What drives the ocean currents?

Friction: transfer of momentum from atmosphere to oceanic Ekman layer

Vorticity dynamics for the ocean and include the wind stress term

\[ D_t u - f v = - \frac{1}{\rho} \frac{\partial p}{\partial x} + \frac{1}{\rho} \partial_z \tau_{xz} \] \[ D_t v + f u = - \frac{1}{\rho} \frac{\partial p}{\partial y} + \frac{1}{\rho} \partial_z \tau_{yz} \]

\[ \frac{D}{Dt} \left( {\zeta+f}\right) - \frac{\left(\zeta+f \right)}{h} \frac{D}{Dt} h \, = \, \frac{1}{\rho} \underbrace{\left( \frac{\partial}{\partial x} \, \partial_z \tau_{yz} - \frac{\partial}{\partial y}\, \partial_z \tau_{xz} \right)}_{curl \, \partial_z \tau} \quad . \]

\[ \frac{D}{Dt} \left( \frac{\zeta+f}{h}\right) = \frac{1}{\rho \, h} \, \mbox{curl} \, \partial_z \tau \, \]

Sverdrup transport

Sverdrup

applied globally using the wind stress from Hellerman and Rosenstein (1983). Contour interval is \(10\) Sverdrups (Tomczak and Godfrey, 1994).

\[ V = \frac{1}{\rho \beta} \, \left( \frac{\partial \tau_{yz} }{\partial x} \, - \frac{\partial \tau_{xz}}{\partial y}\, \right) = \frac{1}{\rho \beta} \, \, \operatorname{curl} \, \tau \]

Ekman Pumping: vertical velocity at the bottom

\(w_E\) as the Ekman vertical velocity the bottom of the Ekman layer \[ w_E = - \int_{-E}^0 \frac{\partial w}{\partial z} dz = \frac{\partial}{\partial x} U_E + \frac{\partial}{\partial y} V_E \]

\(\operatorname{curl} \mathbf{\tau}\) produces a divergence of the Ekman transports leading to \(w_E\)

\[ w_E = \, \frac{\partial }{\partial x} \left( \frac{ \tau_{y}}{\rho \;f }\, \right) - \frac{\partial }{\partial y}\, \left( \frac{ \tau_{x}}{\rho \;f }\, \right) =\operatorname{curl}\left(\frac{\mathbf{\tau}}{\rho\;f}\right) \simeq \frac{1}{\rho\;f} \, \operatorname{curl} \mathbf{\tau} \]

The order of magnitude of the Ekman vertical velocity:

typical wind stress variation of \(0.2 N m^{-2}\) per 2000 km in y-direction:

\[ w_E \simeq - \frac{ \Delta \tau_{x}}{\rho \;f_0 \Delta y}\, \simeq \frac{1 }{10^3 kg m^{-3}} \frac{0.2 N m^{-2} }{10^{-4} s^{-1}\, \, 2 \cdot 10^6 m} \simeq 32 \, \, \frac{m}{yr} \]

Ekman Pumping & Sverdrup Transport

Ekman

The center of a subtropical gyre is a high pressure zone: clockwise on the Northern Hemisphere

Ekman surface currents towards the center of the gyre

The Ekman vertical velocity balanced by \[ w_E=w_g \] vertical geostrophic current in the interior

geostrophic flow towards the equator

returned flow towards the pole in western boundary currents

Ocean layers

THC

Steady winds blowing on the sea surface produce a thin, horizontal boundary layer, the Ekman layer. A similar boundary layer exists at the bottom of the atmosphere just above the sea surface, the planetary boundary layer.

North Atlantic Current & Gulfstream

North Atlantic Current

Gulf Stream & NA Current

brings warm water northward where it cools.

returns southward as a cold, deep, western-boundary current.

Gulf Stream carries 40 Sv of 18°C water northward.

Of this, 15 Sv return southward in the deep western boundary current at a temperature of 2°C.

