Mathematics

Not fully fleshed out, more details to be added.

Pressure Thickness

Pressure Thickness is the measurement of the distance (in metres) between any two constant pressure surfaces.

Pressure Thickness Definition (provided by the NWS of the USA)

One of the most common thickness charts used in meteorology is the 1000-500 hPa thickness, and for the purposes of this project, it will be the sole interest. This is the distance between the elevation of the 1,000 hPa and 500 hPa levels. Typically, the 1,000 hPa surface is used to represent sea level but this is just a generalisation. On pressure charts, the last digit (zero) of a thickness value is typically truncated. So, a 1000-500 thickness value of 570 means the distance between the two surfaces is 5,700 metres. The 1000-500 hPa thickness value of 540 is traditionally used to determine rain versus snow. If precipitation is predicted poleward of this 540-thickness line (if the thickness value is less than 540), it is expected that it will be snow. If precipitation is predicted on the equator side of this line (if the thickness value is greater than 540), then it is expected that the precipitation will be in a liquid form. The reason one is able to make such an expectation is due to the fact that the 540-thickness line closely follows the surface freezing temperature of 273 K [17].

Rain/Snow Line (provided by the NWS of the USA)

To determine the pressure thickness between two constant pressure surfaces, the Hypsometric equation is utilised and as shown in equation (1).

The hypsometric equation relates an atmospheric pressure ratio to the equivalent thickness of an atmospheric layer considering the layer mean of virtual temperature, gravity, and occasionally wind. It is derived from the hydrostatic equation and the ideal gas law.

(1)\[h = \Phi_2 - \Phi_1 = \frac{R \bar{T_v}}{g} \ln{\frac{p_1}{p_2}}\]

Precipitable Water

Precipitable Water is the total atmospheric water vapour contained in a vertical column of unit cross-sectional area extending between any two specified pressure levels.

Based on the definition, precipitable water can be described mathematically as being:

(2)\[W = \int_{0}^{z} \rho_v dz\]

where \(\rho_v\) is the density of water vapour, and where \(\rho_v\) is defined as:

\[\rho_v = \frac{\texttt{mass of vapour}}{\texttt{unit volume}}\]

Following which, the hydrostatic equation can be applied to equation (2) in order to replace \(dz\) with \(dp\). The reason for doing this is that atmospheric pressure is extremely easier to measure, with devices such as weather balloons being readily available.

(3)\[W = -\int_{p_1}^{p_2} \frac{\rho_v}{\rho g} dp\]

Where \(p_1\) and \(p_2\) are constant pressure surfaces, and where \(p_1 > p_2\). Substituting in the definition of density, \(\rho_v = \frac{m_v}{V}; \rho = \frac{m_{air}}{V}\), into equation (3) results in:

\[W = -\int_{p_1}^{p_2} \frac{1}{g} \frac{m_v V}{m_{air} V} dp\]
\[\Rightarrow W = - \frac{1}{g} \int_{p_1}^{p_2} \frac{m_v}{m_{air}} dp\]

The integration term in this particular equation is the definition for the specific humidity, with the units of measurement being \(\frac{kg}{kg}\). The specific humidity can be approximated by the mixing ratio, with an error typically around 4 % [12].

Mixing Ratio is the ratio of the mass of a variable atmospheric constituent to the mass of dry air.

(4)\[\therefore W = -\frac{1}{g} \int_{p_1}^{p_2} m dp\]

The units as given by the equation are \(\frac{kg}{m^2}\) (dimensionless), but, the preferred unit of measurement for rainfall is \(mm\). The conversion between the two units of measurements is one to one (\(1 \frac{kg}{m^2} = 1 mm\)) In actual rainstorms, particularly thunderstorms, amounts of rain very often exceed the total precipitable water of the overlying atmosphere. This results from the action of convergence that brings into the rainstorm the water vapour from a surrounding area that is often quite large. Nevertheless, there is general correlation between precipitation amounts in given storms and the precipitable water of the air masses involved in those storms [18].

For the purposes of numerically calculating the precipitable water for a given column of air, equation :eq: pwv_derive_fin is commonly rewritten as the following:

(5)\[W = -\frac{1}{\rho g} \int_{p_1}^{p_2} \frac{0.622 e}{p - e} dp\]

Vapour Pressure is the pressure exerted by a vapour when the vapour is in equilibrium with the liquid or solid form, or both, of the same substance. In meteorology, vapour pressure is used almost exclusively to denote the partial pressure of water vapour in the atmosphere.

Considering vapor pressure is not one of the three prognostic variables defined in section, it is therefore necessary to express vapor pressure in terms of the already defined variables. This can be done through the utilisation of temperature and relative humidity. First, one must determine the saturated vapour pressure. This can be done using the following equation [2]:

\[e_s = 6.112 \exp(\frac{17.67 T}{T + 243.5})\]

After which, the vapor pressure can be calculated as follows:

(6)\[e = \frac{e_{s} r}{100}\]

In wet periods, the precipitable water is particularly close to the saturated precipitable water. In situations when the precipitable water is close to the saturated precipitable water, the precipitable water changes very little over the day. Saturated precipitable water also makes calculations a whole lot simpler, as specific humidity data is rather difficult to come by.

In regards to the numerical calculation of the precipitable water, the SciPy method, scipy.integrate.quad, is utilised in order to determine the definite integral in equation :eq: pwv. This method integrates the function using a technique from the Fortran library, QUADPACK [16].

Virtual Temperature

The virtual temperature is the temperature at which dry air would have the same density as the moist air, at a given pressure. In other words, two air samples with the same virtual temperature have the same density, regardless of their actual temperature or relative humidity. Because water vapor is less dense than dry air and warm air is less dense than cool air, the virtual temperature is always greater than or equal to the actual temperature.

In relation to this project, the virtual temperature will be extremely useful as it will allow for the calculation of the evolution of relative humidity through the utilisation of the thermodynamic advection equation. This is due to the fact that virtual temperature can be expressed in terms of vapor pressure, which can be converted into relative humidity. This can be seen as follows:

\[T_v = \frac{T}{1 - \frac{0.378 e}{p}}\]

The equation becomes the following, by substituting equation (6) in and rearranging for relative humidity:

\[r = \frac{100 (p T_v - p T)}{0.378 e_s T_v}\]

Primitive Equations

To be added.