = (0.5 * rho_water * water_drag_coef * Ao * rel_water_norm * rel_water_y_vel)
return np.array([F_ocean_x, F_ocean_y])
[docs]
def wind_drag(iceb_vel, wind_vel, Aa, wind_drag_coef):
""" Wind force
Args:
iceb_vel : Iceberg velocity at time t
wind_vel : Wind velocity
Aa : Iceberg's area exposed to the wind (length x sail)
rho_air : Air density
wind_drag_coef : Ca is the drag coefficient applied on the iceberg's sail
Returns:
Wind force
"""
vxa, vya = wind_vel[0], wind_vel[1] # wind velocity
x_vel, y_vel = iceb_vel[0], iceb_vel[1] # iceberg velocity
rel_wind_x_vel = vxa - x_vel
rel_wind_y_vel = vya - y_vel
rel_wind_norm = np.sqrt(rel_wind_x_vel**2 + rel_wind_y_vel**2)
F_wind_x = 0.5 * rho_air * wind_drag_coef * Aa * rel_wind_norm * rel_wind_x_vel
F_wind_y = 0.5 * rho_air * wind_drag_coef * Aa * rel_wind_norm * rel_wind_y_vel
return np.array([F_wind_x, F_wind_y])
[docs]
def wave_radiation_force(rho_water, wave_height, wave_direction, iceb_length):
""" Wave radiation force
Args:
iceb_vel : Iceberg velocity at time t
rho_water : Water density
wave_height : Wave significant height
wave_direction : Wave direction
iceb_length : Iceberg's length
Returns:
Wave radiation force
"""
F_wave_x = (0.5 * rho_water * wave_drag_coef * g * iceb_length * (wave_height / 2) ** 2 * np.sin(np.deg2rad((wave_direction + 180) % 360)))
F_wave_y = (0.5 * rho_water * wave_drag_coef * g * iceb_length * (wave_height / 2) ** 2 * np.cos(np.deg2rad((wave_direction + 180) % 360)))
return np.array([F_wave_x, F_wave_y])
[docs]
def advect_iceberg_no_acc(f, water_vel, wind_vel):
""" Advect iceberg without acceleration
Args:
f : Wind drift factor
water_vel : Ocean current velocity
wind_vel : Wind velocity
Returns:
Iceberg velocity without acceleration
"""
vxo, vyo = water_vel[0], water_vel[1]
vxa, vya = wind_vel[0], wind_vel[1]
no_acc_model_x = (1 - f) * vxo + f * vxa
no_acc_model_y = (1 - f) * vyo + f * vya
V = np.array([no_acc_model_x, no_acc_model_y])
if not np.isfinite(V).all():
logger.error("Infinite value in iceberg's velocity without acceleration: Please check the wind drift factor f ")
return V
[docs]
def sea_ice_force(iceb_vel, sea_ice_conc, sea_ice_thickness, sea_ice_vel, iceb_width, sum_force):
""" Sea ice force
Args:
iceb_vel : Iceberg velocity at time t
sea_ice_conc : Sea ice concentration [%]
sea_ice_thickness : Sea ice thickness [m]
sea_ice_vel : Sea ice velocity [m/s]
rho_ice : Sea ice density
iceb_width : Iceberg width
sum_force : Effect of all other forces exerted on the iceberg (apart from the sea ice force)
Returns:
Sea ice force
"""
ice_x, ice_y = sea_ice_vel
x_vel, y_vel = iceb_vel[0], iceb_vel[1]
diff_vel = np.sqrt((ice_x - x_vel) ** 2 + (ice_y - y_vel) ** 2)
force_x, force_y = sum_force
F_ice_x = np.zeros_like(x_vel)
F_ice_y = np.zeros_like(y_vel)
F_ice_x = (0.5 * (rho_ice * csi * sea_ice_thickness * iceb_width) * diff_vel * (ice_x - x_vel))
F_ice_y = (0.5 * (rho_ice * csi * sea_ice_thickness * iceb_width) * diff_vel * (ice_y - y_vel))
F_ice_x[sea_ice_conc <= 0.15] = 0
F_ice_y[sea_ice_conc <= 0.15] = 0
F_ice_x[sea_ice_conc >= 0.9] = -force_x[sea_ice_conc >= 0.9]
F_ice_y[sea_ice_conc >= 0.9] = -force_y[sea_ice_conc >= 0.9]
return np.array([F_ice_x, F_ice_y])
[docs]
def melwav(iceb_length, iceb_width, x_wind, y_wind, sst, sea_ice_conc, dt):
""" Update the iceberg's dimensions (length and width) due to wave erosion
Parameterization from Keghouche et al. 2009
Args:
iceb_length : Iceberg's length
iceb_width : Iceberg's width
x_wind : Wind speed in the x-direction
y_wind : Wind speed in the y-direction
sst : Sea surface temperature
sea_ice_conc : Sea ice concentration
dt : Timestep of the simulation
"""
Ss = -5 + np.sqrt(32 + 2 * np.sqrt(x_wind**2 + y_wind**2))
Vsst = (1 / 6.0) * (sst + 2) * Ss
