import numpy as np
""" Conversion factor for [:math:`^\\circ` C] to [:math:`^\\circ` K] """
CtoK = 273.16 # 273.15
""" von Karman's constant """
kappa = 0.4 # NOTE: 0.41
# ---------------------------------------------------------------------
def charnock_C35(wind, u10n, usr, seastate, waveage, wcp, sigH, lat):
g = gc(lat, None)
a1, a2 = 0.0017, -0.0050
charnC = np.where(u10n > 19, a1*19+a2, a1*u10n+a2)
A, B = 0.114, 0.622 # wave-age dependent coefficients
Ad, Bd = 0.091, 2.0 # Sea-state/wave-age dependent coefficients
charnW = A*(usr/wcp)**B
zoS = sigH*Ad*(usr/wcp)**Bd
charnS = (zoS*g)/usr**2
charn = np.where(wind > 10, 0.011+(wind-10)/(18-10)*(0.018-0.011),
np.where(wind > 18, 0.018, 0.011*np.ones(np.shape(wind))))
if waveage:
if seastate:
charn = charnS
else:
charn = charnW
else:
charn = charnC
ac = np.zeros((len(wind), 3))
ac[:, 0] = charn
ac[:, 1] = charnS
ac[:, 2] = charnW
return ac
# ---------------------------------------------------------------------
def cd_C35(u10n, wind, usr, charn, monob, Ta, hh_in, lat):
g = gc(lat, None)
zo = charn*usr**2/g+0.11*visc_air(Ta)/usr # surface roughness
rr = zo*usr/visc_air(Ta)
# These thermal roughness lengths give Stanton and
zoq = np.where(5.8e-5/rr**0.72 > 1.6e-4, 1.6e-4, 5.8e-5/rr**0.72)
zot = zoq # Dalton numbers that closely approximate COARE 3.0
cdhf = kappa/(np.log(hh_in[0]/zo)-psiu_26(hh_in[0]/monob))
cthf = kappa/(np.log(hh_in[1]/zot)-psit_26(hh_in[1]/monob))
cqhf = kappa/(np.log(hh_in[2]/zoq)-psit_26(hh_in[2]/monob))
return zo, cdhf, cthf, cqhf
# ---------------------------------------------------------------------
def cdn_calc(u10n, Ta, Tp, method="Smith80"):
if (method == "Smith80"):
cdn = np.where(u10n <= 3, (0.61+0.567/u10n)*0.001,
(0.61+0.063*u10n)*0.001)
elif (method == "LP82"):
cdn = np.where((u10n < 11) & (u10n >= 4), 1.2*0.001,
np.where((u10n <= 25) & (u10n >= 11),
(0.49+0.065*u10n)*0.001, 1.14*0.001))
elif (method == "Smith88" or method == "COARE3.0" or
method == "COARE4.0" or method == "UA" or method == "ERA5"):
cdn = cdn_from_roughness(u10n, Ta, None, method)
elif (method == "HEXOS"):
# Smith et al. 1991 #(0.27 + 0.116*u10n)*0.001 Smith et al. 1992
cdn = (0.5+0.091*u10n)*0.001
elif (method == "HEXOSwave"):
cdn = cdn_from_roughness(u10n, Ta, Tp, method)
elif (method == "YT96"):
# for u<3 same as Smith80
cdn = np.where((u10n < 6) & (u10n >= 3),
(0.29+3.1/u10n+7.7/u10n**2)*0.001,
np.where((u10n <= 26) & (u10n >= 6),
(0.60 + 0.070*u10n)*0.001, (0.61+0.567/u10n)*0.001))
elif (method == "LY04"):
cdn = np.where(u10n >= 0.5,
(0.142+(2.7/u10n)+(u10n/13.09))*0.001, np.nan)
else:
print("unknown method cdn: "+method)
return cdn
# ---------------------------------------------------------------------
def cdn_from_roughness(u10n, Ta, Tp, method="Smith88"):
g, tol = 9.812, 0.000001
