Redfield noted the similarity between the average nitrogen-to-phosphorus ratio in plankton (N:P = 16 by atoms) and in deep oceanic waters (N:P = 15; refs 1, 2). He argued that this was neither a coincidence, nor the result of the plankton adapting to the oceanic stoichiometry, but rather that phytoplankton adjust the N:P stoichiometry of the ocean to meet their requirements through nitrogen fixation, an idea supported by recent modelling studies3,4. But what determines the N:P requirements of phytoplankton? Here we use a stoichiometrically explicit model of phytoplankton physiology and resource competition to derive from first principles the optimal phytoplankton stoichiometry under diverse ecological scenarios. Competitive equilibrium favours greater allocation to P-poor resource-acquisition machinery and therefore a higher N:P ratio; exponential growth favours greater allocation to P-rich assembly machinery and therefore a lower N:P ratio. P-limited environments favour slightly less allocation to assembly than N-limited or light-limited environments. The model predicts that optimal N:P ratios will vary from 8.2 to 45.0, depending on the ecological conditions. Our results show that the canonical Redfield N:P ratio of 16 is not a universal biochemical optimum, but instead represents an average of species-specific N:P ratios.

Optimal nitrogen-to-phosphorus stoichiometry of phytoplankton

Hydrodynamic and hydromagnetic stability

80 years ago, Joseph Larmor planted the seed that grew into today's imposing body of knowledge about how the Earth's magnetic field is created. His simple idea, that the geomagnetic field is the result of dynamo action in the Earth's electrically conducting, fluid core, encountered many difficulties, but these have by now been largely overcome, while alternative proposals have been found to be untenable. The development of the theory and its current status are reviewed below. The basic electrodynamics are summarized, but the main focus is on dynamical questions. A special study is made of the energy and entropy requirements of the dynamo and in particular of how efficient it is, considered as a heat engine. Particular attention is paid to modeling core magnetohydrodynamics in a way that is tractable but nevertheless incorporates the dynamical effects of core turbulence in an approximate way. This theory has been tested by numerical integrations, some results from which are presented. The success of these simulations seems to be considerable, when measured against the known geomagnetic facts summarized here. Obstacles that still remain to be overcome are discussed, and some other future challenges are described.

Geodynamo theory and simulations

In the Earth's fluid outer core, a dynamo process converts thermal and gravitational energy into magnetic energy. The power needed to sustain the geomagnetic field is set by the ohmic losses (dissipation due to electrical resistance). Recent estimates of ohmic losses cover a wide range, from 0.1 to 3.5 TW, or roughly 0.3-10% of the Earth's surface heat flow. The energy requirement of the dynamo puts constraints on the thermal budget and evolution of the core through Earth's history. Here we use a set of numerical dynamo models to derive scaling relations between the core's characteristic dissipation time and the core's magnetic and hydrodynamic Reynolds numbers--dimensionless numbers that measure the ratio of advective transport to magnetic and viscous diffusion, respectively. The ohmic dissipation of the Karlsruhe dynamo experiment supports a simple dependence on the magnetic Reynolds number alone, indicating that flow turbulence in the experiment and in the Earth's core has little influence on its characteristic dissipation time. We use these results to predict moderate ohmic dissipation in the range of 0.2-0.5 TW, which removes the need for strong radioactive heating in the core and allows the age of the solid inner core to exceed 2.5 billion years.

Power requirement of the geodynamo from ohmic losses in numerical and laboratory dynamos

MHD thermofluid issues of liquid-metal blankets: Phenomena and advances

This paper presents an experimental study of the momentum and heat transport in a turbulent magnetohydrodynamic duct flow with strong wall jets at the walls parallel to the magnetic field. Local turbulent flow quantities are measured by a traversable combined temperature-potential-difference probe. The simultaneous measurements of time-dependent velocity and temperature signals facilitates the evaluation of Reynolds stresses and turbulent heat fluxes. Integral quantities such as pressure drop and temperature at the heated wall are evaluated and compared with results from conservative design correlations. At strong enough magnetic fields the destabilizing effect of strong shear generated at the sidewalls wins the competition with the damping effect by Joule's dissipation and turbulent side layers are created. Due to the strong non-isotropic character of the electromagnetic forces, the turbulence structure is characterized by large-scale two-dimensional vortices with their axis aligned in the direction of the magnetic field. As most of the turbulent kinetic energy is concentrated in the near-wall turbulent side layers, the temperatures at the heated wall are governed by the development of the thermal boundary layer in the turbulent flow.

