Assessing the value of frost forecasts involves complicated decision analysis, which uses conditional probabilities and economics. Accurate frost forecasting can potentially reduce frost damage because it provides growers with the opportunity to prepare for frost events. This chapter presents and discusses the value of frost forecasting, some frost forecasting models currently in use, and a simple model for on-farm prediction of minimum temperature during a radiation frost.
While decision analysis is used in many fields, applications to frost forecasting are limited. Papers by Banquet, Halter and Conklin (1976) and Katz, Murphy and Winkler (1982) have discussed using decision analysis to evaluate the cost-effectiveness of frost forecasts. Katz, Murphy and Winkler (1982) thoroughly investigated the value of frost forecasting in the Yakima Valley in Washington State, USA. This valley is well known for production of apples and to a lesser extent pears and peaches. The valley is also noted for a problem with frequent freezing during bud break, flowering and small-fruit stages of these crops. The authors used Markov decision processes in a model that structures the problem into identifying possible actions, events and consequences. Crop sensitivity to freezing changes during bud break, bloom and small-fruit stages, so logistic functions that relate crop loss to minimum temperature were derived for each developmental period where the relationship between damage and temperature was known. Then the utility of the frost forecast was evaluated by calculating the conditional standard deviation in minimum forecast using only climate data, current forecasts from the USA National Weather Service and a perfect forecast where the minimum temperature prediction is always correct. The standard deviation is "conditional" because it is based on an assumed level of forecast accuracy.
Based on climate data alone, Katz, Murphy and Winkler (1982) estimated that the conditional standard deviation of the minimum forecast would be 3.6 °C. By definition, the standard deviation is 0 °C for a perfect forecast. Based on forecaster skill in the 1970s, the "current" forecast conditional standard deviation was about 2.1 °C. Therefore, the National Weather Service forecast skill had improved the conditional standard deviation by 48 percent [i.e.] of the difference between using climate data and a perfect forecast. The relative values (i.e. economic value of the forecast divided by the total value of production), expressed in percentages, are shown in Table 5.1. The economic value of the forecast is the additional net value of production resulting from having the forecast. The table shows that increasing the current forecast skill to a perfect forecast would increase the relative values by an additional 18 percent, 15 percent and 23 percent for apples, pears and peaches. Therefore, except for peaches, the economic benefits from further improvements in forecast skill are smaller than comparable past improvements.
TABLE 5. 1
Relative value (percent of total production) and total value of production ($ per hectare) for apples cv. Red Delicious, pears cv. Bartlett and peaches cv. Elberta in the Yakima Valley of Washington State (USA) using climatology, 1970s' forecasts from the National Weather Service, and a perfect forecast
$ 5 802 ha-1
$ 4 586 ha-1
$ 3 064 ha-1
NOTES: Conditional standard deviations about the true minimum temperature were 3.6 °C for climatology, 2.1 °C for the current forecasts and 0 °C for a perfect forecast (after Katz, Murphy and Winkler, 1982).
Predicting when the temperature falls to a critical value is important for starting active frost protection methods. Starting and stopping protection at the proper temperature is important because it avoids losses resulting from starting too late and it saves energy by reducing the operation time of the various methods. While it is beyond the scope of this book to address minimum temperature forecasting with synoptic or mesoscale models, some guidelines are possible on how to forecast minimum temperature during radiation frost conditions, using local data.
Ideally, one would develop a microscale (i.e. local) temperature forecast model using energy balance calculations. This has been thoroughly reviewed by Kalma et al. (1992). The main conclusion of their review was that "air temperatures cannot be predicted satisfactorily from surface energy balance considerations alone, even if the difference between surface and air temperatures can be specified accurately". They attribute this inability to difficulties with: (1) measuring turbulent sensible heat flux in the range typical of frost nights; (2) accounting for advection; and (3) spatial variations in surface radiation emissivity. Rather than using the energy balance to study the rate of cooling of the ground surface, Kalma et al. (1992) proposed to estimate the rate of cooling of a column of air. However, they recognized that both radiative and turbulent sensible heat fluxes depend on vertical profiles of wind, humidity and temperature, which make the process impractical because of measurement problems.
