Coupling of Wind and Potential Temperature in an Ekman Model in the Stratified Atmospheric Boundary Layer

A linear Ekman model in the stratified atmospheric boundary layer (ABL) is proposed based on the steady state version of the linearized three-dimensional primitive equations with the inclusion of the vertical diffusivity. Due to the inclusion of the potential temperature equation and hydrostatic equation, pressure and potential temperature couple with wind in the proposed model, and thus are not arbitrarily specified variables as in previous studies on the baroclinicity in the Ekman model. The extended thermal wind balance equation and the Ekman potential vorticity equation are derived to describe the coupling. The two equations, along with the equation describing the constraint on potential temperature, are employed to derive the analytical solutions of the proposed Ekman model. Because potential temperature is not a specified variable but part of the solution, the derived analytical solutions have very different forms from those derived in previous studies. The differences illustrate the impact of the inclusion of the potential temperature equation and hydrostatic equation on wind, pressure and potential temperature in the proposed Ekman model. It is found that the computed wind profiles based on the proposed model can capture some important features of the observed wind profiles.

Calculation of evaporation length of a liquid bridge flowing between inclined hot tubes

The bridge consists of liquid held by surface tension forces between two inclined tubes in an LNG heat exchanger. The shape of the bridge is calculated by the hydrostatic equation, which is reduced to a nonlinear integral equation and resolved by the Newton method. The velocity and temperature fields in the bridge are described by the Navier-Stokes and energy equations, respectively. They are reduced to the boundary integral equations and calculated by the method of boundary elements. Heat transfer coefficient is calculated for evaporating bridge and the length of total bridge evaporation is estimated.

Invented Forces in Supercell Models

This paper examines methods used in supercell models to maintain a steady, sheared, horizontally uniform environment with a three-force balance in the planetary boundary layer (PBL) and a two-force balance above it. Steady environments are maintained while ignoring the thermal-wind balance that permits large shear above the PBL. The Taylor-Proudman theorem indicates that wind profiles above the PBL must be unidirectional for balanced environments. In principle, supercell models that do not accommodate thermal advection should not support balanced steady environments with veering horizontally uniform winds.Recent methods add a permanent, pervasive, horizontal external force that varies only with height. By adding two more degrees of freedom, this force circumvents the Taylor-Proudman theorem and enables a static, horizontally uniform environment for any wind profile. It succeeds by adding spurious energy in lieu of flow towards low pressure to offset frictional loss of kinetic energy. However, the artificial force has downsides. It decouples the environmental horizontal equation of motion from the hydrostatic equation and the thermodynamics from the dynamics. It cancels environmental friction and the part of the Coriolis force that acts on the environmental wind. Within the storm, its curl can speciously generate significant horizontal vorticity near the ground. Inaccuracies arise in circulations around material circuits because of modifications by the artificial force and resulting miscalculations of parcel trajectories. Doubt is cast on conclusions about tornadogenesis drawn from recent simulations that contain an invented force.

X-ray scaling relations of elliptical galaxies

We present the X-ray scaling relations of a sample of elliptical galaxies using X-ray Chandra observations. We use a single β-model to reconstruct the density gas profile and the hydrostatic equation to measure the total mass. We build X-ray scaling relations between temperature, luminosity and total mass at 5Re. Our sample in- cludes gas-poor objects that help us to extend already published LX − T , M − T and LX − M scaling relations to lower magnitudes. The derived slopes for all three relations are much steeper than those for galaxy clusters and groups, and are extreme compared to self-similar models considering gravity only. This increase of steepness from clusters to galaxies indicates that non-thermal processes (e.g. AGN feedback) have a stronger influence on the small-scale systems. Our results indicate that the hot interstellar medium of gas-poor early-type galaxies is not in the wind/outflow state as suggested by recent hydrodynamical simulations. We also discuss the possible biases of our calculations and compare our results to recent papers.

Pressure variation in a fluid at rest (hydrostatics)

The three fundamental principles for the variation of static pressure p throughout a body of fluid at rest are (a) the pressure at a point is the same in all directions (Pascal’s law), (b) the pressure is the same at all points on the same horizontal level, and (c) the pressure increases with depth z according to the hydrostatic equation. dp/dz= ρ‎g For a fluid with constant density ρ‎, the increase in pressure over a depth increase h is ρ‎gh, a result which can be used to analyse the response of simple barometers and manometers to applied pressure changes and differences. In situations where very large changes in pressure occur an equation of state may be required to relate pressure and density together with an assumption about the fluid temperature. The hydrostatic equation is still valid but more difficult to integrate, as illustrated by consideration of the earth’s atmosphere.

Two Overlooked Biases of the Advanced Research WRF (ARW) Model in Geopotential Height and Temperature

The authors have discovered two sizeable biases in the Weather Research and Forecasting (WRF) model: a negative bias in geopotential and a warm bias in temperature, appearing both in the initial condition and the forecast. The biases increase with height and thus manifest themselves at the upper part of the model domain. Both biases stem from a common root, which is that vertical structures of specific volume and potential temperature are convex functions. The geopotential bias is caused by the particular discrete hydrostatic equation used in WRF and is proportional to the square of the thickness of model layers. For the vertical levels used in this study, the bias far exceeds the gross 1-day forecast bias combining all other sources. The bias is fixed by revising the discrete hydrostatic equation. WRF interpolates potential temperature from the grids of an external dataset to the WRF grids in generating the initial condition. Associated with the Exner function, this leads to the marked bias in temperature. By interpolating temperature to the WRF grids and then computing potential temperature, the bias is removed. The bias corrections developed in this study are expected to reduce the disparity between the forecast and observations, and eventually to improve the quality of analysis and forecast in the subsequent data assimilation. The bias corrections might be especially beneficial to assimilating height-based observations (e.g., radio occultation data).