Create thermo.py

thermodynamics functions from Bolton (1980), Buck (1981), Chris Bretherton and Simon de Szoeke

eurec4a_snd/thermo.py

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+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+"""
+Thermodynamics functions
+
+From Chris Bretherton, Bolton 1980, etc.
+
+Simon de Szoeke
+Created on Mon Aug 27 21:30:59 2018
+
+@author: sdeszoek
+"""
+
+import numpy as np
+
+#  Important global constants (Emanuel, Appendix 2)
+g     = 9.8
+stefan= 5.67e-8
+r_earth = 6370*1000
+Omega   = 2*np.pi/86400
+
+Rd    = 287.04
+Rv    = 461.50
+Cp    = 1005.7
+Cpv   = 1870
+Cw    = 4190
+L     = 2.501e6
+delta = 0.608 # Rv/Rd - 1
+CtoK  = 273.15
+
+# Blackbody radiative constants
+kb=1.381e-23     # J/K   Boltzmann
+hplanck=6.6261e-34   # J s   Planck
+clight=2.998e8   # m/s   speed of light
+hc=1.98645e-25   # J m   Planck*speed of light
+
+# Usually functions describing water and water vapor depend on T in Celsius
+# and functions of generalized gases, thermodynmics, and physics depend on 
+# absolute T in Kelvin.
+
+def Lv(T=0):
+    #  Latent heat of vaporization as a function of T
+    #  Lv(Temp[C]) = 2.501e6 + (Cpv-Cw)*(Temp)  [J/kg]
+    #  From Bolton, 1980, MWR, 108, 1046-1053.
+    Lvap = L + (Cpv-Cw) * T 
+    return Lvap
+
+def es(T,p=1.0e5):
+    #   es(T[C],p[Pa]) [Pa] is saturation vapor pressure based on Wexler's formula,
+    #   with enhancement factor for moist air rather than water vapor.
+    #   The enhancement factor requires a pressure.
+    #   T [degrees C], p [Pa] (note the reversed input order), es [Pa]
+    #   From A. L. Buck 1981: JAM, 20, 1527-1532.
+    #   SPdeS 7 July 2004
+    esat = 1e2*6.1121*(1.0007 + 3.46e-8*p)*np.exp((17.502*T)/(240.97 + T))
+    return esat
+
+def qs(T,p=1.0e5):
+    #  qs(p,T) is saturation specific humidity based on Wexler's formula for es
+    #   with enhancement factor (see es.m).
+    #   T [degrees C], p [Pa], qs [kg/kg]
+    #   From A. L. Buck 1981: JAM, 20, 1527-1532.
+    #   SPdeS 7 July 2004
+    esat = es(T,p)
+    qsat = (Rd/Rv)*esat/(p+(Rd/Rv-1)*esat)
+    return qsat
+
+# qsea = lambda T,p: 0.98 * qs(T,p)
+def qsea(T,p):
+    # seawater specific humidity [kg/kg]
+    return 0.98*qs(T,p)
+
+def Tdew(e,p):
+    #   Tdew(e,p)
+    #   e [Pa] vapor pressure, p pressure [Pa]
+    #   Dewpoint Temperature [Celsius] from inversion of es formula of
+    #   A. L. Buck 1981: JAM, 20, 1527-1532.
+    #   SPdeS 2 August 2010
+    a  = 17.502
+    b  = 240.97
+    k  = np.log( e/6.1121 / (1.0007+3.46e-6*p) ) # depends only on e,p
+    Td = b*k / (a-k)
+    return Td
+
+def dqsdT(T,p):
+    # dqsdT(T[C],p[Pa]) [(kg/kg) / K]
+    # saturation specific humidity change with T from Clausius-Clapeyron
+    dqsatdT = qs(T,p)*Lv(T) / ( Rv * (T+CtoK)**2 )
+    return dqsatdT
+
+def dqsdp(Temp,p):
+    #  dqsdp(p,Temp) = d(qs)/d(Temp) = -qs(p,Temp)/(p - es(Temp,p))   [(kg/kg) Pa^{-1}]
+    #  p[Pa], Temp[degree C]
+    dqsatdp = -qs(Temp,p) / (p - es(Temp,p));
+    return dqsatdp
+
+def rs(T,p=1.0e5):
+    # rs(p,T) is saturation mixing ratio based on Wexler's formula for es
+    # with enhancement factor (see es.m).
