6.10.1. Broaden Knowledge & Comprehension
B1. Search the web or consult engineering documents that indicate the temperature and water vapor content of exhaust from aircraft jet engines. What ambient temperature and humidity of the atmosphere near the aircraft would be needed such that the mixture of the exhaust with the air would just become saturated and show a jet contrail in the sky. (You might need to utilize a spreadsheet to experiment with different mixing proportions to find the one that would cause saturation first.)
B2. Use the WMO Cloud Atlas website to find images corresponding to the clouds in Figs. 6.3 and 6.5. Copy the best example of each cloud type to your own page to create a cloud chart. Be sure to also record and cite the photographer for each photo.
B3. Search the web for thermo diagrams showing soundings for cases where different cloud types were present (e.g., active, passive, fog). Estimate cloud base and cloud top altitudes from the sounding, and compare with observations. Cloud base observations are available from the METAR surface reports made at airports. Cloud top observations are reported by aircraft pilots in the form of PIREPS.
B4. Search the web to identify some of the instruments that have been devised to determine cloudbase altitude?
B5. Search the web for cloud images corresponding to the “Other” clouds discussed in section 6.2.4. Copy the best example of each cloud type to your own page. Be sure to also record and cite the original web site and photographer for each cloud photo.
B6. Search the web for satellite and ground-based images of cloud streets, open cells, and closed cells. Create a image table of these different types of cloud organization.
B7. Search the web for a surface weather map that plots surface observations for locations near you. For 3 of the stations having clouds, interpret (describe in words) the sky cover and cloud genera symbols (using Tables 6-7 and 6-2) that are plotted.
B8. Table 6-2 shows only a subset of the symbols used to represent clouds on weather maps. Search the web for map legends or tables of symbols that give a more complete list of cloud symbols.
- Search the web for fractal images that you can print. Use box counting methods on these images to find the fractal dimension, and compare with the fractal dimension that was specified by the creator of that fractal image.
- Make a list of the URL web addresses of the ten fractal images that you like the best (try to find ones from different research groups or other people).
B10. Search the web for satellite images that show fog. (Hints: Try near San Francisco, or try major river valleys in morning, or try over snowy ground in spring, or try over unfrozen lakes in Fall). For the fog that you found, determine the type of fog (radiation, advection, steam, etc.) in that image, and justify your decision.
B11. a. Search the web for very high resolution visible satellite images of cumulus clouds or cloud clusters over an ocean. Display or print these images, and then trace the edges of the clouds. Find the fractal dimension of the cloud edge, similar to the procedure that was done with Fig. 6.12.
B12. Search the web for a discussion about which satellite channels can be used to discriminate between clouds and fog. Summarize your findings.
B13. Search the web for methods that operational forecasters use to predict onset and dissipation times of fog. Summarize your findings. (Hint: Try web sites of regional offices of the weather service.)
A1. Given the following temperature and vapor pressure for an air parcel. (i) How much moisture ∆e (kPa) must be added to bring the original air parcel to saturation? (ii) How much cooling ∆T (°C) must be done to bring the original parcel to saturation? Original [T (°C) , e (kPa) ] =
|a. 20, 0.2||b. 20, 0.4||c. 20, 0.6||d. 20, 0.8||e. 20, 1.0|
|f. 20, 1.5||g. 20, 2.0||h. 20, 2.2||i. 10, 0.2||j. 10, 0.4|
|k. 10, 0.6||l. 10, 0.8||m. 10, 1.0||n. 30, 1.0||o. 30, 3.0|
A2. On a winter day, suppose your breath has T = 30°C with Td = 28°C, and it mixes with the ambient air of temperature T = –10°C and Td = –11°C. Will you see your breath? Assume you are at sea level, and that your breath and the environment mix in proportions of (breath : environment):
|a. 1:9||b. 2:8||c. 3:7||d. 4:6||e. 5:5|
|f. 6:4||g. 7:3||h. 8:2||i. 9:1|
A3(§). A jet aircraft is flying at an altitude where the ambient conditions are P = 30 kPa, T = –40°C, and Td = –42°C. Will a visible contrail form? (Hint, assume that all possible mixing proportions occur.)
