import numpy as np
from util_subs import (kappa, visc_air)
# ---------------------------------------------------------------------
def cdn_calc(u10n, usr, Ta, grav, meth):
"""
Calculates neutral drag coefficient
Parameters
----------
u10n : float
neutral 10m wind speed [m/s]
usr : float
friction velocity [m/s]
Ta : float
air temperature [K]
grav : float
gravity [m/s^2]
meth : str
Returns
-------
cdn : float
zo : float
"""
cdn = np.zeros(Ta.shape)*np.nan
if (meth == "S80"): # eq. 14 Smith 1980
cdn = (0.61+0.063*u10n)*0.001
elif (meth == "LP82"):
# Large & Pond 1981 u10n <11m/s & eq. 21 Large & Pond 1982
cdn = np.where(u10n < 11, 1.2*0.001, (0.49+0.065*u10n)*0.001)
elif (meth == "S88" or meth == "UA" or meth == "ecmwf" or meth == "C30" or
meth == "C35" or meth == "Beljaars"): # or meth == "C40"
cdn = cdn_from_roughness(u10n, usr, Ta, grav, meth)
elif (meth == "YT96"):
# convert usr in eq. 21 to cdn to expand for low wind speeds
cdn = np.power((0.10038+u10n*2.17e-3+np.power(u10n, 2)*2.78e-3 -
np.power(u10n, 3)*4.4e-5)/u10n, 2)
elif (meth == "NCAR"): # eq. 11 Large and Yeager 2009
cdn = np.where(u10n > 0.5, (0.142+2.7/u10n+u10n/13.09 -
3.14807e-10*np.power(u10n, 6))*1e-3,
(0.142+2.7/0.5+0.5/13.09 -
3.14807e-10*np.power(0.5, 6))*1e-3)
cdn = np.where(u10n > 33, 2.34e-3, np.copy(cdn))
cdn = np.maximum(np.copy(cdn), 0.1e-3)
else:
raise ValueError("unknown method cdn: "+meth)
zo = 10/np.exp(kappa/np.sqrt(cdn))
return cdn, zo
# ---------------------------------------------------------------------
def cdn_from_roughness(u10n, usr, Ta, grav, meth):
"""
Calculates neutral drag coefficient from roughness length
Parameters
----------
u10n : float
neutral 10m wind speed [m/s]
usr : float
friction velocity [m/s]
Ta : float
air temperature [K]
grav : float [m/s]
gravity
meth : str
Returns
-------
cdn : float
"""
#cdn = (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):
if (meth == "S88"):
# Charnock roughness length (eq. 4 in Smith 88)
zc = 0.011*np.power(usr, 2)/grav
# smooth surface roughness length (eq. 6 in Smith 88)
zs = 0.11*visc_air(Ta)/usr
zo = zc + zs # eq. 7 & 8 in Smith 88
elif (meth == "UA"):
# valid for 0<u<18m/s # Zeng et al. 1998 (24)
zo = 0.013*np.power(usr, 2)/grav+0.11*visc_air(Ta)/usr
elif (meth == "C30"): # eq. 25 Fairall et al. 1996a
a = 0.011*np.ones(Ta.shape)
a = np.where(u10n > 10, 0.011+(u10n-10)*(0.018-0.011)/(18-10),
np.where(u10n > 18, 0.018, a))
zo = a*np.power(usr, 2)/grav+0.11*visc_air(Ta)/usr
elif (meth == "C35"): # eq.6-11 Edson et al. (2013)
zo = (0.11*visc_air(Ta)/usr +
np.minimum(0.0017*19-0.0050, 0.0017*u10n-0.0050) *
np.power(usr, 2)/grav)
elif ((meth == "ecmwf" or meth == "Beljaars")):
# eq. (3.26) p.38 over sea IFS Documentation cy46r1
zo = 0.018*np.power(usr, 2)/grav+0.11*visc_air(Ta)/usr
else:
raise ValueError("unknown method for cdn_from_roughness "+meth)
cdn = np.power(kappa/np.log(10/zo), 2)
return cdn
# ---------------------------------------------------------------------
def cd_calc(cdn, hin, hout, psim):
"""
Calculates drag coefficient at reference height
Parameters
----------
cdn : float
neutral drag coefficient
hin : float
wind speed height [m]
hout : float
reference height [m]
psim : float
momentum stability function
Returns
-------
cd : float
"""
