## Abstract

To correct time lag errors in radiosonde temperatures the sensor time constant has to be known. Time constants are not published for some widely used sensors and, in some cases, available time constants disagree. This study focuses on ML-405, ML-419, VIZ/Sippican Mark II Microsonde and B2, Russian MMT-1, and Chinese GZZ-7 rod thermistors. It measures still air time constants and heat capacities and derives theoretical still air and aerated time constants based on heat transfer involving nonuniform cylinders. With low aeration, such as in the stratosphere, heat conduction by lead wires from the thermistor noticeably shortens the time constant. Some discrepancies in published time constants are explained by researchers not considering the temperature dependence of all relevant variables. Empirical formulas are derived to estimate the aerated time constant of cylindrical temperature sensors based on dimensions. The aerated time constant in soundings is found to be about 6 times as long at 10 hPa as near sea level.

## 1. Introduction

Accurate radiosonde temperature profiles require temperature measurements assigned to correct heights. The temperature sensor responds more slowly than the pressure sensor as the radiosonde ascends, so reported temperatures are averages of a layer below the radiosonde, and the lag increases with height as density decreases. Temperature errors are largest in strong gradients. The boundary layer gradient is systematically underestimated because a radiosonde is supposed to be at equilibrium with surface conditions when launched and inversions and tropopauses are smoothed and displaced upward. Climate trends computed from archived soundings are distorted as radiosonde sensor lag times have systematically decreased.

Most operational soundings have had no lag corrections (e.g., Luers and Eskridge 1995). Only a few documented operational lag corrections are found as follows: Russia RZ-049 from 1957 to the early 1960s (Mordukhovich 1960), U.K. Mark IIB from 1956 to 1977 (Hawson 1956), Graw DFM-06 and DFM-09 starting in 2006 (http://www.isac.cnr.it/wavacs/sites/default/files/WG1%20Helsinki%202010%20report.pdf dated 2010), AN/AMT-3 dropsondes in the 1950s (Air Weather Service 1956), and AN/AMQ-9 radiosondes used at U.S. Air Force launch ranges from ≈1960 to 1980 [S. Spratt 2009, personal communication, National Weather Service (NWS), Melbourne, Florida]. The WMO sounding code format (World Meteorological Organization 2011) provides for listing the radiation correction (Code Table 3849) but no lag correction.

Because radiosondes with rod thermistors have been very widely used since the 1940s, this study focuses on determining lags of cylindrical temperature sensors under cross-flow (in the list above, only AN/AMQ-9 uses a rod thermistor). A time constant *τ* is the time (in seconds) for the temperature difference of a thermistor from the surrounding air to decrease to *e*^{−1} ≈ 37% of its initial value, either in still air or when aerated (ventilated) by the radiosonde’s ascent. For many thermistors, the time constant is unpublished, and published time constants differ for the ML-405 sensor by almost a factor of 2.5.

This study measures time constants of rod thermistors from several obsolete and recent radiosondes. Section 2 uses heat transfer theory to derive still air and aerated time constants. Section 3 describes still air time constant measurement experiments, and reconciles published and computed aerated time constants, to develop best fitting aerated time constant formulas. Section 4 summarizes uses of time constant formulas toward making lag corrections of archived soundings.

Thermistor time constants and heat capacities were determined in still air because a wind tunnel was not available. General formulas are developed to compute the Nusselt number describing heat exchange under natural and forced convection, and the aerated time constant using measured heat capacities in still air and available published aerated time constants. Published aerated time constants are quite well reproduced, so aerated time constants are estimated for thermistors from Mark II Microsonde and B2 radiosondes (made by VIZ, which was acquired by Sippican in 1997), for which published time constants have not been found.

It was found that thermally active thermistor dimensions depend on the amount of convection. Lead wires contribute significantly to heat exchange with low aeration, such as in still air near sea level or while ascending in the stratosphere. Published wind tunnel experiments generally did not reveal this lead wire property because aerated time constants of multiple thermistors were not compared based on physical differences as done here.

## 2. Theoretical considerations

All variables and parameters used in this study are described, including units, in the appendix.

Where Δ*T*(*t*) *= T*(*t*) *− T _{∞}* is the temperature difference between a sensor

*T*(

*t*) and the surrounding air

*T*, the thermal time constant

_{∞}*τ*is defined using Newton’s law of cooling:

For initial sensor temperature *T*(*t =* 0), the sensor approaches *T _{∞}* according to

The time constant of a thermistor is generally related to its physical properties by

where volume *V*, surface area *A*, convective heat transfer coefficient *h*, total sensor heat capacity *C* = Σ *c _{i}*

*ρ*

_{i}*V*(

_{i}*c*,

_{i}*ρ*, and

_{i}*V*are specific heat, density, and volume of thermistor component

_{i}*i*, respectively), and

*c′*=

*C*/

*V*which denotes volumetric heat capacity.

These equations assume the entire thermistor is isothermal at all times. When *τ* is measured in thermistor resistive heating and cooling experiments, heat capacities computed using (3) are less than the total heat capacity of thermistor materials. The unexpectedly small measured heat capacities are caused by factors (discussed separately in section 3.4) such as temperature gradients within the thermistor, so the “apparent” thermistor heat capacity *Ç* and the apparent volumetric heat capacity *ç′* are substituted for *C* and *c′* in (3):

The heat transfer coefficient is

where Nu is the Nusselt number, *k* is the air thermal conductivity, and *d* is the characteristic dimension (relevant thermistor diameter).

Heat transfer occurs by natural convection in still air, and by forced convection in moving air. Hereafter, variables use subscript *o* to denote a still air value or subscript ν for an aerated (ventilated) value. Because air properties vary with pressure and temperature, when variables or measurements are expressed at standard conditions defined below, subscript *o*Std indicates the still air standard condition, and subscript νStd indicates the ventilated standard condition.

Researchers often approximate the Nusselt number as a simple power function of the Rayleigh number Ra with natural convection or the Reynolds number Re with forced convection (e.g., Morgan 1975), having dimensionless proportionality factor *a* and exponent *m*:

where Ra = (*g*Δ*Tβd*^{3})/(*ηα*) with gravity *g*, temperature difference Δ*T*, air expansivity β, kinematic viscosity η, and thermal diffusivity α; and Re = (*ρυd*)/*μ* with air density ρ, air velocity υ, and dynamic viscosity μ. These are approximations because Nu_{ν} approaches Nu* _{o}*, not 0, as

*υ*diminishes. This study seeks to derive a common set of Nusselt parameters

*a*,

_{o}*m*,

_{o}*a*, and

_{v}*m*that applies to all relatively long cylindrical thermistors.

