Module 2

Ocean Circulation & Thermohaline Dynamics

Wind-driven gyres, deep-water formation, and the ocean's role as Earth's heat engine

2.1 Ekman Transport: Wind-Driven Surface Flow

When wind blows over the ocean surface, friction transfers momentum downward. In a rotating frame, the balance between Coriolis force and turbulent friction produces the Ekman spiral: the surface current is deflected 45\(°\) from the wind direction (to the right in the Northern Hemisphere), with each deeper layer deflected further and weaker.

Derivation of the Ekman Spiral

In steady state, the horizontal momentum equations with vertical eddy viscosity \(A_z\):

\( -fv = A_z \frac{\partial^2 u}{\partial z^2} \)

\( fu = A_z \frac{\partial^2 v}{\partial z^2} \)

Define the complex velocity \(W = u + iv\). Adding \(i\) times the second equation to the first:

\( A_z \frac{\partial^2 W}{\partial z^2} = ifW \)

The solution with boundary conditions (wind stress \(\tau_0\) at surface, \(W \to 0\) as\(z \to -\infty\)):

\( W(z) = V_0\,e^{(1+i)z/D_E} \)

\( D_E = \sqrt{\frac{2A_z}{|f|}} \quad \text{(Ekman depth)} \)

\( V_0 = \frac{\tau_0}{\rho \sqrt{A_z |f|}} \cdot e^{-i\pi/4} \)

In component form:

\( u(z) = V_0\,e^{z/D_E}\cos\!\left(\frac{\pi}{4} + \frac{z}{D_E}\right) \)

\( v(z) = V_0\,e^{z/D_E}\sin\!\left(\frac{\pi}{4} + \frac{z}{D_E}\right) \)

Net Ekman Transport

Integrating over the Ekman layer, the net mass transport is perpendicular to the wind stress:

\( \mathbf{M}_E = \frac{1}{f}\,\hat{\mathbf{k}} \times \boldsymbol{\tau} \)

Net Ekman transport: 90\(°\) to the right of the wind (NH)

This perpendicular transport is critical for ocean dynamics: wind patterns create convergence and divergence zones that drive Ekman pumping and upwelling.

Ekman Layer Depth

The Ekman depth \(D_E = \sqrt{2A_z/|f|}\) depends on the eddy viscosity and latitude. Typical values:

Mid-latitudes: \(A_z \sim 0.01{-}0.1\;\text{m}^2/\text{s}\), \(D_E \sim 15{-}50\;\text{m}\)
Tropics: \(f \to 0\), Ekman theory breaks down; inertial oscillations dominate
Surface current: typically 1-3% of wind speed, deflected 45\(°\) from wind

In reality, the Ekman spiral is rarely observed in its pure form because turbulent mixing varies with depth (not constant \(A_z\)), Stokes drift from surface waves modifies the dynamics, and stratification limits the layer depth. Modern observations from drifters and ADCP moorings reveal a compressed spiral with reduced veering.

2.2 Sverdrup Balance & Wind-Driven Gyres

Sverdrup balance describes how the wind stress curl drives the large-scale interior ocean circulation. Starting from the vertically-integrated vorticity equation:

Vorticity equation: \( \frac{\partial \zeta}{\partial t} + \beta v = \frac{1}{\rho}\text{curl}_z(\boldsymbol{\tau}) + \text{friction} \)

Steady state, neglecting friction: \( \beta v = \frac{1}{\rho}\frac{\partial \tau_x}{\partial y} - \frac{1}{\rho}\frac{\partial \tau_y}{\partial x} \)

The Sverdrup relation for the meridional transport:

\( \boxed{\beta V = \frac{1}{\rho}\,\text{curl}_z\!\left(\frac{\boldsymbol{\tau}}{\rho}\right)} \)

Sverdrup balance: wind stress curl drives meridional transport

Key physical insight: a column of fluid moving poleward (\(v > 0\)) must be stretched (to conserve potential vorticity \((\zeta + f)/H\)), and this stretching is provided by Ekman pumping from the wind stress curl.

Western Boundary Currents

Sverdrup balance fails near the western boundary because friction becomes important. The Stommel model shows that the \(\beta\)-effect (variation of \(f\) with latitude) forces the return flow into a narrow western boundary layer:

\( \delta_w = \frac{r}{\beta} \quad \text{(Stommel boundary layer width)} \)

where \(r\) is the bottom friction coefficient

This explains why the Gulf Stream, Kuroshio, and other western boundary currents are narrow (\(\sim 100\) km), swift (\(\sim 2\) m/s), and warm—carrying equatorial heat poleward.

