A1. Polar Amplification in simple models

Note

The materials and scripts are obtained from climlab tutorial: here

%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
import climlab
from climlab import constants as const

EBM with surface and atmosphere layers

ebm = climlab.GreyRadiationModel(num_lev=1, num_lat=90)
insolation = climlab.radiation.AnnualMeanInsolation(domains=ebm.Ts.domain)
ebm.add_subprocess('insolation', insolation)
ebm.subprocess.SW.flux_from_space = ebm.subprocess.insolation.insolation
print(ebm)

#  add a fixed relative humidity process
#  (will only affect surface evaporation)
h2o = climlab.radiation.ManabeWaterVapor(state=ebm.state, **ebm.param)
ebm.add_subprocess('H2O', h2o)

#  Add surface heat fluxes
shf = climlab.surface.SensibleHeatFlux(state=ebm.state, Cd=3E-4)
lhf = climlab.surface.LatentHeatFlux(state=ebm.state, Cd=3E-4)
# couple water vapor to latent heat flux process
lhf.q = h2o.q
ebm.add_subprocess('SHF', shf)
ebm.add_subprocess('LHF', lhf)

ebm.integrate_years(1)

plt.plot(ebm.lat, ebm.Ts)
plt.plot(ebm.lat, ebm.Tatm)

climlab Process of type <class 'climlab.model.column.GreyRadiationModel'>. 
State variables and domain shapes: 
  Ts: (90, 1) 
  Tatm: (90, 1) 
The subprocess tree: 
Untitled: <class 'climlab.model.column.GreyRadiationModel'>
   LW: <class 'climlab.radiation.greygas.GreyGas'>
   SW: <class 'climlab.radiation.greygas.GreyGasSW'>
   insolation: <class 'climlab.radiation.insolation.AnnualMeanInsolation'>

Integrating for 365 steps, 365.2422 days, or 1 years.
Total elapsed time is 0.9993368783782377 years.

[<matplotlib.lines.Line2D at 0x14a377610>]

co2ebm = climlab.process_like(ebm)
co2ebm.subprocess['LW'].absorptivity = ebm.subprocess['LW'].absorptivity*1.1

co2ebm.integrate_years(3.)

#  no heat transport but with evaporation -- no polar amplification
plt.plot(ebm.lat, co2ebm.Ts - ebm.Ts)
plt.plot(ebm.lat, co2ebm.Tatm - ebm.Tatm)

Integrating for 1095 steps, 1095.7266 days, or 3.0 years.

Total elapsed time is 3.997347513512951 years.

[<matplotlib.lines.Line2D at 0x14a468cd0>]

Include meridional heat transport

diffebm = climlab.process_like(ebm)
# thermal diffusivity in W/m**2/degC
D = 0.6
# meridional diffusivity in m**2/s
K = D / diffebm.Tatm.domain.heat_capacity * const.a**2
d = climlab.dynamics.MeridionalDiffusion(K=K, state={'Tatm': diffebm.Tatm}, **diffebm.param)
diffebm.add_subprocess('diffusion', d)
print(diffebm)

diffebm.integrate_years(3)

plt.plot(diffebm.lat, diffebm.Ts)
plt.plot(diffebm.lat, diffebm.Tatm)

climlab Process of type <class 'climlab.model.column.GreyRadiationModel'>. 
State variables and domain shapes: 
  Ts: (90, 1) 
  Tatm: (90, 1) 
The subprocess tree: 
Untitled: <class 'climlab.model.column.GreyRadiationModel'>
   LW: <class 'climlab.radiation.greygas.GreyGas'>
   SW: <class 'climlab.radiation.greygas.GreyGasSW'>
   insolation: <class 'climlab.radiation.insolation.AnnualMeanInsolation'>
   H2O: <class 'climlab.radiation.water_vapor.ManabeWaterVapor'>
   SHF: <class 'climlab.surface.turbulent.SensibleHeatFlux'>
   LHF: <class 'climlab.surface.turbulent.LatentHeatFlux'>
   diffusion: <class 'climlab.dynamics.meridional_advection_diffusion.MeridionalDiffusion'>

Integrating for 1095 steps, 1095.7266 days, or 3 years.

