def __init__(
self,
z=None, # grid (input)
kappa=None, # diffusivity profile (input)
bs=0.025, # surface buoyancy bound. cond (input)
bbot=0.0, # bottom buoyancy boundary condition (input)
bzbot=None, # bottom strat. as alternative boundary condition (input)
b=0.0, # Buoyancy profile (input, output)
Area=None, # Horizontal area (can be function of depth)
N2min=1e-7 # Minimum strat. for conv adjustment
):
r"""
Parameters
----------
z : ndarray
Vertical depth levels of column grid. Units: m
kappa : float, function, or ndarray
Vertical diffusivity profile. Units: m\ :sup:`2`/s
bs : float
Surface level buoyancy boundary condition. Units: m/s\ :sup:`2`
bbot : float; optional
Bottom level buoyancy boundary condition. Units: m/s\ :sup:`2`
bzbot : float; optional
Bottom level buoyancy stratification. Can be used as an alternative to **bbot**. Units: s\ :sup:`-2`
b : float, function, or ndarray
Initial vertical buoyancy profile. Recalculated on model run. Units: m/s
Area : float, function, or ndarray
Horizontal area of basin. Units: m\ :sup:`2`
N2min : float; optional
Minimum stratification for convective adjustment. Units: s\ :sup:`-1`
"""
# initialize grid:
if isinstance(z, np.ndarray) and len(z) > 0:
self.z = z
else:
raise TypeError('z needs to be numpy array providing grid levels')
self.kappa = make_func(kappa, self.z, 'kappa')
self.Area = make_func(Area, self.z, 'Area')
self.bs = bs
self.bbot = bbot
self.bzbot = bzbot
self.N2min = N2min
self.b = make_array(b, self.z, 'b')
if check_numpy_version():
self.bz = np.gradient(self.b, z)
else:
self.bz = 0. * z # notice that this is just for initialization of ode solver
def test_update(self, psi):
b1 = 10.0
b2 = 50.0
z = psi.z
psi.update(b1, b2)
b1_func = make_func(b1, z, 'b1')
b2_func = make_func(b2, z, 'b2')
assert (all(psi.b1(z) == b1_func(z)))
assert (all(psi.b2(z) == b2_func(z)))
def test_update(self, psi_so):
b = 10.0
bs = 50.0
z = psi_so.z
y = psi_so.y
psi_so.update(b, bs)
b_func = make_func(b, z, 'b')
bs_func = make_func(bs, y, 'bs')
assert (all(psi_so.b(z) == b_func(z)))
assert (all(psi_so.bs(y) == bs_func(y)))
def test_psi_thermwind_init(self, psi_config):
if not 'z' in psi_config or not isinstance(psi_config['z'], np.ndarray
) or not len(psi_config['z']):
with pytest.raises(TypeError) as zinfo:
Psi_Thermwind(**psi_config)
assert (
str(zinfo.value) == "z needs to be numpy array providing grid levels"
)
return
psi = Psi_Thermwind(**psi_config)
for k in ['z', 'f', 'sol_init', 'b1', 'b2']:
assert hasattr(psi, k)
psi_signature = funcsigs.signature(Psi_Thermwind)
for k in ['f']:
assert getattr(psi, k) == (
psi_config[k] if k in psi_config and psi_config[k] else
psi_signature.parameters[k].default
)
if not 'sol_init' in psi_config:
testing.assert_array_equal(psi.sol_init, np.zeros((2, len(psi.z))))
for k in ['b1', 'b2']:
f = getattr(psi, k)
ft = make_func(psi_config[k], psi_config['z'], k)
for z in psi.z:
assert f(z) == ft(z)
def update(self, b1=None, b2=None):
r"""
Update the vertical buoyancy profiles from the southern basin and northern region.
Parameters
----------
b1 : float, function, or ndarray; optional
Vertical buoyancy profile from the southern basin. Units: m/s\ :sup:`2`
b2 : float, function, or ndarray; optional
Vertical buoyancy profile from the northern basin, representing the
deepwater formation region. Units: m/s\ :sup:`2`
"""
# update buoyancy profiles
if b1 is not None:
self.b1 = make_func(b1, self.z, 'b1')
if b2 is not None:
self.b2 = make_func(b2, self.z, 'b2')
def update(self, b=None, bs=None):
r"""
Update the vertical buoyancy profile and surface buoyancy, based on changes
in the adjoining basin and/or in the surface boundary conditions.
