8.1 Ocean-Atmosphere Interaction

The ocean and atmosphere exchange momentum, heat, and freshwater through their shared interface. These fluxes drive ocean circulation, modulate global climate, power tropical cyclones, and control the planetary energy budget. Bulk aerodynamic formulas provide the essential parameterizations used in all coupled climate models.

Momentum Flux: Wind Stress

The atmosphere transfers momentum to the ocean through wind stress, the tangential force per unit area exerted on the sea surface. The bulk aerodynamic formula is:

$$\vec{\tau} = \rho_a C_D |\vec{U}_{10}| \vec{U}_{10}$$

where $\rho_a \approx 1.22$ kg/m³ is air density, $C_D \approx (0.6 + 0.07|U_{10}|) \times 10^{-3}$ is the drag coefficient

The drag coefficient $C_D$ increases with wind speed due to increasing sea surface roughness (wave generation). At hurricane wind speeds (>30 m/s), C_D saturates or decreases due to spray effects. Wind stress drives Ekman transport, upwelling/downwelling, and the ocean gyre circulation.

Typical Values

Trade winds: τ ≈ 0.05–0.1 N/m². Westerlies: τ ≈ 0.1–0.3 N/m². Storm conditions: τ > 1 N/m².

Wave-Dependent Drag

Young wind seas have higher drag than old swell. The Charnock parameter $\alpha_c = gz_0/u_*^2$ characterizes surface roughness.

Surface Heat Fluxes

The net heat flux across the air-sea interface is the sum of four components:

$$Q_{\text{net}} = Q_{\text{sw}} + Q_{\text{lw}} + Q_{\text{latent}} + Q_{\text{sensible}}$$

Shortwave Radiation $Q_{\text{sw}}$ (warming, +170 W/m² global avg)

Solar radiation absorbed by the upper ocean. Reduced by cloud cover (albedo) and high solar zenith angles. Penetrates to ~100 m in clear water. Diurnal and seasonal cycles dominate. Maximum in tropics (~240 W/m²).

Net Longwave Radiation $Q_{\text{lw}}$ (cooling, −50 W/m² avg)

Ocean emits infrared radiation proportional to $\sigma T_s^4$ (Stefan-Boltzmann). Partially offset by atmospheric back-radiation. Always a net loss from the ocean. Reduced under cloudy skies (clouds radiate back).

Latent Heat Flux $Q_{\text{latent}}$ (cooling, −80 W/m² avg)

Energy lost through evaporation. Largest heat loss term. Bulk formula:

$Q_E = \rho_a L_v C_E U_{10}(q_s - q_a)$

where $L_v \approx 2.5 \times 10^6$ J/kg, $C_E \approx 1.2 \times 10^{-3}$, and $q_s, q_a$ are specific humidities at the surface and reference height.

Sensible Heat Flux $Q_{\text{sensible}}$ (cooling, −10 W/m² avg)

Direct heat conduction/convection between ocean and atmosphere:

$Q_H = \rho_a c_{pa} C_H U_{10}(T_s - T_a)$

Usually small except during cold air outbreaks (Gulf Stream in winter: $Q_H > 200$ W/m²).

Sea Surface Temperature Budget

The mixed layer temperature evolution is governed by the SST budget equation:

$$\rho c_p h_m \frac{\partial T_s}{\partial t} = Q_{\text{net}} - \rho c_p h_m (\vec{u} \cdot \nabla T_s) + \text{entrainment} + \text{diffusion}$$

$h_m$ is the mixed layer depth, which itself varies seasonally

In the tropics, the seasonal SST cycle is small (~2°C) because solar heating is nearly constant. At mid-latitudes, SST varies by 10–15°C seasonally, lagging air temperature by 1–2 months due to the ocean's thermal inertia.

