Non-unique self-similar blowups in shell models: insights from dynamical systems and machine-learning

Abstract Strong numerical hints exist in favor of a universal blowup scenario in the Sabra shell model, a popular cascade model of 3D turbulence, which features complex velocity variables on a geometric progression of scales <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>∝</mml:mo> <mml:msup> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . The blowup is thought to be of self-similar type and characterized by the finite-time convergence towards a universal profile with non-Kolmogorov (anomalous) small-scale scaling <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>∝</mml:mo> <mml:msubsup> <mml:mi>ℓ</mml:mi> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> . Solving the underlying nonlinear eigenvalue problem has however proven challenging, and prior insights mainly used the Dombre–Gilson renormalization scheme, transforming self-similar solutions into solitons propagating over infinite rescaled time horizon. Here, we further characterize Sabra blowups by implementing two strategies targeting the eigenvalue problem. The first involves formal expansion in terms of the bookkeeping parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>log</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>λ</mml:mi> </mml:mrow> </mml:math> , and interpretes the self-similar solution as a (degenerate) homoclinic bifurcation. Using standard bifurcation toolkits, we show that the homoclinic bifurcations identified under finite-truncation of the series converge to the observed Sabra solution. The second strategy uses machine-learning optimization to solve directly for the Sabra eigenvalue. It reveals an intricate phase space, with the presence of a continuous family of non-universal blowup profiles, characterized by various number N of pulses and exponents <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mtext>⩾</mml:mtext> <mml:mi>x</mml:mi> </mml:mrow> </mml:math> .