Parameters
----------
b : float, function, or ndarray
Vertical buoyancy profile from the adjoining basin, on the north
side of the ACC. Units: m/s\ :sup:`2`
bs : float, function, or ndarray
Surface level buoyancy boundary condition. Can be a constant,
or an array or function in y. Units: m/s\ :sup:`2`
"""
if b is not None:
self.b = make_func(b, self.z, 'b')
if bs is not None:
self.bs = make_func(bs, self.y, 'bs')
def __init__(
self,
f=1.2e-4, # Coriolis parameter (input)
z=None, # grid (input)
sol_init=None, # Initial conditions for ODE solver (input)
b1=None, # Buoyancy in the basin (input, output)
b2=0., # Buoyancy in the deep water formation region (input, output)
):
r"""
Parameters
----------
f : float
Coriolis parameter. Units s\ :sup:`-1`
z : ndarray
Vertical depth levels of overturning grid. Units: m
sol_init : ndarray; optional
Initial guess at the solution to the thermal wind overturning streamfunction. Units: [...]
b1 : float, function, or ndarray; optional
Vertical buoyancy profile from the southern basin. Units: m/s\ :sup:`2`
b2 : float, function, or ndarray; optional
Vertical buoyancy profile from the northern basin, representing the
deepwater formation region. Units: m/s\ :sup:`2`
"""
self.f = f
# initialize grid:
if isinstance(z, np.ndarray):
self.z = z
nz = np.size(z)
else:
raise TypeError('z needs to be numpy array providing grid levels')
self.b1 = make_func(b1, self.z, 'b1')
self.b2 = make_func(b2, self.z, 'b2')
# Set initial conditions for BVP solver
if sol_init is None:
self.sol_init = np.zeros((2, nz))
else:
self.sol_init = sol_init
def calc_GM(self):
r"""
Compute the eddy (Gent & Mcwilliams) transport based on the meridionally
averaged isopycnal slope.
.. math::
\begin{aligned}
\Psi_{GM} &= K_{GM}\cdot s \\
s\left(b\right) &\equiv \frac{z_B\left(b\right)}{L_y - y_{SO}\left(b\right)}
\end{aligned}
Where :math:`z_B\left(b\right)` is the depth of isopycnals of density class :math:`b`
in the adjoining basin, and :math:`y_{SO}\left(b\right)` is the outcropping latitude
of isopycnals of density class :math:`b` in the Southern Ocean, available via
:meth:`pymoc.modules.Psi_SO.ys`.
Returns
-------
Psi_GM : ndarray
An array representing the meridional average of the eddy transport at
each vertical level of the Southern Ocean model.
"""
dy_atz = 0 * self.z
eps = 0.1 # minimum dy (in meters) (to avoid div. by 0)
for ii in range(0, np.size(self.z)):
dy_atz[ii] = max(self.y[-1] - self.ys(self.b(self.z[ii])), eps)
bottaper = self.calc_bottom_taper(self.Htaperbot, self.z)
toptaper = self.calc_top_taper(self.Htapertop, self.z)
if self.c is not None:
temp = make_func(
self.KGM * self.z / dy_atz * self.L * toptaper * bottaper,
self.z, 'psiGM')
N2 = self.calc_N2()
def ode(z, y):
return np.vstack((y[1], N2(z) / self.c**2. * (y[0] - temp(z))))
#Solve the boundary value problem
res = integrate.solve_bvp(ode, self.bc_GM, self.z,
np.zeros((2, np.size(self.z))))
# return solution interpolated onto original grid
temp = res.sol(self.z)[0, :]
else:
temp = self.KGM * np.maximum(
self.z / dy_atz, -self.smax) * self.L * toptaper * bottaper
# limit Psi_GM to -Psi_Ek on isopycnals that don't outcrop:
idx = dy_atz > self.y[-1] - self.y[0]
temp[idx] = np.maximum(temp[idx], -self.Psi_Ek[idx] * 1e6)
return temp
def calc_N2(self):
r"""
Calculate the buouyancy (Brunt-Väisällä) frequency profile for the Southern Ocean
Returns
-------
N2 : function
A depth dependent function that returns the buoyancy frequency :math:`N^2` for a given depth :math:`z`.
"""
dz = self.z[1:] - self.z[:-1]
N2 = np.zeros(np.size(self.z))
b = self.b(self.z)
N2[1:-1] = (b[2:] - b[:-2]) / (dz[1:] + dz[:-1])
N2[0] = (b[1] - b[0]) / dz[0]
N2[-1] = (b[-1] - b[-2]) / dz[-1]
return make_func(N2, self.z, 'N2')
def solve_equi(self, wA):
r"""
Solve for the equilibrium buoyancy profile, given a specified vertical
velocity profile, and pre-set surface and bottom boundary conditions, based
on the system of equations defined by :meth:`pymoc.modules.Column.ode`.
