6.1 Radiation Budget

The Earth's radiation budget describes the balance between incoming shortwave solar radiation and outgoing longwave terrestrial radiation at the top of the atmosphere. This balance is the fundamental driver of global climate, governing planetary temperature, atmospheric circulation, and the hydrological cycle. Understanding the radiation budget requires mastery of blackbody theory, radiative transfer, and the spectral properties of the atmosphere.

Blackbody Radiation Laws

All bodies with temperature above absolute zero emit electromagnetic radiation. A blackbody is an idealized object that absorbs all incident radiation and emits the maximum possible radiation at every wavelength. The spectral radiance of a blackbody is described by the Planck function, which gives the power emitted per unit area, per unit solid angle, per unit frequency:

$$B(\nu, T) = \frac{2h\nu^3}{c^2} \cdot \frac{1}{e^{h\nu / k_B T} - 1}$$

Planck function (frequency form), where $h = 6.626 \times 10^{-34}$ J s, $c = 3 \times 10^8$ m/s, $k_B = 1.381 \times 10^{-23}$ J/K

In wavelength form, the Planck function becomes:

$$B(\lambda, T) = \frac{2hc^2}{\lambda^5} \cdot \frac{1}{e^{hc / \lambda k_B T} - 1}$$

Planck function (wavelength form) in W m⁻² sr⁻¹ m⁻¹

Integrating the Planck function over all wavelengths and over the hemisphere yields the Stefan-Boltzmann law, which gives the total power radiated per unit area:

$$F = \sigma T^4 \quad \text{where} \quad \sigma = \frac{2\pi^5 k_B^4}{15 c^2 h^3} = 5.670 \times 10^{-8} \text{ W m}^{-2}\text{K}^{-4}$$

Stefan-Boltzmann law: total emitted flux scales as the fourth power of temperature

The wavelength at which the Planck function peaks is given by Wien's displacement law:

$$\lambda_{\text{max}} = \frac{b}{T} \quad \text{where} \quad b = 2897.8 \text{ } \mu\text{m K}$$

Wien's displacement law: hotter objects emit at shorter wavelengths

Sun (T = 5778 K)

Peak wavelength: $\lambda_{\max}$ = 0.50 $\mu$m (visible green)

Total flux: F = 6.32 x 10⁷ W/m²

Luminosity: L = 3.846 x 10²⁶ W

Earth (T = 255 K)

Peak wavelength: $\lambda_{\max}$ = 11.4 $\mu$m (thermal infrared)

Total flux: F = 240 W/m² (effective)

Emission overwhelmingly in the infrared

Solar Constant and Inverse Square Law

The Sun emits radiation isotropically from its photosphere at T = 5778 K. As this radiation propagates outward, it spreads over an ever-increasing spherical surface area. The flux at distance d from the Sun is given by the inverse square law:

$$S(d) = \frac{L_\odot}{4\pi d^2} = \sigma T_\odot^4 \left(\frac{R_\odot}{d}\right)^2$$

Solar flux decreases as the square of the distance from the Sun

At Earth's mean orbital distance (d = 1 AU = 1.496 x 10¹¹ m), the solar flux is the solar constant:

$$S_0 = 1361 \text{ W/m}^2$$

Total Solar Irradiance (TSI) measured by SORCE/TIM and TSIS-1 satellites

Due to Earth's slightly elliptical orbit (eccentricity e = 0.0167), the actual TSI varies seasonally between approximately 1321 W/m² at aphelion (July) and 1413 W/m² at perihelion (January), a variation of about 6.7%. The 11-year solar cycle introduces an additional variation of approximately 1 W/m² (0.07%).