cd = (cdn/np.power(1+(np.sqrt(cdn)*(np.log(hin/hout)-psim))/kappa, 2))
return cd
# ---------------------------------------------------------------------
def ctqn_calc(corq, zol, cdn, usr, zo, Ta, meth):
"""
Calculates neutral heat and moisture exchange coefficients
Parameters
----------
corq : flag to select
"ct" or "cq"
zol : float
height over MO length
cdn : float
neutral drag coefficient
usr : float
friction velocity [m/s]
zo : float
surface roughness [m]
Ta : float
air temperature [K]
meth : str
Returns
-------
ctqn : float
neutral heat exchange coefficient
zotq : float
roughness length for t or q
"""
if (meth == "S80" or meth == "S88" or meth == "YT96"):
cqn = np.ones(Ta.shape)*1.20*0.001 # from S88
ctn = np.ones(Ta.shape)*1.00*0.001
zot = 10/(np.exp(np.power(kappa, 2) / (ctn*np.log(10/zo))))
zoq = 10/(np.exp(np.power(kappa, 2) / (cqn*np.log(10/zo))))
elif (meth == "LP82"):
cqn = np.where((zol <= 0), 1.15*0.001, 1*0.001)
ctn = np.where((zol <= 0), 1.13*0.001, 0.66*0.001)
zot = 10/(np.exp(np.power(kappa, 2) / (ctn*np.log(10/zo))))
zoq = 10/(np.exp(np.power(kappa, 2) / (cqn*np.log(10/zo))))
elif (meth == "NCAR"):
cqn = np.maximum(34.6*0.001*np.sqrt(cdn), 0.1e-3)
ctn = np.maximum(np.where(zol <= 0, 32.7*0.001*np.sqrt(cdn),
18*0.001*np.sqrt(cdn)), 0.1e-3)
zot = 10/(np.exp(np.power(kappa, 2) / (ctn*np.log(10/zo))))
zoq = 10/(np.exp(np.power(kappa, 2) / (cqn*np.log(10/zo))))
elif (meth == "UA"):
# Zeng et al. 1998 (25)
rr = usr*zo/visc_air(Ta)
zoq = zo/np.exp(2.67*np.power(rr, 1/4)-2.57)
zot = np.copy(zoq)
cqn = np.power(kappa, 2)/(np.log(10/zo)*np.log(10/zoq))
ctn = np.power(kappa, 2)/(np.log(10/zo)*np.log(10/zoq))
elif (meth == "C30"):
rr = zo*usr/visc_air(Ta)
zoq = np.minimum(5e-5/np.power(rr, 0.6), 1.15e-4) # moisture roughness
zot = np.copy(zoq) # temperature roughness
cqn = np.power(kappa, 2)/np.log(10/zo)/np.log(10/zoq)
ctn = np.power(kappa, 2)/np.log(10/zo)/np.log(10/zot)
elif (meth == "C35"):
rr = zo*usr/visc_air(Ta)
zoq = np.minimum(5.8e-5/np.power(rr, 0.72), 1.6e-4) # moisture roughness
zot = np.copy(zoq) # temperature roughness
cqn = np.power(kappa, 2)/np.log(10/zo)/np.log(10/zoq)
ctn = np.power(kappa, 2)/np.log(10/zo)/np.log(10/zot)
elif (meth == "ecmwf" or meth == "Beljaars"):
# eq. (3.26) p.38 over sea IFS Documentation cy46r1
zot = 0.40*visc_air(Ta)/usr
zoq = 0.62*visc_air(Ta)/usr
cqn = np.power(kappa, 2)/np.log(10/zo)/np.log(10/zoq)
ctn = np.power(kappa, 2)/np.log(10/zo)/np.log(10/zot)
else:
raise ValueError("unknown method ctqn: "+meth)
if corq == "ct":
ctqn = ctn
zotq = zot
elif corq == "cq":
ctqn = cqn
zotq = zoq
else:
raise ValueError("unknown flag - should be ct or cq: "+corq)
return ctqn, zotq
# ---------------------------------------------------------------------
# ---------------------------------------------------------------------
def ctq_calc(cdn, cd, ctqn, hin, hout, psitq):
"""
Calculates heat and moisture exchange coefficients at reference height
Parameters
----------
cdn : float
neutral drag coefficient
cd : float
drag coefficient at reference height
ctqn : float
neutral heat or moisture exchange coefficient
hin : float
original temperature or humidity sensor height [m]
hout : float
reference height [m]
psitq : float
heat or moisture stability function