Munk Model: Wind-Driven Gyre Solution

The Munk model uses lateral eddy viscosity \(A_H\) instead of bottom friction, giving a western boundary layer of width:

\( \delta_M = \left(\frac{A_H}{\beta}\right)^{1/3} \approx 50{-}100\;\text{km} \)

The full vorticity equation for the Munk model:

\( \beta\frac{\partial\psi}{\partial x} = \frac{1}{\rho H}\,\text{curl}(\boldsymbol{\tau}) + A_H\nabla^4\psi \)

The Gulf Stream transports about 30 Sv (\(1\;\text{Sv} = 10^6\;\text{m}^3/\text{s}\)) of warm water northward, releasing heat to the atmosphere over the North Atlantic and significantly moderating European winters. Its separation point near Cape Hatteras is determined by the zero wind stress curl latitude and inertial effects.

2.3 The Atlantic Meridional Overturning Circulation (AMOC)

The thermohaline circulation is driven by density differences arising from temperature (thermo) and salinity (haline) variations. In the North Atlantic, cold salty surface water sinks in the Labrador and Nordic Seas, forming North Atlantic Deep Water (NADW), which flows southward at depth.

Seawater density depends on both temperature and salinity via the equation of state:

\( \rho = \rho_0\left[1 - \alpha_T(T - T_0) + \beta_S(S - S_0)\right] \)

\(\alpha_T \approx 2\times10^{-4}\) K\(^{-1}\) (thermal expansion), \(\beta_S \approx 7.6\times10^{-4}\) (psu)\(^{-1}\) (haline contraction)

Stommel Two-Box Model

Stommel (1961) showed that the thermohaline circulation exhibits bistability using a two-box model. Box 1 (equatorial) and Box 2 (polar) exchange water at a rate proportional to their density difference:

\( q = k|\rho_2 - \rho_1| = k|\alpha_T \Delta T - \beta_S \Delta S| \)

The temperature and salinity contrasts evolve as:

\( \frac{d(\Delta T)}{dt} = \lambda_T(\Delta T^* - \Delta T) - |q|\Delta T \)

\( \frac{d(\Delta S)}{dt} = \lambda_S(\Delta S^* - \Delta S) - |q|\Delta S \)

where \(\lambda_T \gg \lambda_S\) because the atmosphere restores temperature much faster than the hydrological cycle restores salinity (\(\Delta T^*, \Delta S^*\) are forcing values).

The critical freshwater flux for AMOC collapse:

\( F_{\text{crit}} \propto \frac{\alpha_T^2 (\Delta T^*)^2}{4\beta_S} \)

Beyond this threshold, the salinity-driven (haline) mode dominates and AMOC collapses

This bistability is a major concern for climate change: Greenland ice sheet melting adds freshwater to the North Atlantic, potentially weakening or shutting down the AMOC. As discussed in the Climate & Biodiversity Module 3 (Cryosphere), this would have cascading effects on ecosystems and regional climate.

2.4 ENSO: El Ni\(\tilde{n}\)o-Southern Oscillation

ENSO is the dominant mode of interannual climate variability, involving coupled ocean-atmosphere interactions in the tropical Pacific. The Walker circulation normally drives easterly trade winds, piling warm water in the western Pacific (warm pool). During El Nino, the trade winds weaken, warm water sloshes eastward, and the thermocline flattens.

Bjerknes Positive Feedback

The Bjerknes feedback amplifies initial SST anomalies:

Warm SST anomaly in eastern Pacific
Reduced east-west SST gradient weakens trade winds
Less Ekman upwelling of cold water in the east
Further warming of eastern SST (positive feedback loop)

Delayed Oscillator Model

The oscillatory nature of ENSO is captured by the delayed oscillator model (Suarez & Schopf, 1988). The SST anomaly \(T\) in the eastern Pacific evolves as:

\( \frac{dT}{dt} = aT(t) - bT(t - \delta) - cT^3 \)

where:

\(aT(t)\): Bjerknes positive feedback (local amplification)
\(-bT(t-\delta)\): delayed negative feedback from equatorial Rossby waves reflecting off the western boundary as downwelling Kelvin waves (\(\delta \sim 6\) months)
\(-cT^3\): nonlinear damping that limits growth

The period of oscillation is roughly \(P \approx 4\delta \approx 2{-}7\) years, consistent with observed ENSO cycles.

ENSO Teleconnections

ENSO affects climate globally through atmospheric teleconnections. During El Nino:

Pacific: Reduced upwelling, fishery collapse (anchovy), coral bleaching
Americas: Floods in Peru, drought in Amazon, warm winters in North America
Asia/Australia: Drought, weakened monsoon, bushfires in Australia
Africa: Drought in southern and eastern Africa, flooding in East Africa
Global: Temporary global mean temperature increase of \(\sim 0.1{-}0.2\) K

The Pacific Decadal Oscillation (PDO) modulates ENSO on multi-decadal timescales. The PDO positive phase enhances El Nino frequency; negative phase enhances La Nina. Understanding ENSO-PDO interactions is critical for decadal climate prediction.

