1.11: Process Terminology

Processes associated with constant temperature are isothermal. For example, eqs. (1.9a) and (1.13a) apply for an isothermal atmosphere. Those occurring with constant pressure are isobaric. A line on a weather map connecting points of equal temperature is called an isotherm, while one connecting points of equal pressure is an isobar. Table 1-6 summarizes many of the process terms.

Table 1-6.Process names. (tendency = change with time)

Name

Constant or equal

adiabat contour isallobar isallohypse isallotherm isanabat isanomal isentrope isobar isobath isobathytherm isoceraunic isochrone isodop isodrosotherm isoecho isogon isogram isohel isohume isohyet isohypse isoline isoneph isopleth isopycnic isoshear isostere isotach isotherm

entropy (no heat exchange) height pressure tendency height tendency temperature tendency vertical wind speed weather anomaly entropy or potential temp. pressure water depth depth of constant temperature thunderstorm activity or freq. time (Doppler) radial wind speed dew-point temperature radar reflectivity intensity wind direction (generic, for any quantity) sunshine humidity precipitation accumulation height (similar to contour) (generic, for any quantity) cloudiness (generic, for any quantity) density wind shear specific volume (1/ρ) speed temperature

Sample Application

Name the process for constant density.

Find the Answer

From Table 1-6: It is an isopycnal process.

Exposition: Isopycnics are used in oceanography, where both temperature and salinity affect density.

HIGHER MATH • Hypsometric Eq.

To derive eq. (1.26) from the ideal gas law and the hydrostatic equation, one must use calculus. It cannot be done using algebra alone. However, once the equation is derived, the answer is in algebraic form.

The derivation is shown here only to illustrate the need for calculus. Derivations will NOT be given for most of the other equations in this book. Students can take advanced meteorology courses, or read advanced textbooks, to find such derivations.

Derivation of the hypsometric equation:

Given: the hydrostatic eq:

\(\ \begin{align}\frac{dP}{dz}=-\rho\cdot |g|\tag{1.25c}\end{align}\)

and the ideal gas law:

\(\ \begin{align}P=\rho \cdot \Re_{d} \cdot T_{v}\tag{1.23}\end{align}\)

First, rearrange eq. (1.23) to solve for density:

\(\rho=P /\left(\Re_{d} \cdot T_{v}\right)\)

Then substitute this into (1.25c):

\(\frac{\mathrm{d} P}{\mathrm{d} z}=-\frac{P \cdot|g|}{\mathfrak{R}_{d} \cdot T_{v}}\)

One trick for integrating equations is to separate variables. Move all the pressure factors to one side, and all height factors to the other. Therefore, multiply both sides of the above equation by dz, and divide both sides by P.

\(\frac{\mathrm{d} P}{P}=-\frac{|g|}{\Re_{d} \cdot T_{v}} \mathrm{d} z\)

Compared to the other variables, g and ℜd are relatively constant, so we will assume that they are constant and separate them from the other variables. However, usually temperature varies with height: T(z). Thus:

\(\frac{\mathrm{d} P}{P}=-\frac{|g|}{\Re_{d}} \cdot \frac{\mathrm{d} z}{T_{v}(z)}\)

Next, integrate the whole eq. from some lower altitude z₁ where the pressure is P₁, to some higher altitude z₂ where the pressure is P₂:

\(\int_{P_{1}}^{P_{2}} \frac{\mathrm{d} P}{P}=-\frac{|g|}{\Re_{d}} \cdot \int_{z_{1}}^{z_{2}} \frac{\mathrm{d} z}{T_{v}(z)}\)

where |g|/ℜd is pulled out of the integral on the RHS because it is constant.

The left side of that equation integrates to become a natural logarithm (consult tables of integrals).

The right side of that equation is more difficult, because we don’t know the functional form of the vertical temperature profile. On any given day, the profile has a complex shape that is not conveniently described by an equation that can be integrated.

Instead, we will invoke the mean-value theorem of calculus to bring T_v out of the integral. The overbar denotes an average (over height, in this context).

That leaves only dz on the right side. After integrating, we get:

\(\left.\ln (P)\right|_{P_{1}} ^{P_{2}}=-\left.\frac{|g|}{\Re_{d}} \cdot \overline{\left(\frac{1}{T_{v}}\right)} \cdot z\right|_{z_{1}} ^{z_{2}}\)

Plugging in the upper and lower limits gives:

\(\ln \left(P_{2}\right)-\ln \left(P_{1}\right)=-\frac{|g|}{\Re_{d}} \cdot \overline{\left(\frac{1}{T_{v}}\right)} \cdot\left(z_{2}-z_{1}\right)\)

But the difference between two logarithms can be written as the ln of the ratio of their arguments:

\(\ln \left(\frac{P_{2}}{P_{1}}\right)=-\frac{|g|}{\Re_{d}} \cdot \overline{\left(\frac{1}{T_{v}}\right)} \cdot\left(z_{2}-z_{1}\right)\)

Recalling that ln(x) = –ln(1/x), then:

\(\ln \left(\frac{P_{1}}{P_{2}}\right)=\frac{|g|}{\Re_{d}} \cdot \overline{\left(\frac{1}{T_{v}}\right)} \cdot\left(z_{2}-z_{1}\right)\)

Rearranging and approximating \(\overline{1 / T_{v}} \approx 1 / \overline{T_{v}}\) (which is NOT an identity), then one finally gets the hypsometric eq:

\(\ \begin{align} \left(z_{2}-z_{1}\right) \approx \frac{\Re_{d}}{|g|} \cdot \overline{T_{v}} \cdot \ln \left(\frac{P_{1}}{P_{2}}\right)\tag{1.26}\end{align}\)