Kalma et al. (1992) discuss the one-dimensional temperature prediction models of Sutherland (1980) and Cellier (1982, 1993). The Sutherland model uses a surface energy balance equation assuming that latent heat contributions are negligible, a soil heat flux model and a sensible heat flux model for the bottom 9.0 m of the atmosphere. The input variables are temperature at 0, 1.5, 3.0 and 9.0 m, soil temperature at 0.1 and 0.5 m depth, wind speed at 10 m and net radiation. This model was reported to forecast within 3 °C 90 percent of the time and 2 °C 82 percent of the time. Ultimately, the model was combined with a statistical model to improve the forecast over Florida, USA.
The Cellier (1982, 1993) temperature model calculates changes in temperature within eight layers up to a height of about 100 m in the atmosphere and down to 1.0 m in the soil. The input variables include mean soil temperature and wind speed and air temperature at 3.0 m height at the time when net radiation becomes negative, the most expected negative net radiation value and the dew-point temperature when the net radiation is at its most negative value. The model was reported to provide realistic surface energy balance estimates during the night, but improvements were needed in the estimation of soil heat transfer and atmospheric exchange coefficients (Kalma et al., 1992).
A working, deterministic energy balance model to estimate changes in temperature during frost events is preferable; however, no universally applicable model with easily attainable input variables is currently available. Many empirical models for predicting minimum temperature have been reported (Bagdonas, Georg and Gerber, 1978) and some are known to give reasonably accurate forecasts. For example, the equation from Young (1920) has been widely used by the USA National Weather Service (NWS) with considerable success throughout the western USA. However, Young' equation was not used directly but was calibrated for local conditions to account for the time of year and local conditions. These modifications are site specific and they are not widely published. It is unknown whether similar modifications are used to improve the multitude of prediction formulae that are used in various countries (Bagdonas, Georg and Gerber, 1978). Clearly, accounting for time of the year and local conditions should improve minimum temperature forecasting. In fact, Bagdonas, Georg and Gerber (1978) recommended that a forecast model that uses local meteorological factors and site-specific climate data is likely to give the best results. In addition to forecasting the minimum temperature, it is also useful to predict the temperature trend during a frost night. A model based on the original paper by Allen (1957) was used to develop the frost temperature trend model (FTrend.xls) included with this book. Another more complicated model was presented by Krasovitski, Kimmel, and Amir (1996).
For several decades, NWS Service provided fruit-frost forecasts to growers in regions of the USA with high-value crops sensitive to frost. Since the NWS forecasters have more experience forecasting and more and better facilities, they can provide more accurate predictions than a grower can make a day or two in advance of a frost. However, in the late 1980s, these services were dropped from the weather service and growers had to either employ private forecasters or develop their own methods to predict minimum temperatures for their crops.
When the fruit-frost service was operating, weather service meteorologists would forecast for key stations within a region and growers would develop correction factors to predict minimum temperatures in their crops. Generally, the corrections consisted of adding or subtracting a correction to the key station forecast. For example, a grower might subtract 0.5 °C from a key station forecast for a crop located in a low spot. In some cases, growers would use spreadsheets or statistical computer application programs to determine regression equations with the key station minimum temperatures as the independent and minimum temperatures in their crop as the dependent variable.
After the NWS fruit-frost forecasting service ended, large growers and those with serious frost problems would employ private weather forecast services to provide site-specific minimum temperature predictions. In many cases, groups of farmers would cooperate and hire a private forecaster to continue forecasting for the key stations used by the NWS. Then their correction factors could still be used to predict minimum temperatures in their crops. Although using correction factors and key stations for site-specific frost predictions is helpful for two- to three-day planning and management, the direct use of data collected in or near the crop is likely to give better predictions during a particular frost night. A method to develop local forecasts is presented in the next section.
