"""Calculates the equilibrium activated fraction given the details of a population

of aerosol sizes.

NOTE - This works for a *single mode*. In order to study the entire aerosol

population activated in the parcel model, this will need to be called however

many times there are modes for each separate mode.

"""

r_crits, s_crits = zip(*[kohler_crit(T, r_dry, kappa) for r_dry in r_drys])

s_crits = np.array(s_crits)

r_crits = np.array(r_crits)

activated_eq = (Smax >= s_crits)

activated_kn = (rs >= r_crits)

N_tot = np.sum(Nis)

eq_frac = np.sum(Nis[activated_eq])/N_tot

kn_frac = np.sum(Nis[activated_kn])/N_tot

"""Ming activation scheme.

NOTE - right now, the variable names correspond to the FORTRAN implementation

of the routine. Will change in the future.

TODO: rename variables

TODO: docstring

TODO: extend for multiple modes.

"""

Num = aerosol.Nis*1e-6

RpDry = aerosol.distribution.mu*1e-6

kappa = aerosol.kappa

## pre-algorithm

## subroutine Kohler()... calculate things from Kohler theory, particularly critical

## radii and supersaturations for each bin

r_crits, s_crits = zip(*[kohler_crit(T, r_dry, kappa) for r_dry in aerosol.r_drys])

## subroutine CalcAlphaGamma

alpha = (g*Mw*L)/(Cp*R*(T**2)) - (g*Ma)/(R*T)

gamma = (R*T)/(es(T-273.15)*Mw) + (Mw*(L**2))/(Cp*Ma*T*P)

## re-name variables as in Ming scheme

Dpc = 2.*np.array(r_crits)*1e6

Dp0 = r_crits/np.sqrt(3.)

Sc = np.array(s_crits)+1.0

DryDp = aerosol.r_drys*2.

## Begin algorithm

Smax1 = 1.0

Smax2 = 1.1

iter_count = 1

while (Smax2 - Smax1) > 1e-7:

#print "\t", iter_count, Smax1, Smax2

Smax = 0.5*(Smax2 + Smax1)

#print "---", Smax-1.0

## subroutine Grow()

## subroutine CalcG()

# TODO: implement size-dependent effects on Dv, ka, using Dpc

#G_a = (rho_w*R*T)/(es(T-273.15)*Dv_T(T)*Mw)

G_a = (rho_w*R*T)/(es(T-273.15)*dv(T, (Dpc*1e-6)/2.)*Mw)

#G_b = (L*rho_w*((L*Mw/(R*T))-1))/(ka_T(T)*T)

G_b = (L*rho_w*((L*Mw/(R*T))-1))/(ka(T, 1.007e3, (Dpc*1e-6)/2.)*T)

G = 1./(G_a + G_b) # multiply by four since we're doing diameter this time

Smax_large = (Smax > Sc) # if(Smax>Sc(count1,count2))

WetDp = np.zeros_like(Dpc)

#WetDp[Smax_large] = np.sqrt(Dpc[Smax_large]**2. + 1e12*(G[Smax_large]/(alpha*V))*((Smax-.0)**2.4 - (Sc[Smax_large]-.0)**2.4))

WetDp[Smax_large] = 1e6*np.sqrt((Dpc[Smax_large]*1e-6)**2. + (G[Smax_large]/(alpha*V))*((Smax-.0)**2.4 - (Sc[Smax_large]-.0)**2.4))

#print Dpc

#print WetDp/DryDp

#print WetDp

## subroutine Activity()

def Activity(dry, wet, dens, molar_weight):

temp1 = (dry**3)*dens/molar_weight

temp2 = ((wet**3) - (dry**3))*1e3/0.018

act = temp2/(temp1+temp2)*np.exp(0.66/T/wet)

#print dry[0], wet[0], dens, molar_weight, act[0]

return act

# Is this just the Kohler curve?

Act = np.ones_like(WetDp)

WetDp_large = (WetDp > 1e-5) # if(WetDp(i,j)>1e-5)

Act[WetDp_large] = Seq(WetDp[WetDp_large]*1e-6, DryDp[WetDp_large], T, kappa) + 1.0

#Act[WetDp_large] = Activity(DryDp[WetDp_large]*1e6, WetDp[WetDp_large], 1.7418e3, 0.132)

#print Act

## subroutine Conden()

## subroutine CalcG()

# TODO: implement size-dependent effects on Dv, ka, using WetDp

#G_a = (rho_w*R*T)/(es(T-273.15)*Dv_T(T)*Mw)

G_a = (rho_w*R*T)/(es(T-273.15)*dv(T, (WetDp*1e-6)/2.)*Mw)

#G_b = (L*rho_w*((L*Mw/(R*T))-1))/(ka_T(T)*T)

G_b = (L*rho_w*((L*Mw/(R*T))-1))/(ka(T, 1.3e3, (WetDp*1e-6)/2.)*T)

G = 1./(G_a + G_b) # multiply by four since we're doing diameter this time

WetDp_large = (WetDp > Dpc) # (WetDp(count1,count2)>Dpc(count1,count2))

#WetDp_large = (WetDp > 0)

f_stre = lambda x: "%12.12e" % x

f_strf = lambda x: "%1.12f" % x

#for i, a in enumerate(Act):

# if WetDp[i] > Dpc[i]:

# print " ",i+1, Act[i], f_stre(Smax-Act[i])

CondenRate = np.sum((np.pi/2.)*1e3*G[WetDp_large]*(WetDp[WetDp_large]*1e-6)*Num[WetDp_large]*1e6*

(Smax-Act[WetDp_large]))

#print iter_count, "%r %r %r" % (Smax, CondenRate, alpha*V/gamma)

DropletNum = np.sum(Num[WetDp_large])

