3D PeN-model

"PeN-model"is the acronym for a numerical model of the atmosphere that was developed at IMAU (Utrecht University) for education purposes for courses such as, Dynamical Meteorology, Boundary Layers, Transport and Mixing and Simulation of Oceans, Atmospheres and Climate. Pe stands for "primitive equations" and N stands for N layers. A 3D version of the model with N=36 and a 2D version of the model with N=200 are "operational". The 3D version is used to study the life cycles of unstable baroclinic waves and the formation of storm tracks in middle latitudes, while the 2D version is used to study the interaction between adiabatic dynamics and diabatic processes, such as absorption and emission of radiation and latent heat release, in the General Circulation. On this page two simulations with the 3D model of the (unstable) growth of a baroclinic wave are described. Simulations of the zonal mean general circulation are described on the following web page: link.

SIMULATION OF A GROWING BAROCLINIC WAVE ON THE f-PLANE

The model domain is a channel, which is periodic in the zonal direction. Initially a temperature distribution is imposed with a strong meridional gradient in the centre of the domain (see panel 1 to the right). The associated geopotential height is calculated from hydrostatic balance, assuming that surface pressure is constant initially. The associated initial wind is calculated from geostrophic balance.

The thermal state is disturbed slightly by superposing a wave-like perturbation in the temperature. The zonal wave length of this perturbation is 4716 km, which is also the zonal length of the domain (corresponding to 60° longitude at 45° latitude). This wave grows into a realistic cyclone with a cold front, a warm front and a back-bent front, almost exactly as proposed by Shapiro and Keyser in 1990 (link) , who were inspired by the work done before 1919 by the members of the Norwegian school, the most prominent being father, Vilhelm, and son, Jacob, Bjerknes (link), (link) .

Initially the wave in temperature and in geopotential are in phase at all levels and the wave does not tilt westward with height. Instead, the phase shift between the temperature wave and the geopotential wave as well as a westeward tilt with increasing height are created by the growing wave itself (see panel 2), thereby initiating a northward transport of heat .

If the Q-vectors are directed up the temperature gradient (i.e. from cold to warm) this indicates that the temperature gradient is intensifying, i.e. the front is undergoing frontogenesis. The opposite is the case if the Q-vectors are directed down the temperature gradient (i.e. from warm to cold), i.e. the front is undergoing frontolysis. The increase of the absolute temperature gradients at 864 hPa in the later phase of the life cycle of the growing baoclinic wave is quite spectacular (see panel 3 and the following animation: Run200).

During the initial phase of the growth of the wave the Q-vectors are in fact directed mostly parallel to the isotherms (see panel 2). This indicates that the process of frontogenesis is connected to the rotation of isotherms, i.e. the direction of the temperature gradient is changing. This process also disturbs thermal wind balance. The atmosphere responds by creating "secondary" ageostrophic vertical circulations, which are intended to restore thermal wind balance, but which also drive the system further away from zonal symmetry.

If the Q-vectors converge in a certain area, such in the war sector of the mature middle latitude cyclone shown in panel 3, the vertical component of the "secondary" ageostrophic motion should be upward, according to the "omega" equation. This is verified in panel 4.

SIMULATION OF A GROWING BAROCLINIC WAVE ON THE beta-PLANE AND THE SUBSEQUENT BEHAVIOUR IF THE SYSTEM IS FORCED TO RELAX BACK TO THE INITIAL TEMPERATURE DISTRIBUTION

The simulation on the f-plane (constant Coriolis parameter, f=0.0001 /s) is repeated on the beta-plane (constant meridional gradient of the Coriolis parameter, beta=1.648x10^-11 m^-1s^-1 and f=0.0001 /s in the middle of the domain, correponding to the location of the maximum initial meridional temperature gradient). In addition to this change to the set up of the model simulation, a diabatic heating/cooling term is added to the temperature tendency equation, which "relaxes" the atmosphere back to the initial thermal state, shown in the first panel. This parametrisation of the effect of radiation, which is called "Newtonian heating/cooling", assumes that diabatic heating/cooling is proportional to the temperature difference between the actual state and the initial state. The initial state, thus, represents the "radiative equilibrium state". This state is also a dynamic equilibrium of the system under study, because it is associated with a westerly flow and an associated jet stream (see panel 1). The time scale of the relaxation to the radiative equilibrium is 11.6 days, which is significantly longer than the time scale of the life cycle of an unstable baroclinic wave, which is therefore not greatly affected by the "Newtonian heating/cooling".

