Thermohaline

Ocean circulation: Thermohaline

A simplified model of a stratified, rotating (box-shaped) ocean can be cast in terms of the dynamics of the ocean's centre-of-mass. It appears to relate to the basin-averaged angular momentum vector. Simplification of the resulting 6 equations (2 vectors), leads to several well-known systems, such as the Lorenz-1963 equations (for nonrotating fluid under differential heating in the vertical), the Lorenz-1984 equations (for high Prandtl-limit flow under differential heating in the meridional direction), and Stommel-1963 type of equations (with multiple equilibria), under the addition of salt. Of particular interest is the fact that it provides a different interpretation to the Lorenz-1963 equations, which can therefore be said to be 'upside-down'. The reason for saying so is due to the fact that the dynamic variables no longer represent Fourier-amplitudes, but rather have a direct physical meaning. One of them represents the angular mometum, the other two represent the horizontal and vertical coordinates of the centre-of-mass. The latter carries (through the direction of gravity) a natural notion of what is up and what is down. It also helps to rewrite these equations from their conventional form into a set which has an autonomous forcing, making it then obvious that the remaining linear damping terms stem from either diffusion or friction. Tempted to see if one can dispense with either of these damping terms, the simplified, diffusionless Lorenz equations were formulated, without loss of dynamical richness though.
		Diffusionless Lorenz Equations. Y,Z represent horizontal and vertical coordinates of centre-of-mass, X the basin averaged angular momentum. R is the ratio of forcing and friction. In the first equation, angular momentum is driven by the buoyancy torque (Y) and damped by friction. The second and third equation show that the position of the centre-of-mass position changes by angular motion (nonlinear terms), while it is driven purely upwards by the forcing (autonomous term in third equation).
The Lorenz equations also describe the motions of the so-called waterwheel (first conceived by W.V.R. Malkus). It was rebuild by Gerard van der Schrier, see here. A mechanical analog of the (diffusionless) waterwheel is proposed in an upward-driven pendulum which was build by Dirk-Jurjen Buijsman. It consists of a pendulum that is clamped in between two gears that always drive it upwards. It can be seen in motion here. (After it has loaded, turn it on/off with a single mouse click). A somewhat longer pendulum, with a sharper image and taken over a longer timespan is here. This pendulum has blue and red stickers at its ends. Monitoring their position allows one to quantify the motion of the centre of mass. For different upward-driving speeds, a periodic and a chaotic regime were found; see EGS2003-poster.

Publications:
Maas, L.R.M. (2003) Basin scale dynamics of a stratified rotating fluid.� Surveys in Geophysics, in press. ( Download PDF-file ) Schrier, G. van der, Maas, L.R.M., Zimmerman, J.T.F. (2001) Zonal and rotational effects on cross-equatorial thermohaline flow in a simple three-dimensional model.� Submitted to Journal of Marine Research. Schrier, G. van der, Maas, L.R.M. (2000) The diffusionless Lorenz equations: Shil'nikov bifurcations and reduction to an explicit map.�� Physica D, 141: 19-36. ( Download PDF-file ) Schrier, G. van der, Maas, L.R.M. (1998) Chaos in a simple model of the three-dimensional, salt-dominated ocean circulation.� Climate Dynamics, 14: 489-502. Maas, L.R.M. (1994) A simple model for the three-dimensional, thermally and wind-driven ocean circulation.� Tellus, 46A: 671-680.