Assume the jet exhaust has the following conditions: [T (°C) , Td (°C) ] =
|a. 200, 180||b. 200, 160||c. 200, 140||d.200, 120|
|e. 200, 100||f. 200, 80||g. 200, 60||h. 200, 40|
|i. 400, 375||j. 400, 350||k. 400, 325||l. 400, 300|
|m. 400, 275||n. 400, 250||o. 400, 200||p. 400, 150|
A4. Given the following descriptions of ordinary clouds. (i) First classify as cumuliform or stratiform. (ii) Then name the cloud. (iii) Next, draw both the WMO and USA symbols for the cloud. (iv) Indicate if the cloud is made mostly of liquid water or ice (or both). (v) Indicate the likely altitude of its cloud base and top. (vi) Finally, sketch the cloud similar to those in Figs. 6.3 or 6.5.
- Deep vertical towers of cloud shaped like a gigantic mushroom, with an anvil shaped cloud on top. Flat, dark-grey cloud base, with heavy precipitation showers surrounded by nonprecipitating regions. Can have lightning, thunder, hail, strong gusty winds, and tornadoes. Bright white cloud surrounded by blue sky when viewed from the side, but the cloud diameter is so large that when from directly underneath it might cover the whole sky.
- Sheet of light-grey cloud covering most of sky, with sun or moon faintly showing through it, casting diffuse shadows on ground behind buildings and people.
- Isolated clouds that look like large white cotton balls or cauliflower, but with flat bases. Lots of blue sky in between, and no precipitation. Diameter of individual clouds roughly equal to the height of their tops above ground.
- Thin streaks that look like horse tails, with lots of blue sky showing through, allowing bright sun to shine through it with crisp shadows cast on the ground behind trees and people.
- Thick layer of grey cloud with well poorly-defined cloud base relatively close to the ground, and widespread drizzle or light rain or snow. No direct sunlight shining through, and no shadows cast on the ground. Gloomy.
- Isolated clouds that look like small, white cotton balls or popcorn (but with flat bases), with lots of blue sky in between. Size of individual clouds roughly equal to their height above ground.
- Thin uniform veil covering most of sky showing some blue sky through it, with possibly a halo around a bright sun, allowing crisp shadows cast on the ground behind trees and people.
- Layer of grey cloud close to the ground, but lumpy with some darker grey clouds dispersed among thinner light grey or small clear patches in a patchwork or chessboard pattern. Not usually precipitating.
- Thin veil of clouds broken into very small lumps, with blue sky showing through, allowing bright sun to shine through it with crisp shadows cast on the ground behind trees and people.
- Sheet of cloud covering large areas, but broken into flat lumps, with sun or moon faintly showing through the cloudy parts but with small patches of blue sky in between the lumps, casting diffuse shadows behind buildings and people.
- Deep towers of cloud with bright white sides and top during daytime, but grey when viewed from the bottom. Clouds shaped like stacks of ice-cream balls or turrets of cotton balls with tops extending high in the sky, but with flat bases relatively close to the ground. Usually no precipitation.
- Thick layer of grey cloud with well defined cloud base relatively close to the ground. No direct sunlight shining through, and no shadows cast on the ground. No precipitation.
A5. Use a thermo diagram to plot the following environmental sounding:
|P (kPa)||T (°C)|
Determine cloud activity (active, passive, fog, none), cloud-base height, and cloud-top height, for the conditions of near-surface (P = 100 kPa) air parcels given below: [T (°C) , Td (°C)] =
|a. 20, 14||b. 20, 7||c. 20, 17||d. 20, 19|
|e. 15, 15||f. 30, 0||g. 30, 10||h. 30, 24|
|i. 25, 24||j. 25, 20||k. 25, 16||l. 25, 12|
|m. 25, 10||n. 25, 6||o. 15, 10||p. 22, 22|
A6. The buoyancy of a cloudy air parcel depends on its virtual temperature compared to that of its environment. Given a saturated air parcel of temperature and liquid-water mixing ratio as listed below. What is its virtual temperature, and how would it compare to the virtual temperature with no liquid water? [T(°C), rL(g kg–1)] =
|a. 20, 1||b. 20, 2||c. 20, 5||d. 20, 10||e. 10, 1||f. 10, 3|
|g. 10, 7||h. 10, 12||i. 0, 2||j. 0, 4||k. 0, 8||l. 0, 15|
|m. 5, 1||n. 5, 2||o. 5, 4||p. 5, 6||q. 5, 8||r. 5, 10|
A7. On a large thermo diagram from the end of the Atmospheric Stability chapter, plot a hypothetical sounding of temperature and dew-point that would be possible for the following clouds.
|a. cirrus||b. cirrostratus||c. cirrocumulus||d. altostratus|
|e. altocumulus||f. stratus||g. nimbostratus||h. fog|
A8. Name these special clouds.
