11. Convective boundary layer

11.1. Derivation of the mixed-layer model

During daytime, there is a well-mixed convective boundary layer of approximately 1 km in depth. If we assume horizontal homogeneity, no subsidence and high Reynolds numbers, the conservation of energy in this layer can be described as

\[ \def\pd#1#2{\dfrac{\partial #1}{\partial #2}} \def\thetabar{\overline{\theta}} \def\qbar{\overline{q}} \def\thetabulk{\left< \thetabar \right>} \def\qbulk{\left< \qbar \right>} \pd{\overline{\theta}}{t} = - \pd{\overline{w^\prime \theta^\prime}}{z} \]

The well-mixed property of this layer allows us to derive a very simple bulk model (thus no spatial derivatives). We integrate from surface to the depth of the layer.

\[ \int_0^h \pd{\thetabar}{t} \mathrm{d}z = - \int_0^h \pd{\overline{w^\prime \theta^\prime}}{z} \mathrm{d}z \]

We rewrite the lefthand side using Leibniz’ rule:

\[ \dfrac{\mathrm{d}}{\mathrm{d} t} \int_0^h \thetabar \, \mathrm{d}z = \int_0^h \pd{\thetabar}{t} \mathrm{d}z + \dfrac{\mathrm{d} h}{\mathrm{d} t} \thetabar \left(h\right) - \dfrac{\mathrm{d} 0}{\mathrm{d} t} \thetabar \left(0\right) \]

where the last term is equal to zero.

If we define the bulk mean temperature of the convective boundary layer

\[ \thetabulk = \dfrac{1}{h} \int_0^h \thetabar \, \mathrm{d} z \]

and substitute the lefthand side of the integrated equation with the help of the Leibniz rule to get

\[\begin{split} \dfrac{\mathrm{d}}{\mathrm{d} t} \left( \thetabulk h \right) - \dfrac{\mathrm{d} h}{\mathrm{d} t} \thetabar \left(h\right) = - \overline{w^\prime \theta^\prime} \left( h \right) + \overline{w^\prime \theta^\prime} \left( 0 \right) \\ \dfrac{\mathrm{d} \thetabulk}{\mathrm{d} t} h + \dfrac{\mathrm{d} h}{\mathrm{d} t} \thetabulk - \dfrac{\mathrm{d} h}{\mathrm{d} t} \thetabar \left(h\right) = \overline{w^\prime \theta^\prime} \left( 0 \right) - \overline{w^\prime \theta^\prime} \left( h \right) \end{split}\]

If we assume that the boundary layer is well-mixed, the bulk temperature \(\thetabulk\) and the boundary-layer top temperature \(\thetabar \left( h \right)\) equal and after division by \(h\) the final equation can be written as

\[ \dfrac{\mathrm{d} \thetabulk}{\mathrm{d} t} = \dfrac{ \overline{w^\prime \theta^\prime} \left( 0 \right) - \overline{w^\prime \theta^\prime} \left( h \right) }{h} \]

Similarly, we can derive a budget around the mixed-layer top

\[ \int_{h-\epsilon}^{h + \epsilon} \pd{\thetabar}{t} \mathrm{d}z = - \int_{h-\epsilon}^{h + \epsilon} \pd{\overline{w^\prime \theta^\prime}}{z} \mathrm{d}z \]

Again with application of Leibniz rule, we can write

\[ \dfrac{\mathrm{d}}{\mathrm{d} t} \int_{h - \epsilon}^{h + \epsilon} \thetabar \, \mathrm{d}z - \dfrac{\mathrm{d} \left( h + \epsilon \right) }{\mathrm{d} t} \thetabar \left(h + \epsilon \right) + \dfrac{\mathrm{d} \left( h - \epsilon \right) }{\mathrm{d} t} \thetabar \left(h - \epsilon \right) = - \overline{w^\prime \theta^\prime} \left( h + \epsilon \right) + \overline{w^\prime \theta^\prime} \left( h - \epsilon \right) \]

