## 1. Introduction

*F*is assumed to be proportional to the mass of the accumulated ice Δ

*M*(Brown 1982), that is, Δ

*F*∼ Δ

*M.*Variations in the frequency are sensed and converted into a voltage

*V*so that

*M*

*k*

*V.*

Techniques for determining the coefficient *k* have been discussed by Heymsfield and Miloshevich (1989), Baumgardner and Rodi (1989), and Claffey et al. (1995). The difficulties in defining *k* are related to simultaneous, accurate measurements of accumulated ice Δ*M* and frequency changes Δ*F.* Heymsfield and Miloshevich (1989) suggested calibrating the Rosemount probe by using Δ*M* derived from Forward Scattering Spectrometer Probe (FSSP) measurements. The limitation of this method is that the FSSP is known to react to ice particles, and it is difficult to determine what fraction of the counts in supercooled clouds were caused by ice and what fraction were caused by liquid droplets. This leads to uncertainty in measurements of Δ*M.* Baumgardner and Rodi (1989) found that the coefficient *k* varies from probe to probe, and it depends also on the distribution of ice over the cylinder surface. They obtained a different *k* coefficient than that reported by Heymsfield and Miloshevich (1989). Claffey et al. (1995) calibrated three RICE probes using a rotating multicylinder at the Mt. Washington Observatory. They found that each ice detector was unique and should be calibrated individually before being used.

This paper presents a theoretical study of ice accretion on the surface of the cylinder in the flow of supercooled droplets. A steady-state heat balance on the surface of the icing cylinder is considered in detail. A new calibration technique for the RICE probe, based on the measurements of the rate of ice sublimation in cloud-free air, is suggested. The calibration coefficient derived using this sublimation technique is compared with that obtained using the conventional technique, that is, when ice is accreting on the cylinder.

## 2. Heat balance on the surface of a riming cylinder

Consider a cylinder exposed to the flow of air with velocity *U,* temperature *T*_{a}, pressure *P*_{a}, and containing supercooled droplets with radius *r* and liquid water content (LWC) *W.* It is assumed that the axis of the cylinder is perpendicular to the axis of the vector of the air velocity. The temperature of the droplets is considered to be the same as the temperature of the air. In the vicinity of the cylinder the trajectories of the droplets due to their inertia will deviate from the trajectories of the air. As a result the droplets will hit the cylinder within a certain band limited by polar angles −*ϕ*_{0} to *ϕ*_{0} (Fig. 1). The angle *ϕ*_{0} is a function of the droplet radius, cylinder radius, airspeed, air temperature, and pressure (Langmuir and Blodgett 1945; Mazin 1957; Borovikov et al. 1963). At angles *ϕ* > |*ϕ*_{0}| the droplets with radii *r* do not impact the cylinder and they flow around the cylinder with the airflow. Thus in the case of supercooled droplets, ice would accrete on the surface of the cylinder where *ϕ* < |*ϕ*_{0}|. In principle, the shape of the accreted ice depends on the airspeed, droplet size, LWC air temperature, and other parameters that define the thermodynamical processes on the icing surface. The shape of the accreted ice is also a function of time and can vary significantly from that shown in Fig. 1.

The shape of accreted ice on a cylindrical surface has been discussed in a number of studies (e.g., Mazin 1957;Lozowski et. al 1983a,b). The RICE probe automatically de-ices itself with an internal heater after approximately 0.5 mm of ice accumulates on its surface. This value is small compared to the RICE cylinder diameter (≈6.3 mm). Therefore, in this study it is assumed that the ice accreted on the RICE probe surface has a shape close to that of a circular cylinder. The spongy ice formation and runback icing is not considered in the frame of this study.

### a. Local heat balance

The pioneering work on heat balance of an icing surface goes back to the 1950s (e.g., Hardy 1945; Ludlam 1951; Tribus 1951; Messinger 1953; Fraser et al. 1953;Hardy and Brown 1954). A detailed discussion of these papers can be found in Mazin (1957).

Consider a steady-state heat balance at a local point on the surface of the cylinder. The surface temperature is defined by several processes: dynamic heating, freezing of droplets, ice sublimation, heat exchange between the droplets, and the cylinder’s surface. These processes are schematically shown in Fig. 2. Due to axial symmetry of the cylinder, the thermal processes on the surface will be a function of the polar angle *ϕ* (Fig. 1).

*q*

_{i}are the densities of the heat fluxes resulting from different processes. The value of

*q*

_{i}is defined as an amount of heat passing through the unit surface per unit time (J m

^{−2}s

^{−1}). Because of dynamic heating and latent heat release from freezing droplets, the surface temperature of the cylinder will normally be higher than the ambient temperature.