Vwe = Vsst * 0.5 * (1 + np.cos(np.pi * sea_ice_conc**3)) / 86400
new_iceb_length = np.zeros_like(iceb_length)
new_iceb_width = np.zeros_like(iceb_width)
new_iceb_length[iceb_length != 0] = (iceb_length[iceb_length != 0] - Vwe[iceb_length != 0] * dt)
new_iceb_width[iceb_length != 0] = (iceb_width[iceb_length != 0] / iceb_length[iceb_length != 0] * new_iceb_length[iceb_length != 0])
new_iceb_length[new_iceb_length < 0] = 0
new_iceb_width[new_iceb_width < 0] = 0
return new_iceb_length, new_iceb_width
[docs]
def mellat(iceb_length, iceb_width, tempib, salnib, dt):
""" Update the iceberg's dimensions (length and width) due to lateral melting
Parameterization from Keghouche et al. 2009
Args:
iceb_length : Iceberg's length
iceb_width : Iceberg's width
tempib : Water temperature
salnib : Water salinity
dt : Timestep of the simulation
"""
TfS = -0.036 - 0.0499 * salnib - 0.000112 * salnib**2
Tfp = TfS * np.exp(-0.19 * (tempib - TfS))
deltaT = tempib - Tfp
deltaT = np.concatenate([2.78 * deltaT, 0.47 * deltaT**2], axis=0)
sumVb = np.nansum(deltaT, axis=0)
dx = sumVb / 365 / 86400 * dt # Convert sumVb from (meter/year) to (meter per second)
new_iceb_length = np.zeros_like(iceb_length)
new_iceb_width = np.zeros_like(iceb_width)
new_iceb_length[iceb_length != 0] = (iceb_length[iceb_length != 0] - 2 * dx[iceb_length != 0])
new_iceb_width[iceb_length != 0] = (iceb_width[iceb_length != 0] / iceb_length[iceb_length != 0] * new_iceb_length[iceb_length != 0])
new_iceb_length[new_iceb_length < 0] = 0
new_iceb_width[new_iceb_width < 0] = 0
return new_iceb_length, new_iceb_width
[docs]
def melbas(iceb_draft, iceb_sail, iceb_length, salnib, tempib, x_water_vel, y_water_vel, x_iceb_vel, y_iceb_vel, dt):
""" Update the iceberg's dimensions (draft and sail) due to forced convection
Parameterization from Keghouche et al. 2009
"""
# Temperature at the base layer of the iceberg
absv = np.sqrt(((x_water_vel - x_iceb_vel) ** 2 + (y_water_vel - y_iceb_vel) ** 2))
TfS = -0.036 - 0.0499 * salnib - 0.000112 * salnib**2
Tfp = TfS * 2.71828 ** (-0.19 * (tempib - TfS))
deltat = tempib - Tfp
Vf = 0.58 * absv**0.8 * deltat / (iceb_length**0.2)
Vf = Vf / 86400 # converted to m/s
# Update the draft
new_iceb_draft = np.zeros_like(iceb_draft)
new_iceb_draft[iceb_draft != 0] = (abs(iceb_draft[iceb_draft != 0]) - Vf[iceb_draft != 0] * dt)
# Melt at base of the iceberg
new_iceb_draft[iceb_draft < 0] = 0
return new_iceb_draft, iceb_sail
[docs]
class OpenBerg(OceanDrift):
ElementType = IcebergObj
required_variables = {
"x_sea_water_velocity": {"fallback": None, "profiles": True},
"y_sea_water_velocity": {"fallback": None, "profiles": True},
"sea_floor_depth_below_sea_level": {"fallback": 10000},
"x_wind": {"fallback": 0, "important": False},
"y_wind": {"fallback": 0, "important": False},
"sea_surface_wave_significant_height": {"fallback": 0},
"sea_surface_wave_from_direction": {"fallback": 0},
"sea_surface_wave_stokes_drift_x_velocity": {"fallback": 0},
"sea_surface_wave_stokes_drift_y_velocity": {"fallback": 0},
"sea_water_temperature": {"fallback": 2, "profiles": True},
"sea_water_salinity": {"fallback": 35, "profiles": True},
"sea_ice_area_fraction": {"fallback": 0},
"sea_ice_thickness": {"fallback": 0},
"sea_ice_x_velocity": {"fallback": 0, "important": False},
"sea_ice_y_velocity": {"fallback": 0, "important": False},
"land_binary_mask": {"fallback": None},
}
[docs]
def get_profile_masked(self, variable):
""" Apply a mask to extract data from the surface down to the iceberg's draft """
draft = self.elements.draft
profile = self.environment_profiles[variable]
z = self.environment_profiles["z"]
if len(z) == 1:
if z == np.array([None]): # Convert None values to zeros for surface current
z = np.zeros_like(z)
