The stable deterministic steady state periodic cycle around 2.0 showing that the periodic forcing cos(2πt) determines the periodicity of the final solution.
Comparison of the analytical solutions (dashed lines) and the numerical solutions (thick lines) for several key statistical moments: (a) the standard deviation, (b) the deviation of the stochastic means from the deterministic solution x S (t), and (c) the skewness. The case of σ = 0.2 (σ = 0.4) is given by the blue (red) lines.
Comparison of several key statistical moments between the analytic solutions and the numerical solutions in the case of constant additive noise. Thick lines represent the results from the numerical simulations and dashed lines from the analytical solutions. The four colors show the statistical moments for four different ΔF 0, 10 (black), 12 (blue), 14 (green), and 16 (red). The moments are (a) the standard deviation, (b) the deviation of the stochastic mean from the deterministic solution, and (c) the skewness. The value of σ is 0.05.
Same as Figure 3 but with multiplicative noise of magnitude σE.
The memory effect, 1/(e 2γ − 1) (red line) and γ (blue line) for different values of ΔF 0. The memory effect is mainly controlled by γ and hence the two quantities depend similarly on ΔF 0.
The seasonal cycle of the standard deviation of Arctic sea ice thickness with different ΔF 0.
The dependence of the shape of V upon the signs of c and d. The dotted line represents V for d = 0. When c is positive (negative), a given perturbation due to noise forcing grows (decays). The term amplifies the asymmetries as follows. If d(t) is positive (negative), the stochastic solutions tend to move to the positive (negative) side.
For each ΔF 0, d(t) and are shown as the blue solid line and the red dotted line, respectively. The deviation of the stochastic mean is the product of d(t) and at each time. The largest contribution comes from the period between June and November when the standard deviation and the asymmetry d(t) are maximal. The peaks shown in d(t) are approximate representations of delta functions generated by differentiation of a discontinuity. (a) ΔF 0 = 10, (b) ΔF 0 = 13, (c) ΔF 0 = 15, and (d) ΔF 0 = 16.
The deviation of the stochastic means from the deterministic solutions for different values of ΔF 0. For lower ΔF 0, the deviation is negative during the entire year. However, at higher values of ΔF 0, a positive deviation appears during summer.
The skewness of η2 for different values of ΔF 0. For lower ΔF 0, the skewness is negative during the entire year. However, for higher ΔF 0, the skewness becomes positive during the summer.
The seasonal standard deviation of η1 for several values of ΔF 0 at O(σ). As ΔF 0 increases, the overall standard deviation decreases. But, it is reentrant as sea ice-albedo feedback becomes more effective during summer, and the standard deviation begins to grow with ΔF 0.
The seasonal cycle of the deviation of the stochastic mean for different values of ΔF 0 at O(σ2). Relative to the constant noise case, the effect of the multiplicative noise dominates that of the nonlinear asymmetry in the deterministic solution, and the means shift toward negative values. When ΔF 0 is small, the deviation is negative all season, but as ΔF 0 increases, the sea ice albedo feedback becomes more effective and the deviation becomes positive in summer.
The comparison between (blue) and (red) for a ΔF 0 of 15 and 16, where the sea ice-albedo feedback is influential.
The skewness for the multiplicative noise case for several values of ΔF 0 at O(σ). Unlike the constant noise case, the skewness is negative during the entire season as ΔF 0 increases.
The comparison of (blue) and (red) for a ΔF 0 of 15 and 16. Even though , which represents the contributions of the asymmetry embodied in d(t), are positive the contributions from the multiplicative nature of the noise forcing ( ) are negative and of larger magnitude.
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