+    # p [Pa], T [degrees C], rs [kg/kg]
+    # From A. L. Buck 1981: JAM, 20, 1527-1532.
+    # SPdeS 7 July 2004
+    esat = es(T,p)
+    rsat = (Rd/Rv)*esat/(p-esat)
+    return rsat
+
+def Tlcl(Temp,p,qv):
+    #   Tl=Tlcl(p[Pa], Temp[K], qv[kg/kg])
+    #   Temperature at the LCL
+    #   From Bolton, 1980, MWR, 108, 1046-1053.
+    ev = p * qv / (Rd/Rv + qv);
+    Tl = 2840/(3.5*np.log(Temp) - np.log(0.01*ev) - 4.805) + 55;
+    return Tl
+
+
+def Exner(p,qv=0):
+    # Exner function Pi = (p/1.e5)**((Rd/Cp)*(1-0.28*qv))
+    Pi = (p/1.e5)**((Rd/Cp)*(1-0.28*qv))
+    return Pi
+
+def theta(Temp,p,qv=0):
+    #   Potential temperature Temp*(1e5/p)^(Rd/Cp)
+    th = Temp / Exner(p,qv)
+    return th
+
+def temp(th,p,qv=0):
+    # temp(theta[K],p[Pa],qv[kg/kg]=0)
+    return th * Exner(p,qv)
+
+def theta_e(Temp,p,qv):
+    # theta_e(Temp[K],p[Pa],qv[kg/kg]) from eqn. 43 of Bolton, 1980, MWR, 108, 1046-1053.
+    Tl = Tlcl(Temp,p,qv)
+    thetae = Temp/Exner(p,qv) * np.exp((3376/Tl - 2.54) * qv * (1 + 0.81*qv))
+    return thetae
+
+def theta_es(Temp,p):
+    # theta_es(Temp[K],p[Pa],qv[kg/kg]) from eqn. 43 of Bolton, 1980, MWR, 108, 1046-1053.
+    return theta_e(Temp,p,qs(Temp-CtoK,p))
+
+def Twet(T,qv,p=1.0e5,niter=5):
+    # Tw=Twet(Tdry[C], qv[kg/kg], p[Pa], niter=5)
+    # Computes the wet bulb temperature from an isobaric process
+    # by using heat internal energy -Cp*dT=L*dq to evaporate
+    # and raise qv to qs(Twb).
+    #
+    # Simon de Szoeke 2016-04-07
+
+    # initialize Tw, qw
+    Tw=T
+    # problem for temperatures greater than 100 C!
+    qw=min(qs(Tw,p),qv)
+
+    # Steps by 1/4 of saturation deficit and enforces -Cp*dT=L*dq by successive approximations
+    for iter in range(1,niter+1):
+        dq = 0.25 * (qs(p,Tw) - qw)
+        if dq < 0:
+            break
+        dT = -Lv(Tw) / Cp*dq
+        qw = qw + dq
+        Tw = Tw + dT
+
+    return Tw
+
+def theta_w(Temp,p,qv):
+    # theta_w(Temp[K],p[Pa],qv[kg/kg]) from eqn 3.8 of Davies-Jones 2008 and 
+    # eqn. 43 of Bolton, 1980, MWR, 108, 1046-1053.
