Bifurcations and dynamics in convection with temperature-dependent viscosity in the presence of the O(2) symmetry

We focus the study of a convection problem in a 2D setup in the presence of the O(2) symmetry. The viscosity in the fluid depends on the temperature as it changes its value abruptly in an interval around a temperature of transition. The influence of the viscosity law on the morphology of the plumes is examined for several parameter settings, and a variety of shapes ranging from spout to mushroom shaped is found. We explore the impact of the symmetry on the time evolution of this type of fluid, and find solutions which are greatly influenced by its presence: at a large aspect ratio and high Rayleigh numbers, traveling waves, heteroclinic connections and chaotic regimes are found. These solutions, which are due to the symmetry presence, have not been previously described in the context of temperature dependent viscosities. However, similarities are found with solutions described in other contexts such as flame propagation problems or convection problems with constant viscosity also under the presence of the O(2) symmetry, thus confirming the determining role of the symmetry in the dynamics.Comment: 21 pages, 10 figure

Symmetry and plate-like convection in fluids with temperature-dependent viscosity

We explore the instabilities developed in a fluid in which viscosity depends on temperature. In particular, we consider a dependency that models a very viscous (and thus rather rigid) lithosphere over a convecting mantle. To this end, we study a 2D convection problem in which viscosity depends on temperature by abruptly changing its value by a factor of 400 within a narrow temperature gap. We conduct a study which combines bifurcation analysis and time-dependent simulations. Solutions such as limit cycles are found that are fundamentally related to the presence of symmetry. Spontaneous plate-like behaviors that rapidly evolve towards a stagnant lid regime emerge sporadically through abrupt bursts during these cycles. The plate-like evolution alternates motions towards either the right or the left, thereby introducing temporary asymmetries on the convecting styles. Further time-dependent regimes with stagnant and plate-like lids are found and described.Comment: 19 pages, 8 figures. arXiv admin note: text overlap with arXiv:1302.073

Effect of the inclination angle on the transient melting dynamics and heat transfer of a phase change material

“This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in (Curbelo, J.; Madruga, S. Effect of the inclination angle on the transient melting dynamics and heat transfer of a phase change material. "Physics of fluids", 1 Maig 2021, vol. 33, núm. 5.) and may be found at https://aip.scitation.org/doi/10.1063/5.0047367"We report two-dimensional simulations and analytic results on the effect of the inclination on the transient heat transfer, flow, and melting dynamics of a phase change material within a square domain heated from one side. The liquid phase has Prandtl number Pr¿=¿60.8, Stefan number Ste¿=¿0.49, and Rayleigh numbers extend over eight orders of magnitude 0=¿¿=6.6·108 for the largest geometry studied. The tilt determines the stability threshold of the base state. Above a critical inclination, there exists only a laminar flow at the melted phase, irrespective of the Rayleigh number. Below that inclination, the base state destabilizes following two paths according to the inclination: either leading to a turbulent state for angles near the critical inclination or passing through a regime of plume coarsening before reaching the turbulent state for smaller angles. We find that the Nusselt and Reynolds numbers follow a power law as ¿¿~¿¿¿,¿¿¿~¿¿¿ in the turbulent regime. Small inclinations reduce very slightly a and strongly ß. The inclination leads to subduction of the kinematic boundary layer into the thermal boundary layer. The scaling laws of the Nusselt and Reynolds numbers and boundary layers are in agreement with different results at high Rayleigh convection. However, some striking differences appear as the stabilization of turbulent states with further increasing of the Rayleigh number. We find as well that the turbulent regime exhibits a higher dispersion in quantities related to heat transfer and flow dynamics on smaller domains.S.M. acknowledges support by the Spanish Ministerio de Economía y Competitividad under Project Nos. TRA2016‐75075-R and ESP2015‐70458-P. J.C. acknowledges the support of the “Ramon y Cajal” Project No. RYC2018‐025169 and ICMAT Severo Ochoa Project No. SEV-2015‐0554.Postprint (author's final draft

Instabilities in geophysical fluid dynamics: the influence of symmetry and temperature dependent viscosity in convection