Diurnal SST Cycle

Under light winds and strong insolation, a warm diurnal layer forms in the upper 1–5 m with amplitude 1–3°C. At high winds, mixing distributes heat deeper and the diurnal signal is negligible. Satellite SST measurements must account for this skin-bulk temperature difference.

Entrainment

Deepening of the mixed layer by wind stirring and convective overturning entrains cold water from below the thermocline. Entrainment flux $Q_e = -\rho c_p w_e \Delta T$ can be the dominant cooling term in regions of active mixed layer deepening, especially in autumn at mid-latitudes.

Freshwater Flux (E − P)

The net freshwater flux at the ocean surface is the difference between evaporation and precipitation:

$$\text{FW flux} = E - P - R$$

where R includes river runoff and ice melt

E > P Regions

Subtropics (dry): salinity maximum. Evaporation driven by trade winds and descending air. Red Sea, Mediterranean.

P > E Regions

Tropics (ITCZ) and high latitudes: freshening. Low salinity surface waters. Important for deep water formation inhibition.

Tropical Cyclone–Ocean Interaction

Tropical cyclones (hurricanes/typhoons) extract enormous energy from warm ocean water through latent heat flux. The SST threshold for genesis is approximately 26.5°C. The ocean responds with a cold wake:

$$\Delta T_{\text{wake}} \sim -1 \text{ to } -5\text{°C}$$

The cold wake forms from wind-driven mixing and upwelling (Ekman pumping). It is rightward-biased in the Northern Hemisphere due to inertial current asymmetry. This negative SST feedback can limit cyclone intensity, creating a critical ocean-atmosphere coupling.

Python: Air-Sea Fluxes, SST Budget & Seasonal Cycle

Python

!/usr/bin/env python3

script.py114 lines

#!/usr/bin/env python3
"""ocean_atmosphere.py - Air-sea fluxes, SST budget, seasonal cycle"""
import numpy as np
import matplotlib.pyplot as plt

rho_a = 1.22     # air density (kg/m^3)
rho_w = 1025.0   # water density (kg/m^3)
cp_a = 1005.0    # specific heat of air (J/kg/K)
cp_w = 3990.0    # specific heat of seawater (J/kg/K)
Lv = 2.5e6       # latent heat of vaporization (J/kg)
sigma = 5.67e-8  # Stefan-Boltzmann constant

# Bulk transfer coefficients
CD = 1.3e-3; CH = 1.2e-3; CE = 1.2e-3

def wind_stress(U10):
    """Wind stress magnitude (N/m^2)."""
    return rho_a * CD * U10**2

def sensible_heat(U10, Ts, Ta):
    """Sensible heat flux (W/m^2), positive = ocean heating."""
    return -rho_a * cp_a * CH * U10 * (Ts - Ta)

def latent_heat(U10, qs, qa):
    """Latent heat flux (W/m^2), positive = ocean heating."""
    return -rho_a * Lv * CE * U10 * (qs - qa)

def saturation_q(T, P=101325):
    """Saturation specific humidity at temperature T (deg C)."""
    es = 611.2 * np.exp(17.67 * T / (T + 243.5))
    return 0.622 * es / (P - 0.378 * es)

# --- 1. Full heat budget computation ---
fig, axes = plt.subplots(2, 2, figsize=(14, 10))

# Annual cycle at 40N (mid-latitude)
months = np.arange(1, 13)
Ts_40N = np.array([8, 7, 8, 10, 13, 17, 21, 23, 21, 17, 13, 10])
Ta_40N = np.array([5, 4, 6, 10, 14, 18, 22, 22, 19, 14, 9, 6])
U10_40N = np.array([10, 10, 9, 8, 7, 6, 6, 6, 7, 8, 9, 10])
RH_40N = np.array([0.75]*12)

Qsw = np.array([80, 120, 180, 230, 270, 290, 280, 240, 190, 130, 80, 60])
Qlw = -50 * np.ones(12)

