fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasma (physics), plasmas) and the forces on them. Originally applied to water (hydromechanics), it found applications in a wide range of discipl ...

meteorology Meteorology is the scientific study of the Earth's atmosphere and short-term atmospheric phenomena (i.e. weather), with a focus on weather forecasting. It has applications in the military, aviation, energy production, transport, agricultur ...

(weather) and

oceanography Oceanography (), also known as oceanology, sea science, ocean science, and marine science, is the scientific study of the ocean, including its physics, chemistry, biology, and geology. It is an Earth science, which covers a wide range of to ...

, a trajectory traces the motion of a single point, often called a parcel, in the flow. 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. 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 In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives ...

: :

\frac = \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, 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 In the mathematics, mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one ...

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 Overhead in computer systems consists of shared functions that benefit all users or processes but are not directly attributable to any specific task. It is thus similar to overhead in organizations. Computer system overhead shows up as slower pr ...

. 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. 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 = \frac

\frac = \frac

where,

\theta

and

\phi

are, respectively, the longitude and latitude in

radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...

s, ''r'' is the radius of the Earth, ''u'' is the zonal wind and ''v'' is the meridional wind. 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 In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

system close to the poles. Another is to perform the integration on a pair of

Azimuthal equidistant projection The azimuthal equidistant projection is an azimuthal map projection. It has the useful properties that all points on the map are at proportionally correct distances from the center point, and that all points on the map are at the correct azimuth ...

s—one for the N. Hemisphere and one for the S. Hemisphere. Trajectories can be validated by balloons in the

atmosphere An atmosphere () is a layer of gases that envelop an astronomical object, held in place by the gravity of the object. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A stellar atmosph ...

and buoys in the

ocean The ocean is the body of salt water that covers approximately 70.8% of Earth. The ocean is conventionally divided into large bodies of water, which are also referred to as ''oceans'' (the Pacific, Atlantic, Indian Ocean, Indian, Southern Ocean ...

External links

ctraj
A trajectory integrator written in C++.

References

{{Reflist Fluid dynamics Continuum mechanics Meteorological concepts Numerical analysis Numerical climate and weather models