cdn, usr = np.zeros(Ta.shape), np.zeros(Ta.shape)
cdnn = (0.61+0.063*u10n)*0.001
zo, zc, zs = np.zeros(Ta.shape), np.zeros(Ta.shape), np.zeros(Ta.shape)
for it in range(5):
cdn = np.copy(cdnn)
usr = np.sqrt(cdn*u10n**2)
if (method == "Smith88"):
# .....Charnock roughness length (equn 4 in Smith 88)
zc = 0.011*np.power(usr, 2)/g
# .....smooth surface roughness length (equn 6 in Smith 88)
zs = 0.11*visc_air(Ta)/usr
zo = zc + zs # .....equns 7 & 8 in Smith 88 to calculate new CDN
elif (method == "COARE3.0"):
zc = 0.011 + (u10n-10)/(18-10)*(0.018-0.011)
zc = np.where(u10n < 10, 0.011, np.where(u10n > 18, 0.018, zc))
zs = 0.11*visc_air(Ta)/usr
zo = zc*np.power(usr, 2)/g + zs
elif (method == "HEXOSwave"):
if ((Tp is None) or np.nansum(Tp) == 0):
Tp = 0.729*u10n # Taylor and Yelland 2001
cp_wave = g*Tp/2/np.pi # use input wave period
zo = 0.48*np.power(usr, 3)/g/cp_wave # Smith et al. 1992
elif (method == "UA"):
# valid for 0<u<18m/s # Zeng et al. 1998 (24)
zo = 0.013*np.power(usr, 2)/g+0.11*visc_air(Ta)/usr
elif (method == "ERA5"):
# eq. (3.26) p.40 over sea IFS Documentation cy46r1
zo = 0.11*visc_air(Ta)/usr+0.018*np.power(usr, 2)/g
else:
print("unknown method for cdn_from_roughness "+method)
cdnn = (kappa/np.log(10/zo))**2
cdn = np.where(np.abs(cdnn-cdn) < tol, cdnn, np.nan)
return cdnn
# ---------------------------------------------------------------------
def ctcqn_calc(zol, cdn, u10n, zo, Ta, method="Smith80"):
if (method == "Smith80" or method == "Smith88" or method == "YT96"):
cqn = np.ones(Ta.shape)*1.20*0.001 # from Smith88
ctn = np.ones(Ta.shape)*1.00*0.001
elif (method == "LP82"):
cqn = np.where((zol <= 0) & (u10n > 4) & (u10n < 14), 1.15*0.001,
np.nan)
ctn = np.where((zol <= 0) & (u10n > 4) & (u10n < 25), 1.13*0.001,
0.66*0.001)
elif (method == "HEXOS" or method == "HEXOSwave"):
cqn = np.where((u10n <= 23) & (u10n >= 3), 1.1*0.001, np.nan)
ctn = np.where((u10n <= 18) & (u10n >= 3), 1.1*0.001, np.nan)
elif (method == "COARE3.0" or method == "COARE4.0"):
usr = (cdn*u10n**2)**0.5
rr = zo*usr/visc_air(Ta)
zoq = 5.5e-5/rr**0.6
zoq[zoq > 1.15e-4] = 1.15e-4
zot = zoq
cqn = kappa**2/np.log(10/zo)/np.log(10/zoq)
ctn = kappa**2/np.log(10/zo)/np.log(10/zot)
elif (method == "LY04"):
cqn = 34.6*0.001*cdn**0.5
ctn = np.where(zol <= 0, 32.7*0.001*cdn**0.5, 18*0.001*cdn**0.5)
elif (method == "UA"):
usr = (cdn * u10n**2)**0.5
# Zeng et al. 1998 (25)
zoq = zo*np.exp(-(2.67*np.power(usr*zo/visc_air(Ta), 1/4)-2.57))
zot = zoq
cqn = np.where((u10n > 0.5) & (u10n < 18), np.power(kappa, 2) /
(np.log(10/zo)*np.log(10/zoq)), np.nan)
ctn = np.where((u10n > 0.5) & (u10n < 18), np.power(kappa, 2) /
(np.log(10/zo)*np.log(10/zoq)), np.nan)
elif (method == "ERA5"):
# eq. (3.26) p.40 over sea IFS Documentation cy46r1
usr = np.sqrt(cdn*np.power(u10n, 2))
zot = 0.40*visc_air(Ta)/usr
