Piston-Shock model

Synopsis

Piston-shock model is an analytic description of the momentum interchange between the coronal mass ejection (CME) and the ambient solar wind through a shock-structure (plasma sheath). The piston-shock model's equations express the trajectories of both CME leading edge and shock front as functions of time. In order to compute these trajecotries the model requires data available, in principle, since the detection time of the CME or few hours after it.

In the middle of 2015, the SCiESMEX Board decided to implement the Piston-Shock model as an experimental tool for arrival-forecasting of CME/shocks. Since then, the model's predictions have been regularly used in SCiESMEX's space weather reports (weekly and special issues - in Spanish).

SCiESMEX team explored the forecasting capabilities of the model, the test period finished by January 2016. Since January 2016, SCiESMEX began the development of SPArToS, which is an experimental tool for arrival-forecasting of CMEs based on the Piston-Shock model. In the near future, we expect to implement the model to arrival-recasting of shock waves.

The model allows the user to calculate:

the time a CME (or shock) spends in traveling from near the Sun to Earth's orbit (travel time) and the speed with which it arrives (arrival speed),
the in-situ registers (synthetic transits) of CMEs and associated plasma sheath and shocks,
radial trajectories of CMEs and associated shocks,
approximate the CME observed mass (by coronagraph), and
synthetic type II radio burst associated with the CME/shock propagation.
the model has been probed capable to forecast in-situ arrivals (at Earth's orbit) of CMEs.
approximate in-situ transits of shock waves, plasma sheaths and CMEs

The data inputs required by the model are:

initial position and speed of CMEs (from coronagraph images),
rise phase of X-ray emission of solar flares (soft X-ray fluxes from GOES-spaceships),
in-situ conditions of solar wind (density, speed and temperature of protons, as well as magnetic field magnitude), and
geometrical properties of CMEs (from coronograph imagages) when available.

Development of the model

Corona-Romero & Gonzalez-Esparza [2011] used hydrodinamic numerical simulations to study the dynamic relations between fast CMEs (driver) and their associated shocks. They found that the relation between the drivers and the shocks could be addressed by an analytic approach by Canto et.al. [2000]. The analytic approach solves the mass center trajectory of two interacting fluid parcels through interchange of linear momentum and conservation of mass. According the numerical simulations, the masscenter corresponded to the leading edge of the driver (CME).

The numerical simulations pointed out that the trajectories of shocks have two dynamic phases: the first where the shock is driven by the CME, followed by a decaying phase. Although the shock is a non-adiabatic process, during the driving phase a shock wave has a constant speed, due to the driver replaces the energy and momentum spent by the shock wave. In contrast, in the decaying phase the shock evolves as a blast wave because the driver's energy is exausted.

By comparing the results from numerical simulations and analytic model, Corona-Romero & Gonzalez-Esparza [2011] were able to use the analytic model and initial conditions of events to reproduce in-situ arrivals of CMEs and associated shocks. They also found systematic differences between calculated arrivals of shocks, which were provoke by the geometrical assumptions on analytic shock's shape.

Corona-Romero & Gonzalez-Esparza [2012] changed the geometry of the associated shock wave from a spherical shock to a bow-shock. This extended version of the piston-shock model solved the systematic errors in calculated arrivals of shocks. Corona-Romero et.al. [2013] validated this extended version by approximate the trajectories, in-situ arrivals, and type II radio bursts of a number of radio-loudly CME/shock events.

Corona-Romero & Gonzalez-Esparza [2016] probed that piston-shock model is capable to approximate the in-situ transit profiles of shocks and plasma sheaths associated with the arrival of CMEs. Recently, Corona-Romero et.al. [2017], have probed that piston shock model is capable to forecast in-situ arrivals of CMEs.

Pedro Corona-Romero & Pete Riley [2020] probed that piston-shock model is capable to approximate the in-situ transit profiles of CMEs.

We have not finished to develop all the capabillities of piston-shock model. We are still working in finding new ways to use our model to describe solar storms.