ActDp = 0.0

for i in xrange(1, len(WetDp)):

if (WetDp[i] > Dpc[i]) and (WetDp[i-1] < Dpc[i]):

ActDp = DryDp[i]

## Iteration logic

if CondenRate < (alpha*V/gamma):

Smax1 = Smax*1.0

else:

Smax2 = Smax*1.0

iter_count += 1

Smax = Smax-1.0

## Originally from Abdul-Razzak 1998 w/ Ma. Need kappa formulation

alpha = (g*Mw*L)/(Cp*R*(T**2)) - (g*Ma)/(R*T)

gamma = (R*T)/(es(T-273.15)*Mw) + (Mw*(L**2))/(Cp*Ma*T*P)

## Condensation effects

G_a = (rho_w*R*T)/(es(T-273.15)*dv_cont(T, P)*Mw)

G_b = (L*rho_w*((L*Mw/(R*T))-1))/(ka_cont(T)*T)

G = 1./(G_a + G_b)

Smis = []

Sparts = []

for aerosol in aerosols:

sigma = aerosol.distribution.sigma

am = aerosol.distribution.mu*1e-6

N = aerosol.distribution.N*1e6

kappa = aerosol.kappa

fi = 0.5*np.exp(2.5*(np.log(sigma)**2))

gi = 1.0 + 0.25*np.log(sigma)

A = (2.*sigma_w(T)*Mw)/(rho_w*R*T)

_, Smi2 = kohler_crit(T, am, kappa)

zeta = (2./3.)*A*(np.sqrt(alpha*V/G))

etai = ((alpha*V/G)**(3./2.))/(N*gamma*rho_w*2.*np.pi)

##

Spa = fi*((zeta/etai)**(1.5))

Spb = gi*(((Smi2**2)/(etai + 3.*zeta))**(0.75))

S_part = (1./(Smi2**2))*(Spa + Spb)

Smis.append(Smi2)

Sparts.append(S_part)

Smax = 1./np.sqrt(np.sum(Sparts))

act_fracs = []

for Smi, aerosol in zip(Smis, aerosols):

ui = 2.*np.log(Smi/Smax)/(3.*np.sqrt(2.)*np.log(aerosol.distribution.sigma))

N_act = 0.5*aerosol.distribution.N*erfc(ui)

act_fracs.append(N_act/aerosol.distribution.N)

"""

NS 2003 algorithm + FN 2005 corrections for diffusive growth rate

"""

#aer = aerosols[0]

A = (4.*Mw*sigma_w(T))/(R*T*rho_w)

## Compute rho_air by assuming air is saturated at given T, P

# Petty (3.41)

qsat = 0.622*(es(T-273.15)/P)

Tv = T*(1.0 + 0.61*qsat)

rho_air = P/Rd/Tv # air density

#print "rho_air", rho_air

Dp_big = 5e-6

Dp_low = np.min([0.207683*(ac**-0.33048), 5.0])*1e-5

Dp_B = 2.*dv_cont(T, P)*np.sqrt(2*np.pi*Mw/R/T)/ac

Dp_diff = Dp_big - Dp_low

Dv_ave = (dv_cont(T, P)/Dp_diff)*(Dp_diff - Dp_B*np.log((Dp_big + Dp_B)/(Dp_low+Dp_B)))

G_a = (rho_w*R*T)/(es(T-273.15)*Dv_ave*Mw)

G_b = (L*rho_w)*(L*Mw/R/T - 1.0)/(ka_cont(T)*T)

G = 4./(G_a + G_b)

alpha = (g*Mw*L)/(Cp*R*(T**2)) - (g*Ma)/(R*T)

gamma = (P*Ma)/(Mw*es(T-273.15)) + (Mw*L*L)/(Cp*R*T*T)

## Compute sgi of each mode

sgis = []

for aerosol in aerosols:

_, sgi = kohler_crit(T, aerosol.distribution.mu*1e-6, aerosol.kappa)

sgis.append(sgi)

#print "--"*20

def S_integral(Smax):

delta = Smax**4 - (16./9.)*(A*A*alpha*V/G)

#delta = 1.0 - (16./9.)*alpha*V*(1./G)*((A/(Smax**2))**2)

if delta > 0:

sp_sm_sq = 0.5*(1.0 + np.sqrt(delta))

sp_sm = np.sqrt(sp_sm_sq)

else:

arg = (2e7*A/3.)*Smax**(-0.3824)

sp_sm = np.min([arg, 1.0])

Spart = sp_sm*Smax

#print "Spart", Smax, Spart, delta

I1s, I2s = 0.0, 0.0

for aerosol, sgi in zip(aerosols, sgis):

log_sig = np.log(aerosol.distribution.sigma)

Ni = aerosol.distribution.N*1e6

upart = 2.*np.log(sgi/Spart)/(3.*np.sqrt(2)*log_sig)

umax = 2.*np.log(sgi/Smax)/(3.*np.sqrt(2)*log_sig)

def I1(Smax):

A1 = (Ni/2.)*((G/alpha/V)**0.5)

A2 = Smax

C1 = erfc(upart)

C2 = 0.5*((sgi/Smax)**2)

C3a = np.exp(9.*(log_sig**2)/2.)

C3b = erfc(upart + 3.*log_sig/np.sqrt(2.))

return A1*A2*(C1 - C2*C3a*C3b)

def I2(Smax):

A1 = A*Ni/3./sgi

A2 = np.exp(9.*(log_sig**2.)/8.)

C1 = erf(upart - 3.*log_sig/(2.*np.sqrt(2.)))

