Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model

Figure 1.  Shown is a schematic of the computational domain $ \Omega $

Figure 2.  The series of snapshots shows the high-resolution solution $ q^f $ computed on the fine grid $ G^f = 2049\times1025 $ ($ dx\approx dy\approx 1.9\, {\rm km} $), the true solution $ q^a $ computed on the coarse grid $ G^c = 257\times129 $ ($ dx\approx dy\approx 15\, {\rm km} $), and the low-resolution solution $ q^m $ also computed on $ G^c $ by simulating the QG model for the low drag $ \boldsymbol{\mu = 4\times10^{-8}\, {\rm s^{-1}}} $ (heterogeneous flow). All the solutions are given in units of $ [s^{-1}f^{-1}_0] $, where $ f_0 = 0.83\times10^{-4}\, {\rm s^{-1}} $ is the Coriolis parameter. In order to visualize the solutions on the same color scale we have multiplied the ones in the second layer by a factor of 5

Figure 3.  The same as in Figure 2, but for $ G^c = 129\times65 $ ($ dx\approx dy\approx 299\, {\rm km} $)

Figure 4.  The series of snapshots shows the dependence of the solution on the resolution for the low drag $ \boldsymbol{\mu = 4\times10^{-8}\, {\rm s^{-1}}} $ (heterogeneous flow). All the solutions are given in units of $ [s^{-1}f^{-1}_0] $, where $ f_0 = 0.83\times10^{-4}\, {\rm s^{-1}} $ is the Coriolis parameter. In order to visualize the solutions on the same color scale we have multiplied the ones in the second layer by a factor of 5

Figure 5.  The series of snapshots in the figure shows the high-resolution solution $ q^f $ computed on the fine grid $ G^f = 2049\times1025 $ ($ dx\approx dy\approx 1.9\, {\rm km} $), the true solution $ q^a $ computed on the coarse grid $ G^c = 129\times65 $ ($ dx\approx dy\approx 299\, {\rm km} $), and the low-resolution solution $ q^m $ computed on the coarse grid $ G^c $ by simulating the QG model for the high drag $ \boldsymbol{\mu = 4\times10^{-7}\, {\rm s^{-1}}} $ (homogeneous flow). All the solutions are given in units of $ [s^{-1}f^{-1}_0] $, where $ f_0 = 0.83\times10^{-4}\, {\rm s^{-1}} $ is the Coriolis parameter. In order to visualize the solutions on the same color scale we have multiplied the ones in the second layer by a factor of 5

Figure 6.  Shown is a typical dependence of the area of the stochastic cloud $ A^c $ on the size of the stochastic ensemble $ \overline{\bf x} $. The left and right column shows the area of the stochastic cloud (marked in grey color) which consists of $ N = 100 $ and $ N = 400 $ ensemble members, respectively. The stochastic ensemble has been computed for the first 64 leading EOFs capturing 96% of the flow variability (top row) and the first 128 leading EOFs capturing 99% of the flow variability (bottom row). The true solution $ \bar{\bf x}^c $ is marked with a black dot. The plot represents a typical part of the computational domain of size $ [10,45]\times[45,65] $ in the first layer, which can be divided into two regions: a fast flow region (the boundary layer along the northern boundary $ [10,45]\times[60,65] $, the jet occupying the domain $ [10,45]\times[45,52] $) and a slow flow region $ [10,45]\times(52,60) $

Figure 7.  Shown are (a) instantaneous and (b) time-averaged normalized velocity fields for the heterogeneous flow

Figure 8.  Shown is the dependence of the averaged area of the stochastic cloud for the slow, $ \overline{A}^c_s $, and fast, $ \overline{A}^c_f $, flow regions on the number of EOFs, $ K $, and the size of the stochastic ensemble $ N $

Figure 9.  Shown is the dependence of $ \widetilde{R}_{\mathcal{S}} $ for the velocity component $ u^a $ ($ v^a $ is not shown, since it behaves qualitatively similar to $ u^a $), stream function $ \psi^a $, and PV anomaly $ q^a $ on (a) the number of EOFs $ K $ ($ N = 100 $ in this case) and (b) size of the stochastic ensemble $ N $ ($ K = 1 $ in this case) over the time period $ T = [0,21] $ days for the heterogeneous flow in Figure 3; $ G^c = 129\times65 $. Using $ K = \{1,2,4,8,16,32,64\} $ leading EOFs allows to capture 23%, 42%, 60%, 77%, 89%, 96%, and 99% of the flow variability, respectively. The initial conditions for the stochastic model have been computed over the spin up period $ T_{\rm spin} = [-8,0] $ hours

Figure 10.  The same as in Figure 9, but for $ \widetilde{R}_{\mathcal{S}_{\sigma}} $

Figure 11.  The same as in Figure 9, but for $ G^c = 257\times129 $. We show $ v^a $ component of the velocity field, since it behaves differently from $ u^a $. For the higher resolution, using $ K = \{1,2,4,8,16,32,64\} $ leading EOFs allows to capture 39%, 61%, 77%, 90%, 97%, 99%, and 99.8% of the flow variability, respectively

Figure 12.  The same as in Figure 11, but for $ \widetilde{R}_{\mathcal{S}_{\sigma}} $