Parameters
----------
wA : ndarray
Area integrated velocity profile for the equilibrium solution. Units: m\ :sup:`3`/s
"""
self.wA = make_func(wA, self.z, 'w')
sol_init = np.zeros((2, np.size(self.z)))
sol_init[0, :] = self.b
sol_init[1, :] = self.bz
res = integrate.solve_bvp(self.ode, self.bc, self.z, sol_init)
# interpolate solution for b and db/dz onto original grid
self.b = res.sol(self.z)[0, :]
self.bz = res.sol(self.z)[1, :]
def test_psi_so_init(self, psi_so_config):
if not 'z' in psi_so_config or not isinstance(
psi_so_config['z'], np.ndarray
) or not len(psi_so_config['z']):
with pytest.raises(TypeError) as zinfo:
Psi_SO(**psi_so_config)
assert (
str(zinfo.value) == "z needs to be numpy array providing grid levels"
)
return
if not 'y' in psi_so_config or not isinstance(
psi_so_config['y'], np.ndarray
) or not len(psi_so_config['y']):
with pytest.raises(TypeError) as yinfo:
Psi_SO(**psi_so_config)
assert (
str(yinfo.value) ==
"y needs to be numpy array providing horizontal grid (or boundaries) of ACC"
)
return
# Implicit exceptions thrown if these params are missing, should change these to explicit checks
for k in ['b', 'bs', 'tau']:
if not k in psi_so_config:
with pytest.raises(TypeError) as kinfo:
Psi_SO(**psi_so_config)
assert (
str(kinfo.value) == "('" + k +
"', 'needs to be either function, numpy array, or float')"
)
return
psi_so = Psi_SO(**psi_so_config)
for k in [
'z', 'y', 'b', 'bs', 'tau', 'f', 'rho', 'L', 'KGM', 'c', 'bvp_with_Ek',
'Hsill', 'HEk', 'Htapertop', 'Htaperbot', 'smax'
]:
assert hasattr(psi_so, k)
psi_so_signature = funcsigs.signature(Psi_SO)
for k in [
'f', 'rho', 'L', 'KGM', 'c', 'bvp_with_Ek', 'Hsill', 'HEk',
'Htapertop', 'Htaperbot', 'smax'
]:
assert getattr(psi_so, k) == (
psi_so_config[k] if k in psi_so_config and psi_so_config[k] else
psi_so_signature.parameters[k].default
)
for k in ['b']:
f = getattr(psi_so, k)
ft = make_func(psi_so_config[k], psi_so_config['z'], k)
for z in psi_so.z:
assert f(z) == ft(z)
for k in ['bs', 'tau']:
f = getattr(psi_so, k)
ft = make_func(psi_so_config[k], psi_so_config['y'], k)
for y in psi_so.y:
assert f(y) == ft(y)
def test_calc_GM_no_bvp(self, psi_so):
# Test constant isopycnal sloping
dy_atz = 2001000.0
GM = [psi_so.L * psi_so.KGM * z / dy_atz for z in psi_so.z]
psi_so.Psi_Ek = psi_so.calc_Ekman()
assert (
all(
np.around(GM, decimals=3) ==
np.around(psi_so.calc_GM(), decimals=3)
)
)
# Test minimum dy limit
dy_atz = 0.1
psi_so.b = make_func(np.linspace(0.3, 0.2, 81), psi_so.z, 'b')
GM = [-psi_so.L * psi_so.KGM * psi_so.smax for z in psi_so.z]
GM[-1] = 0
assert (
all(
np.around(GM, decimals=3) ==
np.around(psi_so.calc_GM(), decimals=3)
)
)
# Test maximum isopycnal slope limit
dy_atz = 2001000.0
psi_so.smax = -0.01
psi_so.b = make_func(np.linspace(0.03, -0.001, 81), psi_so.z, 'b')
GM = [psi_so.L * psi_so.KGM * 0.01 for _ in psi_so.z]
psi_so.Psi_Ek = psi_so.calc_Ekman()
assert (
all(
np.around(GM, decimals=3) ==
np.around(psi_so.calc_GM(), decimals=3)
)
)
# Test quadratic bottom tapering of streamfunction
psi_so.smax = 0.01
psi_so.Htaperbot = 800.0
dy_atz = 2001000.0
GM = [psi_so.L * psi_so.KGM * z / dy_atz for z in psi_so.z]
GM[0:17] = [(1.0 - (-3200.0 - psi_so.z[i])**2 / 6.4e5) * GM[i]
for i in range(0, 17)]
psi_so.Psi_Ek = psi_so.calc_Ekman()
assert (
all(
np.around(GM, decimals=3) ==
np.around(psi_so.calc_GM(), decimals=3)
)
)
psi_so.Htaperbot = None
# Test quadratic top tapering of streamfunction
psi_so.smax = 0.01
psi_so.Htapertop = 500.0
dy_atz = 2001000.0
GM = [psi_so.L * psi_so.KGM * z / dy_atz for z in psi_so.z]
GM[-11:] = [(1.0 - (500.0 + psi_so.z[i])**2 / 2.5e5) * GM[i]
for i in range(70, 81)]
psi_so.Psi_Ek = psi_so.calc_Ekman()
assert (
all(
np.around(GM, decimals=3) ==
np.around(psi_so.calc_GM(), decimals=3)
)
)
psi_so.Htapertop = None