_{v}Military specification MIL-PRF-23648F (Defense Logistics Agency 2009, paragraph 4.8.11.1) describes how to determine a still air time constant *τ _{o}* by electrically heating the thermistor to 75°C and measuring its cooling. For meteorological purposes, this study defines the standard condition for the still air time constant

*τ*

_{o}_{Std}as 20°C ambient air at pressure 1000 hPa with the thermistor cooling from 35° to 25.5°C (

*e*

^{−1}of the original 15° difference). The standard condition for the aerated time constant

*τ*

_{v}_{Std}is 25°C at 1000 hPa with 5 m s

^{−1}aeration. Since the standard still air condition is a temperature range, temperature-dependent variables used to derive

*τ*

_{o}_{Std}are computed at the relevant time-averaged mean temperature between

*t*= 0 and

*τ*≈ 29.5°C (average of cooling from 35° to 25.5°C) for the thermistor, or ≈24.7°C (average of thermistor and ambient air) for air properties at the thermistor surface.

The heat capacity of typical thermistor materials varies with temperature. Defining relative volumetric heat capacity *f _{ç}* as

*f*= 1 at 25°C,

_{ç}the following function is obtained by fitting Moncur (1969, Fig. 5) measurements for ML-405 between ≈50° and −60°C:

All thermistors analyzed here are assumed to have the same relative temperature dependence. Section 3c(2) results for ML-405 and MMT-1 thermistors are consistent with this assumption. For other thermistors this correction is only applied within smaller temperature ranges.

Since temperature sensing is performed in the main cylinder of a thermistor, initial analysis treated thermistors as long cylinders (*V*/*A* ≈ *d*_{cyl}/4, where *d*_{cyl} = main cylindrical diameter). For different thermistors with similar *ç′* all factors in (10)–(11) except *d* are constant, so and , or in both cases to roughly *d*^{1.5} based on the final Nusselt parameters in this study. ML-405 and MMT-1 thermistors have relatively similar measured *ç′* but MMT-1 is thicker and shorter. As expected, MMT-1 has a longer aerated time constant than ML-405, but their measured still air time constants are unexpectedly almost the same. Apparently, in still air the cylinder ends and thin lead wires rapidly conduct heat from the MMT-1 thermistor, reducing the MMT-1 still air time constant. Huang (1987) found that lead wires cannot be ignored at low convection, and Luers and Eskridge (1995) considered them when deriving radiation corrections.

This effect is modeled by replacing thermistor diameter *d* in (10)–(11) by a thermally active or “apparent” mean diameter *đ*, which is the length-weighted mean of its different cylindrical parts that affect the temperature reading (main cylinder, thicker contacts where lead wires are soldered to the cylinder, and thin lead wires, with diameters *d _{i}* and lengths

*L*):

_{i}Lengths of contacts and lead wires in these equations are sums of both contacts or lead wires attached to the thermistor. The “apparent” lead wire length *L*_{Wire}, which depends on the amount of convection and the wire’s thermal conductivity, is proportional to its cross section *A*_{Wire} using factor *κ*:

Values of *κ* that best fit observed results are estimated later in section 3e as *κ _{o}* = 125 mm

^{−1}in still air and in section 3c as

*κ*= 0 in the standard aerated condition (lead wires appear not to contribute to the aerated heat exchange near sea level). Volumes and surface areas are calculated using measured dimensions, but because

_{v}*L*

_{Wire}is 0 in the standard aerated condition and nonzero in still air,

*V*,

*A*, and

*ç′*(due to differing lead wire contributions) are replaced by

*V*,

_{o}*A*, and in (10) and

_{o}*V*,

_{v}*A*, and in (11). The

_{v}*ç′*values are computed analogously to the definition of

*c′*in (3)–(4) with different volumes by . “Apparent”

*Ç*is measured in still air and is expressed as equivalent “apparent”

_{o}*Ç*by

_{v}*Ç*=

_{v}*Ç*(

_{o}*V*/

_{v}*V*). The resulting is expressed at any temperature using (8)–(9). At standard conditions, is computed at 29.5°C, and is expressed at 25°C by .

_{o}To compute standard time constants, “constant” terms in (10)–(11) are collected into standard heat transfer parameter Φ, assumed to apply to all relatively similar rod thermistors:

## 3. Measurements and data analysis

The objective of this section is to calculate the aerated time constant *τ _{v}* for a rod thermistor using (11) based on its dimensions. Section 3a summarizes dimensions of thermistors considered in this study.

Section 3b describes still air experimental procedures and time constant analyses. These are performed to determine the heat capacity, an input to the aerated time constant Eq. (11), and still air Nusselt parameters.

Section 3c describes validation and reconciliation of published aerated time constants by fitting (11) to determine forced convection Nusselt parameters. Data fitting to (11) required several refinements to improve the parameter estimates. The fit is first performed with published ML-405 time constants, then with MMT-1. Systematic fitting errors were traced to incomplete temperature dependence when the researchers adjusted their observed time constants to the standard atmosphere. This section obtains nearly exact fits after recomputing time constants with complete temperature dependence and derives formulas for estimating the aerated time constants for any similar rod thermistor based on dimensions and heat capacity at any pressure level with (11) or in its simplified form for standard condition only with (18).

Section 3d shows that computed volumetric heat capacities are smaller than expected because only a relatively small core diameter substantially affects indicated temperature readings and derives a better-fitting time constant estimate Eqs. (37)–(39) for thermistors with similar heat capacities.

Two issues are discussed separately, including contributions of lead wires to heat exchange (section 3e) and special GZZ-7 thermistor characteristics (section 3f).

### a. Thermistors and dimensions

Early radiosonde rod thermistors were uncoated and mounted in a duct for protection from radiative heating, but thermistor response was slow owing to reduced ventilation and the thermal mass of the duct. Later rod thermistors were coated with white paint to reduce radiative heating and mounted externally. According to Ratner (1964, p. 10), white-coated thermistors were gradually introduced by the U.S. military starting in 1946 and the United States Weather Bureau (USWB) starting in 1960. Thermistors in this study are as follows. Based on examination, thermistor lengths vary, and diameters vary by ≈0.1 mm because the paint coating is applied by dipping.

The USWB uncoated rod thermistor had a reported diameter of 1.02 mm (made by Bendix) in Sion (1955) or 1.04–1.09 mm in Badgley (1957). A very clear photo of the thermistor and mounting is on p. 73 of U.S. Weather Bureau (1960).

Sion (1955) and Badgley (1957) measured a white coated 0.46-mm diameter and 38-mm-long military thermistor.

ML-405 is a military specification for white-coated thermistors. Also, all VIZ (Sippican starting 1997) radiosondes for the USWB (NWS starting 1970), introduced in 1960, used ML-405 thermistors until at least the early 1990s (Gaffen 1993, 117–123). An ML-405 from ≈1960 appears to be ≈1.4-mm diameter (DuBois et al. 2002, Fig. 59), but all ML-405 thermistors from 1970 to 1995 at Texas A&M University (TAMU) or from H. Richner (2010, personal communication) or R. Guillaume (2010, personal communication) are 1.2–1.3-mm diameter and 42–48 mm long. ML-405 thermistors were used on Australian Astor radiosondes (Moncur 1969), AIR IS-4A (Mahesh et al. 1997), and probably all Canadian, Italian, and Korean models licensed by VIZ from ≈1960 to 1995.