C2 = erf(umax - 3.*log_sig/(2.*np.sqrt(2.)))

return A1*A2*(C1 - C2)

beta = 0.5*np.pi*gamma*rho_w*G/alpha/V#/rho_air

#beta = 0.5*np.pi*gamma*rho_w/bet2_par/alpha/V/rho_air

#print "++", Smax, I1(Smax), I2(Smax)

I1s += I1(Smax)

I2s += I2(Smax)

return Smax*beta*(I1s + I2s) - 1.0

x1 = 1e-5

y1 = S_integral(x1)

x2 = 1.0

y2 = S_integral(x2)

#print (x1, y1), (x2, y2)

#print "BISECTION"

#print "--"*20

for i in xrange(max_iters):

## Bisection

#y1, y2 = S_integral(x1), S_integral(x2)

x3 = 0.5*(x1+x2)

y3 = S_integral(x3)

#print "--", x3, y3, "--"

if np.sign(y1)*np.sign(y3) <= 0.:

x2 = x3

y2 = y3

else:

x1 = x3

y1 = y3

if np.abs(x2-x1) < tol*x1: break

#print i, (x1, y1), (x2, y2)

## Converged ; return

x3 = 0.5*(x1 + x2)

Smax = x3

#print "Smax = %f (%f)" % (Smax, 0.0)

act_fracs = []

for aerosol, sgi in zip(aerosols, sgis):

ui = 2.*np.log(sgi/Smax)/(3.*np.sqrt(2.)*np.log(aerosol.distribution.sigma))

N_act = 0.5*aerosol.distribution.N*erfc(ui)

act_fracs.append(N_act/aerosol.distribution.N)

"""Sketch implementation of Nenes and Seinfeld (2003) parameterization

"""

nmd_par = len(aerosols) # number of modes

vhfi = 3.0 # van't hoff factor (ions/molecule)

## Setup constants

akoh_par = 4.0*Mw*sigma_w(T)/R/T/rho_w

## Compute rho_air by assuming air is saturated at given T, P

# Petty (3.41)

qsat = 0.622*(es(T-273.15)/P)

Tv = T*(1.0 + 0.61*qsat)

rho_air = P/Rd/Tv # air density

alpha = g*Mw*L/Cp/R/T/T - g*Ma/R/T

gamma = P*Ma/es(T-273.15)/Mw + Mw*L*L/Cp/R/T/T

bet2_par = R*T*rho_w/es(T-273.15)/Dv/Mw/4. + L*rho_w/4./Ka/T*(L*Mw/R/T - 1.0)

beta = 0.5*np.pi*gamma*rho_w/bet2_par/alpha/V/rho_air

cf1 = 0.5*(((1/bet2_par)/(alpha*V))**0.5)

cf2 = akoh_par/3.0

sgis = []

for aerosol in aerosols:

_, sgi = kohler_crit(T, aerosol.distribution.mu*1e-6, aerosol.kappa)

sgis.append(sgi)

def sintegral(spar):

## descriminant criterion

descr = 1.0 - (16./9.)*alpha*V*bet2_par*((akoh_par/(spar**2))**2)

if descr <= 0.0:

crit2 = True

ratio = (2e7/3.)*akoh_par*spar**(-0.3824)

if ratio > 1.0: ratio = 1.0

ssplt2 = spar*ratio

else:

crit2 = False

ssplt1 = 0.5*(1.0 - np.sqrt(descr)) # min root of both

ssplt2 = 0.5*(1.0 + np.sqrt(descr)) # max root of both

ssplt1 = np.sqrt(ssplt1)*spar # multiply ratios with smax

ssplt2 = np.sqrt(ssplt2)*spar

ssplt = ssplt2 # store ssplit in common

summ, summat, summa = 0, 0, 0

sqtwo = np.sqrt(2.0)

for aerosol, sgi in zip(aerosols, sgis):

sg_par = sgi

tpi = aerosol.distribution.N*1e6

dlgsg = np.log(aerosol.distribution.sigma)

dlgsp = np.log(sg_par/spar)

orism1 = 2.0*np.log(sg_par/ssplt2)/(3.*sqtwo*dlgsg)

orism2 = orism1 - 3.0*dlgsg/(2.0*sqtwo)

orism3 = 2.0*dlgsp/(3.0*sqtwo*dlgsg) - 3.0*dlgsg/(2.0*sqtwo)

orism4 = orism1 + 3.0*dlgsg/sqtwo

orism5 = 2.0*dlgsp/(3*sqtwo*dlgsg)

ekth = np.exp((9./2.)*dlgsg*dlgsg)

integ1 = tpi*spar*((1-erf(orism1)) - 0.5*((sg_par/spar)**2)*ekth*(1.0-erf(orism4)))

integ2 = (np.exp((9./8.)*dlgsg*dlgsg)*tpi/sg_par)*(erf(orism2) - erf(orism3))

nd = (tpi/2.)*(1.0 - erf(orism5))

summ += integ1

summat += integ2

summa += nd

return summa, summ, summat

## Initial values for bisection

x1 = 1e-5 # min cloud supersaturation

ndrpl, sinteg1, sinteg2 = sintegral(x1)

print ndrpl, sinteg1, sinteg2

y1 = (sinteg1*cf1 + sinteg2*cf2)*beta*x1 - 1.0

x2 = 1.0 # max cloud supersaturation

ndrpl, sinteg1, sinteg2 = sintegral(x2)

print ndrpl, sinteg1, sinteg2

y2 = (sinteg1*cf1 + sinteg2*cf2)*beta*x2 - 1.0

print (x1, y1), (x2, y2)

print "BISECTION"

print "--"*20

## Perform bisection

for i in xrange(max_iters):

x3 = 0.5*(x1 + x2)

ndrpl, sinteg1, sinteg3 = sintegral(x3)

y3 = (sinteg1*cf1 + sinteg2*cf2)*beta*x3 - 1.0

if np.sign(y1)*np.sign(y3) <= 0.:

y2 = y3

x2 = x3

else:

y1 = y3

x1 = x3

if np.abs(x2-x1) <= tol*x1: break

print i, (x1, y1), (x2, y2)

