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Non-cyclic lateral boundary conditions
Figure 1 shows the grid structure for non-cyclic boundary conditions at the left/right boundary LB/RB (bc_lr) and figure 2 for non-cyclic boundary conditions at the north/south boundary NB/SB (bc_ns).
The indices (i,j,k) represent the directions (x,y,z).
The model domain extends from -1:nx+1 in the x-direction, from -1:ny+1 in the y-direction and from 0:nzt+1 in the z-direction.
For the advection scheme of Wicker and Skamarock, two more grid points are added at the lateral boundaries which are not needed for non-cyclic boundary conditions.
The figures display the grid layer of the horizontal velocity components u and v, and scalar s.
The grid points of the vertical velocity w are defined at the scalar position but shifted by one half grid spacing in vertical direction (not shown, detailed information about the grid structure in PALM can be found here).
The prognostic equations are solved at all inner grid points which are marked black.
The grid points at the respective non-cyclic boundaries (blue) are treated as follows.
Figure 1: Grid structure at the lateral boundaries with non-cyclic lateral boundary conditions along the left-right direction.
Figure 2: Grid structure of the lateral boundaries with non-cyclic lateral boundary conditions along the north-south direction.
LB is defined at i = -1 for v, w, s and at i = 0 for u.
SB is defined at j = -1 for u, w, s and at j = 0 for v.
RB is defined at i = nx + 1 and NB at j = ny + 1 for all quantities.
LB and SB are treated this way so that the order and number of grid points for the streamwise velocity component and scalars is the same, independent of the flow direction.
For technical reasons, the prognostic equations are first solved for u at i = 0 (v at j = 0), since these grid points technically belong to the inner grid, but afterwards, these results at i = 0 (j = 0) are replaced by the respective boundary condition in routine boundary_conds.f90.
In case of a Dirichlet condition, the values at i = 0 (j = 0) are taken from i = -1 (j = -1).
In case of a radiation boundary condition, the solution of the Sommerfeld equation overwrites the prognostic values at i = 0 (j = 0).
For non-cyclic lateral boundary conditions, the parameter psolver has to be set to 'multigrid' because the default FFT-solver can only be applied for cyclic boundary conditions.
Inflow boundary
At the inflow boundary, Dirichlet conditions are used for the three velocity components ψ = {u,v,w} as well as for all scalar quantities s and are implemented as follows (here e.g. for s and a flow in positive x direction):
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"This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)\n restricted \\write18 enabled.\nentering extended mode\n(./aea1766d507d4aa5852bc125d8eb9a395d4957c4.tex\nLaTeX2e <2011/06/27>\nBabel <3.9h> and hyphenation patterns for 78 languages loaded.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/article.cls\nDocument Class: article 2007/10/19 v1.4h Standard LaTeX document class\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/size10.clo))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/inputenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/utf8.def\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/ot1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/omsenc.dfu)))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cmap/cmap.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cm-super/type1ec.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1cmr.fd))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/fontenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.def)<<t1.cmap\n>>) (/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsmath.sty\nFor additional information on amsmath, use the `?' option.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amstext.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsgen.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsbsy.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsopn.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amscls/amsthm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amssymb.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amsfonts.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/tools/bm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/preview.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/prtightpage.def)\n)\nNo file aea1766d507d4aa5852bc125d8eb9a395d4957c4.aux.\nPreview: Fontsize 10pt\nPreview: PDFoutput 1\n! Missing $ inserted.\n<inserted text> \n $\nl.14 s^\n {t + \\Delta t}(k,j,-1) = s_{init}(k) \\; . \\quad(1)\n<<ot1.cmap>><<oml.cmap>><<oms.cmap>><<omx.cmap>>\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsa.fd)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsb.fd)\n! Missing $ inserted.\n<inserted text> \n $\nl.15 \\end{preview}\n \nPreview: Tightpage -32891 -32891 32891 32891\n[1{/localdata/software/texlive/2013/texmf-var/fonts/map/pdftex/updmap/pdftex.ma\np}] (./aea1766d507d4aa5852bc125d8eb9a395d4957c4.aux) )\n(see the transcript file for additional information){/localdata/software/texliv\ne/2013/texmf-dist/fonts/enc/dvips/cm-super/cm-super-t1.enc}</localdata/software\n/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></localdata/\nsoftware/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></loc\naldata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pf\nb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/c\nmr7.pfb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfont\ns/cm/cmsy10.pfb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public\n/cm-super/sfrm1000.pfb>\nOutput written on aea1766d507d4aa5852bc125d8eb9a395d4957c4.pdf (1 page, 52571 b\nytes).\nTranscript written on aea1766d507d4aa5852bc125d8eb9a395d4957c4.log.\n"t denotes the time, Δt the time step and sinit the initialization profile of the scalar quantities which is constant in time. The quantities at the inflow are set by the initial vertical profiles (see initializing_actions). A Neumann condition is used for the subgrid-scale turbulent kinetic energy e (here e.g. for a left-right flow):
