def dynmodes(n=6, lat0=5., plot=False, model='Fratantoni_etal1995'):
"""
Computes the discrete eigenmodes (dynamical modes)
for a quasi-geostrophic ocean with n isopycnal layers.
Rigid lids are placed at the surface and the bottom.
Inputs:
-------
n: Number of layers.
lat0: Reference latitude.
H: List of rest depths of each layer.
S: List of potential density values for each layer.
"""
omega = 2*np.pi/86400. # [rad s^{-1}]
f0 = 2*omega*np.sin(lat0*np.pi/180) # [s^{-1}]
f02 = f0**2 # [s^{-2}]
g = 9.81 # [m s^{-1}]
rho0 = 1027. # [kg m^{-3}]
if model=='Fratantoni_etal1995':
H = np.array([80.,170.,175.,250.,325.,3000.])
S = np.array([24.97,26.30,26.83,27.12,27.32,27.77])
tit = ur'Modelo de seis camadas para a CNB de Fratantoni \textit{et al.} (1995)'
figname = 'vmodes_fratantoni_etal1995'
elif model=='Bub_Brown1996':
H = np.array([150.,440.,240.,445.,225.,2500.])
S = np.array([24.13,26.97,27.28,27.48,27.74,27.87])
tit = ur'Modelo de seis camadas para a CNB de Bub e Brown (1996)'
figname = 'vmodes_bub_brown1996'
# Normalized density jumps.
E = (S[1:]-S[:-1])/rho0
# Rigid lids at the surface and the bottom,
# meaning infinite density jumps.
E = np.hstack( (np.inf,E,np.inf) )
# Building the tridiagonal matrix.
A = np.zeros( (n,n) )
for i in xrange(n):
A[i,i] = -f02/(E[i+1]*g*H[i]) -f02/(E[i]*g*H[i]) # The main diagonal.
if i>0:
A[i,i-1] = f02/(E[i]*g*H[i])
if i<(n-1):
A[i,i+1] = f02/(E[i+1]*g*H[i])
# get eigenvalues and convert them
# to internal deformation radii
lam,v = eig(A)
lam = np.abs(lam)
# Baroclinic def. radii in km:
uno = np.ones( (lam.size,lam.size) )
Rd = 1e-3*uno/np.sqrt(lam)
Rd = np.unique(Rd)
Rd = np.flipud(Rd)
np.disp("Deformation radii [km]:")
[np.disp(int(r)) for r in Rd]
# orthonormalize eigenvectors, i.e.,
# find the dynamical mode vertical structure functions.
F = np.zeros( (n,n) )
for i in xrange(n):
mi = v[:,i] # The vertical structure of the i-th vertical mode.
fac = np.sqrt(np.sum(H*mi*mi)/np.sum(H))
F[:,i] = 1/fac*mi
F=-F
F[:,0] = np.abs(F[:,1])
F = np.fliplr(F)
Fi = np.vstack( (F[0,:],F) )
Fi = np.flipud(Fi)
for i in xrange(n-1):
Fi[i,:] = F[i+1,:]
Fi = np.flipud(Fi)
# Plot the vertical modes.
if plot:
plt.close('all')
kw = dict(fontsize=15, fontweight='black')
fig,ax = plt.subplots()
ax.hold(True)
Hc = np.sum(H)
z = np.flipud(np.linspace(-Hc,0,1000))
Hp = np.append(0,H)
Hp = -np.cumsum(Hp)
# build modes for plotting purposes
Fp = np.zeros( (z.size,n) )
fo = 0
for i in xrange(n):
f1 = near(z,Hp[i])[0][0]
for j in xrange(fo,f1):
Fp[j,:] = F[i,:]
fo=f1
for i in xrange(n):
l = 'Modo %s'%str(i)
ax.plot(Fp[:,i], z, label=l)
xl,xr = ax.set_xlim(-5,5)
ax.hlines(Hp,xl,xr,linestyle='dashed')
ax.hold(False)
ax.set_title(tit, **kw)
ax.set_xlabel(ur'Autofunção [adimensional]', **kw)
ax.set_ylabel(ur'Profundidade [m]', **kw)
try:
rstyle(ax)
except:
pass
ax.legend(loc='lower left', fontsize=20, fancybox=True, shadow=True)
fmt='png'
fig.savefig(figname+'.'+fmt, format=fmt, bbox='tight')
def dynmodes(n=6, lat0=5., plot=False, model='Fratantoni_etal1995'):
"""
Computes the discrete eigenmodes (dynamical modes)
for a quasi-geostrophic ocean with n isopycnal layers.
Rigid lids are placed at the surface and the bottom.
Inputs:
-------
n: Number of layers.
lat0: Reference latitude.
H: List of rest depths of each layer.
S: List of potential density values for each layer.