1361 W/m²

Solar constant S₀

340.25 W/m²

Global mean (S₀/4)

~0.07%

Solar cycle variation

Earth's Global Energy Balance

Earth intercepts solar radiation over a cross-sectional area $\pi R_E^2$, but rotates so the average flux over the entire surface area $4\pi R_E^2$ is $S_0/4$. A fraction $\alpha$ (the planetary albedo) is reflected back to space without being absorbed. At radiative equilibrium, the absorbed solar radiation equals the outgoing longwave radiation (OLR):

$$\frac{S_0 (1 - \alpha)}{4} = \sigma T_e^4$$

Planetary energy balance: absorbed shortwave = emitted longwave

Solving for the effective emission temperature T_e:

$$T_e = \left[\frac{S_0(1-\alpha)}{4\sigma}\right]^{1/4} = \left[\frac{1361 \times 0.70}{4 \times 5.67 \times 10^{-8}}\right]^{1/4} \approx 255 \text{ K} \approx -18°\text{C}$$

Earth's effective emission temperature without greenhouse effect

The observed mean surface temperature is approximately 288 K (15 degrees C), which is 33 K warmer than T_e. This 33 K difference is the greenhouse effect. The planetary albedo $\alpha \approx 0.30$ arises from multiple sources:

Clouds

Contribution to $\alpha$: ~0.15

Thick low clouds (stratocumulus) are highly reflective, contributing roughly half of planetary albedo

Surface

Contribution to $\alpha$: ~0.10

Ice sheets, snow cover, deserts; varies strongly with latitude and season

Aerosols & Rayleigh

Contribution to $\alpha$: ~0.05

Sulfate aerosols, dust, and molecular Rayleigh scattering of shortwave radiation

Top-of-Atmosphere (TOA) Radiation Budget

At the top of the atmosphere, the net radiative flux is the difference between absorbed shortwave and outgoing longwave radiation. This net flux determines whether the Earth system is gaining or losing energy:

$$R_{\text{TOA}} = \frac{S_0}{4}(1-\alpha) - F_{\text{OLR}}$$

Net radiation at TOA: positive means Earth gains energy

Satellite missions such as CERES (Clouds and the Earth's Radiant Energy System) on the Aqua and Terra platforms provide global measurements of the TOA radiation budget. Current observations indicate a net positive imbalance of approximately 0.7-1.0 W/m², meaning Earth is currently accumulating energy, primarily in the ocean. The three components measured by CERES are:

340 W/m²

Incoming SW (S₀/4)

Varies with latitude: ~480 W/m² at equator, ~170 W/m² at poles (annual mean)

~100 W/m²

Reflected SW

Reflected by clouds, surface, aerosols; higher at poles due to ice

~240 W/m²

Outgoing LW (OLR)

Emitted by surface and atmosphere; reduced by greenhouse gases and clouds

Surface Radiation and Energy Budget

At the surface, the energy balance includes both radiative and non-radiative components. The surface gains energy from absorbed solar radiation and downwelling longwave radiation from the atmosphere, and loses energy through upwelling longwave emission, sensible heat flux, and latent heat flux (evaporation):

$$R_{\text{net,sfc}} = S_{\downarrow}(1-\alpha_s) + L_{\downarrow} - L_{\uparrow} - H - \lambda E$$

Surface energy balance: net SW + net LW = sensible + latent heat

Radiative Components

$S_{\downarrow}(1-\alpha_s)$: Net surface SW absorption (~185 W/m²)

$L_{\downarrow}$: Downwelling LW from atmosphere (~340 W/m²)

$L_{\uparrow} = \varepsilon \sigma T_s^4$: Surface LW emission (~398 W/m²)

Net surface LW: $L_{\downarrow} - L_{\uparrow}$ = -58 W/m²

Non-Radiative Components

H (Sensible heat): ~20 W/m² (turbulent transfer)

$\lambda E$ (Latent heat): ~84 W/m² (evaporation/transpiration)

Ground heat flux G: ~2 W/m² (conduction into soil/ocean)

Bowen ratio B = H/$\lambda E$ varies by surface type

Atmospheric Window and Absorption

The atmosphere is not uniformly opaque to radiation. The atmospheric window between 8 and 13 micrometers is a spectral region where the atmosphere is relatively transparent to terrestrial infrared radiation. Through this window, surface radiation can escape directly to space without being absorbed. However, this window is partially closed by ozone absorption near 9.6 micrometers and by the continuum absorption of water vapor.

