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The behaviour of warm water discharge at a temperature higher then Tm horizontally into a homogeneous body of cold fresh water at a temperature lower then Tm was investigated by means of a numerical model. Water density here was taken to be a quadratic function of temperature. Thus cabbeling process was inevitable as positively buoyant water form surface current while penetrating the ambient water. The current halted as mixture became dense and sink. These results are very similar to the experimental study of warm discharge into cold water by Marmoush et al. [14] and Bukreev [22]. The results showed an initially sinking water at the point where the two water bodies meets within the first few time interval. Development of Rayleigh-Taylor instabilities was observed at the lower part of the surface current as lighter fluid penetrate further. The frontal head was found to being replenished by a surface flow of warm unadulterated water, but after much entrainment of ambient fluid and cabbeling then, this head halted and sink. On the floor, denser fluid advance in the same direction as the original surface current, with some degree of Kelvin-Helmholtz instability as it penetrates further. Relations were also drawn that describes the speed, the spread length of both surface current were obtained. Relation were also drawn that describes the final spread length of the surface current Lsm and the time taken to reach that final spread length Tsm as a function ϕin. This work as presented here is practical and relevant to many fields of study and also enhances policy making towards the protection of the aquatic ecosystems.

Laminar plumes from a line source of warm water at the base of a shallow, homogeneous body of cold water (below the temperature of maximum density)were simulated by a computational model. The plume water undergoes buoyancy reversal as it mixes with the cold ambient. If this occurs before the plume has reached the ceiling of the domain, the plume flaps from side to side. Otherwise, it spreads along the ceiling and then sinks, with a vortex enclosed between the rising plume and the sinking flow. Some of the dense, mixed water from the sinking flow is re-entrained into the rising plume, while the rest flows outwards along the floor. However, with high source temperatures, a sufficient volume of warm water eventually builds up to also form a positively buoyant gravity current along the ceiling.

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The orienteering route choice problem involves finding the fastest route between two given points, with running speed determined by various properties of the terrain. In this study, I consider only the effect of climbing or descending on running speed. If a runner's pace p (the reciprocal of speed) varies linearly with gradient m, the straight‐line route always is fastest. However, a nonlinear formulation for p(m), with d2p/dm2 > 0, will more accurately model runners’ capabilities. As a result, critical gradients may exist for ascent and/or descent, such that optimal routes will never ascend or descend more steeply than the critical gradient. I review and propose several formulations for the pace function p(m) and calculate their critical gradients. In principle, the Euler–Lagrange equation can be used to find optimal routes between arbitrary points on any topography where the height can be expressed as a smooth function of horizontal coordinates. I obtain first integrals of this equation for idealized landforms: hillsides with straight contours and axisymmetric hills. Next, optimal routes are computed for various combinations of start‐ and endpoints on these landforms based on various pace functions. These routes are classified as either subcritical or maximal steepness: The former ascends or descends less steeply than the critical gradient; the latter takes the line of steepest ascent where it is not steeper than the critical gradient, but follows a curve at the critical gradient where the slope is steeper. In some cases, the optimal route zigzags up or down a hill along sections of a critical‐gradient curve. El problema de trazo y selección de ruta en terrenos accidentados (orienteering route choice problem) consiste en encontrar la ruta más rápida entre dos puntos dados, cuando la velocidad de desplazamiento está en función a varias propiedades del terreno. En este estudio, el autor sólo considera el efecto de subir o bajar sobre la velocidad de desplazamiento. Si el paso p de un individuo (la inversa de la velocidad) varía linealmente con la pendiente m, la ruta en línea recta siempre es la más rápida. Sin embargo, una formulación no lineal de p (m), con D2P / dm 2> 0, resulta en un modelo más preciso de las capacidades de los individuos que se desplazan por el terreno. Como resultado, es posible que existan gradientes críticos tanto para el movimiento de ascenso como el de descenso. En dicho caso, las rutas óptimas nunca ascenderán o descenderán por gradientes más pronunciados que las identificadas comode gradiente crítico. El autor revisa y propone varias fórmulas para una función de paso p (m) y calcula los gradientes críticos respectivos. En principio, la ecuación de Euler‐Lagrange puede ser utilizada para trazar las rutas óptimas entre puntos arbitrarios en cualquier topografía en donde la altura pueda ser expresada como una función suavizada de las coordenadas horizontales. El autor calcula las integrales de esta ecuación para dos formas terrestres idealizadas: laderas con contornos rectos, y colinas c...

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Momentum and energy equations for vertical flow with viscous dissipation are derived and shown to require that the cross-section mean density is taken as the reference density for calculation of buoyancy forces under the Boussinesq approximation. Solutions are obtained for flow between parallel plane walls, with and without the pressure work as an explicit term in the energy equation. Both walls are at the same temperature, so there is no thermal forcing, but solutions are obtained for all admissible values of dynamic pressure gradient. The passive convection condition, whereby the flow is driven entirely by buoyancy forces resulting from heat generated by the flow's own viscous dissipation, is found on one branch of the dual solutions. However, while theoretically possible, passive convection is not physically realisable with any real fluid.