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 25-04-2014Spectral numerical methods are proposed to solve the time evolution of a convection problem in a 2D domain with viscosity strongly dependent on temperature. We have considered periodic boundary conditions along the horizontal coordinate which introduce the O(2) symmetry into the setting. This motivates the use of spectral methods as an approach to the problem. The analysis is assisted by bifurcation techniques such as branch continuation, which has proven to be a useful, and systematic method for gaining insight into the possible stationary solutions satis ed by the basic equations. Several viscosity laws which correspond to di erent dependences of the viscosity with the temperature are investigated. Numerous examples are found along the branching diagrams, in which stable stationary solutions become unstable through a Hopf bifurcation. In the neighborhood of these bifurcation points, the scope of our techniques is examined by exploring transitions from stationary regimes towards time dependent regimes. Our study is mainly focused on viscosity laws that model an abrupt transition of viscosity with temperature. In particular, both a smooth and a sharp transition are explored. Regarding the stationary solutions, the way in which di erent parameters in the viscosity laws a ect the formation and morphology of thermal plumes is discussed. A variety of shapes ranging from spout to mushroom shaped are found. Some stationary stable patterns that break the plume symmetry along their vertical axis are detected, as well as others that correspond to non-uniformly distributed plumes. The main di erence between the solutions observed for the smooth and sharp transition laws is the presence in the latter case of a stagnant lid, which is absent in the rst law. In both cases, we report time-dependent solutions that are greatly in uenced by the presence of the symmetry and which have not previously been described in the context of temperature-dependent viscosities, such as travelling waves, heteroclinic connections and chaotic regimes. Notable solutions are found for the sharp transition viscosity law in which time-dependent solutions alternate an upper stagnant lid with plate-like behaviors that move either towards the right or towards the left. This introduces temporary asymmetries on the convecting styles. This kind of solutions are also related to the presence of the O(2) symmetry and constitute an example of a plate-like convective style which is not linked to a subduction process. These ndings provide an innovative approach to the understanding of convection styles in planetary interiors and suggest that symmetry may play a role in describing how planets work. Finally, the centrifugal and viscosity e ects in a rotating cylinder with large Prandtl number are numerically studied in a regime where the Coriolis force is relatively large. Our focus is on aqueous mixtures of glycerine with mass concentration in the range of 60%-90%, and Rayleigh number values that extend from the onset, where thermal convection is in the so-called wall modes regime, in which pairs of hot and cold thermal plumes ascend and descend in the sidewall boundary layer, to values in which the bulk uid region is also convecting. The mean viscosity, which varies faster than exponentially with variations in the percentage of glycerine, leads to a faster than exponential increase in the Froude number for a xed Coriolis force, and hence an enhancement of the centrifugal buoyancy e ects with signi cant dynamical consequences are described.En esta tesis proponemos métodos numéricos espectrales, para resolver la evolución temporal de un problema de convección en un dominio 2D con viscosidad fuertemente dependiente de la temperatura. Las condiciones de contorno periódicas a lo largo de la coordenada horizontal introducen la simetría O(2) en el problema lo que motiva el uso de métodos espectrales en este contexto. Realizamos un análisis de las soluciones mediante técnicas propias de la teoría de bifurcaciones, y constatamos que son un método útil y sistemático para describir el panorama de las soluciones estacionarias que satisfacen las ecuaciones básicas. Investigamos varias leyes de viscosidad que corresponden a diferentes dependencias de ésta con la temperatura. A lo largo de los diagramas de bifurcación se encuentran numerosos ejemplos en los que la solución estacionaria estable se vuelve inestable a través de una bifurcación Hopf. En las proximidades de esos puntos examinamos el alcance de nuestras técnicas, explorando la transición desde regímenes estacionarios a regímenes dependientes del tiempo. Nuestro estudio se centra principalmente en las leyes de la viscosidad que modelan una transición abrupta de la viscosidad con la temperatura. En particular, se exploran tanto una transición suave como una brusca. En cuanto a las soluciones estacionarias, se discute como los diferentes pará metros en las leyes de viscosidad afectan a la formación y la morfología de las plumas térmicas. Se encuentran una variedad de la formas que van desde forma de protuberancia (\spout") a la forma de seta. Se detectan algunos patrones de soluciones estacionarias estables que rompen la simetría de la pluma a lo largo de su eje vertical y otros que se corresponden con plumas distribuidas de manera no uniforme. La principal diferencia entre las soluciones observadas para las leyes de transición suave y brusca es la presencia, con esta última ley, de una capa estancada que no está presente con la primera. En ambos casos mostramos soluciones dependientes del tiempo que están muy influenciadas por la presencia de la simetría y que no se han descrito previamente en el contexto de convección con viscosidad dependiente de la temperatura. Estas soluciones son por ejemplo ondas viajeras, conexiones heteroclínicas y regímenes caótico. Para transiciones bruscas de la ley de viscosidad destacan soluciones dependientes del tiempo, en las que se alternan una capa superior estancada, con una capa o placa que se mueve rígidamente hacia la derecha o la izquierda. Esto introduce estilos de convección que son asimétricos en el tiempo. Este tipo de soluciones también están relacionadas con la presencia de la simetría O(2) y constituyen un ejemplo de convección en forma de placa que no est a vinculada a un proceso de subducción. Estos resultados aportan un enfoque innovador para la comprensión de estilos de convección en el interior de planetas y sugieren que la simetría puede desempeñar un papel importante en la descripción de como funcionan. Por último, se estudian numéricamente los efectos centrífugos en un cilindro que rota, en un régimen en el que la fuerza de Coriolis es relativamente grande y en el que el fluido tiene un número de Prandtl alto. Nuestra atención se centra en mezclas acuosas de glicerina con concentraciones de masa en el intervalo de 60 %-90% y valores de número de Rayleigh que se extienden desde el inicio de la convección térmica; que son el denominado régimen de modos de pared, donde pares de plumas calientes y frías ascienden y descienden en la capa límite de la pared lateral; hasta valores en los que la convección está completamente desarrollada en toda la celda. El aumento de la viscosidad media, que varía con el porcentaje de glicerina considerado, conduce, para una fuerza de Coriolis ja, a un aumento en el n mero de Froude y por lo tanto, a un incremento de los efectos centrífugos para los que describimos su impacto en la dinámica