ML-419 is another military white-coated thermistor specification, and different manufacturers and sizes are found. Reported dimensions include 0.9-mm diameter and 25 mm long (Ference 1951), 0.69-mm diameter (Badgley 1957), and 0.7 mm × 39 mm from Bendix (Vanik et al. 1963). ML-419 thermistors at TAMU are 0.9–1.0 mm × 31 mm (1960/61, Carborundum Co., with the core of a broken thermistor 0.45-mm diameter) or 0.7–0.8 mm × 39–40 mm (≈1963, Bendix).

Mark II Microsonde thermistors (hereafter Mark II thermistors) at TAMU, made in 1998, are 0.9-mm diameter and 22.5 mm long, with the core of a broken thermistor 0.55-mm diameter.

The NWS used B2 radiosondes starting 1997. No specific thermistor description has been found, but the B2 thermistor may be the same as ML-405 since the thermistor tested here is 1.3-mm diameter and 42 mm long and has about the same resistance (http://www.radiosonde.eu/RS03/RS03U06.html shows the same radiosonde with an approximate thermistor length).

MMT-1 thermistors, with diameter 2.2 ± 0.2 mm and length 12 ± 2 mm (A. Kats 2010, personal communication), have been used in all Russian radiosonde models starting with RKZ (Zaitseva 1993), except recent RF95 and AK2–02 models (Fridzon and Ermoshenko 2009).

This study performs measurements on the following white-coated rod thermistors, with characteristics listed in Table 1: 1) ML-405 from VIZ model 1392 (≈1980) provided by ETH Institute for Atmospheric and Climate Science, Zurich; 2) Two MMT-1 (≈2010), serial numbers 41903 and 43400, from the Russian Central Aerological Observatory; 3) ML-419, unknown VIZ model (≈1980) from R. Guillaume; 4) B2 (≈2010) from Richard Norton, NWS, Caribou, Maine; and 5) Mark II thermistor (1996) from R. Guillaume. Calculations are also performed using dimensions (10-mm length, diameter 0.47 mm uncoated and nominally 1.0 mm including the paint coating) of the Chinese GZZ-7 thermistor (≈1986), which is similar to the China GTS1 operational radiosonde thermistor (B. Huang 2011, personal communication).

### b. Still air experimental procedures and analyses

To use (11) to determine aerated time constants, the volumetric heat content *ç′* is needed, which is obtained here from still air time constant measurements performed as specified in Defense Logistics Agency (2009). The “heating” phase heats the thermistor with a dc voltage for a defined time. In the “cooling” phase, immediately after cutting the voltage a digital multimeter (Chauvin Arnoux model MX-40) is connected to the thermistor for measuring its resistance, and therefore temperature, at defined time intervals. The multimeter requires 2 s for initialization, so the thermistor temperature *T*_{max} at the moment of cutting the voltage is determined by projecting backward from available temperatures.

Only the ML-405 and MMT-1 thermistors were provided with resistance–temperature (*R*–*T*) relationships. Therefore, the *R*–*T* relationship was determined for each thermistor by placing it in a well-stirred nonconductive liquid and measuring its resistance at different temperatures (to 0.1°C by an unknown brand digital thermometer). Resistances were fitted to function *T* = *B*/Ln(*R*/*A*) − *C* with parameters *A*, *B*, and *C*. To adjust for possible multimeter and thermometer biases, the provided *R*–*T* relationships are assumed to be most accurate. Initial fitted *R*–*T* functions overestimated ML-405 and MMT-1 temperatures. Because the measurements relate to temperature *differences* the bias was decreased to ≲1% by reducing measured resistances 1% and temperatures 0.5°C before deriving final *R*–*T* functions for all thermistors.

For each thermistor, two sets of measurements were made (Heating and Cooling) and two analysis procedures were performed (Direct and Indirect). Temperature *T* measurements are resistances converted to temperatures.

#### 1) Heating dataset

Each thermistor was repeatedly heated from *T _{∞}* with voltage

*U*= 25 V for ≈3–60 s, and temperature

*T*was measured once 2 s after cutting the voltage.

#### 2) Cooling dataset

Each thermistor was heated with voltage *U* = 25 V (ML-419: 43 V) for 58 s (Mark II: 60 s), and *T* was measured at 10 defined times during the first minute of the cooling phase, with this procedure performed 14 times per thermistor. (The Mark II thermistor was measured 7 times in 1 min, but was damaged later and could not be retested.)

The data analysis procedures to determine heat capacities are performed using both measurement datasets.

#### 3) Direct method

Time constants *τ _{o}* were determined from cooling data by fitting a second-order polynomial through Ln(Δ

*T*) as a function of Δ

*t*to each of the 14 cooling phases. For each cooling phase

*T*was projected back 2 s using the mean

*τ*to estimate

_{o}*T*

_{max}. Heat capacities

*Ç*

_{o}_{est}were calculated using each measurement as

where, by assuming for the small temperature range a linear relationship between *T* and *R*, power supplied by electric heating is

and power lost by natural convection from (5) is

where 〈Δ*T*_{thermistor}〉 = (*T*_{max} – *T*_{∞})/2, and the mean air temperature difference at the thermistor surface in calculating Ra is 〈Δ*T*_{air}〉 = (*T*_{max} – *T*_{∞})/4. Here, Nu_{o} is estimated with an empirical formula in Churchill and Thelen [1975, their Eq. (4)], using a “first guess” *m _{o}* obtained by fitting (6) to approximate their empirical Nu

_{o}(σ = 0.6%) for 0.1 ≤ Ra ≤ 4.0, giving

*m*= 0.121 (and

_{o}*a*= 0.855), and a preliminary

_{o}*đ*is calculated with (12).

_{o}The final heat capacity *Ç _{o}* was found for each thermistor by extrapolating a linear regression of

*Ç*

_{o}_{est}as a function of Δ

*t*to heating duration Δ

*t*= 0. Figure 1, left panel, shows the heat capacity derived for ML-405. While the empirical Nu

_{o}formula is not strictly valid for nonuniform cylinders, the introduced error is zero at Δ

*t*= 0, corresponding to Δ

*T*= 0 or no heat exchange. As usual, volumetric heat capacity , and (8)–(9) is used to express at 29.5°C.

Nusselt parameters *a _{o}* and

*m*were computed by minimizing squared differences between

_{o}*τ*

_{o}_{Std}measured and calculated with (17) for all thermistors, and were used to calculate mean diameter

*đ*

_{o}_{Std }(12) and factor Φ

_{oStd }(15) for each thermistor in the Direct method. These

*a*,

_{o}*m*, and

_{o}*đ*values replace the first guess values mentioned above and are compared below with Indirect method estimates.