Physical background

Piston-shock model assumes that CMEs are fast and blunt ostacles that interact with ambient solar wind. The initial mass and energy of CMEs are set at the initiation of the event and remain constant along the interplanetary propagation of CMEs. The initial speed of CMEs need to be larger than the local super-magnetosonic speed. Thus, CMEs initially drive shock waves which geometry needs to resemble bow-waves. The ambient solar wind is thought stationary polytropic plasma, which expands at a constant rate.

Perhaps, the main assumption of the model is that CMEs drive for a short period (tens of minutes) their associated shocks. After this driving phase the CME and shock evolve independently each other. On one hand, the shock waves evolve as blast waves, until they discipate into compression waves. On the other hand, the CMEs decelerate due to the interaction with the shocked solar wind (plasma sheath material). This interaction provokes an inertial drag (due to ram pressure), leading CME speeds to asymptotically equilize solar wind expansion speed.

Limitations

Piston-shock model is an hydrodynamic approach to a full MHD phenomenon. The validity of piston-shock model is linked to the local values of sonic (solar wind), alfvenic (solar wind), and magnetosonic (CME-speed) Mach numbers. As long these numbers be shorter than 1, the piston-shock model is reliable.

Piston-shock model obtains the best results for isolated events propagating during periods of non-structurated solar wind. It is preferably that the events source region (associated solar flare's active region) be as near as possible to the center of solar disk (Sun-Earth line of sight).

The error associated to the model's calculations grows with the complexity of ambient solar wind.
Source region far from disk center (>35°) may increase the error or even make the model to fail.
Orientation of CMEs may also affect the model's results.

Piston-Shock model may fail when analzing complex events (CME-CME or CME-streams interactions), save very special cases listed in Corona-Romero et.al. [2017]

For a MHD description of CMEs propagation see "A Semi-empirical Approach to the Dynamic Coupling of CMEs and Solar Wind" [Corona-Romero et.al. 2022].

References

Peer-review published papers:

Corona-Romero, P. & Gonzalez-Esparza, J. A. Numeric and analytic study of interplanetary coronal mass ejection and shock evolution: Driving, decoupling, and decaying. Journal of Geophysical Research, Volume 116, 2011.
Corona-Romero, P. & Gonzalez-Esparza, J. A. Dynamic evolution of interplanetary shock waves driven by CMEs. Comparative Magnetic Minima: Characterizing quiet times in the Sun and Stars, Proceedings of the International Astronomical Union, IAU Symposium, Volume 286, 2012.
Corona-Romero, P.; Gonzalez-Esparza, J. A.; Aguilar-Rodriguez, E. Propagation of Fast Coronal Mass Ejections and Shock Waves Associated with Type II Radio-Burst Emission: An Analytic Study. Solar Physics, Volume 285, 2013.
Corona-Romero, P. & Gonzalez-Esparza, J. A. Development of a formalism for computing transits of Earth-directed CMEs, plasma sheaths, and shocks. Towards a forecasting tool. Advances in Space Research, Volume 58, 2016.
Corona-Romero, P.; et.al. Calculating CME arrivals to Earth's environment: an analytic tool for space weather forecasting. Space Weather Journal, Volume 15, 2017.
Corona-Romero P. & Riley P. Development of a formalism for computing in situ transits of Earth-directed CMEs – Part 2: Towards a forecasting tool. Annales Geophysicae, Volume 38, 2020.

Related research works:

Corona-Romero, P. Master thesis work. Evolución de Eyecciones de Masa Coronal y Ondas de Choque Interplanetarias: Impulsión, Desacoplamiento y Decaimiento, 2009. In Spanish.
Ontiveros, V.; Patsourakos, S.; Corona-Romero, P.; Gonzalez-Esparza, J. A. Arrival Times of Interplanetary CMEs and Shocks into the Earth's Vicinity: STEREO Observations and Analytical Modeling. 10th Hellenic Astronomical Conference, Proceedings of the conference held at Ioannina.
Corona-Romero, P. PhD dissertation. Propagación de Eyecciones de Masa Coronal Rápidas y sus Ondas de Choque en el Medio Interplanetario, 2013. In Spanish.
Kajdic, Primoz; Gonzalez Esparza, Juan-Americo; Aguilar Rodriguez, Ernesto; Corona-Romero, Pedro. Radial speeds of an extreme Interplanetary Coronal Mass Ejection and its shock, 2014.