Returns
-------
ctq : float
heat or moisture exchange coefficient
"""
ctq = (ctqn*np.sqrt(cd/cdn) /
(1+ctqn*((np.log(hin/hout)-psitq)/(kappa*np.sqrt(cdn)))))
return ctq
# ---------------------------------------------------------------------
def get_stabco(meth):
"""
Gives the coefficients \\alpha, \\beta, \\gamma for stability functions
Parameters
----------
meth : str
Returns
-------
coeffs : float
"""
alpha, beta, gamma = 0, 0, 0
if (meth == "S80" or meth == "S88" or meth == "NCAR" or
meth == "UA" or meth == "ecmwf" or meth == "C30" or
meth == "C35" or meth == "Beljaars"):
alpha, beta, gamma = 16, 0.25, 5 # Smith 1980, from Dyer (1974)
elif (meth == "LP82"):
alpha, beta, gamma = 16, 0.25, 7
elif (meth == "YT96"):
alpha, beta, gamma = 20, 0.25, 5
else:
raise ValueError("unknown method stabco: "+meth)
coeffs = np.zeros(3)
coeffs[0] = alpha
coeffs[1] = beta
coeffs[2] = gamma
return coeffs
# ---------------------------------------------------------------------
def psim_calc(zol, meth):
"""
Calculates momentum stability function
Parameters
----------
zol : float
height over MO length
meth : str
Returns
-------
psim : float
"""
if (meth == "ecmwf"):
psim = psim_ecmwf(zol)
elif (meth == "C30" or meth == "C35"):
psim = psiu_26(zol, meth)
elif (meth == "Beljaars"): # Beljaars (1997) eq. 16, 17
psim = np.where(zol < 0, psim_conv(zol, meth), psi_Bel(zol))
else:
psim = np.where(zol < 0, psim_conv(zol, meth),
psim_stab(zol, meth))
return psim
# ---------------------------------------------------------------------
def psit_calc(zol, meth):
"""
Calculates heat stability function
Parameters
----------
zol : float
height over MO length
meth : str
parameterisation method
Returns
-------
psit : float
"""
if (meth == "ecmwf"):
psit = np.where(zol < 0, psi_conv(zol, meth),
psi_ecmwf(zol))
elif (meth == "C30" or meth == "C35"):
psit = psit_26(zol)
elif (meth == "Beljaars"): # Beljaars (1997) eq. 16, 17
psit = np.where(zol < 0, psi_conv(zol, meth), psi_Bel(zol))
else:
psit = np.where(zol < 0, psi_conv(zol, meth),
psi_stab(zol, meth))
return psit
# ---------------------------------------------------------------------
def psi_Bel(zol):
"""
Calculates momentum/heat stability function
Parameters
----------
zol : float
height over MO length
meth : str
parameterisation method
Returns
-------
psit : float
"""
a, b, c, d = 0.7, 0.75, 5, 0.35
psi = -(a*zol+b*(zol-c/d)*np.exp(-d*zol)+b*c/d)
return psi
# ---------------------------------------------------------------------
def psi_ecmwf(zol):
"""
Calculates heat stability function for stable conditions
for method ecmwf
Parameters
----------
zol : float
height over MO length
Returns
-------
psit : float
"""
# eq (3.22) p. 37 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 psit_26(zol):
"""
Computes temperature structure function as in C35
Parameters
----------
zol : float
height over MO length
Returns
-------
psi : float
"""
b, d = 2/3, 0.35
dzol = np.minimum(d*zol, 50)
psi = -1*((1+b*zol)**1.5+b*(zol-14.28)*np.exp(-dzol)+8.525)
k = np.where(zol < 0)
x = np.sqrt(1-15*zol[k])
psik = 2*np.log((1+x)/2)
x = np.power(1-34.15*zol[k], 1/3)
psic = (1.5*np.log((1+x+np.power(x, 2))/3)-np.sqrt(3) *
np.arctan((1+2*x)/np.sqrt(3))+4*np.arctan(1)/np.sqrt(3))
