Leonhard Scheck, Sarah Jones, Martin Baumann, Vincent Heuveline
Forecasting the development and motion of tropical cyclones (TCs) and their impact on the midlatitude flow presents a severe challenge for numerical weather prediction. A major roadblock to progress in this area is that horizontal scales from above several thousand to below one kilometre must be considered. Goal-oriented adaptive methods are a promising way to tackle such multi-scale problems. This approach allows e.g. for an automatic detection and refinement of the grid cells that contribute most stongly to the error in the TC position. In the framework of the research program SPP 1276 Metström IMK-TRO and the Engineering Mathematics and Computing Lab (EMCL) apply goal-oriented adaptive methods to TC-related problems. While at EMCL adaptive techniques are developed and implemented, idealised test cases related to tropical cyclones are defined and investigated at IMK-TRO. Together we set up and evaluate adaptive model runs.
Idealised tropical cyclone scenarios
Idealised problems are not only well-suited as test cases, but provide also a clearer view on selected processes than more realistic, but hard to interpret full-physics model results. We investigated several idealised scenarios in which processes are at work that are important for the evolution and motion of real TCs. In the following we present two examples.
Cyclone-front interaction in a barotropic model
This scenario deals with the interaction between a TC and a jetstream, a common process for TCs that encounter the mid-latitude flow. In a barotropic model a jet-like zonal velocity profile can be generated by adopting a basic state consisting of two regions with constant absolute vorticity separated by a discontinuity. The latter can be interpreted as a tropopause front. For low absolute vorticity values south and high values north of the front a eastward directed jet results, with maximum wind speeds at the discontinuity. To this basic state vorticity field we add the vorticity distribution of a TC-like barotropic vortex. The basic process by which the TC interacts subsequently with the tropopause front (and the associated jet) works as follows (see Fig. 1): The TC circulation deforms the vorticity disconinuity at the front, thereby creating vorticity anomalies. This process can also be interpreted as the excitation of a frontal wave. The circulation associated with the vorticity anomalies causes an advection of the TC (initially northward) and leads to a downstream propagation of the frontal wave.
Fig. 1: Interaction between a tropical cyclone and a tropopause front.
Experiments with different vortex profiles and jet speeds indicate that mainly resonant frontal waves are excited, i.e. waves with a phase speed that matches the zonal translation velocity of the TC. We investigated also the more realistic case, where frontal waves are present in the initial state and modify the cyclone-jet interaction. In particular, we studied the unexpectedly high sensitivity of the cyclone track with respect to its inital position relative to the wave. An example for this extreme sensitivity is shown in Fig. 2. A displacement of the TC in the initial conditions by only 50km leads to vastly different cyclone tracks.
Fig. 2: Absolute vorticity distributions after 7 days (left column) and 14 days (right column) for two model runs (upper / lower row) in which the initial TC position relative to the frontal wave differs by about 50km. Movies showing the full time evolution of the vorticity distribution are available in the following formats:
We were able to show that a bifurcation process takes place for cyclone trajectories that come close to a specific location on the trough axis. The bifurcation point is located at a latitude where the phase speed of the wave matches the zonal translation speed of the cyclone. As the meridional flow component caused by the frontal wave is zero on the trough axis, cyclones close to the bifurcation point move very slowly, relative to the wave. For cyclones that approach the bifurcation point arbitrarily small changes in their initial position determine whether they are advected northwards and accelerated, or advected southwards and decelerated. A second point on the ridge axis (point R in Fig. 3) fulfills the same conditions (wave speed matches translation speed, no meridional flow) but the flow in its vicinity tends to steer all cyclones into similar tracks orbiting around the point. The influence of these two points becomes clear when cyclone tracks are plotted in the frame moving with the wave (Fig. 3).
Fig. 3: Cyclone tracks in the frame comoving with the frontal wave for 44 model runs with different initial cyclone longitudes. The y-axis inidicates the meridional cyclone position in kilometers, the x-axis is the phase of the wave at the zonal position of the cyclone in units of π. The frontal wave (dashed) is thus fixed in this plot. The theoretical position (derived neglecting the excitation of waves by the cyclone) of the bifurcation point is marked with T, the point on the ridge axis with R. The blue tracks are from cyclones separated at the actual bifurcation point, which is located slightly NW of point T. The red tracks are from cyclones that are first advected around point R and then separated, when they approach the bifurcation point from the other side.
