Advancing Eddy Parameterizations: Dynamic Energy Backscatter and the Role of Subgrid Energy Advection and Stochastic Forcing

Viscosity in the momentum equation is needed for numerical stability, as well as to arrest the direct cascade of enstrophy at grid scales. However, a viscous momentum closure tends to over-dissipate eddy kinetic energy. To return excessively dissipated energy to the system, the viscous closure is equipped with what is called dynamic kinetic energy backscatter. The amplitude of backscatter is based on the amount of unresolved kinetic energy (UKE). This energy is tracked through space and time via a prognostic equation. Our study proposes to add advection of UKE by the resolved flow to that equation to explicitly consider the effects of nonlocality on the subgrid energy budget. UKE can consequently be advected by the resolved flow before it is reinjected via backscatter. Furthermore, we suggest incorporating a stochastic element into the UKE equation to account for missing small-scale variability, which is not present in the purely deterministic approach. The implementations are tested on two intermediate complexity setups of the global ocean model FESOM2: an idealized channel setup and a double-gyre setup. The impacts of these additional terms are analyzed, highlighting increased eddy activity and improved flow characteristics when advection and carefully tuned, stochastic sources are incorporated into the UKE budget. Additionally, we provide diagnostics to gain further insights into the effects of scale separation between the viscous dissipation operator and the backscatter operator responsible for the energy injection. Oceanic swirls or "eddies" have a typical size of 10-100 km, which is close to the smallest scales that global ocean models commonly resolve. For physical and numerical reasons, these models require the addition of artificial terms that influence the flow near its smallest scales. Common approaches have the drawback of introducing systematic loss of kinetic energy contained in the eddies, which leads to errors that also affect the oceanic circulation on global scales. In our research, we compensate for this error by returning some of the missing energy back into the simulation, using a so-called kinetic energy backscatter scheme. In this work, we continue the development of an already existing and successful backscatter scheme, adding certain improvements to the way energy is budgeted and returned to the flow: we ensure that the local energy budget is attached to each fluid parcel as it is transported by the large-scale flow, and we also add a random forcing term that mimics unknown sources of such energy to bring its statistical properties closer to reality. We demonstrate that these modifications effectively improve the characteristics of the simulated flow. Extension of the subgrid energy equation of the kinetic energy backscatter parameterization by adding advection and a stochastic term Both additional terms improve several flow characteristics in two idealized test cases, a channel and a double-gyre Scale analysis reveals the necessity of sufficient scale separation between viscous energy dissipation and energy injection via backscatter