A simple, empirical forecast model (FFST.xls), which can easily be calibrated for local conditions, is included with this book. The model, which is based on the method of Allen (1957), uses historical records of air and dew-point temperature at two hours past sunset and the observed minimum temperature during clear sky, calm, frost nights to develop the regression coefficients needed to accurately predict the minimum temperature during a particular period of the year. Two hours past sunset is the starting time (t0) for the model. This time corresponds to when the net radiation has reached its most negative value (Figure 5.1). Assuming there is little or no cloud cover or fog during the night, the net radiation changes little from time t0 until sunrise the next morning. On a night with intermittent cloud cover or fog or variable wind speed, the model may predict a temperature that is lower than observed. The model may predict too high a minimum temperature if a cold front passes or if there is cold air drainage.
Air temperature at 2.0 m height, net radiation and change in net radiation using 20-minute-interval data collected during a frost night (28 February - 1 March 2002) in a walnut orchard near Ladoga, California (USA)
Key: Tss = Time of sunset. T0 = 2 hours after sunset.
FIGURE 5.2 Sample weather data during an advection frost near Zamora, California (USA), in March 2002. Sunset was at about 1942 h. Dates are given in USA notation (mm/dd/yy)
For use in the FFST.xls application program, select data only from radiation frost nights. Avoid including nights with wind speeds greater than 2.0 m s-1 and nights with cloud cover or fog. For example, Figure 5.2 illustrates the data selection problem. On 6 March, there were rainy windy conditions, which continued until near noon on 7 March. Then the rain stopped, but the wind changed from a south to a west-northwest wind and the wind speed was high until about 2100 h. A sharp drop in dew-point temperature is typical of the passage of a cold front. Sunset occurred at about 1742 h, so the wind speed was high for more than three hours past sunset. Net radiation was not measured at this site, so information on cloud cover is unknown. However, intermittent cloud cover often follows a cold front. At two hours past sunset, the air and dew-point temperatures were 5.1 °C and 0.0 °C and the wind speed had just dropped from 2.6 to 1.6 m s-1. There was still a large drop in temperature after this point, which is not characteristic of a radiation frost. Based on this weather data, the weather conditions on 7-8 March were too windy in the evening and they are not typical of a radiation frost. On 8-9 March, the air and dew-point temperatures were 3.9 °C and 1.4 °C at two hours past sunset and the wind speed dropped earlier in the evening (i.e. near sunset). There was no evidence of cold air advection, so the data from 8-9 March can be input into the FFST.xls application program to determine a forecast model. Note that one can use data from nights when the minimum air temperature does not fall below 0 °C as long as the night had clear skies and calm or little wind.
The FFST.xls application program is written in MS Excel for easy input and for graphic as well as tabular output. For as many as 50 nights, the air and dew-point temperature at two hours past sunset are input along with the observed minimum temperature the following morning. A sample input screen with 10 days of input data is shown in Figure 5.3.
In Figure 5.3, the input data were used to determine a linear regression of the observed minimum (Tn) versus the air temperature at two hours past sunset (T0), and the results are shown in the "Prediction from Temperature (Tp')" column. The output equation for T' = b1 × T0 + a1 is shown at the top of the page. To the left of the equation, the root mean square error (RMSE) is shown. This statistic is similar to a standard deviation in that it is a measure of closeness of the predicted and observed values. In Figure 5.3, the RMSE is 0.65 °C for the formula based on the two hours past sunset temperature only. This implies that one deviation about the 1:1 line is approximately 0.65 °C, and two deviations about the 1:1 line would be about 1.3 °C. Assuming that the variation of the RMSE is about the same as a standard deviation, this means that the variation about the 1:1 line would be less than 1.3 °C about 85 percent of the time. After calculating Tp', the residuals (R1 = Tn - Tp') are calculated and displayed. Then, a linear regression of R1 versus the dew-point temperature (Td) is computed and the predicted residual (R1') values are shown. If Tp' = b1 × T0 + a1 and R1'= b2 × Td + a2, then the forecast minimum temperature is given by: Tp = Tp' + R1' = b1 × T0 + b2 × Td + (a1 + a2). In the Excel program, the output equation Tp = b1To + b2Td + a3, where a3 = a1 + a2, is displayed at the top of the input-calculation table for easy viewing. Again, the RMSE comparing observed and predicted minimum temperatures is shown to the left of the equation. In this particular data set, the RMSE values were nearly identical for both prediction equations, so there was no apparent advantage from including the dew-point temperature to predict the minimum temperature with this data set. However, including the dew-point temperature in the model will typically improve the prediction.