f = np.power(zol[k], 2)/(1+np.power(zol[k], 2))
psi[k] = (1-f)*psik+f*psic
return psi
# ---------------------------------------------------------------------
def psi_conv(zol, meth):
"""
Calculates heat stability function for unstable conditions
Parameters
----------
zol : float
height over MO length
meth : str
parameterisation method
Returns
-------
psit : float
"""
coeffs = get_stabco(meth)
alpha, beta = coeffs[0], coeffs[1]
xtmp = np.power(1-alpha*zol, beta)
psit = 2*np.log((1+np.power(xtmp, 2))*0.5)
return psit
# ---------------------------------------------------------------------
def psi_stab(zol, meth):
"""
Calculates heat stability function for stable conditions
Parameters
----------
zol : float
height over MO length
meth : str
parameterisation method
Returns
-------
psit : float
"""
coeffs = get_stabco(meth)
gamma = coeffs[2]
psit = -gamma*zol
return psit
# ---------------------------------------------------------------------
def psim_ecmwf(zol):
"""
Calculates momentum stability function for method ecmwf
Parameters
----------
zol : float
height over MO length
Returns
-------
psim : float
"""
# eq (3.20, 3.22) p. 37 IFS Documentation cy46r1
coeffs = get_stabco("ecmwf")
alpha, beta = coeffs[0], coeffs[1]
xtmp = np.power(1-alpha*zol, beta)
a, b, c, d = 1, 2/3, 5, 0.35
psim = np.where(zol < 0, np.pi/2-2*np.arctan(xtmp) +
np.log((np.power(1+xtmp, 2)*(1+np.power(xtmp, 2)))/8),
-b*(zol-c/d)*np.exp(-d*zol)-a*zol-(b*c)/d)
return psim
# ---------------------------------------------------------------------
def psiu_26(zol, meth):
"""
Computes velocity structure function C35
Parameters
----------
zol : float
height over MO length
Returns
-------
psi : float
"""
if (meth == "C30"):
dzol = np.minimum(0.35*zol, 50) # stable
psi = -1*((1+zol)+0.6667*(zol-14.28)*np.exp(-dzol)+8.525)
k = np.where(zol < 0) # unstable
x = (1-15*zol[k])**0.25
psik = (2*np.log((1+x)/2)+np.log((1+x*x)/2)-2*np.arctan(x) +
2*np.arctan(1))
x = (1-10.15*zol[k])**(1/3)
psic = (1.5*np.log((1+x+x*x)/3) -
np.sqrt(3)*np.arctan((1+2*x)/np.sqrt(3)) +
4*np.arctan(1)/np.sqrt(3))
f = zol[k]**2/(1+zol[k]**2)
psi[k] = (1-f)*psik+f*psic
elif (meth == "C35"): # or meth == "C40"
dzol = np.minimum(50, 0.35*zol) # stable
a, b, c, d = 0.7, 3/4, 5, 0.35
psi = -1*(a*zol+b*(zol-c/d)*np.exp(-dzol)+b*c/d)
k = np.where(zol < 0) # unstable
x = np.power(1-15*zol[k], 1/4)
psik = 2*np.log((1+x)/2)+np.log((1+x*x)/2)-2*np.arctan(x)+2*np.arctan(1)
x = np.power(1-10.15*zol[k], 1/3)
psic = (1.5*np.log((1+x+np.power(x, 2))/3)-np.sqrt(3) *
np.arctan((1+2*x)/np.sqrt(3))+4*np.arctan(1)/np.sqrt(3))
f = np.power(zol[k], 2)/(1+np.power(zol[k], 2))
psi[k] = (1-f)*psik+f*psic
return psi
#------------------------------------------------------------------------------
def psim_conv(zol, meth):
"""
Calculates momentum stability function for unstable conditions
Parameters
----------
zol : float
height over MO length
meth : str
parameterisation method
Returns
-------
psim : float
"""
coeffs = get_stabco(meth)
alpha, beta = coeffs[0], coeffs[1]
xtmp = np.power(1-alpha*zol, beta)
psim = (2*np.log((1+xtmp)*0.5)+np.log((1+np.power(xtmp, 2))*0.5) -
2*np.arctan(xtmp)+np.pi/2)
return psim
# ---------------------------------------------------------------------
def psim_stab(zol, meth):
"""
