Background
Sea-breeze Theory
Later sections of this paper discuss in some detail clear-air echoes associated with the local sea breeze circulation. To aid the understanding of these sections a brief introduction to sea breeze theory is provided below.
In coastal regions, primarily in spring and summer there exists a phenomenon known as the sea breeze. This coastal air-circulation system brings fresh air from the sea in the afternoon. (Hsu, 1988)
This air-circulation system is composed of opposing flows at two different levels of the atmosphere. The onshore component is located in the lower levels of the atmosphere near the surface. This onshore flow is the sea breeze. The offshore return flow occurs higher up in the atmosphere around the 700-800 millibar level. To complete this circulation system, there is rising motion on the landward side of the cell, coupled with sinking motion on the seaward side over the ocean. Figure 1 provides a simplified illustration of a sea-breeze circulation.
The driving force of the sea breeze circulation is the difference between the air temperatures over land and sea. In the morning, the temperatures over land are cooler than those over the sea. At this time, a land breeze at the surface may exist. As the temperatures over the land rise to levels equal to those over the sea, any land breeze that existed along the coast diminishes.
On sunny days the sea surface temperature varies very little but the land warms quickly. As the land warms, so does the air close to the surface. Convection currents of air distribute heat through several thousand meters above the ground. A stable layer limits the upward penetration of this convection. As the air is warmed it expands. Since it cannot expand into the stable layer, sideways expansion produces changes in pressure which are transmitted sideways at the speed of sound (Simpson, 1994). This results in the pressure distribution illustrated in Figure 1. It is this pressure pattern which results in onshore flow at the surface.
In addition, the pressure field combined with the contrast in temperature along the coastal zone creates a baroclinic field over the coastal zone (baroclinicity exists when isobars and isotherms cross each other). The thermal wind equation indicates that in the presence of a baroclinic field, there exists the potential for rising air. This helps to maintain the circulation. A weaker return flow aloft is necessary to balance the system. As the warm land air aloft converges with the cooler sea air, it cools and sinks thus completing the circulation.
The onshore flow of this cell is known as the sea breeze. The head of the sea breeze is a cold frontal type, which is generally called a sea breeze front (see Figure 2). Its penetration can be observed by abrupt changes in air temperature and wind direction when the front passes an observation point. (Chiba, 1992) The general sequence of changes that occur at the surface upon passage of the front, are as follows: a temperature drop of 5°F or greater depending on the magnitude of the land/sea temperature difference; relative humidity first drops about 7% and then rises about 14%; and most noticeably, surface wind rotates clockwise from west to east with a total direction change of up to 180° in one hour (Hsu, 1988).
To interpret radar echoes from a sea breeze front, it is useful to consider the structure of the front.
It has been suggested (Simpson, 1994) that the sea breeze front can be represented as a gravity (density) current. The density changes are largely due to temperature differences, with 3°C representing a density change of roughly 1%. As the denser maritime air meets the warmer land air, a sharp leading edge is formed which is about twice the height of the main circulation. This is illustrated in Figure 3. Kelvin-Helmholtz billows may form due to the interaction of the two fluids of different density moving relative to one another.
Figure 4. Formation of clefts and lobes at the sea breeze front. Some of the warmer air is overrun at low levels and is ingested into the cleft at the center (after Simpson, 1994).
As with turbulent mixing at the interface, the frontal structure is affected by the frictional effect of the ground and the air mass’s inertia. These effects lead to cold air overrunning the warm air at low levels up to about 100 meters. The resulting instability leads to the formation of lobes and clefts by the process shown in Figure 4.
These lobes and clefts are responsible for the scalloped characteristics of fronts on the Plan Position Indicator (PPI) radar plot.
Weather Radar Background
Radar is an acronym standing for radio detection and ranging. Weather radar works using exactly the same principles used in radar detection of other objects like aircraft. That is, detection of electromagnetic radiation transmitted from an antenna which is scattered by objects. Typically radar waves are emitted as pulses of electromagnetic energy of about 1m s duration. The radar then "listens" for any return signal for a period of about 1ms. Because the transmitted pulse is of significantly greater power than the return, the transmitter must be off when the receiver is on and vice versa.
The distance of a target from the radar is determined by the time interval between emission and receipt of the return signal.
The magnitude of the backscattered signal is heavily dependent on the size of the target, the wavelength used and the target’s refractive index. Obviously, the number of targets within the scanned volume is also a factor.
It is not within the scope of this paper to provide a detailed discussion of how radar equations are used to interpret the power of the backscattered signal. However, below is a brief summary of some of the basic principles.
The general approximate form of the radar equation for a single scatterer is:
(1)
where Pr is the received power, Pt is the transmitted power, G is the antenna gain, l the wavelength, r the distance to the scatterer and s i is the radar cross section of the target.
