Degree Of Inflection Research Articles

Estimation of Scalar Source/Sink Distributions in Plant Canopies Using Lagrangian Dispersion Analysis: Corrections for Atmospheric Stability and Comparison with a Multilayer Canopy Model

Source/sink distributions of heat, water vapour and CO 2 within a rice canopy were inferred using an inverse Lagrangian dispersion analysis and measured mean profiles of temperature, specific humidity and CO 2 mixing ratio. Monin-Obukhov similarity theory was used to account for the effects of atmospheric stability on σ w (z). the standard deviation of vertical velocity and τ L (z), the Lagrangian time scale of the turbulence. Classical surface layer scaling was applied in the inertial sublayer (z > z ruf ) using the similarity parameter ζ = (z - d)/L, where z is height above ground, d is the zero plane displacement height for momentum, L is the Obukhov length, and z ruf 2.3h c , where h c is canopy height. A single length scale h c , was used for the stability parameter 3 = h c /L in the height range 0.25 < z/ h c < 2.5. This choice is justified by mixing layer theory, which shows that within the roughness sublayer there is one dominant turbulence length scale determined by the degree of inflection in the wind profile at the canopy top. In the absence of theoretical or experimental evidence for guidance, standard Monin-Obukhov similarity functions, with ζ = h c /L, were used to calculate the stability dependence of σ w (z) and τ L (z) in the roughness sublayer. For z/h c < 0.25 the turbulence length and time scales are influenced by the presence of the lower surface, and stability effects are minimal. With these assumptions there was excellent agreement between eddy covariance flux measurements and deductions from the inverse Lagrangian analysis. Stability corrections were particularly necessary for night time fluxes when the atmosphere was stably stratified. The inverse Lagrangian analysis provides a useful tool for testing and refining multilayer canopy models used to predict radiation absorption, energy partitioning and CO 2 exchanges within the canopy and at the soil surface. Comparison of model predictions with source strengths deduced from the inverse analysis gave good results. Observed discrepancies may be due to incorrect specification of the turbulent time scales and vertical velocity fluctuations close to the ground. Further investigation of turbulence characteristics within plant canopies is required to resolve these issues.

Generalized Frenkel-Kontorova models

We have studied two generalized Frenkel-Kontorova (FK) models. In one of them the interatomic potential is changed from simple harmonic to Toda; in the other the external potential is changed from sinusoidal to nonanalytic. Their critical behaviors are found to be markedly different. In the Toda case, the local critical exponents are the same as those in the standard FK model. However, the global singularity spectrum and the generalized dimension depend on the anharmonicity parameter. In the nonanalytic FK model, the critical exponents are found to vary with the degree of inflection z for 2≤ z<3, but remain the same for z≥3. The degree of inflection thus plays a role quite similar to that of dimensionality in phase transitions with z = 2 and 3 corresponding respectively to the lower and upper critical dimensions. For z>3, a sequence of pinning and depinning transitions, reflecting the recurrence of Kolmogorov-Arnol'd-Moser tori, has also been observed.

TRAJECTORY SCALING FUNCTION OF POLYNOMIAL CIRCLE MAPS

The universality of trajectory scaling function (TSF) for the polynomial form of the circle map in the quasiperiodic-to-chaos transition is discussed. It is found that the TSF gets bumpier with increase in the degree of inflection z, for both Golden and Silver Mean cases, and shows multiple scaling with increase in z, reflecting the monotone decrease of α with increasing z. The effect of noise is also studied.

Critical phenomena of invariant circles.

Some novel critical phenomena are discovered in a class of nonanalytic twist maps. It is found that the degree of inflection {ital z} plays a role reminiscent of that of dimensionality in phase transitions with {ital z}=2 and 3 corresponding to the lower and upper critical dimensions, respectively. Moreover, recurrence of invariant circles has also been observed. An inverse residue criterion,'' complementary to the residue criterion'' for the determination of the disappearance point, is introduced to determine the reappearance point of invariant circles.

Universality and asymptotic limits of the scaling exponents in circle maps

In he quasiperiodic route to chaos as modeled by the circle map, the universal behavior is dictated by the degree of inflection z. Using a polynomial form of the circle map, we have studied the dependence of the scaling exponents α and δ on z. The asymptotic limits of α( z) and δ( z) as z→∞ have also been found.