Calculates momentum stability function for stable conditions
Parameters
----------
zol : float
height over MO length
meth : str
parameterisation method
Returns
-------
psim : float
"""
coeffs = get_stabco(meth)
gamma = coeffs[2]
psim = -gamma*zol
return psim
# ---------------------------------------------------------------------
def get_gust(beta, Ta, usr, tsrv, zi, grav):
"""
Computes gustiness
Parameters
----------
beta : float
constant
Ta : float
air temperature [K]
usr : float
friction velocity [m/s]
tsrv : float
star virtual temperature of air [K]
zi : int
scale height of the boundary layer depth [m]
grav : float
gravity
Returns
-------
ug : float [m/s]
"""
if (np.nanmax(Ta) < 200): # convert to K if in Celsius
Ta = Ta+273.16
# minus sign to allow cube root
Bf = (-grav/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_strs(hin, monob, wind, zo, zot, zoq, dt, dq, cd, ct, cq, meth):
"""
calculates star wind speed, temperature and specific humidity
Parameters
----------
hin : float
sensor heights [m]
monob : float
M-O length [m]
wind : float
wind speed [m/s]
zo : float
momentum roughness length [m]
zot : float
temperature roughness length [m]
zoq : float
moisture roughness length [m]
dt : float
temperature difference [K]
dq : float
specific humidity difference [g/kg]
ct : float
temperature exchange coefficient
cq : float
moisture exchange coefficient
meth : str
bulk parameterisation method option: "S80", "S88", "LP82", "YT96", "UA",
"NCAR", "C30", "C35", "ecmwf", "Beljaars"
Returns
-------
usr : float
friction wind speed [m/s]
tsr : float
star temperature [K]
qsr : float
star specific humidity [g/kg]
"""
if (meth == "UA"):
# Zeng et al. 1998
# away from extremes UA follows e.g. S80
usr = wind*np.power(cd, 1/2)
tsr = ct*wind*dt/usr
qsr = cq*wind*dq/usr
# momentum
hol0 = hin[0]/np.copy(monob)
# very unstable (Zeng et al. 1998 eq 7)
usr = np.where(hol0 <= -1.574, wind*kappa / (np.log(-1.574*monob/zo)-psim_calc(-1.574, meth) + psim_calc(zo/monob, meth) + 1.14*(np.power(-hin[0]/monob, 1/3) - np.power(1.574, 1/3))), usr)
# very stable (Zeng et al. 1998 eq 10)
usr = np.where(hol0 > 1, wind*kappa/(np.log(monob/zo)+5 - 5*zo/monob + 5*np.log(hin[0]/monob) + hin[0]/monob-1), usr)
# temperature
hol1 = hin[1]/np.copy(monob)
# very unstable (Zeng et al. 1998 eq 11)
tsr = np.where(hol1 < -0.465, kappa*dt / (np.log((-0.465*monob)/zot) - psit_calc(-0.465, meth)+0.8*(np.power(0.465, -1/3) - np.power(-hin[1]/monob, -1/3))), tsr)
# very stable (Zeng et al. 1998 eq 14)
tsr = np.where(hol1 > 1, kappa*(dt) / (np.log(monob/zot)+5 - 5*zot/monob+5*np.log(hin[1]/monob) + hin[1]/monob-1), tsr)
# humidity
hol2 = hin[2]/monob
# very unstable (Zeng et al. 1998 eq 11)
qsr = np.where(hol2 < -0.465, kappa*dq / (np.log((-0.465*monob)/zoq) - psit_calc(-0.465, meth)+psit_calc(zoq/monob, meth) + 0.8*(np.power(0.465, -1/3) - np.power(-hin[2]/monob, -1/3))), qsr)
# very stable (Zeng et al. 1998 eq 14)
qsr = np.where(hol2 > 1, kappa*dq / (np.log(monob/zoq)+5-5*zoq/monob + 5*np.log(hin[2]/monob) + hin[2]/monob-1), qsr)
else:
usr = wind*np.power(cd, 1/2)
tsr = ct*wind*dt/usr
qsr = cq*wind*dq/usr
return usr, tsr, qsr