mask = draft[:, np.newaxis] < -z
mask = mask.T
mask[np.argmax(mask, axis=0), np.arange(mask.shape[1])] = False
return np.ma.masked_array(profile, mask, fill_value=np.nan)
[docs]
def get_basal_env(self, variable):
""" Get the basal layer of the variable for the icebergs """
profile = self.get_profile_masked(variable)
last = np.argmin(np.logical_not(profile.mask), axis=0) - 1
return profile[last, np.arange(profile.shape[1])]
# Configuration
def __init__(
self,
add_stokes_drift: bool = True,
wave_rad: bool = True,
grounding: bool = False,
vertical_profile: bool = False,
melting: bool = False,
choose_melting: dict[bool] = {"wave": True, "lateral": True, "basal": True},
*args,
**kwargs,
):
# The constructor of parent class must always be called
# to perform some necessary common initialisation tasks:
super(OpenBerg, self).__init__(*args, **kwargs)
self.wave_rad = wave_rad
self.add_stokes_drift = add_stokes_drift
self.grounding = grounding
self.vertical_profile = vertical_profile
self.melting = (melting) #Activate the melting
self.choose_melting = (choose_melting) #Choose how it melts
[docs]
def advect_iceberg(self, rho_water, stokes_drift=True, wave_rad=True, grounding=False, vertical_profile=False):
draft = self.elements.draft
length = self.elements.length
Ao = abs(draft) * length ### Area_wet
Aa = self.elements.sail * length ### Area_dry
mass = self.elements.width * (Aa + Ao) * rho_iceb
k = (rho_air * self.elements.wind_drag_coeff * Aa / (rho_water * self.elements.water_drag_coeff * Ao))
f = np.sqrt(k) / (1 + np.sqrt(k))
if not vertical_profile:
water_vel = np.array([self.environment.x_sea_water_velocity + int(stokes_drift) * self.environment.sea_surface_wave_stokes_drift_x_velocity,
self.environment.y_sea_water_velocity + int(stokes_drift) * self.environment.sea_surface_wave_stokes_drift_y_velocity])
else:
logger.info("Calculating the Depth Integrated Currents ...")
uprof = self.get_profile_masked("x_sea_water_velocity")
vprof = self.get_profile_masked("y_sea_water_velocity")
z = self.environment_profiles["z"]
thickness = -(z[1:] - z[:-1]).reshape((-1, 1))
mask = uprof.mask
uprof_mean_inter = (uprof[1:] + uprof[:-1]) / 2
vprof_mean_inter = (vprof[1:] + vprof[:-1]) / 2
mask = mask[:-1]
thickness_reshaped = np.tile(thickness, (1, mask.shape[1]))
thickness_reshaped[mask] = np.nan
umean = np.nansum(thickness_reshaped * uprof_mean_inter, axis=0) / np.nansum(thickness_reshaped, axis=0)
vmean = np.nansum(thickness_reshaped * vprof_mean_inter, axis=0) / np.nansum(thickness_reshaped, axis=0)
water_vel = np.array([umean, vmean])
water_depth = self.environment.sea_floor_depth_below_sea_level
wind_vel = np.array([self.environment.x_wind, self.environment.y_wind])
wave_height = self.environment.sea_surface_wave_significant_height
wave_direction = self.environment.sea_surface_wave_from_direction
sea_ice_conc = self.environment.sea_ice_area_fraction
sea_ice_thickness = self.environment.sea_ice_thickness
sea_ice_vel = np.array([self.environment.sea_ice_x_velocity, self.environment.sea_ice_y_velocity])
def dynamic(t,iceb_vel, water_vel, wind_vel, wave_height, wave_direction, Ao,
Aa, rho_water, water_drag_coef, wind_drag_coef, iceb_length, mass):
""" Function required by solve_ivp. The t and iceb_vel parameters are required by solve_ivp, shouldn't be deleted """
sum_force = (water_drag(iceb_vel, water_vel, Ao, rho_water, water_drag_coef)
+ wind_drag(iceb_vel,wind_vel, Aa, wind_drag_coef)
+ int(wave_rad) * wave_radiation_force(rho_water, wave_height, wave_direction, iceb_length))
sum_force = sum_force + sea_ice_force(iceb_vel, sea_ice_conc, sea_ice_thickness, sea_ice_vel, self.elements.width, sum_force)
return 1 / mass * sum_force
