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5.1: Lagrangian and Eulerian descriptions

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    Figure \(\PageIndex{1}\): Cloud patterns reveal complex motions in Jupiter’s atmosphere. The large vortex at the right is the Great Red Spot. Image courtesy of NASA.

    We can observe a flow in two ways, first by focusing on the motion of a specific fluid parcel (see section 1.2), second by stepping back and looking at the pattern as a whole. These are called the Lagrangian and Eulerian descriptions of flow, respectively. Here we will seek to understand the distinction more fully, to become fluent with both points of view, and to translate between them.

    Definitions:

    1. Lagrangian information concerns the nature and behavior of fluid parcels.

    2. Eulerian information concerns fields, i.e., properties like velocity, pressure and temperature that vary in time and space.

    Here are some examples:

    1. Statements made in a weather forecast

    • “A cold air mass is moving in from the North.” (Lagrangian)

    • “Here (your city), the temperature will decrease.” (Eulerian)

    2. Ocean observations

    • Moorings fixed in space (Eulerian)

    • Drifters that move with the current (Lagrangian)

    The Lagrangian perspective is a natural way to describe the motion of solid objects. For example, suppose an apple falls from a tree. Newton taught us to describe the height and velocity of the apple as functions of time. This is a Lagrangian description. To try to describe this event in terms of Eulerian fields would be very awkward. You might define \(A(z,t)\) as the “appliness”: \(A\) = 1 at points in space and time occupied by the apple and 0 everywhere else, then try to derive a differential equation for \(A\). Good luck.

    The Eulerian perspective, while useless for solid objects, is natural for fluids. As we will see, it makes the math easier by providing partial differential equations for fields like velocity \(\vec{u}\left(\vec{x},t\right)\) and temperature \(T\left(\vec{x},t\right)\).

    So for fluid mechanics, why do we not just stick to the Eulerian approach? The reason is that the existing laws of physics apply most naturally to fluid parcels. For example,

    • if you apply heat to a fluid parcel, its temperature will increase: \(dT/dt\) = heating rate;

    • if you apply a force to a fluid parcel, it will accelerate: \(d\vec{u}/dt = \vec{F}/m\).

    So in fluid mechanics we must be bilingual. When we set out to develop the equations of motion (next chapter), we will do it in two steps:

    1. apply the known laws of physics to fluid parcels

    2. translate the results into Eulerian form for mathematical analysis.

    But how do we accomplish this translation between Lagrangian and Eulerian perspectives? The key is an operation called the material derivative, which we discuss next.

    5.1.1 The material derivative

    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:

    \[\frac{d \phi}{d t}=\frac{\partial \phi}{\partial t}+\frac{\partial \phi}{\partial x_{j}} \frac{d x_{j}}{d t}=\frac{\partial \phi}{\partial t}+v_{j} \frac{\partial \phi}{\partial x_{j}}.\]

    (Note that derivatives of \(\phi\) are partial derivatives because \(\phi\) is a function of several variables, whereas derivatives of \(\vec{x}\) are total derivatives because \(\vec{x}\) is a function of time only.)

    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:

    \[\frac{D}{D t} \equiv \frac{\partial}{\partial t}+u_{j} \frac{\partial}{\partial x_{j}} \equiv \frac{\partial}{\partial t}+\vec{u} \cdot \vec{\nabla}\label{eqn:1}\]

    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 solve Equation \(\ref{eqn:1}\) for the partial time derivative. Suppose, for example, that the field in question is air temperature. Then

    \[\underbrace{\frac{\partial T}{\partial t}}_{\text {thermometer reading }}=\overbrace{\frac{D T}{D t}}^{\text {heating/cooling }}-\underbrace{\vec{u} \cdot \vec{\nabla} T}_{\text {advection }}.\label{eqn:2}\]

    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 rise1. This process, whereby local changes result from transport by the flow, is called advection2.

    Test your understanding by doing exercise 21.

    1The minus sign in the advection term of Equation \(\ref{eqn:2}\) indicates that temperature change is determined by the direction the wind is from. This is why meteorologists (and sailors, and folksingers) traditionally name a wind by its origin, e.g., a “west wind” or “westerly wind” blows from the west. Oceanographers (such as the present author) eschew this perverse tradition; they refer instead to the direction a current flows toward, e.g., an eastward current.

    2This is why we call \(\vec{u}\cdot\vec{\nabla}\) the advective derivative, cf. section 4.1.5.