*q*

_{1}is related to the aerodynamic heating due to adiabatic compression of the air. In the following discussion the thermal conductivity of ice is neglected, resulting in no heat flux inside the cylinder. In this case the adiabatic temperature on the cylinder’s surface is (e.g., Hilton 1951):

*C*

_{p}is the specific heat capacity of the air at constant pressure, and

*κ*is the recovery factor. The factor

*κ*takes into account the dissipation of energy due to internal friction. For air the value of the recovery factor can be assumed to be

*κ*≈

*κ*≈ 0.85. Experimental measurements have shown that the recovery factor varies along the surface of the cylinder from 1 to 0.5 (Seban 1960; Lozowski et al. 1983a).

*T*

_{sϕ}may be different from the adiabatic one, that is,

*T*

_{sϕ}≠

*T*

_{sϕA}. The convective heat losses at the cylinder’s surface, defined by a polar angle

*ϕ,*can be presented as

*q*

_{1}

*α*

_{ϕ}

*T*

_{sϕ}

*T*

_{sϕA}

*α*

_{ϕ}is the heat transfer coefficient that describes the amount of heat per unit temperature interval that comes to the unit of surface area during a unit time interval (e.g., Incropera and DeWitt 1985). The coefficient

*α*

_{ϕ}varies along the surface of the cylinder (appendix A). The flux

*q*

_{1}is considered to be positive if the heat is directed toward the cylinder surface from outside; that is, if

*T*

_{sϕA}>

*T*

_{sϕ}.

*e*

_{a}is less than its saturated value with respect to ice

*E*

_{iϕ}at the surface temperature

*T*

_{sϕ}, then the heat flux

*q*

_{2}associated with ice evaporation is

*q*

_{2}

*m*

_{iϕ}

*L*

_{iϕ}

*L*

_{iϕ}is the latent heat of sublimation at temperature

*T*

_{sϕ}, and

*m*

_{iϕ}is the mass of the ice sublimated from the unit surface area during the unit time. Assuming that the water vapor is saturated with respect to ice

*E*

_{iϕ}at the ice surface (Mazin 1957):

*ρ*

_{υs},

*ρ*

_{υa}are the densities of the saturated water vapor at surface and air temperatures, respectively, and

*ρ*

_{a}is the air density. Minervin (1956) showed that the coefficient

*b*deviates from unity by not more than about 1%. Lozowski et al. (1983a) used

*b*= (Pr/Sc)

^{0.63}, where Pr and Sc are the Prandtl and Schmidt numbers, respectively. This coefficient is rather close to unity as well [at

*T*

_{a}= −10°C, (Pr/Sc)

^{0.63}= 1.13]. In the following consideration it is assumed that

*b*= 1 and it is omitted in the equations.

*R*

_{a}and

*R*

_{υ}are the specific gas constants of air and water vapor, respectively, and

*P*

_{a}is the air pressure. The flux

*q*

_{2}is directed outward from the cylinder surface if ice evaporates, that is, when

*E*

_{iϕ}>

*e*

_{a}. Equations (3) and (8) show that the rate and the sign of ice mass changes on the cylinder surface depend on the air temperature

*T*

_{a}, surface temperature

*T*

_{s}, water vapor pressure

*e*

_{a}, and airflow velocity

*U.*

*T*

_{a}to 0°C following:

*q*

_{3}

*m*

_{wϕ}

*C*

_{w}

*T*

_{a}

*m*

_{wϕ}

*C*

_{w}

*T*

_{a}

*q*

_{4}

*m*

_{wϕ}

*L*

_{f}

*T*

_{sϕ}following

*q*

_{5}

*m*

_{wϕ}

*C*

_{i}

*T*

_{sϕ}

*m*

_{wϕ}

*C*

_{i}

*T*

_{sϕ}

*C*

_{w}and

*C*

_{i}are the specific thermal capacity of water and ice, respectively;

*L*

_{f}is the specific latent heat of water freezing at temperature 0°C; and

*m*

_{wϕ}is the mass flux of the supercooled water deposited on a unit surface of the cylinder surface during a unit time (kg m

^{−2}s

^{−1}). The heat flux

*q*

_{3}is negative, while the fluxes

*q*

_{4}and

*q*

_{5}are positive.

*m*

_{wϕ}is associated with the flow of droplets, and can be presented as

*m*

_{wϕ}

_{ϕ}

*UW*

*ϕ,*

*W*is the liquid water content; ɛ

_{ϕ}= ɛ

_{ϕ}(

*ϕ,*Stk, Re) is the local collision efficiency; Stk = (2

*Ur*

^{2})/(9

*μR*

_{c})

*ρ*

_{w}is the Stokes number; Re = (2

*Ur*)/

*υ*is the Reynolds number for droplets of radius

*r*;

*R*

_{c}is the radius of cylinder;

*μ*and

*ν*are the dynamic and kinematic viscosity of the air, respectively; and

*ρ*

_{w}is the water density. The product ɛ

_{ϕ}cos

*ϕ*is the ratio of the distance Δ

*y*between the neighboring trajectories of droplets of radius

*r*far away from the cylinder to the length of the arc Δ

*s*resulting from the intersection of the trajectories and the cylinder (Fig. 1). In case of a polydisperse droplet size distribution