The FFST.xls program also plots the predicted versus observed temperatures for both the temperature only model (Figure 5.4) and for the temperature and dew-point prediction model (not shown).
Sample input and calculations for the FFST.xls application program for predicting minimum temperature (Tp)
The data are from the December 1990 and 1998 frosts in the citrus growing region of Lindcove, California (USA).
Predicted versus observed minimum temperature from the data in Figure 5.3, using only the temperature data from two hours past sunset
In addition to predicting the minimum temperature, it is useful to have the temperature trend during the night to help determine when protection methods should be started and stopped. Knowing temperature trend during the night helps growers to foresee when active methods should be initiated during the night. The FTrend.xls model estimates temperature trends from two hours past sunset until sunrise the following morning. Sunset and sunrise are determined from the input latitude, longitude and date. The program uses an empirical temperature trend model to predict how the temperature will change during the night. This model uses a square root function to predict the air temperature from two hours after sunset (i.e. time t0) until reaching the predicted minimum temperature (Tp) at sunrise (i.e. time tp) the next morning. In addition to the air temperature, the application calculates the change in wet-bulb temperature based on temperature trend and initial dew-point temperature.
The FTrend.xls application contains the worksheets "Title", "Help", "Input", "Plot", "Wet-bulb" and "Forecast". The Title and Help worksheets provide information on the developers and instructions on how to use the program. The Input worksheet is used to input temperature data and to display the results of the trend calculations. The Wet-bulb worksheet is used to calculate the air temperature corresponding to the air and dew-point temperature at a given elevation. It is used to help determine the air temperature to stop sprinklers following a frost night. The Forecast worksheet is used to calculate an estimate of the minimum temperature at sunrise the next morning using an input of the air and dew-point temperatures measured at two hours past sunset. In the following sections, these worksheets and their functions will be discussed.
A forecast of the sunrise temperature is needed for the FTrend.xls application. That forecast can come from a weather forecast service or from the model developed in the FFST.xls program. If a forecast service is used, then the "Forecast" worksheet in the FTrend.xls program is unnecessary. If the minimum temperature forecast comes from the FFST.xls program, then the "Forecast" worksheet in FTrend.xls is used to make the calculation.
Figure 5.5 shows a sample data entry for the "Forecast" worksheet. First the regression coefficients from either the Tp' = b1 × T0 + a1 or the Tp = b1To + b2Td + a3 equations are input into the appropriate cells in the Forecast worksheet (e.g. b1 = 0.494, b2 = 0.027, a1 = -5.872 and a3 = -5.783 in Figure 5.5). The two equations in the Forecast worksheet are completely independent and data can be entered in either one or both to forecast the minimum temperature. In Figure 5.5, the air temperature at two hours past sunset T0 = 9.0 °C was entered in the upper equation and the forecast was Tp = -1.4 °C. The air and dew-point temperatures input into the lower equation were T0 = 9.0 °C and Td = -5.0 °C and the resulting prediction was Tp = -1.5 °C.
Sample minimum temperature forecast coefficient and temperature entry in the "Forecast" worksheet of the FTrend.xls application program
The worksheet Wet-bulb in the FTrend.xls application is for determining the air temperature corresponding to an input value for wet-bulb and dew-point temperature at a specified elevation. This is used to help determine when to start and stop the use of sprinklers for frost protection. A sample of the Wet-bulb worksheet is shown in Figure 5.6. In the example, the elevation was entered as EL = 146 m above mean sea level. If the critical damage temperature for the protected crop is Tc = -1.0 °C, then Tw = -1.0 is input as shown in Figure 5.6. Recall that the critical temperature will vary depending on the crop, variety, phenological stage and hardening. In Figure 5.6, the value Td = -6.0 °C was input for the dew-point temperature. After the elevation, wet-bulb and dew-point temperatures are entered, the program calculates the corresponding air temperature. When using sprinklers for frost protection, they should be started and stopped when the air temperature measured upwind from the protected crop is higher than the air temperature shown in the Wet-bulb worksheet. The Wet-bulb worksheet also calculates the barometric pressure as a function of the elevation and the saturation vapour pressures at the dew-point (ed), wet-bulb (ew) and air temperatures (es). Note that the actual water vapour pressure (e) is equal to ed.