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'/usr/bin/pdflatex -interaction=nonstopmode 1c575cb34a5033432c39aad25d1c7873f5802aa7.tex'failed with the following output:
"This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)\n restricted \\write18 enabled.\nentering extended mode\n(./1c575cb34a5033432c39aad25d1c7873f5802aa7.tex\nLaTeX2e <2011/06/27>\nBabel <3.9h> and hyphenation patterns for 78 languages loaded.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/article.cls\nDocument Class: article 2007/10/19 v1.4h Standard LaTeX document class\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/size10.clo))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/inputenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/utf8.def\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/ot1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/omsenc.dfu)))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cmap/cmap.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cm-super/type1ec.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1cmr.fd))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/fontenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.def)<<t1.cmap\n>>) (/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsmath.sty\nFor additional information on amsmath, use the `?' option.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amstext.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsgen.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsbsy.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsopn.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amscls/amsthm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amssymb.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amsfonts.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/tools/bm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/preview.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/prtightpage.def)\n)\nNo file 1c575cb34a5033432c39aad25d1c7873f5802aa7.aux.\nPreview: Fontsize 10pt\nPreview: PDFoutput 1\n! Missing $ inserted.\n<inserted text> \n $\nl.14 \\theta\n ^{t+1}(d) = ... - \\Delta t \\cdot K(d) \\cdot \\left( \\theta^{t}(d) ...\n\n<<ot1.cmap>><<oml.cmap>><<oms.cmap>><<omx.cmap>>\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsa.fd)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsb.fd)\n! Missing $ inserted.\n<inserted text> \n $\nl.15 \\end{preview}\n \nPreview: Tightpage -32891 -32891 32891 32891\n[1{/localdata/software/texlive/2013/texmf-var/fonts/map/pdftex/updmap/pdftex.ma\np}] (./1c575cb34a5033432c39aad25d1c7873f5802aa7.aux) )\n(see the transcript file for additional information)</localdata/software/texliv\ne/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></localdata/softwar\ne/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></localdata/\nsoftware/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></loc\naldata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb\n></localdata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cm\nsy10.pfb>\nOutput written on 1c575cb34a5033432c39aad25d1c7873f5802aa7.pdf (1 page, 44716 b\nytes).\nTranscript written on 1c575cb34a5033432c39aad25d1c7873f5802aa7.log.\n"Here, d is the distance normal to the wall and θinit the initial value of the potential temperature, which corresponds to the value at the inflow boundary. The damping or relaxation function K depends only on the distance d to the inflow. K is calculated by
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Outflow boundary
At the outflow, an open boundary condition is needed to ensure that disturbances of the mean flow can exit the model domain without effecting the flow upstream. For the scalar quantities, Neumann boundary conditions are used at the outflow boundary which is the simplest way. For the velocity components, a Neumann condition would require to be considered in the solution of the Poisson equation for perturbation pressure, which has not been realized so far, because it requires some technical effort. Instead, PALM offers two types of radiation boundary conditions for the velocity components, which are not in conflict with the pressure solver (see bc_lr and bc_ns). For the radiation condition, the Sommerfeld radiation equation is solved at the outflow
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The command:
'/usr/bin/pdflatex -interaction=nonstopmode 1e4bbababa2c6e8fd3572b0a7c9a558a305b422f.tex'failed with the following output:
"This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)\n restricted \\write18 enabled.\nentering extended mode\n(./1e4bbababa2c6e8fd3572b0a7c9a558a305b422f.tex\nLaTeX2e <2011/06/27>\nBabel <3.9h> and hyphenation patterns for 78 languages loaded.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/article.cls\nDocument Class: article 2007/10/19 v1.4h Standard LaTeX document class\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/size10.clo))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/inputenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/utf8.def\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/ot1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/omsenc.dfu)))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cmap/cmap.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cm-super/type1ec.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1cmr.fd))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/fontenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.def)<<t1.cmap\n>>) (/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsmath.sty\nFor additional information on amsmath, use the `?' option.