Scaling the Vertical-Velocity Variance During the Very Late Afternoon Transition of the Convective Boundary Layer

Abstract In the bulk of the convective boundary layer driven by surface heating, the vertical-velocity variance is known to scale with the convective velocity scale. This scaling relies on the quasi-equilibrium assumption that the surface heat flux ( H , the water-vapor contribution to buoyancy is neglected) varies slowly compared to the adjustment time scale of the large scale convective eddies (or the eddy turnover time), a condition which over land is typically satisfied from the late morning to the early afternoon transition. Later on, when the surface heat flux starts decaying moderately compared to the eddy turnover time, a departure from the quasi-equilibrium regime is expected. A recent idealized large-eddy simulations (LES) study proposed a parameter for describing such departure during the late afternoon transition: $${\widetilde{r}} \equiv |H^{-1}dH/dt|^{-1}/t_*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>r</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>≡</mml:mo> <mml:msup> <mml:mrow> <mml:mo>|</mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>d</mml:mi> <mml:mi>H</mml:mi> <mml:mo>/</mml:mo> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>/</mml:mo> <mml:mmultiscripts> <mml:mi>t</mml:mi> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> <mml:mrow/> </mml:mmultiscripts> </mml:mrow> </mml:math> , where $$t_*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mi>t</mml:mi> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> <mml:mrow/> </mml:mmultiscripts> </mml:math> is the eddy turnover time. The quasi-equilibrium assumption applies when $${\widetilde{r}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>r</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> $$&gt;&gt;$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>&gt;</mml:mo> <mml:mo>&gt;</mml:mo> </mml:mrow> </mml:math> 1, and breaks down when $${\widetilde{r}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>r</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> $$\sim $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>∼</mml:mo> </mml:math> 1. Building on these results, we further investigate the scaling for the vertical-velocity variance during the very late afternoon transition, i.e. in the regime $${\widetilde{r}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>r</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> $$&lt;&lt;$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>&lt;</mml:mo> <mml:mo>&lt;</mml:mo> </mml:mrow> </mml:math> 1 where the surface heat flux decays rapidly compared to the eddy turnover time. In a first step, we use LES to reveal that the regime $${\widetilde{r}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>r</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> $$&lt;&lt;$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>&lt;</mml:mo> <mml:mo>&lt;</mml:mo> </mml:mrow> </mml:math> 1 is characterized by a new velocity scale $$w_{*r} \equiv ((g/\overline{\theta })|dH/dt| z_i^2)^{1/4}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mmultiscripts> <mml:mi>w</mml:mi> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> <mml:mi>r</mml:mi> </mml:mrow> <mml:mrow/> </mml:mmultiscripts> <mml:mrow> <mml:mo>≡</mml:mo> <mml:mo>(</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>/</mml:mo> <mml:mover> <mml:mi>θ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>d</mml:mi> <mml:mi>H</mml:mi> <mml:mo>/</mml:mo> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:msubsup> <mml:mi>z</mml:mi> <mml:mi>i</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:msup> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . Within the specific framework of this study using the parameter $${\widetilde{r}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>r</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> for identifying the out-of-equilibrium regime $${\widetilde{r}}$$ <mml:math xmlns:mml="http://www.w3.org/