*n*(

*r*) the local collision efficiency should be replaced by the integral one

*q*

_{1}and

*q*

_{2}make the major contribution to the equation of heat balance Eq. (2), while the flux

*q*

_{6}is about one–two orders magnitude smaller compared to other terms in Eq. (2). Therefore, the term

*q*

_{6}is neglected. The radiation flux is also neglected and it is assumed that there is no heat transfer along the surface of the cylinder (Lozowski and d’Amours 1980; Makkonen 1981). Combining Eqs. (2)–(4) and Eqs. (8)–(12) gives an equation for the steady-state heat balance:

Equation (15) describes the heat balance at a local point on the surface of the cylinder. The RICE probe response depends on the mass of accreted ice and its distribution on the surface of the cylinder (Baumgardner and Rodi 1989). In the following discussion it is assumed that ice is distributed uniformly along the axis of the cylinder.

### b. Integral heat balance

Here *T*_{s} is the temperature averaged over the surface covered by ice, that is, over the sector limited by the angles −*ϕ*_{0} to *ϕ*_{0}; *L*_{i} = *L*_{i}(*T*_{s}) is the latent heat of evaporation at temperature *T*_{s}; *E*_{i} = *E*_{i}(*T*_{s}) is water vapor pressure saturated over ice at temperature *T*_{s}, ɛ = ∫ ɛ_{ϕ} cos*ϕ* *dϕ* is the integral collision efficiency (appendix B), *α*_{ϕ}*ξα* is the heat transfer coefficient for a cylinder averaged over the angle range −*ϕ*_{0} to *ϕ*_{0}, *α* is the heat transfer coefficient averaged over the whole cylinder, that is, −*π* < *ϕ*_{0} < *π* (appendix A); *ξ* is a coefficient defined by the angle *ϕ*_{0} (appendix A).

These assumptions work with a reasonable accuracy, if changes of the variables along the surface are relatively small. Estimates show that if *U* < 180 m s^{−1} *α*_{ϕ} and *T*_{s} change by not more than ±30%. The averaged product of the above variables deviate from the product of the averaged ones by no more than 10%–15%.

## 3. Threshold liquid water content

*W*

_{e0}is derived, which balances the ice sublimation. The mass of sublimating ice per unit time

*M*

_{e}can be found by integrating Eq. (7) over the icing surface as

*l*is the length of icing cylinder. The liquid water content, which balances the sublimating ice,

*M*

_{e0}, is

*W*

_{e0}from Eq. (20) we require

*E*

_{i}, which is a function of

*T*

_{s}. To find

*T*

_{s}, substitute Eq. (20) in Eq. (16) and take into account that

*L*

_{i}+

*C*

_{i}

*T*

_{s}=

*L*

_{i0}(0°C) to obtain

Equation (21) contains only one unknown variable *T*_{s}, which can be computed using numerical methods.

Figure 3 shows the dependence of the difference *T*_{s} − *T*_{a} versus the air velocity *U* for different: (a) air temperatures *T*_{a}, (b) relative humidity *e*_{a}/*E*_{w}, and (c) air pressure *P*_{a}. As is seen from Fig. 3, the ice sublimation results in additional cooling of the surface compared to a dry cylinder (curve *T*_{s} = *T*_{sA}). The effect of cooling depends on *T*_{a}, *e*_{a}/*E*_{w}, *P*_{a}, and *U,* and it increases with an increase of the air temperature (Fig. 3a), a decrease of relative humidity (Fig. 3b), and a decrease of pressure (Fig. 3c). At *U* = 100 m s^{−1} the effect of cooling is about 2°C (Fig. 3a at *T*_{a} = −5°C, *P*_{a} = 800 mb), while at *U* = 200 m s^{−1} the effect increases up to 9°C (Fig. 3c at *T*_{a} = −10°C, *P*_{a} = 400 mb). At low temperatures (*T*_{a} < −40°C) the cooling effect becomes insignificant (Fig. 3a). Under certain conditions [i.e., when *e*_{a} > *E*_{i}(*T*_{s})] the effect of ice sublimation may exceed the adiabatic heating and the temperature of the surface of the ice becomes lower than the temperature of the air (Fig. 3b). Such situations may occur if the air is dry. This phenomenon is similar to the cooling of a wet thermometer. The analysis of Eq. (21) shows that under certain conditions the deposition of ice on the surface of the cylinder from water vapor is possible. It is worth noting that *T*_{s} in Eq. (22) does not depend on the collection efficiency ɛ and the angle *ϕ*_{0}.

Figure 4 shows the dependence of *W*_{e0} on airspeed using Eq. (20), for different air pressures and humidity. Though the zero relative humidity case is unrealistic for in-cloud regions, the examples of *W*_{e0} calculated for zero humidity give an idea about the behavior of *W*_{e0}. This may be useful for studies of icing in wind tunnels or simulated icing behind air tankers where the humidity may be significantly less 100%. For calculations of *W*_{e0} the radius of the droplets was assumed to be *r* = 10 *μ*m and the collision efficiency ɛ, angle *ϕ*_{0}, and coefficient *ξ* were calculated based on this droplet size (appendixes A and B).

