Heat Extremes Driven by Amplification of Phase‐Locked Circumglobal Waves Forced by Topography in an Idealized Atmospheric Model

Abstract Heatwaves are persistent temperature extremes associated with devastating impacts on human societies and ecosystems. In the midlatitudes, amplified quasi‐stationary Rossby waves have been identified as a key mechanism for heatwave occurrence. Amplified waves with preferred longitudinal locations lead to concurrent extremes in specific locations. It is therefore important to identify the essential components in the climate system that contribute to phase‐locking of wave patterns. Here, we investigate the role of dry atmospheric dynamics and topography in causing concurrent heatwaves by using an idealized general circulation model. Topography is included in the model experiments as a Gaussian mountain. Our results show that amplified Rossby waves exhibit clear phase‐locking behavior and a decrease in the zonal phase speed when a large‐scale localized topographic forcing is imposed, leading to concurrent heat extremes at preferred locations.


Introduction
The supporting information provides more details about the model climatology and the methods employed in this study. It also contains supplementary figures to support some of the conclusions in the main manuscript.
The model zonal wind climatology at 300h Pa (u300) for the 4 model experiments presented in the study this shown alongside with the ERA-Interim reanalysis in Figure S1.
The variability in the wave amplitude is shown as box plots for different zonal wavenum-X -2 : bers for the model runs and reanalysis is shown in Figure S2. We also include the same composite analysis of T1000, v300 and relative change heatwave frequency of Figure 3 in the main manuscript but for the two other topography experiments ( Figures S4 and S5). Figure S6 complements Figure 4 in the main text by displaying the composited time evolution of wavenumbers k=4,7 which are related to weaker anomalies than k=5,6 (shown in the main text). Figure S7 also complements Figure 4 in the main text by comparing the climatology of v300 with the total v300 field during amplification events of wavenumbers 5 and 6. The temporal evolution of the amplitudes of single zonal renumbers for the 8km, 25 • N) simulation is shown in Figure S8. Finally, the duration of the wave amplification events is shown in Figure S9 for the different model simulations. The implications of these figures are discussed in the main manuscript.

Phase-locking index
We use the meridional wind at 300 hPa (v300) to identify Rossby waves in the midlatitude troposphere. To quantify phase-locking we define the index δ m as the narrowest window in the phase space that contains half of the probability density. Numerically we start with a window of width δ = π and calculate the probability density for all the possible windows of width delta: where Φ 0 ∈ [−π, π] is the left limit of the integration window and p(Φ) is the normalized probability density. Then we identify the window that contains the maximum probability P (δ, Φ 0 ). If max[P (δ, Φ 0 )] > 0.5 then we decrease δ by a small amount (2π/100) and repeat the procedure until we obtain δ = δ m , where δ m corresponds to the minimum value of δ, for which max(P (δ m , Φ 0 )) > 0.5. Note that due to the size of the bins we choose, the value of δ m has an accuracy of π/50 ≈ 0.063 radians.
We also quantify the uncertainty in the δ m value by applying a bootstrapping methodology. We select 1000 different random sub-samples consisting of 1079 blocks of 10 consecutive days each (10790 days per sub-sample) and calculate the value of δ m for each of these sub-samples. The minimum size of the blocks is made in order to account for the auto-correlation of the time series. Using just a random subset, without requesting a minimum size of the blocks, leads to an underestimation of the uncertainty of the δ m value. Each of the 1000 estimates of δ m represent a probability distribution function which is used to estimate the maximum and minimum values of the distribution of δ m .
These limits are shown in Figure S3 and represent the uncertainty in δ m in our model experiments.

Hayashi spectra
To estimate the phase speed c of the waves, the frequency -wavenumber spectra of the unfiltered v300 daily anomalies are computed for each latitudinal band using a 2dimensional FFT. Note that for this analysis we do not use the 7-day running mean averaging as in the the previous analysis in order to represent the entire spectrum of X -4 : waves. The Hayashi spectrum (Hayashi, 1979) represents the power density spectrum of v300 in the phase speed (c) -wavenumber (k) phase space. We use the original method by Randel and Held (1991) to linearly interpolate from the wavenumber -frequency to the wavenumber -phase speed space, using a grid of resolution ∆c = 0.3m/s. The obtained values are multiplied by k/a cos θ to conserve the total power, where a is the Earth's radius and θ is latitude. Finally, we average over the the 30-60