with Stephan Juricke, Peter DΓΌben, Tim Palmer
The exact equations are known except for some caveats.

Dynamical core intercomparison to evaluate their performance should also consider their inherent uncertainty.
$$\frac{\partial u_j}{\partial t} + \frac{\phi_{j+1}  \phi_{j1}}{2\Delta x} = 0$$
Radiation and convection parameterized as
$$Q = \kappa(TT_{eq})$$
Model initialised at T = 255 K uniformly everywhere.
U, V = 0 everywhere. Resolution: T159.
Mean winds and temperature over last 1000 days of 1500 days simulation
$$ F_\psi = (1)^{q+1} K_{2q}\nabla^{2q}\psi $$
where $q>1$ is a positive integer, 2q  the order of the diffusion, $K_{2q}$  diffusion coefficient and $\nabla$ is the gradient operator.
We now compare the use of biharmonic (q=2) and hyper (q=4) diffusion in a HeldSuarez testcase.
Uncertainty in the diffusion operator translates into uncertainty in model solution
Uncertainty in the diffusion operator translates into uncertainty in model solution.
700 hPa Kinetic energy spectra (biharmonic vs hyper diffusion)
The kinetic energy spectra are comparable except at large wavenumbers, the hyper diffusion has lesser energy.
The net parameterized physics tendency: $$πΏ= π_U, π_π, π_π, π_π$$ are perturbed with multiplicative noise $$πΏβ² = (1 + \mu π)πΏ$$
SPPT is currently implemented at ECMWF, NCEP, NCAR, UKMO, JMA and others.
Stochastic tendencies of zonal and meridional wind saved from an aquaplanet simulation of ECMWF IFS
$$\frac{\partial V}{\partial t} =  U . \nabla(V) + Mixing + \sigma \mathcal{N}(0,1)$$
!===================================================================================
! Stochastic Perturbations to U, V
!===================================================================================
IF (LHELDSUAREZ_STOCH) THEN
! WRITE(NULOUT,*) 'Starting Loop for stochastic forcing IBL:',IBL
! CALL FLUSH(NULOUT)
DO JK=1,KLEV
DO JL=KIDIA,KFDIA
PTENU(JL,JK) = PTENU(JL,JK) + TRLXQ(JL,JK,1,IBL)
PTENV(JL,JK) = PTENV(JL,JK) + TRLXQL(JL,JK,1,IBL)
ENDDO
ENDDO
! WRITE(NULOUT,*) 'Stoch update sucessfully finished'
! CALL FLUSH(NULOUT)
ENDIF
Stochastic (SPPT) perturbations to dynamical tendencies do not change the mean climate significantly.
Stochastic (SPPT) perturbations to dynamical tendencies do not change the eddy variability significantly either.
Kinetic energy spectra (700 hPa)
Stochastic perturbation run (right) comparable to the deterministic run (left)
$$ F_\psi = (1)^{q+1} K_{2q}\nabla^{2q}\psi $$
where $q>1$ is a positive integer, 2q denotes the order of the diffusion, $K_{2q}$ stands for the diffusion coefficient and $\nabla$ is the gradient operator.
Stochastic diffusion coefficient is given by
$$ F'_\psi = (1 + r \mu) F'_\psi $$ where r and $\mu$ are the random perturbations and standard deviation terms.
Stochastic perturbations to diffusion coefficient do not change the mean climate significantly either. Same order of magnitude as using a higher order diffusion
Stochastic perturbations to diffusion coefficient do not change the eddy variability significantly either. Same order of magnitude as using a higher order diffusion
Kinetic energy spectra (700 hPa)
Stochastic perturbations to $F_\psi$ (right) is comparable to the deterministic run (left)