## Converged ; return

x3 = 0.5*(x1 + x2)

ndrpl, sinteg1, sinteg3 = sintegral(x3)

y3 = (sinteg1*cf1 + sinteg2*cf2)*beta*x3 - 1.0

Smax = x3

print "Smax = %f (%f)" % (Smax, 0.0)

act_fracs = []

for aerosol, sgi in zip(aerosols, sgis):

ui = 2.*np.log(sgi/Smax)/(3.*np.sqrt(2.)*np.log(aerosol.distribution.sigma))

N_act = 0.5*aerosol.distribution.N*erfc(ui)

act_fracs.append(N_act/aerosol.distribution.N)

"""

Map a value from the lognormal distribution with given mu and sigma to the

standard normal distribution with mean 0 and std 1

"""

"""

Map a value from the uniform distribution [a, b] to the normal distribution

"""

Smaxes = []

for aerosol in aerosols:

N = aerosol.distribution.N

mu = aerosol.distribution.mu

sigma = aerosol.distribution.sigma

kappa = aerosol.kappa

Smax = _pce_fit(N, mu, sigma, kappa, V, T, P)

Smaxes.append(Smax)

print "PCE with", N, mu, sigma, kappa, V, T, P, Smax

min_smax = nmin(Smaxes)

if 0. <= min_smax <= 0.5:

Smax = min_smax

else:

return 0., [0.]*len(aerosols)

## Compute scrit of each mode

scrits = []

for aerosol in aerosols:

_, scrit = kohler_crit(T, aerosol.distribution.mu*1e-6, aerosol.kappa)

scrits.append(scrit)

act_fracs = []

for aerosol, scrit in zip(aerosols, scrits):

ui = 2.*np.log(scrit/Smax)/(3.*np.sqrt(2.)*np.log(aerosol.distribution.sigma))

N_act = 0.5*aerosol.distribution.N*erfc(ui)

act_fracs.append(N_act/aerosol.distribution.N)

## P in Pa

dist_bounds = {

'mu': [0.01, 0.25],

#'N': [100., 10000.],

'kappa': [0.1, 1.2],

'sigma': [1.2, 3.0],

'V': [0., 4.0],

'T': [235., 310.],

'P': [50000., 105000.],

}

dist_params = {

'N': [ 7.5, 1.95 ],

}

N = lognorm_to_norm(N, *dist_params['N'])

mu = uni_to_norm(mu, *dist_bounds['mu'])

kappa = uni_to_norm(kappa, *dist_bounds['kappa'])

sigma = uni_to_norm(sigma, *dist_bounds['sigma'])

V = uni_to_norm(V, *dist_bounds['V'])

T = uni_to_norm(T, *dist_bounds['T'])

P = uni_to_norm(P, *dist_bounds['P'])