QH = np.array([sensible_heat(U, Ts, Ta) for U, Ts, Ta
               in zip(U10_40N, Ts_40N, Ta_40N)])
qs_arr = saturation_q(Ts_40N)
qa_arr = RH_40N * saturation_q(Ta_40N)
QE = np.array([latent_heat(U, qs, qa) for U, qs, qa
               in zip(U10_40N, qs_arr, qa_arr)])
Qnet = Qsw + Qlw + QH + QE

ax1 = axes[0, 0]
ax1.plot(months, Qsw, 'y-o', label='SW (solar)')
ax1.plot(months, Qlw, 'b-s', label='LW (net)')
ax1.plot(months, QH, 'g-^', label='Sensible')
ax1.plot(months, QE, 'c-v', label='Latent')
ax1.plot(months, Qnet, 'r-D', lw=2, label='Net')
ax1.axhline(0, color='white', ls=':', alpha=0.3)
ax1.set_xlabel('Month'); ax1.set_ylabel('Heat flux (W/m^2)')
ax1.set_title('Seasonal Heat Budget at 40N')
ax1.legend(fontsize=7); ax1.grid(True, alpha=0.3)

# --- 2. SST evolution model (1D mixed layer) ---
ax2 = axes[0, 1]
dt = 86400.0  # 1 day
hm = 50.0     # mixed layer depth (m)
T = 10.0      # initial SST
T_series = []

for day in range(365*3):
    month_idx = int((day % 365) / 30.4) % 12
    Qn = Qnet[month_idx]
    dTdt = Qn / (rho_w * cp_w * hm)
    T += dTdt * dt
    T_series.append(T)

days = np.arange(len(T_series)) / 365
ax2.plot(days, T_series, 'r-', lw=1.5)
ax2.set_xlabel('Time (years)'); ax2.set_ylabel('SST (C)')
ax2.set_title('Mixed Layer Temperature Evolution')
ax2.grid(True, alpha=0.3)

# --- 3. Wind stress map (zonal) ---
ax3 = axes[1, 0]
lat = np.linspace(-90, 90, 180)
# Idealized zonal wind profile
U_zonal = 8*np.cos(3*np.radians(lat)) - 3*np.sin(np.radians(lat))
tau_x = rho_a * CD * np.abs(U_zonal) * U_zonal

ax3.plot(lat, tau_x, 'orange', lw=2)
ax3.fill_between(lat, tau_x, alpha=0.3, color='orange')
ax3.axhline(0, color='white', ls=':', alpha=0.3)
ax3.set_xlabel('Latitude'); ax3.set_ylabel('Zonal wind stress (N/m^2)')
ax3.set_title('Zonal Mean Wind Stress')
ax3.grid(True, alpha=0.3)

# --- 4. Latent heat vs wind and dq ---
ax4 = axes[1, 1]
U_range = np.linspace(1, 25, 50)
dq_range = np.linspace(0.001, 0.010, 50)
U_grid, DQ_grid = np.meshgrid(U_range, dq_range)
QE_grid = rho_a * Lv * CE * U_grid * DQ_grid

cf = ax4.contourf(U_grid, DQ_grid*1000, QE_grid, levels=20, cmap='YlOrRd')
plt.colorbar(cf, ax=ax4, label='Latent heat flux (W/m^2)')
ax4.set_xlabel('Wind speed (m/s)')
ax4.set_ylabel('q_s - q_a (g/kg)')
ax4.set_title('Latent Heat Flux Sensitivity')

plt.tight_layout()
plt.savefig('ocean_atmosphere.png', dpi=150)
plt.show()

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: COARE Bulk Air-Sea Flux Algorithm