- Parallel bands of clouds perpendicular to the shear vector, at the level of altocumulus.
- Parallel bands of cumulus clouds parallel to the wind vector in the boundary layer.
- Two, long, closely-spaced parallel cloud lines at high altitude.
- Clouds that look like flags or pennants attached to and downwind of mountain peaks.
- Look like altocumulus, but with more vertical development that causes them to look like castles.
- Clouds that look like breaking waves.
- Look like cumulus or cumulus mediocris, but relatively tall and small diameter causing them to look like castles.
- Clouds that look lens shaped when viewed from the side, and which remain relatively stationary in spite of a non-zero wind.
- A low ragged cloud that is relatively stationary and rotating about a horizontal axis.
- Ragged low scattered clouds that are blowing rapidly downwind.
- Low clouds that look like cotton balls, but forming above industrial cooling towers or smoke stacks.
- A curved thin cloud at the top of a mountain. m. A curved thin cloud at the top of a rapidly rising cumulus congestus.
A9. Discuss the difference between cloud genera, species, varieties, supplementary features, accessory clouds, mother clouds, and meteors.
A10. Given the cloud genera abbreviations below, write out the full name of the genus, and give the WMO and USA weather-map symbols.
|a. Cb||b. Cc||c. Ci||d. Cs||e. Cu|
|f. Sc||g. Ac||h. St||i. As||j. Ns|
A11. For the day and time specified by your instructor, fully classify the clouds that you observe in the sky. Include all of the WMO-allowed classification categories: genera, specie, variety, feature, accessory cloud, and mother cloud. Justify your classification, and include photos or sketches if possible.
[HINT: When you take cloud photos, it is often useful to include some ground, buildings, mountains, or trees in the photo, to help others judge the scale of the cloud when they look at your picture. Try to avoid pictures looking straight up. To enhance the cloud image, set the exposure based on the cloud brightness, not on the overall scene brightness. Also, to make the cloud stand out against a blue-sky background, use a polarizing filter and aim the camera at right angles to the sunbeams. Telephoto lenses are extremely helpful to photograph distant clouds, and wide-angle lenses help with widespread nearby clouds. Many cloud photographers use zoom lenses that span a range from wide angle to telephoto. Also, always set the camera to focus on infinity, and never use a flash.]
A12. Draw the cloud coverage symbol for weather maps, and write the METAR abbreviation, for sky cover of the following amount (oktas).
|a. 0||b. 1||c. 2||d. 3||e. 4|
|f. 5||g. 6||h. 7||i. 8||j. obscured|
A13.(§). Use a spreadsheet to find and plot the fraction of clouds vs. size. Use ∆X = 100 m, with X in the range 0 to 5000 m.
(i) For fixed Sx = 0.5, plot curves for LX (m) =
|a. 200||b. 300||c. 400||d. 500||e. 700||f. 1000||g. 2000|
(ii) For fixed LX = 1000 m, plot curves for SX =
|h. 0.1||i. 0.2||j. 0.3||k. 0.4||l. 0.5||m. 0.6||n. 0.8|
A14. In Fig. 6.12, divide each tile into 4 equal quadrants, and count N vs. M for these new smaller tiles to add data points to the solved-example figure for measuring fractal dimension. Do these finer-resolution tiles converge to a different answer? Do this box-counting for Fig. 6.12 part:
|a. (a)||b. (b)||c. (c)||d. (d)|
A15. Use the box counting method to determine the fractal dimensions in Fig. 6.11b. Use M =
|a. 4||b. 5||c. 6||d. 7||e. 8||f. 9|
|g. 10||h. 12||i. 14||j. 16||k. 18||l. 20|
A16. For air starting at sea level:
- How high (km) must it be lifted to form upslope fog?
- How much water (gliq kgair–1) must be added to cause precipitation fog?
- How much cooling (°C) is necessary to cause radiation fog?