Sample data entry and calculations from the Wet-bulb worksheet of the FTrend.xls application program
The "Input" worksheet is used to enter the air temperature (T0) at two hours past sunset and the predicted minimum temperature (Tp) the next morning (Figure 5.7). The latitude, longitude, elevation and local time meridian are entered into the Input worksheet to determine day length, the local standard time and constants that depend on the elevation. Enter positive latitude for the Northern (°N) and negative latitude for the Southern (°S) Hemisphere. Enter positive longitude for locations west of Greenwich (°W) and negative longitude for locations east of Greenwich (°E). Input the elevation in metres above mean sea level. The temperature trend during the night is calculated using a square root function and the results are displayed in the Input worksheet (Figure 5.7) and are plotted in the Plot chart (Figure 5.8) of the FTrend.xls application. A critical temperature Tc = 1.5 °C was entered in this case.
Sample of the "Input" worksheet of the FTrend.xls application with the air temperature at two hours past sunset when T0 = 4.4 °C and the predicted minimum temperature Tp = -4.0 °C
Sample plot resulting from data entry into the "Input" worksheet of the FTrendl.xls application, using the data shown in Figure 5.7
If the dew-point temperature at two hours past sunset is also entered into the Input worksheet, then the application will calculate the change in wet-bulb and dew-point temperature as well as air temperature. A sample of the Input worksheet with Td = -2.8 °C is shown in Figure 5.9 and the plot is shown in Figure 5.10. The dew-point temperature is fixed at the input value during the night unless the air temperature drops below the input dew-point (Figure 5.10). Then the dew-point temperature falls with the air temperature to the predicted minimum temperature. For example, the air and dew-point both fell from T = -2.8 °C, when the air reached the dew-point temperature, to Tp = -4.0 °C at sunrise (Figure 5.10). This commonly occurs on nights when the air becomes saturated during the night.
The wet-bulb temperature curve in the FTrend.xls application is used to estimate when sprinklers need to be started for frost protection. For example, the wet-bulb temperature falls to the critical damage temperature Tc = -1.5 °C at 2300 h in Figure 5.10. In this situation, the sprinklers should be started prior to 2300 h before the wet-bulb temperature (Tw) falls below Tc = -1.5 °C. Assuming that the latitude, longitude and date are input correctly, the temperature trend plots go from two hours past sunset to sunrise (e.g. 1838 h to 0705 h in Figure 5.10).
A sample of the Input worksheet of the FTrend.xls application program with the additional entry of the dew-point temperature (Td) at two hours past sunset
Sample plot resulting from data entry into the "Input" worksheet of the FTrendl.xls application using data from Figure 5.9 with a dew-point temperature of Td = -2.8 °C at two hours past sunset
The Plot chart of the FTrend.xls application is also useful to help decide if the sprinkler should be used or not. For growers without soil waterlogging problems, shortage of water or concerns about cost, it is best to start the sprinklers when the wet-bulb temperature approaches either 0 °C or the critical damage temperature, depending on the value of the crop and concern about losses. However, for growers who are concerned about these problems, using the FTrend.xls application will help to determine when to start the sprinklers to minimize damage, waterlogging, energy usage and loss of water supply.
When using under-plant microsprinklers, the starting temperature is less important than for other sprinkler systems because mainly the ground and not the plants are wetted. When first started, there may be a small short-term temperature drop as the sprayed water evaporates; however, if the application rate is sufficient, the temperature should recover quickly. With under-plant microsprinklers, one can start when the air temperature approaches 0 °C, without too much risk. The same applies to conventional under-plant sprinkler systems that do not wet the lower branches of the trees. If the under-plant sprinklers do wet the lower branches, then the same starting criteria should be used as for over-plant sprinklers.
