Shouhong Wang
Professor
Department of Mathematics
Indiana University
Bloomington, IN 47405.
Phone: 812 855 8350
Fax: 812 855 0046
Email: showang@indiana.eduBooks:
- Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics, AMS Monograph and Mathematical Survey Series, vol. 119, 2005 (with Tian Ma)
- Bifurcation Theory and Applications, World Scientific Series on Nonlinear Science, Series A - Vol. 53, 2005 (with Tian Ma)
- Stability and Bifurcation of Nonlinear Evolutions Equations, Science Press, Beijing, April, 2007, pp. 437 (with Tian Ma)
- Phase Transition Dynamics in Nonlinear Sciences, Springer-Verlag, to appear in May, 2012, about 700pp. (with T. Ma)
Recent Research Highlights
Phase Transitions
It is well-known that the gas-liquid coexistence curve terminates at a critical point, also called the Andrews critical point. It is a longstanding open question why the Andrews critical point exists and what is the order of transition going beyond this critical point. For the first time, we show that 1) the gas-liquid co-existence curve can be extended beyond the Andrews critical point, and 2) the transition is {\it first order} before the critical point, {\it second-order} at the critical point, and {\it third order} beyond the Andrews critical point. This clearly explains why it is hard to observe the gas-liquid phase transition beyond the Andrews critical point. Furthermore, the analysis leads naturally the introduction of a general asymmetry principle of fluctuations and the preferred transition mechanism for a thermodynamic system. The theoretical results derived in this article are in agreement with the experimental results obtained in (K. Nishikawa and T. Morita, Fluid behavior at supercritical states studied by small-angle X-ray scattering, Journal of Supercritical Fluid, 13 (1998), pp. 143-148) and their related articles. Also, the derived second-order transition at the critical point is consistent with the result obtained in (M. Fisher, Specific heat of a gas near the critical point, Physical Review, 136:6A (1964), pp. A1599-A1604).
We have derived new Ginzburg-Landau type of models for liquid helium-3 [78], helium-4 [77] and their mixture [82], leading to various physical predictions, such as the existence of a new phase $C$ for helium-3. Although these predictions need yet to be verified experimentally, they certainly offer new insights to both theoretical and experimental studies for a better understanding of the underlying physical problems.
Geophysical Fluid Dynamics and Climate Dynamics
- New Mechanism of El Nino Southern Oscillation (ENSO) [91, 86]
We discovered a new mechanism of the ENSO, as a self-organizing and self-excitation system, with two highly coupled oscillatory processes: 1) the oscillation between the two metastable warm (El Nino phase) and cold events (La Nina phase), and 2) the spatiotemporal oscillation of the sea surface temperature (SST) field. The interplay between these two processes gives rises the climate variability associated with the ENSO, leads to both the random and deterministic features of the ENSO, and defines a new natural feedback mechanism, which drives the sporadic oscillation of the ENSO. The randomness is closely related to the uncertainty/fluctuations of the initial data between the narrow basins of attractions of the corresponding metastable events, and the deterministic feature is represented by a deterministic coupled atmospheric and oceanic model predicting the basins of attraction and the sea-surface temperature (SST). It is hoped this mechanism based on a rigorous mathematical theory could lead to a better understanding and prediction of the ENSO phenomena.
- Thermohaline Circulation [88]
Oceanic circulation is one of key sources of internal climate variability. One important source of such variability is the thermohaline circulation (THC). Physically speaking, the buoyancy fluxes at the ocean surface give rise to gradients in temperature and salinity, which produce, in turn, density gradients. These gradients are, overall, sharper
in the vertical than in the horizontal and are associated therefore with an overturning or THC. A mathematical theory associated with the thermohaline circulations (THC) is derived in [88], using the Boussinesq system, governing the motion and states of the large-scale ocean circulation. First, it is shown that the first transition is either to multiple steady states or to oscillations (periodic solutions), determined by the sign of a non-dimensional parameter $K$, depending on the geometry of the physical domain and the thermal and saline Rayleigh numbers. Second, for both the multiple equilibria and periodic solutions transitions, both Type-I (continuous) and Type-II (jump) transitions can occur, and precise criteria are derived in terms of two computable nondimensional parameters $b_1$ and $b_2$. Associated with Type-II transitions is the hysteresis phenomena, and the physical reality is represented by either metastable states or by a local attractor away from the basic solution, showing more complex dynamical behavior. Third, a convection scale law is introduced, leading to an introduction of proper friction terms in the model in order to derive the correct circulation length scale. In particular, the dynamic transitions of the model with the derived friction terms suggest that the THC favors the continuous transitions to stable multiple equilibria.