It is seen from Fig. 4 that *W*_{e0} increases with increasing temperature and airspeed in saturated air. The characteristic values of *W*_{e0} in saturated air at *U* = 100 m s^{−1} is of the order of 0.005 g m^{−3}. At *U* = 200 m s^{−1} *W*_{e0} exceeds 0.01 g m^{−3} at *T*_{a} > −20°C.

The value *W*_{e0} is an important parameter, since it defines a theoretical lower threshold for liquid water content measurements by the RICE probe; that is, the RICE probe cannot measure liquid water content less than *W*_{e0}. If *W* = *W*_{e0} the RICE probe will measure zero LWC, and if *W* < *W*_{e0} the RICE signal will decrease, if it was not zero. Another important conclusion is that it is necessary to make corrections for the mass of sublimating ice when the LWC is of the order of 0.01 g m^{−3}, and it becomes comparable with the amount of sublimating ice. This is important for aircraft with speeds higher than 150 m s^{−1}.

## 4. Saturated liquid water content (Ludlam limit)

An increase of the LWC leads to an increase of the heat flux *q*_{4}. At some LWC the heat of freezing will increase the surface temperature *T*_{s} to 0°C. Further increase in the LWC will not result in an increase in the surface temperature. In this case only a fraction of the supercooled water freezes. This fact may be taken into account by introducing the coefficient of freezing (Mazin 1957; Borovikov et al. 1963). The minimum value of the LWC *W* = *W*_{cr} at which *T*_{s} reaches 0°C is called the Ludlam limit (Ludlam 1951). The unfrozen fraction of liquid water may shed away with the airflow or freeze at the backside of the cylinder (runback icing), or may be partly incorporated into a spongy ice structure similar to hail growth (Greenan and List 1995). For the RICE probe an increase in the LWC above *W*_{cr} would cause a dropoff of the rate of change of the output signal, so that for LWCs *W* > *W*_{cr} cannot be accurately measured by the RICE probe. Baumgardner and Rodi (1989) and Cober et al. (2001) demonstrated this effect in RICE measurements when the Ludlam limit is reached. Thus, *W*_{cr} can be considered as an upper limit of supercooled LWC that can be measured by RICE probe.

*W*

_{cr}can be derived from Eq. (16), assuming

*T*

_{s}= 0°C and that the vapor pressure is equal to the saturated value with respect to water

*e*

_{a}=

*E*

_{w}(

*T*

_{a}):

The results of *W*_{cr} calculations are presented in Fig. 5. As seen in Fig. 5, *W*_{cr} increases rapidly with decreasing *T*_{a} and an increasing *U.* At temperatures *T*_{a} > −10°C and *U* > 100 m s^{−1} the situation when *W* > *W*_{cr} is quite typical for clouds and the Ludlam limit may be easily reached.

## 5. Calculation of *T*_{s}, *W*_{e}, *W*

*M*is the total mass of droplets that impact the surface of the cylinder and freeze during the time period Δ

*t.*Some part of the ice

*M*

_{e}will sublimate and another part

*M*

_{a}will stay on the cylinder as accreted ice. The mass balance on the surface yields

*M*

*t*

*M*

_{a}

*M*

_{e}

*t*

*R*

_{c}

*l*

*U*

*W*

_{a}

*W*

_{e}

*t,*

*W*

_{a}and

*W*

_{e}are the fractions of LWCs that accrete as ice on the cylinder and sublimate, respectively. Since only

*M*

_{a}causes the RICE probe response, Eq. (23) using Eq. (1) can be rewritten as

The values of *T*_{s}, *W*_{e}, and *W,* can be found iteratively using Eqs. (16) and (20). For calculation of *W* and *T*_{s} at *U* = 175 m s^{−1}, *P*_{a} = 400 mb, *T*_{a} = −20°C, and *W*_{m} < 0.1 g m^{−3} with accuracy *δT* = 1°C, and *δW* = 0.002 g m^{−3} 12 iterations are required. The number of iterations increases with an increase in temperature and decrease of the air pressure.

## 6. RICE probe calibration

The objective of the calibration is to find the coefficient *k,* which relates the signal *V* and the mass of the accreted ice *M* on the RICE probe cylinder [Eq. (1)].

### a. “Sublimating” technique

The conventional techniques derive the coefficient *k* for the cases when ice is growing on the cylinder, that is, *dM*/*dt* > 0 (Baumgardner and Rodi 1989; Heymsfield and Miloshevich 1989; Claffey et al. 1995). However, Eq. (1) does not imply any limitations on the sign of *dM*/*dt.* Therefore, there is a possibility to calibrate the RICE by measuring the rate of ice sublimation in cloud-free air. Figure 6 shows an example of the time history of the RICE probe signal during ice accretion inside clouds and sublimation in the cloud-free regions. After leaving cloud (Fig. 6c), ice retained on the RICE probe started to sublimate in cloud-free air resulting in a decrease in the RICE probe signal (Fig. 6d). The measured rate of the RICE probe signal reduction Δ*V*/Δ*t* and the rate of ice sublimation calculated theoretically based on the measurements of *U, T*_{a}, *P*_{a}, *e*_{a} can be used for calibration of the probe.

