Trajectory (fluid mechanics)

Last updated March 15, 2019

In fluid mechanics, meteorology and oceanography, a trajectory traces the motion of a single point, often called a parcel, in the flow.

Fluid mechanics is the branch of physics concerned with the mechanics of fluids and the forces on them. It has applications in a wide range of disciplines, including mechanical, civil, chemical and biomedical engineering, geophysics, astrophysics, and biology.

Meteorology is a branch of the atmospheric sciences which includes atmospheric chemistry and atmospheric physics, with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did not occur until the 18th century. The 19th century saw modest progress in the field after weather observation networks were formed across broad regions. Prior attempts at prediction of weather depended on historical data. It was not until after the elucidation of the laws of physics and more particularly, the development of the computer, allowing for the automated solution of a great many equations that model the weather, in the latter half of the 20th century that significant breakthroughs in weather forecasting were achieved.

Oceanography, also known as oceanology, is the study of the physical and biological aspects of the ocean. It is an Earth science, which covers a wide range of topics, including ecosystem dynamics; ocean currents, waves, and geophysical fluid dynamics; plate tectonics and the geology of the sea floor; and fluxes of various chemical substances and physical properties within the ocean and across its boundaries. These diverse topics reflect multiple disciplines that oceanographers blend to further knowledge of the world ocean and understanding of processes within: astronomy, biology, chemistry, climatology, geography, geology, hydrology, meteorology and physics. Paleoceanography studies the history of the oceans in the geologic past.

Trajectories are useful for tracking atmospheric contaminants, such as smoke plumes, and as constituents to Lagrangian simulations, such as contour advection or semi-Lagrangian schemes.

Contour advection is a Lagrangian method of simulating the evolution of one or more contours or isolines of a tracer as it is stirred by a moving fluid. Consider a blob of dye injected into a river or stream: to first order it could be modelled by tracking only the motion of its outlines. It is an excellent method for studying chaotic mixing: even when advected by smooth or finitely-resolved velocity fields, through a continuous process of stretching and folding, these contours often develop into intricate fractals. The tracer is typically passive as in but may also be active as in, representing a dynamical property of the fluid such as vorticity. At present, advection of contours is limited to two dimensions, but generalizations to three dimensions are possible.

The Semi-Lagrangian scheme (SLS) is a numerical method that is widely used in numerical weather prediction models for the integration of the equations governing atmospheric motion. A Lagrangian description of a system focuses on following individual air parcels along their trajectories as opposed to the Eulerian description, which considers the rate of change of system variables fixed at a particular point in space. A semi-Lagrangian scheme uses Eulerian framework but the discrete equations come from the Lagrangian perspective.

Suppose we have a time-varying flow field, ${\vec {v}}({\vec {x}},~t)$ . The motion of a fluid parcel, or trajectory, is given by the following system of ordinary differential equations:

{\frac {d{\vec {x}}}{dt}}={\vec {v}}({\vec {x}},~t)

While the equation looks simple, there are at least three concerns when attempting to solve it numerically. The first is the integration scheme. This is typically a Runge-Kutta,^[1] although others can be useful as well, such as a leapfrog. The second is the method of determining the velocity vector, ${\vec {v}}$ at a given position, ${\vec {x}}$ , and time, t. Normally, it is not known at all positions and times, therefore some method of interpolation is required. If the velocities are gridded in space and time, then bilinear, trilinear or higher-dimensional linear interpolation is appropriate. Bicubic, tricubic, etc., interpolation is used as well, but is probably not worth the extra computational overhead.

In numerical analysis, leapfrog integration is a method for numerically integrating differential equations of the form

In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.

In mathematics, bilinear interpolation is an extension of linear interpolation for interpolating functions of two variables on a rectilinear 2D grid.

Velocity fields can be determined by measurement, e.g. from weather balloons, from numerical models or especially from a combination of the two, e.g. assimilation models.

Data assimilation is a mathematical discipline that seeks to optimally combine theory with observations. There may be a number of different goals sought, for example—to determine the optimal state estimate of a system, to determine initial conditions for a numerical forecast model, to interpolate sparse observation data using knowledge of the system being observed, to train numerical model parameters based on observed data. Depending on the goal, different solution methods may be used. Data assimilation is distinguished from other forms of machine learning, image analysis, and statistical methods in that it utilizes a dynamical model of the system being analyzed.

The final concern is metric corrections. These are necessary for geophysical fluid flows on a spherical Earth. The differential equations for tracing a two-dimensional, atmospheric trajectory in longitude-latitude coordinates are as follows:

{\frac {d\theta }{dt}}={\frac {u}{r\cos \phi }}

{\frac {d\phi }{dt}}={\frac {v}{r}}

where, $\theta$ and $\phi$ are, respectively, the longitude and latitude in radians, r is the radius of the Earth, u is the zonal wind and v is the meridional wind.

The radian is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees. The unit was formerly an SI supplementary unit, but this category was abolished in 1995 and the radian is now considered an SI derived unit.

One problem with this formulation is the polar singularity: notice how the denominator in the first equation goes to zero when the latitude is 90 degrees—plus or minus. One means of overcoming this is to use a locally Cartesian coordinate system close to the poles. Another is to perform the integration on a pair of Azimuthal equidistant projections—one for the N. Hemisphere and one for the S. Hemisphere.^[2]

Trajectories can be validated by balloons in the atmosphere and buoys in the ocean.

External links

ctraj: A trajectory integrator written in C++.

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A great circle, also known as an orthodrome, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere. A great circle is the largest circle that can be drawn on any given sphere. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same center and circumference as each other. This special case of a circle of a sphere is in opposition to a small circle, that is, the intersection of the sphere and a plane that does not pass through the center. Every circle in Euclidean 3-space is a great circle of exactly one sphere.

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In physics, a force is said to do work if, when acting, there is a displacement of the point of application in the direction of the force. For example, when a ball is held above the ground and then dropped, the work done on the ball as it falls is equal to the weight of the ball multiplied by the distance to the ground. When the force is constant and the angle between the force and the displacement is θ, then the work done is given by W = Fs cos θ.

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References

↑ William H. Press; Brian P. Flannery; Saul A. Teukolsky; William T. Vetterling (1992). Numerical Recipes in C: the Art of Scientific Computing (2nd ed.). Cambridge University Press.
↑ Mills, Peter (2012). "Principal component proxy tracer analysis". arXiv: 1202.1999   [physics.ao-ph].

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[Press_etal1992-1] William H. Press; Brian P. Flannery; Saul A. Teukolsky; William T. Vetterling (1992). Numerical Recipes in C: the Art of Scientific Computing (2nd ed.). Cambridge University Press.

[Mills2012-2] Mills, Peter (2012). "Principal component proxy tracer analysis". arXiv: 1202.1999   [physics.ao-ph].