Smax = 6.4584111537e-6*N**6 - \

2.5994976288e-5*N**5 - \

1.7065251097e-7*N**4*P**2 + \

1.3741352226e-5*N**4*P + \

2.8567989557e-5*N**4*T**2 - \

7.4876643038e-5*N**4*T - \

2.0388391982e-6*N**4*V**2 + \

4.3054466907e-5*N**4*V + \

3.6504788687e-6*N**4*kappa**2 + \

8.7165631487e-7*N**4*kappa + \

1.6542743001e-5*N**4*mu**2 + \

4.8195946039e-6*N**4*mu + \

3.9282682647e-6*N**4*sigma**2 + \

1.1137326431e-5*N**4*sigma + \

2.795758112227e-5*N**4 + \

1.5947545697e-6*N**3*P**2 - \

6.9358311166e-5*N**3*P - \

0.00014252420422*N**3*T**2 + \

0.00039466661884*N**3*T + \

2.15368184e-5*N**3*V**2 - \

0.00025279065671*N**3*V + \

4.6142483833e-6*N**3*kappa**2 - \

2.5055687574e-5*N**3*kappa - \

3.0424806654e-6*N**3*mu**2 - \

4.5156027497e-5*N**3*mu - \

1.780917608e-6*N**3*sigma**2 - \

2.516400813e-5*N**3*sigma - \

0.0003567127574296*N**3 + \

5.9696014699e-7*N**2*P**4 - \

1.3472490172e-5*N**2*P**3 - \

1.0610551852e-6*N**2*P**2*T**2 + \

2.0181530448e-6*N**2*P**2*T + \

2.5327194907e-7*N**2*P**2*V**2 - \

1.4006527233e-6*N**2*P**2*V + \

5.4851851852e-7*N**2*P**2*kappa**2 - \

1.320380981e-6*N**2*P**2*kappa + \

1.7644666667e-7*N**2*P**2*mu**2 - \

2.7894950894e-7*N**2*P**2*mu + \

1.8201189815e-7*N**2*P**2*sigma**2 - \

5.0510811394e-7*N**2*P**2*sigma - \

6.88818634103e-6*N**2*P**2 + \

5.0207581099e-5*N**2*P*T**2 - \

0.00013814911722*N**2*P*T - \

6.2792651121e-6*N**2*P*V**2 + \

7.2980075931e-5*N**2*P*V - \

3.7856114614e-6*N**2*P*kappa**2 + \

1.2860228333e-5*N**2*P*kappa - \

1.5691902399e-6*N**2*P*mu**2 + \

8.2376491667e-6*N**2*P*mu - \

1.3435745045e-6*N**2*P*sigma**2 + \

6.0282465278e-6*N**2*P*sigma + \

0.0001877522259389*N**2*P - \

4.0442507595e-5*N**2*T**4 + \

5.6586533058e-5*N**2*T**3 - \

8.9548419306e-6*N**2*T**2*V**2 + \

0.00014183762216*N**2*T**2*V + \

1.7477041667e-7*N**2*T**2*kappa**2 - \

2.2336680774e-5*N**2*T**2*kappa - \

3.9516949861e-5*N**2*T**2*mu**2 - \

1.428384236e-5*N**2*T**2*mu - \

8.1085041667e-6*N**2*T**2*sigma**2 - \

4.4004842538e-5*N**2*T**2*sigma + \

0.00038258884934483*N**2*T**2 + \

2.9970384599e-5*N**2*T*V**2 - \

0.00041049796829*N**2*T*V + \

6.5092115599e-6*N**2*T*kappa**2 + \

3.0809800694e-5*N**2*T*kappa + \

9.9551207477e-5*N**2*T*mu**2 + \

1.0952167639e-5*N**2*T*mu + \

2.1329980047e-5*N**2*T*sigma**2 + \

8.81912525e-5*N**2*T*sigma - \

0.0008911162845737*N**2*T + \

6.9026802931e-6*N**2*V**4 - \

4.7531336217e-5*N**2*V**3 + \

2.5832318241e-6*N**2*V**2*kappa**2 - \

1.2472907784e-6*N**2*V**2*kappa + \

1.1149875079e-5*N**2*V**2*mu**2 - \

2.9708960501e-6*N**2*V**2*mu + \

3.2880035648e-7*N**2*V**2*sigma**2 + \

7.9685785603e-6*N**2*V**2*sigma - \

8.857197689645e-5*N**2*V**2 - \

1.3905780926e-5*N**2*V*kappa**2 + \

2.6425726833e-5*N**2*V*kappa - \

4.4290453362e-5*N**2*V*mu**2 + \

3.4602470958e-5*N**2*V*mu - \

9.497372933e-6*N**2*V*sigma**2 - \

8.4509070972e-6*N**2*V*sigma + \

0.0007493009795633*N**2*V + \

2.8884698866e-6*N**2*kappa**4 + \

1.349739092e-6*N**2*kappa**3 + \

1.7550156389e-5*N**2*kappa**2*mu**2 - \

1.9786638902e-6*N**2*kappa**2*mu + \

5.529520787e-6*N**2*kappa**2*sigma**2 + \

1.2209966835e-5*N**2*kappa**2*sigma - \

5.448370112109e-5*N**2*kappa**2 - \

4.359847391e-5*N**2*kappa*mu**2 - \

2.2737228056e-5*N**2*kappa*mu - \

9.990113266e-6*N**2*kappa*sigma**2 - \

5.9185131528e-5*N**2*kappa*sigma + \

9.22206763018e-6*N**2*kappa + \

3.0424263183e-5*N**2*mu**4 - \

1.9098455668e-5*N**2*mu**3 + \

8.54937625e-6*N**2*mu**2*sigma**2 - \

4.684071842e-6*N**2*mu**2*sigma - \

0.00035110649314667*N**2*mu**2 - \

3.6261121147e-6*N**2*mu*sigma**2 - \

9.2769369028e-5*N**2*mu*sigma + \

0.00011212992202954*N**2*mu + \

5.0891009441e-6*N**2*sigma**4 + \

3.5893477645e-6*N**2*sigma**3 - \

7.197212424173e-5*N**2*sigma**2 - \

0.00011060069230486*N**2*sigma + \

0.00151313669719111*N**2 - \

1.1284287469e-6*N*P**4 + \

3.0704412322e-5*N*P**3 + \

1.6278653855e-6*N*P**2*T**2 - \

3.3672619444e-6*N*P**2*T - \

