Velocity field statistics in the inertial to dissipation range of three-dimensional homogeneous steady turbulent flow are studied using a high-resolution DNS with up to N=10243 grid points. The range of the Taylor microscale Reynolds number is between 38 and 460. Isotropy at the small scales of motion is well satisfied from half the integral scale (L) down to the Kolmogorov scale (η). The Kolmogorov constant is 1.64±0.04, which is close to experimentally determined values. The third order moment of the longitudinal velocity difference scales as the separation distance r, and its coefficient is close to 4/5. A clear inertial range is observed for moments of the velocity difference up to the tenth order, between 2λ≈100η and L/2≈300η, where λ is the Taylor microscale. The scaling exponents are measured directly from the structure functions; the transverse scaling exponents are smaller than the longitudinal exponents when the order is greater than four. The crossover length of the longitudinal velocity structure function increases with the order and approaches 2λ, while that of the transverse function remains approximately constant at λ. The crossover length and importance of the Taylor microscale are discussed.

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We report that in La2−xSrxCuO4 (La214) and Bi2Sr2CaCu2O 8+δ (Bi2212), the resistivity ρ and magnetic susceptibility χ decrease more sharply below T * , corresponding to the onset temperature of a spin gap (pseudogap) at least for Bi2212. In each system, T * scales with the superconducting gap magnitude 2Δ0 (T * ∼ 2Δ0/4.3kB), which increases as the doping level is lowered. We also report that 2Δ0 is almost proportional to Tmax (>T * ), where Tmax is the temperature exhibiting a broad peak of χ and kBTmax has been considered to give the measure of effective exchange energy J eff ; the α factors in 2Δ0 ∼ αkBTmax ∼ αJ eff are nearly one and two for La214 and Bi2212, respectively.KEYWORDS: cuprate superconductor, superconducting gap, pseudogap, STM/STS 2622 in-plane resistivity below T * , according to the spin gap formation. 13, 14) Temperature T * , much higher than T c at low p, decreases with increasing p and seems to connect smoothly to the T c -p curve at an overdoping level. Interestingly, it has recently been demonstrated by scanning tunneling spectroscopy (STS) that in Bi2212 the p dependence of T * correlates with that of the superconducting gap magnitude 2Δ 0 at T T c . 14) In this paper, we report that in La214 and Bi2212, the resistivity and magnetic susceptibility decrease more sharply below T * , comparable to the onset temperature of a spin gap (pseudogap) at least for Bi2212, and T * scales with 2Δ 0 examined by STS and break junction tunneling spectroscopy (BJTS) at T T c . Interestingly, T * ∼ 2Δ 0 /4.3k B in the entire p-range examined, corresponding to the transition temperature T MF c expected in the mean-field theory for d-wave superconductors or the onset temperature T RVB of spin-channel singlet pairing expected in the mean-field RVB theory. This fact suggests that the crossover behavior around T * will be superconducting in origin. We will also report that 2Δ 0 is almost proportional to k B T max (∼J eff ); the α factors in 2Δ 0 ∼ αk B T max ∼ αJ eff are nearly one and two for the typical monolayer and bilayer cuprates, La214 and Bi2212, respectively.Most of the present experiments were performed using single crystals, except for break junction tunneling, resistivity and magnetic susceptibility measurements of La214, which were mainly done using ceramic samples. The doping level p of La214 was defined as the Sr content x. On the other hand, the p value of Bi2212, controlled by annealing in oxygen or by substituting Dy 3+ for Ca 2+ , 14) was estimated from the T c value so that it falls on the T c -p curve determined by Gröen et al. 15) In STM/STS experiments, we used an Olympus LTSTM-300 system with a Pt-Ir alloy tip. The tip was approached along the direction perpendicular to the CuIt has been broadly thought in high-T c cuprates that the electronic properties exhibit two crossover behaviors around T max (or T 0 ) and T * (T max > T * > T c ), which are both reduced by increasing the carrier-doping level p of the Cu-O plane. Around T max , the uniform magnetic susceptibility χ, following th...

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