Climate models project that ENSO will continue but with increased frequency of extreme El Nino events under warming, with profound consequences for global weather patterns and the ecosystems discussed in the Climate & Biodiversity Module 4 (Extreme Weather).

2.5 Ocean Heat Content & Sea Level Rise

The ocean absorbs approximately 93% of the excess heat from anthropogenic greenhouse warming. The ocean heat content change:

\( \Delta Q = \int_0^D \rho c_p \Delta T(z)\,dz \)

\(c_p \approx 3990\) J/(kg K) for seawater, integrated to depth \(D\)

Thermal Expansion & Steric Sea Level Rise

As the ocean warms, it expands. The thermal expansion coefficient:

\( \alpha_T = -\frac{1}{\rho}\frac{\partial \rho}{\partial T}\bigg|_{p,S} \)

The steric sea level rise from thermal expansion:

\( \Delta h = \int_0^D \alpha_T(z) \cdot \Delta T(z)\,dz \)

For \(\alpha_T \approx 2 \times 10^{-4}\) K\(^{-1}\) and \(\Delta T \approx 0.1\) K over \(D = 2000\) m: \(\Delta h \approx 0.04\) m. This thermosteric contribution accounts for about 40% of observed sea level rise, with the remainder from ice melt.

As explored in the Climate & Biodiversity Module 2 (Ocean Chemistry & Life), ocean warming also reduces oxygen solubility, drives stratification, and exacerbates acidification, threatening marine ecosystems.

Global Ocean Conveyor Belt

Schematic of the AMOC showing warm surface currents (red), deep NADW flow (blue), and Antarctic Bottom Water (light blue). Approximate heat transport and volume flux indicated.

2.9 Ocean Stratification & Climate Change

Ocean stratification, measured by the Brunt-Vaisala frequency \(N\), controls vertical mixing, nutrient supply, and the depth of the mixed layer:

\( N^2 = -\frac{g}{\rho_0}\frac{\partial\rho}{\partial z} = g\left(\alpha_T\frac{\partial T}{\partial z} - \beta_S\frac{\partial S}{\partial z}\right) \)

Under global warming, surface waters warm faster than deep waters, increasing \(N^2\)and strengthening stratification. Observed consequences:

Mixed layer shallowing: reduced ventilation of subsurface waters
Reduced nutrient supply: less vertical mixing limits phytoplankton productivity
Deoxygenation: warmer water holds less O\(_2\), and reduced mixing limits O\(_2\) resupply
Weakened carbon pump: less deep water formation reduces CO\(_2\) sequestration

Upper ocean stratification has increased by 5-18% since 1970 (IPCC AR6), with the largest increases in the Arctic and Southern Oceans.

Simulation: Ekman Spiral & Sverdrup Transport

Visualizing the Ekman spiral velocity profile and computing the Sverdrup stream function for a subtropical gyre:

Python

script.py1 lines

import numpy as np\nimport matplotlib.pyplot as plt\n\nfig, axes = plt.subplots(2, 2, figsize=(14, 11))\nfig.patch.set_facecolor('#0a0a1a')\n\n# Panel 1: Ekman Spiral (hodograph)\nax = axes[0, 0]\nf = 1e-4  # Coriolis parameter\nAz = 0.01  # eddy viscosity m^2/s\nD_E = np.sqrt(2*Az/abs(f))  # Ekman depth\ntau0 = 0.1  # wind stress N/m^2\nrho = 1025  # seawater density\nV0 = tau0 / (rho * np.sqrt(Az * abs(f)))\n\nz = np.linspace(0, -5*D_E, 500)  # depth\nu = V0 * np.exp(z/D_E) * np.cos(np.pi/4 + z/D_E)\nv = V0 * np.exp(z/D_E) * np.sin(np.pi/4 + z/D_E)\n\n# Hodograph\nax.plot(u*100, v*100, color='#818cf8', linewidth=2)\n# Mark depths\nfor depth_frac, label, color in [(0, 'Surface', '#f87171'), \n                                   (-1, f'z = -D_E = {D_E:.0f}m', '#fbbf24'),\n                                   (-2, 'z = -2D_E', '#60a5fa')]:\n    idx = np.argmin(abs(z - depth_frac*D_E))\n    ax.plot(u[idx]*100, v[idx]*100, 'o', color=color, markersize=8)\n    ax.annotate(label, xy=(u[idx]*100, v[idx]*100), xytext=(u[idx]*100+0.3, v[idx]*100+0.2),\n                color=color, fontsize=9)\n\n# Wind direction\nax.annotate('Wind', xy=(0, 0), xytext=(0, 2),\n            color='#86efac', fontsize=12, fontweight='bold',\n            arrowprops=dict(arrowstyle='->', color='#86efac', lw=2))\n# Net transport\nax.annotate('Net Ekman\\ntransport', xy=(1.5, 0), xytext=(2.5, -0.5),\n            color='#fbbf24', fontsize=10,\n            arrowprops=dict(arrowstyle='->', color='#fbbf24', lw=2))\n\nax.set_xlabel('u (cm/s)', color='white', fontsize=11)\nax.set_ylabel('v (cm/s)', color='white', fontsize=11)\nax.set_title('Ekman Spiral (Hodograph)', color='#818cf8', fontsize=13, fontweight='bold')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\nax.grid(True, alpha=0.2, color='#4f46e5')\nax.set_aspect('equal')\nax.axhline(y=0, color='#334155', linewidth=0.5)\nax.axvline(x=0, color='#334155', linewidth=0.5)\n\n# Panel 2: Ekman velocity profiles\nax = axes[0, 1]\nax.plot(u*100, z, color='#f87171', linewidth=2, label='u (zonal)')\nax.plot(v*100, z, color='#60a5fa', linewidth=2, label='v (meridional)')\nspeed = np.sqrt(u**2 + v**2)\nax.plot(speed*100, z, color='#fbbf24', linewidth=2, linestyle='--', label='speed')\nax.axhline(y=-D_E, color='#818cf8', linestyle=':', alpha=0.5, label=f'D_E = {D_E:.1f} m')\nax.set_xlabel('Velocity (cm/s)', color='white', fontsize=11)\nax.set_ylabel('Depth (m)', color='white', fontsize=11)\nax.set_title('Ekman Layer Velocity', color='#818cf8', fontsize=13, fontweight='bold')\nax.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#4f46e5', labelcolor='white')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\nax.grid(True, alpha=0.2, color='#4f46e5')\n\n# Panel 3: Sverdrup stream function\nax = axes[1, 0]\nnx, ny = 100, 80\nx_sv = np.linspace(0, 6000e3, nx)  # basin width\ny_sv = np.linspace(0, 4000e3, ny)  # basin length\nX_sv, Y_sv = np.meshgrid(x_sv, y_sv)\n\n# Wind stress: sinusoidal westerlies/trades\nL_y = 4000e3\ntau_x = -0.1 * np.cos(2*np.pi * Y_sv / L_y)\nbeta = 2e-11\ncurl_tau = 0.1 * 2*np.pi/L_y * np.sin(2*np.pi * Y_sv / L_y)\n\n# Sverdrup transport\nV_sv = curl_tau / (rho * beta)\n\n# Stream function (integrating from eastern boundary)\npsi = np.zeros_like(X_sv)\ndx = x_sv[1] - x_sv[0]\nfor i in range(nx-2, -1, -1):\n    psi[:, i] = psi[:, i+1] - V_sv[:, i] * dx\n\ncf = ax.contourf(X_sv/1e3, Y_sv/1e3, psi/1e6, levels=20, cmap='RdBu_r')\ncs = ax.contour(X_sv/1e3, Y_sv/1e3, psi/1e6, levels=10, colors='white', linewidths=0.5, alpha=0.3)\ncb = plt.colorbar(cf, ax=ax, label='Stream function (Sv)')\ncb.ax.yaxis.label.set_color('white')\ncb.ax.tick_params(colors='white')\nax.set_xlabel('Zonal distance (km)', color='white', fontsize=11)\nax.set_ylabel('Meridional distance (km)', color='white', fontsize=11)\nax.set_title('Sverdrup Stream Function', color='#818cf8', fontsize=13, fontweight='bold')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\n\n# Panel 4: Wind stress curl\nax = axes[1, 1]\nax.plot(tau_x[0,:]*0 + tau_x[:,nx//2], y_sv/1e3, color='#60a5fa', linewidth=2, label='\u03c4_x (wind stress)')\nax.plot(curl_tau[:,nx//2]*1e7, y_sv/1e3, color='#f87171', linewidth=2, label='curl(\u03c4) x10\u2077')\nax.axvline(x=0, color='#334155', linewidth=0.5)\nax.set_xlabel('Stress (N/m\u00b2) / Curl (x10\u207b\u2077 N/m\u00b3)', color='white', fontsize=11)\nax.set_ylabel('Meridional distance (km)', color='white', fontsize=11)\nax.set_title('Wind Stress Profile', color='#818cf8', fontsize=13, fontweight='bold')\nax.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#4f46e5', labelcolor='white')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\nax.grid(True, alpha=0.2, color='#4f46e5')\n\nplt.tight_layout()\nplt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')\nprint('Ekman spiral and Sverdrup transport simulation complete')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Simulation: AMOC Bistability & ENSO Oscillator

The Stommel two-box model reveals the bistable nature of the thermohaline circulation, while the delayed oscillator captures ENSO dynamics:

Python

script.py1 lines

import numpy as np\nimport matplotlib.pyplot as plt\n\nfig, axes = plt.subplots(2, 2, figsize=(14, 11))\nfig.patch.set_facecolor('#0a0a1a')\n\n# ===== Stommel Box Model =====\n# Panel 1: AMOC state as function of freshwater flux\nax = axes[0, 0]\nalpha_T = 2e-4  # thermal expansion K^-1\nbeta_S = 7.6e-4  # haline contraction psu^-1\nk = 1.5e-6      # hydraulic constant\nlam_T = 1.0/(30*86400)  # thermal relaxation ~30 days\nlam_S_vals = [1.0/(300*86400), 1.0/(600*86400)]  # salinity relaxation\nDT_star = 20.0  # imposed temp difference K\n\nDS_star_range = np.linspace(0.1, 5.0, 200)\n\nfor lam_S, color, label in zip(lam_S_vals, ['#f87171', '#60a5fa'],\n                                ['Fast FW (300 d)', 'Slow FW (600 d)']):\n    q_vals = []\n    for DS_star in DS_star_range:\n        # Find equilibrium numerically\n        DT = DT_star * lam_T / (lam_T + 2*k*abs(alpha_T*DT_star))  # approximate\n        DS = DS_star * lam_S / (lam_S + 2*k*abs(alpha_T*DT_star))\n        q = k * abs(alpha_T * DT - beta_S * DS) * 1e6  # Sv-like units\n        if alpha_T * DT > beta_S * DS:\n            q_vals.append(q)\n        else:\n            q_vals.append(-q)\n    ax.plot(DS_star_range, q_vals, color=color, linewidth=2, label=label)\nax.axhline(y=0, color='#4f46e5', linestyle='--', alpha=0.5)\nax.set_xlabel('Freshwater forcing \u0394S* (psu)', color='white', fontsize=11)\nax.set_ylabel('Overturning (arb. Sv)', color='white', fontsize=11)\nax.set_title('Stommel Model: AMOC vs Freshwater', color='#818cf8', fontsize=13, fontweight='bold')\nax.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#4f46e5', labelcolor='white')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\nax.grid(True, alpha=0.2, color='#4f46e5')\n\n# Panel 2: Time evolution of AMOC with freshwater hosing\nax = axes[0, 1]\ndt_s = 86400.0  # 1 day\nyears = 500\nsteps = int(years * 365.25)\nlam_T_sim = 1.0/(30*86400)\nlam_S_sim = 1.0/(400*86400)\nDT_star_sim = 20.0\n\nfor DS_ramp, color, label in [(1.0, '#60a5fa', '\u0394S* = 1.0 psu'),\n                               (2.5, '#fbbf24', '\u0394S* = 2.5 psu (ramp)'),\n                               (4.0, '#f87171', '\u0394S* = 4.0 psu (collapse)')]:\n    DT = 15.0\n    DS = 0.5\n    q_history = []\n    for i in range(steps):\n        # Ramping freshwater\n        DS_target = DS_ramp * min(1.0, i/(100*365.25))\n        q = k * (alpha_T * DT - beta_S * DS)\n        dDT = (lam_T_sim * (DT_star_sim - DT) - abs(q)*DT) * dt_s\n        dDS = (lam_S_sim * (DS_target - DS) - abs(q)*DS) * dt_s\n        DT += dDT\n        DS += dDS\n        if i % 365 == 0:\n            q_history.append(q * 1e7)\n    time = np.arange(len(q_history))\n    ax.plot(time, q_history, color=color, linewidth=1.5, label=label, alpha=0.9)\n\nax.axhline(y=0, color='#ef4444', linestyle=':', alpha=0.5, label='AMOC shutdown')\nax.set_xlabel('Time (years)', color='white', fontsize=11)\nax.set_ylabel('AMOC strength (arb.)', color='white', fontsize=11)\nax.set_title('AMOC Response to Freshwater Hosing', color='#818cf8', fontsize=13, fontweight='bold')\nax.legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#4f46e5', labelcolor='white')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\nax.grid(True, alpha=0.2, color='#4f46e5')\n\n# ===== ENSO Delayed Oscillator =====\n# Panel 3: ENSO time series\nax = axes[1, 0]\na_enso = 0.8   # Bjerknes feedback\nb_enso = 1.2   # delayed feedback\nc_enso = 0.5   # nonlinear damping\ndelay = 180    # days\ndt_e = 1.0     # day\ndays = 365 * 30\n\nT_enso = np.zeros(days)\nT_enso[:delay] = 0.1 * np.random.randn(delay)\n\nfor i in range(delay, days):\n    dTdt = a_enso * T_enso[i-1] - b_enso * T_enso[i-1-delay] - c_enso * T_enso[i-1]**3\n    T_enso[i] = T_enso[i-1] + dTdt * dt_e/365\n\ntime_y = np.arange(days) / 365\nax.plot(time_y, T_enso, color='#f87171', linewidth=1.2, alpha=0.8)\nax.fill_between(time_y, T_enso, 0, where=T_enso>0, alpha=0.15, color='#ef4444', label='El Nino (warm)')\nax.fill_between(time_y, T_enso, 0, where=T_enso<0, alpha=0.15, color='#3b82f6', label='La Nina (cool)')\nax.axhline(y=0, color='#4f46e5', linestyle='-', alpha=0.3)\nax.set_xlabel('Time (years)', color='white', fontsize=11)\nax.set_ylabel('SST Anomaly (arb.)', color='white', fontsize=11)\nax.set_title('ENSO Delayed Oscillator', color='#818cf8', fontsize=13, fontweight='bold')\nax.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#4f46e5', labelcolor='white')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\nax.grid(True, alpha=0.2, color='#4f46e5')\nax.set_xlim(0, 30)\n\n# Panel 4: ENSO phase space\nax = axes[1, 1]\nax.plot(T_enso[delay:], T_enso[:-delay], color='#818cf8', linewidth=0.5, alpha=0.6)\nax.scatter(T_enso[delay::365], T_enso[:-delay:365], c=time_y[delay::365],\n           cmap='hot', s=20, zorder=5)\nax.set_xlabel('T(t)', color='white', fontsize=11)\nax.set_ylabel(f'T(t - {delay}d)', color='white', fontsize=11)\nax.set_title('ENSO Phase Space', color='#818cf8', fontsize=13, fontweight='bold')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\nax.grid(True, alpha=0.2, color='#4f46e5')\nax.axhline(y=0, color='#334155', linewidth=0.5)\nax.axvline(x=0, color='#334155', linewidth=0.5)\n\nplt.tight_layout()\nplt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')\nprint('AMOC and ENSO simulation complete')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