The beta-effect damps the growth of the baroclinic wave, thus reducing the meridional amplitude of the wave. The beta-effect also reduces eastward phase propagation of the wave. Nevertheless, the baroclinic life cycle in the first 96 hours in this case is very similar to the baroclinic life cycle in the previous case, i.e on the f-plane and excluding dibatic cooling or heating.

The baroclinic wave transports heat northward, thus maintaining the atmosphere above (below) radiative equilibrium in the north (south) of the domain. Diabatic cooling/heating (cooling in the north and heating in the south) drives the system back to the initial state. However, the system never returns to this initial state, because this state is unstable to small perturbations. Instead, the system produces a train of relatively small growing disturbances, which propagate from west to east along the southern flank (resembling a "storm track") of a large "mother" low pressure system and seem to "feed" this "mother" low, which itself propagates eastward very slowly.

One remarkable lesson from this simulation is the following. Zonally asymmetric forcing by e.g. zonally asymmetric lower boundary conditions (e.g. mountains) is not needed to create zonal asymmetries in the circulation as long as the radiative equilibrium state represents a baroclinically unstable state!

 

CONCLUSION ON 3D-SIMULATIONS

The 3D version of the PeN model reproduces many realistic characteristics of the life cycle of an unstable baroclinic wave, including the recurring formation of relatively small scale "satellite" baroclinic disturbances that propagate eastward along the south flank of a "mother" low. The system produces zonal asymmetries in the total absence of zonally asymmetric boundary conditions.

 

 

 

Background material

LECTURE NOTES

The PeN-model is described in chapters 10 and 12 of the lecture notes on Atmospheric Dynamics: link

See also the page describing the project "Circulation & global change": link

Acknowledgement

I wish to thank Koen Manders, Niels Zweers and Roos de Wit for help in developing and debugging the numerical model and for stimulating scientific input in the early stages of this research project, Yvonne Hinssen and Theo Opsteegh for useful discussions on potential vorticity inversion, Bruce Denby for his large contribution in developing the computer graphics, the students of my course on climate and the water cycle for useful suggestions on incorporating and validating the parametrizations associated with the water cycle, Marcel Portanger for advice and help on computer problems and, finally, all my colleagues at IMAU for allowing me to work on such a large and time-consuming project.

Panel 1: Initial state: meridional cross section of the zonal mean potential temperature and the zonal mean zonal wind, labeled in K and m/s, repectively. The Coriolis parameter is constant (f=0.0001 /s). The y-coordinate increases towards the left.

Panel 2:Geopotential height (m) and potential temperature (°C) after 24 hours of intergration. The wave in the geopotential height and the wave in the temperature are in phase initially, but not after 24 hours! The arrows represent the Q-vectors (shown only if the amplitude exceeds a threshold value). Watch the growth of the wave at 864 hPa during the first 96 hours: Run200

Panel 3:Geopotential height (m) and potential temperature (°C) after 96 hours of intergration. The arrows represent the Q-vectors (shown only if the amplitude exceeds a threshold value). Note that both the cold front (cf) and the warm front (wf) are undergoing frontogenesis, i.e. Q-vectors point toward the warm side of the front.

Panel 4:Vectors of the horizontal wind component and vertical velocity at 864 hPa after 96 hours of integration. Vertcal velocity is labeled in units of hPa/hr.

Panel 5:Vectors of the horizontal wind component and absolute temperature gradient at 864 hPa after 72 hours of integration (labeled in units of 10^-5 K/km; contour interval is 0.4x10^-5 K/km; first contour shown is 1.2x10^-5 K/km). The abreviations, cf, wf and bbf indicate, respectively, cold front, warm front and back-bent front. Watch the intensification of the fronts at 864 hPa during the first 90 hours of the simulation and the formation of a cold front, a warm front and a backbent front: Run200

Panel 6:Geopotential height (m) and potential temperature (°C) after 408 hours of integration in a simulation on a beta-plane, including "Newtonian heating/cooling". The arrows represent the Q-vectors (shown only if the amplitude exceeds a threshold value). The letter "L" between quotation marks indicates a small baroclinic disturbances.

Panel 7:As panel 6, but for t=690 hrs. The The letter "L" between quotation marks indicates a different small baroclinic disturbance than the one indicated in panel 6. An animation of this run can be viewed here: Run2000