  Surface quasi geostrophy (SQG) describes the two-dimensional active transport
of a temperature field in a strongly stratified and rotating environment.
Besides its relevance to geophysics, SQG bears formal resemblance with various
flows of interest for turbulence studies, from passive scalar and Burgers to
incompressible fluids in two and three dimensions. This analogy is here
substantiated by considering the turbulent SQG regime emerging from
deterministic and smooth initial data prescribed by the superposition of a few
Fourier modes. While still unsettled in the inviscid case, the initial value
problem is known to be mathematically well-posed when regularised by a small
viscosity. In practice, numerics reveal that in the presence of viscosity, a
turbulent regime appears in finite time, which features three of the
distinctive anomalies usually observed in three-dimensional developed
turbulence: (i) dissipative anomaly, (ii) multifractal scaling, and (iii)
super-diffusive separation of fluid particles, both backward and forward in
time. These three anomalies point towards three spontaneously broken symmetries
in the vanishing viscosity limit: scale invariance, time reversal and
uniqueness of the Lagrangian flow, a fascinating phenomenon that Krzysztof
Gawedzki dubbed spontaneous stochasticity. In the light of Gawedzki's work on
the passive scalar problem, we argue that spontaneous stochasticity and
irreversibility are intertwined in SQG, and provide numerical evidence for this
connection. Our numerics, though, reveal that the deterministic SQG setting
only features a tempered version of spontaneous stochasticity, characterised in
particular by non-universal statistics.