6.2110532065e-8*N*P**2*V**2 + \

3.2172427639e-6*N*P**2*V - \

2.8443321387e-7*N*P**2*kappa**2 + \

7.4341916667e-7*N*P**2*kappa - \

7.2252756038e-7*N*P**2*mu**2 + \

8.4614527778e-7*N*P**2*mu - \

1.2720654237e-7*N*P**2*sigma**2 + \

2.7419097222e-7*N*P**2*sigma + \

8.764064001385e-6*N*P**2 - \

9.5804124167e-5*N*P*T**2 + \

0.00027775604478*N*P*T + \

1.2225588236e-5*N*P*V**2 - \

0.0001716045343*N*P*V + \

1.8806313889e-6*N*P*kappa**2 - \

1.2701263287e-6*N*P*kappa + \

1.3923449444e-5*N*P*mu**2 - \

1.4186857698e-6*N*P*mu + \

3.9634190278e-6*N*P*sigma**2 + \

9.947051115e-6*N*P*sigma - \

0.0003223815596977*N*P + \

7.5931315992e-5*N*T**4 - \

0.00011373913927*N*T**3 + \

1.5275617964e-5*N*T**2*V**2 - \

0.00027033311532*N*T**2*V - \

9.5487976257e-6*N*T**2*kappa**2 + \

6.1326942361e-5*N*T**2*kappa + \

1.1264628419e-5*N*T**2*mu**2 + \

0.00013788716208*N*T**2*mu + \

8.4656605352e-6*N*T**2*sigma**2 + \

9.7041865833e-5*N*T**2*sigma - \

0.00046604057151*N*T**2 - \

5.2633321153e-5*N*T*V**2 + \

0.00082461082486*N*T*V + \

1.9930343472e-5*N*T*kappa**2 - \

0.00014021885993*N*T*kappa - \

3.2740759028e-5*N*T*mu**2 - \

0.00033633604715*N*T*mu - \

2.7467490833e-5*N*T*sigma**2 - \

0.0002267701814*N*T*sigma + \

0.0010623078872764*N*T - \

1.158262868e-5*N*V**4 + \

0.00010282581987*N*V**3 + \

2.0236346649e-7*N*V**2*kappa**2 - \

7.0861126667e-6*N*V**2*kappa - \

1.8571179464e-7*N*V**2*mu**2 - \

2.9025647069e-5*N*V**2*mu - \

2.8250510694e-6*N*V**2*sigma**2 - \

1.0873061236e-5*N*V**2*sigma + \

9.5035330545615e-5*N*V**2 - \

4.7114408333e-7*N*V*kappa**2 + \

3.5583995899e-5*N*V*kappa + \

4.3931099764e-5*N*V*mu**2 + \

9.4893199047e-5*N*V*mu + \

1.9266056153e-5*N*V*sigma**2 + \

8.172457216e-5*N*V*sigma - \

0.00114623544365757*N*V + \

2.5465455757e-6*N*kappa**4 - \

1.1844938245e-5*N*kappa**3 - \

1.2548361851e-5*N*kappa**2*mu**2 - \

6.6498102778e-6*N*kappa**2*mu - \

2.6413318548e-6*N*kappa**2*sigma**2 - \

1.9743217083e-5*N*kappa**2*sigma - \

5.237116508322e-5*N*kappa**2 + \

2.2628180833e-5*N*kappa*mu**2 + \

6.4409460169e-5*N*kappa*mu + \

2.1659551389e-6*N*kappa*sigma**2 + \

8.2294682962e-5*N*kappa*sigma + \

0.00031141975514413*N*kappa - \

1.1565474947e-5*N*mu**4 - \

2.1450508636e-5*N*mu**3 + \

6.1585477702e-6*N*mu**2*sigma**2 - \

8.815663375e-5*N*mu**2*sigma + \

8.443778184842e-5*N*mu**2 - \

3.4406999306e-5*N*mu*sigma**2 + \

0.00027943018423*N*mu*sigma + \

0.00073132224303402*N*mu - \

8.4378328798e-6*N*sigma**4 - \

2.0928942447e-5*N*sigma**3 + \

9.361717372097e-5*N*sigma**2 + \

0.00042563590086478*N*sigma - \

0.00207631579223133*N - \

1.9577562243e-8*P**6 + \

9.8981784049e-7*P**5 - \

5.8363352597e-8*P**4*T**2 + \

2.6457614122e-8*P**4*T + \

9.0459993866e-8*P**4*V**2 + \

2.1439092975e-7*P**4*V + \

1.7814328446e-7*P**4*kappa**2 - \

4.1686622901e-7*P**4*kappa + \

5.3644855238e-8*P**4*mu**2 - \

2.0156224591e-7*P**4*mu + \

5.8210558734e-8*P**4*sigma**2 - \

1.8962978248e-7*P**4*sigma + \

4.46780827354e-7*P**4 - \

1.1972281072e-6*P**3*T**2 + \

5.3768532472e-6*P**3*T + \

2.2417961995e-7*P**3*V**2 - \

6.4936735747e-6*P**3*V - \

7.4042040112e-7*P**3*kappa**2 + \

3.1988643743e-6*P**3*kappa - \

1.1174867493e-7*P**3*mu**2 + \

4.9097886778e-6*P**3*mu + \

2.5563905537e-7*P**3*sigma**2 + \

3.0112545186e-6*P**3*sigma - \

2.528028422697e-5*P**3 - \

6.2461608542e-8*P**2*T**4 + \

7.2160816962e-9*P**2*T**3 - \

3.9231712963e-8*P**2*T**2*V**2 + \

1.2045330835e-7*P**2*T**2*V - \

1.2065046296e-7*P**2*T**2*kappa**2 + \

5.5655764074e-8*P**2*T**2*kappa - \

7.1769768519e-7*P**2*T**2*mu**2 + \

9.214390015e-7*P**2*T**2*mu - \

1.1352175926e-7*P**2*T**2*sigma**2 - \

7.2727690784e-8*P**2*T**2*sigma + \

9.20245283507e-7*P**2*T**2 + \

8.0179117696e-8*P**2*T*V**2 + \

4.235625e-8*P**2*T*V + \

2.258923022e-7*P**2*T*kappa**2 - \

4.446875e-7*P**2*T*kappa + \

1.319233337e-6*P**2*T*mu**2 - \

2.0462986111e-6*P**2*T*mu + \

8.8850197051e-8*P**2*T*sigma**2 + \

