source: palm/trunk/DOC/misc/prandtl_layer.tex @ 1320

Last change on this file since 1320 was 187, checked in by letzel, 13 years ago
  • new: descriptions of plant canopy model and prandtl layer (trunk/DOC/misc)
  • changed: more consistent flux definitions; modification of the integrated version of the profile function for momentum for unstable stratification (wall_fluxes, production_e)
  • bugfix: change definition of us_wall from 1D to 2D (prandtl_fluxes, wall_fluxes)
File size: 12.9 KB
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1\documentclass[11pt,a4paper,titlepage]{scrreprt}
2\documentstyle[Flow]
3\usepackage{graphics}
4\usepackage{german}
5\usepackage{chimuk}
6\usepackage{bibgerm}
7\usepackage{a4wide}
8\usepackage{amsmath}
9\usepackage{flafter}
10\usepackage[dvips]{epsfig}
11\usepackage{texdraw}
12\usepackage{supertabular}
13\usepackage{longtable}
14\usepackage{scrpage}
15\usepackage{fancyheadings}
16\usepackage{flow}
17\frenchspacing
18\sloppy
19\pagestyle{fancyplain}
20\addtolength{\headheight}{\baselineskip}
21\renewcommand{\chaptermark}[1]{\markboth{\thechapter~~#1}{}}
22\renewcommand{\sectionmark}[1]{\markright{\thesection\ #1}}
23\lhead{\fancyplain{}{\bfseries\leftmark}}
24\rhead{}
25%\begin{titlepage}
26%\author{Gerald Steinfeld}
27%\title{Prandtl layer parameterisation in PALM}
28%\end{titlepage}
29\begin{document}
30%\maketitle
31%\tableofcontents
32\pagebreak
33
34\chapter{Prandtl layer parameterisation in PALM}
35
36The friction velocity $u_{\ast}$ is a velocity scale that is defined by the relation
37\begin{equation}
38u_{\ast}=\left ( \left | \tau / \overline{\rho} \right | \right )^{\frac{1}{2}},
39\end{equation}
40where $\tau$ is the Reynolds stress and $\rho$ is the air density.
41Using the surface kinematic momentum fluxes in the x and y directions
42$\left ( -\overline{u'w'}, -\overline{v'w'} \right )$ to represent
43surface stress, the friction velocity can be written as   
44\begin{equation}
45u_{\ast}=\left ( (-\overline{u'w'})^2 + (-\overline{v'w'})^2 \right )^{\frac{1}{4}}.
46\end{equation}
47Based on the definition of the friction velocity $u_{\ast}$ the vertical turbulent
48momentum flux $\overline{u'w'}$ can be determined from
49\begin{equation}
50{u^2_{\ast}} \cos \left ( \alpha_0 \right ) = - \overline{u'w'},
51\end{equation}
52while the vertical turbulent momentum flux $\overline{v'w'}$ can be determined
53from 
54\begin{equation}
55{u^2_{\ast}} \sin \left ( \alpha_0 \right ) = - \overline{v'w'}.
56\end{equation}
57The angle $\alpha_0$, that is assumed to be constant in the Prandtl layer, is the
58angle between the x-direction and the direction of the mean horizontal wind and can
59be evaluated by   
60\begin{equation} \label{win}
61\alpha_0 = \arctan \left ( \frac{\overline{v}(z_p)}{\overline{u}(z_p)} \right ).
62\end{equation}
63According to the similarity theory of Monin and Obukhov the following relationship for
64the profile of the mean horizontal wind is valid
65\begin{equation} \label{hmpr}
66\begin{split}
67\frac{\partial \left | \overline{\vec{v}} \right |}{\partial z} &= \frac{u_{\ast}}{\kappa z} \phi_m \left ( \frac{z}{L} \right ) \\
68                                                                &= u_{\ast} \frac{1}{\kappa z} \phi_m \left ( \frac{z}{L} \right ).
69\end{split} 
70\end{equation}
71In equation \ref{hmpr} $L$ denotes the Monin Obukhov length and $\kappa$ denotes the von Karman constant, while $\phi_m$ denotes the profile
72or Dyer-Businger functions for momentum:
73\begin{equation}
74\phi_m =
75\begin{cases}
761+5 \text{Rif} & \text{if $\text{Rif} > 0$}, \\
771 & \text{if $\text{Rif} = 0$}, \\
78\left ( 1 - 16 \text{Rif} \right )^{-\frac{1}{4}} & \text{if $\text{Rif} < 0$}.