Figure 7 shows the rate of ice sublimation derived from Eqs. (16) and (19) for the assumption *W* = 0. The rate of sublimation increases rapidly with an increase of the airspeed and decrease of the humidity. For example, at *T*_{a} = −20°C and *e*_{a} = *E*_{w} the rate of ice sublimation at *U* = 200 m s^{−1} is approximately 14 times larger than that at *U* = 100 m s^{−1}. This illustrates how significant ice sublimation can be for high-speed airplanes. It is worth mentioning that ice may grow up on the cylinder at low airspeeds even when flying in cloud-free air. Thus, at temperature *T*_{a} = −20°C and *e*_{a} = *E*_{w} the ice starts to build up at *U* < 60 m s^{−1} (Fig. 7b). It happens when the vapor pressure in the air becomes larger than the saturation pressure over ice at the surface temperature.

*k*is related to the rate of ice evaporation

*M*

_{e}as

The advantage of this technique is that it derives the coefficient *k* from first principles and does not need reference measurements of liquid water content.

For calibration purposes, a flight leg should be chosen in cloud-free air where the pressure, air temperature, humidity, and airspeed stay approximately constant. Measuring the signal changes Δ*V* during time Δ*t* and substituting into Eq. (26) gives the calibrating coefficient *k.* The vapor pressure *E*_{s} is calculated using the temperature *T*_{s}, whereas the temperature *T*_{s} is calculated from Eq. (16) using in situ measurements of *U, T*_{a}, *P*_{a}, *e*_{a}, and assuming *W* = 0.

Figure 8 presents results of calibrations of three different RICE probes mounted on three different aircraft:Convair-580, King Air, and Sabreliner. In the calculations of *k* in Eq. (26) it was assumed *l* = 2.54 cm, *R*_{c} = 0.317 cm, *ξ* = 1.41, *ϕ*_{0} = 1.48 rad (appendix A). The airspeed *U,* temperature *T*_{a}, pressure *P*_{a}, and dewpoint temperature *T*_{dew} were measured by the aircraft instrumentation. Water vapor pressure *e*_{a} was calculated from the dewpoint temperature *T*_{dew} measured by an EG&G dewpoint thermometer. Figure 8a shows the results of calibration of the Meteorological Service of Canada (MSC) RICE probe mounted on the National Research Council (NRC) Convair-580. The measured parameters changed in the ranges 85 m s^{−1} < *U* < 115 m s^{−1}, −23°C < *T*_{a} < −7°C, 0.46 mb < *e*_{a} < 2.6 mb, 525 mb < *P*_{a} < 950 mb. Figure 8b shows the results for the National Center for Atmospheric Research (NCAR) RICE probe installed on the King Air. The measured parameters changed in the ranges 115 m s^{−1} < *U* < 140 m s^{−1}, −29°C < *T*_{a} < −9°C, 0.18 mb < *e*_{a} < 1.5 mb, 390 mb < *P*_{a} < 550 mb. Figure 8c shows the results for another RICE probe installed on the NCAR Sabreliner. The measured parameters changed in the ranges *U* = 180 m s^{−1}, −34°C < *T*_{a} < −30°C, 0.07 mb < *e*_{a} < 0.18 mb, 365 mb *P*_{a} < 375 mb.

The RICE coefficients *k* are found to be noticeably different and equal to 1.40 × 10^{−5} ± 0.14 × 10^{−5} kg V^{−1} (MSC Fig. 8a); 2.13 × 10^{−5} ± 0.39 × 10^{−5} kg V^{−1} (NCAR Fig. 8b); 3.49 × 10^{−5} ± 0.41 × 10^{−5} kg V^{−1} (NCAR Fig. 8c). This result is consistent with the study of Baumgardner and Rodi (1989) who stated that the coefficient *k* may be significantly different from probe to probe. Another explanation may be related to the effect of the RICE probe installation location on the aircraft. If the cylinder is mounted too close to the fuselage or the wing surface, it may cause nonuniform deposition of ice on the cylinder due to the gradient of the airspeed in the vicinity of the airplane surface. This may result in different length *l* of the ice deposition along the cylinder’s axis, so that the length *l* may not necessarily be equal to the length of the cylinder. The errors related to changes of local temperature and, consequently, *E*_{s} are relatively small and cannot explain the observed difference in *k* (see sections 4 and 8).

### b. Retrieval of liquid water content

The adequacy of the proposed method was tested by a comparison of LWC measured from other probes (*W*_{m}) with the LWC retrieved from RICE measurements (*W*_{r}) using calibrating coefficient *k* derived from the “sublimation” technique.

