5.2: The Streamfunction

Figure \(\PageIndex{1}\): (a) Streamfunction for a pair of Gaussian vortices located at \(x = \pm 1\): \(\psi = −e^{−(x−1)^2−y^2} − e^{−(x+1)^2−y^2}\). Blue indicates negative values. The sign of the streamfunction corresponds to positive vorticity (counterclockwise rotation as indicated by arrows). (b) An example in nature: satellite photo of hurricanes Madeline and Lester approaching Hawaii, August 29, 2016 (NASA Earth Observatory).

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: \[\vec{u} \cdot \vec{\nabla} \Psi=u \frac{\partial \Psi}{\partial x}+v \frac{\partial \Psi}{\partial y}=u v-v u=0, \nonumber \] 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:

\[\Psi=-\int u \, d y+f(x), \nonumber \]

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

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

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).