A Degree Theory for Lagrangian Boundary Value Problems

We study those nonlinear partial differential equations which appear as Euler-Lagrange equations of variational problems. On defining weak boundary values of solutions to such equations we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to Lagrangian problems

The Competing Effects of Stability and Humidity on Subtropical Stratocumulus Entrainment and Cloud Evolution from a Lagrangian Perspective

The evolution of subtropical stratocumulus clouds and the boundary layer is studied on daily time scales from the Lagrangian perspective, following the flow. Measures of humidity above the boundary layer and of inversion strength are obtained from reanalysis data, and their effects on the Lagrangian evolution of cloud cover and the boundary layer are compared. An analysis that disentangles these variables and tests their effects independently is developed. Increased inversion strength and increased humidity above the boundary layer lead to anomalously persistent cloud cover and slower Lagrangian deepening of the boundary layer. These parameters affect the stratocumulus boundary layer in different ways: inversion strength controls the buoyancy difference across the inversion, while humidity differences affect both the radiation balance and rate of cloud drop evaporation at cloud top. The relative strengths of the two effects of humidity are compared using two products: the entraining humidity in the layer directly above the inversion and the radiating humidity, which is the mean humidity in the column above the entraining humidity. Results show that the variability in the radiating humidity is the primary driver of Lagrangian boundary layer depth changes, but entraining humidity variation is mostly responsible for altering cloud lifetime.

The genesis of Hurricane Nate and its interaction with a nearby environment of very dry air

The interaction of a tropical disturbance with its environment is thought to play an important role in whether a disturbance will develop of not. Most developing disturbances are somewhat protected from the intrusion of environmental dry air at mid-levels. For African easterly wave (AEW) disturbances, the protective boundary is approximated by closed streamlines in the wave-relative frame, and their interior is called the wave-pouch. The dynamic and thermodynamic processes of spin-up occur inside the pouch. In this study we define the kinematic boundaries for a non-AEW disturbance in the Bay of Campeche that originated along a sharp frontal boundary in a confluent region of low pressure. We examine these boundaries during the genesis of Hurricane Nate (2011) to show how a layer-wise pouch boundary in the Lagrangian frame may allow for some transport into the pouch along the frontal boundary while still protecting the innermost development region. This result illustrates a generic property of weakly unsteady flows, including the time-dependent critical-layer of AEWs, that lateral exchange of air occurs along a segment of the boundary formed by the instantaneous, closed translating streamlines. Transport in the Lagrangian frame is simplest when measured with respect to the stable and unstable manifolds of a hyperbolic trajectory, which are topologically invariant. In this framework, an exact analysis of vorticity transport identifies two sources; i) the advection of vorticity through the entrainment and expulsion of bounded material regions called lobes, and ii) the baroclinic contribution of vorticity transport through the tilting mechanism across the Lagrangian boundary. We also show how these Lagrangian boundaries impact the concentration of moisture, influence convection, and contribute to the pouch vertical structure.

Impact of environmental moisture on tropical cyclone intensification

The impacts of environmental moisture on the intensification of a tropical cyclone (TC) are investigated in the Weather Research and Forecasting (WRF) model, with a focus on the azimuthal asymmetry of the moisture impacts relative to the storm path. A series of sensitivity experiments with varying moisture perturbations in the environment are conducted and the Marsupial Paradigm framework is employed to understand the different moisture impacts. We find that modification of environmental moisture has insignificant impacts on the storm in this case unless it leads to convective activity that deforms the quasi-Lagrangian boundary of the storm and changes the moisture transport into the storm. By facilitating convection and precipitation outside the storm, enhanced environmental moisture ahead of the northwestward-moving storm induces a dry air intrusion to the inner core and limits TC intensification. In contrast, increased moisture in the rear quadrants favors intensification by providing more moisture to the inner core and promoting storm symmetry, with primary contributions coming from moisture increase in the boundary layer. The different impacts of environmental moisture on TC intensification are governed by the relative locations of moisture perturbations and their interactions with the storm Lagrangian structure.

Impact of environmental moisture on tropical cyclone intensification

The impacts of environmental moisture on the intensification of a tropical cyclone (TC) are investigated in the Weather Research and Forecasting (WRF) model, with a focus on the azimuthal asymmetry of the moisture impacts. A series of sensitivity experiments with varying moisture perturbations in the environment are conducted and the Marsupial Paradigm framework is employed to understand the different moisture impacts. We find that modification of environmental moisture has insignificant impacts on the storm in this case unless it leads to convective activity in the environment, which deforms the quasi-Lagrangian boundary of the storm. By facilitating convection and precipitation outside the storm, enhanced environmental moisture ahead of the northwestward-moving storm induces a dry air intrusion to the inner core and limits TC intensification. However, increased moisture in the rear quadrants favors intensification by providing more moisture to the inner core and promoting storm symmetry, with primary contributions coming from moisture increase in the boundary layer. The different impacts of environmental moisture on TC intensification are governed by the relative locations of moisture perturbations and their interactions with the storm Lagrangian structure.