The Cross–Newell phase diffusion equation τ(k)ΘT = −∇ · kB(k), k = ∇Θ, | k| = k, and its regularization describe patterns and defects far from onset in large aspect ratio systems with translational and rotational symmetry. In this paper we show how director field solutions of this equation can be used to describe features of global patterns. The ideas are illustrated in the context of a non-tri...

The concern of this paper is the Cauchy problem for the Prandtl equation. This problem is known to be well-posed for analytic data [13, 10], or for data with monotonicity properties [11, 15]. We prove here that it is linearly ill-posed in Sobolev type spaces. The key of the analysis is the construction, at high tangential frequencies, of unstable quasimodes for the linearization around solution...

Klaus Kirchgässner was born in Mannheim on December 26, 1931, and died in Konstanz on July 9, 2011. He grew up in the small, lovely town of Waldshut, which is located along the Rhine at the extreme south of the Black Forest. With his friends he spoke with enthusiasm about this city, where he did swimming in the Rhine and in the Aare river and where he completed his high-school education (“Gymna...

We study the asymptotic behavior of Oleinik’s solution to steady Prandtl equation when outer flow $$U(x)=1$$ . Serrin proved that converges famous Blasius $${\bar{u}}$$ in $$L^\infty _y$$ sense as $$x\rightarrow +\infty $$ The explicit decay estimates $$u-{\bar{u}}$$ and its derivatives were by Iyer (ARMA 237,2020) initial data is a small localized perturbation profile. In this paper, we prove ...

In this article we apply the machinery developed by Guo and Iyer [Validity of Steady Prandtl Layer Expansions, Comm. Pure Appl. Math. (to appear) (2022)] together with a new compactness estimate quantity, “degree”, in order to prove validity steady layer expansions external forcing. The techniques introduced allow us treat more general background layers than (2022)].

The validity of the anelastic approximation has recently been questioned in the regime of rapidly-rotating compressible convection in low Prandtl number fluids (Calkins, Julien and Marti, Proc. R. Soc. A, 2015, vol. 471, 20140689). Given the broad usage and the high computational efficiency of sound-proof approaches in this astrophysically relevant regime, this paper clarifies the conditions fo...

We study the problem of thermal convection in a laterally heated horizontal cylinder rotating about its axis. A cylinder of aspect ratio Γ=H/2R=2 containing a small Prandtl number fluid (σ=0.01) representative of molten metals and molten semiconductors at high temperature is considered. We focus on a slow rotation regime (Ω<8), where the effects of rotation and buoyancy forces are comparable. T...

Several recent advances in turbulent dynamo theory are reviewed. High resolution simulations of small-scale and large-scale dynamo action in periodic domains are compared with each other and contrasted with similar results at low magnetic Prandtl numbers. It is argued that all the different cases show similarities at intermediate length scales. On the other hand, in the presence of helicity of ...

We aim at comparing computations with asymptotic models issued from incompressible Navier-Stokes at high Reynolds number: the Reduced Navier-Stokes/Prandtl (RNS/P) equations and the Double Deck (DD) equations. We treat the case of the steady two dimensional flow in a constricted pipe. In particular, finite differences and finite element solvers are compared for the RNS/P equations. It results f...

With a non local shell model of magnetohydrodynamic turbulence we investigate numerically the turbulent dynamo action for low and high magnetic Prandtl numbers (Pm). The results obtained in the kinematic regime and along the way to dynamo saturation are understood in terms of a phenomenological approach based on the local (Pm ≪ 1) or non local (Pm ≫ 1) nature of the energy transfers. In both ca...