"""
omega = 2 * np.pi / 86400. # [rad s^{-1}]
f0 = 2 * omega * np.sin(lat0 * np.pi / 180) # [s^{-1}]
f02 = f0**2 # [s^{-2}]
g = 9.81 # [m s^{-1}]
rho0 = 1027. # [kg m^{-3}]
if model == 'Fratantoni_etal1995':
H = np.array([80., 170., 175., 250., 325., 3000.])
S = np.array([24.97, 26.30, 26.83, 27.12, 27.32, 27.77])
tit = 'Modelo de seis camadas para a CNB de Fratantoni \textit{et al.} (1995)'
figname = 'vmodes_fratantoni_etal1995'
elif model == 'Bub_Brown1996':
H = np.array([150., 440., 240., 445., 225., 2500.])
S = np.array([24.13, 26.97, 27.28, 27.48, 27.74, 27.87])
tit = 'Modelo de seis camadas para a CNB de Bub e Brown (1996)'
figname = 'vmodes_bub_brown1996'
# Normalized density jumps.
E = (S[1:] - S[:-1]) / rho0
# Rigid lids at the surface and the bottom,
# meaning infinite density jumps.
E = np.hstack((np.inf, E, np.inf))
# Building the tridiagonal matrix.
A = np.zeros((n, n))
for i in range(n):
A[i, i] = -f02 / (E[i + 1] * g * H[i]) - f02 / (E[i] * g * H[i]
) # The main diagonal.
if i > 0:
A[i, i - 1] = f02 / (E[i] * g * H[i])
if i < (n - 1):
A[i, i + 1] = f02 / (E[i + 1] * g * H[i])
# get eigenvalues and convert them
# to internal deformation radii
lam, v = eig(A)
lam = np.abs(lam)
# Baroclinic def. radii in km:
uno = np.ones((lam.size, lam.size))
Rd = 1e-3 * uno / np.sqrt(lam)
Rd = np.unique(Rd)
Rd = np.flipud(Rd)
np.disp("Deformation radii [km]:")
[np.disp(int(r)) for r in Rd]
# orthonormalize eigenvectors, i.e.,
# find the dynamical mode vertical structure functions.
F = np.zeros((n, n))
for i in range(n):
mi = v[:, i] # The vertical structure of the i-th vertical mode.
fac = np.sqrt(np.sum(H * mi * mi) / np.sum(H))
F[:, i] = 1 / fac * mi
F = -F
F[:, 0] = np.abs(F[:, 1])
F = np.fliplr(F)
Fi = np.vstack((F[0, :], F))
Fi = np.flipud(Fi)
for i in range(n - 1):
Fi[i, :] = F[i + 1, :]
Fi = np.flipud(Fi)
# Plot the vertical modes.
if plot:
plt.close('all')
kw = dict(fontsize=15, fontweight='black')
fig, ax = plt.subplots()
ax.hold(True)
Hc = np.sum(H)
z = np.flipud(np.linspace(-Hc, 0, 1000))
Hp = np.append(0, H)
Hp = -np.cumsum(Hp)
# build modes for plotting purposes
Fp = np.zeros((z.size, n))
fo = 0
for i in range(n):
f1 = near(z, Hp[i])[0][0]
for j in range(fo, f1):
Fp[j, :] = F[i, :]
fo = f1
for i in range(n):
l = 'Modo %s' % str(i)
ax.plot(Fp[:, i], z, label=l)
xl, xr = ax.set_xlim(-5, 5)
ax.hlines(Hp, xl, xr, linestyle='dashed')
ax.hold(False)
ax.set_title(tit, **kw)
ax.set_xlabel('Autofunção [adimensional]', **kw)
ax.set_ylabel('Profundidade [m]', **kw)
try:
rstyle(ax)
except:
pass
ax.legend(loc='lower left', fontsize=20, fancybox=True, shadow=True)
fmt = 'png'
fig.savefig(figname + '.' + fmt, format=fmt, bbox='tight')
def ekman(r=4., plot=True, savefig=False):
"""
Plots the solution of the Ekman problem
in a vertically finite ocean of depth H.
Wind is purely meridional, f-plane, linearized
steady-state linearized solutions. The wind is
imposed impulsively over the initially at rest
water column.
"""
tau0y = 0.08 # Wind stress, meridional-only [Pa]
rho = 1025. # Seawater density [kg m^{-3}]
A = 5e-2 # Eddy viscosity coefficient [m^{2} s^{-1}]
theta0 = -30. # (Reference) latitude. [ ]
omega = 2*np.pi/86400. # Planetary angular frequency. [s^{-1}]
f = 2*omega*np.sin(theta0*np.pi/180.) # Coriolis parameter. [s^{-1}]
d = np.sqrt(2*A/np.abs(f)) # Ekman layer depth. [m]
H = r*d # Bottom depth. [m]
st = "H, d, H/d = %s, %s, %s"%(str(H),str(d),str(r))
np.disp(st)
z = np.linspace(0, H, 1000.)