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amstext.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsgen.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsbsy.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsopn.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amscls/amsthm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amssymb.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amsfonts.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/tools/bm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/preview.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/prtightpage.def)\n)\nNo file 1e4bbababa2c6e8fd3572b0a7c9a558a305b422f.aux.\nPreview: Fontsize 10pt\nPreview: PDFoutput 1\n! Missing $ inserted.\n<inserted text> \n $\nl.14 \\partial\n _t \\psi + c_{\\psi} \\partial_n \\psi = 0 \\; , \\quad (5)\n<<ot1.cmap>><<oml.cmap>><<oms.cmap>><<omx.cmap>>\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsa.fd)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsb.fd)\n! Missing $ inserted.\n<inserted text> \n $\nl.15 \\end{preview}\n \nPreview: Tightpage -32891 -32891 32891 32891\n[1{/localdata/software/texlive/2013/texmf-var/fonts/map/pdftex/updmap/pdftex.ma\np}] (./1e4bbababa2c6e8fd3572b0a7c9a558a305b422f.aux) )\n(see the transcript file for additional information)</localdata/software/texliv\ne/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></localdata/softwar\ne/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></localdata/\nsoftware/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb>\nOutput written on 1e4bbababa2c6e8fd3572b0a7c9a558a305b422f.pdf (1 page, 30105 b\nytes).\nTranscript written on 1e4bbababa2c6e8fd3572b0a7c9a558a305b422f.log.\n"which considers flow disturbances propagating with the mean flow and by waves. Here ψ is the transported quantity and ∂n is the derivative normal to the outflow boundary. In PALM, based on (5), the radiation boundary condition is realized in two ways as follows.
Radiation boundary condition I
The phase velocity cψ (also called convective velocity) is calculated after Orlanski (1976):
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'/usr/bin/pdflatex -interaction=nonstopmode 179dff8bc4721ca10253c1d89fe0211cea33728d.tex'failed with the following output:
"This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)\n restricted \\write18 enabled.\nentering extended mode\n(./179dff8bc4721ca10253c1d89fe0211cea33728d.tex\nLaTeX2e <2011/06/27>\nBabel <3.9h> and hyphenation patterns for 78 languages loaded.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/article.cls\nDocument Class: article 2007/10/19 v1.4h Standard LaTeX document class\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/size10.clo))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/inputenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/utf8.def\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/ot1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/omsenc.dfu)))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cmap/cmap.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cm-super/type1ec.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1cmr.fd))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/fontenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.def)<<t1.cmap\n>>) (/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsmath.sty\nFor additional information on amsmath, use the `?' option.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amstext.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsgen.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsbsy.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsopn.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amscls/amsthm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amssymb.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amsfonts.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/tools/bm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/preview.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/prtightpage.def)\n)\nNo file 179dff8bc4721ca10253c1d89fe0211cea33728d.aux.\nPreview: Fontsize 10pt\nPreview: PDFoutput 1\n! Missing $ inserted.\n<inserted text> \n $\nl.14 c_\n {\\psi} = - \\frac{\\partial_t \\psi}{\\partial_n \\psi} \\; . \\quad (6)\n<<ot1.cmap>><<oml.cmap>><<oms.cmap>><<omx.cmap>>\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsa.fd)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsb.fd)\n! Missing $ inserted.\n<inserted text> \n $\nl.15 \\end{preview}\n \nPreview: Tightpage -32891 -32891 32891 32891\n[1{/localdata/software/texlive/2013/texmf-var/fonts/map/pdftex/updmap/pdftex.ma\np}] (./179dff8bc4721ca10253c1d89fe0211cea33728d.aux) )\n(see the transcript file for additional information){/localdata/software/texliv\ne/2013/texmf-dist/fonts/enc/dvips/cm-super/cm-super-t1.enc}</localdata/software\n/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></localdata/\nsoftware/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb></loc\naldata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pf\nb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/c\nmr10.pfb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfon\nts/cm/cmsy10.pfb></localdata/software/texlive/2013/texmf-dist/fonts/type1/publi\nc/cm-super/sfrm1000.pfb>\nOutput written on 179dff8bc4721ca10253c1d89fe0211cea33728d.pdf (1 page, 48781 b\nytes).\nTranscript written on 179dff8bc4721ca10253c1d89fe0211cea33728d.log.\n"Left-right flow
If the outflow is defined at the right boundary (i = nx + 1, see Fig. 1), the phase velocity for each velocity components is calculated by
TracMath macro processor has detected an error. Please fix the problem before continuing.
