Convective Adjustment and Thermohaline Excitability

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  • 1 Scripps Institution of Oceanography, La Jolla, California
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Abstract

Welander's flip–flop model exhibits oscillations when forced by stochastic white noise (with zero mean) even in the region of parameters where the deterministic system has a globally stable fixed point. Perturbations away from the attracting solutions decay exponentially in time, without any oscillation. Thus, the oscillation that appears when the system is stochastically forced is not related to an eigenfrequency of the linearized system.

The characteristics of this noise-induced oscillation are contrasted with those obtained when the damped harmonic oscillator is forced stochastically. In the case of a stochastically forced damped harmonic oscillator the spectral peak coincides with the frequency of the oscillator, and the amplitude of the oscillations is proportional to the strength of the noise. In the stochastically forced flip-flop model the amplitude of the oscillations is independent of the strength of the noise and the spectral peak moves to lower frequencies as the amplitude of the noise is reduced. Moreover, for noise below a critical threshold, no spectral peak is obtained.

The flip–flop model shares four characteristics with the thermohaline oscillations observed in OGCMs:

  1. The freshwater flux determines whether the system oscillates or settles into a steady state. The period of the oscillations is very sensitive to the freshwater flux and becomes arbitrarily long near the transition from steady to periodic behavior.
  2. The oscillations are of finite amplitude even just past the threshold value of the freshwater flux that separates periodic behavior from a steady equilibrium.
  3. One extremum of the oscillation excursion is close to the value of the steady state that exists below the threshold for transition.
  4. When the deterministic system reaches a steady state, oscillations can be excited by adding a stochastic component to the freshwater flux. The period of the resulting oscillations decreases with increasing noise amplitude, while the amplitude of the oscillations is insensitive to the amplitude of the noise.

Abstract

Welander's flip–flop model exhibits oscillations when forced by stochastic white noise (with zero mean) even in the region of parameters where the deterministic system has a globally stable fixed point. Perturbations away from the attracting solutions decay exponentially in time, without any oscillation. Thus, the oscillation that appears when the system is stochastically forced is not related to an eigenfrequency of the linearized system.

The characteristics of this noise-induced oscillation are contrasted with those obtained when the damped harmonic oscillator is forced stochastically. In the case of a stochastically forced damped harmonic oscillator the spectral peak coincides with the frequency of the oscillator, and the amplitude of the oscillations is proportional to the strength of the noise. In the stochastically forced flip-flop model the amplitude of the oscillations is independent of the strength of the noise and the spectral peak moves to lower frequencies as the amplitude of the noise is reduced. Moreover, for noise below a critical threshold, no spectral peak is obtained.

The flip–flop model shares four characteristics with the thermohaline oscillations observed in OGCMs:

  1. The freshwater flux determines whether the system oscillates or settles into a steady state. The period of the oscillations is very sensitive to the freshwater flux and becomes arbitrarily long near the transition from steady to periodic behavior.
  2. The oscillations are of finite amplitude even just past the threshold value of the freshwater flux that separates periodic behavior from a steady equilibrium.
  3. One extremum of the oscillation excursion is close to the value of the steady state that exists below the threshold for transition.
  4. When the deterministic system reaches a steady state, oscillations can be excited by adding a stochastic component to the freshwater flux. The period of the resulting oscillations decreases with increasing noise amplitude, while the amplitude of the oscillations is insensitive to the amplitude of the noise.
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