Over-plant sprinklers should be started so that they are all operating when the wet-bulb temperature approaches the critical damage temperature (Tc). However, note that the published critical damage temperatures are not always correct, so selecting a Tc slightly higher (e.g. by 0.5 °C) than the published value might be advisable. The choice depends on the risk one is willing to accept. If the minimum temperature (Tp) is forecast to be more than 1.0 °C lower than Tc, it is generally advisable to start the sprinklers as the wet-bulb temperature approaches Tc using the FTrend.xls program, as previously described. The problem arises when Tp is forecast to be near Tc. Even if Tp is slightly higher than Tc, it is possible for the equation to be incorrect on any given night depending on the local conditions. For example, the Tp forecast equation could work well for years and then it might fail completely on one night due to strange conditions on that night (e.g. often it is related to infrequent cold air drainage). This has happened to professional fruit-frost meteorologists in California and it is not so uncommon. However, in most cases, the prediction equations should work well. This is a good reason for close temperature monitoring during a frost night.
When Tp is forecast to be near Ta the decision whether to protect and when to protect depends on the dew-point temperature. If the dew-point temperature is low, then it is often advisable to start the sprinklers before Tw falls below Tc. This is illustrated in Figure 5.11, where Tp = -2.0 °C and Td = -5.0 °C were input. Although Tp is only slightly lower than Tc, because Td is low, Tw falls to Tc before midnight. Consequently, the decision whether or not to use the sprinklers must be made before midnight. In this example, the sprinklers would need to run for more than seven hours. If the sprinklers are not started, there is a good chance that the air temperature will fall slightly below Tc for about two hours. Depending on accuracy of the forecast, hardening of the crop, etc., the crop would probably experience some damage. However, if the forecast is low or Tc is set too high, there might be little or no damage. This makes the sprinkler starting decision difficult. Again, it depends on the amount of risk the grower wants to accept and if there are problems with waterlogging, water shortage, or cost. However, if the sprinklers are used for this example, they should be started before midnight.
Figure 5.12 shows a temperature trend plot with the input dew-point temperature Td = Tp = -2.0. In this case, Tp is below Tc and protection may be needed. However, because the dew-point temperature is relatively high, the grower can wait until later in the night to decide whether or not to protect. If sprinklers are used, the grower should start them at about 0400 h, so they would run for slightly more than three hours. If the sprinklers are operated correctly, it is unlikely that damage would result from the moderate frost on the night depicted in Figure 5.12. If the sprinklers are not used, it is uncertain if there would be damage or how much would occur. Again, it depends on the forecast and other physical and economic factors. Also, some crops that abort fruit or nuts (e.g. apple trees), can lose buds, flowers, fruit or nuts to freeze injury, yet overall production may not be greatly affected. For other crops that lose production due to loss of any buds, flowers, nuts or fruits (e.g. almond trees), damage should be avoided and less risk taken. Another big decision is related to whether or not the conditions are too severe for the sprinkler application rate to provide adequate protection. This is discussed in the chapter on active frost protection.
Temperature trend plot using data from Figure 5.9, but with the predicted minimum temperature Tp = -2.0 °C and the dew-point temperature Td = -5.0 °C
Sprinklers can be stopped after sunrise when the wet-bulb temperature again rises above the critical damage temperature. The temperature increase after sunrise depends on many factors and it is nearly impossible to accurately forecast. To determine when to stop the sprinklers, one should measure the wet-bulb temperature or the dew-point temperature upwind from the protected crop, and then use the Wet-bulb worksheet in the FTrend.xls application program to calculate the minimum air temperature for stopping the sprinklers. Enter the elevation, dew-point temperature and the wet-bulb temperature equal to the critical damage temperature (i.e. Tw = Tc). The sprinklers can be stopped if the sun is up and shining on the crop and the air temperature is higher than the air temperature calculated in the Wet-bulb worksheet. To be absolutely safe, input 0 °C for the wet-bulb temperature and stop the sprinklers when the sun is shining and the air temperature measured upwind from the protected field is higher than the calculated air temperature from the Wet-bulb worksheet.