If we take the limit of \(\epsilon\) going to zero, then the first term goes to zero. The second and third term can be combined, taking into account that temperature at \(h\) approached from below is \(\thetabar\), while from above \(\thetabar + \Delta \theta\). The first term on the righthand side is zero, as the flux at \(h\) approached from above equals zero. All these assumptions give

\[ - \dfrac{\mathrm{d} h}{\mathrm{d} t} \Delta \theta = \overline{w^\prime \theta^\prime} \left( h \right) \]

which can be rewritten into

\[ \dfrac{\mathrm{d} h}{\mathrm{d} t} = - \dfrac{\overline{w^\prime \theta^\prime} \left( h \right)}{\Delta \theta} \]

As a last step, we need an evolution equation for the jump \(\Delta \theta\). This evolution is a competition of boundary layer growth that makes the jump larger, and boundary-layer heating that makes the jump smaller.

\[ \dfrac{\mathrm{d} \Delta \theta}{\mathrm{d} t} = \gamma_{\theta} \dfrac{\mathrm{d} h}{\mathrm{d} t} - \dfrac{\mathrm{d} \thetabulk}{\mathrm{d} t} \]

These three equations combined provide the set of mixed-layer equations. We have, however, a new closure problem, because we do not know the entrainment flux. Analysis of field measurements have revealed that the total entrainment of virtual potential temperature is a fixed fraction \(\beta\) of its surface flux

\[ \dfrac{\mathrm{d} h}{\mathrm{d} t} = \beta \dfrac{\overline{w^\prime \theta_v^\prime} \left( 0 \right)}{\Delta \theta_v} \]

The growth rate of the boundary layer due to entrainment is often also called the entrainment velocity \(w_e\), such that in absense of subsidence \(\mathrm{d}h / \mathrm{d}t = w_e\)

We can now write down the total set

Mixed-layer equations for potential temperature

\[\begin{split} \dfrac{\mathrm{d} \thetabulk}{\mathrm{d} t} &= \dfrac{ \overline{w^\prime \theta^\prime} \left( 0 \right) - \overline{w^\prime \theta^\prime} \left( h \right) }{h} \\ \overline{w^\prime \theta^\prime} \left( h \right) &= - w_e {\Delta \theta} \\ \dfrac{\mathrm{d} \Delta \theta}{\mathrm{d} t} &= \gamma_{\theta} \dfrac{\mathrm{d} h}{\mathrm{d} t} - \dfrac{\mathrm{d} \thetabulk}{\mathrm{d} t} \\ w_e &= \beta \dfrac{\overline{w^\prime \theta_v^\prime} \left( 0 \right)}{\Delta \theta_v} \end{split}\]

In a similar fashion, we can construct the set for \(q\)

Mixed-layer equations for specific humidity

\[\begin{split} \dfrac{\mathrm{d} \qbulk}{\mathrm{d} t} &= \dfrac{ \overline{w^\prime q^\prime} \left( 0 \right) - \overline{w^\prime q^\prime} \left( h \right) }{h} \\ \overline{w^\prime q^\prime} \left( h \right) &= - w_e {\Delta q} \\ \dfrac{\mathrm{d} \Delta q}{\mathrm{d} t} &= \gamma_{q} \dfrac{\mathrm{d} h}{\mathrm{d} t} - \dfrac{\mathrm{d} \qbulk}{\mathrm{d} t} \end{split}\]

where make use of the same entrainment velocity defined earlier.

11.2. The Deardorff velocity scale

The velocities in the convective boundary layer in its most basic form are governed by two variables: the surface buoyancy flux \(\dfrac{g}{\overline{\theta}_\mathrm{v}} \overline{ w^\prime \theta_\mathrm{v}^\prime}_0\) and the boundary layer depth \(h\). From these two variables a velocity scale \(w_*\) can be derived.

The Deardorff velocity scale

\[ w_* \equiv \left( \dfrac{g}{\overline{\theta}_\mathrm{v}} \overline{ w^\prime \theta_\mathrm{v}^\prime}_0 h \right)^\frac{1}{3} \]

The Deardorff velocity scale \(w_*\) is an estimate of the velocity of a typical plume in the convective boundary layer, and can be used to normalize the velocity variances to universal profiles, as shown in the lecture. A typical value for \(w_*\) in a daytime boundary layer is ~1 m s\(^{-1}\)