The attenuation of radiation as it passes through an absorbing medium is described by the Beer-Lambert law:

$$I(z) = I_0 \, e^{-\tau(z)}$$

Beer-Lambert law: exponential attenuation with optical depth

The optical depth $\tau$ is the integral of the absorption coefficient along the path:

$$\tau(\nu, z) = \int_0^{z} \kappa_\nu(z') \, \rho(z') \, dz'$$

Optical depth: $\kappa_\nu$ is mass absorption coefficient (m²/kg), $\rho$ is absorber density (kg/m³)

Physical interpretation: When $\tau = 1$, the intensity has decreased to $1/e \approx 37\%$ of its original value. A medium with $\tau \gg 1$ is optically thick (opaque), while $\tau \ll 1$ is optically thin (transparent). The atmosphere is optically thick at most infrared wavelengths except the 8-13 $\mu$m window.

Schwarzschild's Equation and Radiative Transfer

The complete description of radiation propagating through an absorbing and emitting atmosphere is given by the Schwarzschild equation (also called the equation of radiative transfer). This equation accounts for both the attenuation of radiation by absorption and the addition of radiation by thermal emission:

$$\frac{dI_\nu}{d\tau_\nu} = I_\nu - B_\nu(T)$$

Schwarzschild equation: change in intensity = absorption - emission

The formal solution of this equation for upwelling radiation at the top of the atmosphere is:

$$I_\nu^{\uparrow}(0) = B_\nu(T_s) \, e^{-\tau_\nu^*} + \int_0^{\tau_\nu^*} B_\nu(T(\tau')) \, e^{-\tau'} \, d\tau'$$

Formal solution: surface emission attenuated + atmospheric emission contributions

The first term represents surface emission attenuated by atmospheric absorption. The second term is the integrated contribution from each atmospheric layer, each attenuated by the optical path above it. The weighting function $e^{-\tau'} d\tau'/dz$ peaks at the level where $\tau \approx 1$, defining the effective emission level for each frequency.

Two-Stream Approximation

For practical calculations, the full angular dependence of the radiation field is often simplified using the two-stream approximation, which divides the radiation into upward and downward streams. The equations become:

$$\frac{dF^{\uparrow}}{d\tau} = F^{\uparrow} - \pi B(T) \quad ; \quad \frac{dF^{\downarrow}}{d\tau} = -F^{\downarrow} + \pi B(T)$$

This approximation captures the essential physics while reducing the angular integral to two discrete directions. A diffusivity factor of D = 1.66 is commonly used to account for the non-vertical paths of diffuse radiation: $\tau^* = D \cdot \tau$. This approach is the foundation of most radiative transfer schemes in climate models.

Fortran + Python: TOA Radiation Budget with Visualization

Python

Compiles and runs a Fortran program (gfortran) that computes the zonal-mean radiation budget. Python parses the output and generates 4-panel matplotlib plots: SW/LW vs latitude, net radiation surplus/deficit, albedo/temperature profiles, and SW budget partition.

rad_budget_fortran_plot.py211 lines

import subprocess, os
import numpy as np
import matplotlib.pyplot as plt

# ---- Fortran source: TOA Radiation Budget vs Latitude ----
fortran_src = r"""program toa_radiation_budget
  implicit none
  integer, parameter :: nlat = 181
  real(8), parameter :: S0 = 1361.0d0
  real(8), parameter :: sigma = 5.670d-8
  real(8), parameter :: pi = 3.14159265d0
  real(8) :: lat(nlat), lat_rad, cos_lat
  real(8) :: S_in(nlat), S_ref(nlat), OLR(nlat), R_net(nlat)
  real(8) :: albedo(nlat), T_surf(nlat)
  real(8) :: global_Sin, global_Sref, global_OLR, global_Rnet, weight_sum
  integer :: i