Temperature trend plot using data from Figure 5.9, but with the dew-point temperature Td = -2.0 °C and predicted minimum temperature Tp = -2.0 °C
Another feature of the FTrend.xls program is that the temperature trend can be updated during the night with observed temperatures. For example, if it were cloudy between 2000 and 2200h during the night described in Figure 5.9 and the temperature at 2200 h was measured as T = 1.0 °C rather than the 0.0 °C as predicted in Figure 5.9, then T = 1.0 °C is entered for 2200 h in the Tupdate column (Figure 5.13) and all of the subsequent temperatures are shifted upward to account for the update (Figure 5.14). The predicted minimum temperature and the wet-bulb temperature trend from 2200 h until sunrise were both increased. The change in the wet-bulb temperature trend is significant in that the time when the wet-bulb temperature intersects the critical damage temperature was shifted from 2300 h to 0100 h. Therefore, starting sprinklers for frost protection could be delayed by about two hours. This illustrates the importance of monitoring temperatures and updating the FTrend.xls application model during the night.
A sample of the Input worksheet of the FTrend.xls application program with the 2200 h air temperature updated to Tupdate = 1.0 °C
A sample of the Plot chart from the FTrend.xls worksheet using the input data from Figure 5.12 with the measured air temperature updated at 2200 h
The air temperature trend calculation uses a square root function from two hours after sunset (i.e. time t0) until sunrise (i.e. time tp) the next morning. First a calibration factor b' is calculated from the predicted minimum temperature (Tp) and the temperature at time t0 (T0) as:
where h is the time (hours) between t0 and tp (e.g. h = (24 - t0) + tp). The temperature (Ti) at any time ti hours after t0 is estimated as:
If only the T0 and Tp temperature data are inputted, then the FTrend.xls application calculates only the temperature trend. However, if the dew-point temperature at two hours past sunset (Td) is also input, the application calculates the wet-bulb temperature between t0 and tp as well. During the night, the dew-point is fixed at the initial value Td unless the temperature trend falls below Td.
When the air temperature trend is less than Td, the dew-point temperature is set equal to the air temperature. The wet-bulb temperature is calculated as a function of the corresponding air and dew-point temperatures and the barometric pressure, which is estimated from the elevation.
The wet-bulb (Tw) temperature is calculated from the dew-point (Td) and air temperature (Ta) in °C as:
where es and ed are the saturation vapour pressures (kPa) at the air and dew-point temperature, D is the slope of the saturation vapour pressure curve at the air temperature (Ta) in °C:
and g is approximately equal to the psychrometric constant.
where Pb is the barometric pressure in kPa and l is the latent heat of vaporization:
where Ta is the air temperature in °C. Note that g from Equation 3.6 will give similar results to using Equations 5.5 and 5.6.
Although forecasting temperature trends during frost nights is important for identifying approximately if and when protection is needed, a good temperature monitoring program may be more important. The basic essentials include a frost alarm to wake you in time to start any protection methods before damage occurs and a network of temperature stations throughout the crop. Frost alarms are commercially available from a variety of sources. The cost of an alarm depends on its features. Some alarms have cables with temperature sensors that can be placed outside of your home in a standard shelter while the alarm is inside where the alarm bell can wake you. There are also alarms that can call you on the telephone or that use infrared or radio signals to communicate from a remote station back to your home to operate an alarm. However, as the frost alarm becomes more sophisticated, so the cost goes up.
Commonly, frost damage percentages are based on the plant tissue being exposed to half-an-hour below a critical temperature, whereas air temperatures are measured in a standard (or fruit frost) shelter at a height of 1.5 m. Perry (1994) recommends that thermometers should be placed at the lowest height where protection is desired. Perry (1994) also cautioned that sensors should be set where they will not be directly affected by protection methods (e.g. radiation from heaters). The general recommendation was to place the thermometers lower in short, dense crops and higher in taller, sparse crops. The idea is to have the sheltered air temperature reading as close as possible to the plant temperature that is being protected.
