### Relation between the Distribution of Properties and the Currents in the Sea

Consider any scalar quantity, *s* (temperature, salinity, pressure, oxygen content, and so on), the distribution of which is continuous in space and time, so that it can be represented as a function of time and the three space coordinates, *s = f*(*t,x,y,z*). Let us assume that this scalar quantity can be considered a property of the individual particles of the fluid. A particle in motion after a time *dt* will be in a new locality,

*x + dx, y + dy, z + dz,*where the scalar quantity under consideration has the value

*s + ds = f*(

*t + dt, x + dx, y + dy, z + dz*). The property,

*s*, of the individual particle has therefore been changed by the amount

*ds*in the time

*dt*; that is, the time rate of change is

*ds/dt*. This time rate can also be expressed by the characteristics of the field, because, by means of Taylor's expansion, one has

[Equation]

*s = f(t,x,y,z),*

[Equation]

Dividing by *dt* and considering that *dx/dt, dy/dt* and *dz/dt* represent the components of the velocity, one obtains

[Equation]

*local*change. The last three terms are together called the

*advection*term, because they represent changes that take place in the presence of currents. This relationship is a purely formalistic one and gives no information as to the processes affecting the distribution; it merely states that within a field the

*individual time change can be considered as composed of two terms: the local time change and the advection.*

A few important points can be brought out by means of the above equation: (1) the distribution of any scalar quantity is *stationary*—that is, independent of time if the local change is zero (*∂s/∂ t* = 0); (2) the advection terms disappear if there is no motion or if the field is uniform—that is, if either *v*_{x} = *v*_{y} = *v*_{z} = 0 or ∂ *s*/∂ *x* = ∂ *s*/∂ *y* = ∂ *s*/∂ *z* = 0; (3) when the individual change is zero (*ds/dt* = 0), the local change is equal to the advection but is of opposite sign; (4) if the field of a property is stationary *(∂ s/∂ t* = 0) and if, further, the individual time change is zero (*ds/dt* = 0), equation (V, 4) is reduced to

[Equation]