+
+    # Davies-Jones rational function coefficients
+    a0=   7.101574
+    a1= -20.68208
+    a2=  16.11182
+    a3=   2.574631
+    a4=  -5.205688
+    b1=  -3.552497
+    b2=   3.781782
+    b3=  -0.6899655
+    b4=  -0.5929340
+
+    # compute Theta_e from Bolton as first approx for thw
+    th = Temp / Exner(p,qv)
+    Tl = Tlcl(Temp,p,qv)
+    thw = th * np.exp( (3376/Tl - 2.54) *qv * (1 + 0.81*qv) );
+
+    # refine thw
+    # evaluate Davies-Jones 2008 rational function fit for thw [here in K]
+    if thw >= 173.15:
+        X = thw / 273.15
+        X2 = X  * X
+        X3 = X2 * X
+        X4 = X2 * X2
+        thw = thw - np.exp( (a0+a1*X+a2*X2+a3*X3+a4*X4) / (1+b1*X+b2*X2+b3*X3+b4*X4) );
+
+    return thw
+
+def Tv(T,qv):
+    # Tv(T[K],qv[kg/kg]) = T * (1 + delta*qv); delta = 0.608
+    Tvirt = T * (1 + delta*qv)
+    return Tvirt
+
+def density(T,p,qv=0):
+    # rho = p./(Rd*Tv(T,qv))
+    rho = p / (Rd * Tv(T,qv))
+    return rho
+
+
+# lapse rate and sounding methods
+
+def gamma(Temp,p):
+    # dT/dz = gamma(T[C],p[Pa])
+    # moist adiabatic lapse rate [degree C/m]
+    gam = ( Lv(Temp) / Cp ) * dqsdT(Temp,p)
+    return gam
+
+def Hscale(Temp,qv=0):
+    # Hscale(Temp[K],qv[kh/kh]=0) = Rd*Temp/g [m] is pressure scale height
+    return Rd * Tv(Temp,qv) / g;
+
+def dqsdzs(Temp,p):
+    # dqsdzs(p,Temp) = dqs/dz [K/m] on a moist adiabat.
+    # Temp [C], p [Pa]
+    #
+    # dqsatdzs = dqsdzu(Temp,p) / (1 + gamma(Temp,p))
+    return dqsdzu(Temp,p) / (1 + gamma(Temp,p))
+
+def dqsdzu(p,Temp):
+    #  dqsdzu = dqs/dz [K/m] on a dry adiabat.
+    # Temp [C], p [Pa]
+    dqsatdzu = ((Rd*(Temp+CtoK)/Cp) * dqsdT(Temp,p) + p * dqsdp(Temp,p)) / Hscale(Temp+CtoK)
+
+    return dqsatdzu
+
+
+def buoyancy_flux(shf, lhf, T=25.0, p=1.0e5, q=0):
+    # [bflx, bflx_s]=buoyancy_flux(shf, lhf, T, q, p)  [shf, lhf W/m^2; T °C; q kg/kg; p hPa]
+    # Buoyancy flux from sensible and latent heat flux. Optionally also returns the sensible
+    # part of the buoyancy flux. bflx = bflx_virt + bflx_s
+    #
+    # Simon de Szoeke  2018-09-01
+
+    virtfac = 1.0 + delta*q
+    Tvirt   = (T+CtoK) * virtfac
+    rho     = density(Tvirt,p)
+    wt      = shf / (rho*Cp)
+    wtv     = wt * virtfac + lhf/(rho * Lv(T)) * (T+CtoK)*delta
+    bflx    = g * wtv / Tvirt
+    bflx_s  = g * wt  / Tvirt
+
+    return bflx, bflx_s
+
+
+# Planck functions
+def Planck_la(la,T):
+    # B_la=Planck_la(lambda[m],T[K]) [W/sr/m^3]
+    # Computes Planck's blackbody spectral radiance as a function of wavelength lambda [m].
+    # (c) Simon de Szoeke 2016-04-18
+
+    B_la = 2 * hc * clight / ( (la**5) * ( np.exp(hc / (la*kb*T)) - 1 ) ) # W/sr/m^3
+    return B_la
+
+def Planck_nu(nu,T):
+    # B_nu=Planck_nu(nu[1/s],T[K]) [W/sr/m^3]
+    # Computes Planck's blackbody spectral radiance as a function of wavelength lambda [m].
+    # (c) Simon de Szoeke 2016-04-18
+    B_nu = 2 * hplanck * nu*nu*nu / ( clight*clight * ( np.exp( hplanck*nu / (kb*T) ) - 1 ) ) # W/sr/m^2 s
+    return B_nu