The results of a parameter study with varying frontal wave parameters indicate that bifurcations are possible only for a limited range of wavelengths that depends on the dispersion relation of the frontal waves. The largest effects are obtained for resonant frontal waves propagating with a phase speed matching the initial zonal translation speed of the cyclone.
- Scheck, L., Jones, S.C., Juckes, M.N.: The Resonant Interaction of a Tropical Cyclone and a Tropopause Front in a Barotropic Model - Part I: Zonally-oriented Front, J. Atmos. Sci., 68, 405-419
- Scheck, L., Jones, S.C., Juckes, M.N.: The Resonant Interaction of a Tropical Cyclone and a Tropopause Front in a Barotropic Model - Part II: Frontal Waves, J. Atmos. Sci., 68, 420-429
Cyclone-Cyclone interaction in a barotropic model and a dry 3D model
The binary interaction of two tropical cyclones often leads to complex cyclone tracks and can increase forecast errors significantly. We investigated a strongly idealised version of the interaction process using a non-divergent barotropic model. The initial state consists of two vortices that are placed on a f-plane with periodic boundary conditions. The two cyclones advect each other mutually and the shear of their circulations induces changes in the vortex structure and can lead to the formation of filaments. In the symmetric case (two identical cyclones) the cyclones merge if their inital separation is below a critical value. In our setup the bifurcation between merging and non-merging solutions occurs at a initial separation of di = 380km. For larger initial separations the vortices orbit and deform each other initially but drift apart eventually. For vortex profiles that include negative relative vorticity at larger radii, as in the SUD vortices (Smith et al. 1990) used in our scenario, the fluid with negative relative vorticity is redistributed in the inital orbiting phase such that two cyclone-anticyclone pairs form (Fig. 4, t<96h). The pairs subsequently propagate along straight tracks in opposite directions (Fig. 4, t=96h).
Fig. 4: Evolution of the relative vorticity for the interaction of two SUD vortices with initial separation 400km.
The duration of the orbiting phase and thus also the final propagation direction depends strongly on the initial separation. The solution is also higly sensitive to numerical errors introduced during the model evolution, which makes this setup an interesting test case for grid adaptation. We investigated also a 3D version of this scenario using baroclinic vortices. In this case the vertical shear created by one vortex at the location of the other leads causes a tilt of the vortex axes. As explained in Jones (1995), the vortex axes start to precess. These additional 3D processes have a strong influence on the tracks of the vortices, can lead to abrupt changes in the direction of motion and are also decisive in determining wether the vortices will eventually merge or not (see Fig. 5).
Fig. 5: Vorticity distribution after 72h for the interaction of two baroclinic SUD vortices with initial separation of 560km. Orange colors indicate positive and blue colors negative vorticity values, respectively. The difference between the two cases is the value of the coriolis parameter, which corresponds to a latitude of 40 degree (left) and 20 degree (right) and influences how easily the vortex axes can be tilted.
Publication: A. Richter: Untersuchungen zur Wechselwirkung tropischer Wirbelstürme in einem idealisierten dreidimensionalen numerischen Modell, 2012, Diploma thesis
Linear sensitivity analysis
Linear sensitivity analysis, i.e. determining, wich perturbations of the inital state have the strongest influcence on the solution, is an important component of goal-oriented methods (see below) and is also widely used in meteorology. For instance, sensitivity information is required for targeted observations and is used to define initial perturbations for ensembles. The two commonly used quantities representing sensitivity, adjoint-based sensitivity and singular vectors (SVs), both can be interpreted as optimal perturbations. They are optimal in the sense that for a given time interval they show the fastest growth or have the strongest influence on the final state. Therefore, interpreting sensitivity information requires to understand perturbation growth mechanisms and could thus lead to new insights concerning the evolution of errors in forecasts. Sensitivity information for three-dimensional full-physics models is due to its complex structure and the multitude of possible error growth mechanisms rather hard to interpret. The idealised scenarios investigated here provide the opportunity to compute and interpret sensitivity information for less complex cases that still contain important error growth mechanisms.