79\end{cases}
80\end{equation} 
81Rif denotes the dimensionless Richardson flux number.
82
83By integrating equation \ref{hmpr} over $z$ from $z_0$ to a height $z$ the following relationship for the friction velocity can be deduced:   
84\begin{equation} \label{usb}
85u_{\ast} =
86\begin{cases}
87\frac{\left | \overline{\vec{v}} \right | \kappa}{\left ( \ln \left ( \frac{z}{z_0} \right ) + 5 \text{Rif} \left ( \frac{z-z_0}{z} \right ) \right )} & \text{if $\text{Rif} \ge 0$}, \\
88%\frac{\left | \overline{\vec{v}} \right | \kappa}{\left ( ln \left ( \frac{1+B}{1-B} \frac{1-A}{1+A} \right ) + 2 \left ( arctan(B) - arctan(A) \right ) \right )} & \text{if $Rif<0$}
89\frac{\left | \overline{\vec{v}} \right | \kappa}{\ln \left ( \frac{z}{z_0} \right ) - \ln \left ( \frac{ \left ( 1+A \right )^\left ( 1+A^2 \right ) }{ \left ( 1+B \right )^\left ( 1+B^2 \right ) }\right ) + 2 \left ( \arctan(A) - \arctan(B) \right )} & \text{if $\text{Rif}<0$}
90\end{cases} 
91\end{equation}
92with
93\begin{equation}
94%A=\left ( 1 - 16 Rif \right )^{-\frac{1}{4}}
95A=\left ( 1 - 16 \text{Rif} \right )^{\frac{1}{4}} 
96\end{equation}
97and
98\begin{equation}
99%B=\left ( 1 - 16 Rif \frac{z_0}{z} \right )^{-\frac{1}{4}}.
100B=\left ( 1 - 16 \text{Rif} \frac{z_0}{z} \right )^{\frac{1}{4}}.
101\end{equation}
102In fact, equation \ref{usb} is used in PALM to determine the friction velocity $u_{\ast}$.
103
104The following paragraph is a short digression dealing with the integration of equation \ref{hmpr} that finally leads to equation \ref{usb}. The integration
105is only shown for the profile function for unstable stratification. According to PAULSON (1970) the general result of an integration of $\phi=\frac{\kappa z'}{u_{\ast}}
106\frac{\partial \overline{u}}{\partial z'}$ over $z'$ from $z_0$ to $z$ is
107\begin{equation}
108\overline{u}(z) = \frac{u_{\ast}}{\kappa} \left [ \ln \left ( \frac{z}{z_0} \right ) - \Psi \right ]
109\end{equation}
110with
111\begin{equation}
112\Psi = \int_{\frac{z_0}{L}}^{\frac{z}{L}}d \left ( \frac{z'}{L} \right ) \frac{1-\phi \left ( \frac{z'}{L} \right )}{\frac{z'}{L}}.
113\end{equation} 
114Applying the profile function for unstable stratification, $\phi=\left ( 1 - 16 \frac{z}{L} \right )^{-\frac{1}{4}}$, $\Psi$ can be determined as follows:
115\begin{equation}
116\begin{split}
117\Psi &= \int_{\frac{z_0}{L}}^{\frac{z}{L}}d \left ( \frac{z'}{L} \right ) \frac{1- \left ( 1 - 16 \frac{z'}{L} \right )^{-\frac{1}{4}} }{\frac{z'}{L}} \text{ | substitution: } x=\frac{1}{\phi \left ( \frac{z'}{L} \right ) } \\
118     &= \int_{\left ( 1-16\frac{z_0}{L} \right )^{\frac{1}{4}}}^{\left ( 1-16\frac{z}{L} \right )^{\frac{1}{4}}} dx \left ( 4 \frac{x^2 - x^3}{1 - x^4} \right ) \\
119     &= \int_{\left ( 1-16\frac{z_0}{L} \right )^{\frac{1}{4}}}^{\left ( 1-16\frac{z}{L} \right )^{\frac{1}{4}}} dx \left ( 2 \left ( \frac{1}{1+x} + \frac{x}{1+x^2} - \frac{1}{1+x^2} \right ) \right ) \\
120     &= \left [ 2 \ln \left ( \frac{1+x}{2} \right ) + \ln \left ( \frac{1+x^2}{2} \right ) -2 \arctan(x) \right ]_{\left ( 1-16\frac{z_0}{L} \right )^{\frac{1}{4}}=B}^{\left ( 1-16\frac{z}{L} \right )^{\frac{1}{4}}=A} \\
121     &= \ln \left ( \frac{\left ( 1 + A \right )^2}{\left ( 1 + B \right )^2} \frac{1+A^2}{1+B^2} \right ) - 2 \left ( \arctan(A) - \arctan(B) \right )
122\end{split}