The command:
'/usr/bin/pdflatex -interaction=nonstopmode c087b29362862117034804cc3c4683ea5d95ac3e.tex'failed with the following output:
"This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)\n restricted \\write18 enabled.\nentering extended mode\n(./c087b29362862117034804cc3c4683ea5d95ac3e.tex\nLaTeX2e <2011/06/27>\nBabel <3.9h> and hyphenation patterns for 78 languages loaded.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/article.cls\nDocument Class: article 2007/10/19 v1.4h Standard LaTeX document class\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/size10.clo))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/inputenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/utf8.def\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/ot1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/omsenc.dfu)))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cmap/cmap.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cm-super/type1ec.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1cmr.fd))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/fontenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.def)<<t1.cmap\n>>) (/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsmath.sty\nFor additional information on amsmath, use the `?' option.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amstext.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsgen.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsbsy.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsopn.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amscls/amsthm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amssymb.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amsfonts.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/tools/bm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/preview.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/prtightpage.def)\n)\nNo file c087b29362862117034804cc3c4683ea5d95ac3e.aux.\nPreview: Fontsize 10pt\nPreview: PDFoutput 1\n! Missing $ inserted.\n<inserted text> \n $\nl.14 c_\n {\\psi} = - c_{max} \\frac{\\psi^t_{nx} - \\psi^{t - \\Delta t}_{nx} }{ \\p...\n\n<<ot1.cmap>><<oml.cmap>><<oms.cmap>><<omx.cmap>>\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsa.fd)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsb.fd)\n! Missing $ inserted.\n<inserted text> \n $\nl.15 \\end{preview}\n \nPreview: Tightpage -32891 -32891 32891 32891\n[1{/localdata/software/texlive/2013/texmf-var/fonts/map/pdftex/updmap/pdftex.ma\np}] (./c087b29362862117034804cc3c4683ea5d95ac3e.aux) )\n(see the transcript file for additional information){/localdata/software/texliv\ne/2013/texmf-dist/fonts/enc/dvips/cm-super/cm-super-t1.enc}</localdata/software\n/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></localdata/\nsoftware/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb></loc\naldata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pf\nb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/c\nmr10.pfb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfon\nts/cm/cmr5.pfb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public/\namsfonts/cm/cmsy10.pfb></localdata/software/texlive/2013/texmf-dist/fonts/type1\n/public/amsfonts/cm/cmsy5.pfb></localdata/software/texlive/2013/texmf-dist/font\ns/type1/public/amsfonts/cm/cmsy7.pfb></localdata/software/texlive/2013/texmf-di\nst/fonts/type1/public/cm-super/sfrm1000.pfb>\nOutput written on c087b29362862117034804cc3c4683ea5d95ac3e.pdf (1 page, 71049 b\nytes).\nTranscript written on c087b29362862117034804cc3c4683ea5d95ac3e.log.\n"with the maximum phase velocity
TracMath macro processor has detected an error. Please fix the problem before continuing.
The command:
'/usr/bin/pdflatex -interaction=nonstopmode 6a58035b1b1012118d07ac32f84a2542f8ee979c.tex'failed with the following output:
"This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)\n restricted \\write18 enabled.\nentering extended mode\n(./6a58035b1b1012118d07ac32f84a2542f8ee979c.tex\nLaTeX2e <2011/06/27>\nBabel <3.9h> and hyphenation patterns for 78 languages loaded.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/article.cls\nDocument Class: article 2007/10/19 v1.4h Standard LaTeX document class\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/size10.clo))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/inputenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/utf8.def\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/ot1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/omsenc.dfu)))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cmap/cmap.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cm-super/type1ec.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1cmr.fd))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/fontenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.def)<<t1.cmap\n>>) (/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsmath.sty\nFor additional information on amsmath, use the `?' option.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amstext.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsgen.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsbsy.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsopn.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amscls/amsthm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amssymb.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amsfonts.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/tools/bm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/preview.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/prtightpage.def)\n)\nNo file 6a58035b1b1012118d07ac32f84a2542f8ee979c.aux.\nPreview: Fontsize 10pt\nPreview: PDFoutput 1\n! Missing $ inserted.\n<inserted text> \n $\nl.14 c_\n {max} = \\frac{\\Delta x}{\\Delta t} \\; . \\quad (8)\n<<ot1.cmap>><<oml.cmap>><<oms.cmap>><<omx.cmap>>\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsa.fd)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsb.fd)\n! Missing $ inserted.\n<inserted text> \n $\nl.15 \\end{preview}\n \nPreview: Tightpage -32891 -32891 32891 32891\n[1{/localdata/software/texlive/2013/texmf-var/fonts/map/pdftex/updmap/pdftex.ma\np}] (./6a58035b1b1012118d07ac32f84a2542f8ee979c.aux) )\n(see the transcript file for additional information){/localdata/software/texliv\ne/2013/texmf-dist/fonts/enc/dvips/cm-super/cm-super-t1.enc}</localdata/software\n/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></localdata/\nsoftware/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></loc\naldata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pf\nb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/c\nmr7.pfb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public/cm-supe\nr/sfrm1000.pfb>\nOutput written on 6a58035b1b1012118d07ac32f84a2542f8ee979c.pdf (1 page, 41969 b\nytes).\nTranscript written on 6a58035b1b1012118d07ac32f84a2542f8ee979c.log.\n"The phase velocity has to be in the range of 0 ≤ cψ < cmax because negative values propagate waves back to the inner domain. cmax represents the maximum phase velocity that ensures numerical stability (Courant-Friedrichs-Lewy condition). Both conditions are not ensured by (7), hence cψ is set to zero for negative values and to cmax for larger values than cmax. The velocity components ψ at the outflow boundary are then calculated by
TracMath macro processor has detected an error. Please fix the problem before continuing.
