# pressure bounds derivations

1. pressure ranges from midpoints

symbol

description

unit

variable name

$$p(i)$$

pressure

$$Pa$$

pressure {:,vertical}

$$p^{B}(i,l)$$

pressure boundaries ($$l \in \{1,2\}$$)

$$Pa$$

pressure_bounds {:,vertical,2}

The pattern : for the dimensions can represent {time}, or no dimension at all.

\begin{eqnarray} p^{B}(1,1) & = & e^{\frac{3\ln(p(1)) - \ln(p(2))}{2}} \\ p^{B}(i,1) & = & e^{\frac{\ln(p(i-1)) + \ln(p(i))}{2}}, 1 < i \leq N \\ p^{B}(i,2) & = & p^{B}(i+1,1), 1 \leq i < N \\ p^{B}(N,2) & = & e^{\frac{3\ln(p(N)) - \ln(p(N-1))}{2}} \end{eqnarray}

This formula applies if the harp option regrid_out_of_bounds is set to nan or to extrapolate. If the option is set to edge then the first and last boundary value are set to the midpoints ($$p^{B}(1,1) = p(1)$$, $$p^{B}(N,2) = p(N)$$).