8.1. Definition
In the previous lecture, we derived the equation that describes the conservation of mean turbulence variances.
\[
\def\ubar{\overline{u}}
\def\vbar{\overline{v}}
\def\wbar{\overline{w}}
\def\ebar{\overline{e}}
\def\uprime{u^\prime}
\def\vprime{v^\prime}
\def\wprime{w^\prime}
\def\uvar{\overline{u^{\prime 2}}}
\def\vvar{\overline{v^{\prime 2}}}
\def\wvar{\overline{w^{\prime 2}}}
\def\pd#1#2{\dfrac{\partial #1}{\partial #2}}
\def\rhobar{\overline{\rho}}
\def\pdd#1#2{\dfrac{\partial^2 #1}{\partial#2^2}}
\def\uibar{\overline{u}_i}
\def\ujbar{\overline{u}_j}
\def\uiprime{u_i^\prime}
\def\uiprimesq{u_i^{\prime 2}}
\def\uiprimesqbar{\overline{u_i^{\prime 2}}}
\def\ujprime{u_j^\prime}
\pd{\uiprimesqbar}{t}
+ \pd{\uiprimesqbar \ujbar}{x_j}
=
- \pd{\overline{\uiprimesq \ujprime}}{x_j}
- 2 \overline{\uiprime \ujprime} \pd{\uibar}{x_j}
- \dfrac{2}{\rhobar} \pd{\overline{\uiprime p^\prime}}{x_i}
+ \dfrac{2}{\rhobar} \overline{ p^\prime \pd{\uiprime}{x_i} }
+ 2 f \epsilon_{ij3} \overline{ \uiprime \ujprime }
+ 2 \delta_{i3} \dfrac{g}{\overline{\theta}_\mathrm{v}} \overline{ \uiprime \theta_\mathrm{v}^\prime}
+ \nu \pdd{\uiprimesqbar}{x_j}
- 2 \nu \overline{ \left( \pd{\uiprime}{x_j} \right)^2}
\]
If we define \(e \equiv \frac{1}{2} \uiprimesq\), and assume the viscous transport is small, we can write the general equation for TKE conservation as
Conservation of turbulence kinetic energy (TKE)
\[
\pd{\ebar}{t}
+ \pd{\ebar\,\ujbar}{x_j}
=
- \pd{\overline{e\ujprime}}{x_j}
- \overline{\uiprime \ujprime} \pd{\uibar}{x_j}
- \dfrac{1}{\rhobar} \pd{\overline{\uiprime p^\prime}}{x_i}
+ \delta_{i3} \dfrac{g}{\overline{\theta}_\mathrm{v}} \overline{ \uiprime \theta_\mathrm{v}^\prime}
- \nu \overline{ \left( \pd{\uiprime}{x_j} \right)^2}
\]
If we now assume horizontal homogeneity and no subsidence, and define the dissipation as \(\epsilon \equiv \nu \overline{ \left( \pd{\uiprime}{x_j} \right)^2}\), we can write
Conservation of turbulence kinetic energy (TKE) under horizontal homogeneity
\[
\pd{\ebar}{t}
=
- \pd{\overline{e w^\prime}}{z}
- \overline{u^\prime w^\prime} \pd{\ubar}{z}
- \overline{v^\prime w^\prime} \pd{\vbar}{z}
- \dfrac{1}{\rhobar} \pd{\overline{w^\prime p^\prime}}{z}
+ \dfrac{g}{\overline{\theta}_\mathrm{v}} \overline{ w^\prime \theta_\mathrm{v}^\prime}
- \epsilon
\]