Turbulent transport of momentum and heat in magnetohydrodynamic rectangular duct flow with strong sidewall jets

This article presents an analytical and experimental study of magnetohydrodynamic Rayleigh–Benard convection in a large aspect ratio, 20[ratio ]10[ratio ]1, rectangular box. The test fluid is a eutectic sodium potassium Na22K78 alloy with a small Prandtl number of Pr≈0:02. The experimental setup covers Rayleigh numbers in the range 103< Ra<8×104 and Chandrasekhar numbers 0[les ]Q[les ]1.44×106 or Hartmann numbers 0[les ]M[les ]1200, respectively.When a horizontal magnetic field is imposed on a heated liquid metal layer, the electromagnetic forces give rise to a transition of the three-dimensional convective roll pattern into a quasi-two-dimensional flow pattern in such a way that convective rolls become more and more aligned with the magnetic field. A linear stability analysis based on two-dimensional model equations shows that the critical Rayleigh number for the onset of convection of quasi-two-dimensional flow is shifted to significantly higher values due to Hartmann braking at walls perpendicular to the magnetic field. This finding is experimentally confirmed by measured Nusselt numbers. Moreover, the experiments show that the convective heat transport at supercritical conditions is clearly diminished. Adjacent to the onset of convection there is a significant region of stationary convection with significant convective heat transfer before the flow proceeds to time-dependent convection. However, in spite of the Joule dissipation effect there is a certain range of magnetic field intensities where an enhanced heat transfer is observed. Estimates of the local isotropy properties of the flow by a four-element temperature probe demonstrate that the increase in convective heat transport is accompanied by the formation of strong non-isotropic time-dependent flow in the form of large-scale convective rolls aligned with the magnetic field which exhibit a simpler temporal structure compared to ordinary hydrodynamic flow and which are very effective for the convective heat transport.

Rayleigh-Bénard convection in liquid metal layers under the influence of a horizontal magnetic field

The influence of a vertical magnetic field on the integral heat transfer and the temporal dynamics of liquid metal Rayleigh–Benard convection is studied in an experiment using a small Prandtl number (Pr≈0.02) sodium potassium alloy Na22Kr78 as a test fluid. The test section is a rectangular box of large aspect ratio 20 : 10 : 1 that covers a parameter range of Rayleigh numbers, 103<Ra<105, and Chandrasekhar numbers, 0<Q<1.44×104. The integral heat transfer across the layer is evaluated from the measured temperatures at the upper and the lower boundary and the applied heat flux. Local, time-dependent temperatures are obtained from a four-element temperature probe placed in the middle of the liquid metal layer. The noncoplanar arrangement of the thermocouples enables the evaluation of the time-dependent temperature gradient vector that allows us to estimate the local isotropy properties of the time-dependent flow. From the damping effect of Joule dissipation, the convective heat transport decreases monotoni...

Rayleigh–Bénard convection in liquid metal layers under the influence of a vertical magnetic field

. It has been shown theoretically in the past that homogeneous dynamos may occur in electrically conducting fluids for various vortical velocity fields. Roberts (1972) investigated spatially periodic, infinitely extended fields of vortices which Busse (1978, 1992) confined to a finite cylindrical domain. Based on Busse's vortex arrangement a conceptual design for an experimental homogeneous dynamo has been developed and a test facility was setup at the Forschungszentrum Karlsruhe. The first experiments demonstrated that permanent dynamo action can be generated in a cylindrical container filled with liquid sodium in which by means of guide tubes counterrotating and countercurrent spiral vortices are established. The dynamo is self-exciting and the magnetic field saturates at a mean value for fixed super-critical flow rates. The instantaneous magnetic field fluctuates around this mean value by an order of about 5%. As predicted by theory the mode of the observed magnetic field is non-axisymmetric. In a series of experiments a phase- and a bifurcation diagram has been derived as a function of the spiral and axial flow rates.

The Karlsruhe Dynamo Experiment

It has been shown theoretically that homogeneous kinematic dynamo action is possible for many unconfined and confined velocity fields, but a rigorous experimental validation is still lacking. G. O. Roberts [Philos. Trans. R. Soc. London, Ser. A 266, 535 (1970)] proposed a spatially periodic velocity field capable to generate a dynamo, which Busse [Geophys. J. R. Astron. Soc. 42, 437 (1975)] modified by introducing a second length scale larger in order to obtain a solution for a finite domain. Based on a scale separation approach Busse [Springer Proceedings in Physics, Vol. 69 (Springer Verlag, New York, 1992)] proposed a conceptual design for an experimental homogeneous dynamo. An engineering design was developed and a test facility has been set up. This test facility is described and first experimental results confirming dynamo action are presented.

Ulrich Müller

Papers

Turbulent transport of momentum and heat in magnetohydrodynamic rectangular duct flow with strong sidewall jets

Rayleigh-Bénard convection in liquid metal layers under the influence of a horizontal magnetic field

Rayleigh–Bénard convection in liquid metal layers under the influence of a vertical magnetic field

The Karlsruhe Dynamo Experiment

Experimental Demonstration of a Homogeneous Two-Scale Dynamo