Changes between Version 16 and Version 17 of doc/tec/bc

Timestamp:

Apr 26, 2016 8:59:53 PM (9 years ago)

Author:

Giersch

Comment:

--

doc/tec/bc

@@ +1 @@
+[[NoteBox(warn,This site is currently under construction!)]]
+
 = Boundary conditions =
+== Basics ==
 
 PALM offers a variety of boundary conditions. Dirichlet or Neumann boundary conditions can be chosen for ''u'', ''v'', ''θ'',
@@ -5 +8 @@
 Δz'') instead. Vertical velocity is assumed to be zero at the surface and top boundaries, which implies using Neumann conditions for pressure.
 
-
-
-
-
-== Constant flux layer ==
-=== Basics ===
-
 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.
 
@@ -24 +20 @@
 {{{
 #!Latex
-\begin{align}
+\begin{align*}
   & \frac{\partial u_\mathrm{h}}{\partial z} =
   \frac{u_\ast}{\kappa z}\Phi_\mathrm{m}\left(\frac{z}{L}\right)\;,
-\end{align}
+\end{align*}
 }}}
 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])
 {{{
 #!Latex
-\begin{align}
+\begin{align*}
   & \Phi_\mathrm{m} =
   \begin{cases}
@@ -39 +35 @@
     \frac{z}{L} < 0\;.
   \end{cases}
-\end{align}
+\end{align*}
 }}}
 Here, ''L'' is the Obukhov length, calculated as
 {{{
 #!Latex
-\begin{align}
+\begin{align*}
   & 
 L = \frac{\theta_\mathrm{v}(z) u_\ast^2}{\kappa g
     \left[\theta_\ast + 0.61 \theta(z) q_\ast + 0.61
       q_\mathrm{v}(z) \theta_\ast\right]}\;.
-\end{align}
+\end{align*}
 }}}
 The scaling parameters ''θ'',,*,, and  ''q'',,*,, are defined by MOST
@@ -55 +51 @@
 {{{
 #!Latex
-\begin{align}
+\begin{align*}
   & \theta_\ast = -
   \frac{\overline{w^{\prime\prime}\theta^{\prime\prime}}_0}{u_\ast},~q_\ast
   = -
   \frac{\overline{w^{\prime\prime}q_\mathrm{v}^{\prime\prime}}_0}{u_\ast}\;,
-\end{align}
+\end{align*}
 }}}
 
@@ -66 +62 @@
 {{{
 #!Latex
-\begin{align}
+\begin{align*}
   & u_\ast =
   \left[\left(\overline{u^{\prime\prime} w^{\prime\prime}}_0\right)^2
     + \left(\overline{v^{\prime\prime} w^{\prime\prime}}_0\right)^2
   \right]^{\frac{1}{4}}\;.
-\end{align}
+\end{align*}
 }}}
 
@@ -79 +75 @@
 {{{
 #!Latex
-\begin{align}
+\begin{align*}
   & 
 \frac{\partial u}{\partial z} =
@@ -88 +84 @@
       w^{\prime\prime}}_0}{u_\ast \kappa z}
   \Phi_\mathrm{m}\left(\frac{z}{L}\right)\;.
-\end{align}
-}}}
-Vertical integration of the above equation over ''z'' from ''z'',,0,, to ''z'',,MO,, then yields the surface momentum fluxes
-{{{
-#!Latex
-\begin{equation*}
-\overline{u^{\prime\prime} w^{\prime\prime}}_0,\;\; \overline{v^{\prime\prime} w^{\prime\prime}}_0
-\end{equation*}
+\end{align*}
+}}}
+Vertical integration over ''z'' from ''z'',,0,, to ''z'',,MO,, of the equation above then yields the surface momentum fluxes
+{{{
+#!Latex
+$\overline{u^{\prime\prime} w^{\prime\prime}}_0,\;\; \overline{v^{\prime\prime} w^{\prime\prime}}_0$
 }}}
 
@@ -124 +118 @@
 }}}
 
-=== Implementation ===
 Currently, there are three different options to calculate the Obukhov length and the surface fluxes which are steered via the NAMELIST parameter [wiki:doc/app/inipar#most_method most_method]. 
 
-==== Method 1: circular ====
-The traditional implementation in PALM ({{{most_method = 'circular'}}}) 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.
+=== Method 1: circular ===
+The traditional implementation in PALM ({{{most_method = 'circular'}}}) 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 of the corresponding vertical derivation functions mentioned above using the value of ''z'',,MO,,/L from the previous time step. Second, the new value of ''z'',,MO,,/L is derived from the equation for ''L'' 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 above equations for the scaling parameters are employed to calculate the new surface fluxes by using ''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.
 
 In summary, the following actions are performed in sequential order:
@@ -137 +130 @@
 5. derive surface fluxes
 
-==== Method 2: Newton iteration / lookup table ====
+=== Method 2: Newton iteration / lookup table ===
 Alternatively, the Obukhov length can be calculated by solving an implicit equation relating the ''L'' to the bulk Richardson number. This can be achieved either by a Newton iteration algorithm ({{{most_method = 'newton'}}}) or by using a lookup table ({{{most_method = 'lookup'}}}). Note that the latter is the new default in PALM as it is much faster than the Newton iteration method and the results are more precise compared to the circular method. However, it can only be used when the roughness lengths are homogeneously set on each processor.
 