Ocean heat transport

Heat transport

Conveyor Belt

Conveyor

Ocean: History

von Lenz, 1847

Conveyor Belt

Belt

Conveyor belt: climate change

Variations

halting or reversing the ocean circulation

interpretation of Greenland ice core records

climate states with different ocean modes

Overturning Circulation (model)

Modelled meridional overturning streamfunction in Sv 10^6 = m^3 /s in the Atlantic Ocean. Grey areas represent zonally integrated smoothed bathymetry

Atlantic water masses

Water masses

North

Estimates of overturning

Timescale T to ‘fill up’ the Atlantic basin:

\[ T = \frac{ 80 \cdot 10^{12} \, m^2 \cdot 4000 \, m}{15 \cdot 10^6 \, m^3 s^{-1}} = 2.13 \cdot 10^{10} s = 676 \;years\]

Overturning is balanced by large-scale upwelling:

\[ area \cdot w = 15 \cdot 10^6 \, m^3 s^{-1}\]

\[ w = 0.1875 \cdot 10^{-6} m\;s^{-1} = 5.9 \cdot 10^{-15} m \; y^{-1}. \]

compare to Ekman pumping \[ w_E \simeq 32 \, \, m \; y^{-1}. \]

Vorticity dynamics of meridional overturning (y,z)

\[ \frac{\partial}{\partial t} v \quad = \quad - \frac{1}{\rho_0} \frac{\partial p}{\partial y} \quad - \quad f u \quad - \quad \kappa v \]

\[ \frac{\partial}{\partial t} w \quad = \quad - \frac{1}{\rho_0} \frac{\partial p}{\partial z} \quad - \quad \frac{g}{\rho_0} (\rho -\rho_0) \quad - \quad \kappa w \] \(\kappa\) as parameter for Rayleigh friction.

Using the continuity equation
\[ 0 \quad = \quad \frac{\partial v}{\partial y} \quad + \quad \frac{\partial w}{\partial z} \]

\[ \mbox{ one can introduce a streamfunction } \Phi (y,z): v= \partial_z \Phi; w = - \partial_y \Phi \]

The associated vorticity equation in the (y,z)-plane is therefore

\[ \frac{\partial}{\partial t} \nabla^2 \Phi \, = \, - f \frac{\partial u}{\partial z} \quad + \quad \frac{g}{\rho_0} \frac{\partial \rho}{\partial y} \quad - \quad \kappa \nabla^2 \Phi \]

Galerkin approximation

\[ {\frac{\partial}{\partial t} \nabla^2 \Phi} \, = \, \underbrace{- f \frac{\partial u}{\partial z}}_{2.} \quad + \quad { \frac{g}{\rho_0} \frac{\partial \rho}{\partial y}} \quad - \quad \underbrace{ \kappa \nabla^2 \Phi }_{4.} \]

Term 2. is “absorbed” into the viscous term (4.)

\[ \Phi (y,z,t) \, = \, \sum_{k=1}^\infty \sum_{l=1}^\infty \Phi_{max}^{k,l} (t) \, \, \sin (\pi k y/L) \, \times \, \sin (\pi l z/H) \] yielding a first order differential equation in time for \(\Phi_{max}^{k,l} (t)\).

Simple ansatz satisfying that the normal velocity at the boundary vanishes \[ \Phi (y,z,t) \, = \, \Phi_{max} (t) \, \, \sin \left(\frac{\pi y}{L}\right) \, \times \, \sin \left(\frac{\pi z}{H}\right) \]

