From here.
Consider a function of time and space $\phi\left(t,\vec{x}\right)$. For specificity, this could be some ocean property such as temperature or salinity. Now suppose that we evaluate $\phi\left(t,\vec{x} \right)$ along some arbitrary trajectory $\vec{x}(t)$. That could be the course of an oceanographic vessel from which $\phi$ is measured. Let the velocity of the measurement point (i.e., the ship) be $\vec{v}=d\vec{x}/dt$. At what rate will our measured value of $\phi$ change in time? The answer is given by the chain rule:
$$ \frac{d \phi}{d t}=\frac{\partial \phi}{\partial t}+\frac{\partial \phi}{\partial x_{j}} \frac{d x_{j}}{d t}=\frac{\partial \phi}{\partial t}+v_{j} \frac{\partial \phi}{\partial x_{j}}. $$Now consider the special case in which $\vec{x}(t)$ is the trajectory of a fluid parcel, and the observer is following the same trajectory (e.g., a boat allowed to drift with the current). The velocity $\vec{v}$ is now the velocity of the flow at the parcel’s location at any given time: $\vec{u}(t)$. In this special case, the rate of change we measure will be
$$ \frac{d \phi}{d t}=\frac{\partial \phi}{\partial t}+u_{j} \frac{\partial \phi}{\partial x_{j}}. $$The expression on the right-hand side is called the material derivative, i.e., the time derivative following a material parcel. It is a total time derivative, but is distinguished by the use of an uppercase “D”, e.g., $D\phi/Dt$. It can be written equivalently in index form or in vector form:
$$ \frac{D}{D t} \equiv \frac{\partial}{\partial t}+u_{j} \frac{\partial}{\partial x_{j}} \equiv \frac{\partial}{\partial t}+\vec{u} \cdot \vec{\nabla} $$The material derivative has a dual character: it expresses Lagrangian information (the rate of change following a fluid parcel), but does so in an Eulerian way, i.e., in terms of partial derivatives with respect to space and time.
It is instructive to isolate in the expression above the partial time derivative. Suppose, for example, that the field in question is air temperature. Then
$$ \underbrace{\frac{\partial T}{\partial t}}_{\text{thermometer reading }}=\overbrace{\frac{D T}{D t}}^{\text {heating/cooling }}-\underbrace{\vec{u} \cdot \vec{\nabla} T}_{\text{advection }}. $$This tells us that the temperature at a given location can change for two reasons corresponding to the two terms on the right-hand side. The first term, $DT/Dt$, is nonzero only if the air parcels are actually being heated or cooled (heated by the sun, perhaps). The second term is due to the wind: if the wind is blowing from a warm place, the local temperature will rise. This process, whereby local changes result from transport by the flow, is called advection.
If there is no external factors it should be $\frac{D T}{D t}=(\frac{\partial}{\partial t}+\vec{u} \cdot \vec{\nabla}) T=0$, because if we follow a particle (fluid parcel) the temperature is always the same.
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Author of the notes: Antonio J. Pan-Collantes
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