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25 | %\begin{titlepage} |
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26 | %\author{Gerald Steinfeld} |
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27 | %\title{Prandtl layer parameterisation in PALM} |
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28 | %\end{titlepage} |
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29 | \begin{document} |
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30 | %\maketitle |
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31 | %\tableofcontents |
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32 | \pagebreak |
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33 | |
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34 | \chapter{Plant canopy model in PALM} |
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35 | The basic equations used in the LES are the Navier-Stokes equations using the Boussinesq approximation (conservation of momentum; equation \ref{NSG}), |
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36 | the equation of continuity (conservation of mass; equation \ref{KON}), the first law of thermodynamics (conservation of energy; equation \ref{1HS}) and |
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37 | the conservation equation for a passive scalar (equation \ref{CPS}). All equations are filtered (in PALM by averaging over a discrete grid volume) in |
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38 | order to eliminate small-scale perturbations. Thus, the set of equations on that the LES model PALM is based is formed by the following equations: |
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39 | |
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40 | \begin{equation} \label{NSG} |
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41 | \frac{\partial \overline{u}_i}{\partial t} = - \frac{\partial}{\partial x_k} \overline{u}_k \overline{u}_i - \frac{1}{\rho_0} \frac{\partial \overline{p^{\ast}}}{\partial x_i} - \left ( \epsilon_{ijk} f_j \overline{u}_k - \epsilon_{i3k} f_3 u_{g_k} \right ) + g \frac{\overline{\Theta^{\ast}}}{\Theta_0} \delta_{i3} - \frac{\partial}{\partial x_k} \overline{u_i'u_k'} - c_d a U \overline{u}_i |
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42 | \end{equation} |
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43 | |
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44 | \begin{equation} \label{KON} |
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45 | \frac{\partial \overline{u}}{\partial x_i} = 0 |
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46 | \end{equation} |
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47 | |
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48 | \begin{equation} \label{1HS} |
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49 | \frac{\partial \overline{\Theta}}{\partial t} = - \frac {\partial}{\partial x_k} \overline{u}_k \overline{\Theta} - \frac{\partial}{\partial x_k} \overline{u_k'\Theta'} + S |
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50 | \end{equation} |
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51 | |
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52 | \begin{equation} \label{CPS} |
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53 | \frac{\partial \overline s}{\partial t} = - \frac{\partial}{\partial x_k} \overline{u}_k \overline{s} - \frac{\partial}{\partial x_k} \overline{u_k's'} - c_s a U \left ( \overline{s} - s_l \right ) |
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54 | \end{equation} |
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55 | |
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56 | The overline denotes filtered variables, while the prime denotes sub-gridscale, local spatial fluctuations at time t that are not resolved by the model; |
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57 | the star denotes a departure of the basic state that is marked by a subscript $0$ and that varies only |
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58 | with height; $u_i$ is the velocity in the $x_i$-direction; $\rho$ is the air density; $p$ is the pressure; $f_i=(0,2\Omega cos(\phi),2\Omega sin(\phi))$ |
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59 | is the Coriolis parameter with the angular velocity of the earth $\Omega$; ${u_g}_i$ denotes the components of the geostrophic wind; $g$ is the |
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60 | acceleration due to gravity; $\Theta$ is the potential temperature and $s$ denotes the concentration of a passive scalar. |
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61 | The sub-gridscale fluxes in equations \ref{NSG}, \ref{1HS} and \ref{CPS} are parameterised by a 1.5-order closure model using resolved-scale variables |
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62 | and the sub-gridscale turbulent kinetic energy $\overline{e}$. Thus, an additional prognostic equation has to be solved for the sub-gridscale turbulent |
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63 | kinetic energy $\overline{e}$: |
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64 | |
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65 | \begin{equation} \label{TKE} |
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66 | \frac{\partial \overline e}{\partial t} = - \frac{\partial}{\partial x_k} \overline{u}_k \overline{e} - \tau_{ij} \frac{\partial \overline{u}_i}{\partial x_j} + \frac{g}{\Theta_0} \overline{u_3'\Theta_j'} - \frac{\partial}{\partial x_j} \left ( \overline{u_j' \left ( e + \frac{p'}{\rho_0} \right ) } \right ) - \epsilon - 2 c_d a U \overline{e}. |
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67 | \end{equation} |
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68 | |
