  Fully turbulent flows are characterized by intermittent formation of very
localized and intense velocity gradients. These gradients can be orders of
magnitude larger than their typical value and lead to many unique properties of
turbulence. Using direct numerical simulations of the Navier-Stokes equations
with unprecedented small-scale resolution, we characterize such extreme events
over a significant range of turbulence intensities, parameterized by the
Taylor-scale Reynolds number ($R_\lambda$). Remarkably, we find the strongest
velocity gradients to empirically scale as $\tau_K^{-1} R_\lambda^{\beta}$,
with $\beta \approx 0.775 \pm 0.025$, where $\tau_K$ is the Kolmogorov time
scale (with its inverse, $\tau_K^{-1}$, being the {r.m.s.} of velocity gradient
fluctuations). Additionally, we observe velocity increments across very small
distances $r \le \eta$, where $\eta$ is the Kolmogorov length scale, to be as
large as the {r.m.s.} of the velocity fluctuations. Both observations suggest
that the smallest length scale in the flow behaves as $\eta
R_\lambda^{-\alpha}$, with $\alpha = \beta - \frac{1}{2}$, which is at odds
with predictions from existing phenomenological theories. We find that extreme
gradients are arranged in vortex tubes, such that strain conditioned on
vorticity grows on average slower than vorticity, approximately as a power law
with an exponent $\gamma < 1$, which weakly increases with $R_\lambda$. Using
scaling arguments, we get $\beta=(2-\gamma)^{-1}$, which suggests that $\beta$
would also slowly increase with $R_\lambda$. We conjecture that approaching the
limit of infinite $R_\lambda$, the flow is overall smooth, with intense
velocity gradients over scale $ \eta R_\lambda^{-1/2}$, corresponding to $\beta
= 1$.