L and H dentote the meridional and depth extend: y from 0 to L, z from 0 to H

The remaining terms and integration

\[ \underbrace{\int_0^L dy \int_0^H dz \quad \frac{\partial}{\partial t} \nabla^2 \Phi}_{1.} \, = \, \underbrace{ \int_0^L dy \int_0^H dz \quad \frac{g}{\rho_0} \frac{\partial \rho}{\partial y}}_{2.} \quad - \underbrace{ \int_0^L dy \int_0^H dz \quad \kappa \nabla^2 \Phi }_{3.} \]

\[ \frac{d}{dt} \Phi_{max} \left(\frac{\pi^2}{L^2} + \frac{\pi^2}{H^2}\right) \int\limits_0^L dy \sin \left(\frac{\pi y}{L}\right) \int\limits_0^H dz \sin \left(\frac{\pi z}{H}\right) = 4 LH \left(\frac{1}{L^2} + \frac{1}{H^2}\right) \frac{d}{dt} \Phi_{max} \]

\[ \int\limits_0^L dy \int\limits_0^H dz \frac{g}{\rho_0} \frac{\partial \rho}{\partial y} = \frac{g}{\rho_0} \, H \, (\rho_{north} - \rho_{south}) \quad \mbox{ with } \rho_{north} = \rho (y=L) , \rho_{south} = \rho (y=0) \]

\[ \kappa \Phi_{max} \left(\frac{\pi^2}{L^2} + \frac{\pi^2}{H^2}\right) \int\limits_0^L dy \sin \left(\frac{\pi y}{L}\right) \int\limits_0^H dz \sin \left(\frac{\pi z}{H}\right) = \kappa \, 4 LH \left(\frac{1}{L^2} + \frac{1}{H^2}\right) \Phi_{max} \]

Amplitude of overturning

\[ \frac{d}{dt} \Phi_{max} \, = \, \frac{a}{\rho_0} (\rho_{north} - \rho_{south}) \, \, - \, \, \kappa \Phi_{max} \] with \[ a = g L H^2/4(L^2 + H^2) \, \]

This shows that the overturning circulation depends on the density differences on the right and left boxes.

It is simplified to a diagnostic relation

\[ \Phi_{max} = \frac{a}{\rho_0 \, \kappa} \, \, (\rho_{north} - \rho_{south}) \quad \]

because the adjustment of \(\Phi_{max}\) is quasi-instantaneous due to adjustment processes, e.g. Kelvin waves.

North-south density gradient in an ocean basin

Primary north-south gradient in balance with an eastward geostrophic current: generates a secondary high & low pressure system, northward current

WE to SN

Box Models

Box Model

Schematic picture of the hemispheric two box model (a) and of the interhemispheric box model

Meridional Densities

Density

1. The Atlantic surface density is mainly related to temperature differences.

2. But the pole-to-pole differences are caused by salinity differences.

Box model of the thermohaline circulation

Ocean: multiple equilibria in GCM

THC

Ocean: multiple equilibria in GCM

THC

Ocean response: Hosing the North Atlantic

Hosing

1. AMOC indices of North Atlantic hosing for different hosing areas. Units are Sv. Black line represents the unperturbed LGM experiment. Hosing is for the period 840–990. (b) Annual mean sea surface salinity anomaly between LGM and the perturbation experiment LGM with 0.2 Sv for the model years 900–950.

-> Multi-scale Ocean GCM

Conveyor belt: climate change

THC

halting or reversing the ocean circulation

interpretation of Greenland ice core records

climate states with different ocean modes

Deglaciation

Abrupt climate change, termination of ice sheets, Climate System II

Ocean box model

THC

Ocean: Temperature

Box Temperatures

      Tnln = Tnl + dts * ((Hfnl)/(rcz2)-(Tnl-Tml)*phi/Vnl);
      
      Tmln = Tml + dts * ((Hfml)/(rcz1)-(Tml-Tsl)*phi/Vml);
      
      Tsln = Tsl + dts * ((Hfsl)/(rcz2)-(Tsl-Td)*phi/Vsl);
      
      Tdn = Td + dts * (-(Td-Tnl)*(phi/Vd));

Atmospheric Temperatures

Atmospheric equations

Boxmodel