# 5.2: The Streamfunction

Many flows are approximately two-dimensional. For example, the thickness of the Earth’s atmosphere relative to the planet is comparable to the skin on an apple. Large-scale atmospheric flows are therefore nearly two-dimensional. When a flow is two-dimensional and also incompressible $$\left( \vec{\nabla}\cdot\vec{u}=0 \right)$$, it can represented by means of a streamfunction $$\psi$$. Suppose the two dimensions are $$x$$ and $$y$$, with corresponding velocity components $$u(x, y)$$ and $$v(x, y)$$. Then $$\psi$$ is defined such that

$u=-\frac{\partial \Psi}{\partial y} ; \quad v=\frac{\partial \Psi}{\partial x} \label{eqn:1}$

Curves of constant $$\psi$$ are called streamlines. The simple example of a pair of nearby minima is shown in (Figure 5.2(a)).

Several properties of the streamfunction are noteworthy.

1. The definition Equation $$\ref{eqn:1}$$ guarantees that $$\vec{\nabla}\cdot\vec{u}$$ will be zero. (Check this for yourself.)

2. The sign convention is arbitrary. The present convention leads to $$\omega^{(z)} = +\nabla^2\psi$$.

3. You can add any fixed number to $$Ψ$$ and it makes no difference to the resulting flow. In other words, the streamfunction is defined only up to an additive constant.

4. The direction of the flow vector $$\vec{u}$$ is perpendicular to $$\vec{\nabla}\psi$$ or, equivalently, parallel to the streamlines:

$\vec{u} \cdot \vec{\nabla} \Psi=u \frac{\partial \Psi}{\partial x}+v \frac{\partial \Psi}{\partial y}=u v-v u=0,$

using Equation $$\ref{eqn:1}$$.

5. The flow direction is defined more fully by noting the signs of the derivatives in Equation $$\ref{eqn:1}$$. These require that flow be clockwise around a maximum in $$\psi$$ and counterclockwise around a minimum (such as the pair of minima in Figure $$\PageIndex{1}$$). In (Figure $$\PageIndex{1}$$a), the streamfunction increases (becomes less negative) from point “A” to point “B”, hence $$\partial\psi/\partial x > 0$$, hence $$v > 0$$.

6. The speed (velocity magnitude) is $$\sqrt{u^2+v^2} = |\vec{\nabla}\psi|$$. Therefore, the flow is fastest where streamlines are clustered together, such as just outside the two peaks on (Figure $$\PageIndex{1}$$a). Flow is slow where streamlines are widely spaced.

Given the velocity components $$u$$, $$v$$, one can easily invert Equation $$\ref{eqn:1}$$ to obtain the streamfunction. We can start with either equation; here we’ll pick the first one. Integrating:

$\Psi=-\int u \, d y+f(x),$

where $$f$$ is an unknown function independent of $$y$$. Substituting this into the second of Equation $$\ref{eqn:1}$$ gives

$\frac{\partial \Psi}{\partial x}=-\int \frac{\partial u}{\partial x} d y+f^{\prime}(x)=v.$

This is readily solved for $$f^\prime$$, which we integrate to obtain $$f$$. Note that the constant of integration is arbitrary because of point 3 above.

Test your understanding by doing exercise 23, parts (a) and (b).