# ---------------------------------------------------------------------
# --------------------------------------------------------------------------------
# --------------------------------------------------------------------------------
# ---------------------------------------------------------------------
def get_tsrv(tsr, qsr, Ta, qair):
"""
calculates virtual star temperature
Parameters
----------
tsr : float
star temperature (K)
qsr : float
star specific humidity (g/kg)
Ta : float
air temperature (K)
qair : float
air specific humidity (g/kg)
Returns
-------
tsrv : float
virtual star temperature (K)
"""
# as in aerobulk One_on_L in mod_phymbl.f90
tsrv = tsr*(1+0.6077*qair)+0.6077*Ta*qsr
return tsrv
# ---------------------------------------------------------------------
def get_Rb(grav, usr, hin_u, hin_t, tv, dtv, wind, monob, meth):
"""
calculates bulk Richardson number
Parameters
----------
grav : float
acceleration due to gravity (m/s2)
usr : float
friction wind speed (m/s)
hin_u : float
u sensor height (m)
hin_t : float
t sensor height (m)
tv : float
virtual temperature (K)
dtv : float
virtual temperature difference, air and sea (K)
wind : float
wind speed (m/s)
monob : float
Monin-Obukhov length from previous iteration step (m)
psim : float
momentum stability function
meth : str
bulk parameterisation method option: "S80", "S88", "LP82", "YT96",
"UA", "NCAR", "C30", "C35", "ecmwf", "Beljaars"
Returns
-------
Rb : float
Richardson number
"""
# now input dtv
#tvs = sst*(1+0.6077*qsea) # virtual SST
#dtv = tv - tvs # virtual air - sea temp. diff
# adjust wind to t measurement height
uz = (wind-usr/kappa*(np.log(hin_u/hin_t)-psim_calc(hin_u/monob, meth) +
psim_calc(hin_t/monob, meth)))
Rb = grav*dtv*hin_t/(tv*uz*uz)
return Rb
# ---------------------------------------------------------------------
def get_LRb(Rb, hin_t, monob, zo, zot, meth):
"""
calculates Monin-Obukhov length following ecmwf (IFS Documentation cy46r1)
default for methods ecmwf and Beljaars
Parameters
----------
Rb : float
Richardson number
hin_t : float
t sensor height (m)
monob : float
Monin-Obukhov length from previous iteration step (m)
zo : float
surface roughness (m)
zot : float
temperature roughness length (m)
meth : str
bulk parameterisation method option: "S80", "S88", "LP82", "YT96",
"UA", "NCAR", "C30", "C35", "ecmwf", "Beljaars"
Returns
-------
monob : float
M-O length (m)
"""
zol = (Rb*(np.power(np.log((hin_t+zo)/zo)-psim_calc((hin_t+zo) /
monob, meth) + psim_calc(zo/monob, meth), 2) /
(np.log((hin_t+zo)/zot) -
psit_calc((hin_t+zo)/monob, meth) +
psit_calc(zot/monob, meth))))
monob = hin_t/zol
return monob
# ---------------------------------------------------------------------
def get_Ltsrv(tsrv, grav, tv, usr):
"""
calculates Monin-Obukhov length from tsrv
Parameters
----------
tsrv : float
virtual star temperature (K)
grav : float
acceleration due to gravity (m/s2)
tv : float
virtual temperature (K)
usr : float
friction wind speed (m/s)
Returns
-------
monob : float
M-O length (m)
"""
#tmp = (g*kappa*tsrv /
# np.maximum(np.power(usr, 2)*Ta*(1+0.6077*qair), 1e-9))
#tmp = np.minimum(np.abs(tmp), 200)*np.sign(tmp)
#monob = 1/np.copy(tmp)
tsrv = np.maximum(np.abs(tsrv), 1e-9)*np.sign(tsrv)
monob = (np.power(usr, 2)*tv)/(grav*kappa*tsrv)
return monob
# ---------------------------------------------------------------------