Circulation Around a Constrained Curve: An Alternative Analysis Tool for Diagnosing the Origins of Tornado Rotation in Numerical Supercell Simulations

Fine-resolution computer models of supercell storms generate realistic tornadic vortices. Like real tornadoes, the origins of these virtual vortices are mysterious. To diagnose the origin of a tornado, typically a near-ground material circuit is drawn around it. This circuit is then traced back in time using backward trajectories. The rate of change of the circulation around the circuit is equal to the total force circulation. This circulation theorem is used to deduce the origins of the tornado’s large vorticity. However, there is a well-known problem with this approach; with staggered grids parcel trajectories become uncertain as they dip into the layer next to the ground where horizontal wind cannot be interpolated.To circumvent this dilemma, we obtain a generalized circulation theorem that pertains to any circuit. We apply this theorem either to moving circuits that are constrained to simple surfaces or to a ‘hybrid’ circuit defined next. Let A be the horizontal surface at one grid spacing off the ground. Above A the circuit moves as a material circuit. Horizontal curve segments that move in A with the horizontal wind replace segments of the material circuit that dip below A. The circulation equation for the modified circuit includes the force circulation of the inertial force that is required to keep the curve segments horizontal. This term is easily evaluated on A.Use of planar or circular circuits facilitates explanation of some simple flows. The hybrid-circuit method significantly improves the accuracy of the circulation budget in an idealized supercell simulation.

Stochastic Wave–Current Interaction in Thermal Shallow Water Dynamics

Holm (Proc R Soc A Math Phys Eng Sci 471(2176):20140963, 2015) introduced a variational framework for stochastically parametrising unresolved scales of hydrodynamic motion. This variational framework preserves fundamental features of fluid dynamics, such as Kelvin’s circulation theorem, while also allowing for dispersive nonlinear wave propagation, both within a stratified fluid and at its free surface. The present paper combines asymptotic expansions and vertical averaging with the stochastic variational framework to formulate a new approach for developing stochastic parametrisation schemes for nonlinear waves in fluid dynamics. The approach is applied to two sequences of shallow water models which descend from Euler’s three-dimensional fluid equations with rotation and stratification under approximation by asymptotic expansions and vertical averaging. In the entire family of nonlinear stochastic wave–current interaction equations derived here using this approach, Kelvin’s circulation theorem reveals a barotropic mechanism for wave generation of horizontal circulation or convection (cyclogenesis) which is activated whenever the gradients of wave elevation and/or topography are not aligned with the gradient of the vertically averaged buoyancy.

A Hamiltonian Interacting Particle System for Compressible Flow

The decomposition of the energy of a compressible fluid parcel into slow (deterministic) and fast (stochastic) components is interpreted as a stochastic Hamiltonian interacting particle system (HIPS). It is shown that the McKean–Vlasov equation associated to the mean field limit yields the barotropic Navier–Stokes equation with density-dependent viscosity. Capillary forces can also be treated by this approach. Due to the Hamiltonian structure, the mean field system satisfies a Kelvin circulation theorem along stochastic Lagrangian paths.

CONSERVED QUANTITIES ON MULTISYMPLECTIC MANIFOLDS

Given a vector field on a manifold $M$, we define a globally conserved quantity to be a differential form whose Lie derivative is exact. Integrals of conserved quantities over suitable submanifolds are constant under time evolution, the Kelvin circulation theorem being a well-known special case. More generally, conserved quantities are well behaved under transgression to spaces of maps into $M$. We focus on the case of multisymplectic manifolds and Hamiltonian vector fields. Our main result is that in the presence of a Lie group of symmetries admitting a homotopy co-momentum map, one obtains a whole family of globally conserved quantities. This extends a classical result in symplectic geometry. We carry this out in a general setting, considering several variants of the notion of globally conserved quantity.

A Hamiltonian mean field system for the Navier–Stokes equation

We use a Hamiltonian interacting particle system to derive a stochastic mean field system whose McKean–Vlasov equation yields the incompressible Navier–Stokes equation. Since the system is Hamiltonian, the particle relabeling symmetry implies a Kelvin Circulation Theorem along stochastic Lagrangian paths. Moreover, issues of energy dissipation are discussed and the model is connected to other approaches in the literature.

The Regulation of Tornado Intensity by Updraft Width

Strong to violent tornadoes cause a disproportionate amount of damage, in part because the width and length of a tornado damage track are correlated to tornado intensity (as now estimated through enhanced Fujita scale ratings). The tendency expressed in the observational record is that the most intense tornadoes are often the widest. Herein the authors explore the simple hypothesis that wide intense tornadoes should form more readily out of wide rotating updrafts. This hypothesis is based on an application of Kelvin’s circulation theorem, which is used to argue that the large circulation associated with a wide intense tornado is more plausibly associated with a wide mesocyclone. Because a mesocyclone is, strictly speaking, a rotating updraft, the mesocyclone width should increase with increasing updraft width. A simple mathematical model that is quantified using observations of mesocyclones supports this hypothesis, as do idealized numerical simulations of supercellular thunderstorms.

A variational formulation of vertical slice models

A variational framework is defined for vertical slice models with three-dimensional velocity depending only on x and z . The models that result from this framework are Hamiltonian, and have a Kelvin–Noether circulation theorem that results in a conserved potential vorticity in the slice geometry. These results are demonstrated for the incompressible Euler–Boussinesq equations with a constant temperature gradient in the y -direction (the Eady–Boussinesq model), which is an idealized problem used to study the formation and subsequent evolution of weather fronts. We then introduce a new compressible extension of this model. Unlike the incompressible model, the compressible model does not produce solutions that are also solutions of the three-dimensional equations, but it does reduce to the Eady–Boussinesq model in the low Mach number limit. Hence, the new model could be used in asymptotic limit error testing for compressible weather models running in a vertical slice configuration.