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Next: Bottom Boundary Conditions I: Up: Simple case: Perfect Gas Previous: Boundary Equations

Boundary Conditions

We consider only subsonic outflow and inflow situations. To calculated the derivatives along the characteristic directions, tex2html_wrap_inline2318 , we make the local 1D approximation of keeping oly the derivatives perpendicular to the boundary and gravity, which is called the local 1D inviscid (LODI) approximation. The local 1D inviscid characteristic equations are


The physical boundary conditions are:
1. Minimal reflected waves
2. Entropy of inflowing material given
3. Net mass flux tends toward zero

  1. No Reflected Waves

    For both subsonic inflow and outflow boundaries at least one characteristic is incoming, corresponding to tex2html_wrap_inline2300 at the bottom and tex2html_wrap_inline2308 at the top (assuming z increases downward). The physical boundary condition is NO REFLECTED WAVES. This can be achieved by requiring that the amplitude of incoming waves is constant in time (zero). The characteristic equation for incoming waves is


    at the bottom and top respectively. Thus the appropriate boundary condition for no reflected adiabatic waves is


    At outflow boundaries this is the only condition that can be specified. At inflow boundaries there are three other conditions that can be specified on the entropy and horizontal velocities. These conditions will not be perfectly reflecting, because the waves are actually not normally incident.

    Physically, it is not possible to have no reflections if the boundary state is to be kept from drifting. Some (small) reflections are required to propagate inwards the information on deviations from the reference state.

  2. Given Entropy

    A physical boundary condition on inflowing gas is that its ENTROPY IS SPECIFIED. The characteristic equation for the entropy corresponds to tex2html_wrap_inline2304 ,


    The entropy of inflowing material will remain fixed if we set tex2html_wrap_inline2364 . Thus the appropriate boundary condition for constant entropy is


    If it is necessary to alter the entropy of inflowing material, then a non-zero value can be specified for tex2html_wrap_inline2366 .

  3. Given Mass Flux

    We wish to also specify the physical boundary condition that there is NO NET MASS FLUX. However, we don't want an instantaneous response to a plume hitting the boundary, so we should adjust the mass flux with a long time constant. First calculate the net mass flux


    The required average change in the mass flux is the net mass flux divided by the number of inflow locations,




    where tex2html_wrap_inline2368 is the time constant. Either tex2html_wrap_inline2370 or tex2html_wrap_inline2372 is an incoming wave, but the other is an outgoing wave and must be calculated using one-sided derivatives, not determined externally. tex2html_wrap_inline2366 is determined if the entropy of incoming waves is specified. Hence, it is not possible to impose three conditions specifying the entropy, no reflections and the mass flux simultaneously.

  4. Hydrostatic Equilibrium

    We want all changes in pressure and density to be consistent with maintaining hydrostatic equilibrium,


    To first order we can write this as


    Perturbing P and tex2html_wrap_inline2378 gives




    Thus, to maintain hydrostatic equilibrium, the changes in pressure and density must be related by


    where tex2html_wrap_inline2380 is the pressure scale height.

The way these conditions are used is that the values determined for tex2html_wrap_inline2382 are substituted into the full boundary equations (49-52 or 53-56), including the horizontal terms and gravity, which give the time derivatives of the variables on the boundary. If some variable (e.g. entropy) is given, then the corresponding equation (e.g. energy conservation) is dropped from the boundary conditions.

next up previous
Next: Bottom Boundary Conditions I: Up: Simple case: Perfect Gas Previous: Boundary Equations

Viggo Hansteen
Wed Sep 19 18:16:24 MET DST 2001