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Supplemental Material, sj-pdf-1-ppg-10.1177_03091333221079201 for Tracking the methodological evolution of climate change projections for UK river flows by Wilson CH Chan, Theodore G Shepherd, Katie Facer-Childs, Geoff Darch and Nigel W Arnell in Progress in Physical Geography: Earth and Environment

Supplemental Material, sj-pdf-1-ppg-10.1177_03091333221079201 for Tracking the methodological evolution of climate change projections for UK river flows by Wilson CH Chan, Theodore G Shepherd, Katie Facer-Childs, Geoff Darch and Nigel W Arnell in Progress in Physical Geography: Earth and Environment

We study the inverse energy transfer in forced two-dimensional (2D) Navier--Stokes turbulence in a doubly periodic domain. It is shown that an inverse energy cascade that carries a nonzero fraction of the injected energy to the large scales via a power-law energy spectrum $\propto k^{-\alpha}$ requires that $\alpha\ge5/3$. This result is consistent with the classical theory of 2D turbulence that predicts a $k^{-5/3}$ inverse-cascading range, thus providing for the first time a rigorous basis for this important feature of the theory. We derive bounds for the Kolmogorov constant $C$ in the classical energy spectrum $E(k)=C\epsilon^{2/3}k^{-5/3}$, where $\epsilon$ is the energy injection rate. Issues related to Kraichnan's conjecture of energy condensation and to power-law spectra as the quasi-steady dynamics become steady are discussed., Comment: 14 pages, submitted to JFM

The midlatitude, or eddy-driven jet stream is a strong westerly air current that steers weather systems in the midlatitudes. It is therefore one of the atmospheric circulation features that has the most impact on humankind. Understanding if and how it will change in response to anthropogenic forcing is thus crucially important. However, the spread in its response to global warming in state-of-the-art models is large. At the same time, our theoretical understanding is limited. Past work has therefore examined the possibility of constraining future responses from historical jet behaviour. One approach involves finding an inter-model relationship between an observable historical variable and a future response, often called an emergent constraint. Theoretical grounding for a previously proposed emergent constraint between annular mode timescale and future jet shift is provided by the fluctuation-dissipation theorem (FDT), which links internal variability timescales of an unforced system to its forced response. The contribution of this thesis is twofold. First we revisit the application of the FDT to the midlatitude jet stream response, where we propose a new algorithm to improve predictions made from limited datasets. The second part of this work reexamines two previously proposed emergent constraints. The first of these is derived from a simplified version of the FDT, and links the Southern Annular Mode (SAM) timescale to the jet response. Previous work has reached conflicting conclusions and at present it is unclear whether the SAM timescale can constrain the jet response; our analysis indicates that it cannot, and that the SAM timescale is inflated by a non-stationarity associated with the seasonal breakdown of the stratospheric vortex. The second previously proposed constraint links the climatological jet latitude to the future jet shift in austral winter. We question this constraint by showing that the emergent constraint only holds in the zonal mean, but not in separate longitudinal sectors. Furthermore, we demonstrate that the zonal mean jet latitude does not reflect the position of a physical zonal structure, owing to strong zonal asymmetries. We argue that this can explain the origin of the emergent constraint, but also questions its usefulness in constraining future response. Open Access