zoq = 0.62*visc_air(Ta)/usr
cqn = kappa**2/np.log(10/zo)/np.log(10/zoq)
ctn = kappa**2/np.log(10/zo)/np.log(10/zot)
else:
print("unknown method ctcqn: "+method)
return ctn, cqn
# ---------------------------------------------------------------------
def cd_calc(cdn, height, ref_ht, psim):
cd = (cdn*np.power(1+np.sqrt(cdn)*np.power(kappa, -1) *
(np.log(height/ref_ht)-psim), -2))
return cd
# ---------------------------------------------------------------------
def ctcq_calc(cdn, cd, ctn, cqn, h_t, h_q, ref_ht, psit, psiq):
ct = ctn*(cd/cdn)**0.5/(1+ctn*((np.log(h_t/ref_ht)-psit)/(kappa*cdn**0.5)))
cq = cqn*(cd/cdn)**0.5/(1+cqn*((np.log(h_q/ref_ht)-psiq)/(kappa*cdn**0.5)))
return ct, cq
# ---------------------------------------------------------------------
def psim_calc(zol, method="Smith80"):
coeffs = get_stabco(method)
alpha, beta, gamma = coeffs[0], coeffs[1], coeffs[2]
if (method == "COARE3.0" or method == "COARE4.0"):
psim = np.where(zol < 0, psim_conv_coare3(zol, alpha, beta, gamma),
psim_stab_coare3(zol, alpha, beta, gamma))
elif (method == "ERA5"):
psim = np.where(zol < 0, psim_conv(zol, alpha, beta, gamma),
psim_stab_era5(zol, alpha, beta, gamma))
else:
psim = np.where(zol < 0, psim_conv(zol, alpha, beta, gamma),
psim_stab(zol, alpha, beta, gamma))
return psim
# ---------------------------------------------------------------------
def psit_calc(zol, method="Smith80"):
coeffs = get_stabco(method)
alpha, beta, gamma = coeffs[0], coeffs[1], coeffs[2]
if (method == "COARE3.0" or method == "COARE4.0"):
psit = np.where(zol < 0, psi_conv_coare3(zol, alpha, beta, gamma),
psi_stab_coare3(zol, alpha, beta, gamma))
elif (method == "ERA5"):
psit = np.where(zol < 0, psi_conv(zol, alpha, beta, gamma),
psi_stab_era5(zol, alpha, beta, gamma))
else:
psit = np.where(zol < 0, psi_conv(zol, alpha, beta, gamma),
psi_stab(zol, alpha, beta, gamma))
return psit
# ---------------------------------------------------------------------
def get_stabco(method="Smith80"):
if (method == "Smith80" or method == "Smith88" or method == "LY04" or
method == "UA" or method == "ERA5"):
alpha, beta, gamma = 16, 0.25, 5 # Smith 1980, from Dyer (1974)
elif (method == "LP82"):
alpha, beta, gamma = 16, 0.25, 7
elif (method == "HEXOS" or method == "HEXOSwave"):
alpha, beta, gamma = 16, 0.25, 8
elif (method == "YT96"):
alpha, beta, gamma = 20, 0.25, 5
elif (method == "COARE3.0" or method == "COARE4.0"):
# use separate subroutine
alpha, beta, gamma = 15, 1/3, 5 # not sure about gamma=34.15
else:
print("unknown method stabco: "+method)
coeffs = np.zeros(3)
coeffs[0] = alpha
coeffs[1] = beta
coeffs[2] = gamma
return coeffs
# ---------------------------------------------------------------------
def psi_conv_coare3(zol, alpha, beta, gamma):
x = (1-alpha*zol)**0.5 # Kansas unstable
psik = 2*np.log((1+x)/2.)