Given the following initial state of the air
[T (°C) , RH (%) ] =
|a. 10, 20||b. 10, 40||c. 10, 60||d. 10, 80||e. 10, 90|
|f. 20, 20||g. 20, 40||h. 20, 60||i. 20, 80||j. 20, 90|
|k. 0, 20||l. 0, 40||m. 0, 60||n. 0, 80||o. 0, 90|
A17(§). In spring, humid tropical air of initial temperature and dew point as given below flows over colder land of surface temperature 2 °C. At what downwind distance will advection fog form? Also plot the air temperature vs. distance. Assume zi = 200 m, and CH = 0.005.
The initial state of the air is [T (°C), Td (°C)] =
|a. 20, 15||b. 20, 10||c. 20, 5||d. 20, 0||e. 20, –5|
|f. 20, –10||g. 20, –15||h. 10, 8||i. 10, 5||j. 10, 2|
|k. 10, 0||l. 10, –2||m. 10, –5||n. 10, –8||o. 10, –10|
A18(§). Given FH = –0.02 K·m·s–1 . (i) When will radiation fog form? (ii) Also, plot fog depth vs. time. Given residual-layer initial conditions of
[T (°C) , Td (°C) , M (m s–1)] =
|a. 15, 13, 1||b. 15, 13, 2||c. 15, 13, 3||d. 15, 13, 4|
|e. 15, 10, 1||f. 15, 10, 2||g. 15, 10, 3||h. 15, 10, 4|
|i. 15, 8, 1||j. 15, 8, 2||k. 15, 8, 3||l. 15, 8, 4|
|m. 15, 5, 1||n. 15, 5, 2||o. 15, 5, 3||p. 15, 5, 4|
A19(§). (i) When will a well-mixed fog dissipate? Assume: albedo is 0.5, fog forms 6 h after sunset, and daylight duration is 12 h. (ii) Also, plot the cumulative heat vs. time.
Given the following values of surface kinematic heat flux: [ FH.night (K·m s–1) , FH.max day (K·m s–1) ] =
|a. –0.02, 0.15||b. –0.02, 0.13||c. –0.02, 0.11|
|d. –0.02, 0.09||e. –0.02, 0.07||f. –0.02, 0.17|
|g. –0.015, 0.15||h. –0.015, 0.13||i. –0.015, 0.11|
|j. –0.015, 0.09||k. –0.015, 0.07||l. –0.015, 0.17|
6.10.3. Evaluate & Analyze
E1. What processes in the atmosphere might simultaneously cool and moisturize unsaturated air, causing it to become saturated?
E2. Cumulus humilis clouds often have flat bases at approximately a common altitude over any location at any one time. Cumulus fractus clouds do not have flat bases, and the cloud-base altitudes vary widely over any location at one time. What causes this difference? Explain.
E3. Can you use darkness of the cloud base to estimate the thickness of a cloud? If so, what are some of the errors that might affect such an approach? If not, why not?
E4. Fig. 6.4 shows that the LCL computed from surface air conditions is a good estimate of cloudbase altitude for cumulus humilis clouds, but not for cloud-base altitude of altocumulus castellanus. Why?
E5. What methods could you use to estimate the altitudes of stratiform clouds by eye? What are the pitfalls in those methods? This is a common problem for weather observers.
E6. Should stratocumulus clouds be categorized as cumuliform or stratiform? Why?
E7. List all the clouds that are associated with turbulence (namely, would cause a bumpy ride if an airplane flew through the cloud).
E8. What do pileus clouds and lenticular clouds have and common, and what are some differences?
E9. Cloud streets, bands of lenticular clouds, and Kelvin-Helmholtz billows all consist of parallel rows of clouds. Describe the differences between these clouds, and methods that you can use to discriminate between them.
E10. The discrete cloud morphology and altitude classes of the official cloud classification are just points along a continuum of cloud shapes and altitudes. If you observe a cloud that looks halfway between two official cloud shapes, or which has an altitude between the typical altitudes for high, middle, or low, clouds, then what other info can you use as a weather observer to help classify the cloud?
E11 (full term project). Build your own cloud chart with your own cloud photos taken with a digital camera. Keep an eye on the sky so you can try to capture as many different cloud types as possible. Use only the best one example of each cloud type corresponding to the clouds in Figs. 6.3 and 6.5. Some clouds might not occur during your school term project, so it is OK to have missing photos for some of the cloud types.