do i = 1, nlat
    lat(i) = -90.0d0 + (i - 1) * 1.0d0
  end do

do i = 1, nlat
    lat_rad = lat(i) * pi / 180.0d0
    cos_lat = cos(lat_rad)
    S_in(i) = S0 / pi * cos_lat

if (abs(lat(i)) > 60.0d0) then
      albedo(i) = 0.60d0
    else if (abs(lat(i)) > 30.0d0) then
      albedo(i) = 0.30d0
    else
      albedo(i) = 0.22d0
    end if

S_ref(i) = S_in(i) * albedo(i)
    T_surf(i) = 300.0d0 - 50.0d0 * (lat_rad / (pi/2.0d0))**2
    OLR(i) = 0.62d0 * sigma * T_surf(i)**4
    R_net(i) = S_in(i) * (1.0d0 - albedo(i)) - OLR(i)
  end do

! Global means (area-weighted)
  global_Sin = 0.0d0; global_Sref = 0.0d0
  global_OLR = 0.0d0; global_Rnet = 0.0d0; weight_sum = 0.0d0
  do i = 1, nlat
    lat_rad = lat(i) * pi / 180.0d0
    cos_lat = cos(lat_rad)
    global_Sin  = global_Sin  + S_in(i)  * cos_lat
    global_Sref = global_Sref + S_ref(i) * cos_lat
    global_OLR  = global_OLR  + OLR(i)   * cos_lat
    global_Rnet = global_Rnet + R_net(i)  * cos_lat
    weight_sum  = weight_sum  + cos_lat
  end do
  global_Sin  = global_Sin  / weight_sum
  global_Sref = global_Sref / weight_sum
  global_OLR  = global_OLR  / weight_sum
  global_Rnet = global_Rnet / weight_sum

write(*,'(A,F8.2)') 'GLOBAL_SIN ', global_Sin
  write(*,'(A,F8.2)') 'GLOBAL_SREF ', global_Sref
  write(*,'(A,F8.2)') 'GLOBAL_OLR ', global_OLR
  write(*,'(A,F8.2)') 'GLOBAL_RNET ', global_Rnet

! Output full profile as CSV
  write(*,'(A)') 'DATA_START'
  do i = 1, nlat
    write(*,'(F7.1,A,F8.2,A,F8.2,A,F8.2,A,F8.2,A,F5.2,A,F7.1)') &
      lat(i), ',', S_in(i), ',', S_ref(i), ',', OLR(i), ',', &
      R_net(i), ',', albedo(i), ',', T_surf(i)
  end do
  write(*,'(A)') 'DATA_END'

end program toa_radiation_budget
"""

# ---- Step 1: Compile and run Fortran ----
with open('rad_budget.f90', 'w') as f:
    f.write(fortran_src)

comp = subprocess.run(['gfortran', '-o', './rad_budget', 'rad_budget.f90'],
                      capture_output=True, text=True)
if comp.returncode != 0:
    print("COMPILATION ERROR:"); print(comp.stderr)
    raise SystemExit(1)
print("Fortran compiled successfully with gfortran")

result = subprocess.run(['./rad_budget'], capture_output=True, text=True, timeout=30)
output = result.stdout
print("Fortran execution complete\n")

# ---- Step 2: Parse Fortran output ----
globals_dict = {}
data_rows = []
in_data = False

for line in output.split('\n'):
    line = line.strip()
    if line.startswith('GLOBAL_'):
        key, val = line.split()
        globals_dict[key] = float(val)
    elif line == 'DATA_START':
        in_data = True
    elif line == 'DATA_END':
        in_data = False
    elif in_data and ',' in line:
        parts = line.split(',')
        if len(parts) == 7:
            data_rows.append([float(x) for x in parts])

data = np.array(data_rows)
lat    = data[:, 0]
S_in   = data[:, 1]
S_ref  = data[:, 2]
OLR    = data[:, 3]
R_net  = data[:, 4]
albedo = data[:, 5]
T_surf = data[:, 6]
abs_sw = S_in - S_ref