Figure 9 shows a scatterplot of *W*_{m} versus *W*_{r}. The measurements of LWC *W*_{m} were conducted with the help of the airborne hot-wire Nevzorov probe (Korolev et al. 1998) installed in the NRC Convair-580. The accuracy of the Nevzorov probe in measurements of cloud LWC is estimated as 10%. The data were collected mainly in stratiform clouds associated with frontal systems during the Third Canadian Freezing Drizzle Experiment (Isaac et al. 1998) in December 1997–February 1998.

*W*

_{r}was calculated as

*V*is the RICE probe signal change during time Δ

*t.*The time period Δ

*t*for each measurement was taken after the temperature of the cylinder cools down to the air temperature. This was done to avoid the effect of the cylinder residual overheating after a de-icing cycle (Baumgardner and Rodi 1989). The time periods Δ

*t*were selected manually for each RICE probe icing cycle and they varied from 2 to 75 s. The measured LWC

*W*

_{m}shown in Fig. 9 was averaged over a time period Δ

*t.*Figure 9 shows over 710 different triggering cycles of the RICE probe at temperatures −6°C <

*T*

_{a}< −23°C, airspeeds 80 m s

^{−1}<

*U*< 130 m s

^{−1}, and pressures 400 mb <

*P*< 900 mb, representing a dataset that covers a large variety of cloud situations. Figure 9 indicates that on average

*W*

_{r}and

*W*

_{m}are in good agreement, though the scatter is relatively high. The average ratio of the measured and retrieved LWC

*W*

_{r}/

*W*

_{m}= 1.11 ± 0.35, and the correlation coefficient is 0.85. This comparison supports the “sublimating” technique for the RICE probe calibration.

### c. “Ice accretion” technique

*k*obtained for the MSC RICE probe using the “sublimating” technique was compared to that calculated using the conventional technique, that is, when the ice was growing up on the cylinder’s surface. The coefficient

*k*has been calculated as

Figure 10 shows the scatterplot of the rate of ice accretion *dM*/*dt* = 2*R*_{c}*lU**W*_{m}, calculated from in situ measurements of *U* and *W*_{m} versus the rate of the RICE probe signal changes Δ*V*/Δ*t.* Here *W*_{m} is measured LWC averaged over time Δ*t.*

The average value of the coefficient calculated for the dataset shown in Fig. 10 is *k* = 1.35 × 10^{−5} ± 0.42 × 10^{−5} kg V^{−1}, and the correlation coefficient is 0.88. The value of *k* calculated using the ice accretion technique is rather close to that obtained using the sublimating technique 1.40 × 10^{−5} ± 0.14 × 10^{−5} kg V^{−1}, though the dispersion is three times higher.

## 7. Discussion

### a. Accuracy of the sublimating technique

The accuracy of the RICE probe calibration using the “sublimating” technique depends on the accuracy of measurements of *U, T*_{a}, *P*_{a}, *T*_{dew}, and assumptions about *R*_{c}, *l,* ɛ, *ϕ*_{0}, *α,* and *ξ.* The first four measurable parameters result in random errors and are mainly responsible for the scatter in Fig. 8. The remaining parameters would result in systematic errors. The angle *ϕ*_{0} is defined by Re and Stk numbers and it increases approaching *π*/2 with an increase of droplet size. During sublimation, the angle *ϕ*_{0} decreases as well as the length *l* and the radius *R*_{c} of the iced cylinder. During aircraft measurements, it is not possible to control *R*_{c}, *l,* or *ϕ*_{0}. For the calculations presented in Fig. 8 the following assumptions regarding *R*_{c}, *l, ϕ*_{0}, ɛ, and *ξ* were made.

The length of ice is equal to the length of the RICE probe cylinder, that is,

*l*= 2.54 cm.The radius of curvature of the ice is equal to the cylinders radius

*R*_{c}= 0.317 cm.The collision efficiency ɛ was calculated based on Re and Stk numbers (appendix B). In the calculation it was assumed that

*r*= 10*μ*m. For most cases ɛ was no less than 0.9.The angle is equal to

*ϕ*_{0}= 1.48 rad for ɛ = 0.9. Angle*ϕ*_{0}is a function of Re and Stk.The coefficient

*ξ*= 1.41 for the angle*ϕ*_{0}= 1.48 (appendix A).

*k*:

During ice sublimation, *ϕ*_{0} and *l* decrease, whereas *ξ* and *E*_{i} increase. The net effect of the ice sublimation results in a decrease of the product *R*_{c}*lϕ*_{0}*ξ*(*E*_{is} − *e*_{a}). For calibration purposes, the vague Δ*V*/Δ*t* should be measured at the beginning of sublimation, before *ϕ*_{0}, *l,* and *R*_{c} start to change.

A variation of *r* within 2 *μ*m results in errors *ξϕ*_{0} within 3%–5%. The value of *ξα* changes no more than 5%. The relative effect of uncertainty in *R*_{c}*l* is estimated as 3%–5%. The air pressure *P* can be measured with high accuracy so that the relative errors due to *P* are less than 1%. The errors of 1°C in measurement of *T*_{a} and *T*_{dew} can result in relative errors in the term (*E*_{is} − *e*_{a}) of 10%–20%. The relative error in measurements (*E*_{is} − *e*_{a}) may increase when *e*_{a} is close to saturation. The net effect of all errors may reach easily 20%–30%, which explains the scatter in diagrams in Fig. 8.