_{o}#### 4) Indirect method

For each thermistor, each of the 14 measured cooling phases was simulated by integrating *T _{o}* in 0.1-s time steps, using (1) with

*τ*obtained from (10) and by replacing

_{o}*d*with

*đ*:

_{o}where . Parameters *b _{o}* and

*m*and the 14 maximal temperatures

_{o}*T*

_{max}were estimated in a least squares fit of all 14 simulations together, allowing also estimation of the thermistor still air time constant

*τ*.

_{o}Next, the 14 heating phases were simulated by integrating

from *T _{∞}* to

*T*

_{max}in 0.1-s time steps using

where *dT*_{cooling} /*dt* is calculated with (22) using *b _{o}* and

*m*

_{o}values obtained above. The same convective heat loss is assumed during both heating and cooling.

The thermistor heat capacity *Ç _{o}* and Nusselt factor

*a*were estimated by minimizing squared differences of simulated end temperatures from the 14

_{o}*T*

_{max}values, and volumetric heat capacity was computed and converted to as in the Direct method. Figure 1, right panel, shows the heat capacity derived for ML-405 with the Indirect method.

Because experiment temperatures were nonstandard (thermistors were heated to 30°–54°C with air temperatures 17°–24°C), still air time constants *τ _{o}* are converted from experiment temperature range {

*T*

_{1}} to standard range {

*T*

_{2}} using factor

*λ*by

_{o}Using (10),

The mean temperature of the thermistor from *t* = 0 to *τ _{o}* after cutting the power is

and the mean temperature difference (thermistor minus air) during this time for Ra_{i} is

The air properties apply to the mean temperature above the sensor surface (average of thermistor and ambient air)

As discussed above (8), at standard conditions the mean thermistor temperature for is 29.5°C and 〈*T*_{air}〉 is 24.7°C.

Aerated time constants are similarly converted by

Using (11),

with standard 〈*T*_{air2}〉 = 25°C.

Table 2 compiles measurements and derived mean values from the two analyses. The analysis and computation sequence, described below, differs from the presentation order.

In the Direct method section, *T _{∞}* and

*T*

_{max}are average ambient and computed maximum temperatures for the 14 cooling phases,

*τ*is the average fitted time constant at those temperatures, and and are expressed at 29.5°C. Next,

_{o}*a*and

_{o}*m*shown in the Mean row are computed using all thermistors,

_{o}*đ*and Φ

_{o}_{oStd}are computed for each thermistor using

*a*and

_{o}*m*as stated at the end of the Direct method description, and

_{o}*τ*

_{o}_{Std}is expressed at standard conditions using (25).

In the Indirect method section, the same variables are computed from simulations as stated in the Indirect method description, including deriving *a _{o}* and

*m*for each thermistor. Shorter thermistors have higher

_{o}*m*values than longer thermistors, possibly from attributing all heat loss to convection and neglecting conduction through lead wires.

_{o}In the Best estimate section, all variables except *đ _{o}* and the Nusselt parameters

*a*and

_{o}*m*that are not crossed out are averages of the Direct and Indirect estimates (for individual thermistors or all thermistors, depending on the variable), weighted by 1/σ, thus favoring the expectedly more accurate result of the two independent estimates.

_{o}Despite large scatters of *m _{o}*, all methods produce similar Φ

_{o}values for all thermistors owing to weak dependence on

*m*. Note also that still air time constants correlate better with mean diameter

_{o}*đ*than with main cylinder diameter

_{o}*d*

_{cyl}(shown in Table 1).

The crossed out *m _{o}* value has a large standard deviation because of differences between Direct and Indirect procedure estimates. To improve the

*m*(and also

_{o}*a*) estimate, standard time constants for all thermistors

_{o}*τ*

_{o}_{Std}are computed with (17) using mean Φ

_{oStd}= 57.7, and with

*m*chosen to minimize squared deviations of

_{o}*τ*

_{o}_{Std}from the Best estimates. Equation (15) is used to compute the corresponding

*a*value, and

_{o}*đ*(shown in the

_{o}*đ*column) is computed with (12).

_{o}The final still air Nusselt parameters are

and with (17) the standard still air time constant can then be estimated as

with in mJ mm^{−3} K^{−1}, and *V _{o}*/

*A*and

_{o}*đ*in mm. A sensitivity analysis reveals that the still air time constant is fairly insensitive to

_{o}*m*. This could be related to how the mean diameter

_{o}*đ*is defined (12). More approximately, the standard still air time constant can also be estimated as

_{o}with *Ç _{o}*

_{Std}in mJ K

^{−1}and

*A*in mm

_{o}^{2}.

### c. Measurements under aerated conditions

This study relies entirely on published aerated condition measurements. The agreement above between theoretical and measured still air properties implies that aerated time constants may be similarly reconciled. Here, published aerated time constant studies are fitted to (11) to estimate aerated Nusselt parameters *a _{v}* and

*m*and relate time constants

_{v}*τ*to thermistor dimensions and heat capacities. Since published dimensions include—if at all—only the main cylinder diameter and some measurement details are unknown, an error analysis was not feasible.

_{v}#### 1) Aerated ML-405 thermistor time constants

Nusselt parameters *a _{v}* and

*m*are initially estimated by fitting (11) using three ML-405 thermistor analyses.

_{v}Moncur (1969, his Fig. 3) modified a radiosonde to repeatedly heat the thermistor and allow it to cool when switched by baroswitch contacts, and measured 22 time constants in a sounding to 36 km, with radiosonde altitude and ascent rate obtained from radar.

Phillips et al. (1981, hereafter P1981) measured aerated time constants in a chamber with adjustable pressure, temperature, and ventilation speed using temperatures from actual ascents (H. Richner and J. Joss 2009/10, personal communications). They converted measured time constants to the standard atmosphere at 12 pressure levels.

Mahesh et al. (1997) computed time constants by flying a radiosonde from a kite to determine equilibrated temperatures in strong South Pole inversions before launching the same radiosonde attached to a balloon.

Table 3 derives standard aerated time constants from each set of published ML-405 aerated time constants. Nusselt parameters *a _{v}* and

*m*were derived by fitting with Eq. (11) the estimated aerated time constants

_{v}*τ*

_{v}_{fit}to the published ones

*τ*

_{v}_{pub}. Each dataset fit provides two standard aerated time constant estimates. The first,

*τ*

_{v}_{Std}[Eq. (11)], results directly from substituting

*a*and

_{v}*m*into (11). The second,

_{v}*τ*

_{v}_{Std}[Eqs. (30)–(31)], is obtained by converting the time constant at each pressure level with (30)–(31) to standard condition. For each analysis, the bottom part of Table 3 shows the fitted Nusselt parameters, time constant

*τ*

_{v}_{Std}[Eq. (11)], and the standard deviation of the individual

*τ*

_{v}_{Std}[Eqs. (30)–(31)] values.