# Test ekman streamfunction limiting for non-outcropping isopycnals
psi_so = Psi_SO(
**{
'z': np.asarray(np.linspace(-4000, 0, 81)),
'y': np.asarray(np.linspace(0, 2.0e3, 51)),
'b': np.linspace(0.03, -0.001, 81),
'bs': np.linspace(0.05, 0.10, 51),
'tau': 0.12
}
)
dy_atz = 2001000.0
GM = [-psi_so.L * psi_so.KGM * 0.01 for _ in psi_so.z]
GM[-1] = 0
psi_ek = np.asarray([gm + np.abs(gm / 2.0) for gm in GM])
psi_so.Psi_Ek = psi_ek
assert (
all(
np.around(-psi_ek * 1e6, decimals=3) ==
np.around(psi_so.calc_GM(), decimals=3)
)
)
def __init__(
self,
z=None, # vertical grid (array, in)
y=None, # horizontal grid (array, in)
b=None, # buoyancy profile at northern end of ACC (function, array or float, in)
bs=None, # surface buoyancy (function, array or float, in)
tau=None, # surface wind stress (function, array or float, in)
f=1.2e-4, # Coriolis parameter (in)
rho=1030, # Density of sea water (in)
L=1e7, # Zonal length of the ACC (in)
KGM=1e3, # GM coefficient (in)
c=None, # phase speed for F2010 BVP smoother of GM streamfunction
bvp_with_Ek=False, # if true, apply boundary condition that Psi_GM=-Psi_EK in F2010 BVP smoother
Hsill=None, # height (in m above ocean floor) of the "sill", where Psi_Ek is tapered
HEk=None, # depth of surface Ekman layer
Htapertop=None, # A quadratic tapering of the GM streamfunction at the surface
Htaperbot=None, # A quadratic tapering of the GM streamfunction at the bottom
smax=0.01, # maximum slope for clipping of GM streamfunction
):
r"""
Parameters
----------
z : ndarray
Vertical depth levels of overturning grid. Units: m
y : ndarray
Meridional overturning grid. Units: m
b : float, function, or ndarray
Vertical buoyancy profile from the adjoining basin, on the north
side of the ACC. Units: m/s\ :sup:`2`
bs : float, function, or ndarray
Surface level buoyancy boundary condition. Can be a constant,
or an array or function in y. Units: m/s\ :sup:`2`
tau : float, function, or ndarray
Surface wind stress. Can be a constant, or an array or function
in y. Units: N/m\ :sup:`2`
f : float
Coriolis parameter. Units s\ :sup:`-1`
rho : float
Density of sea water for Boussinesq approximation. Units: kg/m\ :sup:`3`
L : float
Zonal length of the modeled ACC. Units: m
KGM : float
Gent & McWilliams (GM) eddy diffusivity coefficient. Units:
c : float
Phase speed cutoff for smoothing when solving the GM boundary value problem. Units: m/s
bvp_with_Ek : logical
Whether to enforce the boundary condition that Psi_GM=-Psi_Ek at the ocean
surface and bottom when solving the boundary value problem for the GM streamfunction.
Hsill : float
Height above the bottom at which the Ekman streamfunction is tapered. Units: m
Hek : float
Depth of the surface Ekman layer. Units: m
Htapertop : float
Height of the quadratic surface tapering layer for the GM streamfunction. Units: m
Htaperbot : float
Height of the quadratic bottom tapering layer for the GM streamfunction. Units: m
smax : float
Maximum slope of the GM streamfunction, above which Psi_GM is clipped. Units: m\ :sup:`-1`
"""
# initialize grid:
if isinstance(z, np.ndarray):
self.z = z
else:
raise TypeError('z needs to be numpy array providing grid levels')
if isinstance(y, np.ndarray):
self.y = y
else:
raise TypeError(
'y needs to be numpy array providing horizontal grid (or boundaries) of ACC'
)
self.b = make_func(b, self.z, 'b')
self.bs = make_func(bs, self.y, 'bs')
self.tau = make_func(tau, self.y, 'tau')
self.f = f
self.rho = rho
self.L = L
self.KGM = KGM
self.c = c
self.bvp_with_Ek = bvp_with_Ek
self.Hsill = Hsill
self.HEk = HEk
self.Htapertop = Htapertop
self.Htaperbot = Htaperbot
self.smax = smax
def make_func(self, myst, name, xin): # Seems unecessary to define a method that already exists identically as a function, no? return make_func(myst, xin, name)
def make_func(self, myst, name, xin):
return make_func(myst, xin, name)