Monroy Diana L, Naumis Gerardo G
Departamento de Sistemas Complejos, Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000, Ciudad de México, CDMX, Mexico.
Phys Rev E. 2021 Mar;103(3-1):032312. doi: 10.1103/PhysRevE.103.032312.
The time-dependent Ginzburg-Landau (or Allen-Cahn) equation and the Swift-Hohenberg equation, both added with a stochastic term, are proposed to describe cloud pattern formation and cloud regime phase transitions of shallow convective clouds organized in mesoscale systems. The starting point is the Hottovy-Stechmann linear spatiotemporal stochastic model for tropical precipitation, used to describe the dynamics of water vapor and tropical convection. By taking into account that shallow stratiform clouds are close to a self-organized criticality and that water vapor content is the order parameter, it is observed that sources must have nonlinear terms in the equation to include the dynamical feedback due to precipitation and evaporation. The nonlinear terms are derived by using the known mean field of the Ising model, as the Hottovy-Stechmann linear model presents the same probability distribution. The inclusion of this nonlinearity leads to a kind of time-dependent Ginzburg-Landau stochastic equation, originally used to describe superconductivity phases. By performing numerical simulations, pattern formation is observed. These patterns are better compared with real satellite observations than the pure linear model. This is done by comparing the spatial Fourier transform of real and numerical cloud fields. However, for highly ordered cellular convective phases, considered as a form of Rayleigh-Bénard convection in moist atmospheric air, the Ginzburg-Landau model does not allow us to reproduce such patterns. Therefore, a change in the form of the small-scale flux convergence term in the equation for moist atmospheric air is proposed. This allows us to derive a Swift-Hohenberg equation. In the case of closed cellular and roll convection, the resulting patterns are much more organized than the ones obtained from the Ginzburg-Landau equation and better reproduce satellite observations as, for example, horizontal convective fields.
提出了添加随机项的含时金兹堡 - 朗道（或艾伦 - 卡恩）方程和斯威夫特 - 霍恩伯格方程，用于描述中尺度系统中组织的浅对流云的云型形成和云态相变。起点是用于描述水汽和热带对流动力学的热带降水的霍托维 - 施特克曼线性时空随机模型。考虑到浅层状云接近自组织临界状态且水汽含量是序参量，观察到方程中的源项必须有非线性项以包含降水和蒸发引起的动力学反馈。由于霍托维 - 施特克曼线性模型呈现相同的概率分布，非线性项通过使用伊辛模型的已知平均场导出。这种非线性的纳入导致了一种最初用于描述超导相的含时金兹堡 - 朗道随机方程。通过进行数值模拟，观察到了模式形成。与纯线性模型相比，这些模式与实际卫星观测结果的匹配度更好。这是通过比较实际云场和数值云场的空间傅里叶变换来实现的。然而，对于被视为潮湿大气中瑞利 - 贝纳德对流形式的高度有序的细胞对流相，金兹堡 - 朗道模型无法让我们重现此类模式。因此，提出了潮湿大气空气方程中小尺度通量收敛项形式的改变。这使我们能够导出斯威夫特 - 霍恩伯格方程。在封闭细胞和滚动对流的情况下，所得模式比从金兹堡 - 朗道方程获得的模式更有组织性，并且能更好地重现卫星观测结果，例如水平对流场。