Total elapsed time is 3.997347513512951 years.

[<matplotlib.lines.Line2D at 0x14a6e0310>]

def inferred_heat_transport( energy_in, lat_deg ):
    '''Returns the inferred heat transport (in PW) by integrating the net energy imbalance from pole to pole.'''
    from scipy import integrate
    from climlab import constants as const
    lat_rad = np.deg2rad( lat_deg )
    return ( 1E-15 * 2 * np.math.pi * const.a**2 * integrate.cumtrapz( np.cos(lat_rad)*energy_in,
            x=lat_rad, initial=0. ) )

#  Plot the northward heat transport in this model
Rtoa = np.squeeze(diffebm.timeave['ASR'] - diffebm.timeave['OLR'])
plt.plot(diffebm.lat, inferred_heat_transport(Rtoa, diffebm.lat))

/var/folders/z_/36cc8kqx7pqf8lzjc5cp2cf00000gn/T/ipykernel_29629/4142479116.py:6: DeprecationWarning: 'scipy.integrate.cumtrapz' is deprecated in favour of 'scipy.integrate.cumulative_trapezoid' and will be removed in SciPy 1.14.0
  return ( 1E-15 * 2 * np.math.pi * const.a**2 * integrate.cumtrapz( np.cos(lat_rad)*energy_in,

[<matplotlib.lines.Line2D at 0x14a74fe90>]

##  Now warm it up!
co2diffebm = climlab.process_like(diffebm)
co2diffebm.subprocess['LW'].absorptivity = diffebm.subprocess['LW'].absorptivity*1.1

co2diffebm.integrate_years(5)

#  with heat transport and evaporation
#   Get some modest polar amplifcation of surface warming
#    but larger equatorial amplification of atmospheric warming
#    Increased atmospheric gradient = increased poleward flux.
plt.plot(diffebm.lat, co2diffebm.Ts - diffebm.Ts, label='Ts')
plt.plot(diffebm.lat, co2diffebm.Tatm - diffebm.Tatm, label='Tatm')
plt.legend()

Integrating for 1826 steps, 1826.2110000000002 days, or 5 years.

Total elapsed time is 8.99676981465997 years.

<matplotlib.legend.Legend at 0x149f8b4d0>

Rtoa = np.squeeze(diffebm.timeave['ASR'] - diffebm.timeave['OLR'])
Rtoa_co2 = np.squeeze(co2diffebm.timeave['ASR'] - co2diffebm.timeave['OLR'])
plt.plot(diffebm.lat, inferred_heat_transport(Rtoa, diffebm.lat), label='1xCO2')
plt.plot(diffebm.lat, inferred_heat_transport(Rtoa_co2, diffebm.lat), label='2xCO2')
plt.legend()

/var/folders/z_/36cc8kqx7pqf8lzjc5cp2cf00000gn/T/ipykernel_29629/4142479116.py:6: DeprecationWarning: 'scipy.integrate.cumtrapz' is deprecated in favour of 'scipy.integrate.cumulative_trapezoid' and will be removed in SciPy 1.14.0
  return ( 1E-15 * 2 * np.math.pi * const.a**2 * integrate.cumtrapz( np.cos(lat_rad)*energy_in,

<matplotlib.legend.Legend at 0x14b041dd0>

We get polar amplification in surface air temperature!!!