At 25°C Moncur’s ML-405 thermistor has heat capacity *Ç _{o}* = 92 mJ K

^{−1}(±10%) compared to 81.5 mJ K

^{−1}for the ETH specimen, consistent with

*d*

_{cyl}about 1.29 mm (≈0.05 in.) and 1.19 mm, respectively. Mahesh et al. (1997) do not report thermistor diameters, and a mean value of 1.2 mm is assumed based on all measured specimens since ≈1970.

The standard aerated ML-405 time constant estimates vary considerably from 3.2 to 5.1 s. The short time constants derived from Moncur are similar to those of the thinner ML-419, but Australian Astor radiosondes in a collection confirm that the thermistor is ML-405. Moncur’s ascent rates (obtained by radar) may be inaccurate (increasing from 2.9 to 9.3 m s^{−1} between 0 and 36 km), and if the mean ascent rate (6 m s^{−1}) is used for all levels, the resulting standard aerated time constant is 3.8 s. His theoretical time constants using an empirical Nusselt number imply an even shorter 2.3-s standard aerated time constant. Mahesh et al. (1997) obtain an ≈8-s time constant (with much scatter) at the South Pole and they estimate 5.6 s at sea level, almost 2.5 times as long as Moncur’s theoretical time constant for the same thermistor.

The ascent rate uncertainty in Moncur’s measurements and the large scatter in the data of Mahesh et al. leave only those from P1981 to possibly provide reasonable estimates for the Nusselt parameters.

#### 2) Measurements using ML-405 and MMT-1 thermistors

Because only the ML-405 time constants derived from P1981 appear satisfactory, similar data fits were performed using published MMT-1 time constants from Ivanov et al. (2004, hereafter I2004). In Table 3, *τ _{v}*

_{fit}values derived from P1981 are consistently lower than

*τ*

_{v}_{pub}between 700 and 20 hPa and higher at other levels. Similarly (not shown),

*τ*

_{v}_{fit}values derived from I2004 are lower than their

*τ*

_{v}_{pub}values between 700 and 200 hPa and at 5 hPa.

Time constants published by P1981 and I2004 are expressed at the standard atmosphere but their measurements were made at different temperatures (for example, the P1981 chamber could not cool below −25°C). Systematic fitting differences suggest that they did not account for all temperature dependencies in expressing time constants at standard atmosphere temperatures. The I2004 empirical MMT-1 time constant equation *τ*[s] = 17.9(*ρυ*)^{−0.46} with *ρ* in kg m^{−3} and *υ* in m s^{−1} (I2004, their Eq. 2.4.18) accounts for ascent rate *υ* and air density *ρ*, but does not consider temperature dependence of volumetric heat capacity *c′*, air heat conductivity *k*, and air dynamic viscosity *μ*. With more experimentation, it was found that P1981 accounted for variations of *υ*, *ρ*, and *k*, but not *c′* and *μ*.

Because these studies fitted reduced parameter sets to measured time constants over the complete pressure range, it was hypothesized that the error is minimized at some midpressure level. The mean of all values from P1981 is approximately the actual value at 500 hPa, and the mean value from I2004 is appropriate for ≈200 hPa. So, fits to (11) were repeated with constant 500-hPa values of and *μ ^{mv}* for P1981 and 200-hPa values of ,

*μ*, and

^{mv}*k*for I2004.

In the left portion of Table 4, the left three columns of the ML-405 and MMT-1 sections show published time constants *τ*_{pub}, “reproduced” reduced temperature dependency fits *τ*_{rep} at specified pressures (column labeled P) and U.S. standard atmosphere temperatures (I2004 uses Russian Standard Atmosphere GOST4401–81, but temperatures differ only marginally from 30–5 hPa), and the percentage difference *τ*_{rep}–*τ*_{pub}. For P1981 with temperature dependency *τ*_{rep }*= f*(*υ*, *ρ*, *k*) and constant 500-hPa values of and *μ*, their *τ*_{pub} values are nearly exactly fitted with *a _{v}* = 0.280 and

*m*= 0.459. For I2004 with temperature dependency

_{v}*τ*

_{rep }

*= f*(

*υ*,

*ρ*) and constant 200-hPa values of ,

*k*, and

*μ*, their

*τ*

_{pub}values are nearly exactly fitted with

*a*= 0.245 and

_{v}*m*= 0.484. Accurate fits to “wrong” published time constants support the nature of each researcher’s approximation.

_{v}To correct neglected temperature dependence, the published time constant *τ*_{pub} at each pressure is multiplied by the ratio of corrected to uncorrected variable values, which at pressure *P* is for P1981, or for I2004, before fitting (11). The *f*_{corr}(*T*) columns show correction factors as percentage changes in *τ*_{pub} needed to include full temperature dependence. For example, *τ*_{pub} = 3.5 s at 1000 hPa from P1981 is increased by 20% to *τ*_{corr} = 4.2 s. Note that I2004 corrections are small even with less temperature dependency considered than by P1981 because their neglected dependencies partly compensate each other.

Section (b) of Table 4 shows corrected time constants *τ*_{corr} and estimated time constants *τ*_{est} resulting from fitting both datasets simultaneously. For both thermistors, Eq. (11) with final aerated Nusselt parameters

agrees between 1000 and 50 hPa to σ = 1.3%, between 30 and 5 hPa to σ = 4.6% and over the complete pressure range to σ = 2.4%. Note that the aerated time constant is about 6 times as long at 10 hPa as near sea level. The lack of systematic deviations supports the adopted approximation for calculating the aerated time constant at all pressure levels only using Nu_{v}. Substituting into (16) and (18), the standard aerated time constant for thermistors similar to ML-405 and MMT-1 is

#### 3) Aerated time constants compared at standard condition

Table 5 compares standard aerated time constants calculated using (11) and Nusselt parameters (35)—which is equivalent to (36)—with reference time constants based on results from different authors. Reference time constants (leftmost *τ _{v}*

_{Std}column) are authors’ time constants adjusted to standard condition, with corrections for full temperature dependency if information is sufficient. ML-405 and MMT-1 reference time constants are

*τ*

_{corr}values from the 1000 hPa row of Table 4, both adjusted from the standard atmospheric 14.3°C and ML-405 adjusted from

*υ*= 4.5 m s

^{−1}. For ML-419, Vanik et al. (1963) time constants of 3.1–3.4 s are equivalent to 3.24 ± 0.16 s at aerated standard condition when assuming their value results from the same number of heating and cooling measurements.

The upper part of Table 5 summarizes this comparison [section (b) will be explained in section 3d]. Other computation inputs (*d*_{cyl}, *đ _{v}*, and ) may be compared with

*d*

_{cyl},

*đ*, and for the same thermistors in Table 2. Time constants obtained with (11) and Nusselt parameters (35) agree well with the corresponding reference values for thermistors with cylinder diameter 0.9–2.1 mm. The different MMT-1 time constants illustrate typical differences in time constants for individual thermistors of the same type.

