We consider only subsonic outflow and inflow situations. To calculated the derivatives along the characteristic directions, , 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
For both subsonic inflow and outflow boundaries at least one characteristic is incoming, corresponding to at the bottom and 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.
A physical boundary condition on inflowing gas is that its ENTROPY IS SPECIFIED. The characteristic equation for the entropy corresponds to ,
The entropy of inflowing material will remain fixed if we set . 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 .
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 is the time constant. Either or is an incoming wave, but the other is an outgoing wave and must be calculated using one-sided derivatives, not determined externally. 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.
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 gives
Thus, to maintain hydrostatic equilibrium, the changes in pressure and density must be related by
where is the pressure scale height.
The way these conditions are used is that the values determined for 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.