Arbitrary-order intrinsic virtual element method for elliptic equations on surfaces

We develop a geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) on polygonal cells for the numerical solution of elliptic surface partial differential equations (PDEs). The PDE is first written in covariant form using an appropriate local reference system. The knowledge of the local parametrization allows us to consider the two-dimensional VEM scheme, without any explicit approximation of the surface geometry. The theoretical properties of the classical VEM are extended to our framework by taking into consideration the highly anisotropic character of the final discretization. These properties are extensively tested on triangular and polygonal meshes using a manufactured solution. The limitations of the scheme are verified as functions of the regularity of the surface and its approximation.

Validation of a Swimming Direction Model for the Downstream Migration of Atlantic Salmon Smolts

Fish swimming performance is strongly influenced by flow hydrodynamics, but little is known about the relation between fine-scale fish movements and hydrodynamics based on in-situ investigations. In the presented study, we validated the etho-hydraulic fish swimming direction model presented in the River Mandal from Southern Norway, using similar behavioral and hydraulic data on salmon smolts from the River Orkla in Central Norway. The re-parametrized model explained the variation of the swimming direction of fish in the Orkla system in same degree as the original model performed in the Mandal system (R2: 84% in both cases). The transferability of the model when using it from one river to predict swimming direction in the other river was lower (R2: 21% and 26%), but nevertheless relatively high given that the two localities differed in hydraulic conditions. The analyses thus provide support for the fact that the identified hydraulic parameters and their interaction affected smolt behavior in a similar way at the two sites, but that local parametrization of the base model is required. The developed etho-hydraulic models can provide important insights into fish behavior and fish migration trajectories and can be developed into prediction models important for the future development of behavioral downstream migration solutions.

LOCAL COORDINATES FOR COMPLEX AND QUATERNIONIC HYPERBOLIC PAIRS

Let $G(n)={\textrm {Sp}}(n,1)$ or ${\textrm {SU}}(n,1)$ . We classify conjugation orbits of generic pairs of loxodromic elements in $G(n)$ . Such pairs, called ‘nonsingular’, were introduced by Gongopadhyay and Parsad for ${\textrm {SU}}(3,1)$ . We extend this notion and classify $G(n)$ -conjugation orbits of such elements in arbitrary dimension. For $n=3$ , they give a subspace that can be parametrized using a set of coordinates whose local dimension equals the dimension of the underlying group. We further construct twist-bend parameters to glue such representations and obtain local parametrization for generic representations of the fundamental group of a closed (genus $g \geq 2$ ) oriented surface into $G(3)$ .

On conjugation orbits of semisimple pairs in rank one

We consider the Lie groups {\mathrm{SU}(n,1)} and {\mathrm{Sp}(n,1)} that act as isometries of the complex and the quaternionic hyperbolic spaces, respectively. We classify pairs of semisimple elements in {\mathrm{Sp}(n,1)} and {\mathrm{SU}(n,1)} up to conjugacy. This gives local parametrization of the representations ρ in {{\mathrm{Hom}}({\mathrm{F}}_{2},G)/G} such that both {\rho(x)} and {\rho(y)} are semisimple elements in G, where {{\mathrm{F}}_{2}=\langle x,y\rangle}, {G=\mathrm{Sp}(n,1)} or {\mathrm{SU}(n,1)}. We use the {{\mathrm{PSp}}(n,1)}-configuration space {{\mathrm{M}}(n,i,m-i)} of ordered m-tuples of distinct points in {\overline{{\mathbf{H}}_{{\mathbb{H}}}^{n}}}, where the first i points in an m-tuple are boundary points, to classify the semisimple pairs. Further, we also classify points on {{\mathrm{M}}(n,i,m-i)}.