The command:
'/usr/bin/pdflatex -interaction=nonstopmode 7def1d005b217b32a50b89810cfd492bc0fa7fcf.tex'failed with the following output:
"This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)\n restricted \\write18 enabled.\nentering extended mode\n(./7def1d005b217b32a50b89810cfd492bc0fa7fcf.tex\nLaTeX2e <2011/06/27>\nBabel <3.9h> and hyphenation patterns for 78 languages loaded.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/article.cls\nDocument Class: article 2007/10/19 v1.4h Standard LaTeX document class\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/size10.clo))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/inputenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/utf8.def\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/ot1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/omsenc.dfu)))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cmap/cmap.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cm-super/type1ec.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1cmr.fd))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/fontenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.def)<<t1.cmap\n>>) (/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsmath.sty\nFor additional information on amsmath, use the `?' option.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amstext.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsgen.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsbsy.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsopn.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amscls/amsthm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amssymb.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amsfonts.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/tools/bm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/preview.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/prtightpage.def)\n)\nNo file 7def1d005b217b32a50b89810cfd492bc0fa7fcf.aux.\nPreview: Fontsize 10pt\nPreview: PDFoutput 1\n! Missing $ inserted.\n<inserted text> \n $\nl.14 \\psi\n ^{t+\\Delta t}_{nx+1} = \\psi^{t}_{nx+1} - \\frac{\\overline{c}_{\\psi}}...\n\n<<ot1.cmap>><<oml.cmap>><<oms.cmap>><<omx.cmap>>\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsa.fd)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsb.fd)\n! Missing $ inserted.\n<inserted text> \n $\nl.15 \\end{preview}\n \nPreview: Tightpage -32891 -32891 32891 32891\n[1{/localdata/software/texlive/2013/texmf-var/fonts/map/pdftex/updmap/pdftex.ma\np}] (./7def1d005b217b32a50b89810cfd492bc0fa7fcf.aux) )\n(see the transcript file for additional information)</localdata/software/texliv\ne/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></localdata/softwar\ne/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb></localdata/\nsoftware/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></loc\naldata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pf\nb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/c\nmr7.pfb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfont\ns/cm/cmsy10.pfb>\nOutput written on 7def1d005b217b32a50b89810cfd492bc0fa7fcf.pdf (1 page, 52391 b\nytes).\nTranscript written on 7def1d005b217b32a50b89810cfd492bc0fa7fcf.log.\n"with the phase velocity averaged parallel to the outflow:
TracMath macro processor has detected an error. Please fix the problem before continuing.