Characterization of UMD Banach spaces by imaginary powers of Hermite and Laguerre operators

In this paper we characterize the Banach spaces with the UMD property by means of Lp-boundedness properties for the imaginary powers of the Hermite and Laguerre operators. In order to do this we need to obtain pointwise representations for the Laplace transform type multipliers associated with Hermite and Laguerre operators.Comment: 17 page

Lagrangian analysis of the northern stratospheric polar vortex split in april 2020

The present study examines the northern stratosphere during April 2020, when the polar vortex split into two cyclonic vortices during a winter-early spring period with the strongest ozone depletion on record. We investigate the dynamical evolution leading to the split at middle stratospheric levels, including the fate of fluid parcels on the vortex boundary during its rupture and the distribution of ozone between the vortices resulting from the split. We also illustrate the vertical structure of the vortices after the split. The findings obtained with Lagrangian methods confirm the key role for the split played by a flow with a special configuration of barriers to the motion of parcels. A trajectory analysis clarifies how the ozone distribution between vortices was such that ozone poorest air remained in the main vortex. The offspring vortex had a deep structure from the troposphere and later decayed to vanish by the end of April.Peer ReviewedObjectius de Desenvolupament Sostenible::13 - Acció per al ClimaPostprint (published version

Square functions in the Hermite setting for functions with values in UMD spaces

In this paper we characterize the Lebesgue Bochner spaces $L^p(\mathbb{R}^n,B)$, $1<p<\infty$, by using Littlewood-Paley $g$-functions in the Hermite setting, provided that $B$ is a UMD Banach space. We use $\gamma$-radonifying operators $\gamma (H,B)$ where $H=L^2((0,\infty),\frac{dt}{t})$. We also characterize the UMD Banach spaces in terms of $L^p(\mathbb{R}^n,B)$-$L^p(\mathbb{R}^n,\gamma (H,B))$ boundedness of Hermite Littlewood-Paley $g$-functions

γ\gamma-radonifying operators and UMD-valued Littlewood-Paley-Stein functions in the Hermite setting on BMO and Hardy spaces

In this paper we study Littlewood-Paley-Stein functions associated with the Poisson semigroup for the Hermite operator on functions with values in a UMD Banach space \B. If we denote by $H$ the Hilbert space L^2((0,\infty),dt/t),\gamma(H,\B) represents the space of $\gamma$-radonifying operators from $H$ into \B. We prove that the Hermite square function defines bounded operators from BMO_\mathcal{L}(\R,\B) (respectively, H^1_\mathcal{L}(\R, \B)) into BMO_\mathcal{L}(\R,\gamma(H,\B)) (respectively, H^1_\mathcal{L}(\R, \gamma(H,\B))), where $BMO_\mathcal{L}$ and $H^1_\mathcal{L}$ denote $BMO$ and Hardy spaces in the Hermite setting. Also, we obtain equivalent norms in BMO_\mathcal{L}(\R, \B) and H^1_\mathcal{L}(\R,\B) by using Littlewood-Paley-Stein functions. As a consequence of our results, we establish new characterizations of the UMD Banach spaces.Comment: 31 page

Mixing and geometry in the North Atlantic Meridional Overturning Circulation

Vertical motions across the ocean are central to processes, like CO2 fixation, heat removal or pollutant transport, which are essential to the Earth's climate. This work explores 3D conveyor routes associated with the Atlantic Meridional Overturning Circulation (AMOC). Our findings show the geometry of mixing structures in the upper and deep ocean layers by means of Lagrangian Coherent Structures. This tool identifies among others, zones linked to vertical transport and characterizes vertical transport time scales. We focus the study in two regions. The first one is the Flemish Cap region, a zone of interaction between the major AMOC components, where our analysis identifies a domain of deep waters that ascend very rapidly to the ocean surface. The second one is the Irminger Sea, where our analysis confirms the existence of a downwelling zone, and reveals a previously unreported upwelling connection between very deep waters and the ocean surface.RB acknowledges support of a CSIC JAE intro fellowship. AMM and GGS acknowledge the support of a CSIC PIE project Ref. 202250E001 and from Grants PID2021-123348OB-I00 funded by MCIN/AEI/10.13039/501100011033/ and by FEDER A way to make Europe. AMM is an active member of the CSIC Interdisciplinary Thematic Platform POLARCSIC and TELEDETECT and acknowledges the support from Grant EIN2020-112235 funded by MCIN/ AEI/10.13039/501100011033/and by the European Union NextGenerationEU/ PRTR. JC also acknowledges the support of the RyC project RYC2018-025169, the Spanish Grant PID2020-114043GB-I00 and PID2021-122954NB-I00 and the “2022 Leonardo Grant for Researchers and Cultural Creators, BBVA Foundation.Peer ReviewedPostprint (published version