@@ -200 +193 @@
 until ''L'' meets a convergence criterion. 
 
-If {{{most_method = 'lookup'}}} is used, a table of ''Ri'',,b,, against ''z'',,MO,,/''L'' is created at model start, based on the prescribed values of the roughness lengths. During the model run, ''Ri,,b,,'' is calculated and the respectively value of ''z'',,MO,,/''L'' is retrieved from the lookup table and using linear interpolation between the discrete values in the table. In order to speed up this method, the value of ''Ri,,b,,'' from the previous time step is used as initial value. Due to the fact that the lookup table is created at model start, it is essential that the roughness lengths are 1) homogeneous on each processor (limiting this method to homogeneous surface configurations) and should not vary during the simulation (should thus not be used when using dynamic roughness length over water surfaces).
-
-For more details, see [source:palm/trunk/SOURCE/surface_layer_fluxes.f90 surface_layer_fluxes.f90].
+If {{{most_method = 'lookup'}}} is used, a table of ''Ri'',,b,, against ''z'',,MO,,/''L'' is created at model start, based on the prescribed values of the roughness lengths. During the model run, ''Ri,,b,,'' is calculated and the respectively value of ''z'',,MO,,/''L'' is retrieved from the lookup table and using linear interpolation between the discrete values in the table. In order to speed up this method, the value of ''Ri,,b,,'' from the previous time step is used as initial value. Due to the fact that the lookup table is created at model start, it is essential that the roughness lengths are 1) homogeneous on each processor (limiting this method to homogeneous surface configurations) and should not vary during the simulation (should thus not be used when using dynamic roughness length over water surfaces). For more details, see [source:palm/trunk/SOURCE/surface_layer_fluxes.f90 surface_layer_fluxes.f90].
+
+Furthermore, the flat bottom of the model can be replaced by a Cartesian topography (see Sect. [wiki:doc/tec/bc#Topography Topography]).
+
+By default, lateral boundary conditions are set to be cyclic in both directions. Alternatively, it is possible to opt for non-cyclic conditions in one direction, i.e., a laminar or turbulent inflow boundary (see Sect. [wiki:doc/tec/bc#Laminarandturbulentinflowboundaryconditions Laminar and turbulent inflow boundary conditions]) and an open outflow boundary on the opposite site (see Sect. [wiki:doc/tec/bc#Openoutflowboundaryconditions Open outflow boundary conditions]). The boundary conditions for the other direction have to remain cyclic.
+
+In order to prevent gravity waves from being reflected at the top boundary, a sponge layer (Rayleigh damping) can be applied to all prognostic variables in the upper part of the model domain ([#klemp1978 Klemp and Lilly, 1978]). Such a sponge layer should be applied only within the free atmosphere, where no turbulence is present.
+
+The model is initialized by horizontally homogeneous vertical profiles of potential temperature, specific humidity (or a passive scalar), and
+the horizontal wind velocities. The latter can be also provided from a 1-D precursor run (see Sect.[wiki:doc/tec/1dmodel 1-D model for precursor runs]). Uniformly distributed random perturbations with a user-defined amplitude can be imposed to  the fields of the horizontal velocities components to initiate turbulence.
+
+== Laminar and turbulent inflow boundary conditions ==
+
+In case
+
+== Turbulence recycling ==
+
+== Open outflow boundary conditions ==
+
+== Topography ==
 
 = References =
-* [=#holtslag] '''Holtslag AAM, Bruin HARD.''' 1988. Applied modelling of the night-time surface energy balance over land. J. Appl. Meteorol., 27, 689–704.
-
-* [=#panofsky] '''Panofsky HA, Dutton JA.''' 1984. Atmospheric Turbulence, Models and Methods for Engineering Applications, John Wiley & Sons, New York.
-
-* [=#paulson] '''Paulson CA''' 1970. The mathematical representation of wind speed and temperature profiles in the unstable atmospheric surface layer. J. Appl. Meteorol., 9, 857–861.
-
-
-
+* [=#holtslag] '''Holtslag AAM, Bruin HARD.''' 1988. Applied modelling of the night-time surface energy balance over land. J. Appl. Meteorol. 27: 689–704.
+
+* [=#panofsky] '''Panofsky HA, Dutton JA.''' 1984. Atmospheric Turbulence, Models and Methods for Engineering Applications. John Wiley & Sons. New York.
+
+* [=#paulson] '''Paulson CA''' 1970. The mathematical representation of wind speed and temperature profiles in the unstable atmospheric surface layer. J. Appl. Meteorol. 9: 857–861.
+
+
+* [=#klemp1978] '''Klemp JB, Lilly DK.''' 1978. Numerical simulation of hydrostatic mountain waves. J. Atmos. Sci. 35: 78–107.
+
+
+