The command:
'/usr/bin/pdflatex -interaction=nonstopmode 0f1e966656c30dedebe08df842bd90b13ee8b4db.tex'failed with the following output:
"This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)\n restricted \\write18 enabled.\nentering extended mode\n(./0f1e966656c30dedebe08df842bd90b13ee8b4db.tex\nLaTeX2e <2011/06/27>\nBabel <3.9h> and hyphenation patterns for 78 languages loaded.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/article.cls\nDocument Class: article 2007/10/19 v1.4h Standard LaTeX document class\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/size10.clo))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/inputenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/utf8.def\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/ot1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/omsenc.dfu)))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cmap/cmap.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cm-super/type1ec.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1cmr.fd))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/fontenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.def)<<t1.cmap\n>>) (/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsmath.sty\nFor additional information on amsmath, use the `?' option.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amstext.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsgen.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsbsy.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsopn.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amscls/amsthm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amssymb.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amsfonts.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/tools/bm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/preview.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/prtightpage.def)\n)\nNo file 0f1e966656c30dedebe08df842bd90b13ee8b4db.aux.\nPreview: Fontsize 10pt\nPreview: PDFoutput 1\n! Missing $ inserted.\n<inserted text> \n $\nl.14 \\overline{c}\n _{\\psi} = \\frac{1}{ny+1} \\sum_{j=0}^{ny} c_{\\psi, j} \\; . ...\n\n<<ot1.cmap>><<oml.cmap>><<oms.cmap>><<omx.cmap>>\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsa.fd)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsb.fd)\n! Missing $ inserted.\n<inserted text> \n $\nl.15 \\end{preview}\n \nPreview: Tightpage -32891 -32891 32891 32891\n[1{/localdata/software/texlive/2013/texmf-var/fonts/map/pdftex/updmap/pdftex.ma\np}] (./0f1e966656c30dedebe08df842bd90b13ee8b4db.aux) )\n(see the transcript file for additional information)</localdata/software/texliv\ne/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></localdata/softwar\ne/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></localdata\n/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></lo\ncaldata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.p\nfb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/\ncmr7.pfb>\nOutput written on 0f1e966656c30dedebe08df842bd90b13ee8b4db.pdf (1 page, 43804 b\nytes).\nTranscript written on 0f1e966656c30dedebe08df842bd90b13ee8b4db.log.\n"In Orlanskis work, the phase velocity cψ was not averaged along the outflow, which is sufficient for simplified flows as shown by Yoshida and Watanabe (2010). However, we found that in simulations of shear driven convective boundary layer with a strong wind component along the outflow, the approach of Orlanski (1976) was unstable. Raymond and Kuo (1984) argued that locally calculated phase velocities leads to a large variation of phase speeds which can lead to numerical instabilities. Other works used a constant phase velocity or an average over the whole outflow area (see e.g. Fröhlich 2006, p.216) instead of the approach of Orlanski (1976). With this in mind, we tested the average phase velocity calculated from (10), and the solution at the outflow becomes stable. We did not additionally average the phase velocity along the vertical direction, because the background wind usually varies with height.
For the other flow directions, the indices of equations (7)-(10) have to be replaced as follows:
TracMath macro processor has detected an error. Please fix the problem before continuing.
The command:
'/usr/bin/pdflatex -interaction=nonstopmode 8ea9fe3c0e02e93deb6a4bca750fa700f7ffad44.tex'failed with the following output:
"This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)\n restricted \\write18 enabled.\nentering extended mode\n(./8ea9fe3c0e02e93deb6a4bca750fa700f7ffad44.tex\nLaTeX2e <2011/06/27>\nBabel <3.9h> and hyphenation patterns for 78 languages loaded.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/article.cls\nDocument Class: article 2007/10/19 v1.4h Standard LaTeX document class\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/size10.clo))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/inputenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/utf8.def\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/ot1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/omsenc.dfu)))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cmap/cmap.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cm-super/type1ec.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1cmr.fd))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/fontenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.def)<<t1.cmap\n>>) (/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsmath.sty\nFor additional information on amsmath, use the `?' option.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amstext.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsgen.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsbsy.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsopn.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amscls/amsthm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amssymb.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amsfonts.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/tools/bm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/preview.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/prtightpage.def)\n)\nNo file 8ea9fe3c0e02e93deb6a4bca750fa700f7ffad44.aux.\nPreview: Fontsize 10pt\nPreview: PDFoutput 1\n<<ot1.cmap>><<oml.cmap>><<oms.cmap>><<omx.cmap>>\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsa.fd)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsb.fd)\n! Undefined control sequence.\nl.18 ...0$ & $(nx + 1) \\rightarrow -1$ \\multirow\n { }{ }& \\multirow{ }{ } & \\\\\n\n! Undefined control sequence.\nl.18 ...ightarrow -1$ \\multirow{ }{ }& \\multirow\n { }{ } & \\\\\nPreview: Tightpage -32891 -32891 32891 32891\n[1{/localdata/software/texlive/2013/texmf-var/fonts/map/pdftex/updmap/pdftex.ma\np}] (./8ea9fe3c0e02e93deb6a4bca750fa700f7ffad44.aux) )\n(see the transcript file for additional information){/localdata/software/texliv\ne/2013/texmf-dist/fonts/enc/dvips/cm-super/cm-super-t1.enc}</localdata/software\n/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></localdata/\nsoftware/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></loc\naldata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.p\nfb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public/cm-super/sfr\nm1000.pfb>\nOutput written on 8ea9fe3c0e02e93deb6a4bca750fa700f7ffad44.pdf (1 page, 44622 b\nytes).\nTranscript written on 8ea9fe3c0e02e93deb6a4bca750fa700f7ffad44.log.\n"Radiation boundary condition II
This is a simplified version of radiation boundary condition I, where just cψ = cmax is used in (5). This leads to a more simple radiation boundary condition (here e.g. for a left-right flow along positive x-direction):
TracMath macro processor has detected an error. Please fix the problem before continuing.