### b. Accuracy of ice accretion technique

The accuracy of the RICE probe calibration using the rate of the ice growth on the cylinder depends significantly on the accuracy of the measurement of LWC by a reference probe. Usually, either the FSSP or hot-wire probes such as the King or Nevzorov probes are used as a reference. In the best case for the hot-wire probes the accuracy of the measurements of LWC is about 10%–15%. For LWC derived from FSSP measurements the errors may be up to 30% or more for liquid water clouds. For mixed clouds the FSSP measurements may be “contaminated” by ice particles and that may result in much higher errors. The net effect of uncertainties in *R*_{c}, *l,* and ɛ may result in errors about 20%. The measurements of true airspeed are rather accurate compared to other sources of errors and therefore they can be neglected. The resulting accuracy of the conventional RICE probe calibrating technique is estimated as 25%–30%.

However, the above estimate does not explain the scatter of the data in Figs. 9 and 10 where the retrieved and measured LWC may be different by a factor of 2. An additional source of error may be related to the changes of the airflow around the cylinder due to different aircraft maneuvers, such as changes of the angle of attack, climbs, descents, turns, etc. These may affect the local airflow around the cylinder or result in partial aerodynamic shadowing of the cylinder.

It should be noted that the RICE probe has a significant “dead” (or “delay”) time compared to other probes used for the measurements of LWC. During this time period the cylinder is heated to de-ice its surface and then it cools down to the air temperature. The heat balance on the surface of the icing cylinder cannot be considered as a steady state during the dead time, and the theoretical consideration developed above is not applicable for these conditions. Such periods must be excluded from measurements of Δ*V*/Δ*t.* Dead time is a function of *W, T*_{a}, *P*_{a}, *U,* and it may vary from 10% to 90% of the measurement cycle. The fraction of the dead time increases with an increase in *U* and *W* and decrease of *T*_{a}. Similar results were found by Baumgardner and Rodi (1989).

### c. Measurements in mixed and glaciated clouds

Though the RICE probe shows relatively large scatter in measurements of LWC (Figs. 8, 9), it can be effectively used in studies of mixed and glaciated clouds. For now it is the only known aircraft instrument used in cloud physics measurements that responds to the liquid phase and is insensitive to ice particles. It should be emphasized that the imaging probes, such as PMS OAPs or CPI, do not provide phase recognition for circular particles, since frozen droplets would appear in 2D imagery in the same way as liquid ones. At the same time the hot-wire probes like the King or Nevzorov probes may have a residual effect of ice particles on measurements (Korolev et al. 1998). The following example demonstrates the capability of the RICE probe in mixed and glaciated environment. Figure 11 shows synchronous measurements of the RICE probe signal and LWC derived from FSSP data. The measurements were conducted by NCAR Sabreliner in wave clouds in 1990 (Heymsfield and Miloshevich 1993). It is clearly seen that the RICE probe repeatedly triggers in approximately half of each cloud during the penetration through it. These triggers are caused by the portion of cloud containing some liquid. In the other parts of the clouds the RICE signal is decreasing due to ice sublimation, which indicates zero LWC and, consequently, complete glaciation of the cloud. This example also demonstrates that FSSP may give rather erroneous measurements of *liquid* water content in cold clouds. This would create problems in calibrating the RICE probe using the FSSP measurements of LWC.

## 8. Conclusions

The above study results in the following conclusions.

Theoretical examination of the rate of ice accretion on the RICE probe cylinder showed that the minimum measured liquid water is limited by the lower threshold

*W*_{e0}[Eq. (20)]. A liquid water less that*W*_{e0}cannot be detected by the RICE probe. The value of*W*_{e0}is mainly defined by*T*_{a}, and*U.*For high airspeeds (*U*> 200 m s^{−1}) the threshold value*W*_{e0}may well exceed 0.01 g m^{−3}, which may become comparable with the measured cloud liquid water content.The upper limit of the measured LWC by the RICE probe is determined by the Ludlam limit

*W*_{cr}[Eq. (22)]. The value of*W*_{cr}depends on*P, T*_{a}, and*U.*For typical aircraft speeds (80 m s^{−1}<*U*< 200 m s^{−1}) in the temperature range −5°C <*T*_{a}< −15°C the Ludlam limit for the RICE probe ranges from 0 to 0.8 g m^{−3}(Fig. 5) and such conditions can be easily met in clouds. If the liquid water content exceeds*W*_{cr}the RICE probe cannot be used for accurate measurements of LWC.A new technique for the calibration of the RICE probe based on the measurements of the rate of ice sublimation in cloud-free air is proposed. The advantage of the method is that it derives the calibration coefficient from first principles and it does not require reference measurement of LWC by other probes. The conventional RICE probe calibration technique is based on the measurement of the rate of ice growth and it needs in addition measurements of LWC. The comparative analysis in section 7c showed that the accuracy of the sublimating calibration technique is better compared to the conventional ice accretion technique.