E12. List all of the factors that might make it difficult to see and identify all the clouds that exist above your outdoor viewing location. Is there anything that you can do to improve the success of your cloud identification?
- On a spreadsheet, enter the cloud-size parameters (SX, LX, ∆X) from the Sample Application into separate cells. Create a graph of the lognormal
- Next, change the values of each of these parameters to see how the shape of the curve changes. Can you explain why LX is called the location parameter, and SX is called the spread parameter? Is this consistent with an analytical interpretation of the factors in eq. (6.6)? Explain.
E14. The box-counting method can also be used for the number of points on a straight line, such as sketched in Fig. 6.11c. In this case, a “box” is really a fixed-length line segment, such as increments on a ruler or meter stick.
- Using the straight line and dots plotted in Fig. 6.11c, use a centimeter rule to count the number of non-overlapping successive 1 cm segments that contain dots. Repeat for half cm increments. Repeat with ever smaller increments. Then plot the results as in the fractal-dimension Sample Application, and calculate the average fractal dimension. Use the zero-set characteristics to find the fractal dimension of the original wiggly line.
- In Fig. 6.11c, draw a different, nearly vertical, straight line, mark the dots where this line crosses the underlying wiggly line, and then repeat the dotcounting procedure as described in part (a). Repeat this for a number of different straight lines, and then average the resulting fractal dimensions to get a more statistically-robust estimate.
- Crumple a sheet of paper into a ball, and carefully slice it in half. This is easier said than done. (A head of cabbage or lettuce could be used instead.) Place ink on the cut end using an ink pad, or by dipping the paper wad or cabbage into a pan with a thin layer of red juice or diluted food coloring in the bottom. Then make a print of the result onto a flat piece of paper, to create a pattern such as shown in Fig. 6.11b. Use the box counting method to find the fractal dimension of the wiggly line that was printed. Using the zero-set characteristic, estimate the fractal dimension of the crumpled paper or vegetable.
- Repeat the experiment, using crumpled paper wads that are more tightly packed, or more loosely packed. Compare their fractal dimensions.
(P.S. Don’t throw the crumpled wads at the instructor. Instructors tend to get annoyed easily.)
E16. Can precipitation fog form when cold raindrops fall through warmer air? Explain.
E17. A fog often forms near large waterfalls. What type of fog is it usually, and how does it form?
E18. In this chapter we listed some locations and seasons where advection fog is likely. Describe 3 other situations (locations and seasons) when advection fog would be likely. (If possible, use locations close to home.)
E19. Derive eq. (6.10), based on the exponential temperature profile for a stable boundary layer (see the Atmospheric Boundary Layer chapter). State and justify all assumptions. [This requires calculus.] Be critical of any simplifications that might not be appropriate for real fogs.
E20. Derive eq. (6.11) from (6.10), and justify all assumptions. Be critical of any simplifications that might not be appropriate for real fogs.
E21. Derive eq. (6.12) using info from the Atmospheric Boundary Layer chapter, and justify all assumptions. Be critical of any simplifications that might not be appropriate for real fogs.
E22. Use the data in the Sample Application for fog dissipation, but find the critical albedo at which fog will just barely dissipate.
E23. During possible frost events, some orchard and vineyard owners try to protect their fruit from freezing by spraying a mist of water droplets or burning smudge pots to make smoke in and above their plants. Why does this method work, and what are its limitations?
S1. What if the saturation curve in Fig. 6.1 was concave down instead of concave up, but that saturation vapor pressure still increased with increasing temperature. Describe how the cooling, moisturizing, and mixing to reach saturation would be different.
S2. Suppose that descending (not ascending) air cools adiabatically. How would cloud shapes be different, if at all? Justify.
S3. Cloud classification is based on morphology; namely, how the cloud looks. A different way to classify clouds is by the processes that make the clouds. Devise a new scheme to classify clouds based on cloud processes; name this scheme after yourself; and make a table showing which traditional cloud names fall into each category of your new scheme, and justify.
S4. Clouds in the atmospheres of other planets in our solar system have various compositions:
- S4. Clouds in the atmospheres of other planets in our solar system have various compositions:
- Mars: water
- Jupiter: ammonia, sulfur, water
- Saturn: ammonia, ammonia hydrosulfide, water
- Uranus & Neptune: methane.
