Numerical simulation of shear-induced instabilities in internal solitary waves

A numerical method that employs a combination of contour advection and pseudo-spectral techniques is used to simulate shear-induced instabilities in an internal solitary wave (ISW). A three-layer configuration for the background stratification, in which a linearly stratified intermediate layer is sandwiched between two homogeneous ones, is considered throughout. The flow is assumed to satisfy the inviscid, incompressible, Oberbeck–Boussinesq equations in two dimensions. Simulations are initialized by fully nonlinear, steady-state, ISWs. The results of the simulations show that the instability takes place in the pycnocline and manifests itself as Kelvin–Helmholtz billows. The billows form near the trough of the wave, subsequently grow and disturb the tail. Both the critical Richardson number (${\mathit{Ri}}_{c} $) and the critical amplitude required for instability are found to be functions of the ratio of the undisturbed layer thicknesses. It is shown, therefore, that the constant, critical bound for instability in ISWs given in Barad & Fringer (J. Fluid Mech., vol. 644, 2010, pp. 61–95), namely ${\mathit{Ri}}_{c} = 0. 1\pm 0. 01$, is not a sufficient condition for instability. It is also shown that the critical value of ${L}_{x} / \lambda $ required for instability, where ${L}_{x} $ is the length of the region in a wave in which $\mathit{Ri}\lt 1/ 4$ and $\lambda $ is the half-width of the wave, is sensitive to the ratio of the layer thicknesses. Similarly, a linear stability analysis reveals that ${\bar {\sigma } }_{i} {T}_{w} $ (where ${\bar {\sigma } }_{i} $ is the growth rate of the instability averaged over ${T}_{w} $, the period in which parcels of fluid are subjected to $\mathit{Ri}\lt 1/ 4$) is very sensitive to the transition between the undisturbed pycnocline and the homogeneous layers, and the amplitude of the wave. Therefore, the alternative tests for instability presented in Fructus et al. (J. Fluid Mech., vol. 620, 2009, pp. 1–29) and Barad & Fringer (J. Fluid Mech., vol. 644, 2010, pp. 61–95), respectively, namely ${L}_{x} / \lambda \geq 0. 86$ and ${\bar {\sigma } }_{i} {T}_{w} \gt 5$, are shown to be valid only for a limited parameter range.

The Diabatic Contour-Advective Semi-Lagrangian Algorithms for the Spherical Shallow Water Equations

The diabatic contour-advective semi-Lagrangian (DCASL) algorithm is extended to the thermally forced shallow water equations on the sphere. DCASL rests on the partitioning of potential vorticity (PV) to adiabatic and diabatic parts solved, respectively, by contour advection and a grid-based conventional algorithm. The presence of PV in the source term for diabatic PV makes the shallow water equations distinct from the quasigeostrophic model previously studied. To address the more rapid generation of finescale structures in diabatic PV, two new features are added to DCASL: (i) the use of multiple sets of contours with successively finer contour intervals and (ii) the application of the underlying method of DCASL at a higher level to diabatic PV. That is, the diabatic PV is allowed to have both contour and grid parts. The added features make it possible to make the grid part of diabatic PV arbitrarily small and thus pave the way for a fully Lagrangian DCASL in the presence of forcing. The DCASL algorithms are constructed using a standard semi-Lagrangian (SL) algorithm to solve for the grid-based part of diabatic PV. The 25-day time evolution of an unstable midlatitude jet triggered by the action of thermal forcing is used as a test case to examine and compare the properties of the DCASL algorithms with a pure SL algorithm for PV. Diagnostic measures of vortical and unbalanced activity as well as of the relative strength of the grid and contour parts of the solution for PV indicate that the superiority of contour advection can be maintained even in the presence of strong, nonsmooth forcing.