2.6 Ekman Pumping & Coastal Upwelling

The convergence/divergence of Ekman transport drives vertical motion at the base of the Ekman layer, known as Ekman pumping:

\( w_E = \frac{1}{\rho f}\,\text{curl}_z\!\left(\boldsymbol{\tau}\right) = \frac{1}{\rho f}\left(\frac{\partial \tau_y}{\partial x} - \frac{\partial \tau_x}{\partial y}\right) \)

Ekman pumping velocity (positive = upwelling)

In the subtropical gyre, the wind stress curl is negative (Northern Hemisphere), producing Ekman downwelling (\(w_E < 0\)) that depresses the thermocline. In subpolar regions, positive curl drives upwelling.

Coastal Upwelling

Along eastern boundaries (California, Peru, Benguela, Canary currents), equatorward winds drive offshore Ekman transport, drawing cold, nutrient-rich water from depth:

Upwelling velocity: \(w \sim 10^{-5}\) m/s (\(\sim 1\) m/day)
Temperature drop: SST 5-10°C cooler than surroundings
Biological productivity: supplies nutrients (NO\(_3^-\), PO\(_4^{3-}\), Si) to the photic zone
Upwelling index: proportional to alongshore wind stress \(\tau_{\parallel}/(\rho f)\)

Climate models project changes to upwelling-favorable winds under warming, with implications for fisheries and marine ecosystems discussed in the Climate & Biodiversity Module 2.

2.7 Diapycnal Mixing & Abyssal Circulation

The deep ocean circulation requires mixing to close the overturning: cold deep water must be slowly warmed by diapycnal (cross-density) diffusion to upwell. Munk (1966) estimated the required global-mean diffusivity:

\( w \frac{\partial T}{\partial z} = \kappa \frac{\partial^2 T}{\partial z^2} \)

\( \kappa_{\text{Munk}} \approx 10^{-4}\;\text{m}^2/\text{s} \)

However, measured open-ocean diffusivities are only \(\kappa \sim 10^{-5}\;\text{m}^2/\text{s}\). The discrepancy is resolved by recognizing that enhanced mixing occurs over rough topography (mid-ocean ridges, seamounts) where internal tides break:

\( \kappa_{\text{near ridges}} \sim 10^{-3}\;\text{m}^2/\text{s} \gg \kappa_{\text{interior}} \)

The spatial distribution of mixing has profound implications for the structure of the deep circulation, the distribution of tracers (radiocarbon, CFCs), and the ocean's capacity to sequester anthropogenic CO\(_2\) and heat.

Internal Tides & Mixing Hotspots

Barotropic tides flowing over rough topography generate internal tides (internal gravity waves at tidal frequency). These waves propagate, steepen, and eventually break, providing the mechanical energy for deep mixing. The global tidal dissipation budget:

\( \dot{E}_{\text{tidal}} \approx 3.5\;\text{TW (total)} \)

\( \sim 1\;\text{TW dissipated in deep ocean} \implies \kappa_{\text{avg}} \sim 10^{-4}\;\text{m}^2/\text{s} \)

The energy available for mixing \(\varepsilon\) connects to diffusivity via the Osborn relation:

\( \kappa = \Gamma \frac{\varepsilon}{N^2} \)

\(\Gamma \approx 0.2\) is the mixing efficiency

Major mixing hotspots include the Mid-Atlantic Ridge, the Hawaiian Ridge, and the Kerguelen Plateau, where microstructure measurements confirm enhanced diffusivities of \(10^{-3}\) m\(^2\)/s or greater.