_{o}The middle part of Table 5 shows standard aerated time constants computed using (11) and (35) for Mark II and B2 (apparently previously not published) and GZZ-7 thermistors (based on 1-mm nominal diameter). For the GZZ-7 thermistor, section 3f explains why its standard aerated time constant is probably somewhat higher than the published 1.7 s. Its apparent volumetric heat capacity results from recomputing its time constant.

The lower part of Table 5 shows aerated time constant estimates where only thermistor diameters are known. They are calculated with (11) assuming (mean of values in the upper part of this table, except Mark II and GZZ-7), and using *V _{v}*/

*A*=

_{v}*d*

_{cyl}/4. Time constants are stated by the authors and some were approximate. Digitizing Sion and Badgley’s measurements and converting them with (30)–(31) results in standard aerated time constants of 3.4 ± 0.7, 3.9 ± 0.6, 2.0 ± 0.1, and 2.1 s, compared to calculated

*τ*

_{v}_{Std}values 3.1, 3.3, 0.9, and 0.9 s, respectively. The last thermistor has a possible diameter ambiguity. Brasefield (1948) added a 0.25-mm paint coating to the 1.4-mm-diameter uncoated thermistor, and claimed a time constant increase from 5 to 6 s (but apparently performed no experiments). It is unclear whether the added 0.25 mm is the paint thickness or diameter increase (but the photo of the modified radiosonde shows a quite thick thermistor, possibly >2-mm diameter), so the last two lines depict both possibilities. Time constant estimates for thin thermistors using a mean have particularly poor agreement with the reported time constants. The next section shows that apparent volumetric heat capacities are higher with smaller diameter thermistors.

### d. Volumetric heat capacities

The volumetric heat capacities in Table 5 are considerably lower than expected from the product *c′* = *cρ* of the likely materials. Using Huang (1987), who quotes specific heat *c* = 0.66 J g^{−1} K^{−1} for the thermistor core (mixed clay and iron oxide) from Badgley (1957) and assumes the same *c* for the paint coating (mostly titanium dioxide), with specific gravity *ρ* = 4.8 and 2.3 g cm^{−3} for the core and the coating (average 2.85 g cm^{−3} based on dimensions in section 3a above), volumetric heat capacity *c′* is computed to be 3.17, 1.52, and 1.88 mJ mm^{−3} K^{−1} for the core, coating, and coated thermistor. Nickel-plated copper lead wires have *c′* ≈ 3.45 mJ mm^{−3} K^{−1}. Based on this limited published data, it appears that in a “typical” thermistor, *c′* of the paint coating is about half as large as for the other components.

The Best estimate apparent volumetric heat capacities based on measurements (Table 2) are smaller than expected and range from 0.81 to 1.37 mJ mm^{−3} K^{−1}. Since the defined heat capacity is the number of joules to change a complete mass by 1 K, there are three possible explanations for unexpectedly low heat capacity.

With internal resistance heating, outer parts of the thermistor do not warm as much as the interior and there is a radial temperature gradient.

In still air experiments, all heat loss is attributed to convection but some occurs by conduction through lead wires, causing additional cooling near the contacts and an uneven temperature distribution along the length of the thermistor.

The temperature distribution within the thermistor is different in the cooling phase (when temperature measurements are made) than during resistance heating.

However, time constant estimations with (33), (34), or (35)–(36), are unaffected by underestimated volumetric heat capacities if they are determined as done here.

Here, measured heat capacities are designated as apparent heat capacity *Ç _{o}* and apparent volumetric heat capacity in all computations starting with (4). The geometric volume and area of each thermistor is unaltered. (This approach is mathematically equivalent to using physical or “true” heat capacities with thermally active or reduced “apparent” volumes, but avoids the difficulty of determining the “apparent” volumes from the unknown mean temperature within the thermistor.) Figure 2 plots from Table 5 versus the square of the cylinder diameter for all measured thermistors (except Mark II and GZZ-7, which appear to be made of different materials), and also the thin 0.46-mm-diameter thermistor, which Sion (1955) measured. Hypothesizing that converges at some central region of the thermistor,

*d*

_{central}, to corresponding to

*c′*of the ceramic semiconductor material at small diameters [here, from Badgley (1957)], but for larger diameters converges to , a least squares fit estimates by

The best fit is obtained with

Approximating *V _{v}*/

*A*with

_{v}*d*

_{cyl}/4 and with in (36), for rod thermistors with similar materials (except Mark II and GZZ 7),

### e. Thermally active lead wire length

Standard aerated time constants fitted using (11) or (18) correlate well with main cylinder diameter *d*_{cyl}, but as mentioned in sections 2 and 3b, still air time constants fitted with (10) or (17) do not correlate well with the same *d*_{cyl} because lead wires contribute noticeably to heat exchange in low convection. Lead wires appear to act as a heat source/sink or like “caloric antennas” that shorten stratospheric response times. In (14), the “thermally active” length of the lead wires *L*_{Wire} is a function of factor *κ* depending on aeration. Heat exchange from lead wires beyond *L*_{Wire} does not affect thermistor temperature readings. To some extent, a varying *L*_{Wire} parameterizes more complex heat flow changes throughout the thermistor. This section determines *κ* values best fitting time constant measurements. As *κ* varies, several variables including *V*, *A*, *ç′*, and *đ* change.

Because lead wires contribute to heat transfer in still air but not with standard aeration, it is hypothesized that *κ* is a function of the air heat transfer coefficient *h*, or *κ* = *f*(*h*). In Fig. 3, *h _{v}* is computed for various pressures and

*h*is computed at 970 hPa, showing that

_{o}*h*

_{o}_{Std}is equivalent to

*h*with standard 5 m s

_{v}^{−1}aeration at about 29–62 hPa, averaging ≈40 hPa. The varying relationship between

*L*

_{Wire}and

*h*at different pressure levels is unknown, so

*κ*is determined only for the standard still air and aerated cases.

The optimal still air *κ _{o}*

_{Std}value was obtained by performing the Table 2 analysis process for multiple

*κ*values, including Direct and Indirect methods using

_{o}*V*,

_{o}*A*,

_{o}*đ*,

_{o}*Ç*

_{o}_{Std}, and values implied by that

*κ*value. The optimal

_{o}*κ*

_{o}_{Std}minimizes the sum of squared differences Σ(ΔΦ

_{oStd})

^{2}= Σ(Φ

_{oi}–〈Φ

_{oStd}〉)

^{2}, where 〈Φ

_{oStd}〉 is the Best estimate Φ

_{oStd}(equivalent to lower right cell of Table 2 for that

*κ*value), and Φ

_{oi}is Φ

_{oStd}from the Direct or Indirect method (2 values for each thermistor, equivalent to last column of Table 2 for individual thermistors). Figure 4 shows that the minimum Σ(ΔΦ

_{oStd})

^{2}is at

*κ*≈ 125 mm

_{o}^{−1}, so Table 2 and its discussion are based on

*κ*

_{o}_{Std}= 125 mm

^{−1}.