The command:
'/usr/bin/pdflatex -interaction=nonstopmode d035fec03d6a3e1e197cfdd4aabd0ba9a5b55af3.tex'failed with the following output:
"This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)\n restricted \\write18 enabled.\nentering extended mode\n(./d035fec03d6a3e1e197cfdd4aabd0ba9a5b55af3.tex\nLaTeX2e <2011/06/27>\nBabel <3.9h> and hyphenation patterns for 78 languages loaded.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/article.cls\nDocument Class: article 2007/10/19 v1.4h Standard LaTeX document class\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/size10.clo))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/inputenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/utf8.def\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/ot1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/omsenc.dfu)))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cmap/cmap.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cm-super/type1ec.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1cmr.fd))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/fontenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.def)<<t1.cmap\n>>) (/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsmath.sty\nFor additional information on amsmath, use the `?' option.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amstext.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsgen.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsbsy.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsopn.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amscls/amsthm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amssymb.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amsfonts.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/tools/bm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/preview.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/prtightpage.def)\n)\nNo file d035fec03d6a3e1e197cfdd4aabd0ba9a5b55af3.aux.\nPreview: Fontsize 10pt\nPreview: PDFoutput 1\n! Missing $ inserted.\n<inserted text> \n $\nl.14 \\psi\n ^{t + \\Delta t}(k,j,nx+1) = \\psi^{t}(k,j,nx) \\; , \\quad (11)\n<<ot1.cmap>><<oml.cmap>><<oms.cmap>><<omx.cmap>>\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsa.fd)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsb.fd)\n! Missing $ inserted.\n<inserted text> \n $\nl.15 \\end{preview}\n \nPreview: Tightpage -32891 -32891 32891 32891\n[1{/localdata/software/texlive/2013/texmf-var/fonts/map/pdftex/updmap/pdftex.ma\np}] (./d035fec03d6a3e1e197cfdd4aabd0ba9a5b55af3.aux) )\n(see the transcript file for additional information)</localdata/software/texliv\ne/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></localdata/softwar\ne/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></localdata/\nsoftware/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></loc\naldata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb\n>\nOutput written on d035fec03d6a3e1e197cfdd4aabd0ba9a5b55af3.pdf (1 page, 36445 b\nytes).\nTranscript written on d035fec03d6a3e1e197cfdd4aabd0ba9a5b55af3.log.\n"with ψ = {u,v,w}. This formulation of the radiation boundary condtions saves computational time compared to the formulation of equations (7) to (10). Although, Orlanski (1976) suggested that this approach of radiation boundary condition leads to reflection for waves smaller than cmax which may occur in complex geophysical flows, our simulations of a convective boundary layer with background wind have been stable so far.
Mass flux correction
PALM offers the possibility of a mass flux correction at the outflow (e.g. Tian, 2004). If parameter conserve_volume_flow is set true, the mass flux at the inflow and outflow is calculated by:
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'/usr/bin/pdflatex -interaction=nonstopmode e7c04c91af9b117c06a57d332937d269f414f2c0.tex'failed with the following output:
"This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)\n restricted \\write18 enabled.\nentering extended mode\n(./e7c04c91af9b117c06a57d332937d269f414f2c0.tex\nLaTeX2e <2011/06/27>\nBabel <3.9h> and hyphenation patterns for 78 languages loaded.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/article.cls\nDocument Class: article 2007/10/19 v1.4h Standard LaTeX document class\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/size10.clo))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/inputenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/utf8.def\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/ot1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/omsenc.dfu)))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cmap/cmap.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cm-super/type1ec.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1cmr.fd))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/fontenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.def)<<t1.cmap\n>>) (/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsmath.sty\nFor additional information on amsmath, use the `?' option.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amstext.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsgen.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsbsy.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsopn.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amscls/amsthm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amssymb.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amsfonts.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/tools/bm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/preview.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/prtightpage.def)\n)\nNo file e7c04c91af9b117c06a57d332937d269f414f2c0.aux.