3.2751388889e-8*P**2*T*sigma - \

3.083990086676e-7*P**2*T + \

1.3373206908e-7*P**2*V**4 + \

9.0363593389e-8*P**2*V**3 + \

2.4463467593e-7*P**2*V**2*kappa**2 - \

3.2486785978e-7*P**2*V**2*kappa + \

2.7125416667e-7*P**2*V**2*mu**2 - \

7.8996752457e-8*P**2*V**2*mu + \

2.3869884259e-7*P**2*V**2*sigma**2 - \

3.9030882886e-8*P**2*V**2*sigma - \

1.676552162853e-6*P**2*V**2 - \

3.2961961287e-7*P**2*V*kappa**2 + \

1.0572459722e-6*P**2*V*kappa + \

5.8846025249e-7*P**2*V*mu**2 - \

1.2420694444e-7*P**2*V*mu + \

1.1084042637e-7*P**2*V*sigma**2 + \

5.9708680556e-7*P**2*V*sigma - \

2.731802676607e-6*P**2*V + \

1.6946466336e-7*P**2*kappa**4 - \

1.697455568e-7*P**2*kappa**3 + \

4.3087824074e-7*P**2*kappa**2*mu**2 - \

2.6231749106e-8*P**2*kappa**2*mu + \

4.1886990741e-7*P**2*kappa**2*sigma**2 - \

2.1180014438e-7*P**2*kappa**2*sigma - \

2.68356980142e-6*P**2*kappa**2 - \

1.1445592205e-6*P**2*kappa*mu**2 + \

2.838125e-7*P**2*kappa*mu - \

7.6035506889e-7*P**2*kappa*sigma**2 + \

3.2736111111e-9*P**2*kappa*sigma + \

5.198700314056e-6*P**2*kappa + \

2.570161515e-7*P**2*mu**4 - \

4.2080255264e-7*P**2*mu**3 + \

1.7831666667e-7*P**2*mu**2*sigma**2 - \

1.3384887703e-6*P**2*mu**2*sigma - \

2.364220102728e-6*P**2*mu**2 + \

8.8640907579e-8*P**2*mu*sigma**2 + \

1.2368444444e-6*P**2*mu*sigma + \

4.855707265804e-6*P**2*mu + \

9.059626753e-8*P**2*sigma**4 - \

1.5254956604e-7*P**2*sigma**3 - \

1.126043894964e-6*P**2*sigma**2 + \

2.8315063203e-6*P**2*sigma - \

2.778055205468e-6*P**2 - \

2.1400747816e-6*P*T**4 + \

4.9258105417e-6*P*T**3 + \

5.6327936107e-7*P*T**2*V**2 + \

9.3250965278e-6*P*T**2*V + \

3.2710316757e-6*P*T**2*kappa**2 - \

8.2942513889e-6*P*T**2*kappa + \

7.9343747988e-6*P*T**2*mu**2 - \

1.9181326389e-5*P*T**2*mu + \

1.5642303199e-6*P*T**2*sigma**2 - \

7.5503763889e-6*P*T**2*sigma + \

2.871260752173e-5*P*T**2 + \

1.4432569444e-7*P*T*V**2 - \

3.6214883811e-5*P*T*V - \

7.5203763889e-6*P*T*kappa**2 + \

2.3241170134e-5*P*T*kappa - \

1.7993693056e-5*P*T*mu**2 + \

5.5177810267e-5*P*T*mu - \

1.9631597222e-6*P*T*sigma**2 + \

2.2474684728e-5*P*T*sigma - \

0.00010697897078404*P*T + \

4.3918577044e-7*P*V**4 - \

6.5398867255e-6*P*V**3 - \

1.6642529664e-6*P*V**2*kappa**2 + \

4.5294820833e-6*P*V**2*kappa - \

8.3413225416e-7*P*V**2*mu**2 + \

3.3331961111e-6*P*V**2*mu - \

5.4879252019e-7*P*V**2*sigma**2 + \

2.5988245833e-6*P*V**2*sigma - \

6.06167138571e-6*P*V**2 + \

1.76840125e-6*P*V*kappa**2 - \

1.2335091147e-5*P*V*kappa - \

7.172135e-6*P*V*mu**2 - \

1.4280271047e-5*P*V*mu - \

1.7709023611e-6*P*V*sigma**2 - \

1.5558439144e-5*P*V*sigma + \

0.0001341572007329*P*V - \

1.3847755834e-6*P*kappa**4 + \

2.9663988051e-6*P*kappa**3 - \

6.701873906e-7*P*kappa**2*mu**2 + \

1.2602319444e-6*P*kappa**2*mu - \

1.6684083648e-6*P*kappa**2*sigma**2 + \

4.8915402778e-6*P*kappa**2*sigma + \

1.830572029266e-5*P*kappa**2 + \

7.2758208333e-6*P*kappa*mu**2 - \

4.4294673496e-6*P*kappa*mu + \

5.7967319444e-6*P*kappa*sigma**2 - \

7.5561654674e-6*P*kappa*sigma - \

6.41037679493e-5*P*kappa + \

1.1303131108e-7*P*mu**4 + \

4.477378242e-6*P*mu**3 - \

8.1756005619e-8*P*mu**2*sigma**2 + \

1.8019847222e-5*P*mu**2*sigma + \

3.591958513889e-6*P*mu**2 + \

3.5776194444e-6*P*mu*sigma**2 - \

2.1832827595e-5*P*mu*sigma - \

0.000104836086236*P*mu + \

4.4303746065e-7*P*sigma**4 + \

3.5105365953e-6*P*sigma**3 - \

5.903007579301e-6*P*sigma**2 - \

6.46457674367e-5*P*sigma + \

0.000248152385516871*P + \

9.3486231047e-7*T**6 - \

1.8168892496e-6*T**5 - \

1.2017117971e-7*T**4*V**2 - \

6.0623117829e-6*T**4*V - \

2.2138037142e-6*T**4*kappa**2 + \

5.3907485972e-6*T**4*kappa - \

1.5012731379e-5*T**4*mu**2 + \

2.9320254519e-5*T**4*mu - \

9.9591672455e-7*T**4*sigma**2 + \

4.5407528726e-6*T**4*sigma - \

2.1701625406048e-5*T**4 - \

8.86080478e-8*T**3*V**2 + \

1.4786332237e-5*T**3*V + \

3.9370511477e-6*T**3*kappa**2 - \

1.1123904654e-5*T**3*kappa + \

