Before a plume can be modelled, the air it is going into has to be described. Two profiles do almost all of that work: how the temperature falls with height, and how the pressure does.

The environmental lapse rate is $\Gamma = \dfrac{T_1 - T_2}{\Delta z}$ — read aloud gamma equals T-one minus T-two, over delta-z. Say the subscripts out loud once and they will never trouble you again: subscript 1 is the LOWER reading and subscript 2 the upper. $T_1$ and $T_2$ are the two air temperatures, $\Delta z$ is the height between them in metres, and $\Gamma$ is the answer, quoted by long convention in °C per kilometre — so a height in metres owes a ×1000 on the way out. Only the DIFFERENCE of the temperatures enters, which means a Celsius interval and a kelvin interval are the same number and the 273.15 cancels itself.

The sign convention is the standing trap of this lesson, and it is worth stating as a rule: lower minus upper, so an atmosphere that cools with height comes out POSITIVE. The dry adiabatic rate is 9.8 °C/km and it is the yardstick — above it the air is superadiabatic and unstable, plumes loop and briefly touch down; below it, stable; below zero it is an inversion, and a plume caught under one goes nowhere at all.

The barometric equation for an isothermal layer is $P = P_0\,e^{-Mgz/RT}$ — P equals P-nought e to the minus M g z over R T. $P_0$ is the pressure at the reference level, $P$ the pressure a height $z$ above it, $M$ the molar mass of dry air (0.0290 kg/mol), $g$ gravity, $R$ the gas constant (8.314 J/mol·K) and $T$ the layer's absolute temperature in kelvin. That last word is not a formality: this is a gas law with a hill in it, and Celsius breaks it outright.

Read the shape of it. $RT/Mg$ is the scale height, about 8.4 km, the height over which pressure falls by a factor of e. And know the assumption you just made: the real atmosphere is not isothermal, so at 1000 m this form gives 90.0 kPa where the true standard atmosphere gives 89.9. That 0.1 kPa is the assumption, not an arithmetic slip — and saying which is which out loud is most of what separates an engineer from a calculator.