# ---- Step 3: Print summary ----
print("=" * 56)
print("  TOA Radiation Budget (Fortran computation)")
print("=" * 56)
print(f"  Global mean incoming SW:   {globals_dict.get('GLOBAL_SIN', 0):8.2f} W/m^2")
print(f"  Global mean reflected SW:  {globals_dict.get('GLOBAL_SREF', 0):8.2f} W/m^2")
print(f"  Global mean absorbed SW:   {globals_dict.get('GLOBAL_SIN', 0) - globals_dict.get('GLOBAL_SREF', 0):8.2f} W/m^2")
print(f"  Global mean OLR:           {globals_dict.get('GLOBAL_OLR', 0):8.2f} W/m^2")
print(f"  Global mean net radiation: {globals_dict.get('GLOBAL_RNET', 0):8.2f} W/m^2")
print()
print("  Lat     S_in    Abs_SW    OLR    R_net   Albedo   T_surf")
print("  (deg)  (W/m2)  (W/m2)   (W/m2)  (W/m2)    (-)     (K)")
print("-" * 60)
for i in range(0, len(lat), 20):
    print(f"  {lat[i]:5.0f}  {S_in[i]:7.1f} {abs_sw[i]:7.1f}  {OLR[i]:7.1f} {R_net[i]:7.1f}   {albedo[i]:5.2f}  {T_surf[i]:7.1f}")

# ---- Step 4: Plot with matplotlib ----
fig, axes = plt.subplots(2, 2, figsize=(14, 10))
fig.patch.set_facecolor('#0f172a')
for ax in axes.flat:
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#cbd5e1')
    for spine in ax.spines.values():
        spine.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.4)

# Panel 1: Incoming, Absorbed, OLR vs latitude
ax = axes[0, 0]
ax.plot(lat, S_in, color='#fbbf24', lw=2.5, label='Incoming SW')
ax.plot(lat, abs_sw, color='#f97316', lw=2.5, label='Absorbed SW')
ax.plot(lat, OLR, color='#f87171', lw=2.5, label='OLR')
ax.fill_between(lat, abs_sw, OLR, where=(abs_sw > OLR),
                alpha=0.2, color='#22d3ee', label='Energy surplus')
ax.fill_between(lat, abs_sw, OLR, where=(abs_sw < OLR),
                alpha=0.2, color='#a78bfa', label='Energy deficit')
ax.axhline(y=0, color='#475569', lw=0.5)
ax.set_xlabel('Latitude (°)', color='#cbd5e1', fontsize=11)
ax.set_ylabel('Flux (W/m²)', color='#cbd5e1', fontsize=11)
ax.set_title('TOA Radiation Budget vs Latitude', color='#cbd5e1', fontsize=12)
ax.legend(fontsize=8, facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

# Panel 2: Net radiation
ax = axes[0, 1]
ax.fill_between(lat, R_net, 0, where=(R_net >= 0),
                alpha=0.6, color='#f97316', label='Net surplus (tropics)')
ax.fill_between(lat, R_net, 0, where=(R_net < 0),
                alpha=0.6, color='#60a5fa', label='Net deficit (poles)')
ax.plot(lat, R_net, color='#cbd5e1', lw=2)
ax.axhline(y=0, color='#94a3b8', lw=1, ls='--')
cross = lat[np.where(np.diff(np.sign(R_net)))[0]]
for c in cross:
    ax.axvline(x=c, color='#22d3ee', lw=1, ls=':', alpha=0.7)
    ax.text(c+2, max(R_net)*0.8, f'{c:.0f}°', color='#22d3ee', fontsize=9)
ax.set_xlabel('Latitude (°)', color='#cbd5e1', fontsize=11)
ax.set_ylabel('Net Flux (W/m²)', color='#cbd5e1', fontsize=11)
ax.set_title('Net TOA Radiation (drives circulation)', color='#cbd5e1', fontsize=12)
ax.legend(fontsize=9, facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