Fortran

Simplified COARE bulk algorithm for air-sea flux computation

program.f9088 lines

program coare_bulk_fluxes
  ! Simplified COARE bulk algorithm for air-sea flux computation
  ! from meteorological observations
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  real(dp), parameter :: g = 9.81_dp, vonk = 0.4_dp
  real(dp), parameter :: Rv = 461.5_dp, Rd = 287.05_dp

real(dp) :: U10, Ta, Ts, RH, Pa, Rl_down
  real(dp) :: rho_a, cp_a, Lv, sigma_sb
  real(dp) :: CD, CH, CE, u_star, tau
  real(dp) :: es_sea, es_air, qa, qs, dq
  real(dp) :: QH, QE, Ql_net, Qs_net, Q_net
  real(dp) :: z0, z0t, z0q, psi_m, psi_h, L_MO
  integer :: iter

! Input meteorological data
  U10 = 8.0_dp          ! wind speed at 10 m (m/s)
  Ta = 25.0_dp           ! air temperature at 10 m (C)
  Ts = 28.0_dp           ! sea surface temperature (C)
  RH = 0.75_dp           ! relative humidity (fraction)
  Pa = 101325.0_dp       ! atmospheric pressure (Pa)
  Rl_down = 370.0_dp     ! downwelling longwave (W/m^2)

! Physical constants
  rho_a = Pa / (Rd * (Ta + 273.15_dp))
  cp_a = 1005.0_dp
  Lv = (2.501_dp - 0.00237_dp * Ts) * 1.0e6_dp
  sigma_sb = 5.67e-8_dp

! Saturation specific humidity (Clausius-Clapeyron)
  es_sea = 611.2_dp * exp(17.67_dp * Ts / (Ts + 243.5_dp))
  es_air = 611.2_dp * exp(17.67_dp * Ta / (Ta + 243.5_dp))
  qs = 0.622_dp * es_sea / (Pa - 0.378_dp * es_sea)
  qa = 0.622_dp * RH * es_air / (Pa - 0.378_dp * RH * es_air)
  dq = qs - qa

! Initial neutral transfer coefficients
  CD = 1.1e-3_dp + 0.07e-3_dp * U10
  CH = 1.1e-3_dp
  CE = 1.15e-3_dp

! Iterative stability correction (Monin-Obukhov)
  do iter = 1, 10
    u_star = sqrt(CD) * U10
    tau = rho_a * CD * U10**2

! Fluxes
    QH = -rho_a * cp_a * CH * U10 * (Ts - Ta)
    QE = -rho_a * Lv * CE * U10 * dq

! Monin-Obukhov length
    L_MO = -rho_a * cp_a * (Ta + 273.15_dp) * u_star**3 / &
           (vonk * g * (QH + 0.61_dp * cp_a * (Ta+273.15_dp) * QE / Lv))
    if (abs(L_MO) < 1.0_dp) L_MO = sign(1.0_dp, L_MO)

! Roughness lengths
    z0 = 0.011_dp * u_star**2 / g + 0.11_dp * 1.5e-5_dp / u_star
    z0t = min(1.1e-4_dp, 5.5e-5_dp / (u_star * z0)**0.6_dp)
    z0q = z0t

! Update coefficients with stability
    psi_m = -5.0_dp * min(10.0_dp / L_MO, 1.0_dp)
    psi_h = psi_m
    CD = (vonk / (log(10.0_dp/z0) - psi_m))**2
    CH = vonk**2 / ((log(10.0_dp/z0)-psi_m)*(log(10.0_dp/z0t)-psi_h))
    CE = vonk**2 / ((log(10.0_dp/z0)-psi_m)*(log(10.0_dp/z0q)-psi_h))
  end do

! Radiation fluxes
  Qs_net = 200.0_dp * (1.0_dp - 0.06_dp)  ! net shortwave (assumed)
  Ql_net = Rl_down - sigma_sb * (Ts + 273.15_dp)**4