123\end{equation} 
124
125Making use of equation \ref{win} and \ref{hmpr}, it is also possible to deduce a relationship for the profile of the mean u-component of the wind velocity
126\begin{equation} \label{upr}
127\begin{split}
128\frac{\partial \overline{u} }{\partial z} &= \frac{\partial \left | \overline{\vec{v}} \right |}{\partial z} \cos \left ( \alpha_0 \right ) \\
129                                          &= \frac{u_{\ast}}{\kappa z} \phi_m \left ( \frac{z}{L} \right ) \cos \left ( \alpha_0 \right ) \\
130                                          &= \frac{1}{u_{\ast}} {u^2_{\ast}} \cos \left ( \alpha_0 \right ) \frac{1}{\kappa z} \phi_m \left ( \frac{z}{L} \right ) \\
131                                          &= \frac{-\overline{u'w'}}{u_{\ast}} \frac{1}{\kappa z} \phi_m \left ( \frac{z}{L} \right )
132\end{split} 
133\end{equation}
134and accordingly a relationship for the profile of the mean v-component of the wind velocity
135\begin{equation} \label{vpr}
136\begin{split}
137\frac{\partial \overline{v} }{\partial z} &= \frac{\partial \left | \overline{\vec{v}} \right |}{\partial z} \sin \left ( \alpha_0 \right ) \\
138                                          &= \frac{u_{\ast}}{\kappa z} \phi_m \left ( \frac{z}{L} \right ) \sin \left ( \alpha_0 \right ) \\
139                                          &= \frac{1}{u_{\ast}} {u^2_{\ast}} \sin \left ( \alpha_0 \right ) \frac{1}{\kappa z} \phi_m \left ( \frac{z}{L} \right ) \\
140                                          &= \frac{-\overline{v'w'}}{u_{\ast}} \frac{1}{\kappa z} \phi_m \left ( \frac{z}{L} \right ).
141\end{split} 
142\end{equation}
143As the right-hand sides of equation \ref{upr} and \ref{vpr} differ only by prefactors that are (assumed to be) constant within the Prandtl layer from the
144right-hand side of equation \ref{hmpr}, we can directly make use of the integration that led to equation \ref{usb} in order to obtain   
145\begin{equation} \label{uswsb}
146C=\frac{-\overline{u'w'}}{u_{\ast}} =
147\begin{cases}
148\frac{\overline{u} \kappa}{\left ( \ln \left ( \frac{z}{z_0} \right ) + 5 \text{Rif} \left ( \frac{z-z_0}{z} \right ) \right )} & \text{if $\text{Rif} \ge 0$}, \\
149%\frac{\overline{u} \kappa}{\left ( ln \left ( \frac{1+B}{1-B} \frac{1-A}{1+A} \right ) + 2 \left ( arctan(B) - arctan(A) \right ) \right )} & \text{if $Rif<0$}.
150\frac{ \overline{u} \kappa}{\ln \left ( \frac{z}{z_0} \right ) - \ln \left ( \frac{ \left ( 1+A \right )^\left ( 1+A^2 \right ) }{ \left ( 1+B \right )^\left ( 1+B^2 \right ) }\right ) + 2 \left ( \arctan(A) - \arctan(B) \right )} & \text{if $\text{Rif}<0$}.
151
152\end{cases} 
153\end{equation}
154and
155\begin{equation} \label{vswsb}
156D=\frac{-\overline{v'w'}}{u_{\ast}} =
157\begin{cases}
158\frac{\overline{v} \kappa}{\left ( \ln \left ( \frac{z}{z_0} \right ) + 5 \text{Rif} \left ( \frac{z-z_0}{z} \right ) \right )} & \text{if $\text{Rif} \ge 0$}, \\
159%\frac{\overline{v} \kappa}{\left ( ln \left ( \frac{1+B}{1-B} \frac{1-A}{1+A} \right ) + 2 \left ( arctan(B) - arctan(A) \right ) \right )} & \text{if $Rif<0$}.
160\frac{\overline{v} \kappa}{\ln \left ( \frac{z}{z_0} \right ) - \ln \left ( \frac{ \left ( 1+A \right )^\left ( 1+A^2 \right ) }{ \left ( 1+B \right )^\left ( 1+B^2 \right ) }\right ) + 2 \left ( \arctan(A) - \arctan(B) \right )} & \text{if $\text{Rif}<0$}.