In reality the temperature of a leaf, bud, or small fruit or nut is likely to be lower than the shelter temperature. Similar to the boundary layer over a cropped surface, there is also a boundary layer over micro surfaces (e.g. leaves, buds, fruit or nuts). Due to long-wave radiation losses, exposed leaves, buds, flowers, etc. will typically be colder than air temperature during a frost night. Sensible heat diffuses from the air to the colder surface through the boundary layer, but the diffusion rate is insufficient to replace radiational heat losses. As a result, sensible heat content of the plant tissues and air near the surface causes temperatures to fall and leads to an inversion condition over the plant tissues. The depth of this micro-scale boundary layer and the gradient of sensible heat help to determine how fast sensible heat transfers to the surface.
The importance of a microscale boundary layer can be illustrated by considering what happens to your skin in a hot environment (e.g. in a dry sauna). If you stand in a "dry" sauna and do not move, you will feel hot because the ambient temperature is higher than your skin temperature. Sensible heat transfers from the ambient air through the small boundary layer to your skin mainly by diffusion. However, if you start to exercise (e.g. do callisthenics), you will quickly get much hotter. This happens because your exercising will ventilate the skin and reduce the thickness of the boundary layer, which enhances sensible heat transfer to the colder surface (i.e. to your skin). The energy balance of a leaf, bud, fruit or nut is similar. Increasing ventilation (e.g. higher wind speed) will reduce the thickness of the boundary layer and enhance sensible heat transfer. During a frost night, the plant parts tend to be colder than the air, so a higher wind speed will warm the plants to nearly as high as the ambient temperature. If the ambient air temperature is sufficiently high, then little or no damage may occur.
Some problems arise from using shelter air temperature (Ta) for critical damage temperature (Tc). Plant temperature can be quite different from air temperature depending on net radiation, exposure to the sky, and ground, leaf and ventilation (wind) conditions. Critical damage temperatures are often determined by placing excised branches in a cold chamber. In the chamber, the temperature is slowly lowered and held below a specific temperature for 30 minutes and later the branch is evaluated for the percentage damage to the buds, blossoms, fruit or nuts. There is no easy solution for comparing published Tc values with what really happens during a frost night. In practice, one should only use Tc values as a guideline and recall that the temperature of exposed branches is likely to be below temperature measured in a shelter.
Knowing the relationship between the temperature of sensitive plant tissues and shelter temperature will help with protection decisions. For example, it is well known that citrus leaves freeze at about -5.8 °C (Powell and Himelrick, 2000). However, measuring leaf temperature is labour intensive and not widely practiced. Therefore, estimating leaf temperature from shelter temperature is desirable. In addition to citrus, the relationship between leaf temperature and shelter temperature is unavailable for bud, blossom, small-fruit and small-nut stages of most stone fruit and small-fruit crops (Powell and Himelrick, 2000).
Supercooling of plant parts makes identification of critical temperatures difficult. For example, citrus has relatively low concentrations of ice-nucleating bacteria, and this might explain why the Tc for citrus leaves was consistently found to be about -5.8 °C. In many deciduous crops, identifying a clear critical temperature is more difficult because super-cooling varies with the concentration of ice-nucleating bacteria.
The presence of water on plant surfaces will also affect frost damage. Powell and Himelrick (2000) noted that dry plant surfaces freeze at lower air temperature than wet surfaces. They mentioned work in California that showed that wet citrus fruit cooled more rapidly than dry fruit during frost events. At the same air temperature, wet fruit is colder than dry fruit because the water evaporates and removes sensible heat. The wet fruit can cool to the wet-bulb temperature, which is always less than or equal to the air temperature. Spots of water on the peel of citrus fruit going into a frost night can result in spot damage because the peel under the water spots can cool to the wet-bulb temperature while the dry parts of the fruit are warmer. Similar damage can occur to the peel of other fruits if wet going into a frost night. Consequently, it is unadvisable to wet plants before a frost night unless sprinklers will be used during the night.