Exclude evaporation

diffebm2 = climlab.process_like(diffebm)
diffebm2.remove_subprocess('LHF')
diffebm2.integrate_years(3)
co2diffebm2 = climlab.process_like(co2diffebm)
co2diffebm2.remove_subprocess('LHF')
co2diffebm2.integrate_years(3)

#  With transport and no evaporation...
#  No polar amplification, either of surface or air temperature!
plt.plot(diffebm2.lat, co2diffebm2.Ts - diffebm2.Ts, label='Ts')
plt.plot(diffebm2.lat, co2diffebm2.Tatm[:,0] - diffebm2.Tatm[:,0], label='Tatm')
plt.legend()
plt.figure()

#  And in this case, the lack of polar amplification is DESPITE an increase in the poleward heat transport.
Rtoa = np.squeeze(diffebm2.timeave['ASR'] - diffebm2.timeave['OLR'])
Rtoa_co2 = np.squeeze(co2diffebm2.timeave['ASR'] - co2diffebm2.timeave['OLR'])
plt.plot(diffebm2.lat, inferred_heat_transport(Rtoa, diffebm2.lat), label='1xCO2')
plt.plot(diffebm2.lat, inferred_heat_transport(Rtoa_co2, diffebm2.lat), label='2xCO2')
plt.legend()

Integrating for 1095 steps, 1095.7266 days, or 3 years.

Total elapsed time is 6.995358148647664 years.
Integrating for 1095 steps, 1095.7266 days, or 3 years.

Total elapsed time is 11.994780449794684 years.

/var/folders/z_/36cc8kqx7pqf8lzjc5cp2cf00000gn/T/ipykernel_29629/4142479116.py:6: DeprecationWarning: 'scipy.integrate.cumtrapz' is deprecated in favour of 'scipy.integrate.cumulative_trapezoid' and will be removed in SciPy 1.14.0
  return ( 1E-15 * 2 * np.math.pi * const.a**2 * integrate.cumtrapz( np.cos(lat_rad)*energy_in,

<matplotlib.legend.Legend at 0x14b1ed9d0>

No polar amplification this time :(

Column model

model = climlab.GreyRadiationModel(num_lev=30, num_lat=90, abs_coeff=1.6E-4)
insolation = climlab.radiation.AnnualMeanInsolation(domains=model.Ts.domain)
model.add_subprocess('insolation', insolation)
model.subprocess.SW.flux_from_space = model.subprocess.insolation.insolation
print(model)

#  Convective adjustment for atmosphere only
conv = climlab.convection.ConvectiveAdjustment(state={'Tatm':model.Tatm}, adj_lapse_rate=6.5,
                                                       **model.param)
model.add_subprocess('convective adjustment', conv)

#  add a fixed relative humidity process
#  (will only affect surface evaporation)
h2o = climlab.radiation.water_vapor.ManabeWaterVapor(state=model.state, **model.param)
model.add_subprocess('H2O', h2o)

#  Add surface heat fluxes
shf = climlab.surface.SensibleHeatFlux(state=model.state, Cd=1E-3)
lhf = climlab.surface.LatentHeatFlux(state=model.state, Cd=1E-3)
lhf.q = model.subprocess.H2O.q
model.add_subprocess('SHF', shf)
model.add_subprocess('LHF', lhf)

model.integrate_years(3.)

def plot_temp_section(model, timeave=True):
    fig = plt.figure()
    ax = fig.add_subplot(111)
    if timeave:
        field = model.timeave['Tatm'].transpose()
    else:
        field = model.Tatm.transpose()
    cax = ax.contourf(model.lat, model.lev, field)
    ax.invert_yaxis()
    ax.set_xlim(-90,90)
    ax.set_xticks([-90, -60, -30, 0, 30, 60, 90])
    fig.colorbar(cax)

plot_temp_section(model, timeave=False)

climlab Process of type <class 'climlab.model.column.GreyRadiationModel'>. 
State variables and domain shapes: 
  Ts: (90, 1) 
  Tatm: (90, 30) 
The subprocess tree: 
Untitled: <class 'climlab.model.column.GreyRadiationModel'>
   LW: <class 'climlab.radiation.greygas.GreyGas'>
   SW: <class 'climlab.radiation.greygas.GreyGasSW'>
   insolation: <class 'climlab.radiation.insolation.AnnualMeanInsolation'>

Integrating for 1095 steps, 1095.7266 days, or 3.0 years.