Figure 13.  Evolution of $ \overline{<\widetilde{T}_{\mathcal{S}_{\sigma}}>} $ for the heterogeneous flow and for different number of EOFs: $ K = 1 $ (solid line), $ K = 2 $ (solid line marked by a cross), $ K = 4 $ (solid line marked by an circle), as well as for different sizes of the stochastic ensemble: $ N = 100 $ (solid line), $ N = 200 $ (solid line marked by an asterisk), $ N = 400 $ (solid line marked by a square)

Figure 14.  Evolution of the root mean square error $ \overline{<RMSE(|\boldsymbol{u}|)>} $ (solid line) and standard deviation $ \overline{<\sigma(|\boldsymbol{u}|)>} $ (dashed line) of the ensemble mean for the heterogeneous flow and different number of EOFs: $ K = 1 $ (solid/dashed line), $ K = 2 $ (solid/dashed line marked by a cross), $ K = 4 $ (solid/dashed line marked by an circle); and for different sizes of ensemble: $ N = 100 $ (solid/dashed line), $ N = 200 $ (solid/dashed line marked by an asterisk), $ N = 400 $ (solid/dashed line marked by a square). Note that the initial ensemble is biased

Figure 15.  The same as in Figure 14, but for an unbiased ensemble. The results for the homogeneous flow look very similar

Figure 16.  Evolution of the root mean square error, $ \overline{<RMSE(|\boldsymbol{u}|)>} $ (solid line), and standard deviation, $ \overline{<\sigma(|\boldsymbol{u}|)>} $ (dashed line), of the ensemble mean for the heterogeneous flow computed for $ K = 1 $, $ N = 100 $ and different amplitudes of the noise $ dW $: $ |dW| = 1 $ (solid/dashed line), $ |dW| = 5 $ (solid/dashed line marked by a cross), $ |dW| = 10 $ (solid/dashed line marked by an circle). Not that the initial ensemble is unbiased. The results for the homogeneous flow look very similar

Figure 17.  Shown is the dependence of $ R_{\mathcal{S}} $ (deterministic case) for the velocity component $ u^a $ ($ v^a $ is not shown, since it behaves qualitatively similar to $ u^a $), stream function $ \psi^a $, and PV anomaly $ q^a $ on (a) the number of EOFs $ K $ ($ N = 100 $ in this case) and (b) size of the stochastic ensemble $ N $ ($ K = 1 $ in this case) over the time period $ T = [0,21] $ days for the heterogeneous flow presented in Figure 3; $ G^c = 129\times65 $. The initial conditions for the deterministic model have been computed over the spin up period $ T_{\rm spin} = [-8,0] $ hours. The results for the homogeneous flow look very similar

Figure 18.  The same as in Figure 17, but for $ G^c = 257\times129 $. The results for the homogeneous flow look very similar

Figure 19.  Evolution of $ \overline{<\widetilde{T}_{\mathcal{S}_{\sigma}}>} $ (solid line; stochastic case) and $ \overline{<T_{\mathcal{S}_{\sigma}}>} $ (dashed line; deterministic case) for the heterogeneous flow; the number of leading EOFs is $ K = 1 $, and the size of the ensemble is $ N = 100 $. The results for the homogeneous flow look very similar

Figure 20.  Typical rank histograms for heterogeneous flow velocity $ \mathbf{u} = (u,v) $ at different locations (not shown) and resolutions. Note that a pair of histograms at different resolutions shares the same location in the computation domain. Each histogram is based on 1000 forecast-observation pairs generated by solving the stochastic QG model. For simulating the stochastic QG model we use 100 stochastic solutions and 32 leading EOFs. Each stochastic solution for the rank histogram is selected randomly from the ensemble every 4 hours. The rank histograms at the higher resolution are flatter and t hus the forecasts are more reliable, although this resolution is still much less than the reference simulation

Figure 21.  The series of snapshots shows the true deterministic solution $ q^a_1 $ computed on $ G^c = \{129\times65 $, $ 257\times129 \}$, and the low-resolution parameterised solution $ \bar{q}^p_1 $ (averaged over the stochastic ensemble of size $ N = 100 $) computed on the same coarse grids by simulating the stochastic QG model from a randomly perturbed zero initial condition for the heterogeneous flow; the parameterised solution uses 32 leading EOFs. All solutions are given in units of $ [s^{-1}f^{-1}_0] $, where $ f_0 = 0.83\times10^{-4}\, {\rm s^{-1}} $ is the Coriolis parameter. The true solutions correspond to day 1 and presented here just to show the structure of the flow

Figure 22.  The series of snapshots shows the true deterministic solution $ q^a_1 $, modelled solution $ q^m_1 $ computed with the deterministic QG model, and parameterised solution $ \bar{q}^p_1 $ (averaged over the stochastic ensemble of size $ N = 100 $) computed with the stochastic QG model. All solutions were computed on the same grid $ G^c = 129\times65 $, have the same initial condition, and the parameterised solution uses 32 leading EOFs. All fields are given in units of $ [s^{-1}f^{-1}_0] $, where $ f_0 = 0.83\times10^{-4}\, {\rm s^{-1}} $ is the Coriolis parameter

Figure 23.  The same as in Figure 22, but for $ G^c = 257\times129 $

Figure 24.  Shown is the relative $ l_2 $-norm error between the true deterministic solution, $ {\bf u}^a $, and modelled solution $ {\bf u}^m $ (red line) and parameterised solution $ {\bf u}^p $ (blue line) as a function of time