2.8 Meridional Ocean Heat Transport

The ocean transports approximately 2 PW (petawatts) of heat poleward, complementing the atmospheric heat transport of \(\sim 3\) PW. The ocean heat transport is computed from:

\( H(\phi) = \rho c_p \int_0^{x_E} \int_{-D}^{0} v\,T\,dz\,dx \)

The AMOC contributes about \(1.3\) PW of northward heat transport at \(26°\)N, measured by the RAPID array since 2004. This heat transport maintains the relatively mild climate of Western Europe compared to similar latitudes in North America.

Key decomposition of ocean heat transport:

\( H = \underbrace{\overline{v}\,\overline{T}}_{\text{Eulerian mean}} + \underbrace{\overline{v'T'}}_{\text{eddy}} + \underbrace{v_{\text{gyre}} T_{\text{gyre}}}_{\text{gyre}} \)

In the Atlantic, the overturning component dominates (warm surface water flows north, cold deep water flows south). In the Pacific and Indian Oceans, the subtropical gyre circulation is the primary mechanism.

Ocean Carbon Pump

The ocean circulation also drives the biological pump and solubility pump that sequester atmospheric CO\(_2\):

Solubility pump: cold polar waters dissolve more CO\(_2\) (Henry's law \(K_H \propto 1/T\)), then sink, transporting carbon to depth
Biological pump: phytoplankton fix CO\(_2\) in surface waters; sinking organic matter exports carbon to depth (\(\sim 10\) GtC/yr)
Carbonate pump: CaCO\(_3\) shell formation and dissolution, countered by the lysocline depth

The ocean has absorbed about 30% of anthropogenic CO\(_2\) emissions since industrialization, but this capacity may weaken as surface waters warm and stratification increases, reducing deep water formation and carbon export. This connects directly to the ocean acidification discussed in the Climate & Biodiversity Module 2.

Simulation: Ekman Pumping & Ocean Heat Content

Computing Ekman pumping from wind stress fields and tracking ocean heat content changes:

Python

script.py1 lines

import numpy as np\nimport matplotlib.pyplot as plt\n\nfig, axes = plt.subplots(2, 2, figsize=(14, 11))\nfig.patch.set_facecolor('#0a0a1a')\n\n# Panel 1: Ekman pumping map\nax = axes[0, 0]\nnx, ny = 80, 60\nlat = np.linspace(-60, 60, ny)\nlon = np.linspace(-80, 0, nx)  # Atlantic\nLON, LAT_grid = np.meshgrid(lon, lat)\n\n# Idealized wind stress (trades + westerlies)\ntau_x = -0.05 * np.cos(3 * np.pi * LAT_grid / 180)\ntau_y = 0.01 * np.sin(2 * np.pi * LAT_grid / 180) * np.cos(np.pi * LON / 80)\n\n# Ekman pumping\nOmega = 7.292e-5\nrho = 1025\nf = 2 * Omega * np.sin(LAT_grid * np.pi / 180)\nf = np.where(np.abs(f) < 1e-6, np.sign(f) * 1e-6, f)  # avoid division by zero\n\ndx = (lon[1]-lon[0]) * np.pi/180 * 6.371e6 * np.cos(LAT_grid * np.pi/180)\ndy = (lat[1]-lat[0]) * np.pi/180 * 6.371e6\n\ncurl_tau = np.gradient(tau_y, axis=1) / dx - np.gradient(tau_x, axis=0) / dy\nw_E = curl_tau / (rho * f) * 86400 * 365  # m/year\n\ncf = ax.contourf(LON, LAT_grid, w_E, levels=np.linspace(-50, 50, 21), cmap='RdBu_r', extend='both')\ncb = plt.colorbar(cf, ax=ax, label='Ekman pumping (m/yr)')\ncb.ax.yaxis.label.set_color('white')\ncb.ax.tick_params(colors='white')\nax.contour(LON, LAT_grid, w_E, levels=[0], colors='white', linewidths=1)\nax.set_xlabel('Longitude', color='white', fontsize=11)\nax.set_ylabel('Latitude', color='white', fontsize=11)\nax.set_title('Ekman Pumping (Atlantic)', color='#818cf8', fontsize=13, fontweight='bold')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\n\n# Panel 2: Thermocline depth from Ekman pumping\nax = axes[0, 1]\n# Reduced gravity model: h^2 = h_E^2 + 2*g_red * int(w_E dx) / beta\ng_red = 0.02  # reduced gravity m/s^2\nh_therm = np.sqrt(np.maximum(100**2 + 2/(g_red) * np.cumsum(w_E * dx, axis=1), 0))\ncf2 = ax.contourf(LON, LAT_grid, h_therm, levels=20, cmap='viridis_r')\ncb2 = plt.colorbar(cf2, ax=ax, label='Thermocline depth (m)')\ncb2.ax.yaxis.label.set_color('white')\ncb2.ax.tick_params(colors='white')\nax.set_xlabel('Longitude', color='white', fontsize=11)\nax.set_ylabel('Latitude', color='white', fontsize=11)\nax.set_title('Thermocline Depth', color='#818cf8', fontsize=13, fontweight='bold')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\n\n# Panel 3: Ocean heat content time series\nax = axes[1, 0]\nyears_ohc = np.arange(1960, 2025)\n# Simplified OHC: accelerating heat uptake\nohc_0_700 = 0.5 * (years_ohc - 1970)**2 * 0.003 + 2*np.sin(0.3*(years_ohc-1960))\nohc_0_700 = np.where(years_ohc < 1970, 0, ohc_0_700 - ohc_0_700[10])\nohc_700_2000 = 0.3 * ohc_0_700 + np.random.RandomState(42).randn(len(years_ohc)) * 1\nohc_total = ohc_0_700 + ohc_700_2000\n\nax.plot(years_ohc, ohc_0_700, color='#f87171', linewidth=2, label='0-700 m')\nax.plot(years_ohc, ohc_700_2000, color='#60a5fa', linewidth=2, label='700-2000 m')\nax.plot(years_ohc, ohc_total, color='#fbbf24', linewidth=2.5, label='Total 0-2000 m')\nax.fill_between(years_ohc, 0, ohc_0_700, alpha=0.1, color='#f87171')\nax.fill_between(years_ohc, 0, ohc_700_2000, alpha=0.1, color='#60a5fa')\nax.axhline(y=0, color='#334155', linewidth=0.5)\nax.set_xlabel('Year', color='white', fontsize=11)\nax.set_ylabel('OHC Anomaly (10\u00b2\u00b2 J)', color='white', fontsize=11)\nax.set_title('Ocean Heat Content', color='#818cf8', fontsize=13, fontweight='bold')\nax.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#4f46e5', labelcolor='white')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\nax.grid(True, alpha=0.2, color='#4f46e5')\n\n# Panel 4: Steric sea level rise\nax = axes[1, 1]\nalpha_T_coeff = 2e-4  # K^-1\nD_mix = 2000  # depth m\nT_warming = np.linspace(0, 3, 200)  # surface warming K\n\n# Steric rise for different penetration profiles\nfor scale_depth, color, label in [(500, '#60a5fa', 'd = 500 m'),\n                                    (1000, '#fbbf24', 'd = 1000 m'),\n                                    (2000, '#f87171', 'd = 2000 m')]:\n    # Exponential temperature profile\n    z_int = np.linspace(0, D_mix, 500)\n    dh = []\n    for dT_surf in T_warming:\n        dT_z = dT_surf * np.exp(-z_int/scale_depth)\n        delta_h = np.trapezoid(alpha_T_coeff * dT_z, z_int) * 100  # cm\n        dh.append(delta_h)\n    ax.plot(T_warming, dh, color=color, linewidth=2, label=label)\n\nax.set_xlabel('Surface Warming (K)', color='white', fontsize=11)\nax.set_ylabel('Steric Sea Level Rise (cm)', color='white', fontsize=11)\nax.set_title('Thermosteric Sea Level Rise', color='#818cf8', fontsize=13, fontweight='bold')\nax.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#4f46e5', labelcolor='white')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\nax.grid(True, alpha=0.2, color='#4f46e5')\n\nplt.tight_layout()\nplt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')\nprint('Ocean dynamics simulation complete')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

References

Vallis, G. K. (2017). Atmospheric and Oceanic Fluid Dynamics (2nd ed.). Cambridge University Press.
Stommel, H. (1961). Thermohaline convection with two stable regimes of flow. Tellus, 13(2), 224-230.
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Suarez, M. J. & Schopf, P. S. (1988). A delayed action oscillator for ENSO. Journal of the Atmospheric Sciences, 45(21), 3283-3287.
Ekman, V. W. (1905). On the influence of the earth's rotation on ocean-currents. Arkiv for Matematik, Astronomi och Fysik, 2(11), 1-52.
Rahmstorf, S. (2002). Ocean circulation and climate during the past 120,000 years. Nature, 419(6903), 207-214.
Trenberth, K. E. & Fasullo, J. T. (2013). An apparent hiatus in global warming? Earth's Future, 1(1), 19-32.
Kuhlbrodt, T. et al. (2007). On the driving processes of the Atlantic meridional overturning circulation. Reviews of Geophysics, 45(2), RG2001.