| 1 | | [[NoteBox(warn,This site is currently under construction!)]] |
| | 1 | = Boundary conditions = |
| | 2 | |
| | 3 | == Constant flux layer == |
| | 4 | Following Monin-Obukhov similarity theory (MOST) a constant flux layer can be assumed as boundary condition between the surface and the first grid level where scalars and horizontal velocities are defined (''k'' = 1, ''z'',,MO,, = 0.5 ''Δz''). It is then required to provide the roughness lengths for momentum ''z'',,0,, and heat ''z'',,0,h,,. Momentum and heat fluxes as well as the horizontal velocity components are calculated using the following framework. The formulation is theoretically only valid for horizontally-averaged quantities. In PALM we assume that MOST can be also applied locally and we therefore calculate local fluxes, velocities, and scaling parameters. |
| | 5 | |
| | 6 | Following MOST, the vertical profile of the horizontal wind velocity |
| | 7 | {{{ |
| | 8 | #!Latex |
| | 9 | \begin{equation*} |
| | 10 | u_\mathrm{h} = (u^2 + v^2)^{\frac{1}{2}} |
| | 11 | \end{equation*} |
| | 12 | }}} |
| | 13 | is given in the surface layer by |
| | 14 | {{{ |
| | 15 | #!Latex |
| | 16 | \begin{align} |
| | 17 | & \frac{\partial u_\mathrm{h}}{\partial z} = |
| | 18 | \frac{u_\ast}{\kappa z}\Phi_\mathrm{m}\left(\frac{z}{L}\right)\;, |
| | 19 | \end{align} |
| | 20 | }}} |
| | 21 | where ''κ'' = 0.4 is the Von Kármán constant and Φ,,m,, is the similarity function for momentum in the formulation of Businger-Dyer (see e.g. [#panofsky Panofsky and Dutton 1984]) |
| | 22 | {{{ |
| | 23 | #!Latex |
| | 24 | \begin{align} |
| | 25 | & \Phi_\mathrm{m} = |
| | 26 | \begin{cases} |
| | 27 | 1 + 5 \frac{z}{L} & \text{for~} \frac{z}{L} \geq 0 \\ |
| | 28 | \left(1 - 16 \frac{z}{L}\right)^{-\frac{1}{4}} & \text{for~} |
| | 29 | \frac{z}{L} < 0\;. |
| | 30 | \end{cases} |
| | 31 | \end{align} |
| | 32 | }}} |
| | 33 | Here, ''L'' is the Obukhov length, calculated as |
| | 34 | {{{ |
| | 35 | #!Latex |
| | 36 | \begin{align} |
| | 37 | & \label{eq:L} |
| | 38 | L = \frac{\theta_\mathrm{v}(z) u_\ast^2}{\kappa g |
| | 39 | \left[\theta_\ast + 0.61 \theta(z) q_\ast + 0.61 |
| | 40 | q_\mathrm{v}(z) \theta_\ast\right]}\;. |
| | 41 | \end{align} |
| | 42 | }}} |
| | 43 | The scaling parameters ''θ'',,*,, and ''q'',,*,, are defined by MOST |
| | 44 | as: |
| | 45 | {{{ |
| | 46 | #!Latex |
| | 47 | \begin{align} |
| | 48 | & \label{eq:scales} \theta_\ast = - |
| | 49 | \frac{\overline{w^{\prime\prime}\theta^{\prime\prime}}_0}{u_\ast},~q_\ast |
| | 50 | = - |
| | 51 | \frac{\overline{w^{\prime\prime}q_\mathrm{v}^{\prime\prime}}_0}{u_\ast}\;, |
| | 52 | \end{align} |
| | 53 | }}} |
| | 54 | |
| | 55 | with the friction velocity ''u'',,*,, defined as |
| | 56 | {{{ |
| | 57 | #!Latex |
| | 58 | \begin{align} |
| | 59 | & u_\ast = |
| | 60 | \left[\left(\overline{u^{\prime\prime} w^{\prime\prime}}_0\right)^2 |
| | 61 | + \left(\overline{v^{\prime\prime} w^{\prime\prime}}_0\right)^2 |
| | 62 | \right]^{\frac{1}{4}}\;. |
| | 63 | \end{align} |
| | 64 | }}} |