# Panel 3: Albedo and surface temperature
ax = axes[1, 0]
color1 = '#22d3ee'
ax.plot(lat, albedo, color=color1, lw=2.5, label='Albedo')
ax.set_ylabel('Albedo', color=color1, fontsize=11)
ax.set_ylim(0, 0.8)
ax.tick_params(axis='y', colors=color1)
ax2 = ax.twinx()
color2 = '#f87171'
ax2.plot(lat, T_surf, color=color2, lw=2.5, label='T_surf')
ax2.set_ylabel('Surface Temperature (K)', color=color2, fontsize=11)
ax2.tick_params(axis='y', colors=color2)
ax2.spines['right'].set_color(color2)
ax2.spines['left'].set_color(color1)
ax.set_xlabel('Latitude (°)', color='#cbd5e1', fontsize=11)
ax.set_title('Albedo & Surface Temperature', color='#cbd5e1', fontsize=12)
lines1, labels1 = ax.get_legend_handles_labels()
lines2, labels2 = ax2.get_legend_handles_labels()
ax.legend(lines1+lines2, labels1+labels2, fontsize=9, facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

# Panel 4: Reflected SW breakdown
ax = axes[1, 1]
ax.stackplot(lat, S_ref, abs_sw,
             colors=['#60a5fa', '#f97316'], alpha=0.8,
             labels=['Reflected SW', 'Absorbed SW'])
ax.plot(lat, S_in, color='#fbbf24', lw=2, ls='--', label='Total incoming')
ax.set_xlabel('Latitude (°)', color='#cbd5e1', fontsize=11)
ax.set_ylabel('Flux (W/m²)', color='#cbd5e1', fontsize=11)
ax.set_title('SW Budget Partition', color='#cbd5e1', fontsize=12)
ax.legend(fontsize=9, facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor=fig.get_facecolor())
print("\nPlots generated from Fortran radiation budget output.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Interactive Simulation: Earth's Energy Budget

Python

Visualize Earth's global radiation budget with stacked bar charts showing incoming solar (341 W/m2), reflected shortwave, absorbed surface and atmosphere, longwave emission, and the greenhouse effect. Prints the full budget balance at TOA and surface.

earth_energy_budget.py91 lines

import numpy as np
import matplotlib.pyplot as plt

# Earth's Radiation Budget Components (W/m^2)
# Based on Trenberth, Fasullo & Kiehl (2009) updated values
components = {
    'Incoming Solar':       {'value': 341, 'type': 'SW_in'},
    'Reflected by\nClouds': {'value': 79,  'type': 'SW_out'},
    'Reflected by\nSurface':{'value': 23,  'type': 'SW_out'},
    'Absorbed by\nAtmosphere': {'value': 79, 'type': 'SW_abs'},
    'Absorbed by\nSurface':    {'value': 161, 'type': 'SW_abs'},
    'Surface LW\nEmission':    {'value': 396, 'type': 'LW_up'},
    'Atmospheric\nWindow':     {'value': 40,  'type': 'LW_out'},
    'Back Radiation':           {'value': 340, 'type': 'LW_down'},
    'OLR from\nAtmosphere':    {'value': 199, 'type': 'LW_out'},
    'Sensible Heat':            {'value': 20,  'type': 'non_rad'},
    'Latent Heat':              {'value': 84,  'type': 'non_rad'},
}

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6))
fig.patch.set_facecolor('#0f172a')
for ax in [ax1, ax2]:
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#cbd5e1')
    for spine in ax.spines.values():
        spine.set_color('#334155')

