How does it work? Physical interpretation of restoring fluxes
The Restoring schemes actually serves as a artificial fluxes, which depends on the model dynamics and the relevent physics, thus it's a derived flux term in addition to nuding process. To be more specific, it does involve in nudging process, but eventually the nudging value will vanish and the coverged restoring fluxes are derived: when model reach the climatology, the nudging term `T^{n\udg\e}` tends to be small, and `Q` term, in turn, represents the fluxes that closing the gap between the missing physical process and reality. To further illuminate the algorithm, consider the simplest case where `m=1` and `n=0`. Recall that when `n=0`:
`q_{1+1/2,1}=Q_{1+1/2,0}`
`Q_{1+1/2,1}=(1/{0+1})sum_{j=1}^{0+1}q_{1+1/2,j} = q_{1+1/2,1}`
Note that for first year, nudging term `a=0`. For simplicity, we will omit the subscript `m`. Then, the `Q` at `n=1` will become:
`Q_{1} = q_{1} `
For the following year `n=1`:
`q_{2}=Q_{1}+a_{1}`
`a_{1}=[T^{cli}-T+sumT^{n\udg\e}]/N`
`Q_{2}=(1/{1+1})sum_{j=1}^{1+1}q_{j} = 1/{1+1}* (q_{1} + q_{2}) = Q_{1} + 1/2*a_{1}`
Next, forward to `n=2`:
`q_{3}=Q_{2}+a_{2}`
`a_{2}=[T^{cli}-T+sumT^{n\udg\e}]/N`
`Q_{3}=(1/{2+1})sum_{j=1}^{2+1}q_{j} = 1/{2+1}* (q_{1} + q_{2} +q_{3}) = Q_{1} + 1/2*a_{1} + 1/3*a_{2}`
The above equations show that the `Q` term will coverge regarding `n`, with the formula:
` Q_{m+1/2,1} = (T_{m+1}^{cli}-T_{m}^{cli})/N`
`"for " n > 1," " Q_{m+1/2,n} = Q_{m+1/2,1} + sum_{j=1}^{n-1}1/{j+1}*a_{m+1/2,j}`
To sum up, the restoring scheme uses instantaneous nudings (from `a = [T^{cli}-T+sumT^{n\udg\e}]/N`) to keep track with the climatology while accumulating the relevent added fluxes, recording these "bias" and further uses this nuding value to construct the artificial fluxes. When model reachs quasi-equilibrium, ideally, the model state will approach climatology and thus the instantaneous nudings tends to vanish, replaced by constructed artificial fluxes which helps model keep track with climatology, representing the missing physical process.
Nevertheless, the fluxes not necessaryly compensate for all the physical process since in the calculation we also have to consider the time scale `tau` for nudging process. Logically, with higher resolution and more detail physics configuration, the `tau` should become smaller, merely as a aid for fixing the unresolved process.