# Solutions for u and v.
C = tau0y/(rho*A)
Denom = np.sinh(r)*np.sin(r) + np.cosh(r)*np.cos(r)
u = C*np.sinh(z/d)*np.cos(z/d)/Denom
v = C*np.cosh(z/d)*np.sin(z/d)/Denom
# Veering angle at the surface.
ang = np.arctan(v[-1]/u[-1])*180/np.pi
st = "Veering angle at the surface: %s"%str(ang)
np.disp(st)
if plot:
# u and v profiles.
plt.close('all')
fig,ax = plt.subplots()
ax.hold(True)
ax.plot(u,z,'k-',label=ur'Zonal velocity, u$_{Ek}$')
ax.plot(v,z,'k--',label=ur'Meridional velocity, v$_{Ek}$')
ax.set_xlabel(ur'Velocity [cm s$^{-1}$]')
ax.set_ylabel(ur'Depth [m]')
ax.legend(loc='best',fontsize=13)
ax.hold(False)
try:
rstyle(ax)
except:
pass
if savefig:
plt.savefig('uv.png',bbox='tight')
# Hodograph plot.
fig,ax = plt.subplots()
ax.hold(True)
ax.plot(u,v,'k-',linewidth=1.5)
# sx = ax.scatter(u,v,c=z,cmap=plt.cm.binary,linewidths=0.,marker='o')
# cb = plt.colorbar(mappable=sx)
# cb.set_label(ur'Depth [m]')
ax.set_xlabel(ur'Zonal velocity [cm s$^{-1}$]')
ax.set_ylabel(ur'Meridional velocity [cm s$^{-1}$]')
ax.hold(False)
try:
rstyle(ax)
except:
pass
if savefig:
plt.savefig('hodo.png',bbox='tight')
return z,u,v
def ekman(r=4., plot=True, savefig=False):
"""
Plots the solution of the Ekman problem
in a vertically finite ocean of depth H.
Wind is purely meridional, f-plane, linearized
steady-state linearized solutions. The wind is
imposed impulsively over the initially at rest
water column.
"""
tau0y = 0.08 # Wind stress, meridional-only [Pa]
rho = 1025. # Seawater density [kg m^{-3}]
A = 5e-2 # Eddy viscosity coefficient [m^{2} s^{-1}]
theta0 = -30. # (Reference) latitude. [ ]
omega = 2 * np.pi / 86400. # Planetary angular frequency. [s^{-1}]
f = 2 * omega * np.sin(
theta0 * np.pi /
180.) # Coriolis parameter. [s^{-1}]
d = np.sqrt(
2 * A /
np.abs(f)) # Ekman layer depth. [m]
H = r * d # Bottom depth. [m]
st = "H, d, H/d = %s, %s, %s" % (str(H), str(d), str(r))
np.disp(st)
z = np.linspace(0, H, 1000.)
# Solutions for u and v.
C = tau0y / (rho * A)
Denom = np.sinh(r) * np.sin(r) + np.cosh(r) * np.cos(r)
u = C * np.sinh(z / d) * np.cos(z / d) / Denom
v = C * np.cosh(z / d) * np.sin(z / d) / Denom
# Veering angle at the surface.
ang = np.arctan(v[-1] / u[-1]) * 180 / np.pi
st = "Veering angle at the surface: %s" % str(ang)
np.disp(st)
if plot:
# u and v profiles.
plt.close('all')
fig, ax = plt.subplots()
ax.hold(True)
ax.plot(u, z, 'k-', label='Zonal velocity, u$_{Ek}$')
ax.plot(v, z, 'k--', label='Meridional velocity, v$_{Ek}$')
ax.set_xlabel('Velocity [cm s$^{-1}$]')
ax.set_ylabel('Depth [m]')
ax.legend(loc='best', fontsize=13)
ax.hold(False)
try:
rstyle(ax)
except:
pass
if savefig:
plt.savefig('uv.png', bbox='tight')
# Hodograph plot.
fig, ax = plt.subplots()
ax.hold(True)
ax.plot(u, v, 'k-', linewidth=1.5)
# sx = ax.scatter(u,v,c=z,cmap=plt.cm.binary,linewidths=0.,marker='o')
# cb = plt.colorbar(mappable=sx)
# cb.set_label('Depth [m]')
ax.set_xlabel('Zonal velocity [cm s$^{-1}$]')
ax.set_ylabel('Meridional velocity [cm s$^{-1}$]')
ax.hold(False)
try:
rstyle(ax)
except:
pass
if savefig:
plt.savefig('hodo.png', bbox='tight')
return z, u, v