In the standard aerated condition (5 m s^{−1} at 1000 hPa), small and unsystematic errors in fitted aerated time constants in section 3c indicate that lead wires do not contribute to the heat transfer with mean diameter *đ _{v}*. At standard condition lead wires would therefore best be completely ignored by setting

*κ*= 0 mm

_{v}^{−1}in (14), corresponding to

*L*

_{Wire}= 0, and

*A*and

_{v}*V*include only the thermistor cylinder and contacts. As discussed following (14),

_{v}*Ç*

_{v}_{Std}is derived from

*Ç*

_{o}_{Std}. However, aeration diminishes with decreasing density so, according to Fig. 3, at ≈40 hPa the heat transfer coefficient

*h*is equivalent to the 1000 hPa still air heat transfer coefficient

_{v}*h*, so

_{o}*κ*should be nonzero at high altitudes.

_{v}To correct the introduced error at higher altitudes the volumetric heat capacity computed using *κ _{v}* = 0 for use in (11) and (18), is replaced by a mean value =

*Ç*/〈

_{o}*V*

_{VHC}〉 valid for all pressure levels, also at standard aerated condition with 〈

*V*

_{VHC}〉 being a mean volume computed with a

*κ*value (

*κ*

_{VHC}), which implies a different thermally active lead wire length. Note that

*κ*already affects fitted aerated time constants through its effect on volume

_{o}*V*when is derived from still air measurements as described following (14). The mean that, with neglected lead wires, best fits aerated time constant measurements at all pressure levels, is not necessarily produced using the optimal still air

_{o}*κ*= 125 and has therefore to be determined separately.

_{o}The *κ*_{VHC} value producing the optimal mean minimizes squared differences between measured aerated time constants and those calculated with (11) using *κ*_{VHC}. For multiple *κ*_{VHC} values, the Best estimate is calculated as in Table 2 (hereafter ). Then, the Table 4 analysis is performed using (with *V _{v}* and

*A*based on

_{v}*κ*= 0) to fit (11) to corrected ML-405 and MMT-1 time constants (full temperature dependence considered). In Fig. 4, the minimum fitting error occurs with calculated with

_{v}*κ*

_{VHC}≈ 75 (the apparent volumetric heat content in aerated conditions includes ≈60% as much of the lead wire length as in still air), so all references to and in this paper correspond to using

*κ*

_{VHC }= 75 mm

^{−1}in computation, such as when fitting time constants in Tables 3 and 4 or when comparing in Table 5 aerated time constants at standard condition.

### f. GZZ-7 thermistor

To determine GZZ-7 thermistor time constants, Huang (1987) performed wind tunnel experiments (probably ≈1000 hPa) with low aeration (0.12–2.6 m s^{−1}, Re range 8–170) to simulate stratospheric conditions. Extrapolating his results to standard aeration (5 m s^{−1} at 1000 hPa, Re ≈ 300), *τ _{v}*

_{Std}averages to his published value 1.7 s. However, with low aeration, heat exchange by lead wires reduces mean diameter

*đ*and time constant

*τ*. Extrapolation to 1000 hPa assumes that the lead wire heat exchange effect does not disappear, and

_{v}*τ*

_{v}_{Std}is underestimated.

To remove lead wire effects, *h _{v}* is first computed with (5) and Huang’s Reynolds numbers, assuming 〈

*T*〉 = 39°C (the thermistor was heated above 60°C) and 1000-hPa pressure, resulting in 16 ≤

_{air}*h*≤ 80 or 〈

_{v}*h*〉 ≈ 50. In Fig. 3 for the GZZ-7 diameter,

_{v}*h*≈

_{o}*h*≈ 20 for stratospheric aerated or surface still air measurements. Assuming air heat transfer

_{v}*h*is inversely proportional to

_{v}*κ*, then in still air

*κ*≈ 125 (20/50) ≈ 50 mm

_{o}^{−1}. Recomputing Huang’s time constants with lead wire effects removed, and then projecting fitted lines, the standard aerated time constant becomes

*τ*

_{v}_{Std}≈ 1.9 s for measured specimens, or

*τ*

_{v}_{Std}≈ 2.1 s for 1-mm nominal diameter. Using (36) with corrected

*τ*

_{v}_{Std}, the apparent volumetric heat capacity is ≈ 0.81 mJ mm

^{−3}K

^{−1}. These

*τ*

_{v}_{Std}and are shown in Table 5 for GZZ-7, and is similar to Mark II.

## 4. Concluding remarks

This study measures still air time constants for several radiosonde rod thermistors and develops data-fitting equations that closely match published aerated time constant measurements after correcting for incomplete temperature dependence of variables in the original studies. The close data fits give confidence in the theoretical development and analysis procedures.

To correct artificial trends in archived sounding data, it is necessary to know the radiosonde types used at each station including sensor types and dates of changes. Developing station radiosonde histories is beyond the scope of this study, but the most widely used, although incomplete, radiosonde metadata file is available online (at http://www.ncdc.noaa.gov/oa/climate/igra/index.php). Likewise, separate studies are needed to determine time constants for sensors other than rod thermistors.

The main use of the fitted equations is to determine time constants to adjust historical radiosonde soundings for temperature lag errors. For a radiosonde thermistor with known dimensions and measured apparent heat capacity, the aerated time constant *τ _{v}* can be computed with (11) and (35) for any condition of a sounding or at standard aerated condition with (36). If the cylindrical diameter and standard still air time constant are known,

*τ*

_{v}_{Std}is approximated to σ = 4.5% by

with 0.167 being ratio Φ_{v}*/*Φ_{o} = 9.64/57.7 from (36) and (33), *d*_{cyl} is in millimeters, and exponent ⅙ is a best data fit. If only *d*_{cyl} is known, but thermistor materials are assumed similar to those in Table 5 (except Mark II and GZZ-7), aerated time constants can be estimated using (37)–(39). The fitting equation time constant uncertainty is to a large extent determined by the variability of individual thermistors owing to factors such as varying paint thickness. Time constants of individual thermistors of the same type are likely to vary by up to ≈ ±7%–12%.

The standard aerated time constant can be adjusted to any other condition of a sounding using (30)–(31), or approximately according to the ratio of *τ*_{est} at a pressure level to *τ*_{est} at 1000 hPa for the most similar thermistor in Table 4 (either ML-405 or MMT-1).