Total elapsed time is 2.998010635134713 years.

co2model = climlab.process_like(model)
co2model.subprocess['LW'].absorptivity = model.subprocess['LW'].absorptivity*1.1

co2model.integrate_years(3)

plot_temp_section(co2model, timeave=False)

#  Without transport, get equatorial amplification
plt.figure()
plt.plot(model.lat, co2model.Ts - model.Ts, label='Ts')
plt.plot(model.lat, co2model.Tatm[:,0] - model.Tatm[:,0], label='Tatm')
plt.legend()

Integrating for 1095 steps, 1095.7266 days, or 3 years.

Total elapsed time is 5.996021270269426 years.

<matplotlib.legend.Legend at 0x14b832690>

Include meridional heat transport

diffmodel = climlab.process_like(model)

# thermal diffusivity in W/m**2/degC
D = 0.05
# meridional diffusivity in m**2/s
K = D / diffmodel.Tatm.domain.heat_capacity[0] * const.a**2
print(K)

d = climlab.dynamics.MeridionalDiffusion(K=K, state={'Tatm':diffmodel.Tatm}, **diffmodel.param)
diffmodel.add_subprocess('diffusion', d)
print(diffmodel)

diffmodel.integrate_years(3)

plot_temp_section(diffmodel)

#  Plot the northward heat transport in this model
plt.figure()
Rtoa = np.squeeze(diffmodel.timeave['ASR'] - diffmodel.timeave['OLR'])
plt.plot(diffmodel.lat, inferred_heat_transport(Rtoa, diffmodel.lat))

5946637.413346613
climlab Process of type <class 'climlab.model.column.GreyRadiationModel'>. 
State variables and domain shapes: 
  Ts: (90, 1) 
  Tatm: (90, 30) 
The subprocess tree: 
Untitled: <class 'climlab.model.column.GreyRadiationModel'>
   LW: <class 'climlab.radiation.greygas.GreyGas'>
   SW: <class 'climlab.radiation.greygas.GreyGasSW'>
   insolation: <class 'climlab.radiation.insolation.AnnualMeanInsolation'>
   convective adjustment: <class 'climlab.convection.convadj.ConvectiveAdjustment'>
   H2O: <class 'climlab.radiation.water_vapor.ManabeWaterVapor'>
   SHF: <class 'climlab.surface.turbulent.SensibleHeatFlux'>
   LHF: <class 'climlab.surface.turbulent.LatentHeatFlux'>
   diffusion: <class 'climlab.dynamics.meridional_advection_diffusion.MeridionalDiffusion'>

Integrating for 1095 steps, 1095.7266 days, or 3 years.

Total elapsed time is 5.996021270269426 years.

/var/folders/z_/36cc8kqx7pqf8lzjc5cp2cf00000gn/T/ipykernel_29629/4142479116.py:6: DeprecationWarning: 'scipy.integrate.cumtrapz' is deprecated in favour of 'scipy.integrate.cumulative_trapezoid' and will be removed in SciPy 1.14.0
  return ( 1E-15 * 2 * np.math.pi * const.a**2 * integrate.cumtrapz( np.cos(lat_rad)*energy_in,

[<matplotlib.lines.Line2D at 0x14ba58150>]

##  Now warm it up!
co2diffmodel = climlab.process_like(diffmodel)
co2diffmodel.subprocess['LW'].absorptivity = diffmodel.subprocess['LW'].absorptivity*1.1

co2diffmodel.integrate_years(3)

#  With transport, get polar amplification...
#   of surface temperature, but not of air temperature!
plt.plot(diffmodel.lat, co2diffmodel.Ts - diffmodel.Ts, label='Ts')
plt.plot(diffmodel.lat, co2diffmodel.Tatm[:,0] - diffmodel.Tatm[:,0], label='Tatm')
plt.legend()


Rtoa = np.squeeze(diffmodel.timeave['ASR'] - diffmodel.timeave['OLR'])
plt.figure()
Rtoa_co2 = np.squeeze(co2diffmodel.timeave['ASR'] - co2diffmodel.timeave['OLR'])
plt.plot(diffmodel.lat, inferred_heat_transport(Rtoa, diffmodel.lat), label='1xCO2')
plt.plot(diffmodel.lat, inferred_heat_transport(Rtoa_co2, diffmodel.lat), label='2xCO2')

Integrating for 1095 steps, 1095.7266 days, or 3 years.

