Convective adjustment in box models |
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| Institution: | 1. Department of Physics and Astronomy, University of Western Ontario, London, Ont., Canada N6A 3K7;2. Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA;3. Department of Atmospheric Sciences, University of Arizona, Tucson, AZ 85721, USA;4. Department of Physics, University of Arizona, Tucson, AZ 85721, USA;1. Univ. Lyon, ENISE, LTDS, UMR 5513 CNRS, 58 rue Jean Parot, 42023, Saint-Etienne cedex 2, France;2. Haulotte Group, 27 rue d’Onzion, 42152, L''Horme, France;1. Department of Physics, Vignana Bharathi Institute of Technology, Aushapur(v) Ghatkesar (M), Hyderabad 501301, Telangana, India;2. Department of Physics, Regency Institute of Technology, Adivipolam Yanam 533464, Pondicherry, India;3. Department of Physics, School of Physical, Chemical and Applied Sciences, Pondicherry University, R.V.Nagar, Kalapet, Pondicherry 605014, India;4. Department of Physics, Andhra University, Visakhapatnam 530003, AndhraPradesh, India;5. Indian Institute of Chemical Technology (CSIR), Tarnaka, Hyderabad 500007, India;6. Department of Physics, University College of Engineering, Jawaharlal Nehru Technological University: Kakinada, Kakinada 533003, India;1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China;2. Computational Science Hubei Key Laboratory, Wuhan University, Wuhan 430072, China;3. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China |
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| Abstract: | A convective adjustment (CA) algorithm is thought to be responsible for grid-scale oceanic-state sustained oscillations seen in oceanic general circulation models (OGCM), an effect that is most evident in simulations with coarse spatio-temporal scales. The CA algorithm is thought to inadvertently create a salt oscillator. Several studies have confirmed that a flip-flop type salt oscillator, which is reminiscent in some respects of simple CA schemes, can develop sustained oscillations. Subsequently, several researchers were able to show how coupled salt oscillators, reacting in a particular temporal sequence, are capable of producing large-scale oscillations not unlike those found in the OGCM simulations. However, the proxy models used to study how CA can create these oscillations in large-scale simulations were never directly related to OGCM results.Here we couple hydrodynamics to the CA and look at zonally-driven flows in the low-frequency, large-scale limit. Adding flow is a step in the direction of developing an analytically tractable model with which to understand the basics of OGCMs. We analytically determine whether, and under what circumstances, the CA scheme is responsible for sustained oscillations.We carry out this program for four basic box-model configurations, each inspired by the general shape of the eigenfunctions and constraints of the large-scale zonally-averaged forced flow over a hemisphere. Furthermore, in order to make our results relevant to the Meridional Ocean Circulation, we also investigate the effect of replacing the usual assumption of a linear relation between thermohaline flow rate and horizontal density gradient with a nonlinear hydraulic relationship.We find that a salt oscillator does not occur in the most common box-model configurations. In one of our models, however, we find wide parameter ranges in which all steady states calculated for the model fail to satisfy the CA scheme, the situation which is expected to result in CA-induced oscillations. The model in question corresponds to a hemispheric shallow thermohaline flow over a deep reservoir. However, we find that oscillations occur in these parameter ranges only if the density threshold for convection is negative, i.e., if the CA scheme turns on convection between vertically adjacent boxes when the density stratification between them is still slightly stable. In this situation, the amplitude and period of the oscillations depend strongly on the size of the density threshold, both vanishing as the threshold is taken toward zero. We also show that the same is true in the Welander flip-flop model of a single salt oscillator. For positive values of the threshold, that is, when the CA scheme is allowed to ignore small unstable stratification changes, oscillations do not occur in the limit of integration time step going to zero, but can still be seen when the time step is finite, even if small. Moreover, the system evolves toward a new steady state, one in which the stratification in one box is exactly the threshold value itself. We show how to calculate these new steady states, and explain why they give way to oscillations when the density threshold is negative. |
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