import fridom.nonhydro as nh
[docs]
class KelvinWave(nh.State):
"""
Initial condition with a kelvin wave at the boundary.
TODO: Add some more details about the math, add proper gallery example.
Description
-----------
Lets consider a Kelvin wave at the southern boundary of the domain with
the horizontal wavenumber `kh` and the vertical wavenumber `kz`. Searching
in the linearized nonhydrostatic equations for a solution of the form:
.. math::
(U, V, W, B) \\exp(- k_n y) \\exp(i(k_h x + k_z z - \\omega t))
yields
.. math::
\\omega = \\sqrt{\\frac{k_h^2 N^2}{k_h^2 + k_z^2}} \\quad
k_n = \\frac{k_h f_0}{\\omega}
.. math::
U = - k_z \\quad
V = 0 \\quad
W = k_h \\quad
B = -i N^2 k_h / \\omega
Similar polarizations can be found for the other boundaries.
Parameters
----------
`mset` : `ModelSettings`
The model settings object.
`side` : `str`
The side of the domain where the wave is located.
Possible values are 'N': North, 'S': South, 'E': East, 'W': West.
`kz` : `int`
The vertical wavenumber. (Can be positive or negative)
`kh` : `int`
The horizontal wavenumber in the direction of the boundary.
(Must be positive)
Examples
--------
.. code-block:: python
import fridom.nonhydro as nh
import numpy as np
grid = nh.grid.cartesian.Grid(
N=[128]*3, L=[1]*3, periodic_bounds=(True, False, True))
mset = nh.ModelSettings(grid=grid)
mset.time_stepper.dt = np.timedelta64(10, 'ms')
mset.tendencies.advection.disable()
mset.setup()
z = nh.initial_conditions.KelvinWave(mset, 'N', kh=1, kz=2)
model = nh.Model(mset)
model.z = z
model.run(runlen=np.timedelta64(5, 's'))
A vertical mode may be constructed with
.. code-block:: python
z = nh.initial_conditions.kelvin_wave(mset, 'N', kh=1, kz=2)
z += nh.initial_conditions.kelvin_wave(mset, 'N', kh=1, kz=-2, phase=np.pi)
"""
[docs]
def __init__(self,
mset: nh.ModelSettings,
side: str,
kz: int,
k_parallel: int,
phase: float = 0):
super().__init__(mset, is_spectral=False)
ncp = nh.config.ncp
# convert the wavenumbers
Lx, Ly, Lz = mset.grid.L
kz = 2 * ncp.pi * kz / Lz
if side == "N" or side == "S":
L_parallel = Lx
if side == "E" or side == "W":
L_parallel = Ly
k_parallel = 2 * ncp.pi * k_parallel / L_parallel
# calculate the frequency
om = ncp.sqrt((k_parallel**2 * mset.N2) /
(kz**2 + mset.dsqr * k_parallel**2))
# calculate the polarizations
pol_u_normal = - kz
pol_w = k_parallel
pol_b = - 1j * mset.N2 * k_parallel / om
# define function to get the exponential decay and the wave pattern
def wave(x_parallel, x_normal, z):
wave = ncp.exp(1j * (k_parallel * x_parallel + kz * z + phase))
wave *= ncp.exp(- mset.f0 * k_parallel / om * x_normal)
return wave
if side == "N":
def get_wave(f: nh.FieldVariable):
x, y, z = f.get_mesh()
x_normal = Ly - y
x_parallel = Lx - x
return wave(x_parallel, x_normal, z)
self.u.arr = (- pol_u_normal * get_wave(self.u)).imag
elif side == "S":
def get_wave(f: nh.FieldVariable):
x, y, z = f.get_mesh()
x_normal = y
x_parallel = x
return wave(x_parallel, x_normal, z)
self.u.arr = (pol_u_normal * get_wave(self.u)).imag
elif side == "E":
def get_wave(f: nh.FieldVariable):
x, y, z = f.get_mesh()
x_normal = Lx - x
x_parallel = y
return wave(x_parallel, x_normal, z)
self.v.arr = (pol_u_normal * get_wave(self.v)).imag
elif side == "W":
def get_wave(f: nh.FieldVariable):
x, y, z = f.get_mesh()
x_normal = x
x_parallel = Ly - y
return wave(x_parallel, x_normal, z)
self.v.arr = (- pol_u_normal * get_wave(self.v)).imag
self.w.arr = (pol_w * get_wave(self.w)).imag
self.b.arr = (pol_b * get_wave(self.b)).imag
# save some parameters
self.k_parallel = k_parallel
self.kz = kz
self.om = om