1.9629299957e-5*T**3*mu**2 - \

4.3941049344e-5*T**3*mu + \

1.1089594981e-6*T**3*sigma**2 - \

9.8896573442e-6*T**3*sigma + \

4.92291899313038e-5*T**3 - \

1.8208011851e-7*T**2*V**4 - \

3.3830247075e-6*T**2*V**3 + \

1.5875634259e-7*T**2*V**2*kappa**2 - \

8.2594310191e-7*T**2*V**2*kappa - \

7.0372356481e-6*T**2*V**2*mu**2 + \

1.1314601854e-5*T**2*V**2*mu + \

2.6109148148e-7*T**2*V**2*sigma**2 - \

1.4817938429e-6*T**2*V**2*sigma + \

2.584581978913e-6*T**2*V**2 + \

8.1820206167e-6*T**2*V*kappa**2 - \

2.152989125e-5*T**2*V*kappa + \

3.5087114657e-5*T**2*V*mu**2 - \

7.62298625e-5*T**2*V*mu + \

3.7441553065e-6*T**2*V*sigma**2 - \

1.9480860556e-5*T**2*V*sigma + \

8.078026266135e-5*T**2*V - \

1.4079168102e-6*T**2*kappa**4 + \

8.494577264e-7*T**2*kappa**3 - \

1.9940069444e-6*T**2*kappa**2*mu**2 - \

3.7981316228e-6*T**2*kappa**2*mu + \

1.2927453704e-7*T**2*kappa**2*sigma**2 - \

5.9931564037e-6*T**2*kappa**2*sigma + \

2.918584228186e-5*T**2*kappa**2 - \

5.4461049955e-7*T**2*kappa*mu**2 + \

1.7234859722e-5*T**2*kappa*mu - \

1.9437451044e-6*T**2*kappa*sigma**2 + \

1.7031693056e-5*T**2*kappa*sigma - \

6.0924394269614e-5*T**2*kappa - \

1.3344139356e-5*T**2*mu**4 + \

6.9440170146e-6*T**2*mu**3 - \

6.7769083333e-6*T**2*mu**2*sigma**2 + \

3.2599345224e-6*T**2*mu**2*sigma + \

0.00022442210764479*T**2*mu**2 + \

7.6319402846e-6*T**2*mu*sigma**2 + \

7.2359722222e-6*T**2*mu*sigma - \

0.0003491716663951*T**2*mu - \

1.1631111786e-6*T**2*sigma**4 + \

2.6724248621e-6*T**2*sigma**3 + \

1.649502809964e-5*T**2*sigma**2 - \

6.1536219106916e-5*T**2*sigma + \

0.000140143705596224*T**2 + \

2.7736623456e-7*T*V**4 + \

1.5654708427e-5*T*V**3 + \

2.1280663429e-6*T*V**2*kappa**2 - \

3.8522365278e-6*T*V**2*kappa + \

2.1368239286e-5*T*V**2*mu**2 - \

3.6494785e-5*T*V**2*mu + \

2.4667129876e-8*T*V**2*sigma**2 + \

8.2984694444e-7*T*V**2*sigma - \

1.16151114743199e-6*T*V**2 - \

2.4386513472e-5*T*V*kappa**2 + \

7.2626637568e-5*T*V*kappa - \

8.6385334444e-5*T*V*mu**2 + \

0.00022158505878*T*V*mu - \

6.7549275e-6*T*V*sigma**2 + \

6.7690826574e-5*T*V*sigma - \

0.000319030815886*T*V + \

5.7579921563e-6*T*kappa**4 - \

5.9128382699e-6*T*kappa**3 + \

5.8599504703e-6*T*kappa**2*mu**2 + \

1.0919901389e-5*T*kappa**2*mu + \

2.3570428983e-6*T*kappa**2*sigma**2 + \

1.1143115278e-5*T*kappa**2*sigma - \

9.05515382865e-5*T*kappa**2 - \

5.1905069444e-6*T*kappa*mu**2 - \

4.8460056021e-5*T*kappa*mu - \

1.9954263889e-6*T*kappa*sigma**2 - \

4.0364821013e-5*T*kappa*sigma + \

0.0002112574003288*T*kappa + \

3.6831785874e-5*T*mu**4 - \

2.2679927293e-5*T*mu**3 + \

1.3165213945e-5*T*mu**2*sigma**2 - \

1.5411466667e-5*T*mu**2*sigma - \

0.0005305662075903*T*mu**2 - \

1.4521058333e-5*T*mu*sigma**2 - \

2.36356423e-5*T*mu*sigma + \

0.0008959089906671*T*mu + \

3.1144363024e-6*T*sigma**4 - \

1.1900713105e-5*T*sigma**3 - \

3.6444491206927e-5*T*sigma**2 + \

0.000221944115643271*T*sigma - \

0.000541037097804642*T + \

1.3872452626e-7*V**6 + \

2.6280928855e-6*V**5 + \

1.4116924747e-6*V**4*kappa**2 - \

2.6532439544e-6*V**4*kappa + \

4.0826945223e-6*V**4*mu**2 - \

6.7305962741e-6*V**4*mu + \

3.2377252949e-7*V**4*sigma**2 - \

2.3853637883e-7*V**4*sigma - \

2.12902319506e-6*V**4 - \

3.4146924781e-6*V**3*kappa**2 + \

1.1308634888e-5*V**3*kappa - \

4.3203910556e-6*V**3*mu**2 + \

2.0522142146e-5*V**3*mu - \

1.1130033126e-7*V**3*sigma**2 + \

9.6641202763e-6*V**3*sigma - \

6.9565263098929e-5*V**3 + \

9.124821437e-7*V**2*kappa**4 - \

2.374870717e-7*V**2*kappa**3 + \

2.0551150926e-6*V**2*kappa**2*mu**2 + \

2.1737964134e-6*V**2*kappa**2*mu + \

1.5400502778e-6*V**2*kappa**2*sigma**2 + \

2.1762389258e-6*V**2*kappa**2*sigma - \

1.908307985662e-5*V**2*kappa**2 - \

2.2644466603e-6*V**2*kappa*mu**2 - \

7.2844616667e-6*V**2*kappa*mu - \

1.327008481e-6*V**2*kappa*sigma**2 - \

7.3647294444e-6*V**2*kappa*sigma + \

3.083805354799e-5*V**2*kappa + \

6.376029485e-6*V**2*mu**4 - \