Total elapsed time is 8.994031905404139 years.

/var/folders/z_/36cc8kqx7pqf8lzjc5cp2cf00000gn/T/ipykernel_29629/4142479116.py:6: DeprecationWarning: 'scipy.integrate.cumtrapz' is deprecated in favour of 'scipy.integrate.cumulative_trapezoid' and will be removed in SciPy 1.14.0
  return ( 1E-15 * 2 * np.math.pi * const.a**2 * integrate.cumtrapz( np.cos(lat_rad)*energy_in,

[<matplotlib.lines.Line2D at 0x14bb4b3d0>]

We get polar amplification!!!

Exclude evaporation

diffmodel2 = climlab.process_like(diffmodel)
diffmodel2.remove_subprocess('LHF')
print(diffmodel2)

diffmodel2.integrate_years(3)

climlab Process of type <class 'climlab.model.column.GreyRadiationModel'>. 
State variables and domain shapes: 
  Ts: (90, 1) 
  Tatm: (90, 30) 
The subprocess tree: 
Untitled: <class 'climlab.model.column.GreyRadiationModel'>
   LW: <class 'climlab.radiation.greygas.GreyGas'>
   SW: <class 'climlab.radiation.greygas.GreyGasSW'>
   insolation: <class 'climlab.radiation.insolation.AnnualMeanInsolation'>
   convective adjustment: <class 'climlab.convection.convadj.ConvectiveAdjustment'>
   H2O: <class 'climlab.radiation.water_vapor.ManabeWaterVapor'>
   SHF: <class 'climlab.surface.turbulent.SensibleHeatFlux'>
   diffusion: <class 'climlab.dynamics.meridional_advection_diffusion.MeridionalDiffusion'>

Integrating for 1095 steps, 1095.7266 days, or 3 years.

Total elapsed time is 8.994031905404139 years.

co2diffmodel2 = climlab.process_like(co2diffmodel)
co2diffmodel2.remove_subprocess('LHF')
co2diffmodel2.integrate_years(3)

Integrating for 1095 steps, 1095.7266 days, or 3 years.

Total elapsed time is 11.992042540538852 years.

#  With transport and no evaporation...
#  No polar amplification, either of surface or air temperature!
plt.plot(diffmodel2.lat, co2diffmodel2.Ts - diffmodel2.Ts, label='Ts')
plt.plot(diffmodel2.lat, co2diffmodel2.Tatm[:,0] - diffmodel2.Tatm[:,0], label='Tatm')
plt.legend()


Rtoa = np.squeeze(diffmodel2.timeave['ASR'] - diffmodel2.timeave['OLR'])
Rtoa_co2 = np.squeeze(co2diffmodel2.timeave['ASR'] - co2diffmodel2.timeave['OLR'])
plt.figure()
plt.plot(diffmodel2.lat, inferred_heat_transport(Rtoa, diffmodel2.lat), label='1xCO2')
plt.plot(diffmodel2.lat, inferred_heat_transport(Rtoa_co2, diffmodel2.lat), label='2xCO2')

/var/folders/z_/36cc8kqx7pqf8lzjc5cp2cf00000gn/T/ipykernel_29629/4142479116.py:6: DeprecationWarning: 'scipy.integrate.cumtrapz' is deprecated in favour of 'scipy.integrate.cumulative_trapezoid' and will be removed in SciPy 1.14.0
  return ( 1E-15 * 2 * np.math.pi * const.a**2 * integrate.cumtrapz( np.cos(lat_rad)*energy_in,

[<matplotlib.lines.Line2D at 0x14ccb7c90>]

Excluding evaporation reduces polar amplification. Why?