161
162\end{cases} 
163\end{equation}
164Both equations, \ref{uswsb} and \ref{vswsb}, are used in PALM. In order to get an information on the turbulent vertical momentum fluxes within the Prandtl layer,
165equation \ref{uswsb} and equation \ref{vswsb} need only to be multiplied by $-1$ and the friction velocity $u_{\ast}$, so that finally the followings two steps
166have to be executed in PALM:
167\begin{equation}
168\overline{u'w'} = -C u_{\ast}
169\end{equation}
170and
171\begin{equation}
172\overline{v'w'} = -D u_{\ast}.
173\end{equation}
174
175In case that no near-surface heat flux has been prescribed by the user of PALM, the near-surface heat flux $\overline{w'\Theta'}_0$ is evaluated from
176\begin{equation} \label{hfe}
177\overline{w'\Theta'} = - u_{\ast} \theta_{\ast}.
178\end{equation}
179Here, $\theta_{\ast}$ is the so-called characteristic temperature for the Prandtl layer. In case of no preset near-surface heat flux the characteristic
180temperature $\theta_{\ast}$ is determined in PALM from the integrated version of the profile function for the potential temperature. According to the
181similarity theory of Monin and Obukhov the following relationship for the vertical gradient of the potential temperature is valid:
182\begin{equation} \label{hmpt}
183\frac{\partial \overline{\Theta}}{\partial z} = \frac{\theta_{\ast}}{\kappa z} \phi_h.
184\end{equation} 
185In equation \ref{hmpt} $\phi_h$ denotes the profile or Dyer-Businger functions for temperature:
186\begin{equation}
187\phi_h =
188\begin{cases}
1891+5 \text{Rif} & \text{if $\text{Rif} > 0$}, \\
1901 & \text{if $\text{Rif} = 0$}, \\
191\left ( 1 - 16 \text{Rif} \right )^{-\frac{1}{2}} & \text{if $\text{Rif} < 0$}.
192\end{cases}
193\end{equation} 
194By integrating equation \ref{hmpt} over $z$ from $z_0$ to a height $z$ the following relationship for the characteristic temperature in the Prandtl
195layer can be deduced:   
196\begin{equation} \label{tsb}
197\theta_{\ast} =
198\begin{cases}
199\frac{\kappa \left ( \overline{\Theta}(z) - \overline{\Theta}(z_0) \right )}{\ln \left ( \frac{z}{z_0} \right ) + 5 \text{Rif} \left ( \frac{z-z_0}{z} \right )} & \text{if $\text{Rif} \ge 0$}, \\
200\frac{\kappa \left ( \overline{\Theta}(z) - \overline{\Theta}(z_0) \right )}{\ln \left ( \frac{z}{z_0} - 2 \ln \left ( \frac{1+A}{1+B} \right )\right )} & \text{if $\text{Rif}<0$}
201\end{cases} 
202\end{equation}
203with
204\begin{equation}
205%A=\left ( 1 - 16 \text{Rif} \right )^{-\frac{1}{4}}
206A=\sqrt{1 - 16 \text{Rif}} 
207\end{equation}
208and
209\begin{equation}
210%B=\left ( 1 - 16 \text{Rif} \frac{z_0}{z} \right )^{-\frac{1}{4}}.
211B=\sqrt{1 - 16 \text{Rif} \frac{z_0}{z}}.
212\end{equation}
213Note, that the temperature at the height of the roughness length that is required for the evaluation of $\theta_{\ast}$ in equation \ref{tsb} is saved
214in PALM at the first vertical grid level of the temperature with index k=0. The height that is assigned to this vertical grid level is -0.5$\Delta z$,
215where $\Delta z$ is the grid length in the vertical direction.   
216In case of a near-surface heat flux $\overline{w'\Theta'}_0$ that has been prescribed by the user, the evaluation of $\theta_{\ast}$ is not needed for
217the integration of the model. Instead $\theta_{\ast}$ is only determined for the evaluation of statistics of the turbulent flow. In that case it is
218simply derived by a transformation of equation \ref{hfe} into 
219\begin{equation}
220\theta_{\ast} = - \frac{\overline{w'\Theta'}_0}{u_{\ast}}.
221\end{equation}   
222In case that PALM is run in its moist version, the evaluation of the characteristic humidity for the Prandtl layer is evaluated in a way correspondent to
223that of the determination of the characteristic temperature.
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249\end{document}
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