Assessing the Numerical Accuracy of Complex Spherical Shallow-Water Flows

The representation of nonlinear shallow-water flows poses severe challenges for numerical modeling. The use of contour advection with contour surgery for potential vorticity (PV) within the contour-advective semi-Lagrangian (CASL) algorithm makes it possible to handle near-discontinuous distributions of PV with an accuracy beyond what is accessible to conventional algorithms used in numerical weather and climate prediction. The emergence of complex distributions of the materially conserved quantity PV, in the absence of forcing and dissipation, results from large-scale shearing and deformation and is a common feature of high Reynolds number flows in the atmosphere and oceans away from boundary layers. The near-discontinuous PV in CASL sets a limit on the actual numerical accuracy of the Eulerian, grid-based part of CASL. For the spherical shallow-water equations, the limit is studied by comparing the accuracy of CASL algorithms with second-order-centered, fourth-order-compact, and sixth-order-supercompact finite differencing in latitude in conjunction with a spectral treatment in longitude. The comparison is carried out on an unstable midlatitude jet at order one Rossby number and low Froude number that evolves into complex vortical structures with sharp gradients of PV. Quantitative measures of global conservation of energy and angular momentum, and of imbalance as diagnosed using PV inversion by means of Bolin–Charney balance, indicate that fourth-order differencing attains the highest numerical accuracy achievable for such nonlinear, advectively dominated flows.

Description of a coupled atmosphere - fire model

This paper describes a coupled fire–atmosphere model that uses a sophisticated high-resolution non-hydrostatic numerical mesoscale model to predict the local winds which are then used as input to the prediction of fire spread. The heat and moisture fluxes from the fire are then fed back to the dynamics, allowing the fire to influence its own mesoscale winds that in turn affect the fire behavior. This model is viewed as a research model and as such requires a fireline propagation scheme that systematically converges with increasing spatial and temporal resolution. To achieve this, a local contour advection scheme was developed to track the fireline using four tracer particles per fuel cell, which define the area of burning fuel. Using the dynamically predicted winds along with the terrain slope and fuel characteristics, algorithms from the BEHAVE system are used to predict the spread rates. A mass loss rate calculation, based on results of the BURNUP fuel burnout model, is used to treat heat exchange between the fire and atmosphere. Tests were conducted with the uncoupled model to test the fire-spread algorithm under specified wind conditions for both spot and line fires. Using tall grass and chaparral, line fires were simulated employing the full fire–atmosphere coupling. Results from two of these experiments show the effects of fire propagation over a small hill. As with previous coupled experiments, the present results show a number of features common to real fires. For example, we show how the well-recognized elliptical fireline shape is a direct result of fire–atmosphere interactions that produce the ‘heading’, ‘flanking’, and ‘backing’ regions of a wind-driven fire with their expected behavior. And, we see how perturbations upon this shape sometimes amplify to become fire whirls along the flanks, which are transported to the head of the fire where they may interact to produce erratic fire behavior.

Stratospheric water vapour as tracer for vortex filamentation in the Arctic winter 2002/2003

During winter 2002/2003, three balloon-borne frost point hygrometers measured high-resolution profiles of stratospheric water vapour above Ny-Ålesund, Spitsbergen. All measurements reveal a high H2O mixing ratio of about 7 ppmv above 24 km, thus differing significantly from the 5 ppmv that are commonly assumed for the calculation of polar stratospheric cloud existence temperatures. The profiles obtained on 12 December 2002 and on 17 January 2003 provide an insight into the vertical distribution of water vapour in the core of the polar vortex. Unlike the earlier profiles, the water vapour sounding on 11 February 2003 detected the vortex edge region in the lower part of the stratosphere. Here, a striking diminuition in H2O mixing ratio stands out between 16 and 19 km. The according stratospheric temperatures clarify that this dehydration can not be caused by the presence of polar stratospheric clouds or earlier PSC particle sedimentation. On the same day, ozone observations by lidar indicate a large scale movement of the polar vortex, while an ozone sonde measurement even shows laminae in the same altitude range as in the water vapour profile. Tracer lamination in the vortex edge region is caused by filamentation of the vortex. The link between the observed water vapour diminuition and filaments in the vortex edge region is highlighted by results of the MIMOSA contour advection model. In the altitude of interest, adjoined filaments of polar and mid-latitudinal air can be identified above the Spitsbergen region. A vertical cross-section reveals that the water vapour sonde has flown through polar air in the lowest part of the stratosphere. Where the low water vapour mixing ratio was detected, the balloon passed through air from a mid-latitudinal filament from about 425 to 445 K, before it finally entered the polar vortex above 450 K. The MIMOSA model results elucidate the correlation that on 11 February 2003 the frost point hygrometer measured strongly variable water vapour concentrations as the sonde detected air with different origins, respectively. Instead of being linked to dehydration due to PSC particle sedimentation, the local diminuition in the stratospheric water vapour profile of 11 February 2003 has been found to be caused by dynamical processes in the polar stratosphere.