Turbulence kinetic energy budget during the afternoon transition – Part 1: Observed surface TKE budget and boundary layer description for 10 intensive observation period days

The decay of turbulence kinetic energy (TKE) and its budget in the afternoon period from midday until zero-buoyancy flux at the surface is studied in a two-part paper by means of measurements from the Boundary Layer Late Afternoon and Sunset Turbulence (BLLAST) field campaign for 10 intensive observation period days. Here, in Part 1, near-surface measurements from a small tower are used to estimate a TKE budget. The overall boundary layer characteristics and mesoscale situation at the site are also described based upon taller tower measurements, radiosoundings and remote sensing instrumentation. Analysis of the TKE budget during the afternoon transition reveals a variety of different surface layer dynamics in terms of TKE and TKE decay. This is largely attributed to variations in the 8 m wind speed, which is responsible for different amounts of near-surface shear production on different afternoons and variations within some of the afternoon periods. The partitioning of near-surface production into local dissipation and transport in neutral and unstably stratified conditions was investigated. Although variations exist both between and within afternoons, as a rule of thumb, our results suggest that about 50 % of the near-surface production of TKE is compensated for by local dissipation near the surface, leaving about 50 % available for transport. This result indicates that it is important to also consider TKE transport as a factor influencing the near-surface TKE decay rate, which in many earlier studies has mainly been linked with the production terms of TKE by buoyancy and wind shear. We also conclude that the TKE tendency is smaller than the other budget terms, indicating a quasi-stationary evolution of TKE in the afternoon transition. Even though the TKE tendency was observed to be small, a strong correlation to mean buoyancy production of −0.69 was found for the afternoon period. For comparison with previous results, the TKE budget terms are normalized with friction velocity and measurement height and discussed in the framework of Monin–Obukhov similarity theory. Empirically fitted expressions are presented. Alternatively, we also suggest a non-local parametrization of dissipation using a TKE–length scale model which takes into account the boundary layer depth in addition to distance above the ground. The non-local formulation is shown to give a better description of dissipation compared to a local parametrization.

Turbulence Kinetic Energy budget during the afternoon transition – Part 1: Observed surface TKE budget and boundary layer description for 10 intensive observation period days

The decay of turbulence kinetic energy (TKE) and its budget in the afternoon period from mid-day until zero buoyancy flux at the surface is studied in a two-part paper by means of measurements from the Boundary Layer Late Afternoon and Sunset Turbulence (BLLAST) field campaign for 10 Intensive Observation Period days. Here, in Part 1, near-surface measurements from a small tower are used to estimate a TKE budget. The overall boundary layer characteristics and meso-scale situation at the site are also described based upon taller tower measurements, radiosoundings and remote sensing instrumentation. Analysis of the TKE budget during the afternoon transition reveals a variety of different surface layer dynamics in terms of TKE and TKE decay. This is largely attributed to variations in the 8 m wind speed, which is responsible for different amounts of near-surface shear production on different afternoons and variations within some of the afternoon periods. The partitioning of near surface production into local dissipation and transport in neutral and unstably stratified conditions was investigated. Although variations exist both between and within afternoons, as a rule of thumb, our results suggest that about 50 % of the near surface production of TKE is compensated by local dissipation near the surface, leaving about 50 % available for transport. This result indicates that it is important to also consider TKE transport as a factor influencing the near-surface TKE decay rate, which in many earlier studies has mainly been linked with the production terms of TKE by buoyancy and wind shear. We also conclude that the TKE tendency is smaller than the other budget terms, indicating a quasi-stationary evolution of TKE in the afternoon transition. Even though the TKE tendency was observed to be small, a strong correlation to mean buoyancy production of −0.69 was found for the afternoon period. For comparison with previous results, the TKE budget terms are normalized with friction velocity and measurement height and discussed in the framework of Monin–Obukhov similarity theory. Empirically fitted expressions are presented. Alternatively, we also suggest a non-local parametrization of dissipation using a TKE-length scale model which takes into account the boundary layer depth in addition to distance above the ground. The non-local formulation is shown to give a better description of dissipation compared to a local parametrization.