y = (1-34.15*zol)**beta
psic = (1.5*np.log((1+y+y*y)/3.)-(3)**0.5*np.arctan((1+2*y)/(3)**0.5) +
4*np.arctan(1)/(3)**0.5)
f = zol*zol/(1.+zol*zol)
psit = (1-f)*psik+f*psic
return psit
# ---------------------------------------------------------------------
def psi_stab_coare3(zol, alpha, beta, gamma): # Stable
c = np.where(0.35*zol > 50, 50, 0.35*zol) # Stable
psit = -((1+2*zol/3)**1.5+0.6667*(zol-14.28)/np.exp(c)+8.525)
return psit
# ---------------------------------------------------------------------
def psi_stab_era5(zol, alpha, beta, gamma):
# eq (3.22) p. 39 IFS Documentation cy46r1
a, b, c, d = 1, 2/3, 5, 0.35
psit = -b*(zol-c/d)*np.exp(-d*zol)-np.power(1+(2/3)*a*zol, 1.5)-(b*c)/d+1
return psit
# ---------------------------------------------------------------------
def psi_conv(zol, alpha, beta, gamma):
xtmp = (1-alpha*zol)**beta
psit = 2*np.log((1+xtmp**2)*0.5)
return psit
# ---------------------------------------------------------------------
def psi_stab(zol, alpha, beta, gamma):
psit = -gamma*zol
return psit
# ---------------------------------------------------------------------
def psit_26(zet):
"""
computes temperature structure function as in COARE3.5
"""
dzet = np.where(0.35*zet > 50, 50, 0.35*zet) # stable
psi = -((1+0.6667*zet)**1.5+0.6667*(zet-14.28)*np.exp(-dzet)+8.525)
k = np.where(zet < 0) # unstable
x = (1-15*zet[k])**0.5
psik = 2*np.log((1+x)/2)
x = (1-34.15*zet[k])**0.3333
psic = (1.5*np.log((1+x+x**2)/3)-np.sqrt(3)*np.arctan((1+2*x) /
np.sqrt(3))+4*np.arctan(1)/np.sqrt(3))
f = zet[k]**2/(1+zet[k]**2)
psi[k] = (1-f)*psik+f*psic
return psi
# ---------------------------------------------------------------------
def psim_conv_coare3(zol, alpha, beta, gamma):
x = (1-15*zol)**0.25 # Kansas unstable
psik = 2*np.log((1+x)/2)+np.log((1+x*x)/2)-2*np.arctan(x)+2*np.arctan(1)
y = (1-10.15*zol)**0.3333 # Convective
psic = (1.5*np.log((1+y+y*y)/3.)-np.sqrt(3)*np.arctan((1+2*y)/np.sqrt(3)) +
4.*np.arctan(1)/np.sqrt(3))
f = zol*zol/(1+zol*zol)
psim = (1-f)*psik+f*psic
return psim
# ---------------------------------------------------------------------
def psim_stab_coare3(zol, alpha, beta, gamma):
c = np.where(0.35*zol > 50, 50, 0.35*zol) # Stable
psim = -((1+1*zol)**1.0+0.6667*(zol-14.28)/np.exp(-c)+8.525)
return psim
# ---------------------------------------------------------------------
def psim_stab_era5(zol, alpha, beta, gamma):
# eq (3.22) p. 39 IFS Documentation cy46r1
a, b, c, d = 1, 2/3, 5, 0.35
psim = -b*(zol-c/d)*np.exp(-d*zol)-a*zol-(b*c)/d
return psim
# ---------------------------------------------------------------------
def psim_conv(zol, alpha, beta, gamma):
xtmp = (1-alpha*zol)**beta
psim = (2*np.log((1+xtmp)*0.5)+np.log((1+xtmp**2)*0.5) -
2*np.arctan(xtmp)+np.pi/2)
return psim
# ---------------------------------------------------------------------
def psim_stab(zol, alpha, beta, gamma):
psim = -gamma*zol
return psim
# ---------------------------------------------------------------------
def psiu_26(zet):
"""
computes velocity structure function COARE3.5