| | 65 | |
| | 66 | In PALM, ''u'',,*,, is calculated from ''u'',,h,, at ''z'',,MO,, by vertical integration over ''z'' from ''z'',,0,, to ''z'',,MO,,. |
| | 67 | |
| | 68 | From the equations above it is possible to derive a formulation for the horizontal wind components, viz. |
| | 69 | {{{ |
| | 70 | #!Latex |
| | 71 | \begin{align} |
| | 72 | & |
| | 73 | \frac{\partial u}{\partial z} = |
| | 74 | \frac{-\overline{u^{\prime\prime} w^{\prime\prime}}_0}{u_\ast \kappa |
| | 75 | z} |
| | 76 | \Phi_\mathrm{m}\left(\frac{z}{L}\right)\,\text{and~}\,\frac{\partial |
| | 77 | v}{\partial z} = \frac{-\overline{v^{\prime\prime} |
| | 78 | w^{\prime\prime}}_0}{u_\ast \kappa z} |
| | 79 | \Phi_\mathrm{m}\left(\frac{z}{L}\right)\;. |
| | 80 | \end{align} |
| | 81 | }}} |
| | 82 | Vertical integration of the above equation over ''z'' from ''z'',,0,, to ''z'',,MO,, then yields the surface momentum fluxes |
| | 83 | {{{ |
| | 84 | #!Latex |
| | 85 | \begin{equation*} |
| | 86 | \overline{u^{\prime\prime} w^{\prime\prime}}_0,\;\; \overline{v^{\prime\prime} w^{\prime\prime}}_0 |
| | 87 | \end{equation*} |
| | 88 | }}} |
| | 89 | |
| | 90 | The formulations above all require knowledge of the scaling parameters ''θ'',,*,, and ''q'',,*,,. These are deduced from vertical |
| | 91 | integration of |
| | 92 | {{{ |
| | 93 | #!Latex |
| | 94 | \begin{align} |
| | 95 | & |
| | 96 | \frac{\partial \theta}{\partial z} = |
| | 97 | \frac{\theta_\ast}{\kappa z} |
| | 98 | \Phi_\mathrm{h}\left(\frac{z}{L}\right)~{\text{and}}~\frac{\partial |
| | 99 | q_\mathrm{v}}{\partial z} = \frac{q_\ast}{\kappa z} |
| | 100 | \Phi_\mathrm{h}\left(\frac{z}{L}\right) |
| | 101 | \end{align} |
| | 102 | }}} |
| | 103 | over ''z'' from ''z'',,0,h,, to ''z'',,MO,,. The similarity function Φ,,h,, is given by |
| | 104 | {{{ |
| | 105 | #!Latex |
| | 106 | \begin{align} |
| | 107 | & \Phi_\mathrm{h} = |
| | 108 | \begin{cases} |
| | 109 | 1 + 5 \frac{z}{L} & \text{for~} \frac{z}{L} \geq 0 \\ |
| | 110 | \left(1 - 16 \frac{z}{L}\right)^{-1/2} & \text{for~} \frac{z}{L} |
| | 111 | < 0\;. |
| | 112 | \end{cases} |
| | 113 | \end{align} |
| | 114 | }}} |
| | 115 | |
| | 116 | Note that this implementation of MOST in PALM requires the use of data from the previous time step. The following steps are thus carried out in sequential order. First of all, ''θ'',,*,, and ''q'',,*,, are calculated by integration using the value of ''z'',,MO,,/L from the previous time step. Second, the new value of ''z'',,MO,,/L is derived using the new values of ''θ'',,*,, and ''q'',,*,, but using ''u'',,*,, from the previous time step. Then, the new values of ''u'',,*,,, and subsequently the momentum fluxes are calculated by integration, respectively. At last, the new surface fluxes are derived from ''θ'',,*,, and ''q'',,*,,, and ''u'',,*,,. In the special case, when surface fluxes are prescribed instead of surface temperature and humidity, the first and last steps are omitted and ''θ'',,*,, and ''q'',,*,, are directly calculated from ''u'',,*,, and the surface fluxes. |