# Panel 1: Stacked bar chart of radiation budget
categories = ['Incoming\nSolar', 'Reflected\nSW', 'Absorbed\nSW', 'Surface\nLW Up',
              'Back\nRadiation', 'OLR to\nSpace', 'Sensible\n+ Latent']
values = [341, 79+23, 79+161, 396, 340, 199+40, 20+84]
bar_colors = ['#fbbf24', '#60a5fa', '#f97316', '#f87171', '#a78bfa', '#22d3ee', '#2dd4bf']

bars = ax1.bar(categories, values, color=bar_colors, edgecolor='#334155', width=0.7)
for bar, val in zip(bars, values):
    ax1.text(bar.get_x() + bar.get_width()/2, bar.get_height() + 5,
             f'{val}', ha='center', va='bottom', color='#cbd5e1', fontsize=10, fontweight='bold')

ax1.set_ylabel('Energy Flux (W/m\u00b2)', color='#cbd5e1', fontsize=12)
ax1.set_title("Earth's Radiation Budget Components", color='#cbd5e1', fontsize=13)
ax1.grid(True, color='#334155', alpha=0.4, axis='y')

# Panel 2: TOA and Surface balance comparison
balance_labels = ['TOA\nAbsorbed SW', 'TOA\nOLR', 'TOA\nNet',
                  'Surface\nNet SW', 'Surface\nNet LW', 'Surface\nNon-Rad', 'Surface\nNet']
toa_abs_sw = 341 - 102  # 239
toa_olr = 239           # 239
toa_net = toa_abs_sw - toa_olr  # ~0 (actually +0.9)
sfc_net_sw = 161
sfc_net_lw = 340 - 396  # -56
sfc_non_rad = -(20 + 84)  # -104
sfc_net = sfc_net_sw + sfc_net_lw + sfc_non_rad  # ~1

balance_vals = [toa_abs_sw, -toa_olr, 0.9, sfc_net_sw, sfc_net_lw, sfc_non_rad, 1.0]
balance_colors = ['#fbbf24', '#f87171', '#2dd4bf', '#f97316', '#a78bfa', '#22d3ee', '#2dd4bf']

bars2 = ax2.bar(balance_labels, balance_vals, color=balance_colors, edgecolor='#334155', width=0.7)
for bar, val in zip(bars2, balance_vals):
    y = bar.get_height()
    offset = 8 if y >= 0 else -15
    ax2.text(bar.get_x() + bar.get_width()/2, y + offset,
             f'{val:+.0f}', ha='center', va='bottom', color='#cbd5e1', fontsize=9, fontweight='bold')

ax2.axhline(y=0, color='#94a3b8', lw=1)
ax2.set_ylabel('Energy Flux (W/m\u00b2)', color='#cbd5e1', fontsize=12)
ax2.set_title('Energy Balance: TOA and Surface', color='#cbd5e1', fontsize=13)
ax2.grid(True, color='#334155', alpha=0.4, axis='y')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor=fig.get_facecolor())

print("Earth's Radiation Budget Summary")
print("=" * 50)
print("TOA Budget:")
print(f"  Incoming solar (S0/4):       341 W/m^2")
print(f"  Reflected SW (clouds+sfc):   102 W/m^2")
print(f"  Absorbed SW:                 239 W/m^2")
print(f"  Outgoing LW (OLR):           239 W/m^2")
print(f"  Net TOA imbalance:           +0.9 W/m^2")
print()
print("Surface Budget:")
print(f"  Absorbed SW:                 161 W/m^2")
print(f"  Back radiation (LW down):    340 W/m^2")
print(f"  Surface LW emission:         396 W/m^2")
print(f"  Sensible heat:                20 W/m^2")
print(f"  Latent heat (evaporation):    84 W/m^2")
print(f"  Net surface balance:         +1 W/m^2")
print()
print("The +0.9 W/m^2 TOA imbalance drives ongoing warming.")
print("Greenhouse effect: 396 - 239 = 157 W/m^2 trapped by atmosphere.")

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Code will be executed with Python 3 on the server

← Part 6 6.2 Greenhouse Effect →