V0 = advect_iceberg_no_acc(f, water_vel, wind_vel) # Approximation of the solution of the dynamic equation for the iceberg velocity
V0[:, sea_ice_conc >= 0.9] = sea_ice_vel[:, sea_ice_conc >= 0.9] # With this criterium, the iceberg moves with the sea ice
V0 = V0.flatten()
hwall = draft - water_depth
grounded = np.logical_and(hwall >= 0, grounding)
if any(grounded) and grounding:
logger.info(f"Grounding condition : Icebergs grounded = {len(hwall[hwall>0])}, hwall={np.round(hwall[hwall>0],3)} meters")
sol = solve_ivp(dynamic, [0, self.time_step.total_seconds()], V0,
args=(water_vel, wind_vel, wave_height, wave_direction,
Ao, Aa, rho_water,
self.elements.water_drag_coeff,
self.elements.wind_drag_coeff,
self.elements.length,
mass),
vectorized=True,
t_eval=np.array([self.time_step.total_seconds()]),
)
V = sol.y.reshape((2, -1))
Vx, Vy = V[0], V[1]
Vx[grounded] = 0
Vy[grounded] = 0
self.update_positions(Vx, Vy)
self.elements.iceb_x_velocity, self.elements.iceb_y_velocity = Vx, Vy
[docs]
def melt(self):
""" Activate melting """
x_wind = self.environment.x_wind
y_wind = self.environment.y_wind
uoib = self.get_basal_env("x_sea_water_velocity")
voib = self.get_basal_env("y_sea_water_velocity")
T_profile = self.environment_profiles["sea_water_temperature"]
S_profile = self.environment_profiles["sea_water_salinity"]
Tn = self.get_basal_env("sea_water_temperature")
Sn = self.get_basal_env("sea_water_salinity")
sea_ice_conc = self.environment.sea_ice_area_fraction
# Wave melting
if self.choose_melting["wave"]:
self.elements.length, self.elements.width = melwav(self.elements.length, self.elements.width, x_wind, y_wind, T_profile[0], sea_ice_conc, self.time_step.total_seconds())
# Lateral melting
if self.choose_melting["lateral"]:
self.elements.length, self.elements.width = mellat(self.elements.length, self.elements.width, T_profile, S_profile, self.time_step.total_seconds())
# Basal melting
if self.choose_melting["basal"]:
self.elements.draft, self.elements.sail = melbas(self.elements.draft, self.elements.sail, self.elements.length, Sn, Tn, uoib, voib, self.elements.iceb_x_velocity, self.elements.iceb_y_velocity, self.time_step.total_seconds())
# Deactivate elements less than 1 meter
self.deactivate_elements(self.elements.draft < 1, "Iceberg melted")
self.deactivate_elements(self.elements.length < 1, "Iceberg melted")
self.deactivate_elements(self.elements.width < 1, "Iceberg melted")
self.deactivate_elements(self.elements.sail < 1, "Iceberg melted")
[docs]
def roll_over(self, rho_water):
""" Iceberg's stability criterium: Parameterization from Keghouche et al. 2009 with a correction from Wagner et al. 2017 """
L = self.elements.length
W = self.elements.width
H = self.elements.draft + self.elements.sail
alpha = rho_iceb / rho_water
crit = np.sqrt(6 * alpha * (1 - alpha))
W, L = np.min([L, W], axis=0), np.max([L, W], axis=0)
mask = (W / H) < crit
if any(mask):
logger.info(f"Rolling over : {np.sum(mask)} icebergs ...")
nL, nW, nH = (np.max([L[mask], H[mask]], axis=0), np.min([L[mask], H[mask]], axis=0), W[mask])
L[mask], W[mask], H[mask] = nL, nW, nH
depthib = H * alpha
sailib = H - depthib
self.elements.length = L
self.elements.width = W
self.elements.sail = sailib
self.elements.draft = depthib
[docs]
def prepare_run(self):
self.profiles_depth = self.elements_scheduled.draft.max()
logger.info(f"Icebergs max draft is : {self.profiles_depth} meters")
[docs]
def update(self):
""" Update positions and properties of particles """
T = self.environment.sea_water_temperature
S = self.environment.sea_water_salinity
rho_water = PhysicsMethods.sea_water_density(T, S)
self.roll_over(rho_water)
if self.melting:
self.melt()
self.advect_iceberg(rho_water, self.add_stokes_drift, self.wave_rad, self.grounding, self.vertical_profile)