Fig. 6: Vorticity distribution of the initial (left) and evolved (right) leading singular vector for a hurricane-like barotropic vortex in a zonal anticyclonic shear background flow. The optimization time interval is 48 hours. The lines are streamlines and at the crossing points of the thick lines the velocity is zero, because the cyclonic flow of the vortex cancels the anticyclonic shear.
An example is shown in Fig. 6, which displays the leading initial and evolved SVs for a cyclonic, barotropic vortex in anticyclonic horizontal shear, as experienced by recurving TCs. In a optimization time interval of 48 hours, the energy of the perturbation displayed in the left panel of Fig. 6 (i.e. the initial SV) grows faster than any other perturbation. For the inital SV the perturbation vorticity is aligned with streamlines that lead to stagnation points north and south of the cyclone (the points where the thick lines cross). The evolved SV is characterized by a dipole pattern indicating a displacement of the cyclone. The perturbation grows by a factor of 47, much stronger than the factor 22 that is obtained without the shear background. Thus, background shear facilitates the displacement of the cyclone considerably, but only in the preferred direction visible in Fig. 6. Enhanced growth rates and sensitive regions far from the cyclone are also seen in full-physics model runs. For the strongly idealised case shown in Fig. 6, we were able to understand how the growth mechanisms benefit from shear and why there is a preferred direction of displacement. A publication on these topics is in preparation.
Publication: Scheck et al.: Singular vectors for barotropic vortices in horizontal shear (in preparation)
Adaptive model runs
In the goal-oriented grid adaptation approach a linear sensitivity analysis is performed to determine in which regions the grid should be refined. For this purpose a user-defined goal functional, in most cases fromulated as an integral over the solution, has to be provided. The goal functional can e.g. be chosen such that it is strongly correlated with the cyclone position at the end of the model run. The sensitivity of the value of the goal functional with respect to changes in the solution is computed for each grid cell in space and time. The quantity representing sensitivity, the dual solution, is very similar to the adjoint-based sensitivity used in meteorology.
For the finite-element based model used in this study it is possible to estimate the current discretization error for each grid cell in time and space. Together with the dual solution this allows for an estimation of the contribution of each grid cell to the error in the goal functional. By refining the grid cells with the largest error contributions and coarsening grid cells with low contributions the grid can be optimized such that for a constant number of grid cells the error in the goal functional is minimized.
Fig. 7: Vorticity (left column) and dual velocity (right column) at the initial (top row) and final (bottom row) time of an adaptive model run with the aim to minimize the error in the position of the left cyclone at the final time.
Figure 7 shows an example based on the binary cyclone interaction scenario discussed above. For demonstration purposes, the goal functional in this case is chosen as the vorticity integral over the core of the left cyclone at the final time. It is thus strongly correlated with the final position of this cyclone, but not sensitive to the final position of the second cyclone. Initially, the second cyclone has a strong influence on the track of the first cyclone. Consequently, the dual solution is large in both cyclones (Fig. 7, top right panel) and the grid resolution is high around both cyclones. In the late phases of the model run the second cyclone is located too far from the first one to have a strong influence on its track. Therefore, the dual solution and the grid resolution at the final location of the right cyclone is rather low (Fig. 7, bottom right panel). The right cyclone shows some numerical noise at the final time (Fig. 7, bottom left panel), but the error in the position of the left cyclone is much lower than it would be for a model run on a uniform grid with the same number of grid cells.
More information on the mathematical and numerical aspects of this project can be found on the EMCL web page.
- Baumann, Heuveline: Evaluation of different strategies for goal oriented adaptivity in CFD - Part I: The stationary case, Preprint Series EMCL, No. 2010-06, 2010; submitted to GEM
- Baumann: Numerical Simulation of Tropical Cyclones using Goal-Oriented Adaptivity, Phd thesis, Karlsruhe Institute of Technology (KIT), 2011
- Bauer, Baumann, Scheck, Gassmann, Heuveline, Jones: Simulation of TC-like vortices in shallow-water ICON-hex using goal-oriented r-adaptivity, submitted to Theor. Comp. Fluid. Dyn.
- Beck: Adaptive Simulation idealisierter Wirbelsturm-Probleme, 2012, Diploma thesis