Would the clouds on these other planets have shapes different than the clouds on Earth? If so, then develop a cloud classification for them, and explain why. If not, why not?
S5. Utilize the information in (a) and (b) below to explain why cloud sizes might have a lognormal distribution.
- The central-limit theorem of statistics states that if you repeat an experiment of adding N random numbers (using different random numbers each time), then there will be more values of the sum in the middle of the range than at the extremes. That is, there is a greater probability of getting a middle value for the sum than of getting a small or large value. This probability distribution has the shape of a Gaussian curve (i.e., a bell curve or a “normal” distribution; see the Air Pollution chapter for examples).
For anyone who has rolled dice, this is well known. Namely, if you roll one die (consider it to be a random number generator with a uniform distribution) you will have an equal chance of getting any of the numbers on the die (1, 2, 3, 4, 5, or 6).
However, if you roll two dice and sum the numbers from each die, you have a much greater chance of getting a sum of 7 than of any other sum (which is exactly half way between the smallest possible sum of 2, and the largest possible sum of 12). You have slightly less chance of rolling a sum of 6 or 8. Even less chance of rolling a sum of 5 or 9, etc.
The reason is that 7 has the most ways (6 ways) of being created from the two dice (1+6, 2+5, 3+4, 4+3, 5+2, 6+1). The sums of 6 and 8 have only 5 ways of being generated. Namely, 1+5, 2+4, 3+3, 4+2, and 5+1 all sum to 6, while 2+6, 3+5, 4+4, 5+3, 6+2 all sum to 8. The other sums are even less likely.
- The logarithm of a product of numbers equals the sum of the logarithms of those numbers.
S6. Build an instrument to measure relative sizes of cumulus clouds as follows. On a small piece of clear plastic, draw a fine-mesh square grid like graph paper. Or take existing fine-mesh graph paper and make a transparency of it using a copy machine. Cut the result to a size to fit on the end of a short tube, such as a toilet-paper tube.
Hold the open end of the tube to your eye, and look through the tube toward cumulus clouds. [CAUTION: Do NOT look toward the sun.] Do this over relatively flat, level ground, perhaps from the roof of a building or from a window just at tree-top level. Pick clouds of medium range, such that the whole cloud is visible through the tube.
For each cloud, record the relative diameter (i.e., the number of grid lines spanned horizontally by the cloud), and the relative height of each cloud base above the horizon (also in terms of number of grid lines). Then, for each cloud, divide the diameter by the cloud-base height to give a normalized diameter. This corrects for perspective, assuming that cumulus cloud bases are all at the same height.
Do this for a relatively large number of clouds that you can see during a relatively short time interval (such as half an hour), and then count the clouds in each bin of normalized cloud diameter.
- Plot the result, and compare it with the lognormal distribution of Fig. 6.10.
- Find the LX and SX parameters of the lognormal distribution that best fit your data. (Hint: Use trial and error on a spreadsheet that has both your measured size distribution and the theoretical distribution on the same graph. Otherwise, if you know statistics, you can use a method such a Maximum Likelihood to find the best-fit parameters.)
S7. Suppose that you extend Euclidian space to 4 dimensions, to include time as well as the 3 space dimensions. Speculate and describe the physical nature of something that has fractal dimension of 3.4 .
S8. For advection fog, eq. (6.8) is based on a wellmixed fog layer. However, it is more likely that advection fog would initially have a temperature profile similar to that for a stable boundary layer (Atmospheric Boundary Layer chapter). Derive a substitute equation in place of eq. (6.8) that includes this better representation of the temperature profile. Assume for simplicity that the wind speed is constant with height, and state any other assumptions you must make to get an answer. Remember that any reasonable answer is better than no answer, so be creative.
S9. Suppose that fog was transparent to infrared (longwave) radiation. Describe how radiation fog formation, growth, and dissipation would be different (if at all) from real radiation fogs?
S10. Suppose that clouds were transparent to solar radiation, and couldn’t shade the ground. Describe and explain the possible differences in cloud morphology, coverage, duration, and their effects on weather and climate, if any.
S11. Find a current weather map (showing only fronts and winds) for your continent. Use your knowledge to circle regions on the map where you expect to find different types of fog and different types of clouds, and label the fog and cloud types.