This study does not develop or apply any specific lag adjustments. However, Mahesh et al. (1997) compares two adjustment methods and finds that shifting reported temperatures downward by the thermistor time constant performs as well as complex lag correction procedures, by comparing the fit of adjusted profiles to the equilibrium profile just obtained by attaching the radiosonde to a kite. Even though their soundings have a large scatter of measured time constants, they conclude that the simple lag correction procedure is justified.

Finally, since height resolution Δ*H* is proportional to ascent rate, one might hope to improve the resolution correspondingly by reducing the ascent rate. However, the aerated time constant *τ _{v}* also depends on the ascent rate:

so resolution only improves approximately with the square root of the velocity reduction.

## Acknowledgments

This study is self-funded and evolved from research to improve estimates of optical refraction near the horizon. We thank Roland Guillaume for performing all measurements in his spare time, and Alexander Kats, Richard Norton, Hans Richner, Jürg Joss, Jean-Pierre Godet, Andrew T. Young, and reviewers for advice, thermistors, and translation help.

### APPENDIX

#### List of Variables and Parameters

##### a. Variables with still air or aerated (ventilated) dependence

Note that parameters related to still air condition have subscript *o* and those related to aerated condition (ventilated) have subscript *v*. Parameters related to standard condition as adopted here have in the case of still air subscript *o*Std and in the aerated case *v*Std.

*A* [mm^{2}] Thermally active surface area calculated with diameters *d _{i}* and lengths

*L*

_{i}*a* Nondimensional fitted Nusselt number proportionality factor

*b _{o}* [mJ mm

^{−3}K

^{−1}] =

*Ç*/

_{o}*a*=

_{o}*ç′V*/

_{o}*a*parameter used in Eq. (22) for determining the heat capacity by the Indirect method

_{o}*C* [mJ K^{−1}] “True” total heat capacity of thermistor material components

*Ç* [mJ K^{−1}] Measured or “apparent” thermistor heat capacity

*đ* [mm] Thermally “effective” mean diameter

*L*_{Wire} [mm] Length of “thermally active” portion of both lead wires, varying with ventilation

*m* Nondimensional fitted Nusselt number power

Nu Nondimensional Nusselt number

*V* [mm^{3}] Thermally active volume calculated with diameters *d _{i}* and lengths

*L*

_{i}〈*V*_{VHC}〉 [mm^{3}] Mean volume calculated with *κ*_{VHC} to obtain the mean volumetric heat capacity

*κ* [mm^{−1}] Factor describing the thermally active or “effective” length of both lead wires (*L*_{Wire}); note that *κ _{o}* = 125,

*κ*= 0, and

_{v}*κ*

_{VHC}= 75

Φ_{oStd} [mJ^{−1} s mm^{1+3mo} K] = 1/[*a _{o }*

*k*

_{Std}(Ra

_{Std })

^{mo}] Standard still air heat transfer parameter

Φ_{vStd} [mJ^{−1} s mm^{1+mv} K] = 1/[*a _{v }*

*k*

_{Std}(Re

_{Std })

^{mv}] Standard aerated heat transfer parameter

*τ* [s] Time constant

##### b. Physical property variables and parameters

Note that computation formulas for these variables and parameters are given in the U.S. Committee on Extension to the Standard Atmosphere (1976), and values at a specified temperature and pressure can be calculated using an online calculator (at http://www.aerospaceweb.org/design/scripts/atmosphere/). (For parameters at 25°C and 1000 hPa, enter altitude 110.7 m and “temperature increment” 10.7195°C.)

*g* [=9.8063 m s^{−2}] Gravity, standard value at 1000 hPa

*k* [W m^{−1} K^{−1}] Air thermal conductivity

Ra = (*g*Δ*Tβd*^{3})/ (*ηα*) Nondimensional Rayleigh number

Re = (*ρυd*)/*μ* Nondimensional Reynolds number

*α* [m^{2} s^{−1}] Air thermal diffusivity

*β* [K^{−1}] Air expansivity

*η* [m^{2} s^{−1}] Air kinematic viscosity

*μ* [N s m^{−2}] Air dynamic viscosity

*ρ* [kg m^{−3}] Air density

##### c. Other variables

*A*_{Wire} [mm^{2}] Cross-sectional area of a lead wire

*c* [J kg^{−1} K^{−1}] Specific heat, or heat capacity per unit mass

*c _{i}* [mJ mg

^{−1}K

^{−1}] Specific heat of thermistor material component

*i*.

*c′* [mJ mm^{−3} K^{−1}] = *C*/*V* Volumetric heat capacity

*ç′*, , , , , [mJ mm^{−3} K^{−1}] = *Ç*/*V* “Apparent” volumetric heat capacity for entire thermistor (*ç′*), in still air (), in aerated case (), estimated (), inferred for temperature-sensing portion (), asymptotic value for large-diameter thermistor ()

[mJ mm^{−3} K^{−1}] Mean “apparent” volumetric heat capacity defining mean for all pressure levels

*d*, *d*_{cyl}, *d _{i}* [mm] Characteristic dimension, (geometrical) diameter of thermistor (

*d*), main cylinder (

*d*

_{cyl}), or component

*i*(

*d*)

_{i}*f _{ç}* Nondimensional thermistor heat capacity relative to

*f*= 1 at 25°C

_{ç}Δ*H* [m] Height resolution of thermistor

*h* [W m^{−2} K^{−1}] Air heat transfer coefficient

*L _{i}* [mm] Length of cylindrical thermistor part

*i*

*P*_{electric} [W] or [J s^{−1}] Rate of heat gain by thermistor from electric heating

*P*_{convection} [W] or [J s^{−1}] Rate of heat loss from thermistor by natural convection

*R*_{(T)} [Ω] Resistance of thermistor at temperature *T*

*T* [°C] or [K] Temperature

*T*_{max} [°C] Temperature of thermistor when heating it at moment of cutting the applied voltage.

*T _{∞}* [°C] Steady state (or ambient air) temperature

Δ*T* [K] Temperature difference from ambient air

Δ*T*_{thermistor} [K] Temperature difference between thermistor and ambient air

Δ*T*_{air} [K] Temperature difference between air layer next to the thermistor’s surface and ambient air

*U* [V] Applied dc voltage

*υ* [m s^{−1}] Air velocity over thermistor from radiosonde ascent

*σ* Relative (percentage) standard deviation

*λ _{o}*,

*λ*Nondimensional factors to convert time constants to a different (usually standard) condition

_{v}*ρ _{i}* [mg mm

^{−3}] Density of thermistor material component

*i*

## REFERENCES

*Compendium of Meteorology,*T. F. Malone, Ed., American Meteorological Society, 1207–1222.

*World of Measurement*. [Available online at http://ria-stk.ru/mi/adetail.php?ID=30717.]

*Radiozondirovanie Atmosfery (Radio Sounding of the Atmosphere).*Ural’skago Otdelenie Rossiya Akademie Nauk, Yekaterinburg, 590 pp.

*U.S. Standard Atmosphere 1976.*U.S. Government Printing Office, 227 pp.