A steady-flow system on an
f-plane whose velocity is uniform in one horizontal direction, say
y direction, is considered. The velocity far upstream in the other horizontal direction, say minus
x direction, is assumed to be horizontal and dependent only on the vertical coordinate, say
z. The potential temperature, Bernoulli function or potential vorticity is not necessarily conserved along the stream lines projected onto the 2-dimensional (
z, x) vertical plane, although they are conserved along the true stream lines in the 3-dimensional (
z, x, y) space. In this note, a temperature-like quantity Θ, an energy-like quantity
B and a vorticity-like quantity
Q, which are conserved along the projected stream lines on the (
z, x) vertical plane, are constructed. They become functions of the 2-dimensional stream function ψ=ψ(
z, x). As in the case of the true Bernoulli function and potential vorticity,
Q=
Q[ψ] is the derivative of
B=
B[ψ],
i. e.,
Q[ψ]=
dB[ψ]/
dψ.
As an application of the conservation laws, a problem of hydrostatic neutral flow with a horizontal rigid lid over a mountain is exactly solved. In particular, both the
z and
x components of velocity are shown not to be affected by the rotation.
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