"""
dzet = np.where(0.35*zet > 50, 50, 0.35*zet) # stable
a, b, c, d = 0.7, 3/4, 5, 0.35
psi = -(a*zet+b*(zet-c/d)*np.exp(-dzet)+b*c/d)
k = np.where(zet < 0) # unstable
x = (1-15*zet[k])**0.25
psik = 2*np.log((1+x)/2)+np.log((1+x**2)/2)-2*np.arctan(x)+2*np.arctan(1)
x = (1-10.15*zet[k])**0.3333
psic = (1.5*np.log((1+x+x**2)/3)-np.sqrt(3)*np.arctan((1+2*x)/np.sqrt(3)) +
4*np.arctan(1)/np.sqrt(3))
f = zet[k]**2/(1+zet[k]**2)
psi[k] = (1-f)*psik+f*psic
return psi
# ------------------------------------------------------------------------------
def psiu_40(zet):
"""
computes velocity structure function COARE3.5
"""
dzet = np.where(0.35*zet > 50, 50, 0.35*zet) # stable
a, b, c, d = 1, 3/4, 5, 0.35
psi = -(a*zet+b*(zet-c/d)*np.exp(-dzet)+b*c/d)
k = np.where(zet < 0) # unstable
x = (1-18*zet[k])**0.25
psik = 2*np.log((1+x)/2)+np.log((1+x**2)/2)-2*np.arctan(x)+2*np.arctan(1)
x = (1-10*zet[k])**0.3333
psic = (1.5*np.log((1+x+x**2)/3)-np.sqrt(3)*np.arctan((1+2*x)/np.sqrt(3)) +
4*np.arctan(1)/np.sqrt(3))
f = zet[k]**2/(1+zet[k]**2)
psi[k] = (1-f)*psik+f*psic
return psi
# ---------------------------------------------------------------------
def get_skin(sst, qsea, rho, jcool, Rl, Rs, Rnl, cp, lv, usr, tsr, qsr, lat):
# coded following Saunders (1967) with lambda = 6
g = gc(lat, None)
if (np.nanmin(sst) > 200): # if Ta in Kelvin convert to Celsius
sst = sst-273.16
# ************ cool skin constants *******
# density of water, specific heat capacity of water, water viscosity,
# thermal conductivity of water
rhow, cpw, visw, tcw = 1022, 4000, 1e-6, 0.6
Al = 2.1e-5*(sst+3.2)**0.79
be = 0.026
bigc = 16*g*cpw*(rhow*visw)**3/(tcw*tcw*rho*rho)
wetc = 0.622*lv*qsea/(287.1*(sst+273.16)**2)
Rns = 0.945*Rs # albedo correction
hsb = -rho*cp*usr*tsr
hlb = -rho*lv*usr*qsr
qout = Rnl+hsb+hlb
tkt = 0.001*np.ones(np.shape(sst))
dels = Rns*(0.065+11*tkt-6.6e-5/tkt*(1-np.exp(-tkt/8.0e-4)))
qcol = qout-dels
alq = Al*qcol+be*hlb*cpw/lv
xlamx = np.where(alq > 0, 6/(1+(bigc*alq/usr**4)**0.75)**0.333, 6)
tkt = xlamx*visw/(np.sqrt(rho/rhow)*usr)
tkt = np.where(alq > 0, np.where(tkt > 0.01, 0.01, tkt), tkt)
dter = qcol*tkt/tcw
dqer = wetc*dter
return dter, dqer
# ---------------------------------------------------------------------
def get_gust(beta, Ta, usr, tsrv, zi, lat):
if (np.max(Ta) < 200): # convert to K if in Celsius
Ta = Ta+273.16
if np.isnan(zi):
zi = 600
g = gc(lat, None)
Bf = -g/Ta*usr*tsrv
ug = np.ones(np.shape(Ta))*0.2
ug = np.where(Bf > 0, beta*np.power(Bf*zi, 1/3), 0.2)
return ug
# ---------------------------------------------------------------------
def get_heights(h):
hh = np.zeros(3)
if (type(h) == float or type(h) == int):
hh[0], hh[1], hh[2] = h, h, h
elif len(h) == 2:
hh[0], hh[1], hh[2] = h[0], h[1], h[1]
else:
hh[0], hh[1], hh[2] = h[0], h[1], h[2]
return hh
# ---------------------------------------------------------------------
def svp_calc(T):
"""
calculates saturation vapour pressure
T is in Kelvin
svp in mb, pure water
"""
if (np.nanmin(T) < 200): # if T in Celsius convert to Kelvin
