  The Ladyzhenskaya-Smagorinsky equations model turbulence phenomena, and are
given by $$\frac{\partial \boldsymbol{u}}{\partial t}-\mu
\mathrm{div}\left(\left(1+|\nabla\boldsymbol{u}|^2\right)^{\frac{p-2}{2}}\nabla\boldsymbol{u}\right)+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\nabla
p=\boldsymbol{f}, \ \nabla\cdot\boldsymbol{u}=0,$$ for $p\geq 2,$ in a bounded
domain $\mathcal{O}\subset\mathbb{R}^d$ ($2\leq d\leq 4$). In this work, we
consider the stochastic Ladyzhenskaya-Smagorinsky equations with the damping
$\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-2}\boldsymbol{u},$ for $r\geq 2$
($\alpha,\beta\geq 0$), subjected to multiplicative Gaussian noise. We show the
local monotoincity ($p\geq \frac{d}{2}+1,\ r\geq 2$) as well as global
monotonicity ($p\geq 2,\ r\geq 4$) properties of the linear and nonlinear
operators, which along with an application of stochastic version of
Minty-Browder technique imply the existence of a unique pathwise strong
solution. Then, we discuss the small time asymptotics by studying the effect of
small, highly nonlinear, unbounded drifts (small time large deviation
principle) for the stochastic Ladyzhenskaya-Smagorinsky equations with damping.
