Milankovitch Theory “as an Initial Value Problem”

Abstract. The dynamics of large ice sheets is fundamentally defined by the advection of mass and temperature. The timescale of these processes is critically dependent on the surface mass balance. Because of the ice-climate system's nonlinearity, its response to the orbital forcing in terms of engagement of negative and positive feedbacks is not symmetrical. This asymmetry may reduce the effective mass influx, and the resultant advection timescale may become longer, which is equivalent to a longer system’s memory of its initial conditions. In this case the Milankovitch theory becomes an initial value problem: Depending on initial conditions, for the same orbital forcing and for the same balance between terrestrial positive and negative feedbacks, the historical glacial rhythmicity could have been dominated either by the eccentricity period of ~100 kyr, or by the doubled obliquity period of ~80 kyr, or by a combination of both. In fact, empirical records demonstrate that the dominant period of the Late Pleistocene ice ages evolved from ~80-kyr to ~100-kyr rhythmicity. The quantitative similarity of this dominant-period trajectory and the one, made by the long-memory model, suggests that the records of the Late Pleistocene glacial rhythmicity could have been produced by a long-memory initial-value-dependent climate system, or, in other words, the slopes in empirical dominant-period trajectories are signatures of a long memory.

The scaling law of the dominant-period trajectory provides a theoretical insight into the discovered phenomenon. It reveals that this trajectory is dependent on the memory duration that is sensitive to initial conditions. The sensitivity of the memory duration to initial values emerges as the result of system’s incomplete similarity in two similarity parameters colliding into one conglomerate similarity parameter that is the ratio of the advection timescale and the orbital period. The critical dependence of this similarity parameter on poorly defined accumulation-minus-ablation mass balance as well as its dependence on initial values makes ice ages to be hardly predictable and disambiguation of paleo-records to be extremely challenging. The quasi-eccentricity periods produced by the long-memory system in response to pure obliquity forcing make a remarkable example of this challenge because in the time series they may be naively attributed to the eccentricity modulated precession forcing.