T = T+273.16
svp = np.where(np.isnan(T), np.nan, 2.1718e08*np.exp(-4157/(T-33.91-0.16)))
return svp
# ---------------------------------------------------------------------
def qsea_calc(sst, pres):
"""
sst in Kelvin
pres in mb
qsea in kg/kg
"""
if (np.nanmin(sst) < 200): # if sst in Celsius convert to Kelvin
sst = sst+273.16
ed = svp_calc(sst)
e = 0.98*ed
qsea = (0.622*e)/(pres-0.378*e)
qsea = np.where(~np.isnan(sst+pres), qsea, np.nan)
return qsea
# ---------------------------------------------------------------------
def q_calc(Ta, rh, pres):
"""
rh in %
air in K, if not it will be converted to K
pres in mb
qair in kg/kg, as in Haltiner and Martin p.24
"""
if (np.nanmin(Ta) < 200): # if sst in Celsius convert to Kelvin
Ta = Ta+273.15
e = np.where(np.isnan(Ta+rh+pres), np.nan, svp_calc(Ta)*rh*0.01)
qair = np.where(np.isnan(e), np.nan, ((0.62197*e)/(pres-0.378*e)))
return qair
# ------------------------------------------------------------------------------
def bucksat(T, P):
"""
computes saturation vapor pressure [mb] as in COARE3.5
given T [degC] and P [mb]
"""
T = np.asarray(T)
if (np.nanmin(T) > 200): # if Ta in Kelvin convert to Celsius
T = T-CtoK
exx = 6.1121*np.exp(17.502*T/(T+240.97))*(1.0007+3.46e-6*P)
return exx
# ------------------------------------------------------------------------------
def qsat26sea(T, P):
"""
computes surface saturation specific humidity [g/kg] as in COARE3.5
given T [degC] and P [mb]
"""
T = np.asarray(T)
if (np.nanmin(T) > 200): # if Ta in Kelvin convert to Celsius
T = T-CtoK
ex = bucksat(T, P)
es = 0.98*ex # reduction at sea surface
qs = 622*es/(P-0.378*es)
return qs
# ------------------------------------------------------------------------------
def qsat26air(T, P, rh):
"""
computes saturation specific humidity [g/kg] as in COARE3.5
given T [degC] and P [mb]
"""
T = np.asarray(T)
if (np.nanmin(T) > 200): # if Ta in Kelvin convert to Celsius
T = T-CtoK
es = bucksat(T, P)
em = 0.01*rh*es
q = 622*em/(P-0.378*em)
return q, em
# ---------------------------------------------------------------------
def gc(lat, lon=None):
"""
computes gravity relative to latitude
inputs:
lat : latitudes in deg
lon : longitudes (optional)
output:
gc: gravity constant
"""
gamma = 9.7803267715
c1 = 0.0052790414
c2 = 0.0000232718
c3 = 0.0000001262
c4 = 0.0000000007
if lon is not None:
lon_m, lat_m = np.meshgrid(lon, lat)
else:
lat_m = lat
phi = lat_m*np.pi/180.
xx = np.sin(phi)
gc = (gamma*(1+c1*np.power(xx, 2)+c2*np.power(xx, 4)+c3*np.power(xx, 6) +
c4*np.power(xx, 8)))
return gc
# ---------------------------------------------------------------------
def visc_air(Ta):
"""
Computes the kinematic viscosity of dry air as a function of air temp.
following Andreas (1989), CRREL Report 89-11.
input:
Ta : air temperature [Celsius]
output
visa : kinematic viscosity [m^2/s]
"""
Ta = np.asarray(Ta)
if (np.nanmin(Ta) > 200): # if Ta in Kelvin convert to Celsius
Ta = Ta-273.16
visa = 1.326e-5 * (1 + 6.542e-3*Ta + 8.301e-6*Ta**2 - 4.84e-9*Ta**3)
return visa