The range of possible supercooled LWCs measured by the RICE probe is limited by

*W*_{e0}and*W*_{cr}. Since*W*_{e0}increases and*W*_{cr}decreases with an increase of the airspeed*U,*the range of measured LWC decreases with the increase of*U.*Therefore, the use of the RICE probe for measurements of supercooled LWC from slow moving platforms would be more effective compared to high-speed ones. For example at*U*> 200 m s^{−1},*T*_{a}> −10°C, and*P*> 800 mb the RICE probe cannot be used for proper measurements of LWC since*W*_{cr}= 0 (Fig. 5). Though the RICE probe will trigger under these conditions due to partial freezing of water on the upwind surface of the cylinder and runback icing. The use of the RICE at low airspeeds is also preferable because of a smaller fraction of the dead time compared to high speed under the same conditions.Low accuracy, large dead times, and complex data postprocessing may significantly limit the capability of RICE probe for measurements of supercooled cloud liquid water. That is why in aircraft cloud microphysics studies the RICE probe is mainly used as a detector of supercooled liquid water. This is one of the most advantageous properties of the RICE probe, since the phase discriminating capability of hot-wire probes are limited due to the residual effect of ice particles (Korolev et al. 1998) and FSSP does not provide reliable LWC measurements in mixed-phase conditions. Nevertheless the RICE probe could be used as a useful additional tool for estimation of LWC values.

## Acknowledgments

The authors would like to thank the financial support of the Canadian National Search and Rescue Secretariat and Transport Canada. In addition, we are grateful for the support from colleagues at the National Research Council of Canada and others at the Meteorological Service of Canada. Special thanks for Dave Rogers and two anonymous reviewers for valuable comments that certainly helped to improve the manuscript.

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## APPENDIX A

### Heat Transfer Coefficient

*α*

_{ϕ}is a function of the polar angle

*ϕ.*Figure A1 shows the dependence of

*α*

_{ϕ}/

*α*versus

*ϕ*(Mikheev 1949), here

*α*is the heat transfer coefficient averaged over the whole cylinder, that is, −

*π*<

*ϕ*<

*π.*The average value of the heat transfer coefficient

*α*can be expressed in terms of the Nusselt number

*λ*is thermal conductivity of air. For the flow around circular cylinder the Nusselt number is related to the Reynolds number as (e.g., Mikheev 1949; Kreith and Bohn 1986)

*A*

^{n}

*ν*is kinematic viscosity. Due to Mikheev (1949) for Re > 5 × 10

^{3}the coefficients

*A*= 0.197 and

*n*= 0.6. For the RICE probe Re > 5 × 10

^{3}corresponds to the airspeeds

*U*> 10 m s

^{−1}at

*P*

_{a}= 1000 mb and

*T*

_{a}= 0°C. Therefore, the values of coefficients

*a*and

*n*are applicable for the following calculations.

*α*

_{ϕ}for the range of angles −

*ϕ*to

*ϕ*can be found as

*α*

_{ϕ}

*ξα.*

The coefficient *ξ* can be obtained by integrating the curve in Fig. A1 for the angle range 0 to *ϕ.* For *ϕ* = 1.48 rad the integrating results in *ξ* = 1.41.

## APPENDIX B

### Collision Efficiency

The ratio of the distance between glancing droplet trajectories hitting the cylinder in undisturbed air to the cross-section of a body is called the total collision efficiency ɛ. The collision efficiency for cylinder is a function of Stokes Stk = (2*Ur*^{2})/(9*μR*_{c})*ρ*_{w} and Reynolds Re = (2*Ur*)/*υ* numbers, where *R*_{c} is the radius of cylinder;*μ* and *υ* are the dynamic and kinematic viscosity, respectively; and *ρ*_{w} is the water density. Figure B1a shows the diagram of ɛ isolines in the Re–Stk coordinates calculated from Langmuir and Blodgett (1945), based on solution of equation of droplet motion in the air flowing around the circular cylinder (Mazin 1957). The polar angle *ϕ* defines the sector within which the droplets impact with the cylinder. At angles larger |*ϕ*| droplets do not hit the surface of the cylinder and flow away with the air. The angle *ϕ* is a function of Re and Stk numbers. Figure B1b show the isolines of *ϕ* in the Re–Stk coordinates (Mazin 1957). The diagrams in Fig. B1 were used for calculation *ϕ* and ɛ. For example, for *U* = 100 m s^{−1}, *r* = 7 *μ*m, *T*_{a} = −20°C, *P*_{a} = 600 mb, we find Stk ≅ 21, Re ≅ 72, ɛ ≅ 0.84, *ϕ* = 82°. For *U* = 100 m s^{−1}, *r*_{3} = 10 *μ*m*t*_{a} = −10° C, *P*_{a} = 800 mb, and Stk ≅ 42, Re ≅ 130, ɛ ≅ 0.88, and *ϕ*_{0} = 84°.