The total return from a pulse is the summation of Pr for all targets over the volume defined by the vertical and horizontal beam width and the pulse length of the radar. Considering this, it can be shown that the average power scattered back to the antenna from a beam with a horizontal beam width of q radians, vertical beam width of f radians and a pulse length (determined by multiplying pulse duration by the speed of light) in meters is:
(2)
The derivation for the above equation is relatively straightforward. However, it should be noted that a number of simplifications (in particular in the way gain is represented) are made which result in this equation over estimating the power backscattered to the radar. Until 1962, this over estimate was compensated for by an empirical correction factor equal to (2ln2)-1. Probert-Jones then derived theoretically a radar equation which incorporated this same factor.
From Mie theory we can show that the radar cross section of a spherical drop is:
(3)
where a is the drop radius and a =2p a/l . The coefficients an and bn describe the effects of magnetic and electric dipoles, (and quadrupoles etc.) respectively in the scattering (Battan, 1973). They can be expressed in terms of spherical Bessel and Hankel functions.
In cases where D < 0.1l , the expansion of this equation simplifies to the Rayleigh approximation viz.:
(4)
where m is the complex index of refraction, D is the target diameter and K=(m2-1)/(m2+2).
Applying the correction factor to equation 2 as discussed above, substituting equation 4 for s i combining all the system constants into C yields:
(5)
where Z is the Radar Reflectivity Factor. This equation is valid within
the Rayleigh scattering range and can be used to determine Z, given measurements
of 
and r. The more familiar
unit dBZ is simply 10logZ, where Z is measured in mm6m-3.
It is more convenient since backscatter is directly proportional to the
dBZ value. Usually the type of scattering that is occurring cannot be determined
so the quantity effective radar reflectivity factor, Ze
is used.
For weather radar, l is selected such that the radar is sensitive to precipitation-size particles and so that these particles for the most part fall within the Rayleigh scattering range. The actual wavelength used is between about 0.8 cm to 10 cm. When detecting precipitation within this range, any discrepancy between the Rayleigh value of s and the actual value will only lead to an error of about 1.5db, except for the case of large hail (Battan, 1973).
The different wavelengths at which radar operates are divided into a number of bands. Table 1 shows details this. Typically operational weather radar use the S or C bands. For example, the wavelength used in the WSR-88D is 10cm, which is in the S-band.
|
Band |
Frequency (MHz) |
Wavelength (cm) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Table 1. Adapted from Radar Observation of the Atmosphere, Battan (1973).
As the wavelength becomes shorter, the radar becomes more sensitive to smaller particles, as the relative fraction of the power returned for these particles increases. However, scattering enters the Mie region for larger particles and interpretation of data becomes much more complex. Thus, the band used must be carefully selected to provide best benefit for the given application.
In addition to measuring reflectivity, Doppler radars are also able
to determine the radial velocity at which the target is moving by analyzing
the Doppler shift in frequency between the transmitted and returned signal.
Note, that the actual velocity of the target can be determined very accurately
by using two radars with differently oriented beams from which the
X and Y components of velocity can then be calculated. It can also be estimated
by using the Velocity Azimuth Display (VAD) algorithm (Heiss and McGrew,.1990).
If the target is moving radially towards the radar at a speed v, then the shifted frequency of the signal is:
(6)
and hence the change in frequency is 
.
The same process occurs on the return trip, so 
.
This shift in frequency is easily measured and thus the radial velocity
of the targets determined.
With this basic framework in place, it becomes possible to see how returns from weather radars can be used to analyze the composition of a volume of the atmosphere. However, this analysis is none trivial. In order to perform it the refractive index and size distribution of each type of scatterer must be known. Determination of these variables is often a difficult task.
Weather radars are most commonly used to detect precipitation. In this case it possible to determine the type of scatterers and their size distribution from atmospheric parameters observed or derived by other methods. It is important to determine the type of precipitation since rain, snow and melting ice all have markedly different indices of refractive index. For example, liquid water droplets have a refractive index of 8, while ice crystals have a refractive index of only 1.8. This is because the absorption and subsequent re-emission of rotational energy is much larger for liquid water than for ice. This affects the complex refractive index, m, in equation 4. Melting ice produces the largest radar return due to the combination of large refractive index of the liquid component and the large size of the ice crystal. The size distribution of the precipitation will also vary depending on the meteorological conditions. Both these factors lead to large differences in reflectivity. The radar meteorologist must be aware of these differences if he is to accurately interpret the returned signal.
While the primary function of operational weather radar is to detect precipitation, it has been found that other airborne objects and atmospheric phenomena also produce measurable return signals at the frequencies used by weather radar. Some of these returns can be a nuisance. Indeed many algorithms exist to filter out such returns. However, it has become increasing apparent that weather radars are extremely useful in studying much more than just precipitation.
When analyzing clear-air echoes the task of interpreting the radar data is even more complex than for precipitation echoes. This is due to the fact that the size distributions of the scatterers and even the type of scatterer itself, and hence its refractive index, are often not known. This has made clear air echoes very difficult to analyze quantitatively.