Q_net = Qs_net + Ql_net + QH + QE

print *, '=== COARE Bulk Air-Sea Flux Results ==='
  print '(A,F8.2,A)', ' Wind stress:       ', tau, ' N/m^2'
  print '(A,F8.2,A)', ' Friction velocity: ', u_star, ' m/s'
  print '(A,F8.2,A)', ' Sensible heat:     ', QH, ' W/m^2'
  print '(A,F8.2,A)', ' Latent heat:       ', QE, ' W/m^2'
  print '(A,F8.2,A)', ' Net shortwave:     ', Qs_net, ' W/m^2'
  print '(A,F8.2,A)', ' Net longwave:      ', Ql_net, ' W/m^2'
  print '(A,F8.2,A)', ' NET heat flux:     ', Q_net, ' W/m^2'
  print '(A,E12.4)',  ' CD = ', CD
  print '(A,E12.4)',  ' CH = ', CH
  print '(A,E12.4)',  ' CE = ', CE

end program coare_bulk_fluxes

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

Monin-Obukhov Similarity Theory

The atmospheric surface layer above the ocean is described by Monin-Obukhov similarity theory, which relates mean profiles of wind, temperature, and humidity to surface fluxes. The key scaling parameter is the Monin-Obukhov length:

$$L = -\frac{\rho c_p T_v u_*^3}{\kappa g \left(Q_H + 0.61 c_p T_v \frac{Q_E}{L_v}\right)}$$

$u_* = \sqrt{|\tau|/\rho_a}$ is friction velocity, $T_v$ is virtual temperature, κ = 0.4 is von Kármán constant

The stability parameter $\zeta = z/L$ determines the form of the universal functions $\Psi_m(\zeta)$ and $\Psi_h(\zeta)$ that modify the logarithmic wind and temperature profiles:

$$U(z) = \frac{u_*}{\kappa}\left[\ln\frac{z}{z_0} - \Psi_m\left(\frac{z}{L}\right)\right]$$

ζ < 0 (Unstable)

SST warmer than air. Enhanced mixing and turbulence. Increased drag and heat transfer. Common in cold air outbreaks and trade wind regions.

ζ ≈ 0 (Neutral)

Near-neutral stability. Logarithmic profiles. Bulk coefficients close to neutral values. Common at moderate wind speeds.

ζ > 0 (Stable)

Air warmer than SST. Suppressed turbulence and reduced fluxes. Common in fog and advection cooling scenarios. Low-level jets can form.

Air-Sea Gas Exchange

The ocean and atmosphere exchange gases (CO', O', DMS, N'O) through the air-sea interface. The gas flux is parameterized as:

$$F = k_w (C_w - C_a/K_H)$$

where $k_w$ is the gas transfer velocity, $C_w$ is dissolved concentration, $K_H$ is Henry's law constant

The gas transfer velocity depends primarily on wind speed, with enhancement from bubble injection at high winds:

$$k_w = 0.251 \cdot U_{10}^2 \cdot \left(\frac{Sc}{660}\right)^{-1/2} \quad \text{(Wanninkhof 2014)}$$

Sc is the Schmidt number (ratio of kinematic viscosity to gas diffusivity)

The ocean absorbs approximately 2.5 GtC/yr of anthropogenic CO', representing about 25% of total emissions. This uptake is modulated by wind speed, SST, and the ocean's carbonate chemistry (buffer capacity).

Key Concepts Summary

Heat Budget

$Q_{\text{net}} = Q_{\text{sw}} + Q_{\text{lw}} + Q_{\text{latent}} + Q_{\text{sensible}}$. Latent heat (evaporation) is the dominant cooling term (~80 W/m² global average).

Bulk Formulas

All fluxes parameterized as product of transfer coefficient, wind speed, and air-sea difference. COARE algorithm standard for climate models.

Freshwater Flux

E − P controls surface salinity. Subtropical excess evaporation; ITCZ and high-latitude excess precipitation. Drives thermohaline circulation.

Tropical Cyclones

Extract energy from warm ocean (>26.5°C SST). Generate cold wake via mixing and upwelling. Negative SST feedback limits intensity.

← Part 8 8.2 El Niño & La Niña →