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69 | Here, $\tau_{ij}$ denotes the sub-gridscale stress tensor. The final terms in equations \ref{NSG}, \ref{1HS}, \ref{CPS} and \ref{TKE} represent the sink |
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70 | or source of momentum, heat and the passive scalar due to canopy elements, respectively. Thus, these terms do only occur within the plant canopy, while |
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71 | they vanish outside the plant canopy (in PALM this distinction between canopy layer and the remaining area is realised by the specification of a height |
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72 | index that contains the information up to which vertical grid level the plant canopy extends; the additional terms for the plant canopy layer are |
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73 | only evaluated for grid levels smaller or equal that plant canopy height index). The additional, canopy related source and sink sources, are considered in |
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74 | PALM in the same way as in the models of WATANABE (2004, Boundary-Layer Meteorol., 112, 307-341) and SHAW AND SCHUMANN |
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75 | (1992, Boundary-Layer Meteorol., 61, 47-64). |
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76 | $c_d$ and $c_s$ are the leaf drag coefficient and the scalar exchange |
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77 | coefficient for a leaf, respectively; $a$ is the leaf area density; $U$ is the instantaneous local wind speed, defined as |
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78 | \begin{equation} |
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79 | U = \left ( \overline{u}^2 + \overline{v}^2 + \overline{w}^2 \right )^{\frac{1}{2}}, |
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80 | \end{equation} |
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81 | and $s_l$ is the scalar concentration at a leaf surface. |
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82 | The heat source distribution $S(z)$ in convective conditions has been given by BROWN AND COVEY (1966, Agric. Meteorol., 3, 73-96) and SHAW AND SCHUMANN |
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83 | (1992). The assumption that leads to the heat source distribution is that short-wave solar radiation penetrates the plant canopy and warms the foliage |
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84 | which, in turn, warms the air in contact with it. The heat source is distributed in such a way that the heat flux $Q$ at the top of the canopy, $z=h$, is |
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85 | prescribed and that the heat flux within the canopy follows a declining exponential function of the downward cumulative leaf area index. The decline of |
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86 | heat flux with decreasing height is similar to the decline of net radiation that has been described by BROWN AND COVEY (1966). The heat source $S(z)$ that |
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87 | is included in equation \ref{1HS} is the vertical derivative of the upward kinematic vertical heat flux $Q$ |
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88 | \begin{equation} |
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89 | S = \frac{\partial Q}{\partial z}. |
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90 | \end{equation} |
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91 | According to the explanations given above the upward kinematic vertical heat flux $Q$ is evaluated as |
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92 | \begin{equation} \label{kvh} |
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93 | Q(z) = Q(h) exp(- \alpha F(z)), |
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94 | \end{equation} |
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95 | where $F$ is the downward cumulative leaf area index (non-dimensional) determined by integration over the leaf area density $a$ |
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96 | \begin{equation} |
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97 | F = \int_z^h a dz |
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98 | \end{equation} |
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99 | and $\alpha$ is an extinction coefficient that is set by default to 0.6 in PALM. |
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100 | |
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101 | The final term in equation \ref{TKE} represents the sink for sub-gridscale turbulent kinetic energy |
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102 | due to the rapid dissipation of wake turbulence formed in the lee of canopy elements. This term is included |
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103 | on the assumption that individual wake motions behind canopy elements are of even smaller scale than those |
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104 | making up the bulk of sub-gridscale kinetic energy. For simulations in that the grid length is of the |
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105 | order of the scale of wake turbulence modifications within equation \ref{TKE} might be necessary. KANDA |
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106 | AND HINO (1994, Boundary-Layer Meteorol., 68, 237-257) made a suggestion how equation \ref{TKE} has to be |
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107 | modified for such cases. |
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108 | |
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109 | \end{document} |
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