\nPreview: Fontsize 10pt\nPreview: PDFoutput 1\n\n! Package amsmath Error: \\dot allowed only in math mode.\n\nSee the amsmath package documentation for explanation.\nType H <return> for immediate help.\n ... \n \nl.14 \\dot{m}\n = \\sum_{k=1}^{nz-1} \\Delta z(k) \\sum_{l=0}^{nx_i} \\psi(l,k) \\De...\n\n! Missing $ inserted.\n<inserted text> \n $\nl.14 \\dot{m} = \\sum\n _{k=1}^{nz-1} \\Delta z(k) \\sum_{l=0}^{nx_i} \\psi(l,k) \\De...\n\n<<ot1.cmap>><<oml.cmap>><<oms.cmap>><<omx.cmap>>\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsa.fd)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsb.fd)\n! Missing $ inserted.\n<inserted text> \n $\nl.15 \\end{preview}\n \nPreview: Tightpage -32891 -32891 32891 32891\n[1{/localdata/software/texlive/2013/texmf-var/fonts/map/pdftex/updmap/pdftex.ma\np}] (./e7c04c91af9b117c06a57d332937d269f414f2c0.aux) )\n(see the transcript file for additional information){/localdata/software/texliv\ne/2013/texmf-dist/fonts/enc/dvips/cm-super/cm-super-t1.enc}</localdata/software\n/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></localdata/\nsoftware/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></lo\ncaldata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.p\nfb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/\ncmmi7.pfb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfo\nnts/cm/cmr10.pfb></localdata/software/texlive/2013/texmf-dist/fonts/type1/publi\nc/amsfonts/cm/cmr7.pfb></localdata/software/texlive/2013/texmf-dist/fonts/type1\n/public/amsfonts/cm/cmsy7.pfb></localdata/software/texlive/2013/texmf-dist/font\ns/type1/public/cm-super/sfrm1000.pfb>\nOutput written on e7c04c91af9b117c06a57d332937d269f414f2c0.pdf (1 page, 67020 b\nytes).\nTranscript written on e7c04c91af9b117c06a57d332937d269f414f2c0.log.\n"where Δxi and ψ is equal to Δy (Δx) and u (v) in case of bc_lr (bc_ns). The correction factor for the outflow velocity, which is necessary due to different mass fluxes at inflow and outflow, can be calculated by
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The command:
'/usr/bin/pdflatex -interaction=nonstopmode a31fe11b300e958f787b015f2bb5ae9ff8254f88.tex'failed with the following output:
"This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)\n restricted \\write18 enabled.\nentering extended mode\n(./a31fe11b300e958f787b015f2bb5ae9ff8254f88.tex\nLaTeX2e <2011/06/27>\nBabel <3.9h> and hyphenation patterns for 78 languages loaded.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/article.cls\nDocument Class: article 2007/10/19 v1.4h Standard LaTeX document class\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/size10.clo))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/inputenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/utf8.def\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/ot1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/omsenc.dfu)))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cmap/cmap.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cm-super/type1ec.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1cmr.fd))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/fontenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.def)<<t1.cmap\n>>) (/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsmath.sty\nFor additional information on amsmath, use the `?' option.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amstext.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsgen.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsbsy.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsopn.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amscls/amsthm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amssymb.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amsfonts.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/tools/bm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/preview.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/prtightpage.def)\n)\nNo file a31fe11b300e958f787b015f2bb5ae9ff8254f88.aux.\nPreview: Fontsize 10pt\nPreview: PDFoutput 1\n! Missing $ inserted.\n<inserted text> \n $\nl.14 \\psi\n _{corr} = \\frac{\\dot{m}_{inflow} - \\dot{m}_{outflow}}{A} \\; ,\n<<ot1.cmap>><<oml.cmap>><<oms.cmap>><<omx.cmap>>\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsa.fd)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsb.fd)\n! Missing $ inserted.\n<inserted text> \n $\nl.15 \\end{preview}\n \nPreview: Tightpage -32891 -32891 32891 32891\n[1{/localdata/software/texlive/2013/texmf-var/fonts/map/pdftex/updmap/pdftex.ma\np}] (./a31fe11b300e958f787b015f2bb5ae9ff8254f88.aux) )\n(see the transcript file for additional information)</localdata/software/texliv\ne/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></localdata/softwar\ne/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb></localdata/\nsoftware/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></loc\naldata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pf\nb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/c\nmr7.pfb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfont\ns/cm/cmsy7.pfb>\nOutput written on a31fe11b300e958f787b015f2bb5ae9ff8254f88.pdf (1 page, 51464 b\nytes).\nTranscript written on a31fe11b300e958f787b015f2bb5ae9ff8254f88.log.\n"where A is the area of the boundary
TracMath macro processor has detected an error. Please fix the problem before continuing.
The command:
'/usr/bin/pdflatex -interaction=nonstopmode 6503a73189ad01e9393abec1b20988537b55b594.tex'failed with the following output:
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References
- Davies, H. C., 1976: A lateral boundary formulation for multi-level prediction models. Quart. J. R. Met. Soc., 102, 405-418.
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